Magnetism and Magnetic Materials (More Advanced)

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					Preface to Volume 17



The Handbook series Magnetic Materials is a continuation of the Handbook series
Ferromagnetic Materials. When Peter Wohlfarth started the latter series, his orig-
inal aim was to combine new developments in magnetism with the achievements
of earlier compilations of monographs, producing a worthy successor to Bozorth’s
classical and monumental book Ferromagnetism. This is the main reason that Ferro-
magnetic Materials was initially chosen as title for the Handbook series, although the
latter aimed at giving a more complete cross-section of magnetism than Bozorth’s
book.
    In the last few decades magnetism has seen an enormous expansion into a va-
riety of different areas of research, comprising the magnetism of several classes of
novel materials that share with truly ferromagnetic materials only the presence of
magnetic moments. For this reason the Editor and Publisher of this Handbook se-
ries have carefully reconsidered the title of the Handbook series and changed it
into Magnetic Materials. It is with much pleasure that I can introduce to you now
Volume 17 of this Handbook series.
    Magnetic tunnel junctions form part of the exciting field of spintronics. In
this field, nanostructured magnetic materials are employed for functional devices
where both the charge and the spin are explicitly exploited in electron transport.
Magnetic junctions offer a number of unique opportunities for investigating novel
effects in physics and have led to several new research directions in spintronics.
Equally important is the fact that magnetic junctions represent excellent materials
for exploring novel and superior types of devices. The physics of spin-dependent
tunneling in magnetic tunnel junctions is reviewed in Chapter 1, concentrating on
ferromagnetic layers separated by an ultrathin insulating barrier. The tunneling cur-
rent between the ferromagnetic electrodes in these junctions depends strongly on
an external magnetic field and as such lends itself to novel applications in the fields
of magnetic media and data storage. Followed by a short introduction on the back-
ground and the elementary principles of magnetoresistance and spin polarization in
magnetic tunnel junctions, the author discusses basic and magnetic transport phe-
nomena, emphasizing the critical role of the preparation and properties of the tunnel
barriers. Later on, key ingredients to understand tunneling spin polarization are in-
troduced in relation to experiments using superconducting probe layers. The author
also discusses a number of crucial results directly addressing the underlying physics
of spin tunneling and the role played by the polarization of the ferromagnetic elec-
trodes. Apart from Al2 O3 , the successful use of alternative crystalline barriers such
as SrTiO3 and MgO is discussed.
    With decreasing size of magnetic elements in magnetic storage media, read
heads, and MRAM elements, the time and energy necessary for reading and writing
vi                                                                    Preface to Volume 17



magnetic domains have become of paramount importance and are studied inten-
sively worldwide. A concept of substantial impact is that of spin-accumulation, i.e.
a non-equilibrium magnetization that is injected electrically into a non-magnetic
material from a ferromagnetic contact by an applied voltage. A breakthrough in
magnetoelectronics is the observation of current-induced magnetization reversal in
several types of layered structures. This effect finds its origin in the transfer of spin
angular momentum by the applied current. On the other hand, magnetization dy-
namics induces spin currents into a conducting heterostructure. These novel effects
couple the magnetization dynamics in hybrid devices with internal and applied spin
and charge currents. The time-dependent properties become non-local, meaning
that they are not a property of a single ferromagnetic element, but depend on the
whole magnetically active region of the device. Recent progress in understand-
ing the magnetization dynamics in ferromagnetic hybrid structures is presented in
Chapter 2.
    Magnetic properties of 3d-4f intermetallic compounds have been reviewed in
several previous Volumes of this Handbook. This includes reviews on magnetically
hard materials and related compounds (Volumes 6, 9 and 10). In these materials, the
magnetocrystalline anisotropy invariably plays a central role. Somewhat apart stands
the literature on experimental studies of the crystal field effects in intermetallics
of rare earths. Results obtained by means of inelastic neutron scattering have been
reviewed in Volume 11. The separation between the topics of magnetic anisotropy
and crystal field effects seems somewhat artificial. In view of the general acceptance
of the single-ion model, little doubt remains about the intimate connection between
the two phenomena. The origin of the apparent splitting between the two topics
mentioned can most likely be found in the fact that the theoretical activity in the
area has been lagging behind experiments ever since the appearance of the last
major review written four decades ago by Callen and Callen, in 1966. However,
one has to realize that theoretical advance on magnetic anisotropy and crystal field
effects did not cease in the meantime. These topics just progressed in different
directions, stimulated by the advent of the density functional theory (Volume 13).
As regards the single-ion model proper, work on it proceeded at a rather slow pace.
Nonetheless, a fair amount of new results has been published between the late
1960s and more recent times. Chapter 3 reviews the progress made in the theory,
filling the gap in the literature between the anisotropy and the crystal field effects.
In this Chapter the authors aim at reasserting the statement that magnetocrystalline
anisotropy is the most important manifestation of the crystal field effects.
    Magnetocaloric effects in the vicinity of phase transitions were already discussed
by Tishin in Volume 12 of this Handbook, published in 1999. Since then there has
been a strong proliferation in research on magnetocaloric materials and their appli-
cation, mainly dealing with the option of magnetocaloric refrigeration at ambient
temperature. A comprehensive review dealing with this latter aspect is presented
in Chapter 4 of the present Volume. The design of a refrigeration system involves
many problems which are far from simple. Its design invariably requires a critical
evaluation of possible solutions by considering factors such as economics, safety,
reliability, and environmental impact. The vapor compression cycle has dominated
the refrigeration market to date because of its advantages: high efficiency, low toxi-
Preface to Volume 17                                                              vii


city, low cost, and simple mechanical embodiments. Perhaps this is because as much
as 90% of the worlds heat pumping power; i.e. refrigeration, water chilling, air con-
ditioning, various industrial heating and cooling processes among others, is based
on the vapor compression cycle principle. However, in recent years environmental
aspects have become an increasingly important issue in the design and development
of refrigeration systems. Especially in vapor compression systems, the banning of
CFCs and HCFCs because of their environmental disadvantages has opened the
way for other refrigeration technologies which until now have been largely ignored
by the refrigeration market. As environmental concerns grow, alternative technolo-
gies which use either inert gasses or no fluid at all become attractive solutions to
the environment problem. A significant part of the refrigeration industry R&D
expenditures worldwide is now oriented towards the development of such alterna-
tive technologies in order to be able to achieve replacement of vapor compression
systems in a mid- to long-term perspective. One of these alternatives is magnetic
refrigeration which is discussed in Chapter 4. In this chapter the author empha-
sizes the many novel experimental results obtained on magnetocaloric materials,
placing them in the proper physical and thermodynamic background. Also measur-
ing systems as well as demonstrators and prototypes for magnetic refrigeration are
discussed.
    Intermetallic compounds in which 3d metals (particularly Mn, Fe, Co and Ni)
are combined with rare earth elements exhibit a large variety of interesting physical
properties. The magnetic properties of these intermetallics are a matter of interest
for two main reasons: Firstly their study helps to elucidate some of the funda-
mental principles of magnetism. Secondly they are of technical interest, because
several compounds were found to be a suitable basis for high performance perma-
nent magnets. More recently the unique soft magnetic properties made amorphous
metal-metalloid alloys to a further class of materials which has attained consider-
able importance with regard to industrial application. In Chapter 5 the hydrides of
such compounds and alloys are discussed. In fact, this chapter can be regarded as an
updating of Chapter 6 in Volume 6 of this Handbook, published in 1991. In order
to reach a self-contained form of this chapter, the authors and the editor agreed to
incorporate the most important results of the previous chapter into the present one.
In this way the novel results can be viewed in the right perspective, not requiring
the interested reader to go back to the previous chapter in Volume 6 at regular
intervals. Here it should be mentioned that a large variety of novel techniques has
been employed more recently in order to elucidate the mechanism and effects of
hydrogen uptake which is particularly complex in intermetallic compounds. They
can roughly be devided into surface sensitive methods such as photo emission and
related spectroscopies, X-ray absorption (XANES, EXAFS), X-ray magnetic cir-
cular dichroism (XMCD), transmission electron microscopy, conversion electron
Mössbauer spectroscopy and to some extent susceptibility measurements. The re-
sults of such investigations are discussed in Chapter 5 together with results of NMR
and ESR and surface insensitive experiments, where only the bulk properties can be
studied (magnetic measurements, neutron and X-ray diffraction, X-ray absorption,
transmission Mössbauer spectroscopy).
viii                                                                  Preface to Volume 17



    It is well known that there have been many new developments in the field of
magnetic sensing and actuation, including new forms of magnetic material. Apart
from this has been much progress in the development of microelectromechanical
systems (MEMS). Hand in hand with this has gone the advance in density of elec-
tronic components on a chip, expressed by the so-called Moore’s Law, where areal
density has doubled every eighteen months. Much of MEMS technology is sil-
icon based, with three-dimensional structures being manufactured from a silicon
platform by means of various lithographic techniques. It is common practice to
include functionality in to MEMS, opening the possibility of sensing or actuation.
Frequently piezoresistive materials are used which requires current and voltage con-
nections to the sensor element, the measured quantity being the strain dependence
of electrical resistivity in the active film. Of special interest is the incorporation of
magnetic materials in to MEMS making it possible to use inductive coupling for
sensing or activation. The major advantage to be gained from this is the possibility
to avoid the requirement for connections, and that it allows packaging and de-
ployment in remote or hostile environments. In Chapter 6 the authors address the
integration of magnetic components into MEMS as a way of providing additional
functionality. They present an overview of advances in thin film magnetic materials
that make the use of MagMEMS a viable option.
    Volume 17 of the Handbook on the Properties of Magnetic Materials, as the
preceding volumes, has a dual purpose. As a textbook it is intended to be of assis-
tance to those who wish to be introduced to a given topic in the field of magnetism
without the need to read the vast amount of literature published. As a work of
reference it is intended for scientists active in magnetism research. To this dual pur-
pose, Volume 17 of the Handbook is composed of topical review articles written
by leading authorities. In each of these articles an extensive description is given in
graphical as well as in tabular form, much emphasis being placed on the discussion
of the experimental material in the framework of physics, chemistry and material
science.
    The task to provide the readership with novel trends and achievements in mag-
netism would have been extremely difficult without the professionalism of the
North Holland Physics Division of Elsevier Science B.V.

                                                               K.H.J. B USCHOW
                                          VAN   DER  WAALS -Z EEMAN I NSTITUTE
                                                    U NIVERSITY OF A MSTERDAM
Contents




Preface to Volume 17                                                            v
Contents                                                                       ix
Contents of Volumes 1–16                                                       xi
Contributors                                                                   xv


1. Spin-Dependent Tunneling in Magnetic Junctions                               1
    H.J.M. Swagten

    1. Introduction                                                             2
    2. Basis Phenomena in MTJs                                                 14
    3. Tunneling Spin Polarization                                             52
    4. Crucial Experiments on Spin-Dependent Tunneling                         71
    5. Outlook                                                                102
    Acknowledgements                                                          106
    References                                                                106

2. Magnetic Nanostructures: Currents and Dynamics                             123
    Gerrit E.W. Bauer, Yaroslav Tserkovnyak, Arne Brataas and Paul J. Kelly

    1. Introduction                                                           124
    2. Ferromagnets and Magnetization Dynamics                                125
    3. Magnetic Multilayers and Spin Valves                                   127
    4. Non-Local Magnetization Dynamics                                       135
    5. The Standard Model                                                     139
    6. Related Topics                                                         142
    7. Outlook                                                                144
    Acknowledgements                                                          144
    References                                                                145

3. Theory of Crystal-Field Effects in 3 d-4 f Intermetallic Compounds         149
    M.D. Kuz’min and A.M. Tishin

    Foreword                                                                  149
    1. Formal Description of the Crystal Field on Rare Earths                 150
    2. The Single-Ion Anisotropy Model for 3 d-4 f Intermetallic Compounds    166
x                                                                         Contents



    3. Spin Reorientation Transitions                                         210
    4. Conclusion                                                             228
    References                                                                229


4. Magnetocaloric Refrigeration at Ambient Temperature                       235
    Ekkes Brück

    List of Symbols and Abbreviations                                        237
    1. Brief Review of Current Refrigeration Technology                      237
    2. Introduction to Magnetic Refrigeration                                239
    3. Thermodynamics                                                        241
    4. Materials                                                             247
    5. Comparison of Different Materials and Miscellaneous Measurements      270
    6. Demonstrators and Prototypes                                          274
    7. Outlook                                                               280
    Acknowledgements                                                         281
    References                                                               281

5. Magnetism of Hydrides                                                     293
    Günter Wiesinger and Gerfried Hilscher

    1. Introduction                                                           293
    2. Formation of Stable Hydrides                                           295
    3. Electronic Properties                                                  296
    4. Basic Aspects of Magnetism                                             300
    5. Review of Experimental and Theoretical Results                         304
    Acknowledgement                                                           422
    References                                                                422


6. Magnetic Microelectromechanical Systems: MagMEMS                           457
    M.R.J. Gibbs, E.W. Hill and P. Wright

    1. Introduction                                                           458
    2. MEMS Fabrication                                                       466
    3. Magnetic Materials for MEMS                                            485
    4. Magnetoresistive Materials and Sensors                                 491
    5. Magnetic MEMS Based Devices                                             511
    References                                                                521


Author Index                                                                  527
Subject Index                                                                 579
Materials Index                                                               583
Contents of Volumes 1–16




Volume 1
1.                                   .
     Iron, Cobalt and Nickel, by E. P Wohlfarth     . . . . . . . . . . . . .                    .   .   .   .   .   .   .     1
2.   Dilute Transition Metal Alloys: Spin Glasses, by J. A. Mydosh and G. J. Nieuwenhuys         .   .   .   .   .   .   .    71
3.   Rare Earth Metals and Alloys, by S. Legvold . . . . . . . . . . . . .                       .   .   .   .   .   .   .   183
4.   Rare Earth Compounds, by K. H. J. Buschow . . . . . . . . . . . . .                         .   .   .   .   .   .   .   297
5.                                             .
     Actinide Elements and Compounds, by W Trzebiatowski . . . . . . . . .                       .   .   .   .   .   .   .   415
6.                                  .
     Amorphous Ferromagnets, by F E. Luborsky . . . . . . . . . . . . .                          .   .   .   .   .   .   .   451
7.   Magnetostrictive Rare Earth–Fe2 Compounds, by A. E. Clark . . . . . . .                     .   .   .   .   .   .   .   531



Volume 2
1.   Ferromagnetic Insulators: Garnets, by M. A. Gilleo . . . . .        .   .   .   .   .   .   .   .   .   .   .   .   .     1
2.   Soft Magnetic Metallic Materials, by G. Y. Chin and J. H. Wernick   .   .   .   .   .   .   .   .   .   .   .   .   .    55
3.                                                   .
     Ferrites for Non-Microwave Applications, by P I. Slick     . . .    .   .   .   .   .   .   .   .   .   .   .   .   .   189
4.   Microwave Ferrites, by J. Nicolas . . . . . . . . . . .             .   .   .   .   .   .   .   .   .   .   .   .   .   243
5.   Crystalline Films for Bubbles, by A. H. Eschenfelder . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .   297
6.   Amorphous Films for Bubbles, by A. H. Eschenfelder . . . .          .   .   .   .   .   .   .   .   .   .   .   .   .   345
7.   Recording Materials, by G. Bate       . . . . . . . . . .           .   .   .   .   .   .   .   .   .   .   .   .   .   381
8.                                   .
     Ferromagnetic Liquids, by S. W Charles and J. Popplewell . . .      .   .   .   .   .   .   .   .   .   .   .   .   .   509



Volume 3
1. Magnetism and Magnetic Materials: Historical Developments and Present Role in Industry
   and Technology, by U. Enz       . . . . . . . . . . . . . . . . . . . . .                                 .   .   .   .   1
2. Permanent Magnets; Theory, by H. Zijlstra . . . . . . . . . . . . . . . .                                 .   .   .   . 37
3. The Structure and Properties of Alnico Permanent Magnet Alloys, by R. A. McCurrie . .                     .   .   .   . 107
                             c         .
4. Oxide Spinels, by S. Krupiˇka and P Novák . . . . . . . . . . . . . . . .                                 .   .   .   . 189
5. Fundamental Properties of Hexagonal Ferrites with Magnetoplumbite Structure,
   by H. Kojima . . . . . . . . . . . . . . . . . . . . . . . . . .                                          .   .   .   .   305
6. Properties of Ferroxplana-Type Hexagonal Ferrites, by M. Sugimoto . . . . . . . .                         .   .   .   .   393
7. Hard Ferrites and Plastoferrites, by H. Stäblein . . . . . . . . . . . . . . .                            .   .   .   .   441
                         .
8. Sulphospinels, by R. P van Stapele    . . . . . . . . . . . . . . . . . . .                               .   .   .   .   603
9. Transport Properties of Ferromagnets, by I. A. Campbell and A. Fert . . . . . . . .                       .   .   .   .   747



Volume 4
1. Permanent Magnet Materials Based on 3d-rich Ternary Compounds, by K. H. J. Buschow . . . . . 1
2. Rare Earth–Cobalt Permanent Magnets, by K. J. Strnat  . . . . . . . . . . . . . . . . 131
3. Ferromagnetic Transition Metal Intermetallic Compounds, by J. G. Booth . . . . . . . . . . 211
xii                                                                                Contents of Volumes 1–16



                                             .
4. Intermetallic Compounds of Actinides, by V Sechovský and L. Havela . . . . . . . . . . . . 309
5. Magneto-Optical Properties of Alloys and Intermetallic Compounds, by K. H. J. Buschow . . . . . 493



Volume 5
1. Quadrupolar Interactions and Magneto-Elastic Effects in Rare-Earth Intermetallic Compounds,
       .
   by P Morin and D. Schmitt . . . . . . . . . . . . . . . . . . . . . . .                          .   .   .     1
                                                               .
2. Magneto-Optical Spectroscopy of f-Electron Systems, by W Reim and J. Schoenes . . . . .          .   .   .   133
                                                                                   .
3. INVAR: Moment-Volume Instabilities in Transition Metals and Alloys, by E. F Wasserman        .   .   .   .   237
                                                              .
4. Strongly Enhanced Itinerant Intermetallics and Alloys, by P E. Brommer and J. J. M. Franse . .   .   .   .   323
5. First-Order Magnetic Processes, by G. Asti . . . . . . . . . . . . . . . . .                     .   .   .   397
6. Magnetic Superconductors, by Ø. Fischer . . . . . . . . . . . . . . . . . .                      .   .   .   465


Volume 6
1. Magnetic Properties of Ternary Rare-Earth Transition-Metal Compounds, by H.-S. Li and
   J. M. D. Coey . . . . . . . . . . . . . . . . . . . . . . . . .                          .   .   .   .   .     1
2. Magnetic Properties of Ternary Intermetallic Rare-Earth Compounds, by A. Szytula   .     .   .   .   .   .    85
3. Compounds of Transition Elements with Nonmetals, by O. Beckman and L. Lundgren     .     .   .   .   .   .   181
                                    .
4. Magnetic Amorphous Alloys, by P Hansen       . . . . . . . . . . . . . . .               .   .   .   .   .   289
5. Magnetism and Quasicrystals, by R. C. O’Handley, R. A. Dunlap and M. E. McHenry . .      .   .   .   .   .   453
6. Magnetism of Hydrides, by G. Wiesinger and G. Hilscher . . . . . . . . . . .             .   .   .   .   .   511


Volume 7
1. Magnetism in Ultrathin Transition Metal Films, by U. Gradmann . . . . . . . . . . .                  . .      1
                                                                   .
2. Energy Band Theory of Metallic Magnetism in the Elements, by V L. Moruzzi and
    .
   P M. Marcus . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                  . .     97
3. Density Functional Theory of the Ground State Magnetic Properties of Rare Earths and Actinides,
   by M. S. S. Brooks and B. Johansson . . . . . . . . . . . . . . . . . . . . .                        . . 139
                                                       .
4. Diluted Magnetic Semiconductors, by J. Kossut and W Dobrowolski    . . . . . . . . . .               . . 231
5. Magnetic Properties of Binary Rare-Earth 3d-Transition-Metal Intermetallic Compounds,
                                     n
   by J. J. M. Franse and R. J. Radwa´ ski . . . . . . . . . . . . . . . . . . . .                      . . 307
6. Neutron Scattering on Heavy Fermion and Valence Fluctuation 4f-systems,
   by M. Loewenhaupt and K. H. Fischer     . . . . . . . . . . . . . . . . . . . .                      . . 503


Volume 8
1. Magnetism in Artificial Metallic Superlattices of Rare Earth Metals, by J. J. Rhyne and
          .
   R. W Erwin      . . . . . . . . . . . . . . . . . . . . . . . . . . .                            . . .        1
2. Thermal Expansion Anomalies and Spontaneous Magnetostriction in Rare-Earth Intermetallics
                                .
   with Cobalt and Iron, by A. V Andreev      . . . . . . . . . . . . . . . . . .                   . . . 59
                                            .
3. Progress in Spinel Ferrite Research, by V A. M. Brabers   . . . . . . . . . . . . .              . . . 189
4. Anisotropy in Iron-Based Soft Magnetic Materials, by M. Soinski and A. J. Moses . . . . .        . . . 325
5. Magnetic Properties of Rare Earth–Cu2 Compounds, by Nguyen Hoang Luong and
   J. J. M. Franse . . . . . . . . . . . . . . . . . . . . . . . . . . .                            . . . 415


Volume 9
1. Heavy Fermions and Related Compounds, by G.J. Nieuwenhuys . . . . . . . . . . . . .                            1
                                                                                     .N.
2. Magnetic Materials Studied by Muon Spin Rotation Spectroscopy, by A. Schenck and F Gygax . .                  57
Contents of Volumes 1–16                                                                                  xiii


3. Interstitially Modified Intermetallics of Rare Earth and 3d Elements, by H. Fujii and H. Sun . . . . 303
4. Field Induced Phase Transitions in Ferrimagnets, by A.K. Zvezdin      . . . . . . . . . . . . 405
                                                        .
5. Photon Beam Studies of Magnetic Materials, by S.W Lovesey . . . . . . . . . . . . . . 545



Volume 10
1. Normal-State Magnetic Properties of Single-Layer Cuprate High-Temperature Superconductors
   and Related Materials, by D.C. Johnston . . . . . . . . . . . . . . . . . . . . .                1
2. Magnetism of Compounds of Rare Earths with Non-Magnetic Metals, by D. Gignoux and D. Schmitt . 239
3. Nanocrystalline Soft Magnetic Alloys, by G. Herzer . . . . . . . . . . . . . . . . . 415
4. Magnetism and Processing of Permanent Magnet Materials, by K.H.J. Buschow . . . . . . . . 463



Volume 11
                                                                     .
1. Magnetism of Ternary Intermetallic Compounds of Uranium, by V Sechovský and L. Havela . . . .          1
2. Magnetic Recording Hard Disk Thin Film Media, by J.C. Lodder . . . . . . . . . . . . . 291
3. Magnetism of Permanent Magnet Materials and Related Compounds as Studied by NMR,
                     .C.
   by Cz. Kapusta, P Riedi and G.J. Tomka       . . . . . . . . . . . . . . . . . . . . 407
4. Crystal Field Effects in Intermetallic Compounds Studied by Inelastic Neutron Scattering, by O. Moze 493



Volume 12
                                                                                   .
1. Giant Magnetoresistance in Magnetic Multilayers, by A. Barthélémy, A. Fert and F Petroff .   . . . .     1
                                                        .C.
2. NMR of Thin Magnetic Films and Superlattices, by P Riedi, T. Thomson and G.J. Tomka          . . . .    97
3. Formation of 3d-Moments and Spin Fluctuations in Some Rare-Earth–Cobalt Compounds,
                    .E.
   by N.H. Duc and P Brommer        . . . . . . . . . . . . . . . . . . . .                     . . . . 259
4. Magnetocaloric Effect in the Vicinity of Phase Transitions, by A.M. Tishin . . . . . .       . . . . 395



Volume 13
1. Interlayer Exchange Coupling in Layered Magnetic Structures, by D.E. Bürgler,
    .
   P Grünberg, S.O. Demokritov and M.T. Johnson . . . . . . . . . . . . . . . . .                   .   .   1
2. Density Functional Theory Applied to 4f and 5f Elements and Metallic Compounds, by M. Richter    .   . 87
3. Magneto-Optical Kerr Spectra, by P.M. Oppeneer . . . . . . . . . . . . . . . .                   .   . 229
                                  .
4. Geometrical Frustration, by A.P Ramirez . . . . . . . . . . . . . . . . . . .                    .   . 423



Volume 14
                                               .
1. III-V Ferromagnetic Semiconductors, by F Matsukura, H. Ohno and T. Dietl . . . . . . .           . .     1
                                                                                     .E.
2. Magnetoelasticity in Nanoscale Heterogeneous Magnetic Materials, by N.H. Duc and P Brommer       . .    89
3. Magnetic and Superconducting Properties of Rare Earth Borocarbides of the Type RNi2 B2 C,
                                                   .N.
   by K.-H. Müller, G. Fuchs, S.-L. Drechsler and V Narozhnyi  . . . . . . . . . . . .              . . 199
4. Spontaneous Magnetoelastic Effects in Gadolinium Compounds, by A. Lindbaum and M. Rotter .       . . 307



Volume 15
1. Giant Magnetoresistance and Magnetic Interactions in Exchange-Biased Spin-Valves, by R. Coehoorn . 1
2. Electronic Structure Calculations of Low-dimensional Transition Metals, by A. Vega, J.C. Parlebas
   and C. Demangeat     . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
xiv                                                                                  Contents of Volumes 1–16


3. II–VI and IV–VI Diluted Magnetic Semiconductors – New Bulk Materials and Low-Dimensional
                             .
   Quantum Structures, by W Dobrowolski, J. Kossut and T. Story . . . . . . . . . . . . . . 289
4. Magnetic Ordering Phenomena and Dynamic Fluctuations in Cuprate Superconductors and Insulating
   Nickelates, by H.B. Brom and J. Zaanen   . . . . . . . . . . . . . . . . . . . . . 379
5. Giant Magnetoimpedance, by M. Knobel, M. Vázquez and L. Kraus . . . . . . . . . . . . 497



Volume 16
1.                                                                           .             .K.
   Giant Magnetostrictive Materials, by O. Söderberg, A. Sozinov, Y. Ge, S.-P Hannula and V Lindroos       .      1
2. Micromagnetic Simulation of Magnetic Materials, by D. Suess, J. Fidler and Th. Schrefl . . . . . .             41
3. Ferrofluids, by S. Odenbach       . . . . . . . . . . . . . . . . . . . . . . . . .                           127
4. Magnetic and Electrical Properties of Practical Antiferromagnetic Mn Alloys, by K. Fukamichi, R.Y. Umetsu,
   A. Sakuma and C. Mitsumata . . . . . . . . . . . . . . . . . . . . . . . . .                                 209
                                                                                       .            .
5. Synthesis, Properties and Biomedical Applications of Magnetic Nanoparticles, by P Tartaj, M.P Morales,
   S. Veintemillas-Verdaguer, T. Gonzalez-Carreño and C.J. Serna . . . . . . . . . . . . . . .                  403
Contributors



Gerrit E.W. Bauer
Centre for Advanced Study at the Norwegian Academy of Science and Letters,
Drammensveien 78, NO-0271 Oslo, Norway; Kavli Institute of NanoScience, Delft
University of Technology, 2628 CJ Delft, The Netherlands

Arne Brataas
Centre for Advanced Study at the Norwegian Academy of Science and Letters,
Drammensveien 78, NO-0271 Oslo, Norway; Department of Physics, Norwegian
University of Science and Technology, N-7491 Trondheim, Norway

Ekkes Brück
Department of Mechanical Engineering, Federal University of Santa Catarina,
Florianopolis SC, Brazil and Van der Waals-Zeeman Instituut, Universiteit van
Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

M.R.J. Gibbs
Sheffield Centre for Advanced Magnetic Materials & Devices, Department of En-
gineering Materials, University of Sheffield, Sheffield, S1 3JD, UK

E.W. Hill
School of Computer Science, Information Technology Building, University of
Manchester, Oxford Road, Manchester, M13 9PL, UK

Gerfried Hilscher
Institute for Solid State Physics, Vienna University of Technology, Wiedner Haupt-
strasse 8-10, A-1040 Wien, Austria

Paul J. Kelly
Faculty of Science and Technology and Mesa+ Research Institute, University of
Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

M.D. Kuz’min
Leibniz-Institut für Festkörper- und Werkstofforschung, Postfach 270116, D-01171
Dresden, Germany

H.J.M. Swagten
Eindhoven University of Technology, Department of Applied Physics, COBRA
xvi                                                                     Contributors



Research Institute and center for NanoMaterials (cNM), P.O. box 513, 5600 MB
Eindhoven, The Netherlands

A.M. Tishin
Department of Physics, M.V. Lomonosov Moscow State University, 119992
Moscow, Russia

Yaroslav Tserkovnyak
Centre for Advanced Study at the Norwegian Academy of Science and Letters,
Drammensveien 78, NO-0271 Oslo, Norway; Department of Physics and Astron-
omy, University of California, Los Angeles, California 90095, USA

Günter Wiesinger
Institute for Solid State Physics, Vienna University of Technology, Wiedner Haupt-
strasse 8-10, A-1040 Wien, Austria

P. Wright
QinetiQ Ltd, Malvern Technology Centre, St Andrews Road, Malvern, WR14
3PS, UK
        CHAPTER             ONE



        Spin-Dependent Tunneling in Magnetic
        Junctions
        H.J.M. Swagten *



        Contents
        1. Introduction                                                                                         2
           1.1 From GMR to tunnel magnetoresistance                                                             2
           1.2 Elementary model for tunnel magnetoresistance                                                    6
           1.3 Beyond the elementary approach                                                                  11
           1.4 Scope of this review                                                                           13
        2. Basis Phenomena in MTJs                                                                            14
           2.1 Basic magneto-transport properties                                                             16
           2.2 Oxidation methods for Al2 O3 barriers                                                          33
           2.3 Towards optimized barriers                                                                     41
        3. Tunneling Spin Polarization                                                                        52
           3.1 How to measure spin polarization?                                                              53
           3.2 Data on tunneling spin polarization                                                            56
           3.3 Ingredients of tunneling spin polarization                                                     62
        4. Crucial Experiments on Spin-Dependent Tunneling                                                    71
           4.1 The relevance of interfaces: using nonmagnetic dusting layers                                  72
           4.2 Quantum-well oscillations in MTJs                                                              75
           4.3 Role of the ferromagnetic electrode for TMR                                                    78
           4.4 Towards infinite TMR with half-metallic electrodes                                              82
           4.5 Role of the barrier for TMR                                                                    87
           4.6 Coherent tunneling in MgO junctions                                                            90
        5. Outlook                                                                                           102
        Acknowledgements                                                                                     106
        References                                                                                           106

        Abstract
        This chapter reviews the physics of spin-dependent tunneling in magnetic tunnel junc-
        tions, i.e. ferromagnetic layers separated by an ultrathin, insulating barrier. In magnetic
        junctions the tunneling current between the ferromagnetic electrodes depends strongly
        on an external magnetic field, facilitating a wealth of applications in the field of magnetic
   *   Eindhoven University of Technology, Department of Applied Physics, COBRA Research Institute and center for
       NanoMaterials (cNM), P.O. box 513, 5600 MB Eindhoven, The Netherlands
       E-mail: h.j.m.swagten@tue.nl

Handbook of Magnetic Materials, edited by K.H.J. Buschow                                     © 2008 Elsevier B.V.
Volume 17 ISSN 1567-2719 DOI 10.1016/S1567-2719(07)17001-3                                    All rights reserved.



                                                                                                                1
2                                                                                 H.J.M. Swagten



     media and storage. After a short introduction on the background and elementary princi-
     ples of magnetoresistance and tunneling spin polarization in magnetic tunnel junctions,
     the basic magnetic and transport phenomena are discussed emphasizing the critical role
     of the preparation and properties of (mostly Al2 O3 ) tunneling barriers. Next, key ingredi-
     ents to understand tunneling spin polarization are introduced in relation to experiments
     using superconducting probe layers. This is followed by discussing a number of crucial
     results directly addressing the physics of spin tunneling, including the role of the po-
     larization of the ferromagnetic electrodes, the interfaces between barrier and electrodes
     and quantum-well formation, and the successful use of alternative crystalline barriers
     such as SrTiO3 and MgO.

     Key Words: magnetic tunnel junctions, magnetoresistance, spin polarization, spin tun-
     neling, spintronics




     1. Introduction
      This review is focusing on the fundamental aspects of magnetic tunnel junctions or
shortly MTJs. It will cover the preparation and experimental aspects of MTJs, and
most of the crucial experiments that were performed to unravel their basic physics.
In the last section, new promising directions for further research will be reviewed.
    In this introductory section the following subjects will be covered:
• the breakthrough towards magnetoresistance in layered magnetic structures, more
  specifically in metallic multilayers and subsequently in magnetic tunnel junctions
• phenomenology of magnetoresistance in MTJs using the Julliere model, includ-
  ing the concept of so-called tunneling spin polarization
• the shortcomings of elementary models via an introduction to some crucial ex-
  perimental observations and advanced theoretical approaches.
   It should be noted that several other reviews exist also partially covering
the physics and applications of spin-polarized tunneling in tunnel junctions; see
Meservey and Tedrow (1994), Moodera et al. (1999a, 2000), Moodera and Mathon
(1999), Dennis et al. (2002), Ziese (2002), Maekawa et al. (2002), Miyazaki (2002),
Tsymbal et al. (2003), Zhang and Butler (2003), Zutic et al. (2004), Shi (2005), and
LeClair et al. (2005). In most cases, however, the focus is different as compared to
the present paper, and some recent developments in this rapidly evolving field may
not be included. To assist the reader, the last part of this introduction will briefly
explain the scope of the present review.

1.1 From GMR to tunnel magnetoresistance
Magnetic tunnel junctions are within the florishing field of magnetoelectronics or
spin electronics, shortly spintronics. In this area, nanostructured magnetic materials
are used for functional devices explicitly exploiting both charge and spin in electron
transport, the so-called spin-polarized transport. As we will see later on, magnetic
Spin-Dependent Tunneling in Magnetic Junctions                                        3


junctions are offering several unique opportunities for studying new, sometimes
unexpected effects in physics, and, furthermore, they have opened up a number
of new research directions within spintronics. Apart from that, magnetic junctions
are superb materials for exploring novel device options, such as improved read-
head sensors, magnetic memories or magnetic biosensors. Before a more detailed
insight in the principles of magnetic tunneling will be given, it is instructive to first
shortly review the field of spin-polarized transport and the ongoing increasing role
of tunneling transport.
     In the mid-eighties the first crucial steps are made towards the exploitation
of magnetic nanostructures for new electrical effects. These breakthroughs were
strongly stimulated by the progress in ultra-high vacuum deposition and character-
ization techniques, enabling full control of layer-by-layer growth of metallic mag-
netic (multi-)layers. One of the first intriguing observations by Carcia et al. (1985)
is the presence of perpendicular magnetic anisotropy in ultrathin magnetic (multi-)-
layers due to strong magnetic surface anisotropies, see, e.g., also Parkin (1994) and
Johnson et al. (1996). Due to perpendicular anisotropy, the magnetization can be
pointing out of the plane of a magnetic thin film, a novel way of engineering
the direction of magnetization in ferromagnetic films. To illustrate the technolog-
ical relevance, this phenomenon is now used in magnetic media to increase the
data density as compared to in-plane magnetized (longitudinal) magnetic disks.
The subsequent discovery of magnetic interaction across ultrathin nonmagnetic
spacers has been critically important for the field of spin-polarized transport. It
is shown by Grünberg et al. (1986) that this so-called interlayer coupling may favor
an antiparallel, in-plane alignment of two neighboring magnetic layers separated
by only a few atomic planes of a nonmagnetic element. It is now well accepted
that the driving mechanism for the interaction is spin-dependent electron reflec-
tion and transmission at the interfaces between the magnetic and nonmagnetic
layers (for a review, see Bürgler et al., 1999). The first observation of remark-
able, unexpected electrical effects in these magnetic nanostructures is indepen-
dently reported by the research groups of Fert and Grünberg (Baibich et al., 1988;
Binasch et al., 1989). They have demonstrated that the resistance of a multilayered
stack of magnetic layers separated by nonmagnetic spacers strongly depends on the
mutual orientation of the layer magnetization. Due to the presence of antiferro-
magnetic coupling, the magnetization of these layers can be engineered between
parallel and anti-parallel via an externally applied magnetic field. The enormous
magnitude of the magnetoresistance at room temperature explains the term giant
magnetoresistance or GMR used since then.
     The observation of GMR has initiated an intensive research effort. Fundamen-
tally, the physics of the underlying spin-polarized transport is studied extensively
using magnetic engineering tools, novel material combinations, and a variety of
theoretical approaches (Coehoorn, 2003). Along with the fundamental interest, the
application potential of this effect has been immediately recognized by the mag-
netic recording industries. As a well-known achievement in this area, the concerted
scientific and industrial effort led to the introduction of a GMR read head al-
ready in 1997, just nine years after the pioneering, curiosity-driven experiments.
A similar strong interplay between scientific discovery and subsequent device im-
4                                                                                  H.J.M. Swagten




Figure 1.1 The development of room-temperature magnetoresistance in layered magnetic
structures. Giant magnetoresistance (GMR) data are restricted to spin valves, where the active
part is consisting of two ferromagnetic layers separated by a metallic spacer. The data for tunnel
magnetoresistance (TMR) are shown since 1995 for tunneling across Al2 O3 barriers, as well
as for MgO, showing a huge rise of TMR in recent years. Note that only a limited number of
the available data have been collected in the graph just to give a representative illustration of
the developments.


plementation can be observed in the field of magnetic tunnel junctions (MTJs).
Although junctions were already studied for a long time (e.g. in the case of one
superconducting and one metallic electrode), especially in the beginning of the
nineties an increasing number of contributions are devoted to full magnetic junc-
tions with two ferromagnetic electrodes. Although these experiments are certainly
inspired by the original work of Julliere (1975) and Maekawa and Gafvert (1982) on
Fe-Ge-Co and Ni-NiO-Ni(Co,Fe), respectively, the booming interest for GMR in
metallic systems has also fuelled the renewed interest. For some of these pioneering
experiments on MTJs in the beginning of the nineties, see Miyazaki et al. (1991),
Nowak and Raułuszkiewicz (1992), Suezawa et al. (1992), Yaoi et al. (1993), and
Plaskett et al. (1994). The final breakthrough in this field takes place in 1995 when
unprecedented large magnetoresistance effects are discovered at room temperature.
Moodera et al. (1995) as well as Miyazaki and Tezuka (1995a) are the first to show
that a system of two magnetic layers separated by a very thin nonmagnetic oxide
layer displays a huge tunnel magnetoresistance or TMR effect, substantially larger than
GMR in a similar system with a metal spacer (for a review on exchange-biased spin-
valves, see, e.g., Coehoorn, 2003). To illustrate the order of magnitude of GMR
versus TMR, Fig. 1.1 shows the chronology of these developments. It is clear from
the graph that the TMR data on Al2 O3 -based MTJs have shown a steady increase
and are always well above GMR data. In more recent years, the use of MgO as a
barrier (as well as other oxide and ferromagnetic material combinations) have un-
Spin-Dependent Tunneling in Magnetic Junctions                                                 5




Figure 1.2 The magnetoresistance (a), expressed as V /I , as a function of external mag-
netic field H at low temperature (T = 10 K) of an evaporated 80 Å Co/14 Å
Al + oxidation/150 Å NiFe junction; see the schematics in (b). The arrows indicate the direc-
tion of magnetization of the two ferromagnetic electrodes. Antiparallel alignment between the
layers is facilitated by different coercivities of the Co and NiFe layer; see also section 2.1.2.
From Moodera (1997).


doubtedly demonstrated the record-high magnitude of TMR effects. In comparing
these data, one should realize that the physics behind the magnetoresistance in tun-
nel junctions is completely different from that in all-metallic GMR structures, since
quantum-mechanical tunneling is now the fundamental process governing the elec-
trical transport. We will return to that in section 1.2. In Fig. 1.2 an experimental
example of tunnel magnetoresistance is shown from the group of Moodera, us-
ing two magnetic layers of different coercivity separated by a thin alumina barrier.
It clearly demonstrates a large resistance change when the two magnetic layers are
switched from a parallel to an anti-parallel orientation by an external magnetic field.
    The magnetoresistance in MTJ’s can be exploited in a novel solid-state mem-
ory. It consists of (sub)micron-sized tunneling elements connected via word and bit
lines in a two-dimensional architecture, a similar layout as in macroscopic ferrite
core memories invented in the fifties; see Livingston (1997) and references therein.
The fact that the electrical current flows perpendicular to the layers in an MTJ (due
to the quantum-mechanical tunneling process across the insulator) rather than in
the plane of the layers (as in GMR) allows for an efficient use of word and bit lines
addressing individual bits. This, together with the huge magnetoresistances of MTJs
paved the way to a fast implementation in memory applications. In fact, new non-
volatile solid-state memories based on magnetic tunnel junctions have entered the
market in the beginning of the new millennium. In Fig. 1.3a a schematics is shown
of one bit cell within a so-called magnetic random access memory or MRAM. It
is shown how to use the magnetoresistance effect (as displayed in Fig. 1.2) to store
information in a solid-state device. In this system one of the layers, the reference
layer, is always pointing in one direction (in Fig. 1.3b to the right), which means
that the applied magnetic fields created by the orthogonal word and bit line should
never exceed its coercivity. On the other hand, the softer magnetic layer is used to
6                                                                                 H.J.M. Swagten




Figure 1.3 (a) Schematics of a magnetic tunnel junction incorporated in a single cell of a
magnetic random access memory (MRAM). Orthogonal word and bit lines create a magnetic
field that is able to set the free layer magnetization direction of the MTJ. Semiconductor
(transistor) elements are used as a switch for read-out. In (b) the memory function of an MTJ
is illustrated by the magnetoresistance of a Co/Al2 O3 /NiFe junction (see Fig. 1.2 for the full
curve of a similar MTJ). The arrows indicate the direction of the in-plane magnetization. To
write a “0” or “1”, a magnetic field is applied by the word/bit line that is just large enough to
switch the softest (storage) magnetic layer, but small enough not to switch the (magnetically
harder) reference layer. To read a bit, the resistance is measured at zero magnetic field.



actually store the information, and is switched by a small magnetic field to create a
zero-field state with low or high resistivity, corresponding to a logical “0” or “1”.
The reader is referred to Tehrani et al. (2000, 2003), de Boeck et al. (2002), Parkin
et al. (2003), DeBrosse et al. (2004), Shi (2005), and references therein, for papers
on MRAM technology.
    Although the magnetoresistance effects in MTJs have been reproducibly re-
ported by many groups, and applications are being developed since then, the
fundamental issues in explaining the observed effects are far from fully understood,
and need a careful introduction. In the following, it is explained how the existence
of TMR can be predicted in the most elementary phenomenological model cap-
turing some of the basic fundamental properties of these devices. This will serve as a
starting point for a further exploration of the underlying physics, which is addressed
later on in the review.

1.2 Elementary model for tunnel magnetoresistance
In elementary textbooks on quantum mechanics, the tunneling current through
a potential barrier is extensively treated, illustrating the finite probability for an
electron to tunnel through energetically forbidden barriers. Within the Wentzel-
Kramers-Brillouin (WKB) approximation, which is valid for potentials U varying
slowly on the scale of the electron wavelength, the transmission probability across
Spin-Dependent Tunneling in Magnetic Junctions                                                 7




Figure 1.4 The wave function in a metal-oxide-metal tunnel structure schematically shows the
concept of quantum-mechanical tunneling for electrons with an energy close to the Fermi en-
ergy EF . The barrier height at the interface between metal and oxide is given by φ. A nonzero
tunneling current is flowing when a bias voltage V is applied between the metallic electrodes.
The grey areas in the metal regions represents the occupied density-of-states; in the barrier the
energy gap of the insulator is indicated in white.


a potential barrier is in one dimension proportional to:
                                                t
                     T (E) ≈ exp –2                 2me [U (x) – E]/h2 dx
                                                                    ¯                        (1)
                                            0

with E the electron energy, me the electron mass, and x the direction perpendicu-
lar to the barrier plane. This equation directly shows the well-known exponential
dependence of tunnel transmission on the thickness t and energy barrier U (x) – E.
Note that the electron momentum in the plane of the layers is assumed to be absent,
i.e., k = 0. In fact, when electrons are impinging the barrier under an off-normal
angle (k = 0), the tunneling probability rapidly decreases with increasing k since
in that case the term 2m[U (x) – E]/h2 in the exponent of the transmission should
                                        ¯
be replaced by 2m[U (x) – E]/h2 + k 2 .
                                   ¯
     In an experimental situation, this tunneling process can be measured in a metal-
oxide-metal structure, a trilayered structure of two metals or electrodes separated by
an insulating spacer. The thickness of the spacer is in the order of just 1 nanome-
ter, a few atomic distances, otherwise the exponentially decaying tunneling current
(proportional to the transmission in Eq. (1)) becomes immeasurably small. The
metal-oxide-metal junction is drawn in Fig. 1.4 where the potential of the barrier
U (x) is assumed to be constant across the barrier and located at an energy φ above
the Fermi energy EF of the metals. Without a voltage difference between the met-
als layers, the Fermi levels will be equal on either side of the barrier, and the tunnel
current is zero. When a finite bias voltage V is applied, the Fermi level is lowered
at the right-hand side of the barrier, and electrons are now able to elastically tunnel
from filled electron states (left) towards unoccupied states in the second (right) elec-
trode. Note that in this case the electrode at right is at a higher electrical potential
as compared to the left electrode, yielding a net electrical current from right to
left. As a result, the amount of current will be proportional to the product of the
available, occupied electron states on the left, and the number of empty states at the
right electrode, multiplied by the barrier transmission probability. Therefore, the
8                                                                         H.J.M. Swagten



tunneling current is directly proportional to the density-of-states of each electrode
(at a specific energy E) multiplied by the Fermi–Dirac factors f (E) and 1 – f (E)
to account for the amount of occupied and unoccupied electron states, respectively.
    To analytically calculate the net tunneling current in the metal-oxide-metal
structure, we first write the current due to electrons tunneling from left to right
assuming an elastic (energy-conserving) electron tunneling process from occupied
states on the left to empty states at the right (see the figure):
       IL→R (E) ∝ NL (E – eV )f (E – eV )T (E, V , φ, t)NR (E)[1 – f (E)].          (2)
As indicated by Eq. (1), the transmission T (E, V , φ, t) depends on the electron
energy and barrier thickness and potential, but it is also affected by the bias voltage
V that effectively reduces the barrier height φ. For the opposite current we write a
similar equation, by which the total current I is obtained by integrating IL→R –IR→L
over all energies:
              +∞
       I∝          NL (E – eV )T (E, V , φ, t)NR (E)[f (E – eV ) – f (E)] dE.       (3)
             –∞
For small voltages eV     φ only the electrons at (or close to) the Fermi level EF
contribute to the tunneling current, by which the transmission no longer depends
on energy E. Moreover, in this limit also the density-of-states factors are in princi-
ple independent of E, which reduces the current to:
                                             +∞
            I ∝ NL (EF )NR (EF )T (φ, t)          [f (E – eV ) – f (E)] dE.         (4)
                                           –∞
For low enough temperature (kB T      eV ) the integral over the Fermi functions
simply yields eV , by which we end up with a transparant expression for the tunnel
conductance:
                      G ≡ dI /dV ∝ NL (EF )NR (EF )T (φ, t).                        (5)
    It shows that in this simple model the tunnel conductance is proportional to
the transmission probability and the density-of-states of the two electron systems.
The explicit dependence of the density-of-states factors is originally proposed by
the pioneering theoretical work of Bardeen (1961), now referred to the transfer-
Hamiltonian method (see Wolf, 1985). Note that usually in this method the prob-
ability T (φ, t) is written as |M|2 , which is the squared transfer matrix element that
determines the tunneling transition rate between an initial and final state.
    Now we can proceed with evaluating the current in a magnetic junction, that
is, two magnetic electrodes separated by a nonmagnetic insulator (see Fig. 1.5).
The density-of-states of a ferromagnetic material is represented by a simple ma-
jority and minority electron band, shifted in energy due to exchange interactions.
First, we consider two identical ferromagnetic electrodes with parallel magnetiza-
tion orientations, separated by an insulating barrier. Assuming that the electron spin
is conserved in these processes (Tedrow and Meservey, 1971a), tunneling may only
occur between bands of the same spin orientation in either electrode, i.e., from
a spin majority band to a spin majority band, and similar for the minorities. Us-
ing Eq. (5) and assuming equal transmission for both spin species, we write the
Spin-Dependent Tunneling in Magnetic Junctions                                                    9




Figure 1.5 Spin-resolved tunneling conductivity G for parallel (top panel) and antiparallel
magnetization (bottom), as indicated at right, is proportional to the product of the den-
sity-of-states factors at the Fermi level EF . The total current in parallel orientation is governed
      2             2
by Nmaj (EF ) + Nmin (EF ), in the antiparallel case by 2Nmaj (EF )Nmin (EF ). The voltage that
introduces a net tunneling current across the barrier (indicated by the grey bar) is negligible in
this schematics.


conductance for parallel magnetization as:
                        GP = G↑ + G↓ ∝ Nmaj (EF ) + Nmin (EF ),
                                        2            2
                                                                                                (6)
where G↑(↓) is the conductance in the up- (down-) spin channel, and Nmaj (EF )
(Nmin (EF )) is the majority (minority) density-of-states at EF . When we switch the
magnetization orientation of one ferromagnetic electrode relative to that of the
other ferromagnetic electrode, the axis of spin quantization is also changed in that
electrode. Tunneling between like spin orientations now means tunneling from
a majority to a minority band, and vice versa. The conductance for antiparallel
aligned magnetization is then simply:
                         GAP = G↑ + G↓ ∝ 2Nmaj (EF )Nmin (EF ).                                 (7)
It is immediately clear that conductances are different for parallel and antiparallel
magnetizations. In other words, ferromagnetic tunnel junctions display a mag-
netoresistance when an external field is used to switch between these magnetic
orientations. This tunnel magnetoresistance (TMR) is usually defined as the differ-
ence in conductance between parallel and antiparallel magnetizations, normalized
by the antiparallel conductance, or, alternatively, as the resistance change normalized by
the parallel resistance:
                                    GP – GAP     RAP – RP
                              TMR ≡            =            .                     (8)
                                       GAP           RP
Note that the equality of the two definitions for TMR is only valid for very small
bias voltage, since in that case the inverse tunnel resistance R –1 = I /V is identi-
10                                                                        H.J.M. Swagten



cal to the conductance dI /dV . In literature on MTJs, another, more pessimistic
definition of TMR is used as well, normalizing the resistance change by the re-
sistance in antiparallel instead of parallel orientation. However, throughout the
review, Eq. (8) will be strictly applied to quantify the magnetoresistance ratio in
magnetic junctions. Using Eqs. (6) and (7), it is easily derived that TMR is equal
to [Nmaj (EF ) – Nmin (EF )]2 /[2Nmaj (EF )Nmin (EF )]. We can generalize this for two
different magnetic electrodes, resulting in the well-known Julliere-formula for the
magnetoresistance of MTJ’s (Julliere, 1975):
                                             2PL PR
                                 TMR =               ,                             (9)
                                           1 – PL PR
where PL(R) is the tunneling spin polarization in the left (right) ferromagnetic elec-
trode. The tunneling spin polarization of each electrode is defined as
                                  Nmaj (EF ) – Nmin (EF )
                            P =                           ,                        (10)
                                  Nmaj (EF ) + Nmin (EF )
and is simply the normalized difference in majority and minority density-of-states
at the Fermi level. From these equations it is immediately seen that in the limit
of zero polarization of one of the electrodes, no TMR is expected. On the other
hand, for a full polarization of ±1, the TMR becomes infinitely high. These fully
polarized materials (one spin channel is absent at the Fermi level) are referred to
as being half-metallic, and have been intensively investigated in this field; see also
section 4.4.
    In an experimental study, Julliere (1975) is the first to use Eqs. (9) and (10) for
TMR in Fe-Ge-Co junctions, although in principal with a different interpretation
of tunneling spin polarization. N(EF ) is defined as an effective number of tunnel-
ing electrons to stress the fact that the tunneling process is not only governed by
the (static) density-of-states at EF . We will return to this crucial point later on.
Nevertheless, it should be emphasized that the Julliere equation in its simplest form
demonstrates the fundamental role of the tunneling spin polarization of the fer-
romagnetic electrode in understanding the observed TMR in magnetic junctions.
The tunneling spin polarization of individual magnetic electrodes can be measured
with a so-called superconducting tunneling spectroscopy (STS) technique that uses a su-
perconductor (in most cases Al) to probe the spin imbalance in tunneling currents.
In more detail, in a ferromagnetic-Al2 O3 -Al junction a magnetic field splits up
the sharply-peaked density-of-states of the superconducting Al electrode, which
leads to an asymmetry in the conductance G(V ) that reflects the amount of spin
polarization. In section 3 this will be further introduced, here only a numerical
example will be given. The tunneling spin polarization for Co is experimentally
determined to be around +0.42, which via Eq. (9) corresponds to a TMR effect of
more than 40% for Co-Al2 O3 -Co MTJs. This is only slightly above the observed
(low-temperature) value. For the moment, it seems that we can use this formula as
a phenomenological equation that nicely connects tunneling polarization P to the
magnitude of the magnetoresistance. However, as we will see below, the physics of
spin-polarized tunneling is much more complex and needs a dramatic reconsidera-
tion of these phenomena.
Spin-Dependent Tunneling in Magnetic Junctions                                            11


1.3 Beyond the elementary approach
Although the model we have introduced captures some of the basic physics in mag-
netic tunnel junctions and is rather illustrative on a tutorial level, it fails to predict
a number of experimental observations. These observations for TMR include, for
instance:
• strong dependence of TMR on the applied bias voltage V and temperature T
• sensitivity of TMR on the electronic structure of the barrier-ferromagnetic inter-
  face region, not just the bulk density-of-states (as suggested by Eqs. (9) and (10))
• relevance of the electronic structure of the barrier, in some cases even leading to
  an inversion of TMR.
     Here we will briefly introduce some of the advanced theories to better appreci-
ate these observations, focusing at this point on the tunneling spin polarization for
its fundamental role in the physics of magnetic tunnel junctions. A more detailed
treatment will be postponed for sections 3 and 4.
     Later on in this review (Table 1.2 in section 3) we will show that the tun-
neling spin polarization of the 3d ferromagnetic metals are all positive, and in the
range of 40–60%. According to the definition of Eq. (10), the positive sign of the
polarization relates to a dominant majority density-of-states at the Fermi level. If
one considers the band structure and density-of-states of the 3d metals, however,
the situation is completely reversed. As an example, Fig. 1.6 shows the (calculated)
density-of-states of Co and Ni, both having a surplus of minority states of the
Fermi level. This would suggest a negative tunneling spin polarization, and com-
pletely contradicts the experimental observations. This dichotomy was recognized
already in the seventies when pioneering experiments in the field of superconduct-
ing tunneling spectroscopy were reported on ferromagnetic-superconducting junc-
tions (Tedrow and Meservey, 1971a, 1971b, 1975). Theoretically, Stearns (1977) has
shown that the conductance in a tunnel junction is not simply determined by the
electron density-of-states at the Fermi level, but should include the probability for
them to tunnel across an ultrathin barrier. Especially the most mobile s-like electron
states are able to tunnel with a much larger probability as compared to the d elec-
trons due to their different effective mass. Based on this, Stearns could explain the
positive spin polarization by considering the spin asymmetry of the s-like energy




Figure 1.6 Density-of-states of the elemental metals fcc Cu (a), fcc Ni (b), and hcp Co (c),
obtained from self-consistent band-structure calculations using the Augmented Spherical Wave
(ASW) method. From Coehoorn (2000).
12                                                                              H.J.M. Swagten



bands, thereby neglecting the contribution from the rapidly decaying d-like wave
functions in tunneling experiments.
    More recently, another advanced aspect of spin-polarized tunneling is reported.
Slonczewski (1989) emphasizes that spin-dependent tunneling is not a process solely
related to the (complex) electronic properties of the ferromagnetic electrodes. He
has analytically calculated the tunneling current between free-electron ferromag-
netic metals within the WKB approximation (see Eq. (1)), assuming that tunneling
electrons have a very small parallel wave vector, close to k = 0. By explicitly
matching the electron wave functions at the barrier interfaces, the tunneling spin
polarization is calculated as:
                                           κ 2 – kF ,maj kF ,min
                              P = P0 ×                           ,                       (11)
                                           κ 2 + kF ,maj kF ,min
where kF ,maj and kF ,min are the Fermi wave vectors, and κ the imaginary component
of the wave vector of electrons in the barrier with k = 0 at the Fermi level, cor-
responding to κ = (2me φ /h2 )1/2 with φ the height of the barrier. The first term
                                ¯
P0 is equal to the earlier result in Eq. (10). The second term, however, contains
the properties of the barrier as well, and is due to the discontinuous change of the
potential at the interface with the barrier. As a result of this interface factor, the
polarization becomes greatly dependent on the band parameters in relation to the
height of the barrier, with the possibility to even change the sign of P . This is in fact
a first demonstration that tunneling spin polarization is not an intrinsic property solely de-
termined by the ferromagnetic electrode. A similar conclusion is reached in free-electron
calculations where the conductance is analytically obtained by matching the free-
electron wave functions (and its derivatives) at the two interfaces (MacLaren et al.,
1997). In this free-electron calculation, also electrons with k = 0 are considered,
although k is assumed to be strictly conserved upon tunneling. In Fig. 1.7 the free-
electron magnetoresistance calculated by Slonczewski (1989) and MacLaren et al.
(1997) is plotted as a function of polarization P = (kF ,maj – kF ,min )/(kF ,maj + kF ,min ),
which is equivalent to P0 in Eq. (11). For thick barriers, the solutions in the cal-
culation of MacLaren et al. (1997) approach the model of Slonczewksi based on
the WKB approximation, whereas no correspondence is found with the Julliere
expression. However, it should be stressed that the predictability of this elementary,
simplified free-electron model is rather poor. As already pointed out by Harrison
(1961), this is related to the suspicious absence of density-of-states factors in the
transport characteristics. MacLaren et al. (1997) and Zhang and Levy (1999) em-
phasize that, generally, these free-electron calculations (including the Julliere model)
fail to predict the observed magnetoresistance behavior in magnetic junctions, and
its dependencies on, e.g., barrier thickness, barrier height, and bias voltage. Never-
theless, there are some attempts to directly use Slonczewski’s or other free-electron
calculations to investigate how TMR behaves as a function of the model parame-
ters. For an example, see the work of Tezuka and Miyazaki (1998) on the variation
of TMR with the Al2 O3 barrier height.
     After the work of Slonczewski (and free-electron calculations by others), a great
number of advanced theoretical investigations have been published to further ex-
plore the physics of TMR and tunneling spin polarization; see for example the
Spin-Dependent Tunneling in Magnetic Junctions                                                     13




Figure 1.7 (a) Calculations of the magnetoresistance (RAP – RP )/RAP as a function of the tun-
neling spin polarization P = (kF ,maj – kF ,min )/(kF ,maj + kF ,min ). The Julliere curve is based on
Eq. (9) although using the pessimistic definition of TMR = 2P 2 /(1 + P 2 ). Calculations within
the model of Slonczewski are performed for a barrier height φ of 3 eV In the free-electron
                                                                                .
calculations (labelled MacLaren), the thickness of the barrier t is 5 Å, 20 Å, and 200 Å (same
barrier height). The inset schematically shows k conservation used in the free-electron model.
(b) Energy versus density-of-states used in free-electron calculations, showing the parabolic
bands for majority and minority electrons on each side of the insulating barrier. EF is the
Fermi level. In the calculations the bias voltage is assumed to be small, eV         φ. Adapted from
MacLaren et al. (1997).


review paper of Zhang and Butler (2003). Along with that, experimental evidence
has become gradually available that shows, e.g., the decisive role of the barrier-
electrode combination for spin-polarized tunneling. Other exciting observations
have been reported, such as the role of crystallinity and orientation of the magnetic
electrode, oscillations in TMR due to the presence of nonmagnetic layers favoring
quantum well states, and unprecedented, giant TMR in junctions when incorporat-
ing half-metallic electrodes or, more recently, crystalline MgO barriers. Especially
in sections 3 and 4 these developments will be extensively addressed.

1.4 Scope of this review
It is the purpose of this review to introduce the reader to the most important as-
pects of spin-polarized tunneling. We have just seen that spin polarization in MTJs
is a complex parameter heavily dependent on the details of the potential the elec-
trons experience when crossing the barrier region. This in turn strongly influences
the fabrication process of MTJs, where obviously utmost care should be taken in
designing and characterizing the barrier and the interface regions with the ferro-
magnetic metals. Barriers in MTJs are traditionally made out of oxidized Al for their
relative ease to create superior coverage of the metallic electrode, together with the
observation of large magnetoresistances. A huge research effort could be witnessed
in the late nineties to optimize the oxidation process for enhancing the function-
ality and reliability of MTJs. Furthermore, a number of oxidation methods have
been explored in great detail, in particular the use of an oxygen plasma to gradually
14                                                                        H.J.M. Swagten



oxidize a previously deposited Al layer. The physical properties and optimization of
the barrier and adjacent ferromagnetic layers are the topic of section 2. Also in this
section the basic design rules for a magnetic junction will be discussed together with
elementary transport properties, such as the dependence of TMR on bias voltage,
barrier thickness, and temperature.
    In section 3 we return to the physics of tunneling spin polarization. Details
will be given on the experimental method involving superconducting probe lay-
ers, followed by a more in-depth discussion on the basic fundamental ingredients.
Topics of interest are the relation between tunneling spin polarization and the fer-
romagnetic magnetization, the relevance of the barrier-electrode interface region
including the local chemical bonding, and the relevance of the symmetry of the
wave functions of tunneling electrons. Section 4 reviews a number of crucial exper-
iments in the field of TMR in magnetic junctions. Especially those topics will be
highlighted that have contributed to the understanding of the underlying physics of
spin tunneling, e.g., addressing the role of the interfaces with the barrier, the (lo-
cal) density-of-states of the magnetic layers, half-metallic or epitaxial ferromagnetic
electrodes, and tunneling across crystalline barriers (such as MgO or SrTiO3 ). The
review will be concluded by briefly considering some of the promising directions
within this field of magnetic junctions or, in a wider perspective, the field of hy-
brid devices where tunnel barriers are often combined with new materials to create
new physics or functionality. This includes, for example, the development of all-
semiconductor MTJs, the use of magnetic semiconductors as (spin-filter) barriers,
and the realization of three-terminal magnetic tunnel transistors.



     2. Basis Phenomena in MTJs
      The fabrication of a properly operating insulating tunnel barrier, separating
the magnetic electrodes, has developed as a wide and very active research field
where many aspects on oxide growth, characterization, magnetism, and transport
are being considered. Although there are several ways to fabricate barriers for MTJs,
a clear distinction can be noticed between crystalline and amorphous barriers. The
amorphous Al2 O3 barriers are most extensively studied due to the ability to serve
as an excellent barrier with a sufficiently small density of pinholes (i.e., electrical
shorts between top and bottom metallic electrode). Usually, alumina barriers are
created by depositing a thin Al layer that is subsequently oxidized by thermal (nat-
ural) or plasma-enhanced oxidation. Figure 1.8 shows a prototypical example of the
magnetic-field dependence of the resistance in a magnetic junction consisting basi-
cally of FeMn-Co-Al2 O3 -Co, with the alumina barrier formed by plasma-oxidizing
an Al layer. The tunneling resistance or current across the Al2 O3 barrier is measured
in the so-called 4-point geometry by contacting the bottom and top electrodes as
indicated in Fig. 1.8c. Recently, there is an increasing amount of studies focusing
on junctions with crystalline or even epitaxial barriers, such as the widely investi-
gated MgO and SrTiO3 . In some cases, this yields magnetoresistance ratios superior
to those with alumina barriers with the added advantage to be able to accurately
Spin-Dependent Tunneling in Magnetic Junctions                                          15




Figure 1.8 In (a) the room-temperature magnetoresistance is shown of an 0.4 × 0.4 mm
MTJ fabricated with UHV magnetron sputtering through metal shadow masks. The ar-
rows indicate the orientation of magnetization. The structure is schematically shown in (b):
Si(100)/SiO2 /50 Å Ta/50 Å Co/100 Å FeMn/35 Å Co/23 Å Al + oxidation/150 Å Co/50 Å Ta.
The top-view layout of these junctions in (c) indicates the actual 4-point geometry for the
resistance measurement. After LeClair (2002).


model the transport processes in these better defined systems. The discussion on
other, crystalline barriers materials will be postponed for section 4. On the other
hand, Al2 O3 -based junctions are a perfect playground to address a great number of
basic physics in magnetic junctions, let alone the huge interest from industrial labs
for the incorporation of these barriers in MTJ-based sensors and magnetic memo-
ries (see, e.g., Parkin et al., 2003).
    In this section, a number of basic phenomena in Al2 O3 -based MTJs will be
reviewed, including the most relevant fabrication and characterization tools. The
topics are:
• basic properties of MTJs, emphasizing general tunneling transport characteristics,
  methods for switching the magnetization in junctions, and the basic behavior of
  TMR
• oxidation of ultrathin metal layers, such as plasma and natural oxidation, in rela-
  tion to the performance of TMR devices
• optimizing barriers for TMR: under- and over-oxidation, pinholes, dielectric
  breakdown, thermal stability, and alternative amorphous barriers.
    In somewhat more detail, the first part (section 2.1) introduces the basic voltage
dependence of tunneling current in relation to the thickness and electron potential
of the insulating barrier, supplemented with a few experimental examples. We will
also shortly focus on the magnetization reversal of the magnetic layers, aiming at
the realization of two macroscopic, magnetically stable states of the ferromagnetic
layers: antiparallel versus parallel. As we have seen in section 1, in these two states
the total tunneling current (sum of spin-up and spin-down current) is essentially
different in a magnetic junction. In the example of Fig. 1.8 the magnitude of the
resistance change is more than 25% (using [RAP – RP ]/RP , Eq. (8)) when switching
16                                                                       H.J.M. Swagten



from the parallel to the antiparallel state, which is accomplished by using so-called
exchange biasing, one of the most widespread magnetic engineering tools. The
basic behavior of the magnitude of the magnetoresistance effect will be discussed
next in section 2.1, for instance focusing on how TMR depends on oxide thickness,
temperature, and bias voltage.
    The second part of this section is devoted to the oxide layer that is sandwiched
between the ferromagnetic layers (section 2.2). In view of the fact that the current
is exponentially dependent on thickness and height of the barrier (apart from many
other details), the preparation and characterization of the oxide layers is the most
critical step in the junction fabrication. The available oxidation procedures will be
reviewed mainly in relation to the magnitude of TMR and the resistance R of a
magnetic tunnel junction, both rather crucial when assessing device applications for
MTJs (see for instance the introduction on MRAM in section 1). This is followed
in section 2.3 by considering a number of key issues in this area, including over-
and underoxidation of ultrathin Al layers, the role of metallic shorts or pinholes and
dielectric breakdown when the barrier becomes extremely thin, thermal stability of
MTJs for processing or operation at elevated T , and the use of alternative barriers
to further tune the device (magneto)resistance. Finally, it is worth mentioning that
the use of a great number of experimental tools will be discussed in this section
(in particular in section 2.3.1), such as X-ray photoelectron spectroscopy (XPS),
Rutherford backscattering spectroscopy (RBS), transmission electron microscopy
(TEM), ballistic electron emission microscopy (BEEM), and optical or ellipsometric
characterization. All these tools have added considerably to the understanding of
how physical or chemical properties of the barrier are related to the tunneling
transport.

2.1 Basic magneto-transport properties
This section reviews the basic experimental observations in electrical transport and
magnetic behavior of magnetic tunnel junctions, and is aiming at explanations
mostly on a phenomenological level. These observations can be summarized as
follows:
• tunneling current I is nonlinear in applied bias voltage V
• conductance dI /dV is approximately parabolic in voltage V , expect for small
  bias
• resistance R at low bias scales inversely with junction area A, and grows expo-
  nentially with barrier thickness
• magnetization M of two ferromagnetic layers adjacent to the barrier can be
  switched independently by several magnetic engineering methods
• TMR is rather independent of barrier thickness t, except for extremely thin
  barriers
• TMR decays with temperature T and with applied bias voltage V .
  As mentioned before, more advanced approaches to address the underlying
mechanisms for spin-polarized tunneling will be reviewed in section 3 and section 4.
Spin-Dependent Tunneling in Magnetic Junctions                                          17


2.1.1 Tunneling transport in junctions
It is shown in Eqs. (3)–(5) that a net tunneling current is induced across a tunnel
junction when applying a finite bias voltage between the ferromagnetic electrodes
of an MTJ. A straightforward I (V ) measurement is a useful tool to directly assess the
existence and properties of the tunneling barrier. The Ohmic behavior as derived in
Eqs. (3)–(5) is only valid for small applied bias voltage, and should be reconsidered
for higher voltages where the I (V ) curve becomes essentially non-linear. For sym-
metric tunnel junctions with identical electrodes, Simmons (1963) has analytically
calculated the tunneling current using the WKB approximation (see Eq. (2)) which
is valid for thick and high barriers:
                               αA      eV                       eV
                    I (V ) =      2
                                    φ–            exp –βt φ –
                                t       2                        2

                                   αA      eV                        eV
                               –      2
                                        φ+          exp –βt φ +         ,            (12)
                                    t       2                         2
with, α = e/(2πh), β = 4π 2m∗ /h (m∗ the effective electron mass in the barrier
                                    e       e
conduction band), V the applied voltage, t the barrier thickness, A the barrier area,
                                                               t
and φ the average barrier height above the Fermi level 0 [V (x) – EF ] dx /t. Here
we neglect the effect of the image charges on the shape of the barrier potential
(Simmons, 1963), which, due to the tendency to round off the potential at the
outer edges of the insulator, leads to an increase of tunneling current; see Hirai et
al. (2002) for data on MTJs.
     The Simmons equation is later adapted by Brinkman et al. (1970) to include
an asymmetry in the barrier potential, with φ the potential difference between
right and left electrode. Generally speaking, the potentials the electrons experi-
ence when transported across a junction is not automatically symmetric in space.
First of all, when employing two different metallic electrodes, their nonequal work
functions will create an electrical field across the barrier, leading to an intrinsically
tilted barrier potential. Apart from that, the barrier itself is often intrinsically asym-
metric related to the preparation. For instance, when oxidizing an Al thin film by
a post-growth oxidation process, the stoichiometry of the oxide may vary in the
direction perpendicular to the layer planes due to over- or under-oxidation. More-
over, this oxidation procedure may create different interfaces with the electrodes,
by which, even when using the same electrode materials, the asymmetry almost
naturally arises. We will come back to this in section 2.3. The tunneling current
for asymmetric barriers is approximated by a Taylor expansion to the third power
(Brinkman et al., 1970):
                                        βe t φ             β 2 e2 t 2
                     I = R0 V –
                          –1
                                                 V2 +                 V3 .           (13)
                                        48 φ 3/2            96 φ
R0 is the Ohmic low-bias resistance of the junction given by:
                                            2t exp(βt φ)
                                     R0 =                  ,                         (14)
                                                 eAαβ φ
18                                                                       H.J.M. Swagten



which, as expected, scales inversely with area, and rapidly grows with thickness
and height of the barrier. In the case of φ = 0, the current is cubic in applied
voltage V , equivalent to a parabolic conductance, one of the basic properties of
transport across tunneling barriers:
                               dI   1          β 2 e2 t 2
                         G≡       =    +                  V 2.                    (15)
                               dV   R0         32R0 φ
The quadratic increase in conductance is in principle valid only for small V and sim-
ply reflects the fact that the effective barrier height becomes smaller when a voltage
is applied across the junction. At higher voltages, however, higher order terms in
Eq. (13) have to be included, and eventually at energies eV exceeding the barrier
height, also the effect of a reduction of the effective barrier width (Rottlander et
al., 2002).
     In experimental studies on MTJs, these formulas given by Simmons (1963) and
Brinkman et al. (1970) have been extensively used to characterize the barrier char-
acteristics, viz. the barrier height, including its asymmetry, and the thickness of
the barrier. It should be kept in mind, however, that these formulas are based on
free-electron-like calculations using single parabolic bands for the metallic elec-
trodes. This means that the spin-dependence of the density-of-states of the magnetic
electrodes is not explicitly incorporated, which is clear from the absence of these
density-of-states factors in Eqs. (12)–(15). One can show that this is related to the
fact that the group velocity of electrons at EF (which determines the rate of attempts
to penetrate the barrier) decreases inversely proportional to the density-of-states at
the Fermi level; see also the discussion by Harrison (1961).
     The widespread use of these equations can be explained by the possibility to at
least compare the barrier parameters of junctions grown in different laboratories,
and offers a first-order indication of the quality of the tunneling transport of an
MTJ. One example out of the rich existing literature is given in Fig. 1.9a, showing
the predicted parabolic conductance, in this case of a CoFe-Al2 O3 -CoFe junction
(Oliver and Nowak, 2004). At low bias an additional anomalous conductance is ob-
served especially at low temperatures, which we will discuss further in section 2.1.4.
The extracted barrier thickness and barrier height are shown in Figs. 1.9b and 1.9c,
respectively. The different parameters in parallel and anti-parallel case are directly
related to the presence of spin-dependent tunneling, since, as we argued before, the
Simmons or Brinkman equations do not contain density-of-states factors, and the
conductance is entirely determined by t and φ. The deviations in t and φ when
temperature is beyond 200 K are indicative for the presence of additional conduc-
tion processes, such as an inelastic, spin-independent hopping conductance that is
dependent on both voltage and temperature (Oliver and Nowak, 2004). In another
case (Dorneles et al., 2003), the fitted barrier thickness and area of a Al-Al2 O3 -Al
junction is found to deviate considerably from the actual nominal values (e.g. ob-
tained from X-ray diffraction or transmission electron microscopy). This hints to a
tunneling process governed predominantly by so-called hot spots, small areas where
the barrier thickness or barrier height is effectively much smaller than for the re-
maining part of the junction. Due to the exponential growth of tunnel resistance
with t (see Eq. (14)), the slightest corrugation at the barrier-electrode interfaces
Spin-Dependent Tunneling in Magnetic Junctions                                              19




Figure 1.9 (a) Parallel and anti-parallel conductance G = dI /dV at T = 5 K of a junction
consisting of 50 Å Ta/250 Å PtMn/22 Å CoFe/9 Å Ru/22 Å CoFe/5 Å Al + oxidation/10 Å
CoFe/25 Å NiFe/150 Å Ta. The solid lines are parabolic fits using the Brinkman expression
(Eq. (13)). From these fits the temperature dependence is extracted of (b) the barrier thickness
and (c) the average barrier height. After Oliver and Nowak (2004).


leads to lateral fluctuations in the barrier thickness, by which the current will be
almost completely dominated by these hot spots. It was shown theoretically by
Bardou (1997) that even when a barrier is controlled in the Ångstrom regime the
tunneling transport can be governed by just a few probable paths due to statisti-
cal fluctuations. Moreover, when metallic shorts (pinholes) are present in junctions
with extremely thin oxides, the barrier parameters are further obscured by a parallel
metallic-like current shunting the true tunneling processes (Akerman et al., 2001).
In that case extracting parameters by fitting to Eqs. (12) or (13) is clearly losing its
physical significance (see also section 2.3.3).
    A clear demonstration of the ambiguities involved in extracting barrier parame-
ters is facilitated by internal photoemission studies, from which the barrier height in
thin-film tunneling structures can be adequately extracted. For early experiments in
this direction see, e.g., Kadlec and Gundlach (1976), Nelson and Anderson (1966),
and Crowell et al. (1962). Conceptually, the technique is rather straightforward, see
Fig. 1.10b. One shines monochromatic light onto a junction structure, and mea-
sures the resulting photocurrent. The incident photons will excite electrons in the
electrodes, gaining an amount of energy equal to the photon energy. When elec-
trons are photo-excited to an energy higher than the internal barrier height φ, some
of the electrons will be able to enter the conduction band of the insulator. After
leaving the barrier at the other side, they will be responsible for a net photocur-
rent when the opposite contributions from the two electrodes do not cancel. From
the onset of this current as a function of the photon energy, the barrier height can
be accurately determined (shown in Fig. 1.10a), and is strongly deviating from the
barrier potential as derived from fits to the Brinkman equation (Koller et al., 2003).
    Lateral fluctuations of the tunneling current can be adequately addressed by
scanning probe microscopies. Costa et al. (1998) are the first to use the atomic
force microscope with a conducting tip in contact with a naturally oxidized epi-
20                                                                                H.J.M. Swagten




Figure 1.10 (a) Photoconductance as a function of photon energy for a structure of
glass/35 Å Ta/30 Å NiFe/100 Å IrMn/25 Å NiFe/15 Å CoFe/17 Å Al + oxidation/40 Å
CoFe/100 Å NiFe/35 Å Ta, plasma oxidizing the Al for 200 sec. The light is incident on
the top electrode; no additional bias voltage is applied. The inset shows the (average) barrier
height as extracted from fitting to the Brinkman equation as well as from photoconductance, as
a function of oxidation time. (b) Schematics of photocurrent generation, showing the energy
across a tunnel junction. Electron excitation by light is indicated with hν. EF is the Fermi
energy, φL,R is the barrier height for the left and right electrode. Adapted from Koller (2004).


taxial Co layer to map the strong fluctuations in tunneling current. Ando et al.
(1999, 2000a) have used a more realistic junction without top electrode (Ta-NiFe-
IrMn-Co-Al2 O3 ), from which a wide distribution in barrier height can be directly
determined, favoring tunneling only from a few hot spots in the barrier. In the
atomic-force-microscope studies of Luo et al. (2001) on Co-Al2 O3 , the observed
current fluctuations are attributed to thickness inhomogeneities on a nanometer
scale. In a more advanced approach using ballistic electron emission microscopy
or BEEM (Kaiser and Bell, 1988), the Al2 O3 barrier height can be directly mea-
sured on a local scale. Electrons emitted from a conductive tip are injected into
the metal–insulator–metal system at variable energy as determined by the voltage
between tip and surface. These injected, hot electrons can only pass the Al2 O3 bar-
rier potential when their energy exceeds the barrier height (Rippard et al., 2001;
Kurnosikov et al., 2002). Rippard et al. (2001) use structures containing Al2 O3
grown on top of Si substrates to create a well-defined Schottky barrier of typically
0.8 eV for selecting the hot electrons. In this work, the alumina barrier height
(around 1.22 eV), is found to be rather independent of the deposition method
(sputtering versus evaporation), the nominal Al thickness, and the oxidation con-
ditions. In Fig. 1.11, BEEM images directly demonstrate that local barrier height
fluctuations emerge upon thinning down of the Al thickness from 6.5 Å to around
4.5 Å (before oxidation). Note that BEEM is only sensitive to the local height of
the barrier potential; lateral variations in the barrier thickness are not resolved.
    In follow-up studies, Rippard et al. (2002) combined BEEM with scanning
tunneling microscopy (STM) and scanning tunneling spectroscopy (STS). They
have demonstrated that due to variations in the local atomic structure of ultrathin
Spin-Dependent Tunneling in Magnetic Junctions                                            21




Figure 1.11 Ballistic electron emission microscopy (BEEM) image (a) of an evaporated junction
consisting of Si(111)/75 Å Au/12 Å Co/6–7 Å Al + oxidation/12 Å Co/30 Å Cu. The grey
scale is proportional to the collected current of hot electrons. (b) The BEEM current for a
thinner barrier (4–5 Å Al + oxidation) shows much stronger variations due to an increase of
barrier height fluctuations. Adapted from Rippard et al. (2001).


barriers, low-energy extended electron states may support conduction channels at
energies below the alumina barrier height. This contradicts the common belief that
for ultrathin barriers only metallic pinholes are an important issue for the collapse of
spin-polarized transport properties (see section 2.3.3 for more details on the effects
of pinholes). In another combined BEEM-STM study, Perrella et al. (2002) have
found that mobile O– adsorbates are present on the surface of an oxidized Al layer,
                       2
having localized energy states located 1–2 eV above the Fermi level. By thermal
annealing (or by electron bombardment) it is possible to drive the adsorbates into
the oxide thereby reducing the local transport via these low-energy channels (see
Mather et al., 2005, and also the X-ray photoelectron spectroscopy results of Tan
et al., 2005). This may be important for the fabrication of high-quality MTJs, since
a thermal treatment of a full junction with two electrodes could homogenize the
chemisorbed oxygen that is trapped close to the interface with the oxide layer. As a
consequence, this could increase the effective barrier height and reduce the (unde-
22                                                                              H.J.M. Swagten




Figure 1.12 Junction resistance R versus the area A. (a) Results on Si(100)/200 Å Pt/40 Å
NiFe/100 Å FeMn/80 Å NiFe/10–30 Å Al + oxidation/80 Å Co/200 Å Pt structured with
e-beam and optical lithography. Adapted from Gallagher et al. (1997). (b) Structured junctions
of glass/10 Å Si/100 Å Co/10–14 Å Al+oxidation/170 Å NiFe/40 Å Al by optical lithography
and with shadow evaporation (Boeve et al. (1998)).


sired) oxidation of the top electrode. For optimization studies on MTJs, including
the effects of over-oxidation and annealing; see section 2.3.
     We now return again to the Simmons and Brinkman equations (12)–(15), which
also show that the current I in an MTJ is, obviously, linearly scaling with lateral area
A of a junction. In other words, the resistance should be inversely proportional to
the junction area. This is successfully tested by Gallagher et al. (1997) and Boeve
et al. (1998) for Ni80 Fe20 -Al2 O3 -Co and Co-Al2 O3 -Ni80 Fe20 junctions, by varying
the junction area over up to 5 orders of magnitude using micro-fabrication with e-
beam or optical lithography, or during evaporation with the help of shadow masks.
The area scaling is illustrated in Fig. 1.12. As a natural consequence of this, the
resistance-area product R × A can be considered as an area-independent property
of a magnetic junction, by which different junctions (from various laboratories)
can be compared; see again Eq. (14). In the application of MTJ’s this product of
resistance and area plays a crucial role, since it determines the resistance noise of
the device. When devices are progressively reduced in lateral dimensions (smaller
A), the resistance will naturally rise as well as the thermal or Johnson noise that is
                  √
proportional to RT . Results on noise characterization in MTJ devices, including
the role of low-frequency 1/f noise and the relation to the magnetic switching
can be found in a number of publications; see Nowak et al. (1999); Ingvarsson
et al. (1999, 2000); Smits (2001); Nazarov et al. (2002); Park et al. (2003); Jiang
et al. (2004a). Apart from the noise issue in devices, it is also important that the
read-out speed of a memory or sensor (as determined by the RC time) does not
increase due to a further reduction of the junction area A (Tehrani et al., 2003;
Das, 2003). Furthermore, for an optimal read-out of an MRAM cell, the resistance
of the MTJ should match with the underlying transistor (see Fig. 1.3a). For future
CMOS technology nodes, the required reduction of R × A is roughly scaling with
the typical feature size within CMOS (Das, 2003). These considerations explain
Spin-Dependent Tunneling in Magnetic Junctions                                         23




Figure 1.13 (a) Resistance times junction area R × A and (b) TMR as a function
of nominal Al thickness. The tunnel junctions consist of 90 Å Ta/70 Å NiFe/40 Å
CoFe/t Al + oxidation/30 Å CoFe/250 Å IrMn/30 Å Ta, in which the Al layer is optimally
oxidized by a remote oxygen plasma. The junctions are patterned down to 1 × 2 µm. After de
Freitas (2001).


the huge effort in reducing R × A, e.g. by either reducing the barrier width, or by
exploring alternative (energetically lower) barriers; see section 2.3.5 and section 4.
    Another implication of the Simmons and Brinkman equation is the fundamental
exponential decay of the current (or, equivalently, exponential growth in resis-
tance) with the thickness of the tunneling barrier (Eq. (14)). This is experimentally
demonstrated in Fig. 1.13a, where R × A is plotted against the barrier thickness
for a large number of CoFe-Al2 O3 -CoFe junctions (de Freitas, 2001). Note that
these junctions have also been annealed after the deposition process, which, how-
ever, does not considerable affect the junction resistance. The main purpose of the
post-deposition anneal step is to enhance the TMR as seen in Fig. 1.13b. We will
come back to this later on, see section 2.3.4.

2.1.2 Engineering and switching the magnetic constituents
The central part of an MTJ is a sandwich of two ferromagnetic layers, separated
by a barrier with a thickness usually below 20–30 Å. The ferromagnetic layers
adjacent to the insulating barrier are typically a few nanometer in thickness, and
should be backed with one or more layers to manipulate the magnetic switching.
This is necessary to switch the magnetic orientation from a parallel state of the
two magnetization vectors to an antiparallel state, which is the basic requirement
to observe tunnel magnetoresistance in a ferromagnetic-insulating-ferromagnetic
junction (see section 1.2).
    Creating an antiparallel magnetization state can be realized in several ways,
as shown schematically in Fig. 1.14 for three important magnetic engineering
schemes. The most straightforward realization is the use of two ferromagnetic ma-
terials having different magnetic anisotropy, of which an experimental example has
been shown earlier in Fig. 1.2. The magnetization will be antiparallel in a field
24                                                                                H.J.M. Swagten




Figure 1.14 Magnetic engineering in magnetic tunnel junctions. In (a) two layers are used
with different coercivities HC . Using an antiferromagnetic layer in (b) creates a wide range of
antiparallel orientation governed by Hex . In (c) exchange biasing is combined with antiferro-
magnetic coupling across a metallic spacer to further improve the field range of antiparallel
orientation, together with the magnetic and thermal stability. Note that the schematic behav-
ior of M is shown over a much wider field range as compared to (a) and (b), to fully show
the decoupling of the artificial antiferromagnet governed by the antiferromagnetic coupling
strength JAF .


range HC1 < H < HC2 with HC1,2 the coercivities of the soft and hard magnetic
layer, respectively (Fig. 1.14a). A serious drawback of this engineering scheme has
been reported by Gider et al. (1999). When the magnetization of the softest mag-
netic layer is repeatedly reversed by magnetic field cycling, the other, magnetically
harder layer is progressively demagnetized, equivalent to erasing the MTJ memory
when used in an MRAM. Using Lorentz electron microscopy and micromagnetic
simulations, the hard-layer magnetization decay is found to result from large fringe
fields surrounding magnetic domain walls in the magnetically soft layer (McCartney
et al., 1999). To avoid domain-wall formation and motion, the soft layer can be re-
versed by coherent rotation (Gider et al., 1999), which, in an MRAM architecture,
can be simulated by subsequent switching with two current pulses from two orthog-
onal conduction lines below and above the junction cell (Schmalhorst et al., 2000a).
In another study on Co80 Fe20 -Co-Al2 O3 -Ni80 Fe20 junctions, magnetic interactions
between domains in the soft and hard magnetic lead to the effect of domain du-
plication, which in turn affects the magnetoresistance in these MTJs (Rottlander et
al., 2004).
     Generally, however, in most cases the use of an antiferromagnet in direct ex-
change contact with one of the ferromagnetic layers is preferred above the hard-soft
system. Due to unidirectional anisotropy induced by the antiferromagnetic layer,
the hysteresis loop of the exchange-biased (pinned) magnetic layer will be shifted
in field with respect to the free magnetic layer. This naturally creates an antiparallel
field range between 0 < H < Hex , with Hex the strength of the exchange bias field
Spin-Dependent Tunneling in Magnetic Junctions                                        25


(neglecting the coercivities of the two layers); see Fig. 1.14b. In Fig. 1.8 an example
is given of an elementary exchange-biased MTJ, consisting of a stack of the follow-
ing sequence: SiO2 -Ta-Co-Fe50 Mn50 -Co-Al2 O3 -Co-Ta. The Ta-Co layers grown
directly on top of the substrate reduce the roughness and provide a proper (111)
texture for the FeMn layer, by which an exchange bias field of more than 10 kA/m
is established in this case. Usually these systems are additionally heated above the
blocking temperature of the antiferromagnet, and subsequently cooled in the pres-
ence of an external magnetic field to enhance the interface interactions between
ferro- and antiferromagnet. Other antiferromagnetic layers such as metallic PtMn
or IrMn compounds are frequently used as exchange-biasing materials, in particu-
lar for their better thermal stability (higher blocking temperature). Also insulating,
antiferromagnetic NiO films can be applied to exchange-bias one of the ferromag-
netic electrodes (Shang et al., 1998a), in this case grown by reactive evaporation of
Ni in an oxygen environment. Further details on these procedures as well as on the
physics of magnetic engineering by exchange biasing can be found in review papers
by Nogues and Schuller (1999) and Coehoorn (2003).
    In a third engineering method, the single exchange-biased magnetic layer is re-
placed by an antiferromagnetically coupled sandwich of two ferromagnetic layers
separated by an ultrathin metallic spacer. This is the so-called artificial antiferro-
magnet (AAF) or synthetic antiferromagnet (Sy-AF) as proposed by Parkin (1995)
and used, e.g., by Willekens et al. (1995); see also the review paper of Parkin et al.
(2003). The magnetization behavior of the three ferromagnetic layers is schemat-
ically shown in Fig. 1.14c. Antiparallel orientation of the free and fixed layer on
each side of the barrier is induced when the coupled, fixed layer and exchange-
biased layer are approximately of equal thickness (Strijkers et al., 2000). In that case,
not only the antiparallel field range is superior to the exchange biasing scheme in
Fig. 1.14b, but also the two antiferromagnetically coupled layers are magnetically
stable with minimal stray field that could affect the magnetization of the free layer.
Especially when the lateral dimensions of MTJs become very small in sensor or
memory applications, this magnetic rigidity is crucial. Moreover, when the free
layer and the pinned layer are ferromagnetically coupled due to their correlated
roughness (so-called orange-peel coupling, see Néel, 1962), the antiferromagneti-
cally coupled sandwich of pinned and exchange-biased layer can be tuned (by layer
thicknesses and coupling strength) to optimally control the switching of the free
layer; see, for example, Vanhelmont and Boeve (2004).
    Although these advanced modifications in the junction stack are crucial for engi-
neering the field sensitivity of MTJ-based sensors and memories (Engel et al., 2002;
Parkin et al., 2003; Tehrani et al., 2003; Pietambaram et al., 2004), we will not fur-
ther explain more details here. Issues in the magnetic behavior of (sub)micrometer
MTJs are generally related to the size dependence of the switching (Gallagher
et al., 1997; Lu et al., 1997; Koch et al., 1998; Kubota et al., 2003), the effect
of boundary roughness of small magnetic elements due to the patterning process
(Meyners et al., 2003), dipolar interactions between MRAM cells (Janesky et al.,
2001), thermal stability of the magnetization (Pietambaram et al., 2004), and so
on. Regarding the implementation of MTJs in MRAM technology, the dynam-
ics of magnetization reversal is obviously also of utmost importance. Strategies to
26                                                                                 H.J.M. Swagten




Figure 1.15 In (a) the Stoner–Wohlfarth astroid shows the switching stability as a function
of the normalized hard-axis and easy-axis applied magnetic fields for coherent rotation of
ellipsoidal particles. Measurements of the easy-axis switching field versus hard-axis applied
field of 0.6 × 1.2 µm2 MTJ cells are shown in (b). Open circles are the average switching fields
with applied fields swept quasi-statically. Closed circles are data on the same bit cells, but now
with magnetic field pulses with 20 ns duration. Adapted from Slaughter et al. (2002).


switch the magnetization of the free layer by sending current through the word
and bit lines of the MRAM array (see Fig. 1.3) are being widely developed by
a number of research groups, see, e.g., Lu et al. (1999); Boeve et al. (1999);
Sousa and Freitas (2000); Engel et al. (2002); Slaughter et al. (2002); de Boeck
et al. (2002); Gerrits et al. (2002); Tehrani et al. (2003); Parkin et al. (2003). As
a typical example within this research area, Fig. 1.15 shows the switching fields
of micrometer-size patterned MTJ cells, using both quasi-static and fast current
pulses (Slaughter et al., 2002). Apparently, in the regime of 20 ns pulses, coherent
Stoner–Wohlfarth rotation is still applicable, without the need to consider more
complex dynamical behavior (Koch et al., 1998). To improve the magnetic sta-
bility of switching one particular MRAM cell, without affecting the other cells
along a row, a new scheme has been developed recently. This so-called toggle
MRAM uses an artificial antiferromagnet (see earlier) as the free magnetic system,
leading to a remarkably improved robustness of cell switching (Engel et al., 2005;
Yamamoto et al., 2005).

2.1.3 Electrical measurement of TMR
Usually the resistance or conductance of a magnetic junction is determined from
a 4-terminal measurement. A power supply (or current source) is connected to
the bottom and top electrode and one measures the tunneling current (or voltage
difference) between the other two terminals; see Fig. 1.8c. An important possible
pitfall of such a conductance measurement on a MTJ is related to a laterally inho-
mogeneous current flowing through the barrier when the resistance of the barrier
Spin-Dependent Tunneling in Magnetic Junctions                                       27


is too low as compared to the resistance of the electrode. This is first recognized by
Moodera et al. (1996) demonstrating a strong artificial increase of TMR, and is later
verified by van de Veerdonk et al. (1997b) using a finite-element approach to model
the current crowding in the barrier region of a tunneling device. As discussed by
Moodera et al. (1996), also the pioneering data by Miyazaki and Tezuka (1995a)
are suffering from an apparent amplification of TMR. Sun et al. (1998a) report on
geometrically enhanced TMR in mm2 -size junctions when the junction resistance
is less than 5 times the resistance of the electrode over the junction area. In another
regime, when the junction radius is much smaller than the width and length of the
leads, Chen et al. (2002) have developed an analytical method to correct for the
artificial changes of R × A and TMR in such a device-like geometry.
     It is interesting to mention that the tunnel conductance or resistance can be
measured also without the need for two electrodes defining the tunneling area for
electrical transport. By applying a voltage across in-line contacts touching the top
of a planar (unpatterned) tunneling structure, a current will not only flow through
the top conducting layers, but partially also via the tunneling barrier through the
bottom part of the stack. It is shown by Worledge and Trouilloud (2003) that four
micrometer-spaced probes can be used to reliably determine the (field-dependent)
resistance of an MTJ, which is further refined to reduce the experimental errors
involved in the positioning of the probes (Worledge, 2004). Especially for testing
MTJ devices on a full wafer level, this method is believed to be extremely fast and
convenient in assessing, e.g., the uniformity of the resistance or switching fields over
a large area.

2.1.4 TMR: basic behavior, role of bias voltage and temperature
In this part three subjects will be treated. First, some basic characteristics of TMR
will be shortly reviewed on a phenomenological level, such as the experimental
relation with the tunneling spin polarization P , the dependence of TMR on the
thickness of the barrier, and the effect of annealing. The use of CoFeB compounds
as an alternative magnetic electrode material will be discussed in some detail for
its intriguing capability to considerably enhance TMR. Thereafter both the tem-
perature dependence and the bias voltage dependence of TMR will be considered
along with an introductory survey of the mechanisms proposed to explain these
experimental data.
Basic behavior of TMR, including the use of CoFeB In section 1 it is derived that
the magnitude of the magnetoresistance in MTJs is directly determined by the
tunneling spin polarization via TMR = 2P1 P2 /(1 – P1 P2 ). Although the physics
behind the polarization P is far from understood (and will be further explored in
section 3), it is clear that tunneling spin polarization of the electrodes at the inter-
face with the barrier offers a direct way to tune TMR. For instance, Cox Fe1–x and
Nix Fe1–x are frequently applied in actual devices because of their high values of po-
larization and TMR (see Kikuchi et al., 2000 for the effect of CoFe composition on
TMR). In Fig. 1.13b it is shown that Co80 Fe20 -Al2 O3 -Co80 Fe20 junctions display
TMR of 40% at room temperature for nominal Al thicknesses above approximately
8 Å (before oxidation). The rather constant TMR for increasing alumina thick-
ness is a common observation in amorphous Al2 O3 -based MTJs, also when using
28                                                                         H.J.M. Swagten



other ferromagnetic electrodes. For thinner barriers it is generally observed that
TMR is suppressed, see again Fig. 1.13b, most likely due to the increasing density
of metallic shorts (see section 2.3.3). It is also observed that annealing of junc-
tions, up to roughly 200–300°C, greatly improves the TMR (Parkin et al., 1999b;
Freitas et al., 2000). This effect has been attributed to a redistribution of the oxygen
in the alumina barrier (Sousa et al., 1998) possible combined with a change of the
interface structure. We will get back to this later on (section 2.3.4).
     Recently, the magnetoresistance for Al2 O3 -based junctions is significantly im-
proved by the use of CoFeB as a soft ferromagnetic electrode material, for instance
by sputtering it from a target with composition Co73.8 Fe16.2 B10 (Cardoso et al.,
2004; Ferreira et al., 2005a), or from a Co60 Fe20 B20 target (Wang et al., 2004;
Dimopoulos et al., 2004a; Wiese et al., 2004). In the paper of Wang et al. (2004),
a room-temperature TMR of 70.4% is achieved which would translate to a tun-
neling spin polarization of around 51%, probably even higher at low temperatures.
Indeed, for Co72 Fe20 B8 , Paluskar et al. (2005b) measure a tunneling spin polariza-
tion of +53.5% using superconducting junctions which is above the polarization of
all other 3d elements or compounds (for more details see section 3). Currently, stud-
ies are aiming at understanding these high TMR effects, which may stem from the
as-deposited amorphous character of CoFeB, possibly reducing the roughness of the
bottom electrode and improving the interface quality (Dimopoulos et al., 2004b;
Bae et al., 2005). Upon annealing up to around 300–400°C, it is observed that
these systems may become crystalline depending, e.g., on the composition (B con-
tent), film thickness (Wiese et al., 2004; Cardoso et al., 2005), or on the character
of the adjacent layers (Bae et al., 2005). However, it is shown by Paluskar et al.
(2005b) that the tunneling spin polarization of their thick CoFeB films remains
almost unaffected by annealing in ultra-high vacuum conditions. It could be that
the electronic structure of CoFeB is not very sensitive to (amorphous–crystalline)
structural changes as suggested by first-principle calculations on Fe-B alloys (Hafner
et al., 1994). Bae et al. (2005) have used MTJs with three different bottom pinned
electrodes, Co32 Fe48 B20 , CoFe-Co32 Fe48 B20 -CoFe, and CoFe. In the former two
B-containing electrodes, the TMR appears to be higher than for the electrode with
only CoFe (after annealing). Given the fact that the tunneling spin polarization is
determined by the interface with Al2 O3 only (section 3), the authors suggest that
the surface flatness and interface quality may be rather important for obtaining high
TMR with CoFeB electrodes.
     In section 4, junctions combining CoFeB electrodes with crystalline MgO bar-
riers will be further discussed. These materials turn out to be superior for their
enormous magnitude of TMR. The remainder of the data described in this section
will be dealing with electrodes not containing these CoFeB electrodes, but rather
traditional 3d elements or compounds such as Co, CoFe, and NiFe covering the
majority of existing papers in this field.
Temperature dependence of TMR The temperature dependence of the (mag-
neto)resistance in MTJs has received enormous attention, both for fundamental
interest as well as for applications in sensors and MRAMs operating almost ex-
clusively at room temperature. In a similar way, also the transport behavior at a
non-zero, finite applied bias voltage is extremely relevant, which will be the topic
Spin-Dependent Tunneling in Magnetic Junctions                                              29




Figure 1.16 Temperature and voltage dependence of TMR in a junction consisting of
Si(100)/SiO2 /50 Å Ta/50 Å Co/100 Å FeMn/35 Å Co/23 Å Al + oxidation/150 Å Co/50
Å Ta. In (a) the low-bias normalized TMR (with respect to low T ) is shown as a function
of temperature, together with the tunnel resistances in parallel and anti-parallel orientation.
Panel (b) shows the voltage-dependence of TMR at T = 5 K. Both the resistance change
(RAP – RP )/RP and conductance change (GP – GAP )/GAP are shown. V1/2 corresponds to the
bias voltage where TMR has dropped to 50% of its zero-bias value. After LeClair (2002).


of the following subsection. Generally, three processes are believed to somehow
contribute to the T dependence of TMR:
• a thermal reduction of magnetic moment (polarization) at the barrier interface,
  directly affecting the magnitude of TMR
• inelastic tunneling due to electron-magnon (spin-wave) scattering at the barrier
  interfaces
• thermally-assisted hopping conductance via impurities or defect states located in
  the barrier region.
    In Al2 O3 -based magnetic junctions, the temperature dependence of the mag-
netoresistance is intensively studied, and it is generally seen that TMR gradually
decreases with temperature. Figure 1.16a shows that TMR (for low bias voltages) is
reduced by more than 25% when heating the junction from T = 5 K to T = 300 K,
which is derived from the change in the parallel and antiparallel resistance (see the
figure).
    To understand why TMR is reduced for higher T , one should first of all real-
ize that an increasing temperature broadens the Fermi distribution of the tunneling
electrons, which allows electrons with higher energies to tunnel across the barrier.
As long as kB T       φ, a criterion well fulfilled at room temperature, this leads to an
increase of the low-bias tunnel conductance as G(T )/G0 = CT / sin(CT ), with
C = (2π 2 kB t /h)(2me /φ)1/2 , and with G0 equal to 1/R0 in Eq. (15). This, how-
ever, corresponds to a resistance drop of only a few percent between 0 and 300 K
for realistic values of barrier thickness and height, in contrast to experimentally ob-
served changes in RP and RAP (see, for example, Fig. 1.16a). A first approach in
further understanding the decaying TMR is proposed by Shang et al. (1998b). The
30                                                                                 H.J.M. Swagten



zero-bias conductance of a magnetic junction is written as:
                                    G0 CT
                    GP,AP (T ) =            [1 ± PL PR ] + Ginelastic (T ).                 (16)
                                   sin(CT )
The prefactor of the first term G0 CT / sin(CT ) is the aforementioned enhanced
tunneling conductance by smearing of the Fermi functions, PL,R is the tunneling
spin polarization of the left and right electrode, where the “+” refers to parallel
oriented magnetizations, “–” to antiparallel.
     The second term in Eq. (16) is representing a spin-independent inelastic con-
tribution to the current, and is believed to originate from hopping conductance
via imperfections in the Al2 O3 barrier. The role of scattering by impurities in the
barrier is separately studied by Jansen and Moodera (2000) in artificially doped
barriers, e.g. by plasma oxidizing an Al-Si-Al trilayer, with a Si thickness of 0.5–
2.0 Å. When using magnetic ions (Ni instead of Si), the inelastic nature of spin
scattering is reflected in a more pronounced temperature dependence of TMR.
Returning to the analysis of Shang et al. (1998b), it is instructive to calculate the
magnetoresistance from Eq. (16) using the definition given in Eq. (8), yielding
TMR = 2PL PR /[1 – PL PR + Ginelastic (T )/G(T )]. This explains the reduction of
TMR with temperature whenever a nonzero inelastic tunneling term is present.
Apart from that, also the polarization PL,R itself is depending on temperature which
is shown theoretically by MacDonald et al. (1998). Due to the presence of thermally
excited spin waves the polarization in the Julliere formula can be effectively writ-
ten as P (T ) = P0 [m(T )/m0 ] with m(T ) the saturation moment at the interface
of the ferromagnetic layer with the barrier, and P0 and m0 the zero-temperature
polarization and magnetic moment, respectively. Using the approach captured by
Eq. (16) including the polarization suppression by thermally excited spin waves,
a good agreement with temperature-dependent experiments has been reported by
Shang et al. (1998b).
     Another approach to model the temperature dependence of tunnel magnetore-
sistance is given by Davis et al. (2001). In this case, tunneling is treated purely
elastically within a free-electron model (Slonczewski, 1989, see also section 1.3)
without incorporating additional inelastic conduction channels. In a free-electron
model the tunneling spin polarization of the (Fermi) electrons can be expressed as
(kF ,maj –kF ,min )/(kF ,maj +kF ,min ), with kF ,maj,min = [2m∗                       ¯ 1/2 , m∗
                                                               maj,min (EF –Umaj,min )/h ]
                                                                                        2

the effective electron mass, and Umaj,min the bottom of the exchange-split parabolic
bands. The temperature dependence of TMR now arises from the T dependence
of the exchange splitting Umaj – Umin and is reported to be nearly proportional to
M(T ) (Shimizu et al., 1966). From fitting the model calculations to experimental
data (Davis et al., 2001), it is shown that a small drop in magnetization between
0 and 300 K may lead to a substantial variation of TMR in accordance with the
experiments. This implies a rather prominent role of intrinsic band structure effects
in understanding the T -dependence of transport in MTJs.
     In an alternative theoretical approach (Zhang et al., 1997a), the reduction of
TMR with temperature is described in terms of inelastic magnon (spin wave) scat-
tering. By the emission or absorption of magnons during the tunneling process
Spin-Dependent Tunneling in Magnetic Junctions                                        31


across the insulating barrier (involving a reversal of spin), TMR is more efficiently
reduced with temperature than for elastic tunneling only. Using a detailed analysis
of the conductance of exchange-biased junctions, Han et al. (2001) have found an
excellent agreement with the magnon-assisted inelastic excitation model, which in-
cludes a proper description of the bias voltage dependence of TMR. We will return
to the model of Zhang et al. (1997a) below.
Bias-voltage dependence of TMR Since the discovery of magnetoresistance in
alumina-based junctions, the significant suppression of TMR with increasing bias
voltage V has been subject of a great number of experimental and theoretical stud-
ies. In Fig. 1.16b the typical reduction of TMR with applied bias voltage in a
Co-Al2 O3 -Co junction is shown using two different representations, viz. as R /RP
and as G/GAP . Obviously, the resistance and conductance change only coincide
at sufficiently small bias voltage when R = V /I is identical to 1/G = dV /dI .
The suppression of TMR with voltage is critically important when operating MTJs
devices at finite voltage, and a huge research effort is seen in optimizing and under-
standing the decay of TMR. Usually the voltage where TMR = R /RP is reduced
by 50%, indicated in the figure by V1/2 , is taken as a representative fingerprint of
the bias-voltage dependence. From the huge amount of reports on the bias-voltage
dependence of TMR, it is seen that V1/2 is typically in the order 0.3–0.6 V in
Al2 O3 -based magnetic junctions.
    As to the explanations of the V dependence, several mechanisms have been
proposed so far:
• spin-mixing due to electron-magnon scattering in the magnetic electrodes, at the
  interfaces with the barrier
• additional tunnel conductance channels provided by defect and impurity states in
  the barrier region
• intrinsic modification of the barrier shape, combined with the spin-dependent
  band structure of the magnetic electrodes.
    In Eq. (15), it is shown that in the WKB approximation the conductance
of a tunnel junction is quadratic in voltage, as experimentally observed for high
enough voltages (see Fig. 1.9). At low bias voltage, however, both the conductances
in parallel and anti-parallel configuration strongly deviate from the parabolic law,
and a quasi-linear, so-called zero-bias anomaly is universally observed in Al2 O3 -
containing MTJs. Zhang et al. (1997b) and Bratkovsky (1998) have shown that the
excess energy of the tunneling electrons as provided by the applied voltage is capable
of collectively excite magnons at the ferromagnet-barrier interface, thereby induc-
ing an additional inelastic conductance contribution that is linear in bias voltage, viz.
G(V ) ∼ V for voltages V        kB TC /e with TC the Curie temperature of the mag-
netic electrode. Due to the reversal of the electron spin associated with the creation
of a magnon, TMR is naturally decaying with voltage in this regime. For higher
voltages, the lifetime of the magnons becomes too short and the additional inelastic
conductance levels off upon further increase of V . Han et al. (2001) have carefully
measured the bias dependence of exchange-biased Co75 Fe25 -Al2 O3 -Co75 Fe25 junc-
tions, not only measuring I (V ) and dI /dV curves, but also measuring d2 I /dV 2 ,
32                                                                          H.J.M. Swagten



so-called inelastic tunneling (IET) spectra. Using the magnon-assisted inelastic exci-
tation model of Zhang et al. (1997b), their data are reasonable well captured by the
calculations, provided that the wavelength-cutoff energy of the spin-wave spectrum
is different for parallel and anti-parallel magnetization (Han et al., 2001).
    From a theoretical point of view, also mechanisms other than (interface) mag-
netic excitations have been proposed to explain the suppression of TMR with
voltage. This includes the effect of the intrinsic band structure, and impurities in the
barrier. The latter contribution is specifically addressed by intentionally adding a δ-
doped ultrathin layer within the Al2 O3 barrier (Jansen and Moodera, 1998, 2000).
Only in the case of magnetic impurities, a stronger bias dependence has been ob-
served and is attributed to spin-exchange scattering. Intrinsic band structure effects
can be understood by realizing that already in elementary free-electron calcula-
tions the TMR is decaying with voltage (Zhang et al., 1997b); see section 1.3 and
Fig. 1.7 for details on the free-electron model. This is due to the fact that the overall
conductance is enhanced by applying a significant bias, simply due to an effectively
reduced barrier height by tilting the barrier potential with voltage. On the other
hand, the difference between conductance in the parallel and anti-parallel orienta-
tion is only slightly affected, since the voltage adds additional energy dependencies
to the density-of-states (or spin polarization) thereby diminishing the imbalance
between the number of majority and minority tunneling states. Assuming that tun-
neling in Fe-Al2 O3 -Fe is dominated by a single free-electron-like spin-resolved d
band, Davis and MacLaren (2000) have found a fair agreement with the data of
Zhang and White (1998), suggesting that the behavior of TMR with applied volt-
age has an intrinsic component resulting purely from the underlying electronic
structure. This is corroborated by free-electron calculations and experiments on the
bias dependence of Co-Al2 O3 -Co junctions (Xiang et al., 2002, 2003), in which a
reasonable variation of the Co density-of-states over energy is required to describe
the resistance and TMR over a broad range of bias voltages. Again this hints to the
relevance of intrinsic electronic properties for the bias dependence of spin tunneling
(see section 4.3 for a further discussion of other experimental results).
    Finally, it is expected that by applying a bias voltage across a magnetic junction,
it should be possible to extract specific density-of-states features of the ferromag-
netic electrodes from the conductance or TMR. However, this turns out to be far
from trivial, and only a limited number of experimental studies are available. Excel-
lent examples are reported for junctions containing, e.g., epitaxial La2/3 Sr1/3 MnO3
electrodes, or Fe combined with MgO barriers. This will be discussed in section 4.
Another point of interest for the bias dependence is raised by the experiment of
Valenzuela et al. (2005). They have produced a lateral double-barrier tunneling
device basically consisting of CoFe-Al2 O3 -Al-Al2 O3 -NiFe, where the Al is later-
ally extended, separating the ferromagnetic electrodes and tunnel barriers over a
distance between 1500 and 10 000 Å. Due to this, the spin-dependence of the elec-
trons tunneling out of one electrode and tunneling into the other electrode can be
disentangled. From their experiments it is suggested that tunneling into the empty
states of the ferromagnetic electrode is dominating the reduction of TMR with
increasing bias voltage, probably due to the intrinsically reduced polarization of the
Spin-Dependent Tunneling in Magnetic Junctions                                      33


hot electrons and the matching of the wave functions at the interfaces with the
tunneling barrier (Valenzuela et al., 2005).

2.2 Oxidation methods for Al2 O3 barriers
The breakthrough of high room-temperature magnetoresistance in MTJs as re-
ported by Miyazaki and Tezuka (1995a) and Moodera et al. (1995) is strongly related
to the successful fabrication of well-controlled, uniform tunneling barriers. Many
investigations related to the search for MTJs with improved properties like high
TMR, low RA product, large V1/2 and breakdown strength, and strong thermal
stability, are intimately connected to improved control over the barrier region. As
we have seen in the introduction, this is due to the physics of TMR and tunneling
spin polarization, determined primarily by the barrier and the interfaces with the
ferromagnetic electrodes. Consequently, a careful control over the barrier and in-
terface regions is indispensable. Related to this, a wide variety of barrier oxidation
and preparation techniques has been explored since the pioneering experiments,
which includes:
•   plasma oxidation, using a DC or RF-generated O-plasma
•   ion-beam oxidation
•   thermal or natural oxidation in an O2 atmosphere
•   UV-light assisted oxidation
•   oxidation by ozone or by O radicals
•   direct Al2 O3 deposition.
    In this subsection, we will focus on these oxidation processes mainly in relation
to the magnitude of TMR and R × A, since both are, among other properties such
as electrical noise (section 2.1.1), decisive parameters for future device implemen-
tation of progressively down-scaled junctions. In Fig. 1.17, a compilation of some
of the existing data is shown for Al2 O3 -based junctions prepared by these different
oxidation techniques; see also Table 1.1. In this section we will restrict ourselves
to alumina-type junctions since these are predominantly studied in this field. In
later sections other barrier materials (such as SrTiO3 and MgO) will be discussed
separately.
    With very few exceptions, it is clear that plasma oxidation, indicated by the solid
symbols in Fig. 1.17, distinguishes itself from all other techniques by the highest
values of TMR, but also with a characteristically high value of R × A. Thermally
oxidized junctions, grouped mostly in the smaller circle, have in general a low
RA product but also a low(er) magnetoresistance. Other techniques, which are
explored for their potential of making junctions with both a high TMR and lower
R × A, are situated between those extremes. Even with such a variety of techniques
it currently appears to be practically impossible to enter the upper-left part (low
R × A, high TMR) of Fig. 1.17 with alumina-based junctions. However, the results
obtained with ion-beam oxidation (Ferreira et al., 2005a) are promising for their
very low RA combined with reasonably high TMR ratios; see also section 2.2.1.
Interestingly, junctions with crystalline MgO barriers are reported to exhibit much
34                                                                            H.J.M. Swagten




Figure 1.17 TMR versus the RA product (at room temperature) of junctions made by various
barrier production techniques. The larger and small circle roughly indicate the plasma and
thermally oxidized junctions, respectively. Note that ion-beam oxidation (resembling the use
of a regular DC plasma) seems superior for their low RA and high TMR. For the underlying
data including references, see Table 1.1.



higher magnetoresistances combined with a relatively low resistance-area product
(section 4.6).
     Before discussing the results reported in literature in more detail, the reader
must bear in mind that the observed spread in TMR or R × A as seen in Fig. 1.17
may naturally stem from lab-to-lab variations of the structure of the junctions, and,
in particular, the barrier region. The structure and morphology of the unoxidized
aluminum layer influences the oxidation process and therefore the quality of the
resulting barrier. The oxide growth can easily be imagined to be affected when
oxidizing an aluminum layer with a grain-like structure. Such a growth mode is
intrinsically induced by the layer on which the aluminum is deposited and can
vary with the bottom layer material and with deposition technique. For example,
Ando et al. (2000b) have shown by atomic-force-microscopy measurements that the
roughness of aluminum can be reduced with 80% by replacing the aluminum buffer
layer under the bottom electrode by Pt. The deposition parameters and character-
istics of the deposition facility can play a huge role. For instance, a small amount of
surface contamination can induce a different growth mode of the aluminum layer.
Fujikata et al. (2001) report a considerable improvement of TMR in junctions with
an intentionally contaminated Ta buffer layer in their junctions. Furthermore, in
the case of plasma oxidation and UV-oxidation, the exact lay-out and operation of
the oxidation setup can be crucial for the quality of the barrier layer. Therefore, the
comparison between oxidation techniques found in literature should be considered
with great care.
Spin-Dependent Tunneling in Magnetic Junctions                                                 35




Table 1.1 A selection out of the vast literature on room-temperature low-bias TMR and R ×A for
alumina-based magnetic tunnel junctions. Oxidation methods are categorized in thermal oxida-
tion, plasma oxidation, ion beam oxidation, UV-assisted oxidation, ozone-enhanced oxidation,
oxidation by radicals, and reactive deposition. Only the electrodes next to the Al2 O3 barrier are
indicated. In several cases the results are obtained after a post-deposition anneal. Data by Li
and Wang (2002), indicated with †, are taken at T = 18 K. TMR with ‡ is measured at a 0.3 V
bias voltage

Method       Electrodes       TMR (%)        R × A ( µm2 )         Reference

Thermal      Fe–CoFe          5              2 × 103               Tsuge and Mitsuzuka (1997)
Thermal      NiFe–NiFe        13             2 × 103               Matsuda et al. (1999)
Thermal      Co–NiFe          20–23          60                    Parkin et al. (1999b)
Thermal      NiFe–NiFe        16             230 × 103             Chen et al. (2000)
Thermal      NiFe–NiFe        14             14                    Ohashi et al. (2000)
Thermal      CoFe–CoFe        32             30–40                 Sun et al. (2000a)
Thermal      CoFe–CoFe        18             140                   Song et al. (2000)
Thermal      CoFe–CoFe        25–30          30–70                 Zhang et al. (2001b)
Thermal      CoFe–CoFe        14–17          10–12                 Zhang et al. (2001b)
Thermal      CoFe–CoFe        29             34                    Moon et al. (2002)
Thermal      Co–Co            20             68 × 106              Diouf et al. (2003)
Thermal      CoFe–CoFe        22             8                     Wang et al. (2003)
Thermal      CoFe–CoFe        23             580                   Das (2003)
Thermal      CoFe–CoFe        18–25          8–14                  Zhang et al. (2003b)
Thermal      CoFe–CoFe        11             4.4                   Zhang et al. (2003b)
Thermal      CoFe–CoFe        29             60                    Shang et al. (2003)
Plasma       Co–CoFe          20             150 × 106             Moodera et al. (1996)
Plasma       Co–NiFe          6              200 × 106             Nassar et al. (1998)
Plasma       NiFe–NiFe        17             60 × 106              Wee et al. (1999b)
Plasma       Co–Co            19             200 × 106             Gillies et al. (1999)
Plasma       CoFe–CoFe        32             11 × 106              Parkin et al. (1999b)
Plasma       CoFe–CoFe        25–27          (10–20) × 103         Sun et al. (1999)
Plasma       CoFe–CoFe        15             2 × 103               Sun et al. (1999)
Plasma       Co–Co            28             50 × 106              LeClair et al. (2000a)
Plasma       CoFe–CoFe        31             230                   Ando et al. (2000b)
Plasma       CoFe–CoFe        50             (1–10) × 103          Ando et al. (2000b)
Plasma       NiFe–NiFe        30             (10–100) × 103        Chen et al. (2000)
Plasma       CoFe–CoFe        32             160 × 106             Park and Lee (2001)
Plasma       Co–Co            20–25          (1–100) × 103         Kuiper et al. (2001a)
Plasma       CoFe–CoFe        26             6 × 103               Dimopoulos et al. (2001a)
Plasma       CoFe–CoFe        59             1 × 106               Tsunoda et al. (2002)
Plasma       CoFe–CoFe        48             (100–500) × 103       Tsunoda et al. (2002)
Plasma       CoFe–CoFe        48             40 × 103              Lohndorf et al. (2002)
                                                                          (continued on next page)
36                                                                         H.J.M. Swagten


Table 1.1   (Continued)

Method            Electrodes      TMR (%) R × A ( µm2 )        Reference

Plasma            NiFeCo–NiFeCo   45        1 × 103            Engel et al. (2002)
Plasma            NiFe–NiFe       26        4.0 × 106          Song et al. (2003)
Plasma            Co–NiFe         34        2.3 × 106          Song et al. (2003)
Plasma            CoFe–CoFe       20        20 × 103           Das (2003)
Plasma            Co–Co           29        300 × 106          Koller et al. (2003)
Plasma            CoFe–CoFe       37        (5–10) × 103       Kim et al. (2003)
Plasma            CoFeB–CoFe      61        25 × 106           Wang et al. (2004)
Plasma            CoFeB–CoFeB     70        24 × 106           Wang et al. (2004)
Ion beam          CoFe–CoFe       40        (500–800) × 103    Cardoso et al. (1999)
Ion beam          NiFe–Co         8–14      2 × 106 –2 × 109   Roos et al. (2001)
Ion beam          CoFeB–CoFeB     20        2–15               Ferreira et al. (2005a)
Ion beam          CoFeB–CoFeB     40–45     60–150             Ferreira et al. (2005a)
UV-assisted       NiFe–NiFe       13        2 × 103            Song et al. (2000)
UV-assisted       NiFe–NiFe       8         300                Song et al. (2000)
UV-assisted       Co–NiFe         10–15     51 × 103           Girgis et al. (2000)
UV-assisted       NiFe–NiFe       14–21     102 –104           Covington et al. (2000)
UV-assisted       Co–NiFe         20        60 × 103           Boeve et al. (2000)
UV-assisted       NiFe–Co         23        1 × 103            Rottlander et al. (2000)
UV-assisted       Co–Co           20        160 × 103          Rudiger et al. (2001)
UV-assisted†      NiFe–NiFe       2–8       0.6–6              Li and Wang (2002)
UV-assisted       CoFe–CoFe       30        15 × 103           Das (2003)
Ozone             CoFe–CoFe       30        (8–24) × 106       Park and Lee (2001)
Ozone             CoFe–CoFe       33        11 × 103           Park and Lee (2001)
Radicals          Co–Co           11–17     350–200 × 103      Shimazawa et al. (2000)
Radicals‡         CoFe–NiFe       40        (1–3) × 103        Kula et al. (2003)
Reactive depo.    NiFe–NiFe       15–20     > 1 × 106          Chen et al. (2000)
Reactive depo.    Fe(211)–CoFe    35–45     103 –109           Yuasa et al. (2000)


2.2.1 Plasma oxidation
Plasma oxidation is currently the most widely applied method for producing alu-
minum oxide for MTJs with the highest values of TMR for amorphous barriers. As
mentioned above, Moodera et al. (1995) are the first to reproducibly produce MTJs
using plasma oxidation, and many groups have followed using numerous variations
on plasma oxidation. A DC glow plasma is easy to set up (see Fig. 1.18b) and is
therefore most commonly applied. Nassar et al. (1998) have applied an AC O2 /Ar
rf-plasma for the production of MTJs. A TMR of 6% is found with an R × A of
200 M µm2 . The low TMR and high RA product suggest that the bottom elec-
trode is oxidized in the process. An inductively coupled plasma (ICP) is generated
without electrodes by Ando et al. (2002) and Song et al. (2003), which means that
there is no contamination by sputtering of electrode material. This method is there-
fore thought to produce less impurities in the tunnel barrier. However, there is no
Spin-Dependent Tunneling in Magnetic Junctions                                         37




Figure 1.18 (a) Differential ellipsometry to in-situ monitor the amount of oxidizing metal
as a function of time. The bottom two curves are taken on a Si/SiO2 /10 Å Al sam-
ple using natural oxidation followed by plasma oxidation (open symbols), and for plasma
oxidation only (closed). The upper curve represents plasma oxidation of a full stack of
Si/SiO2 /50 Å Ta/70 Å Co/100 Å FeMn/35 Å Co/23 Å Al. In (b) a picture of an oxida-
tion chamber as taken from Knechten (2005) shows the DC glow discharge due to the high
(negative) potential of the ring-shaped electrode. See Knechten et al. (2001).


conclusive evidence that impurities in the barrier due to sputtering of the electrode
are causing a degradation of MTJ properties. A plasma generated by radio-frequency
or microwave radiation has been successful in plasma oxidation of silicon and is also
applied for aluminum oxidation, for example by Sun et al. (1999) and Yoon et al.
(2001).
     Plasma oxidation is very fast as compared to many other oxidation methods. For
example, Park and Lee (2001) optimally oxidize 18 Å of aluminum in approximately
40 seconds, and Kuiper et al. (2001a) need only 20 seconds of plasma oxidation to
optimally oxidize 15 Å of aluminum. To monitor these dynamical processes, in-
situ characterization techniques have been developed. Wee et al. (1999a, 1999b)
use the Van der Pauw method to in-situ measure the electrical resistance of the
Al layer during plasma oxidation from which the tunneling barrier thickness can
be estimated. Optical, ellipsometric techniques have been reported by LeClair et
al. (2000c), Lindmark et al. (2000), and Knechten et al. (2001), using the extreme
contrast between the dielectric constant of a metal and that of its oxide. In Fig. 1.18a
it is illustrated that the growth of the oxide from a 10 Å and 23 Å Al layer can
be monitored with high temporal resolution and with sub-monolayer sensitivity,
offering the possibility to investigate the oxidation dynamics in great detail (for
more details, see the thesis work of Knechten, 2005).
     Variations in plasma pressure and plasma composition can improve the perfor-
mance of MTJs. Following the success of a krypton–oxygen mixture (97%:3%) in
silicon oxidation (Sekine et al., 2001), junctions with a TMR of 59% have been
obtained by Tsunoda et al. (2002). Lee et al. (2003) have found that the rough-
38                                                                       H.J.M. Swagten



ness of the barrier interfaces can be tuned by adding a small amount of Zr to the
aluminum. At a doping level of 9.9% Zr, the barrier interfaces, as observed with
transmission electron microscopy, are the smoothest and the TMR is highest. Tun-
nel junctions made by plasma oxidation with a low resistance-area product are, e.g.,
fabricated by Ando et al. (2000b). The barrier layer in their tunnel junctions is made
by inductively-coupled-plasma oxidation of 8 Å Al. After annealing, the junctions
with a 10 seconds oxidation time display an RA product of 230 µm2 combined
with a TMR of 31%. When the oxidation time of the Al layer is longer (30–60 sec-
onds) the maximal TMR is raised to roughly 50%, although now with a higher RA
of typically a few k µm2 . Using CoFeB compounds as ferromagnetic electrodes
(see also section 2.1.4) and a Ar/O2 plasma for Al oxidation, very large magne-
toresistances of more than 70% are reported by Wang et al. (2004). However, the
resistance of these junctions is very high, around 24 M µm2 . Other investigations
are concentrating on further optimizing these CoFeB-based alumina junctions; see
for example Pietambaram et al. (2004), Wiese et al. (2004), and Cardoso et al.
(2005).
Ionized atom-beam oxidation Roos et al. (2001) use an ionized oxygen atom
beam of low energy (30–80 eV) to oxidize the aluminum. MTJs with a 8–14%
TMR and a high R × A of 1–1000 M µm2 are produced. Cardoso et al. (1999)
and Freitas et al. (2000) use an ion beam for the deposition of the layers and for
the oxidation process as well. The ion beam is created by inserting a grid between
a high-power (80 W) Ar/O plasma and the sample, and by applying a voltage (typ-
ically 30 V) over the grid, accelerating oxygen ions towards the sample. With this
technique, junctions with a TMR of 40% and an RA product of around 500 µm2
were created. The high quality is explained by the better layer-by-layer growth as
compared to the more frequently applied sputter deposition. The oxidation times
for optimal TMR are comparable to plasma oxidation, in the order of 60 seconds.
Recently, an even smaller RA product of around 2–15 µm2 has been established
together with a TMR of around 20% (Ferreira et al., 2005a, 2005b). When an
artificial antiferromagnet is present to engineer the top CoFeB electrode (see sec-
tions 2.1.4 and 2.1.2), R × A values of 60–150 µm2 are combined with a TMR
of 40–45%. The ion-beam oxidation is performed in three steps with increasing
plasma reactivity and leads to under-oxidized barriers from an initial 9 Å Al layer.
As can be seen from the data points on ion-beam oxidation in Fig. 1.17, these junc-
tions are very promising for device applications in sensors and memories due to the
unique combination of low RA and high TMR (Ferreira et al., 2005a).

2.2.2 Thermal and natural oxidation
In order to create junctions with lower specific resistances for device applications,
the barriers in MTJs have become progressively thinner. For the oxidation of Al
layers of 10 Å or less, plasma oxidation is thought to be too aggressive (due to
the high-energy particles involved) and not well controllable, possibly resulting in
damage to the interface with the bottom electrode. Therefore, for very thin layers
often natural or thermal oxidation is chosen. We note that for oxidation in an oxy-
gen atmosphere at room temperature normally the term natural oxidation is used,
Spin-Dependent Tunneling in Magnetic Junctions                                   39


whereas thermal oxidation refers to oxidation in an oxygen atmosphere where the
sample is usually but not necessarily at an elevated temperature. For most junctions
reported in literature, oxidation at room temperature is used. Tsuge and Mitsuzuka
(1997) and Matsuda et al. (1999) use pure natural oxidation, and a TMR of 13% is
found with an RA product of only 1.5 k µm2 . The barrier is made by exposure of
20 Å Al to 0.27 bar of pure oxygen for one hour. The fact that a barrier made from
an initial 20 Å Al results in an RA product that is three orders of magnitude lower
than plasma-oxidized junctions starting with the same initial Al thickness, suggests
that the barrier has an inhomogeneous thickness and the tunneling current runs
through the thinnest parts of the barrier. The low TMR is probably due to unoxi-
dized aluminum suggesting that the oxidation in this case is rather inhomogeneous.
Generally speaking, it is observed that the resistance of plasma-oxidized junctions
is much higher than their naturally oxidized counterparts. The explanation of this
striking difference is not yet clear, and maybe directly related to the much higher
oxidation rates for plasma oxidation, combined with the fact that the energetic O
atoms in the plasma more easily oxidize the pinholes (Knechten, 2005).
    Natural oxidation is a slow process if the aluminum layer is typically thicker
than 5 Å. For instance, the optimal oxidation of 10 Å Al takes 15 hours at room
temperature, as reported by Das (2003). In order to reduce processing time, two
cycles of deposition of Al and subsequent oxidation are used. The oxidation time
resulting in optimum TMR is thereby reduced from 15 hours to 2 × 2 hours (Das,
2003). An identical technique is used by Moon et al. (2002) who report MTJs with
30% TMR and an RA product of only 140 µm2 . Extremely small values of R ×A
are reported by Wang et al. (2003) (see also Zhang et al., 2001b, 2002, 2003b), and
is set to 8 µm2 while these junctions have a TMR of 22%. Ohashi et al. (2000)
have produced a functional low-resistance tunnel magnetoresistive sensor for use as
hard disk read-head. The MTJ is made by natural oxidation of 8.5 Å Al in a pure
oxygen atmosphere for 20 minutes, resulting in a TMR of 14% and an R × A value
of 14 µm2 .

2.2.3 UV-light assisted oxidation
Oxidation assisted by UV-light irradiation has been tried as a faster method of
aluminum oxidation as compared to natural oxidation which is due to an increased
reactivity by ozone generation. Since the damage due to high-energetic species is
avoided as compared to plasma oxidation, UV-oxidized junctions possibly combine
a low R × A with a high TMR. Boeve et al. (2000) and Girgis et al. (2000) are
the first to report results on MTJs made with UV-assisted oxidation. They oxidize a
sputter-deposited 13 Å Al layer for one hour in an oxygen atmosphere of 100 mbar,
assisted by an in-situ ultraviolet lamp. They find an RA product of 60 k µm2
and a TMR of about 20%. Compared with their naturally oxidized junctions, UV-
oxidation results in higher TMR but also in higher R × A. Their plasma-oxidized
junctions, which are identical except for the oxidation method, give a higher TMR
but similar R × A values. Later experiments have resulted in junctions with a TMR
of 10%, with a typical RA product of 1 k µm2 (Rottlander et al., 2000). Li and
Wang (2002) have prepared junctions by UV-assisted oxidation of only 5 Å Al,
40                                                                         H.J.M. Swagten



resulting in a few percent TMR but an R × A as low as 3.2 µm2 . By oxidation of
4 Å Al, an extremely low R × A of approximately 0.6 µm2 is found. The TMR
in these junctions has, however, dropped to 2%. Probably this is due to metallic
shorts between the ferromagnetic layers.

2.2.4 Other oxidation and deposition processes
Oxidation using ozone The high reactivity of ozone suggests that by using ozone
for aluminum oxidation, shorter oxidation times with respect to natural oxidation
are possible, and that possibly larger oxide thicknesses as compared to natural oxi-
dation are attainable within reasonable oxidation times. An oxygen–ozone mixture
to oxidize aluminum for MTJs is used by, for instance, Park and Lee (2001) and
Park et al. (2002). In a comparison between ozone-oxidized and plasma-oxidized
junctions, they report slightly higher TMR values (33%) for junctions oxidized by
ozone, whereas the RA product of ozone-oxidized junctions is one order lower
with a lowest value of 10.5 k µm2 . The process is still very slow; 50 minutes of
oxidation are necessary to produce the junction described here. Junctions made
by thermal oxidation have TMR values close to that of the ozone-oxidized junc-
tions (30%), but with considerable lower R × A of 140 µm2 (Moon et al., 2002).
Radical oxidation Shimazawa et al. (2000) report on experiments with oxidation
using a beam of oxygen radicals, arguing that this can be an energetically low and
slower process as compared to plasma oxidation, thus suitable for the oxidation of
ultrathin Al layers. The radicals are produced by a microwave (electron cyclotron
resonance) in approximately 10–3 mbar of oxygen. Junctions are created with R × A
values of 350 µm2 , and with a TMR of about 11%. Kula et al. (2003) applied
radical oxidation as well, resulting in junctions that have a TMR of 40% and a min-
imal R × A of only 2 k µm2 . In a comparison with natural and plasma oxidation,
for radical oxidation a higher TMR is reported as well as a medium RA product in
between thermal and plasma oxidized junctions. The possible advantage of radical
oxidation over plasma oxidation might be the absence of more energetic particles
such as ions.
Direct deposition of Al2 O3 , atomic layer deposition As an alternative to oxidation
processes, direct deposition of Al2 O3 layers has been tried in various forms. First
of all, reactive sputtering has been applied to create Al2 O3 layers. By sputtering alu-
minum in an argon atmosphere which contains a few percent oxygen, in principle
a homogeneous and stoichiometric Al2 O3 layer can be obtained as shown by Koski
et al. (1999). The roughness of thick films (≈ 1 µm) grown on pure Si was reported
to be about 7 Å. Chen et al. (2000) have tried to improve the method by prevent-
ing the oxidation of the bottom electrode by first depositing 7 Å of pure Al before
adding 5 Å of reactively sputtered Al2 O3 . Since their method resembles plasma oxi-
dation, the results are comparable to their plasma oxidation results, with a TMR of
18% and an RA product of 1 M µm2 . Yuasa et al. (2000) have produced reactively
sputtered, amorphous alumina barriers on top of crystalline Fe electrodes (see sec-
tion 4.3). The Al2 O3 is created by evaporation of Al at an O2 pressure of around
7 × 10–6 mbar. Due to the direct deposition method, it is possible to use wedges of
variable alumina thickness, facilitating the study of magneto-transport as a function
Spin-Dependent Tunneling in Magnetic Junctions                                      41


of the thickness of the barrier layer in a single sample (Yuasa et al., 2000). Another
method of direct Al2 O3 deposition is atomic layer deposition (ALD). In principle, this
technique allows for deposition of very thin dielectric films with excellent confor-
mality, uniformity, and atomic-level thickness control; see, for example, Paranjpe
et al. (2001). Although tunneling transport has been demonstrated across sputtered
exchange-biased magnetic junctions incorporating ALD-based Al2 O3 barriers, no
appreciable TMR has been measured (Bubber et al., 2002).

2.3 Towards optimized barriers
In the following subsections, experiments on the optimization of the oxidation
process will be briefly outlined. In many studies, the optimization is performed
mainly in relation to the magnitude of TMR, the resistance R of the junction, the
temperature dependence of TMR and R, and, finally the bias voltage dependence.
These properties have been reviewed in the previous sections. Here we will focus
on optimization schemes in relation to the following prominent issues:
•   over- and under-oxidized barriers
•   barrier pinholes and dielectric breakdown
•   thermal stability upon junction annealing
•   the use of alternative (amorphous) oxides.
   We will start the discussion on these items with a short overview of the exper-
imental tools that have been successfully applied in this area, focusing again on the
properties of Al2 O3 barriers.

2.3.1 Tools for oxidation monitoring and optimization
Apart from the direct measurement of (tunneling) transport characteristics, a num-
ber of diagnostic tools have been used to examine and further optimize the ox-
idation processes. Electrical and optical tools to in-situ monitor the oxidation
dynamics have been mentioned in section 2.2.1, with which the transformation
of Al into its oxide can be followed with submonolayer precision. Surface sensi-
tive techniques are frequently applied to study the chemical composition of the
junction and in particular the alumina barrier, e.g. using X-ray photoelectron spec-
troscopy (Mitsuzuka et al., 1999; LeClair et al., 2000c; de Gronckel et al., 2000;
Kottler et al., 2001), Rutherford backscattering spectroscopy (Sousa et al., 1999;
Gillies et al., 1999), and electron recoil detection (Gillies et al., 2000).
    In Fig. 1.19a the latter two techniques have been applied to measure the oxygen
content in junctions of Ta-NiFe-IrMn-NiFe-Co-Al2 O3 -Co-NiFe-Ta for variable
oxidation time. The observed ln(t) behavior hints to a simple logarithmic growth
law which is proposed by Mott (1947) to describe the oxidation of thin metal
films, typically below 40 Å. In this type of oxidation model, it is assumed that
the aluminium ions are diffusing through the growing oxide and react with the
oxygen at the outer interface. This has been confirmed by Kuiper et al. (2001a)
using an isotope technique in which an Al layer is shortly oxidized with 16 O before
continuing with 18 O; see Fig. 1.19a. In this study, secondary ion mass spectrometry
depth profiles indicate that 16 O moves to larger depth with increasing 18 O oxidation
42                                                                                 H.J.M. Swagten




Figure 1.19 (a) O content of junctions consisting of 35 Å Ta/30 Å NiFe/100 Å IrMn/25 Å
NiFe/15 Å Co/15 Å Al + oxidation/40 Å Co/100 Å NiFe/35 Å Ta for variable oxidation time,
as measured by Rutherford backscattering spectrometry (RBS) and elastic recoil detection
analysis (ERD). Two-step oxidation using 18 O2 and 16 O2 oxygen isotopes is performed to
establish that Al is the moving species during oxidation. (b) Oxide thickness for comparable
junctions extracted from cross-section transmission electron microscopy (XTEM) as shown
in panel (c) for a multilayer of 50 Å Co/15 Å oxidized Al. This suggests an intermediate
oxidation step with increasing O content at constant oxide thickness. After Gillies et al. (1999)
and Kuiper et al. (2001a).


time, while 18 O is incorporated close to the surface. This points to Al as the moving
species during plasma oxidation. Cross-section transmission electron microscopy
(XTEM), used for instance by Boeve et al. (2001) and Bruckl et al. (2001), is
used to further characterize the growth and morphology of the films. For instance,
XTEM on Co-Al2 O3 multilayers by Gillies et al. (2000) shows an excellent contrast
between the metal and oxide layers, see Fig. 1.19c, from which the oxide thickness
can be followed as a function of oxidation time as shown in Fig. 1.19b. When
comparing this to the left panel of the figure, it is suggested that oxidation of the
barrier is governed roughly by distinct steps. In the first stage the oxygen rapidly
penetrates through the total Al layer, then a homogenization stage follows where
the O content steadily increases at a fixed oxide thickness, until, at the third step,
the Co electrode starts to oxidize.
    To continue the discussion on characterization tools, Shen et al. (2003) have
combined XTEM with electron holography to directly measure the shape of the
barrier and its interfaces. An ac-impedance technique is applied by Gillies et al.
(2000) in order to characterize the dielectric properties of the barrier layer. Ana-
lyzing the data by modelling the tunneling across the oxide with an RC network,
complementary information on the structure of the barrier and its evolution with
plasma oxidation time has been extracted. Similarly, Landry et al. (2001) have used
ac-impedance data on NiFe-Al2 O3 -NiFe junctions to determine an interfacial con-
tribution to the capacitance and to extract the electron screening length in the NiFe
electrodes. The complex capacitance of magnetic junctions has been measured also
by Huang and Hsu (2004), in their case over a frequency range from 102 to 108 Hz
in CoFe-Al2 O3 -CoFe junctions. From the analysis of the so-called Cole–Cole di-
agrams, a significant sensitivity to the oxidation process of the metallic Al layers is
reported, being able to clearly discriminate between different stages in the oxidation
process.
Spin-Dependent Tunneling in Magnetic Junctions                                             43


2.3.2 Over- and under-oxidation
In order to study the effect of the oxidation process or just to find the optimum
oxidation parameters, often a series of samples at various stages of oxidation is pro-
duced. This is usually done in one of the following two ways: either by depositing
a number of identical samples, oxidized with various oxidation times (see for exam-
ple Sun et al., 1999; van de Veerdonk, 1999; Gillies et al., 1999, 2000; Song et al.,
2000; Tehrani et al., 2000; Park et al., 2002), or by depositing a series of samples
with a range of Al thicknesses, all oxidized at once (see Moodera et al., 1997; Song
et al., 2000; Tehrani et al., 2000; Boeve et al., 2001; Freeland et al., 2003).
    The first method requires a rather time-consuming experiment due to the num-
ber of oxidation steps, since each oxidation includes a long pump-down stage
and possibly sample transport. An example of such an optimization is shown in
Fig. 1.20a. The second method, making one batch of samples with a range of Al
thicknesses, usually allows for more rapid experiments because only one oxidation
step is necessary, see Fig. 1.20b. The use of a wedge (a film with a lateral variation
in thickness by linear displacement of a shutter during deposition) is even faster and
yields detailed and accurate information (Fig. 1.20c), see for example the work of
LeClair et al. (2000c), Song et al. (2000), and Covington et al. (2000). In another
method, applied by Nowak et al. (2000) and Song et al. (2000), a series of samples at
various stages of oxidation is made by a single deposition and a single oxidation step
in a plasma that is not uniform over the wafer. The interpretation of the results in
terms of oxidation conditions of this method is of course much more complicated.
    From the data shown in Fig. 1.20 it is evident that oxidation of the barrier is a
critical process, where both over- and underoxidation are detrimental to the TMR
effect. Underoxidation leaves the Al layer partially in its metallic state which effec-
tively reduces the tunneling spin polarization of the carriers. The detrimental effect




Figure 1.20 (a) TMR as a function of oxidation time for as-deposited junctions com-
prised of Co90 Fe10 /17 Å Al + oxidation/Co90 Fe10 (Koller, 2004). (b) Results for
Ni80 Fe20 /t Al + oxidation/Ni80 Fe20 (Moodera et al., 1997), and (c) for as-deposited and
annealed (250°C) junctions of Ni80 Fe20 /t Al (wedge) + oxidation/Ni80 Fe20 (Covington et al.,
2000). Note that the original data from Moodera et al. (1997) have been corrected to match
the definition of TMR in Eq. (8). See the references for the composition of the full junction
stack as well as for details of the Al oxidation.
44                                                                         H.J.M. Swagten



of foreign impurities has been addressed by Jansen and Moodera (1998, 2000) by
δ-doping the Al2 O3 barrier with nonmagnetic and magnetic elements. Generally,
a strong reduction of TMR is observed and interpreted via the effect of additional
impurity-assisted tunneling channels combined with spin-flip processes when mag-
netic impurities are involved. Theoretically, in a number of papers the effects of
impurity-assisted tunneling in magnetic junctions have been addressed; see for ex-
ample Bratkovsky (1997), Zhang and White (1998), and Jansen and Lodder (2000).
Parkin (1998) and LeClair et al. (2000d) alternatively demonstrate this effect of re-
duced TMR by adding a very thin nonmagnetic film at the barrier interface. As an
example, the addition of 1 monolayer of Cu at the interface between the bottom
electrode and the barrier reduces TMR by more than a factor of 2; see also sec-
tion 4.1. The reduction of TMR is also observed for overoxidation, although now
the quenching of polarization will be induced by the presence of antiferromagnetic
oxides at the barrier interfaces (Moodera and Mathon, 1999). This leads to addi-
tional spin flip conductance channels, by which the spin polarization (and TMR)
will be suppressed.
     Using the diagnostic techniques mentioned above, it is established by several
groups that an asymmetry in the barrier potential, easily detected by fitting IV -data
with the Simmons or Brinkman formula (Eqs. (12) and (13), respectively), is ac-
companied by a low TMR. Correspondingly, the maximum TMR is found when
the junction has a minimal asymmetry, see the work of Sun et al. (1999), Covington
et al. (2000), and Oepts et al. (2001). Both over- and under-oxidation cause this
asymmetry by creating different electrode-barrier interfaces. An asymmetry in the
barrier height is directly observed by Koller et al. (2003) in photoconductance mea-
surements (see section 2.1.1). Due to the large sensitivity of the technique to the
presence of Al close to the tunnel barrier, the disappearance of a negative contribu-
tion to the photocurrent is correlated to the complete oxidation of the barrier layer
and the corresponding maximum in TMR.
     In order to increase the TMR by preventing such an asymmetry, some modi-
fications to the simple oxidation process are often applied. In several cases, this is
accomplished by creating a reservoir of oxygen at the interface with the bottom-
electrode. In a later step, this reservoir is supposed to fill the Al-Al2 O3 from the
bottom up in order to create a more homogeneous barrier. A first method in which
such a reservoir is used is to first slightly oxidize the surface of the bottom electrode
before deposition of Al. The subsequent deposition of Al will cause an unstable sit-
uation since generally the Gibbs free energy of ferromagnetic oxides is larger than
that of Al2 O3 (Dean, 1992). Since Al2 O3 has a lower energy, the oxygen will nat-
urally move into the aluminum. This is applied by Sun et al. (1999) and Kuiper et
al. (2001b). In the latter case, the authors find that, when the Co bottom electrode
is partially oxidized, up to 10 Å of Al can be completely oxidized by oxygen from
this reservoir.
     A second method involves a small amount of intentional over-oxidation. A gra-
dient of Al or O now exists in the barrier layer. After the top electrode and capping
layers are deposited, the junction is subjected to an annealing step, i.e. the sample
is brought to higher temperatures in a non-reactive argon (or high vacuum) at-
mosphere. This causes the oxygen to leave the bottom electrode and move into the
Spin-Dependent Tunneling in Magnetic Junctions                                     45


barrier, often resulting in a more homogeneous barrier layer. This procedure was
performed by, for instance, Song et al. (2000) and Dimopoulos et al. (2001b). In
all cases, the barrier is homogenized and the TMR is increased. From photocon-
ductance experiments on tantalum-oxide MTJs with variable oxidation time of Ta
(Koller et al., 2005b), the shift of the maximum TMR with anneal temperature is
accompanied by a similar shift of the oxidation time where the asymmetry of the
barrier potential is absent. The experiments of Koller et al. (2005b) directly support
the idea that for obtaining the highest magnetoresistance ratio one should anneal
MTJs that would be characterized as slightly over-oxidized in the as-deposited state.
It is suggested that this result can be understood by a homogenization of the oxy-
gen distribution in the barrier, possibly combined with a change of the bottom
barrier-electrode interface.
     Several groups have attempted to create more homogeneous barriers by applying
a two-step process, each step being comprised of Al deposition and oxidation. Yoon
et al. (2001) applied this technique with plasma oxidation, Das (2003) with natural
oxidation, and Zhang et al. (2003b) with a slow ion-beam process. For all methods,
it is reported that the TMR increases slightly in junctions with a two-step process
with respect to a one-step process. The RA product increases, often by as much
as one order of magnitude (see for example Zhang et al., 2003b). In a comparison
between single-step and two-step oxidation experiments, Yoon et al. (2001) have
found from X-ray photoelectron spectroscopy that indeed the concentration of O
in the barrier is more homogeneous than from single-step plasma oxidation. The
TMR of the two-step oxidized junctions is higher, although not much.

2.3.3 Pinholes and dielectric breakdown
A pinhole is a path of relatively high conduction between the two electrodes,
through the barrier layer. Often this is a metallic short due to inhomogeneous
oxidation, or a very thin part of the barrier due to inhomogeneous deposition of
the aluminum prior to oxidation. Generally, pinholes can decrease the TMR in
two ways. First, for very thin barriers (or for barriers created from a too rough Al)
a strong magnetic coupling between the two electrodes may be present in regions
where the electrodes are in direct contact. In that case, the free layer no longer
fully switches independently from the bottom electrode, and results in a decrease of
TMR (Wang et al., 2003; Zhang et al., 2003b). The second, more generally ob-
served, detrimental aspect of pinholes is that the largest part of the current through
the junction will run via a normal metallic contact instead of the desired spin-
dependent tunneling across the insulator. This shunting of the tunneling current
via metallic shorts has been widely observed and is analyzed by simply modelling a
tunnel resistor with an ohmic resistor in parallel (Oliver et al., 2002).
     Direct visualization of pinholes is usually difficult due to the extremely small
dimensions, probably down to a few atomic distances. With the use of a nematic
liquid crystal deposited on the sample, pinholes can be directly visualized (Oepts et
al., 1998). Upon heating their liquid crystal above the clearing point (56.5°C), the
material changes from the nematic state showing optical anisotropy, to the isotropic
state. By operating the junction just below the clearing point, the power dissipa-
tion due to the pinholes are then visualized as black spots; see Fig. 1.21a. Schad
46                                                                            H.J.M. Swagten




Figure 1.21 (a) Polarized light picture of liquid crystal on top of a shadow-evaporated
Co/20–22 Å Al2 O3 /Co50 Fe50 junction. The black spot in the middle of the junction surface
is the location of a pinhole at a breakdown site (after Oepts et al. (1998)). In (b) a scan-
ning-electron-microscopy image is made after the electrochemical growth of cauliflower-like
Cu islands on an oxidized 12 Å Al layer grown on top of 125 Å NiFeCo. After Schad et al.
(2000).


et al. (2000) have developed a method for pinhole imaging using electrodeposition
of Cu. Selective nucleation at the metallic pinhole locations produces characteris-
tic cauliflower-like structures that can be easily visualized. An example is shown in
Fig. 1.21b. An indirect way to detect pinholes is based on the magnetic field gener-
ated by the large current density flowing through the pinholes. Due to the vertical
direction of the current (⊥ to the plane of the layers), these magnetic fields are in
the plane of the free magnetic layer and are thus able to shift or deform the switch-
ing behavior from which the location of the pinhole may be extracted (Oliver et
al., 2002, 2004).
     Indirect indications for pinholes are widely reported. E.g., Zhang et al. (2003b)
have found evidence for pinholes from resistance and TMR data, combined with
the observation of magnetic coupling between the free and fixed magnetic layers.
Whereas junctions with a barrier of nominal thickness tAl ≥ 6.5 Å are reported to
be pinhole-free, thinner nominal Al layers are clearly suffering from the presence of
pinholes. Han and Yu (2004) report on junctions in which the aluminum (9 Å) is
underoxidized. The as-deposited junctions show a very low resistance (820 µm2 )
and practically no TMR (0.5%), consistent with transmission-electron-microscopy
measurements showing pinholes with a diameter of a few nm. However, after an
annealing step, both the junction resistance and the TMR increase enormously to
30 k µm2 and 43%, respectively, which is related to the reduction of pinholes and
an improvement of the interfaces with the barrier. Moon et al. (2002) show that in
otherwise identical conditions, the usage of a two-step oxidation process instead of
a single-step process increases the TMR as well as the RA product. Together with
the observation that the magnetic coupling between the bottom and top electrode
is reduced in junctions fabricated by the two-step process, they conclude that two-
step oxidation creates tunnel barriers with a much lower pinhole density.
     Pinholes in magnetic tunnel junctions have also triggered a reconsideration
of the criteria formulated by Rowell and others (see, e.g., Brinkman et al.,
1970) to discriminate true tunneling conductivity from other metallic-like cur-
rent paths (Garcia, 2000; Jonsson-Akerman et al., 2000; Rabson et al., 2001;
Spin-Dependent Tunneling in Magnetic Junctions                                        47




Figure 1.22 Ramped current stress measurements (a) and a constant voltage stress measure-
ment (b) on junctions consisting of 30 Å Ta/30 Å NiFe/100 Å IrMn/100 Å CoFe/10 Å
Al + oxidation/40 Å CoFe/60 Å NiFe/50 Å Cu. The arrows indicate the average breakdown
voltage VBD and the moment of breakdown tBD . Adapted from Das (2003).


Akerman et al., 2001). According to these Rowell-criteria the conductance G
(1) should vary exponentially with barrier thickness, (2) is parabolic in bias voltage,
(3) scales with the junction area, and (4) displays a weak insulator-like temperature
dependence; see also section 2.1. Although these are necessary criteria for tunnel-
ing, it is shown by Oliver et al. (2002) that they do not rule out the existence of
pinholes, especially for junctions with ultrathin (<10 Å Al) barriers. Since the de-
tection of pinholes is generally not a straightforward experiment and may not be
inferred from the presence of all tunneling criteria, it is suggested that the exam-
ination of the insulator breakdown mechanisms will reveal the true nature of the
barrier quality including the presence of pinholes (Oliver et al., 2002). Breakdown
of tunneling barriers is crucial in the assessment of the lifetime of MTJ-based de-
vices and has been studied by electrically stressing the system until the oxide breaks.
Experimentally, this is achieved by, for instance, ramping the bias voltage between
the electrodes. At the breakdown voltage, highly conductive paths (pinholes) are
created that shunt the remaining tunnel resistance, thereby quenching the TMR
(Oepts et al., 1998, 1999; Shimazawa et al., 2000; Rao et al., 2001; Schmalhorst et
al., 2001; Oliver and Nowak, 2004).
     Several mechanisms have been proposed to explain breakdown of the oxide
layer, and these are related to the presence or generation of traps or defects in
the barrier that percolate at the point of breakdown. From these voltage-ramp
experiments the so-called E-model for dielectrical breakdown in MTJs is suc-
cessfully confirmed (see, e.g., Oepts et al., 1999), in which it is assumed that
traps are generated when the electric field breaks the dipoles in the oxide. More
recently, constant voltage (or constant current) tests for MTJs are performed,
where the current (voltage) changes are monitored during time (Das et al., 2003;
Nakajima et al., 2003). A typical experiment of breakdown via current ramping
or at constant voltage is shown in Fig. 1.22. It is found by Das et al. (2003) that
48                                                                        H.J.M. Swagten



breakdown (or pre-breakdown) in UV-light assisted and plasma-oxidized barriers
intrinsically occurs due to electric field-induced generation of single traps in the
oxide, similar to the breakdown mechanism in SiO2 . At the position of the trap, the
tunnel barrier potential is locally distorted, leading to dramatic changes in the local
conductivity.
    Extrinsically, breakdown is also strongly related to the quality of the barrier in
terms of post-deposition processing defects, e.g., at the perimeter of the junction
(Nakajima et al., 2003), or via imperfections induced already during deposition.
Especially when the barrier becomes very thin, the inevitable growth-related pin-
holes may grow due to strong Joule heating at the pinhole area upon electrically
stressing the barrier, eventually leading to a junction breakdown as reported by
Oliver et al. (2002). In these naturally oxidized junctions a clear distinction can be
observed between extrinsic breakdown due to the presence of pinholes and more
robust junctions that exhibit intrinsic breakdown. The latter show the highest TMR
rather independent of RA product, in contrast to pinhole-richer samples with lower
TMR and a strong dependence on R × A. This strongly suggests that the tendency
for lower TMR and RA in naturally oxidized junctions (see Fig. 1.17) may be re-
lated to the presence of pinholes with, generally, a much higher density than for
plasma-oxidized junctions.

2.3.4 Thermal stability
In Fig. 1.13, shown earlier in this section, it is demonstrated that TMR in magnetic
junctions can be enhanced by a thermal treatment of the system. This generally ap-
plies to junctions annealed at temperatures up to around 200–300°C, see, e.g., Sato
et al. (1998); Sousa et al. (1998); Parkin et al. (1999b); Cardoso et al. (2000b). The
enhancement may be attributed to an improvement of the active ferromagnetic-
barrier-ferromagnetic region of the junction due to structural changes or diffusion
processes, for instance due to homogenization of the oxygen in the as-deposited
alumina (see section 2.3.2). Also oxygen at the interface with the bottom elec-
trode, due to a slight over-oxidation, may be released by a thermal treatment which
increases the tunneling spin polarization and TMR. The impact of annealing on
the barrier properties has been directly determined from photoconductance mea-
surements (Koller et al., 2004), suggesting that highest TMR can be obtained by
annealing MTJs that are slightly over-oxidized in the as-deposited state.
     Apart from the barrier-related thermal effects, also structural and magnetic
changes in the frequently used antiferromagnetic layers are believed to play an im-
portant role (Cardoso et al., 2000b; Koller et al., 2004), which is reflected in the
thermal stability of the exchange bias field, similar as reported for exchange-biased
GMR systems (Coehoorn, 2003). Indeed, when not using exchange biasing in
junctions with artificial antiferromagnets or with only two layers with different co-
ercivities, the rise of TMR with annealing temperature is only modest (Parkin et
al., 1999a; Schmalhorst et al., 2000b).
     Contrary to the rise of TMR at relatively low anneal temperatures, annealing
above 200–300°C leads to a severe degradation of the tunnel magnetoresistance
(Parkin et al., 1999b; Cardoso et al., 2000b, 2000c). In Fig. 1.23a an example of the
TMR collapse is shown for a CoFe-Al2 O3 -CoFe-IrMn junction. Since incorpora-
Spin-Dependent Tunneling in Magnetic Junctions                                              49




Figure 1.23 (a) TMR measured at T = 300 K as a function of post-deposition annealing
temperature for Ni80 Fe20 /Co82 Fe18 /15 Å Al + oxidation/Co82 Fe18 /Ir26 Mn74 (Cardoso et al.,
2000c) annealed in a vacuum of ≈ 10–6 mbar. In (b) the open symbols show tunneling spin
polarization for Al/Al2 O3 /Co (CoFe)/FeMn superconducting junctions annealed in a vacuum
at a base pressure of < 10–9 mbar. Closed symbols are identical junctions but now without the
top FeMn layer. Adapted from Kant et al. (2004b) and Paluskar et al. (2005a).


tion of MTJs into existing semiconductor technology requires thermal stability up
to at least 400°C (especially in view of the development of MRAM), the suppres-
sion of TMR in that regime has attracted enormous attention. The current belief
is that one of the main reasons for the collapse of TMR is related to the diffusion
of Mn out of the antiferromagnetic layer (such as FeMn, IrMn, PtMn). The impact
of this is twofold. First, the strength of the exchange biasing may be reduced and
the adjacent ferromagnetic layer may suffer from a reduction of the magnetization
or at least an effect on the switching behavior (Cardoso et al., 2000b). Secondly,
it has been demonstrated that the Mn diffuses over considerable distances as de-
termined by Rutherford back-scattering (Cardoso et al., 2001). When reaching
the ferromagnetic-barrier interface, it can obviously deteriorate the tunneling spin
polarization.
    To prevent the Mn from diffusing, anti-diffusion barriers have been imple-
mented in particular between the barrier and the antiferromagnetic layer. However,
no conclusive picture emerges from these experiments. As an example, Ta barriers
do indeed stop the Mn from diffusing towards the barrier (Cardoso et al., 2000a)
although it does not avoid the TMR to collapse. Annealing studies with barriers
of variable thickness (tAl between 7 Å and 15 Å) demonstrate that only changes at
the CoFe-Al2 O3 interfaces (e.g. the roughness) or in the barrier itself can explain
the observed TMR degradation (Cardoso et al., 2001). When exposing the bottom
electrode to a nitrogen plasma prior to the deposition of the Al layer, Shim et al.
(2003) observe a reduction of the Mn diffusion along with a better TMR at an
anneal temperature of 270°C. Oxidic CoFeTaOx diffusion barriers introduced by
Fukumoto et al. (2004) shift the TMR collapse towards higher anneal temperatures
as well, although they simultaneously use thin alumina diffusion layers to separate
the CoFe from the NiFe in the Co90 Fe10 -Ni81 Fe19 top electrode. This suggests
50                                                                       H.J.M. Swagten



that also Ni migration (in this case in the top electrode) may be relevant for the
TMR collapse. By a rapid thermal anneal process of only 10 seconds as compared
to conventional anneals of hours, Lee et al. (2002) have shown that TMR has be-
come thermally more robust probably due to an abrupt change of the oxide barrier
parameters. The important role of structural changes in the barrier is particularly
relevant for ultrathin barriers (Cardoso et al., 2001). In that regime, TMR is most
sensitive to changes in the interfacial region at the bottom of the barrier as deter-
mined from Rutherford backscattering and atomic-force microscopy, and leads to
a reduced thermal robustness of TMR.
    To directly study the possible degradation of the Al2 O3 barrier or its interfaces
in relation to TMR, Kant et al. (2004b) have measured the tunneling spin polariza-
tion as a function of anneal temperature by superconducting tunneling spectroscopy
(which will be explained in more detail in section 3). As mentioned in the intro-
duction, the tunneling spin polarization P is directly responsible for the magnitude
of the magnetoresistance effect via the simple equation TMR = 2P /(1 – P 2 ). It is
shown that annealing of Al-Al2 O3 -Co junctions does not affect the tunneling polar-
ization up to anneals at T = 500°C, demonstrating the intrinsic thermal stability of
the barriers and its interfaces. Also when FeMn is additionally grown on top of the
Co layer, the polarization is still not degrading with anneal temperature, despite the
fact that Mn strongly diffuses above T = 300°C as independently shown by X-ray
photoelectron spectroscopy (Paluskar et al., 2005a). It is suggested by the authors
that this is in qualitative agreement with the work of Kim and Moodera (2002),
who report that Mn concentrations as high as x = 30% in Al-Al2 O3 -Co1–x Mnx
junctions have only a weak negative effect on the tunneling spin polarization (sec-
tion 3).

2.3.5 Alternative barriers for MTJs
As we will show in the following sections 3 and 4, there is an increasing number
of papers devoted to alternative barriers for magnetic tunnel junctions, leading to
intriguing phenomena such as extremely large magnetoresistance ratios and unusual
bias voltage dependencies. In the early years of tunnel magnetoresistance, the search
for alternative barriers has been inspired, among other aspects, by the possibility
to tune the performance of these devices by the insulator band gap and therefore
the potential energy of the barrier. Not only is this directly affecting the RA prod-
uct of a magnetic junction (Eq. (14)), also within free-electron models (see, e.g.,
Slonczewski, 1989) the barrier height is a critical parameter that determines the
magnitude and even the sign of TMR; see also the work of Tezuka and Miyazaki
(1998). Also within more recent models the insulator, and in particular the inter-
faces and bonding with the ferromagnetic electrodes, are critically important for
the spin-dependent tunneling processes, as extensively discussed in the following
sections.
    Another aspect of considering alternative barriers is related to the issue of
overoxidation (section 2.3.2). To prevent degradation of the underlying ferromag-
netic electrode by overoxidation, the metal atoms of the barrier materials should
behave sufficiently electronegative and react preferentially with the supplied oxy-
gen. It appears from experimental studies that, apart from Al2 O3 , a number of
Spin-Dependent Tunneling in Magnetic Junctions                                        51


candidates are available in this respect. However, as we will see below, the results for
alternative amorphous (or partially polycrystalline) barriers are not dramatically dif-
ferent from the observations when regular Al2 O3 is used, although for tuning tunnel
device properties in specific applications these studies may be extremely valuable.
This is in striking contrast to the use of crystalline or epitaxial barriers, leading to
crucial modifications of the tunneling properties (section 4).
    One of the first attempts to use alternatives for alumina as a barrier has been
reported by Platt et al. (1996, 1997), and Smith et al. (1998) using a number of
reactively sputtered oxides. In case of oxidized Hf, Mg, and Ta, a sizable mag-
netoresistance has been observed, although only when the junctions are cooled
down to liquid-nitrogen temperature. The lack of TMR at ambient conditions is
attributed to the vacuum break necessary to change their shadow masks. Hafnium is
used by Wang et al. (2002) in ultrathin Hf/Al bilayers that are naturally oxidized in
pure O2 . Magnetoresistances of more than 10% are reported at room temperature
together with low RA products. The presence of unoxidized Hf (2.5%) close to the
bottom electrode as determined by X-ray photoelectron spectroscopy, has improved
the continuity and conformality of these amorphous barriers.
    As discussed later on in section 3.3.3, in the work of Sharma et al. (1999) the
use of composite Ta2 O5 -Al2 O3 barriers may change the sign of TMR, an effect
that strongly depends on the bias voltage. More recently, a positive 2.5% TMR
at room temperature is reported by Rottlander et al. (2001) for junctions with
plasma-oxidized Ta2 O5 barriers. Up to 10% room-temperature magnetoresistance
has been found in exchange-biased Ta2 O5 junctions, not further improving after
post-deposition annealing up to almost 300°C, see Gillies et al. (2001). Oxidation
by oxygen release during the anneal of a partially oxidized Co electrode also does
not improve this figure, although the RA product is somewhat higher than for the
regular plasma-oxidized barriers. Direct information on the properties of the Ta-
oxidized barrier can be obtained via photoconductance measurements as shown by
Koller et al. (2004), which is facilitated by the low band gap of Ta2 O5 (≈ 4.2 eV)
as compared to alumina. Due to optical electron-hole pair generation in the barrier
itself and subsequent transport in the electric field, the sign and magnitude of the
barrier asymmetry can be determined quite accurately. Moreover, the oxidation
time where the asymmetry becomes zero is found to coincide with a maximum in
the magnetoresistance ratio. This is argued to be due to the complete oxidation of
the barrier material, resulting in a symmetric tunnel barrier. In a follow-up study,
Koller et al. (2005a) have shown that TMR strongly depends on the thickness of
the Ta2 O5 layer, possibly due to structural modifications in the barrier or at the
interfaces.
    Barriers of AlNy and AlOx Ny have been produced by Sharma et al. (2000)
yielding up to 18% magnetoresistance at room temperature when using a mixture
of O2 and N2 for plasma oxidation (nitridation). This magnitude is similar to those
produced with the regular Al2 O3 barriers although with a lower RA product. When
using pure N2 , however, TMR is significantly lower, never exceeding 16% (see
also Shang et al., 2001). Wang et al. (2001a) have combined nitrogen-containing
barriers (AlN) and ferromagnetic electrodes (Fe93.8 Ta2.4 N3.8 ) to minimize possible
tunneling spin polarization losses during post-deposition anneal of their structures.
52                                                                          H.J.M. Swagten



After annealing at 225°C, magnetoresistances of 17% are reported, degrading after
anneals above 250°C. From Rutherford backscattering it is found that a significant
amount of O2 (< 10%) is incorporated in the barrier, which is generally a major
concern in preparing barriers by nitridation due to the extremely high reactivity of
oxygen. For comparable junctions using CoFe alloys a TMR of more than 30% is
observed after annealing (Wang et al., 2001a). Schwickert et al. (2001) have used
AlN and AlOx Ny barriers, as well as AlN-Al that is naturally oxidized. Although
typically more than 10% of magnetoresistance is obtained, the use of ultrathin Al2 O3
is still superior in terms of TMR and RA product. Boron nitride (BN) barriers are
used by Lukaszew et al. (1999) on top of an epitaxial structure consisting of Si(100)-
Cu(100)-Co(100). When combined with a polycrystalline Co or Ni layer on top
of the BN barrier, room-temperature TMR of up to 25% has been observed.
     Wang et al. (2001b) have fabricated junctions with crystalline ZrOx barriers by
plasma oxidation of thin Zr layers (typically 5 Å). The as-deposited barriers appear
to consist of both ZrO and ZrO2 phases. After annealing, interfacial oxygen incor-
porated partially at the bottom CoFe electrode is released into the barrier, resulting
in a considerable increase of TMR, up to a magnitude of around 20%. Upon nat-
ural oxidation of Zr-Al bilayers, the barrier becomes amorphous and smoother than
for crystalline ZrOx or pure amorphous Al2 O3 , with TMR ratios that are still ex-
ceeding 15% after annealing. In another approach, the addition of impurities in an
Al2 O3 barrier is thought to create another microstructure within the insulator and
at its interfaces with the electrodes. Lee et al. (2003) find that the addition of Zr
to the Al, prior to oxidation, severely affects the structural properties of the barrier.
At a 9.9 at.% Zr-alloyed Al-oxide barrier, a very smooth amorphous alloy phase
is established with excellent TMR of almost 40%. Yttrium oxide (YOx ) barriers
obtained from plasma oxidation of an Y film are also reported to be well-defined,
smooth, and amorphous, giving rise to around 25% TMR at room temperature
(Dimopoulos et al., 2003).
     As mentioned earlier, also in section 4 alternative barriers for magnetic tunnel
junctions will be treated. In that case the tunnel barriers are no longer structurally
amorphous (or at most polycrystalline), but are designed to become crystalline and
epitaxially matched to the electrode(s), e.g. by employing molecular beam epitaxy
or pulsed laser deposition.



     3. Tunneling Spin Polarization
      As we have seen in the introduction (section 1), the degree of spin polar-
ization is the key ingredient for the magnetoresistance effect in magnetic junc-
tions. Generally, however, in literature the physical property of spin polarization
is defined in several different ways. To start with, spin polarization is some-
times related directly to magnetization of ferromagnetic metals, i.e. the difference
between the number of spin-up and spin-down electrons. In transport experi-
ments, it is clear that another definition needs to be used, and electrons at the
Fermi level are ruling the spin polarization. However, our first-order definition
Spin-Dependent Tunneling in Magnetic Junctions                                       53


P = [Nmaj (EF ) – Nmin (EF )]/[Nmaj (EF ) + Nmin (EF )], see Eq. (10), is still far from
realistic. We are dealing with 3d ferromagnetic materials having both heavy d elec-
trons as well as light s electrons at the Fermi level. This seriously complicates our
view on spin polarization, since generally d states in these metals are dominating the
magnitude of the density-of-states, whereas the mobile s states are responsible for
electrical transport. Moreover, one should carefully define spin polarization in rela-
tion to the actual geometry and electrical properties of the device. As an example,
in studying giant magnetoresistance (GMR) in all-metallic multilayers it is a useful
concept to consider the spin polarization of current in the bulk of the magnetic lay-
ers. It is believed that Andreev reflection spectroscopy on a (nano)contact between
a ferromagnetic metal and a superconductor is a valuable technique to directly
measure this current spin polarization of bulk ferromagnets (Soulen et al., 1998;
Upadhyay et al., 1998; Mazin, 1999; Nadgorny et al., 2000; Strijkers et al., 2001;
Kant et al., 2003). In a tunneling experiment, obviously a completely different
regime of current polarization is considered: tunneling spin polarization is the po-
larization in electrical current when electrons are tunneling from a ferromagnetic
metal through a nonmagnetic barrier layer. As shown by Mazin (1999), this drasti-
cally changes the physics behind the polarization of electrical current and should not
be confused with the polarization obtained from Andreev reflection spectroscopy or
from other techniques, such as photo-emission studies (see, e.g., Sicot et al., 2003).
    In this section a further introduction will be presented on the wide range of
complexities in understanding the underlying physics of tunneling spin polariza-
tion and its intimate relation to TMR. This will be preceded by the experimental
procedure to determine the polarization via transport in magnetic-superconducting
junctions at low temperatures, so-called superconducting tunneling spectroscopy or
STS. Also an overview will be presented of existing data on tunneling spin polar-
ization and the relation with the physics involved.

3.1 How to measure spin polarization?
The tunneling spin polarization as introduced in Eq. (10) can be measured straight-
forwardly by superconducting tunneling spectroscopy. For an extended review of
this technique, see Meservey and Tedrow (1994), or the thesis work of Worledge
(2000), Kaiser (2004), and Kant (2005). In STS one uses a superconducting elec-
trode as a detector for the tunneling spin polarization in the following way. The
tunneling current in junctions at a finite, low bias voltage is in first order governed
by the density-of-states factors and the tunneling probability, see Eqs. (4) and (5).
In the case of one superconducting electrode this reads:
                                 G ∝ N(EF )T (φ, t)ρ(eV ),                         (17)
with N(EF ) the metal density-of-states at the Fermi level, ρ(eV ) the supercon-
ducting density-of-states at an energy eV , and with V the bias voltage between
superconductor and magnetic metal. A measurement of the conductance is there-
fore directly reflecting the density-of-states of a superconductor, having sharp peaks
at an energy ± , the energy difference between the energy level of the Cooper
pairs and the single-electron states (see Fig. 1.24a). Only at voltages exceeding
54                                                                               H.J.M. Swagten




Figure 1.24 Calculated conductance of a superconductor-metal tunnel junction as sketched in
the bottom right panel. In part (a) the zero-field conductance is shown both at 0 K (thin line)
and at T = 0.1Tc (rounded curve). In panel (b)–(d) the spin-up and spin-down density-of-states
in the superconductor are Zeeman split by a magnetic field B (μB B = 0.6 ) at T = 0.1Tc .
The polarization of the tunneling electrons is zero in (b), +40% in (c), and –80% in (d). The
thin dashed and solid line in graph (b) represent the conductance due to the individual spin-up
and spin-down density-of-states, respectively, both at 0 K.


± /e the electrons can tunnel into the empty single-electron states of the su-
perconductor or vice versa, from the superconductor towards the metal. It is crucial
to note that the magnitude of         is typically around 1 meV, whereas the density-
of-states of a metal shows variations on an energy scale of eV’s. As a result, the
conductance in Eq. (17) exclusively reflects the peak-shaped density-of-states of
the superconductor. At finite temperatures, thermal broadening of the Fermi level
reduces the sharpness of the peaks as can be seen in Fig. 1.24a.
    The density-of-states of a metal N(E) as observed in the conductance mea-
surement is the sum of the density of spin-up states and the density of spin-down
states. In absence of a magnetic field, no energy is required to flip the spin of an
electron, and, accordingly, the contributions to the conductance of the spin-up and
spin-down density-of-states coincide. This situation changes when a magnetic field
is applied parallel to the plane of the tunnel junction. The magnetic field penetrates
the superconductor uniformly since the thickness of the superconducting electrode
is much smaller than the penetration depth of the magnetic field. In presence of the
field, the spin-up electrons (assumed to have their moment parallel to the field) are
lowered in energy with respect to spin-down electrons, corresponding to an energy
difference of 2μB B, where μB is the electron magnetic moment, and B the mag-
Spin-Dependent Tunneling in Magnetic Junctions                                     55


netic induction. Consequently, the magnetic field shifts the density of spin-up states
of the superconductor to lower energy and the density of spin-down states to higher
energy. Figure 1.24b shows how these energy shifts lead to four maxima in the
conductance. The maxima are clearly resolved when the Zeeman splitting 2μB B is
large as compared to kB T , defining the sharpness of the maxima. The maximum
applicable field is limited by the critical field Bc of the superconducting electrode.
Critical fields larger than 4 T can be obtained with aluminum superconducting
electrodes, and, typically, fields of 2 to 3 T are applied. With μB ≈ 58 µeV/T
and kB ≈ 86 µeV/K one finds that a temperature below 1 K is required to clearly
resolve the Zeeman splitting.
    The two conductance maxima at low bias, those numbered 1 and 2 in Fig. 1.24b,
give a direct indication of the spin polarization of the tunneling electrons. Since at
the position of maximum 1 the density of spin-up states is zero, maximum 1 is
a direct measure of the spin-down conductance. Likewise, maximum 2 is a direct
measure of the spin-up conductance. In the example of Fig. 1.24b the spin-up and
spin-down conductance are equal, i.e., the tunneling spin polarization is zero. When
dealing with ferromagnetic metals with a positive tunneling spin polarization, there
are more majority (spin-up) electrons available for tunneling than minority (spin-
down) electrons, by which maximum 2 becomes larger than 1 as shown by the
calculated curve in Fig. 1.24c. In this particular example, the polarization is 40%
and positive, since we assumed that tunneling is dominated by majority electrons.
To a good approximation, the tunneling spin polarization P is given by the relative
difference between the height of maxima 2 and 1,
                                         G 2 – G1
                                        P ≈       ,                               (18)
                                         G 2 + G1
as indicated in the figure. A most accurate extraction of the polarization is obtained
by fitting a model to the measured conductance curve, which will be discussed
later on. To finish the introduction of the spin-polarized tunneling technique, we
consider the case of negative polarization as shown by the calculation in Fig. 1.24d.
Here maximum 1 is larger than 2, which means that the tunnel current is dominated
by minority electrons.
    The superconductor used in the tunnel junctions for spin-polarized tunneling
measurements is usually aluminum. There are two main reasons responsible for
this important role of aluminium. First, aluminum is a superconductor with a low
atomic number. Clear observation of the Zeeman split spin-up and spin-down su-
perconducting density-of-states is possible only when the spin–orbit scattering rate
in the superconductor is low and thus requires a low atomic number. Consequently,
other common superconductors such as niobium, lead and tantalum are not suit-
able and only a few superconductors other than aluminum can be used (Meservey
and Tedrow, 1994). The second reason for the almost exclusive role of aluminum
in spin-polarized tunneling is the defect and pinhole-free amorphous Al2 O3 tun-
nel barrier obtained by exposing metallic aluminum to oxygen; see section 2.2.
Consequently, most published work is based on Al-Al2 O3 -metallic tunnel junctions
obtained by oxidation of the top part of an aluminum electrode followed by de-
position of the normal metal on top of the Al2 O3 barrier. However, also inverted
56                                                                        H.J.M. Swagten



structures of metallic-Al2 O3 -Al have been successfully used in STS experiments
(see, e.g., Kaiser and Parkin, 2004), showing that the tunneling spin polarization
can be different as compared to the system with Al underneath the oxide. In the
inverted structures, especially reactive Ni-containing bottom electrodes (Ni1–x Fex )
are susceptible to oxidation when Al is plasma oxidized prior to deposition of the
top (superconducting) Al layer. These oxidized Ni-Al interfaces then lead to a sig-
nificant reduction of spin polarization.
    Zeeman splitting in the conductance of Al(superconducting)-Al2 O3 -metallic
tunnel junctions is observed for the first time by Tedrow and Meservey (1971b)
in the early 70’s (see also Meservey and Tedrow, 1994). Soon thereafter values
of the tunneling polarization in junctions with various magnetic top electrodes
are obtained. These early polarizations are determined simply from the differences
in the conductance maxima using a procedure similar as given in Eq. (18). This
procedure, however, leads to a small but significant underestimation of the polar-
ization since it does not take into account the effect of orbital depairing and a finite
spin–orbit scattering rate on the spin-up and spin-down superconducting density-
of-states (Monsma and Parkin, 2000a; Worledge and Geballe, 2000a). Depairing in
a superconductor is caused by the presence of a magnetic field. The field can orig-
inate from different sources, but, most relevant for our purpose, it is the external
applied field that induces an orbital motion of the electrons by the Lorentz force
breaking up the Cooper pairs (Tinkham, 1996). In experiments, a small out-of-
plane external magnetic field can seriously broaden the conductance curves shown
in Fig. 1.24. It is estimated that typically the field should be aligned with the plane
of the superconducting layer better than 0.05° to avoid too much broadening to
accurately extract spin polarization (Kant, 2005). As mentioned before, to reduce
the effect of spin–orbit coupling Al is almost exclusively used in STS due to its
low atomic mass, which minimizes an effective mixing between the spin-up and
spin-down channels. However, also in the case of Al, spin–orbit interaction should
be taken into account in analyzing conductance curves to extract tunneling spin
polarization unambiguously (Worledge and Geballe, 2000a).
    In Fig. 1.25 an example is given of a measurement of the tunneling spin po-
larization using STS in a system of Al-Al2 O3 -Co and Al-Al2 O3 -Co90 Fe10 , yielding
in both cases a positive spin polarization (note the similarity with the conductances
shown in Fig. 1.24c). The theoretical, solid curves in Fig. 1.25 are based on the so-
called Maki-theory that takes into account the required corrections for spin–orbit
interaction and orbital depairing in the superconductor; see Maki (1964), Merservey
et al. (1980), and Worledge and Geballe (2000a).

3.2 Data on tunneling spin polarization
An overview of available data on tunneling spin polarization is listed in Table 1.2
and Table 1.3. Generally, the 3d metallic ferromagnets have a positive polarization
between +30% and +55% when tunneling across amorphous Al2 O3 is consid-
ered. The polarization does not scale with the magnetic moment μ↑ – μ↓ of Ni,
Co, and Fe, being 0.62μB , 1.75μB , and 2.22μB per atom, respectively. Although
this is not surprising, even using the first-order definition of tunneling spin po-
Spin-Dependent Tunneling in Magnetic Junctions                                              57




Figure 1.25 Conductance G as a function of bias voltage V of a UHV-sputtered junction
consisting of (a) Al/Al2 O3 /Co and (b) Al/Al2 O3 /Co90 Fe10 , taken at T = 0.3 K in zero field
and in an external field of several Tesla (as indicated). The total thickness of superconducting
Al electrode and Al2 O3 barrier is typically smaller than 50 Å. The top electrodes are 200 Å in
thickness and capped with 60 Å Ta. The solid lines are theoretical fits using the Maki-theory
(Worledge and Geballe, 2000a) yielding the indicated tunneling spin polarization. Reproduced
from Kant (2005).



larization involving the density-of-states at the Fermi level (Eq. (10)), there has
been some debate on this issue in literature, in particular in the beginnings of spin-
polarized tunneling experiments; see Meservey and Tedrow (1994), and also Tezuka
and Miyazaki (1996). The absence of correlation between magnetic moment and
tunneling spin polarization also applies to alloys between Ni, Fe and Co, again
in contrast to the early experiments (Paraskevopoulos et al., 1977)). The data on
Ni1–x Fex alloys as shown in Table 1.3 have been used by van de Veerdonk et al.
(1997a) to verify that magnetic moment μ↑ – μ↓ is not linearly related to tunneling
spin polarization P ; see Fig. 1.26. Also in the case of spin polarization obtained
by Andreev reflection spectroscopy using metal-superconducting contacts, data on
Ni1–x Fex are almost independent of x, again demonstrating the absence of correla-
tion with magnetic moment or magnetization. A similar conclusion is reached by
Kim and Moodera (2002) and Kaiser et al. (2005b) in their STS studies of Co1–x Mnx
and Co1–x Ptx binary alloys, respectively. In both cases, the spin polarization is almost
insensitive to the Mn (Pt) concentration up to x ≈ 0.40 (see Table 1.3), whereas the
magnetization in that regime is linearly suppressed with increasing Mn (Pt) content.
Surprisingly, however, the Co1–x Vx systems studied by Kaiser et al. (2005b) exhibit
an approximately linear relationship between tunneling spin polarization and mag-
netization for x roughly below 0.35.
58                                                                                 H.J.M. Swagten


Table 1.2 Tunneling spin polarization (P ) values obtained with the STS technique for a number
of elementary ferromagnetic materials, as well as for some crystalline ferromagnetic electrodes
or barriers indicated by an asterisk (∗). Data with a dagger (†) are not corrected for depairing
and spin–orbit coupling. The Al2 O3 indicated with the double dagger (‡) has a variable thick-
ness, between 6.4 Å and 16 Å (Munzenberg and Moodera, 2004). Polarization data obtained by
Monsma and Parkin (2000a) and Panchula et al. (2003) are for junctions where the Al sputter
target is slightly doped with Si

System                                   P (%)             Reference
Al/Al2 O3 /Ni                             32–46            Moodera and Mathon (1999),
                                                           Kim and Moodera (2004),
                                                           Monsma and Parkin (2000a)
Ni∗ /Al2 O3 /Al                           25 ± 2           Kim and Moodera (2004)
Al/Al2 O3 /Co                             40 ± 2           Moodera and Mathon (1999),
                                                           Monsma and Parkin (2000a),
                                                           Kant et al. (2004c)
Al/Al2 O3 /Fe                             42 ± 2           Moodera and Mathon (1999),
                                                           Monsma and Parkin (2000a),
                                                           Kant et al. (2004c)
Al/Al2 O3 ‡ /Fe                           22–43            Munzenberg and Moodera (2004)
Al/Al2 O3 /Gd                             13 ± 4†          Merservey et al. (1980)
Al/Al2 O3 /Gd                             13               Kaiser et al. (2005a)
Al/Al2 O3 /Tb                              5 ± 2†          Merservey et al. (1980)
Al/Al2 O3 /Dy                              6 ± 2†          Merservey et al. (1980)
Al/Al2 O3 /Ho                              6 ± 2†          Merservey et al. (1980)
Al/Al2 O3 /Er                              5 ± 2†          Merservey et al. (1980)
Al/Al2 O3 /Tm                              3 ± 2†          Merservey et al. (1980)
NiMnSb∗ /Al2 O3 /Al                        28 ± 2          Tanaka et al. (1999)
Al/Al2 O3 /Mn45 Sb55                      30.5             Panchula et al. (2003)
La0.67 Sr0.33 MnO∗ /SrTiO∗ /Al
                  3      3                72 ± 1           Worledge and Geballe (2000b)
SrRuO∗ /SrTiO∗ /Al
         3       3                       –10 ± 1           Worledge and Geballe (2000c)
Co/SrTiO3 /Al                             31               Thomas et al. (2005)
CrO∗ /Cr2 O3 /Al
     2                                   100               Parker et al. (2002)
Al/MgO/Co                                 30 ± 2           Kant et al. (2004c)
Al/MgO∗ /Co                               39               Kaiser et al. (2005a)
Al/MgO/Fe                                 30 ± 2           Kant et al. (2004c)
Fe∗ /MgO∗ /Al                             57–74            Parkin et al. (2004)
Co70 Fe∗ /MgO∗ /Al
       30                                 52–85            Parkin et al. (2004)


   An intriguing STS experiment addressing the relation between magnetization
and tunneling polarization is performed with ferrimagnetic alloys between Co and
Gd (Kaiser et al., 2005a); see Table 1.3. At the compensation point of the alloy,
the magnetization is absent due to an equal sublattice contribution. Nevertheless,
the tunneling spin polarization at this point can still be very large and can even
Spin-Dependent Tunneling in Magnetic Junctions                                                59




Table 1.3 Tunneling spin polarization (P ) values obtained with the STS technique for a num-
ber of alloys. Data with a dagger (†) are not corrected for depairing and spin–orbit coupling.
Crystallinity of MgO barriers is indicated by an asterisk (∗). Barriers in junctions by Monsma
and Parkin (2000a), Kaiser et al. (2005a), and Kaiser et al. (2005b) are made with a Si-doped Al
sputter target. For data on CoV and CoPt alloys employing AlN as a barrier, we refer to Kaiser et
al. (2005b)

System                                    P (%)                 Reference
Al/Al2 O3 /Co90 Fe10                      48 ± 1                Paluskar et al. (2005a)
Al/Al2 O3 /Co84 Fe16                      50 ± 2                Moodera and Mathon (1999),
                                                                Monsma and Parkin (2000a)
Al/Al2 O3 /Co60 Fe40                      50                    Monsma and Parkin (2000a)
Al/Al2 O3 /Co50 Fe50                      50 ± 1                Moodera and Mathon (1999),
                                                                Monsma and Parkin (2000a)
Al/Al2 O3 /Co40 Fe60                      51                    Monsma and Parkin (2000a)
Al/Al2 O3 /Co72 Fe20 B8                  53.5                   Paluskar et al. (2005b)
Al/Al2 O3 /Ni4 Fe96                       45†                   van de Veerdonk et al. (1997a)
Al/Al2 O3 /Ni12 Fe88                      50†                   van de Veerdonk et al. (1997a)
Al/Al2 O3 /Ni17 Fe83                      49†                   van de Veerdonk et al. (1997a)
Al/Al2 O3 /Ni25 Fe75                      40†                   van de Veerdonk et al. (1997a)
Al/Al2 O3 /Ni30 Fe70                      51†                   van de Veerdonk et al. (1997a)
Al/Al2 O3 /Ni40 Fe60                      55†                   Monsma and Parkin (2000a)
Al/Al2 O3 /Ni47 Fe53                      52†                   van de Veerdonk et al. (1997a)
Al/Al2 O3 /Ni60 Fe40                      53†                   Monsma and Parkin (2000a)
Al/Al2 O3 /Ni74 Fe26                      46†                   van de Veerdonk et al. (1997a)
Al/Al2 O3 /Ni78 Fe22                      45†                   van de Veerdonk et al. (1997a)
Al/Al2 O3 /Ni81 Fe19                      45                    Monsma and Parkin (2000a)
Al/Al2 O3 /Ni86 Fe14                      33†                   van de Veerdonk et al. (1997a)
Al/Al2 O3 /Ni90 Fe10                      36                    Monsma and Parkin (2000a)
Al/Al2 O3 /Ni95 Fe5                       34                    Monsma and Parkin (2000a)
Al/Al2 O3 /Co90 Mn10                      37                    Kim and Moodera (2002)
Al/Al2 O3 /Co73 Mn27                      33                    Kim and Moodera (2002)
Al/Al2 O3 /Co68 Mn32                      33                    Kim and Moodera (2002)
Al/Al2 O3 /Co30 Mn70                       9                    Kim and Moodera (2002)
Al/Al2 O3 /Co95 V5                        30                    Kaiser et al. (2005b)
Al/Al2 O3 /Co89 V11                       22                    Kaiser et al. (2005b)
Al/Al2 O3 /Co87 V13                       19                    Kaiser et al. (2005b)
Al/Al2 O3 /Co83 V17                       16                    Kaiser et al. (2005b)
Al/Al2 O3 /Co77 V23                        8                    Kaiser et al. (2005b)
Al/Al2 O3 /Co74 V26                        6                    Kaiser et al. (2005b)
                                                                         (continued on next page)
60                                                                               H.J.M. Swagten


Table 1.3   (Continued)

System                                       P (%)                        Reference
Al/Al2 O3 /Co90 Pt10                         37                           Kaiser et al. (2005b)
Al/Al2 O3 /Co81 Pt19                         40                           Kaiser et al. (2005b)
Al/Al2 O3 /Co65 Pt35                         42                           Kaiser et al. (2005b)
Al/Al2 O3 /Co60 Pt40                         41                           Kaiser et al. (2005b)
Al/Al2 O3 /Co44 Pt56                         29                           Kaiser et al. (2005b)
Al/Al2 O3 /Co34 Pt66                         28                           Kaiser et al. (2005b)
Al/Al2 O3 /Co26 Pt74                         18                           Kaiser et al. (2005b)
Al/Al2 O3 /Co95 Gd5                         26                            Kaiser et al. (2005a)
Al/Al2 O3 /Co93 Gd7                         31                            Kaiser et al. (2005a)
Al/Al2 O3 /Co79 Gd21                       –12, –8, 0, +9                 Kaiser et al. (2005a)
Al/Al2 O3 /Co70 Gd30                       –17                            Kaiser et al. (2005a)
Al/Al2 O3 /Co62 Gd38                       –20                            Kaiser et al. (2005a)
Al/Al2 O3 /Co50 Gd50                       –20                            Kaiser et al. (2005a)
Al/Al2 O3 /Co39 Gd61                       –14                            Kaiser et al. (2005a)
Al/Al2 O3 /Co31 Gd69                       –10                            Kaiser et al. (2005a)
Al/Al2 O3 /Co24 Gd76                         2                            Kaiser et al. (2005a)
Co77 Gd23 /MgO∗ /Al                        –28, +23                       Kaiser et al. (2005a)
Co68 Gd32 /MgO∗ /Al                        –28                            Kaiser et al. (2005a)
Co40 Gd60 /MgO∗ /Al                        –22                            Kaiser et al. (2005a)
Al/MgO∗ /Co77 Gd23                         –15                            Kaiser et al. (2005a)
Al/MgO∗ /Co68 Gd32                         –15                            Kaiser et al. (2005a)




Figure 1.26 (a) Tunneling spin polarization P measured by STS at T = 0.4 K and (b)
low-temperature saturation magnetization μSAT of Ni1–x Fex compounds. NiFe is either evapo-
rated from a tungsten boat (open symbols) or from an e-gun (solid). The results for Ni samples
prepared under ultra-high vacuum conditions (squares) are taken from Moodera and Mathon
(1999), and Kim and Moodera (2004). Note in (a) the lower polarization of Ni25 Fe75 , prob-
ably due to a structural transformation in this range of composition. The solid curve in (b) is
taken from Bozorth (1993). Adapted from van de Veerdonk et al. (1997a).
Spin-Dependent Tunneling in Magnetic Junctions                                         61


change sign (see section 3.3.2 for a more detailed discussion). As a final remark
to the ongoing debate on the linearity between tunneling spin polarization and
magnetization, Hindmarch et al. (2005a) have used Co-Al2 O3 -Cu38 Ni62 junctions
using CuNi as an electrode with a low Curie temperature of around 240 K. Due
to this, the temperature-dependent magnetization of CuNi can be directly related
to the tunneling spin polarization of CuNi as a function of temperature. The lat-
ter is extracted from TMR(T ) = 2PCo (T )PCuNi (T )/[1 – PCo (T )PCuNi (T )] with
PCo (T ) separately determined from Co-Al2 O3 -Co junctions. Although the rela-
tion between P and M is found to be strictly nonlinear, it can be reproduced by
a theoretical model that incorporates tunneling via multiple s and d (hybridized)
bands. This will be further addressed in section 3.3.
    At present, there is no complete theoretical picture enabling realistic predictions
of tunneling spin polarization for the 3d metals as listed in Table 1.2. Nevertheless,
there are a number of arguments that lead to a qualitative understanding of sign,
and to some extent, the magnitude of P . This will be further discussed in the next
section (3.3), where the ingredients for tunneling spin polarization will be discussed
in some detail. Also shown in Table 1.2 are the 3f and 4f ferromagnetic materials,
showing generally a rather low polarization. This, combined with the low Curie
temperature of these elements, explains that only a few papers have addressed these
materials for implementation in magnetic junctions. Data have also been gathered
to test the half-metallicity of some materials (such as La0.67 Sr0.33 MnO3 , NiMnSb or
CrO2 ) in tunneling experiments, which will be the topic of section 4.4. Only in
the case of CrO2 , a true 100% tunneling spin polarization is found. Related to this,
the table shows the tunneling spin polarization when using crystalline ferromag-
netic materials, in some cases combined with a crystalline insulating barrier as well.
Surprisingly, in SrRuO3 -SrTiO3 -Al epitaxial junctions a negative spin polarization
is measured, only rarely reported in STS experiments. Epitaxial junctions will be
treated extensively in section 4, which includes a detailed discussion on the use of
MgO as a barrier material. As can be seen from Table 1.2, the crystallinity of MgO
is crucial for obtaining a very high tunneling spin polarization, in some cases up
to +85%.
    It is instructive at this point to recall the intimate relation between tunneling spin
polarization and magnetoresistance as suggested by Eq. (9), which for equal mag-
netic electrodes reads: TMR = 2P 2 /(1 – P 2 ). In Fig. 1.27 a compilation is shown
of TMR in a ferromagnet-insulator-ferromagnet junction with equal electrodes as
a function of the tunneling spin polarization measured with STS; see Table 1.2 and
Table 1.3. Generally, it is observed that the Julliere formula (solid curve) is reason-
ably well describing the data (see, however, also the earlier discussions in Miyazaki
and Tezuka, 1995b and Lu et al., 1998). This seems to justify the use of the Julliere
equation, despite a number of theoretical contributions addressing its validity (for
example, see MacLaren et al., 1997; Qi et al., 1998; Tsymbal and Pettifor, 1998;
Mathon and Umerski, 1999; Belashchenko et al., 2004). However, it should be
emphasized that the number of data in Fig. 1.27 is rather limited and is in some
cases obtained by a correction of TMR for the electrodes being unequal. Further-
more, it obviously passes over the rich physics behind spin polarization and TMR,
62                                                                               H.J.M. Swagten




Figure 1.27 TMR measured in magnetic junctions with equal electrodes as a function of the
tunneling spin polarization determined by STS. Some of the TMR data are based on MTJs
with different electrodes, and have been corrected accordingly. See Table 1.2 and Table 1.3 for
references on tunneling spin polarization.

and suggests that the Julliere formula should be used only as a phenomenological
equation that links these physical quantities.
     It is important to mention that some of the tunneling spin polarization data
collected in Table 1.2 and Table 1.3 have been subject to significant changes
over the past decades. This can be partially explained by the progress in de-
position and surface characterization tools, in particular after the realization of
high room-temperature TMR in the mid-nineties. As an example, the polar-
ization of Ni has increased from 5% to 33% over the years, as reported in a
number of review papers (Meservey and Tedrow, 1994; Moodera et al., 1999a;
Moodera and Mathon, 1999). As we will show below, the tunneling spin polar-
ization is quite sensitive to the ferromagnet-insulator interface conditions, which
can obviously vary from lab to lab, and may significantly improve over the years.
Recently, an even higher value for PNi of 46% has been reported, now obtained by
employing cleaner interfaces between polycrystalline Ni and amorphous alumina
using ultra-high vacuum conditions (Kim and Moodera, 2004). The extreme sen-
sitivity of the polarization for tiny structural changes close to the barrier is clearly
demonstrated by Monsma and Parkin (2000b). They observe a suppression of tun-
neling spin polarization of Ni when measured over a large number of weeks, from
28% directly after deposition to 16% after almost a year, which is the result of a
slowly evolving chemical reaction between Ni and the alumina barrier.

3.3 Ingredients of tunneling spin polarization
As discussed before, it is crucial to have a solid interpretation of tunneling spin
polarization measured in an STS experiment. In this section we will introduce the
ingredients for tunneling spin polarization not covered by the elementary formula
given by Eq. (10). This includes the following aspects:
Spin-Dependent Tunneling in Magnetic Junctions                                         63


• the dominant role of the ferromagnet-insulator interfaces, instead of bulk prop-
  erties of the ferromagnetic electrodes
• variations in the mobility or Fermi velocity of tunneling electrons (sp- or d-like),
  and dissimilarity in tunneling transmission coefficients, being dependent on the
  character of the electron wave functions
• relevance of the tunneling barrier due to the chemical bonding in the interface
  region between ferromagnet and insulator.

3.3.1 Density-of-states at the barrier interfaces
The spin polarization of electrical current measured in metal-superconductor junc-
tions is generally defined as the relative difference in the spin-up and spin-down
current or conductance:
                                                 G ↑ – G↓
                                        P =               .                          (19)
                                                 G ↑ + G↓
When using Eq. (17) with an equal transmission across the barrier for both spin-
up and spin-down electrons, the tunneling spin polarization reads [Nmaj (EF ) –
Nmin (EF )]/[Nmaj (EF ) + Nmin (EF )]. This is identical to what is derived in sec-
tion 1, Eq. (10). It suggests that the tunneling spin polarization is determined by
the Fermi electrons of the bulk ferromagnetic material which is in striking contrast
with existing experimental observations. In fact, a crucial point in understanding
tunneling polarization and TMR is that it is not governed by the bulk density-of-
states but rather by the local density-of-states at the interfaces with the barrier. Quite
generally, the tunneling process samples the density-of-states only over a few Fermi
wavelengths, in particular in the strongly perturbed region at the metal-insulator in-
terface. Due to the strong screening of the Fermi sea in a metal, bulk metal electrons
are simply not aware of the interface with the barrier until they are approaching it
at a few monolayers. This has been theoretically recognized already in the early days
of electron tunneling in superconducting as well as in normal metal junctions (see,
e.g., Appelbaum and Brinkman, 1969, 1970; Mezei and Zawadowski, 1971). Along
with experimental data on the relevance of interfaces (see below and also in sec-
tion 4), a number of calculations have appeared to provide a more solid theoretical
basis for the interfacial sensitivity of spin-polarized tunneling (Tsymbal and Petti-
for, 1997; de Boer et al., 1998; Itoh et al. 1999a, 1999b; Zhang and Levy, 1999;
Vedyayev et al., 1999; Uiberacker et al., 2001; Uiberacker and Levy, 2001).
    The crucial role of the interfaces with the barrier for tunneling spin polariza-
tion is experimentally first seen by Tedrow and Meservey (1975) in Al-Al2 O3 -Al
junctions when a thin layer of Co is inserted at the interface between the top non-
superconducting Al and the Al2 O3 ; see Fig. 1.28a. In this STS measurement it is
observed that only one or two monolayers of ferromagnetic metal at the interface
with the alumina barrier induces a finite spin polarization, saturating to the bulk
value at 3 to 5 monolayers only. Also in a more recent experiment a similar inter-
facial effect is observed, although now in a reversed way (Moodera et al., 1989).
The strong tunneling spin polarization in the Al-Al2 O3 -Fe system is efficiently
suppressed by inserting an ultrathin nonmagnetic layer at the barrier-ferromagnetic
64                                                                          H.J.M. Swagten




Figure 1.28 (a) Tunneling spin polarization P versus thickness t of the ferromagnetic Co
layer in a junction consisting of 40 Å Al (superconducting)/15 Å Al2 O3 /t Co/50 Å Al
(normal). (b) P versus thickness of a nonmagnetic Au interlayer in 40 Å Al + oxidation/t
Au/300–500 Å Fe. After Tedrow and Meservey (1975) and Moodera et al. (1989).


interface, as shown in the right panel of Fig. 1.28. This proofs the interface sensitiv-
ity of (spin-dependent) electron tunneling, and suggests that we should consider the
density-of-states at the Fermi level in the tunneling spin polarization as a local inter-
facial density-of-states rather than a bulk density-of-states. Later on (section 4), we
will extensively come back to the role of interfaces and interfacial density-of-states
for TMR.

3.3.2 Weighted density-of-states factors
Apart from the role of the interfaces, there is another ingredient for better under-
standing the tunneling spin polarization, which is obvious when considering the
positive sign of the polarization for traditional ferromagnetic metals such as Fe, Co,
and Ni, as we have compiled in Table 1.2. For example, in the case of Co and Ni
the dominance of minority electrons at the Fermi level, see Fig. 1.6 in section 1,
would result in a negative spin polarization, whereas in the table it is shown that
experimentally they have a positive sign, i.e. tunneling is most efficient for majority
electrons.
    The observation that not just the (interfacial) density-of-states is decisive for
polarization can be explained by reconsidering the definition of Eq. (10): P =
[Nmaj (EF )–Nmin (EF )]/[Nmaj (EF )+Nmin (EF )]. It is theoretically analyzed by Mazin
(1999) that the density-of-states factors Nmin,maj (EF ) should be weighted with spin-
dependent factors reflecting the possibility that electrons of different symmetry and
Fermi velocity are coupled differently to the states in the barrier. This yields an
alternative, more physically justified expression of tunneling spin polarization:
                              wmaj Nmaj (EF ) – wmin Nmin (EF )
                        P =                                     ,                    (20)
                              wmaj Nmaj (EF ) + wmin Nmin (EF )
Spin-Dependent Tunneling in Magnetic Junctions                                         65


with wmin(maj) the spin-resolved weighting coefficients, and, following the earlier
discussion in this subsection, with N(EF ) now referring to density-of-states at
the interfaces. The weighting factors are suggested to depend on (i) the Fermi
velocity of the electrons travelling perpendicular to the tunneling barrier, and
(ii) on transmission coefficients for electron tunneling. As an example, the latter
dependence on transmission matrix elements can be understood from our ear-
lier expression of conductance in STS experiments, see Eq. (17). When inserting
this in the definition of tunneling spin polarization in Eq. (18), it is found that
P = [Tmaj Nmaj (EF ) – Tmin Nmin (EF )]/[Tmaj Nmaj (EF ) + Tmin Nmin (EF )], i.e. density-
of-states factors clearly weighted with transmission coefficients. Mazin (1999) has
pointed out that for a simple model of a delta-type tunnel barrier with a large bar-
rier height (V (x) = W δ(x) with W the barrier height approaching infinity) the
                                             2
weighting factors are proportional to vF ,x , with vF the Fermi velocity. This again
stresses the relevance of the orbital character of the tunneling electrons, not only the
static density-of-states of Fermi electrons. It is subsequently pointed out by Butler
et al. (2001a) from calculations using the layer Korringa Kohn Rostoker technique
(MacLaren et al., 1990) that also the electron wave functions parallel to the plane of
the layers are critically important for spin-dependent tunneling processes. Match-
ing of the wave functions at the interfaces is influenced by the lateral variations
of wave functions of different symmetry, and can dramatically change their de-
cay rate in the barrier region, by which s electrons seem to tunnel more readily
than d-like electrons. Note that this is contrary to the free-electron models, where
the decay rate for a given k and energy E is uniquely given by exp(–2κt) with
κ = [2m(U (x) – E)/h2 + k 2 ]1/2 ; see section 1.
                         ¯
     In view of the relevance of barrier transmission probabilities and Fermi veloci-
ties of tunneling electrons, it is instructive to have a closer look at the band structure
of the 3d ferromagnetic metals commonly used in MTJs. Due to the multi-orbital
character of the band structure, both s- and d-type electrons are active around at
the Fermi level and may therefore contribute to a (tunneling) current. Especially the
dispersive s-like electrons are believed to have the largest tunneling probability due
to their small effective mass, see Eq. (1), whereas d electron wave functions are more
localized and more rapidly decay in the barrier region. In the pioneering theoret-
ical work of Stearns (1977), it is pointed out that this is consistent with a positive
spin polarization of Co and Ni in contrast to the (negative) polarization of the full
density-of-states. The sign change between the tunneling polarization from s- and
d-states is related to the hybridization between these bands as shown more recently
by more sophisticated calculations; see, e.g., Butler et al. (2001a), Mathon (1997),
and Mazin (1999). As an example, in the tight-binding calculations of Mathon
(1997), the current polarization between two Co electrodes changes from negative
to positive when the tight-binding hopping integral is gradually turned off, sim-
ulating the transition from metallic GMR-type conduction to tunneling across an
insulating interlayer.
     An interesting consequence of these ideas is that it may be possible to select
s or d tunneling states by varying the thickness of the insulator as pointed out in
these calculations. When the barrier is sufficiently thick, the itinerant s-like elec-
trons dominate the current. However, the d electrons take over for sufficiently thin
66                                                                         H.J.M. Swagten



barriers, which could turn the spin polarization from positive to negative. The
predicted sign change has not been observed experimentally. Nevertheless, a sig-
nificant decrease of the positive tunneling spin polarization of Fe is seen when
barriers are thinned down to below ≈ 10 Å in Al-Al2 O3 -Fe superconducting junc-
tions (Munzenberg and Moodera, 2004). Although these findings are attributed to
the increasing role of d-like electron states, it should be mentioned that the data
for the thinnest barriers are not very well described by the Maki-theory, possi-
bly related to the extremely small junction resistance in this regime, in some cases
only 1       or less. Also it is important to realize that a similar trend of decreas-
ing polarization is observed in the magnetoresistance of MTJs (Freitas et al., 2000;
Oliver and Nowak, 2004); see also Fig. 1.13. In these data, the suppression of
TMR at lower barrier thickness is related to, e.g., an incomplete coverage of the
ultrathin Al leading to more pinholes in extremely thin barriers. In the STS data
of Munzenberg and Moodera (2004) this is, however, ruled out by the absence of
leakage current around zero bias voltage.
     In section 3.2, it is pointed out that a number of STS studies show that there is
no simple proportionality between tunneling spin polarization and magnetization of
the ferromagnetic electrode (see also Fig. 1.26). Moreover, in Co1–x Mnx (Kim and
Moodera, 2002) and Co1–x Ptx (Kaiser et al., 2005b) the spin polarization remains
almost constant up to x = 0.40, in contrast to the magnetization that is linearly
suppressed. Kaiser et al. (2005b) conjecture that this can be explained by assuming
that in Co1–x Ptx the tunneling rate from Pt atomic sites is much lower than from
the highly polarized Co sites, an argument that the authors derive from scanning
tunneling microscopy studies (Hofer et al., 2003). In fact, it is expected that the
strong bonding of Co to the oxygen at the barrier interface as compared to Pt leads
to the enhanced tunneling transmission from Co sites. In a follow-up study, addi-
tional evidence for this site-dependent tunneling transmission is found from data on
Co1–x Gdx ferrimagnetic alloys of heavy rare earth and 3d transition metals (Kaiser
et al., 2005a). As shown in Fig. 1.29a, the tunneling spin polarization is positive
for low x due to the dominant tunneling probability from Co sites, combined with
the fact that the Co sublattice is aligned with the field direction in this regime (see
Fig. 1.29b). However, at the compensation composition of the ferrimagnetic alloy,
i.e. at x ≈ 0.20, the polarization abruptly changes sign, since now the Co sublattice
magnetization is antiparallel to the field due to the much larger magnetic moment
of Gd as compared to Co. Upon further increase of x, the positive contribution
from the Gd polarization leads to a sign reversal of P from negative to positive at
x ≈ 0.75, finally saturating at P = +13% for pure Gd. Using a phenomenological
model that simply weights the tunneling spin polarization associated with Co and
Gd sites (see the curves in Fig. 1.29a), a tunneling probability is found that is around
20% higher from Co sites than from Gd.

3.3.3 Bonding across the metal-insulator interface
So far, the role of the insulator for tunneling spin polarization is not considered.
However, due to the prominent role of the interfacial density-of-states as well as
the electron waves of different symmetry decaying in the barrier, it is conceivable
that the insulator and in particular the metal-interface region is also crucial for tun-
Spin-Dependent Tunneling in Magnetic Junctions                                            67




Figure 1.29 (a) Tunneling spin polarization P of Co1–x Gdx versus the Gd concentration
x measured by STS at T ≈ 0.3 K across Al2 O3 barriers (open symbols) and MgO (solid
symbols). The curves represent a model calculation that weights the contribution to P
from the available Co and Gd sites. (b) Saturation magnetization MSAT measured with
a superconducting-quantum-interference-device (SQUID) magnetometer at T = 10 K of
100 Å Ta/1000 Å Co1–x Gdx /100 Å Ta (open symbols), together with data taken from Hansen
et al. (1989) (solid curve). The arrows indicate the alignment of the Co and Gd subsystems in
the magnetic field H at the compensation point (vertical line in grey at x ≈ 0.20), as well as
for concentrations below and above this point. Adapted from Kaiser et al. (2005a).



neling spin polarization and TMR. Experimentally, first evidence has been found
by a sign reversal of the tunneling spin polarization when replacing Al2 O3 for a
composite Ta2 O5 -Al2 O3 barrier combined with Ni80 Fe20 electrodes (Sharma et al.,
1999). The use of composite Al2 O3 -Ta2 O5 barriers is inspired by the Julliere for-
mula, TMR = 2PL PR /(1 – PL PR ). A full Ta2 O5 or Al2 O3 barrier will lead to
positive TMR due to the product of PL and PR , irrespective of the sign of the indi-
vidual tunneling spin polarization. Only when the polarization at the left and right
side of the barrier are of opposite sign, a negative TMR will be measured. The
observed negative TMR for Al2 O3 -Ta2 O5 is tentatively attributed to the difference
between s and d dominated electron tunneling across either a Al2 O3 or Ta2 O5 bar-
rier when keeping the ferromagnetic electrodes equal. Moreover, the polarization
strongly depends on the applied bias voltage in the composite Al2 O3 -Ta2 O5 junc-
tions, as well as in single-barrier Ta2 O5 junctions, hinting to the relevance of the
more pronounced d-like density-of-states as compared to the s-electrons (Sharma
et al., 1999). A point of concern regarding these data is raised by Montaigne et
al. (2001). By calculations within a free-electron model incorporating a composite
barrier they also observe a strong and asymmetric variation of TMR with applied
bias voltage; see also similar results by Li et al. (2004). Moreover, it should be noted
that the observed sign reversal in these experiments of Sharma et al. (1999) is only
indirectly inferred from TMR in full magnetic junctions. Unfortunately, a direct de-
termination of P (including its sign) when tunneling across Ta2 O5 is still lacking
due to quenching of the required Zeeman splitting by a large spin–orbit scattering
in Ta-oxide based superconducting junctions (Kant et al., 2004a).
68                                                                                H.J.M. Swagten




Figure 1.30 Calculated density-of-states (DOS) averaged over the first two layers of a Co(001)
surface as a function of energy. Both the total and s-electron partial density-of-states are
plotted for majority electrons (a) and minority electrons (b), showing the opposite sign of the
polarization of s- and d-states at the Fermi level. For clarity, the occupied s-based states are
shaded and multiplied with a factor of 10. After Tsymbal and Pettifor (1997).


     After these initial experiments from Sharma et al. (1999), a number of new
experiments have been launched using epitaxial oxides grown mostly with pulsed
laser deposition (e.g. SrTiO3 ), which will be reviewed in section 4. All together,
strong evidence is found for a critical role of the barrier and its interfaces with the
ferromagnetic layers. Putting it differently, by choosing appropriate barriers in spin-
tunneling experiments, one is able to probe wave functions of different symmetry
related to the ferromagnetic electrodes. This makes spin tunneling a unique tech-
nique for studying specific features of the complex band structure of ferromagnetic
thin films and interface regions.
     From a theoretical point of view, the role of the density-of-states and chemical
bonding at the ferromagnet-insulator interface region has been widely addressed.
The chemical bonding at the ferromagnet-insulator interface determines the effec-
tiveness of transmission at the interface, which, for electrons of different character,
may be markedly different. By combining a tight-binding approximation with bal-
listic quantum-mechanical transport calculations, Tsymbal and Pettifor (1997) have
found that the conductance in a magnetic tunnel junction strongly depends on the
type of covalent bonding between the ferromagnet and the insulator, as represented
by ss, sp and dd hopping integrals. Figure 1.30 shows the calculated density-of-
states of Co surface layers showing the dominance of minority d-like electrons at
the Fermi level, just like the situation for bulk Co (Fig. 1.6). However, for electrons
with s (and also p) character the situation is reversed due to s(p)-d hybridization;
now the majority electrons are outnumbering the minority electrons at the Fermi
level. This has important consequences for the conductance between a ferromag-
netic electrode and a non-magnetic metal represented by an s band, when tunneling
across an insulator having s-bands separating the energy gap; see Fig. 1.31. In case
of sd bonding at the interface between ferromagnet and insulator, there is a large
transmission of the d electrons across the MTJ. Due to the negative d-electron
density-of-states, this consequently leads to a negative polarization of the current.
Spin-Dependent Tunneling in Magnetic Junctions                                                     69




Figure 1.31 Calculated spin-polarized tunneling conductance between a ferromagnetic metal
(using the density-of-states as shown in Fig. 1.30) and a nonmagnetic s-like metal across s-like
tight-binding bands separated by an energy gap. The conductances are plotted for majority
electrons (a) and minority electrons (b) when only ss bonding between ferromagnet and insu-
lator is taken into account (labelled as “ss only”), as well as for the case of ss, sp, and sd bonding
(“full”). The polarization of the tunneling current due to the s states is positive (+34%) in
contrast to that of the full conductance (–11%). After Tsymbal and Pettifor (1997).


In contrast, when interfacial ss bonding is dominant, only s states of the ferromag-
netic film are coupled with those of the insulator. These s states have a strongly
reduced minority density-of-states at the Fermi level, which then leads to a pos-
itive tunneling spin polarization, in the case of Co with a magnitude of +34%.
This is perfectly in line with the experimental observations for alumina barriers
(see Table 1.2).
    To address the interface bonding in relation to the oxidic character of the in-
sulating barriers usually employed, a number of calculations have been reported
to address this in detail. However, an accurate prediction of spin polarization (and
TMR) for amorphous Al2 O3 barriers in MTJs is still lacking due to the enormous
theoretical complexities involved, and therefore, in all cases, the barrier is assumed
to be atomically ordered. Ab-initio calculations by de Boer et al. (1998) have shown
that the spin polarization at crystalline Co-HfO2 interfaces changes sign with re-
spect to the bulk of Co. Tsymbal et al. (2000) show that by covering a Fe(001)
surface with one oxygen overlayer, the spin polarization can be inverted with re-
spect to that of the clean Fe surface. Due to hybridization of the iron 3d levels with
the O 2p orbitals and the strong exchange splitting of the antibonding oxygen states,
a positive spin polarization in the density-of-states of the oxygen atoms is found at
the Fermi level, from there on propagating into the vacuum barrier. In the same
spirit, the tunneling spin polarization of ferromagnetic-Al2 O3 interfaces turns out
to be positive when imperfectly oxidized Al (or off-stoichiometric O ions) are as-
sumed to be present at the interface with the amorphous barrier; see Itoh and Inoue
(2001). Furthermore, this is only weakly dependent on the choice of ferromagnetic
electrodes (viz. bcc Fe, fcc Co, or fcc Ni).
    Oleynik et al. (2000) have studied the bonding at O- and Al-terminated in-
terfaces between fcc Co(111) and crystalline α-Al2 O3 with [0001] orientation to
better understand the atomic and electronic structure of an alumina-based system
70                                                                                   H.J.M. Swagten




Figure 1.32 (a) Local density-of-states (DOS) of [0001] oriented α-Al2 O3 at the Fermi energy
for majority (closed symbols) and minority electrons (open symbols), as a function of the dis-
tance from the interface with ferromagnetic Co. In (b) spin polarization of the density-of-states
is plotted using Eq. (10) for the data in (a), showing a sign reversal to positive spin polarization
at a distance of 10 Å. After Oleynik et al. (2000).


from first principles. As shown in Fig. 1.32a, the alumina density-of-states at the
Fermi level decays exponentially with distance from the Co interface, the average
decay length being larger for the majority electrons than for the minorities. Close
to the interface with Co the spin polarization on the Al and O atoms (using the
definition given by Eq. (10) for the local density-of-states) is negative reflecting the
negative spin polarization of the density-of-states of Co. Most interestingly, at the
interior atoms within the alumina barrier, more specifically beyond 10 Å, the spin
polarization becomes positive in line with the STS experiments as discussed earlier
(see Table 1.2). The crucial role of interface bonding is further corroborated by
calculating the effect of oxidizing a clean Co(111) surface on the tunneling spin
polarization (Belashchenko et al., 2004). For sufficiently thick barriers, the trans-
mission function can be factorized into a product of surface transmission functions
and a decay factor for the barrier, from which the tunneling current can be de-
termined in a system of Co-vacuum-Al. It is demonstrated that one monolayer of
oxygen bonded to the Co(111) surface changes the spin polarization from nega-
tive for a barrier of 20 Å to almost +100% due to the creation of an additional
tunneling barrier in the minority spin channel.
    The relation between oxygen adsorption and tunneling spin polarization is fur-
ther explored by a first-principles Green’s function technique applied to crystalline
(111)-oriented Co-Al2 O3 -Co junctions where O atoms are located at the interface
region (Belashchenko et al., 2005a; Tsymbal and Belashchenko, 2005). When the
three interface O atoms are bonded to two adjacent Al atoms the spin polarization
is found to be negative. In line with the earlier predictions (Belashchenko et al.,
2004), a very strong Co–O bonding of O positioned inside the large pores at the
fcc cobalt interfaces leads to a remarkable enhancement of the tunneling current
in the majority channel, thus reversing the tunneling spin polarization. In actual
experiments, the positive P found for tunneling across Al2 O3 may be fully ruled
by the details of interfacial adsorption of oxygen. This is supported by X-ray ab-
Spin-Dependent Tunneling in Magnetic Junctions                                       71


sorption spectroscopy and X-ray magnetic circular dichroism experiments (Telling
et al., 2004), showing that the polarization at the Co-Al2 O3 interfaces increases
when the bonding changes from Co–Al to Co–O. Also the STS experiments of
Munzenberg and Moodera (2004) on Al/Al2 O3 /Fe junctions as mentioned already
in section 3.3.2 could be qualitatively interpreted in the light of the interface bond-
ing model of Belashchenko et al. (2005a), since the positive spin polarization for
the case of strong Co–O bonding is shown to increase with the thickness of the
barrier.
    The above results all show that tunneling spin polarization is a very complex and
delicate parameter in MTJs given the variety of ingredients discussed in this section.
Nevertheless, the role of the electronic structure and chemistry of the interfacial re-
gions is probably most critical for the sign and magnitude of P , and simple rules of
thumb given by s- and d-dominated tunneling as raised earlier may not be entirely
justified for amorphous Al2 O3 -based junctions. In this respect, experiments using
crystalline barriers are much more promising for revealing the true mechanisms be-
hind spin polarization due to the advantage of a more realistic theoretical treatment.
In section 4, in particular the use of crystalline barriers such as SrTiO3 and MgO
will be introduced in relation to theoretical analyses. In the case of MgO barri-
ers, the tunnel magnetoresistance appears to be highly sensitive to the symmetry of
the propagating states in the electrodes in relation to the way they couple to the
evanescent states in the barrier layer. This is in fact another important ingredient
for tunneling spin polarization for which strong experimental evidence is currently
available, e.g. in epitaxial Fe-MgO-Fe MTJs (section 4).



      4. Crucial Experiments on Spin-Dependent Tunneling

       This section deals with a number of key experiments in the area of magnetic
tunnel junctions. Although the field of magnetic tunneling is not settled and is
still further developing, contributions are selected that are believed to be critically
important for better understanding the physics behind magnetoresistance effects in
MTJs. Through the direct relation between tunneling spin polarization and TMR,
experiments are to a great extent focused on similar aspects as introduced in the
previous section 3. In this section the following themes will be discriminated:
• the application of thin nonmagnetic layers in MTJs to address the relevance of
  the ferromagnet-barrier interface region
• quantum-well observations in TMR by incorporating ultrathin layers for spin-
  dependent confinement of tunneling electrons
• role of the ferromagnetic electrodes, including the use of half-metallic ferromag-
  nets to achieve extremely large TMR
• role of the tunneling barrier using alternative insulators, including coherent tun-
  neling across crystalline MgO barriers.
72                                                                        H.J.M. Swagten



4.1 The relevance of interfaces: using nonmagnetic dusting layers
In the previous sections, we have emphasized that the magnetoresistance effect in
magnetic tunnel junctions can be phenomenologically explained by a simple Jul-
liere formula, viz. TMR = 2PL PR /(1 – PL PR ), illustrating the leading role of the
tunneling spin polarization of the (left and right) electrodes. However, it was ar-
gued before that this tunneling spin polarization should be considered with great
care (section 3). It is not just related to the electronic properties of the electrodes
alone, but sensitively depends on the combined system of (magnetic) electrode and
barrier material.
     In an elegant type of experiment addressing the delicate properties of spin po-
larization, thin nonmagnetic so-called dusting layers are inserted at the interface
between the magnetic electrode and the insulating barrier. When the Julliere for-
mula would be used in a naive manner, the zero polarization of the nonmagnetic
interlayer (see Eq. (20) with density-of-states factors at the interfaces) would im-
mediately lead to a vanishing TMR. Considering the subtle role of the interfaces
for TMR, however, it is conceivable that this may result in a rich spectrum of
interesting physics by engineering structures with nonmagnetic elements incorpo-
rated in an MTJ. In one of the earliest experiments in this field, Moodera et al.
(1989) have directly measured the spin polarization in Al-Al2 O3 -Au-Fe using su-
perconducting tunneling spectroscopy (see section 3, Fig. 1.28), finding that the
polarization rapidly decreases with the thickness of the Au layer. Nevertheless, at
larger Au thickness the polarization still persists and decreases roughly as 1/tAu . In
contrast to this observation, later experiments showed an oscillation of the TMR in
Co-Au-Al2 O3 -NiFe with increasing Au thickness, suggesting that the spin polar-
ization could change sign by inserting the nonmagnetic interlayer (Moodera et al.,
1999b).
     To unravel these and other inconsistencies, a number of experiments with dust-
ing layers were performed using standard exchange-biased Co-Al2 O3 -Co junctions.
Generally, a decaying TMR has been found for all nonmagnetic materials employed
so far (Moodera et al., 2000). However, the location of the interlayer, either grown
on top of the bottom electrode or grown on top of the Al2 O3 barrier, is crucial in
the suppression of TMR, which is shown in Fig. 1.33 for data reported by LeClair et
al. (2000d) using Cu layers up to a thickness of about 10 Å. In the case of Cu on top
of the barrier, the decay length ξ (when assuming a phenomenological exponential
decay function TMR ∝ exp[–tCu /ξ ]) is roughly 7.0 Å, and is much larger than ξ
for the Cu layers on top of the bottom electrode. This is attributed to the non-ideal,
cluster-like growth of metal layers on top of the amorphous Al2 O3 barrier, as veri-
fied by X-ray photoelectron spectroscopy and Auger electron spectroscopy (LeClair
et al., 2000d), and by analyzing the voltage dependence of the conductance (LeClair
et al., 2000b). This non-ideal growth also explains the rather long decay lengths re-
ported by Sun and Freitas (1999), Parkin (1998), and Yamanaka et al. (1999) when
nonmagnetic layers are grown on top of the insulating barrier. In the case of the Cu
on top the bottom electrode, a nearly layer-by-layer growth has been established
(LeClair et al., 2000d), by which the observed length scale ξ = 2.6 Å is now more
intrinsically related to the decaying tunneling spin polarization.
Spin-Dependent Tunneling in Magnetic Junctions                                            73




Figure 1.33 The effect of nonmagnetic dusting layers at the interface between the ferromag-
netic electrode and the barrier. TMR, normalized to the magnetoresistance in Co/Al2 O3 /Co,
is shown as a function of the thickness of the nonmagnetic layer tNM . The data la-
belled with “Al2 O3 /Cu” refer to Cu layers grown on top of alumina as indicated at
right. The other interlayers, viz. Cu, Cr, and Ru, are grown on top of Co (under-
neath the Al2 O3 ), see again the panel at right. The full junction stack is composed of
Si(100)/SiO2 /50 Å Ta/70–80 Å Co/100 Å FeMn/35–50 Å Co/Al2 O3 /150 Å Co, capped
with Ta or Al, with dusting layers at one of the interfaces with alumina. From LeClair et al.
(2000d).


    In the case of dusting with Cr, an even faster intrinsic decay length has been
reported by LeClair et al. (2001b). At room temperature it is measured that
ξ = 1.25 Å (see Fig. 1.33) which means that the addition of ≈ 1.5 monolayer
of Cr reduces TMR to only 10% of a control junction without the spacer layer.
As an additional proof of the interface-sensitivity of spin-dependent tunneling, a
thin Co layer was subsequently deposited on top of the Cr dusting layer, by which
the TMR is recovered almost completely. In the case of Cr dusting, the authors
argue that the Co-Cr interfaces induce a strong spin-dependent modification of the
interfacial density-of-states which enhances scattering of the majority electrons and
thereby strongly reduces TMR (LeClair et al., 2001b). This is further substanti-
ated by an anomalous suppression of the low-temperature conductance at small bias
voltage, a so-called zero-bias anomaly, again related to the density-of-states modifi-
cations at the Cr-Co interface. Additionally, these zero-bias anomalies are used in
multiple dusting experiments (employing dusting with Cr as well as Cu) to iden-
tify that Cr in contact with Co is the driving source for the additional scattering.
A similarly fast decay of TMR has been reported for dusting with Ru, see again
Fig. 1.33. However, in this case the polarization changes sign when tRu 2 Å, and,
after reaching a minimum around tRu = 3 Å, it gradually decays to zero (LeClair
et al., 2001a). It is hypothesized that this is due to the presence of an interfacial
Co-Ru alloy as evidenced by nuclear-magnetic-resonance experiments on Co-Ru
multilayers by Wieldraaijer (2006). Above a critical alloy composition the s electrons
are then believed to have a negative tunneling spin polarization (Itoh et al., 1993;
74                                                                         H.J.M. Swagten



Stepanyuk et al., 1994; Rahmouni et al., 1999), thus reversing the sign of TMR. In
the following (section 4.2), Ru interfacial layers will be employed in epitaxial junc-
tions, leading to an oscillatory TMR upon variation of the dusting layer thickness.
    As to the explanation of the extremely fast decay of spin polarization by dusting
with non-ferromagnetic elements, Zhang and Levy (1998) find theoretically that
for uniform nonmagnetic layers a rather long (30–100 Å) length scale is expected
from coherent transmission, but that for nonmagnetic layers with thickness fluctua-
tions, only a few monolayers are required to completely quench the TMR. This is
in striking contrast to other calculations (Vedyaev et al., 1997; Zhang et al., 1998;
Mathon and Umerski, 1999; Itoh et al., 2003) showing oscillatory behavior of
TMR when tunneling is fully coherent with a strict conservation of k upon tunnel-
ing through the barrier. From the absence of experimental evidence for oscillatory
features in the aforementioned dusting experiments (except for Ru dusting with
a negative TMR attributed to a Co-Ru alloy) it is assumed that the assumption
of k conservation is not very realistic for these structures, even when the bottom
electrode is well-grown in a nearly layer-by-layer fashion. The presence of a small
amount of roughness, combined with the use of (poly)crystalline electrodes and
amorphous Al2 O3 , prevents k conservation and quenches TMR. Nevertheless, in
the case of dusting with Au, Moodera et al. (1999b) have found first indications
for the presence of an oscillation in TMR that are explained in terms of a simple
free-electron tunneling model. The presence of only one sign reversal of TMR and
the strong resemblance with the Ru data of LeClair et al. (2001a) could, however,
point to alternative explanations as well. Shim et al. (2003) have reported an unex-
pected field-dependence of the magnetoresistance when using a bottom Au dusting
layer in an exchange-biased Co-Al2 O3 -NiFe system, although no indications for
quantum-well formation are detected. In section 4.2 the phenomenon of quantum
effects in MTJs will be further discussed.
    We now return to dusting experiments using Cr as a dusting layer. Although
Cr has no macroscopic magnetization, it is known to exhibit a layered antiferro-
magnetic structure when grown on Fe(001), i.e. the spins in each monolayer are
opposite to those in the neighboring layers. When spin-polarized tunneling would
be intrinsically related to the interface between electrode and barrier (as suggested
by the experiments described earlier in this section), this should result in an os-
cillating magnetoresistance with the thickness of a well-defined Cr dusting layer.
Nagahama et al. (2005) have prepared Fe(001)-(t)Cr(001)-Al2 O3 -CoFe by molec-
ular beam epitaxy, with a variable thickness t of the (001)-oriented Cr layer. Indeed,
after an initial suppression from +15% to less than +1% in agreement with LeClair
et al. (2001b), TMR is rapidly changing sign from 3 to up to more than 30 mono-
layers of Cr, with an oscillation period of 2 monolayers and an amplitude that slowly
decays with tCr . In Fig. 1.34a this is shown for T = 50 K. At room temperature
the effect is still present, only the oscillation amplitude is somewhat smaller; see the
inset of the figure. The extreme interface sensitivity can be explained by scatter-
ing of s-type tunneling electrons of 1 symmetry at the interfacial Cr layer (see
the schematics in Fig. 1.34b) due to the absence of a 1 band at the Fermi level
of Cr. In passing, we note that 1 bands are extremely important for coherent tun-
neling across epitaxial MgO barriers as will be discussed in section 4.6. Although
Spin-Dependent Tunneling in Magnetic Junctions                                              75




Figure 1.34 (a) TMR ratio at T = 50 K and at T = 300 K (inset) as a function of the
thickness t of the Cr(001) interlayer in units of Cr monolayers (ML). The junctions consist of
MgO(001)/400 Å Cr(001)/1000 Å Au/300 Å Fe(001)/t Cr(001)/17 Å Al2 O3 /200 Å CoFe.
(b) Schematic illustration of the junction structure showing the antiparallel arrangement of
neighboring Cr layers, and the scattering of electrons at the Cr/Al2 O3 interface. Adapted from
Nagahama et al. (2005).


in principle also quantum-well states could be formed in the Cr(001) layer, the
2 monolayer oscillation is found not to depend on bias voltage which excludes this
possibility. The slight shift of the oscillation phase with bias is again attributed to
band-structure effects in the Cr(001) interlayer (Nagahama et al., 2005).

4.2 Quantum-well oscillations in MTJs
The absence of convincing evidence for electron-confinement effects in an MTJ
could be related to the loss of quantum coherence of tunneling electrons when
polycrystalline thin films are embedded instead of well-defined single-crystalline
entities. Using epitaxial junctions, a crucial experiment addressing the quantum
confinement of electrons in dusting layers has been reported by Yuasa et al.
(2002). They have produced Co(100)-Cu(100)-Al2 O3 -NiFe junctions where the
Co electrode and Cu dusting layer are essentially epitaxial, the alumina barrier
is amorphous, and the top electrode polycrystalline. In this case also the TMR
is quenched considerably already for 1 or 2 monolayers of Cu, as shown in
Fig. 1.35a. However, for thicker Cu spacers, the TMR is clearly changing sign
several times, up to thicknesses of more than 20 Å. Although single sign reversals
of TMR have been observed for Ru and Au dusting layers (LeClair et al., 2001a;
Moodera et al., 1999b), this oscillatory behavior with tCu is a first convincing ev-
idence for resonant tunneling of spin-polarized electrons in the Cu quantum well
as schematically shown in Fig. 1.35b. In Fig. 1.36b it is further illustrated that only
for minority electrons a Cu quantum well exists due to the difference in potential
energy between Cu and minority Co. On the other hand, majority electrons in
Co and Cu have a very similar electronic structure and are therefore not subject
76                                                                               H.J.M. Swagten




Figure 1.35 (a) TMR at T = 2 K and T = 300 K at low bias voltage (10 mV) as a function
of the Cu thickness tCu in a junction consisting of MgO(001)/buffer/200 Å Co(001)/0–32 Å
Cu(001)/12 Å Al2 O3 /100 Å Ni80 Fe20 /Au-cap. The inset shows the saturation magnetic field
HSAT for a 50 Å Co(001)/0–45 Å Cu(001)/50 Å Co(001) structure as a function of tCu as
obtained from room-temperature magneto-optical Kerr-effect measurements. (b) Schematics of
quantum-well reflections for minority electrons in the Cu layer, only when propagating along
k = 0 as indicated in the Fermi surface of fcc Cu in (c). Due to the confinement in z direction,
quantum-well states with scattering vectors q1 and q2 can be formed in the [001] direction.
After Yuasa et al. (2002).




Figure 1.36 (a) Period of oscillation from TMR data as a function of bias voltage V (open
symbols) together with the theoretical curve obtained from the energy dispersion of the 1
band of Cu along the -X axis (Segall, 1962). The solid circle is the period estimated from data
on interlayer coupling (see Fig. 1.35a). (b) The sign convention for the bias voltage, showing
the possibility to trace the quantum-well energies only at positive bias. See Fig. 1.35 for the
actual junction structure. After Yuasa et al. (2002).
Spin-Dependent Tunneling in Magnetic Junctions                                         77




Figure 1.37 (a) Room-temperature TMR as a function of the Ru thickness tRu of
UHV-sputtered 12 Å Co90 Fe10 /t Ru/11 Å Al + oxidation/30 Å Co90 Fe10 grown
on MgO(110). The applied bias voltage is 15 mV In (b) the saturation field HSAT of a
                                                   .
trilayer 150 Å Co90 Fe10 /t Ru/50 Å Co90 Fe10 is shown as measured with a vibrating sample
magnetometer at room temperature. After Nozaki et al. (2004).


to considerable confinement. The TMR oscillation period of about 11 Å almost
perfectly agrees with one of the extremal k-vectors in the [001] direction corre-
sponding to electrons tunneling with k = 0 (see Fig. 1.35c), showing that these
junctions are close to an ideal magnetic junction where electrons can be injected
into the Cu quantum well only when k = 0. Moreover, the same period of os-
cillation is found by the authors when measuring the interlayer coupling fields in
Co(001)-Cu(001)-Co(001) structures, as shown in the inset of Fig. 1.35a. Indeed,
in explaining magnetic interlayer coupling an identical interpretation of resonant
spin-dependent reflection and transmission of electron waves is used to describe os-
cillatory exchange fields (see Bruno, 1995 and Bürgler et al., 1999). In contrast to
the interlayer studies, tunneling offers the unique opportunity to scan the energy
dependence of the resonantly tunneling electrons by applying a variable bias voltage
across the barrier as we discussed already in section 2.1.4. In Fig. 1.36a it shown by
Yuasa et al. (2002) that the oscillation period is considerably enlarged for positive
bias, corresponding to electrons tunneling into the Cu quantum well formed by the
minority electrons. Only in that case a variable bias will probe energy dispersion
along the [001] direction thereby changing the period of oscillation as indicated in
the calculation of Fig. 1.36a.
    In the same spirit, Ru interlayers have been grown on epitaxial CoFe bottom
electrodes by Nozaki et al. (2004). The partially occupied d band at the Fermi
level of Ru is almost equal to the hcp Co minority band, by which only majority
electrons are confined to the Ru quantum well (contrary to the aforementioned
Cu quantum well). In structures of Co90 Fe10 -(t)Ru-Al2 O3 -Co90 Fe10 the bottom
CoFe layer has a hcp(1010) orientation due to growth on MgO(110). The TMR
of the dusted MTJs is displayed in Fig. 1.37a. Although a similar negative TMR has
been observed also by LeClair et al. (2001a) as discussed in section 4.1, at higher
Ru thickness the magnetoresistance again changes sign, and an oscillation seems to
78                                                                         H.J.M. Swagten



persist up to tRu ≈ 20 Å. Moreover, these oscillations are nicely correlated with
the oscillatory interlayer coupling across Ru, as measured separately by the satu-
ration field of antiferromagnetically coupled trilayers of Co90 Fe10 -(t)Ru-Co90 Fe10 ,
see Fig. 1.37b. The TMR ratio of the Ru system changes considerably upon the
application of bias voltages. For thick enough Ru interlayers, roughly beyond 10 Å,
the asymmetry in the bias dependence becomes very strong, leading to sign changes
of TMR for both positive and negative bias voltage. This is in contrast to the obser-
vations by Yuasa et al. (2002) with Cu dusting, showing TMR modulations only for
the bias direction corresponding to an electron flow from the ferromagnetic elec-
trode into the quantum well. Nozaki et al. (2004) have qualitatively explained this
by the contribution of a series of discrete energy levels for the majority electrons
confined in the Ru spacer, and a continuous energy spectrum for the minorities.

4.3 Role of the ferromagnetic electrode for TMR
Although a number of experiments have clearly pointed out the relevance of
interfaces for spin-polarized tunneling, they do not necessarily rule out a (spin-
dependent tunneling) contribution from ferromagnetic material located just behind
the interfacial region. In Fig. 1.28 it is observed that a certain thickness of the
ferromagnetic layer is required to saturate the tunneling spin polarization in a
Al-Al2 O3 -(t)Co-Al superconducting junction. Partially this can be explained by
a development of the full ferromagnetic moment which is obviously linked to the
spin polarization. To avoid these ambiguities, Zhu et al. (2002) have grown a wedge-
shaped Co50 Fe50 layer inserted between the alumina barrier and a bottom Ni81 Fe19
reference layer to eliminate finite-size effects in the magnetization. As shown in
Fig. 1.38a, TMR is developing rather slowly with a characteristic length scale of
≈ 8 Å (from a fit to the data), demonstrating that apart from interface contribu-
tions also a few deeper layers may be relevant. Magnetization measurements (see
Fig. 1.38b) show that with increasing CoFe thickness the slope of the moment per
area is similar when a CoFe wedge is grown on top of an underlying CoFe layer or
on top of a NiFe layer. This confirms that the magnetic moment of the thin CoFe
layer is not seriously quenched by strong interface intermixing.
     In sections 3 and 4.1 it is illustrated that tunneling spin polarization in a
ferromagnetic-insulator-ferromagnetic junction is not just a static density-of-states
parameter of the ferromagnetic electrode, but depends on the interaction of the
mobile electrons with the barrier wave functions and in particular the electronic
modifications at the ferromagnetic-barrier interface region. However, in terms of
the definition of tunneling spin polarization in Eq. (20), the weighting factors
wmin(maj) explicitly depend on the Fermi velocities vF , in limiting cases even in a
quadratic way (section 3.3.2). It is therefore expected that modifications in the
(bulk) ferromagnetic electrodes would affect the spin-dependent tunneling proper-
ties, assuming that the interface region should, although certainly altered, resemble
the bulk electronic properties. Moreover, from the experiments of Zhu et al. (2002)
it is suggested that the interface region is extending into the bulk, at least for a few
monolayers away from the interfaces.
Spin-Dependent Tunneling in Magnetic Junctions                                          79




Figure 1.38 (a) CoFe thickness dependence of room-temperature TMR in Si/200 Å
Ni81 Fe19 /60 Å Cu/120 Å FeMn/80 Å Ni81 Fe19 /t Co50 Fe50 (wedge)/Al2 O3 /130 Å
Ni81 Fe19 /500 Å Cu (open circles). The solid circles are reference data on 80 Å
Ni81 Fe19 /Al2 O3 /130 Å Ni81 Fe19 without the CoFe, scanned along the same direction to
check the uniformity of deposition. (b) Magnetic moment per area (μ/A) measured with a
superconducting-quantum-interference-device (SQUID) magnetometer of a Co50 Fe50 wedge
grown on top of a layer of 150 Å Co50 Fe50 (open) or 200 Å Ni81 Fe19 (closed). After Zhu et
al. (2002).


    A first observation of the dependence of the interface region of ferromagnetic
electrodes relates again to quantum-well formation and is reported by Nagahama
et al. (2001). When the coherence of the electron wave function is conserved,
quantum-well states can be formed in thin epitaxial ferromagnetic layers located
at the interface with the alumina barrier, thereby affecting TMR. By employing
epitaxial junctions similar to those shown in Fig. 1.35, no oscillations in TMR
have been observed when changing the thickness of the interface layer, probably
due to a too small amplitude. However, by inspection of the conductance versus
bias voltage, a clear oscillatory behavior is reported for epitaxial Fe(001) layers of
2 to 9 monolayers in junctions of Cr(001)-Fe(001)-Al2 O3 -Fe50 Co50 . In Fig. 1.39
a selection of these data is presented, where one should focus on voltages beyond
≈ 0.2 V, outside the regime of magnon and phonon-assisted tunneling. Note that
these effects are only observed for one bias direction, since the CoFe layers are
polycrystalline and thick enough to exclude quantum-well states in the CoFe. In
this regime of V      0.2 V, the maxima in conductance are shifting towards lower
bias for thicker Fe layers, as expected for a quantum-size effect. The phase of the
oscillations is found to be identical for the anti-parallel and parallel configuration,
suggesting that only one of the spin bands is active in the quantum-well formation.
Indeed, as shown in the insets of Figs. 1.39a and 1.39b, it is known that the Fe
minority band is very similar to the band structure of Cr, leading to quantum-
well states only in the Fe majority band. Unlike the resistance change R /RP ,
the conductance change defined as G/GAP does show quantum-well oscillations
upon a variation of the applied voltage; see Fig. 1.39b. The arrows in this graph are
the bias voltages with maximum conductance. Despite these convincing qualitative
observations, no calculations are yet available to explain the observed oscillations,
80                                                                             H.J.M. Swagten




Figure 1.39 (a) Conductance dI /dV versus bias voltage V at T = 2 K after subtraction of
a background conductance in junctions consisting of MgO(001)/buffer/200 Å Cr(001)/5, 7,
9 monolayers Fe(001)/17Å Al2 O3 /200 Å Fe50 Co50 /Au-cap. The inset shows the possibility of
quantum-well formation for positive bias. In (b) the conductance change (GP – GAP )/GAP is
shown versus voltage. The arrows indicate the position of the maxima in dI /dV at low bias as
indicated in (a). The inset shows quantum-well reflections for majority electrons along k = 0.
After Nagahama et al. (2001).


and probably require the application of advanced transport theory (Nagahama et al.,
2001).
    Now we will review the (few) existing examples of the influence of the crystal-
lographic orientation of the ferromagnetic electrodes on TMR. In a tunnel junction,
electrons with a momentum vector perpendicular to the barrier plane are strongly
selected by the tunneling process. Due to the anisotropy of the Fermi surface of
ferromagnetic electrodes, this momentum filtering should cause the TMR to de-
pend on the orientation of the ferromagnetic electrodes. This is also reflected by
the tunneling spin polarization (Eq. (20)), where the weighting of the Fermi veloc-
ity is expected to be strongly anisotropic in epitaxial junctions. Yuasa et al. (2000)
have prepared Fe-Al2 O3 -CoFe MTJs with molecular beam epitaxy in which (only)
the bottom electrode is epitaxial with three different orientations: Fe(100), Fe(110),
and Fe(211). A wide variation in thickness of the amorphous Al2 O3 is created by
evaporation of Al in an O2 atmosphere (section 2.2), combined with the use of a
moving shutter during deposition. As shown in Fig. 1.40 there is a distinct differ-
ence between the orientations as well as a significant dependence on the thickness
of the Al2 O3 spacer. As an attempt to explain this anisotropy, the authors have cal-
culated the polarization of the bulk density-of-states in the direction normal to the
interfaces, using the so-called layer Korringa–Kohn–Rostoker approach; see also
MacLaren et al. (1997, 1999). This yields 4% for Fe(100), 31% for Fe(110), and
34% for Fe(211). This qualitative agreement with the trend observed in Fig. 1.40
may be somewhat fortuitous since the interfacial modification of the density-of-
states may dominate the polarization anisotropy. Also the tunneling current across
amorphous Al2 O3 may be dominated by s-like states (see section 3) which is not
taken into account in the calculation. Furthermore, the variation of TMR with
barrier thickness is unexplained, and might be intrinsically related to the complex
Spin-Dependent Tunneling in Magnetic Junctions                                              81




Figure 1.40 (a) TMR as a function of the thickness of the Al2 O3 barrier in Fe/t
Al2 O3 /200 Å Fe50 Co50 junctions. The Fe layer (either 100 Å or 200 Å) is single crystalline
with (100), (110), and (211) orientation and has been obtained by epitaxial growth on proper
crystals and seed layers. Data were taken at T = 2 K using a bias voltage of 20 mV. Note
that the alumina layers are optimized for all thicknesses (with minimal oxidation at the inter-
faces) due to reactive deposition of alumina in ultra-high vacuum. The curves are guides to the
eye. (b) Cross-sectional transmission-electron-microscopy image of a junction with a nominal
barrier thickness of 20 Å. After Yuasa et al. (2000).


interplay between electron wave functions of different character decaying differently
in different orientations. As a final remark, MacLaren et al. (1999) have theoretically
shown that due to the symmetry of the Bloch states at the Fermi level the TMR is
expected to be highest for Fe(100) and should increase with the barrier thickness.
Both these predictions are in contrast with the experiments of Yuasa et al. (2000).
As to the barrier-thickness dependence of TMR, it is suggested by Mizuguchi et al.
(2005) that also thinner barriers (below 10 Å) are experimentally accessible in these
epitaxial junctions. In-situ scanning tunneling microscopy of alumina on top of
epitaxial Fe(100) has revealed that the naturally oxidized Al layer is surprisingly flat
showing mono-atomic steps with a 3 Å step height corresponding to one monolayer
of Al2 O3 .
    Another clear evidence for tunneling spin polarization reflecting the density-of-
states of the ferromagnetic electrode has been reported in Co-Al2 O3 -Co junctions
in which the buffer layer grown underneath induces a particular growth mode of the
Co electrode (LeClair et al., 2002b). When grown on top of a single Ta buffer, their
Co grows in a random, polycrystalline fashion with a mixture of fcc, hcp and stack-
ing faults. However, when a Ta-Co-FeMn buffer is used, the Co located at the in-
terface region with Al2 O3 is (111) textured and predominantly fcc-structured. The
difference in tunneling transport between these two cases is best visible in the (nor-
malized) voltage-dependent conductance change G/GAP (V ). Note that as in the
case of quantum confinement in Fe, see Fig. 1.39, the TMR itself is not conclusive
for finding the spin-dependent transport features. To eliminate effects of magnons
and spin-independent excitations from e.g. phonons which are symmetric in bias
voltage, the odd part of the conductance change G/GAP (V > 0) – G/GAP (V <
0) is analyzed and shown in Fig. 1.41b. The original data are presented in the left
panel of the figure. The strong minimum seen in the fcc data can be qualitatively
explained by a modified elastic tunneling model using free-electron like bands de-
82                                                                                 H.J.M. Swagten




Figure 1.41 (a) Conductance dI /dV versus voltage V in parallel (P) orientation for
Si(100)/SiO2 /buffer + Co/23 Å Al + oxidation/150 Å Co/50 Å Ta. The open symbols refer to
a buffer and magnetic bottom electrode composed of 50 Å Ta/50 Å Co/100 Å FeMn/50 Å
fcc(111) Co. Closed symbols refer to 50 Å Ta/poly 50 Å Co where poly relates to polycrys-
talline and polyphase Co as determined by nuclear magnetic resonance (Wieldraaijer, 2006).
(b) Odd part of G/GAP versus voltage for fcc Co and poly Co. The solid line is based on
a calculation with the modified elastic tunneling model (Davis and MacLaren, 2002). Data are
all taken at T = 5 K. In all cases V > 0 refers to electrons tunneling from the top to the bottom
electrode. After LeClair et al. (2002b).


rived from ab-initio electronic-structure calculations. In this model, the conductance
is dominated by the contribution of the highly dispersive, s-hybridized density-of-
states of fcc Co to reflect the fact that in these Al2 O3 junctions electrons with s
character are decisive for spin-dependent tunneling (corresponding to positive tun-
neling spin polarization; see section 3). In particular, the presence of two sharp peaks
in the s-derived density-of-states above and below EF , as well as a dispersive mi-
nority band just above EF , are key to the observed behavior (LeClair et al., 2002b;
Davis and MacLaren, 2002).
    Inspired by these results, Hindmarch et al. (2005b) have measured the odd part
of the conductance and TMR in junctions with a Cu38 Ni62 magnetic electrode
having a Curie temperature of around 240 K. Due to the low TC of this alloy, the
energy of the bottom of the minority spin bands close to the Fermi energy can be
followed for temperatures up to the magnetic phase transition. From the odd part
of G/GAP versus bias voltage, it is observed that slightly above the Fermi level the
band minimum remains fixed in energy until the temperature is raised to around
T = 190 K. Beyond this point it abruptly drops below the Fermi level, which is
consistent with a Stoner-like collapse of the effective exchange splitting of energy
bands responsible for tunneling.

4.4 Towards infinite TMR with half-metallic electrodes
The implementation of electrodes with a (nearly) 100% tunneling spin polarization,
the so-called half-metallic materials, is expected to yield infinite TMR as indicated by
the Julliere formula 2PL PR /(1–PL PR ) with PL and PR equal to ±1. Experimentally
Spin-Dependent Tunneling in Magnetic Junctions                                      83


as well as theoretically, an ongoing intensive research effort is devoted to these ma-
terials and their implementation; see, e.g., Pickett and Moodera (2001). Although
many predictions of half-metallic behavior have been reported this is verified ex-
perimentally only in a few cases, including La0.7 Sr0.3 MnO3 (Park et al., 1998a;
Soulen et al., 1998), NiMnSb (Ristoiu et al., 2000), and CrO2 (Ji et al., 2001).
In the latter case of CrO2 , Parker et al. (2002) verified the near +100% tun-
neling spin polarization directly in a superconducting-tunneling-spectroscopy ex-
periment on CrO2 -Cr2 O3 -Al and CrO2 -Cr2 O3 -Pb junctions (see Table 1.2). For
La2/3 Sr1/3 MnO3 a polarization of +72% is measured using the same technique
(Worledge and Geballe, 2000b).
    The use of these materials in ferromagnetic-insulator-ferromagnetic junc-
tions is obviously extremely tedious due to the crucial control of two barrier
interfaces. Indeed, for junctions employing one or two half-metallic Heusler-
alloy electrodes such as NiMnSb (Tanaka et al. 1997, 1999) and related MnSb
(Panchula et al., 2003), Co2 MnSi (Kammerer et al., 2004; Schmallhorst et al., 2004;
Nakajima et al., 2005), Co2 MnAl (Kubota et al., 2004), Co2 Cr0.6 Fe0.4 Al (Inomata
et al., 2004), and Co2 FeAl (Okamura et al., 2005), the TMR remains relatively
low and may result from oxidation at the Heusler-barrier interfaces or from site-
disordering and structural defects close to the barrier. More promising, Sakuraba
et al. (2005a, 2005b) observe magnetoresistances of up to 70% at room tempera-
ture and 159% at T = 2 K in UHV-sputtered Co2 MnSi-Al2 O3 -Co75 Fe25 , which
corresponds to a low-temperature tunneling spin polarization of +89%, closely ap-
proaching the theoretical prediction of half-metallicity. CrO2 -based junctions are
not successful in terms of high TMR. As an example, Gupta et al. (2001) have
grown CrO2 -Cr2 O3 -Co(Ni81 Fe19 ) tunnel junctions in which CrO2 is epitaxially
grown on top of TiO2 , and Cr2 O3 (or a composition close to this) is stabilized
by exposing the bottom electrode to an oxygen plasma. In this case, a TMR of
only –8% and –2.3% has been achieved at T = 4 K for Co and permalloy, respec-
tively.
    Magnetite (Fe3 O4 ) is also predicted to be half-metallic due to a gap for the
majority band at the Fermi level. Junctions consisting of Fe3 O4 -MgO-Fe3 O4 show,
however, only a very small TMR for all temperatures (Li et al., 1998), maybe related
to a combination of spin scattering in a magnetically dead interface layer, a distorted
spin structure due to a specific interface termination, or due to a reduced oxide such
as antiferromagnetic Fe1–δ O present at the interface with the MgO barrier. Also in
junctions consisting basically of NiFe-Al2 O3 -Fe3 O4 (with the magnetite fabricated
by plasma oxidizing a thin Fe film) only a very small TMR has been reported (Park
et al., 2005). The authors suggest that the observed negative sign of TMR is con-
sistent with the expected gap for majority Fermi electrons in Fe3 O4 . On the other
hand, Seneor et al. (1999) have reported a positive TMR of +43% at low tempera-
ture and +13% at room temperature in sputtered Co-Al2 O3 -Fe3–δ O4 -Al junctions
where the iron oxide is sputtered from a Fe2 O3 target. This relatively large TMR is
ascribed to the presence of a phase close to magnetite, although the data suggest that
the TMR originates predominantly from conduction channels active only above
and below the Fermi level. In epitaxial La0.7 Sr0.3 MnO3 -CoCr2 O4 -Fe3 O4 junctions
a negative TMR of up to –25% is in qualitative agreement with the theoretically
84                                                                        H.J.M. Swagten



predicted negative spin polarization of Fe3 O4 (Hu and Suzuki, 2002). The observed
maximum TMR at T ≈ 60 K is attributed to the paramagnetic to ferrimagnetic
transition in the CoCr2 O4 barrier. Zhang et al. (2001a) have introduced an Fe-
oxide layer at the barrier interface of CoFe-Al2 O3 -CoFe junctions to improve the
thermal stability when annealing up to temperatures of around 400°C (see sec-
tion 2.3.4). The large TMR measured after annealing is attributed to the formation
of Fe3 O4 in the interfacial region, which is confirmed by a follow-up study using
transmission electron microscopy combined with electron-energy-loss spectroscopy
(Snoeck et al., 2004). In Co-Al2 O3 -NiFe junctions, it is shown that TMR is en-
hanced by roughly a factor of 1.25 due to δ doping the oxide barrier with an Fe
layer with a thickness of less than 2 Å (Jansen and Moodera, 1999). Apart from
other explanations, also in this case the possibility of half-metallic Fe3 O4 formation
is hypothesized by the authors.
    Especially in the perovskite materials, a lot of progress has been witnessed as
described in the review paper by Ziese (2002). Pioneering experiments are done
by Sun et al. (1996, 1997, 1998) on junctions with La2/3 Ca1/3 MnO3 (LCMO) and
La2/3 Sr1/3 MnO3 (LSMO) electrodes and SrTiO3 barriers, later also combined with
ferromagnetic 3d transition metals (Sun et al., 2000b). Jo et al. (2000a, 2000b)
use La2/3 Ca1/3 MnO3 as electrodes in LCMO-NdGaO3 -LCMO junctions reach-
ing TMR magnitudes of more than 500% at low T . La2/3 Sr1/3 MnO3 is used by
Lu et al. (1996) and Viret et al. (1997) in combination with oxide barriers such
as SrTiO3 , yielding low-temperature magnetoresistances of more than 400%. This
reasonably well corresponds to the measured La2/3 Sr1/3 MnO3 tunneling spin polar-
ization of approximately +72% (Worledge and Geballe, 2000b) as discussed before
(see also section 3). In the latter superconducting-tunneling-spectroscopy exper-
iment, the authors use La2/3 Sr1/3 MnO3 -SrTiO3 -Al junctions with a thick layer
of YBa2 Cu3 O7 grown as a buffer layer on the SrTiO3 substrate to prevent cur-
rent crowding in the bottom electrode (see section 2.1.3). Junctions consisting
of La0.7 Ce0.3 MnO3 -SrTiO3 -La0.7 Ca0.3 MnO3 exhibit a large positive TMR at low
temperatures, whereas at intermediate temperatures below TC the sign of the ob-
served TMR is dependent on the bias voltage, suggesting a high degree of tunneling
spin polarization dominated by minority spins (Mitra et al., 2003). In the double-
perovskite Sr2 FeMoO6 the predicted half-metallicity has triggered spin-tunneling
experiments in Sr2 FeMoO6 -SrTiO3 -Co junctions (Bibes et al., 2003a). The authors
have reported a TMR of 50% at low temperature that corresponds to a tunneling
spin polarization of more than 85% at the Sr2 FeMoO6 -SrTiO3 interface (see also
section 4.5). To a great extent, this confirms the half-metallic character of this
double-perovskite compound.
    Bowen et al. (2003) have convincingly demonstrated the impact of half-metals in
MTJs. Epitaxial LSMO-SrTiO3 -LSMO junctions have been grown by pulsed laser
deposition and careful post-deposition lithographic processing, yielding a TMR of
1850% at T = 4 K (see Fig. 1.42). This corresponds to a tunneling spin polar-
ization of ≈ 95% when both LSMO-SrTiO3 interfaces are assumed to be equal.
At higher temperatures though, the TMR is gradually suppressed and disappears
at 280 K, below the Curie temperature of bulk LSMO, which is related to the
interface structure and specifically the LSMO termination at the barrier interface
Spin-Dependent Tunneling in Magnetic Junctions                                           85




Figure 1.42 Magnetoresistance measurements of 350 Å La2/3 Sr1/3 MnO3 /28 Å SrTiO3 /100 Å
La2/3 Sr1/3 MnO3 epitaxial junctions using SrTiO3 substrates. On top a 150 Å Co layer is
deposited and subsequently oxidized for magnetically pinning the top electrode. (a) Relative
change in resistance ([R – RP ]/RP ) versus applied magnetic field H at T = 4.2 K and a bias
                .
voltage of 1 mV In (b) and (c) the temperature dependence of TMR is shown for two junctions
with different area, using V = 10 mV. Solid curves are guides to the eye. After Bowen et al.
(2003).


(Pailloux et al., 2002). In a follow-up study by Garcia et al. (2004), the relatively
low value of TC in LSMO could be exploited to measure how the temperature
dependence of tunneling spin polarization is related to M(T ), a similar approach as
followed by Hindmarch et al. (2005a) for ferromagnetic Cu38 Ni62 . In section 2.1.4
such a relation between tunneling spin polarization and the (surface) magnetic mo-
ment has been suggested to describe the temperature dependence of TMR for
regular Al2 O3 -based MTJs. In Figs. 1.43a and 1.43b the TMR of La2/3 Sr1/3 MnO3 -
TiO2 -La2/3 Sr1/3 MnO3 and La2/3 Sr1/3 MnO3 -LaAlO3 -La2/3 Sr1/3 MnO3 junctions is
plotted versus temperature. From this the tunneling spin polarization is calculated
via the Julliere formula P (T ) = [TMR(T )/(2 + TMR(T ))]1/2 and is plotted in
Fig. 1.43c and Fig. 1.43d for TiO2 and LaAlO3 , respectively. A close resemblance
with the bulk magnetization of separately grown trilayers is observed, although the
Curie temperature deduced from P (T ) is roughly 60 K lower than the temperature
where M(T ) vanishes (350 K). Apparently, the magnetism of interfacial LSMO
is well preserved at the interfaces with TiO2 , LaAlO3 , and SrTiO3 (not shown),
certainly when comparing it with the polarization of a free surface of LSMO mea-
sured with spin-polarized photoemission (Park et al., 1998b). In that case a much
stronger decay with temperature is observed, evidencing that free surfaces and em-
bedded interfaces have strongly different properties in manganites (Garcia et al.,
2004).
    The use of half-metallic electrodes is particularly attractive for directly extract-
ing density-of-states or band-structure features from the bias dependence of the
tunneling transport. When only majority electrons are tunneling from half-metallic
LSMO, one is able to the directly probe the majority (minority) density-of-states
86                                                                             H.J.M. Swagten




Figure 1.43 Temperature dependence of TMR of epitaxial junctions contain-
ing (a) 350 Å La2/3 Sr1/3 MnO3 /32 Å TiO2 /100 Å La2/3 Sr1/3 MnO3 and (b)
350 Å La2/3 Sr1/3 MnO3 /28 Å LaAlO3 /100 Å La2/3 Sr1/3 MnO3 . The lines in (a) and
(b) are guides to the eye. The normalized tunneling spin polarization deduced from TMR
is shown as a function of temperature normalized to TC for the junction with TiO2 (c) and
LaAlO3 (d). The solid line in (c) and (d) is the normalized magnetization measured on similar
trilayers. After Garcia et al. (2004).


of the counter electrode when the magnetizations are (anti)parallel oriented. This
idea is exploited by Bowen et al. (2005a, 2005b). They find a quantitative con-
firmation of the half-metallic band structure of La2/3 Sr1/3 MnO3 by measuring the
conductance and TMR of LSMO-SrTiO3 -LSMO junctions for variable bias volt-
ages. First of all, it is observed that the conductance dI /dV in one bias direction
for parallel oriented magnetization of the LSMO layers shows a dramatic collapse
at V ≈ 0.82 V, whereas the antiparallel conductance continues to increase; see
Fig. 1.44a. The collapse in parallel conductance proves that no minority band
is available at EF from which electrons can tunnel into the minority t2g band,
demonstrating the half-metallic nature of the LSMO. It also proves that the ma-
jority electrons available at EF do not find any empty majority states at EF + Eg
with Eg ≈ 0.82 eV; see the schematic diagram in Fig. 1.44c. This is consistent
with a pseudo-gap in the majority density-of-states of the eg bands as predicted by
Pickett and Singh (1998) for a distorted oxygen environment of Mn ions in man-
ganites. In a related paper, the energy difference δ between the Fermi energy and
the bottom of the minority t2g band is accurately extracted from TMR, conduc-
tance and conductance derivative measurements in these junctions (Bowen et al.,
2005a). In Fig. 1.44b, d2 I /dV 2 reveals a sudden upturn of the antiparallel conduc-
tance at V ≈ 0.34 V. This marks the onset of a conduction channel for majority
electrons tunneling into the minority t2g band at EF + δ (see the schematics in
Fig. 1.44d). This turns out to be in good agreement with data obtained from spin
polarized inverse photoemission experiments, yielding δ = 0.38 ± 0.05 eV. When
the bias voltage across these LSMO junctions exceeds δ /e, the low-temperature
TMR is seen to rapidly decrease with voltage (not shown). This is again due to the
opening of a new conduction channel in the antiparallel orientation, corroborating
Spin-Dependent Tunneling in Magnetic Junctions                                               87




Figure 1.44 (a)        Conductance      dI /dV   versus    applied   bias   voltage    V      of
350 Å La2/3 Sr1/3 MnO3 /28 Å SrTiO3 /100 Å La2/3 Sr1/3 MnO3 epitaxial junctions, for
both antiparallel and parallel oriented magnetization. The conductance collapse in the parallel
case (at ≈ 0.82 V) is due to the absence of conduction channels at V = Eg /e, as shown in
the schematic band diagram in (c); adapted from Bowen et al. (2005b). In (b) the derivative
of the conductance d2 I /dV 2 is shown for bias voltages below 0.5 V. The increase of d2 I /dV 2
in antiparallel orientation observed at V = δ/e ≈ 0.34 V marks the onset of tunneling into
the minority (t2g ) spin band; see the schematics in (d). The lines in (b) are added to better
visualize the conductance upturn. Adapted from Bowen et al. (2005a).


the predictions by Bratkovsky (1997) for the bias-voltage dependence of magnetic
tunnel junctions with half-metallic electrodes.

4.5 Role of the barrier for TMR
Now that we have seen that TMR may be tuned towards very large numbers by
a proper choice of ferromagnetic materials, one should realize that the combined
system of (magnetic) electrodes and barrier material is decisive for the magnitude of
TMR and tunneling spin polarization (section 3). In a series of remarkable exper-
iments on La0.7 Sr0.3 MnO3 -insulator-Co junctions, de Teresa et al. (1999a, 1999b)
have used the full polarization of the half-metallic LSMO as a detector of the spin
polarization of Co adjacent to tunnel barriers of a different character. When using
traditional alumina in LSMO-Al2 O3 -Co, a positive TMR is found at temperatures
well below room temperature, which, via the Julliere formula, reflects a positive
spin polarization of Co. Although this is contrary to what is expected from the
smaller density-of-states at EF for the Co majority spin channel, this is believed to
reflect the positive polarization of s electrons that dominate the tunneling process
(see also the more elaborate discussions in section 3).
    A striking sign reversal of TMR is observed when replacing the alumina by
SrTiO3 or Ce0.69 La0.31 O1.845 . In this case, it appears that electrons with a d-like char-
acter are now preferentially transmitted at the Co-SrTiO3 or Co-Ce0.69 La0.31 O1.845
interfaces (Fig. 1.45a). Moreover, when using a double barrier in a LSMO-SrTiO3 -
Al2 O3 -Co junction the TMR is positive again, see Fig. 1.45b. Apparently, the
electronic structure and chemical bonding at the Co-insulator interface is deci-
sive for the tunneling spin polarization rather than the electron tunneling processes
88                                                                                H.J.M. Swagten




Figure 1.45 (a) TMR as a function of bias voltage V of SrTiO3 (001)/350 Å
La2/3 Sr1/3 MnO3 /25 Å SrTiO3 /300 Å Co measured at T = 5 K and T = 30 K. The in-
set shows the resistance change ([R – RP ]/RP ) with applied magnetic field H at 5 K using a
bias of –0.4 V (b) TMR versus bias voltage at T = 40 K as in (a), but now with a composite
              .
barrier: SrTiO3 (001)/350 Å La2/3 Sr1/3 MnO3 /10 Å SrTiO3 /15 Å Al2 O3 /300 Å Co. (c) Rel-
ative position of the DOS in La2/3 Sr1/3 MnO3 and the d DOS at an fcc Co(001) surface for
a bias around zero. The arrow indicates the high tunneling probability between the majority
band of LSMO and the minority band of Co, when the magnetization is antiparallel (AP). For
V < 0 electrons tunnel into the empty states of Co above the Fermi level EF . After de Teresa et
al. (1999a, 1999b); note that in these papers TMR is alternatively defined as ([RP –RAP ]/RAP ).


in the full barrier. The dependence of TMR on bias voltage is another interesting
aspect of the LSMO-SrTiO3 -Co junctions; see Fig. 1.45a. Since the conductance
is determined by only one spin channel, the variations with bias are found to be
easily correlated with the d-character density-of-states of a Co(001) surface; see
Fig. 1.45c. At a negative bias voltage of around –0.4 V, the majority electrons are
tunneling into the predicted peak in the (unoccupied) minority density-of-states of
Co above the Fermi level, leading to a maximum in negative TMR. In a more gen-
eral perspective, these experiments show that the interfacial bonding is of critical
relevance for spin-dependent tunneling of electrons. When d–d bonding is allowed
by using barriers with d-orbitals such as in SrTiO3 , it is possible to observe the
d-dominated spin polarization of Co (P < 0). In the opposite case of alumina bar-
riers, the absence of d orbitals apparently favors an s-dominated tunneling current
(P > 0). Although these arguments are helpful to qualitatively understand the role
of the barrier for tunneling spin polarization, it is evident that a more solid theo-
retical basis is required to substantiate this. This will be further discussed later on in
this section.
    Thomas et al. (2005) have directly measured the tunneling spin polarization
of Co-SrTiO3 -Al by superconducting tunneling spectroscopy, yielding a positive
spin polarization of +31% (see Table 1.2) in striking contrast to the aforemen-
tioned results at low bias voltage (de Teresa et al., 1999a, 1999b). This could be
explained by the thermal evaporation of the barrier on polycrystalline Co, leading
Spin-Dependent Tunneling in Magnetic Junctions                                      89


to an amorphous SrTiO3 layer as seen by high-resolution transmission electron mi-
croscopy (instead of the epitaxial barriers in the work of de Teresa et al. (1999a,
1999b)). Correspondingly, also a rather small (positive) TMR of around +1% has
been measured at low temperatures in Co-SrTiO3 -Co, Co-SrTiO3 -Ni80 Fe20 , and
Co-TiO2 -Co-Ni80 Fe20 junctions (Thomas et al., 2005). The tedious role of the
chemical structure of the barrier and the interfaces with the ferromagnetic layers in
these LSMO-based junctions is also recognized in other experimental studies, see
for example Sun et al. (2000b) and Hayakawa et al. (2002), showing both negative
and positive TMR in CoFe-SrTiO3 -LSMO and Fe-SrTiO3 -LSMO, strongly and
asymmetrically dependent on the bias voltage.
    In the work of Oleynik et al. (2002), first-principles density-functional calcu-
lations of the atomic and electronic structure of Co-SrTiO3 -Co(001) MTJs have
established the key importance of the atomic arrangement at the barrier interfaces.
It is found that the most stable structure represents the TiO2 -terminated inter-
face with the O atoms lying on top of the Co atoms. At the interface with Co,
an induced magnetic moment of 0.25 μB on the interfacial Ti atoms is aligned
antiparallel to the magnetic moment of the Co layer, which may indeed lead to a
negative tunneling spin polarization of the Co-SrTiO3 barrier (Oleynik et al., 2002;
Oleynik and Tsymbal, 2003). Using ab initio transport calculations including first-
principles band structure methods, Velev et al. (2005) predict a very large TMR
(1000% and more) in Co-SrTiO3 -Co junctions with bcc Co(001) electrodes and
barriers typically 7 to 11 monolayers in thickness. The complex band structure of
SrTiO3 enables an extremely efficient tunneling of minority d electrons from the
Co, causing the tunneling spin polarization to be negative. From the calculations it
is estimated that a single Co-SrTiO3 interface carries a tunneling spin polarization
of –50% that is rather independent of the barrier thickness. This is roughly a factor
of 2 higher than P derived from the experiments of de Teresa et al. (1999a), and
may be explained by effects of interface disorder, e.g. locally affecting the structure
of bcc Co. It should be emphasized that these results show that a spin-polarized
tunneling current across SrTiO3 is carried by minority d electrons. This is essentially
different as compared to sp-bonded insulators such as Al2 O3 (sections 2 and 3) and
MgO (section 4.6), where tunneling is dominated by electrons from majority bands.
    The argument of interface (chemical) bonding has also been used to explain
the sign reversal of TMR observed in junctions with others barriers containing d-
type ions. Experiments by Sharma et al. (1999) on Ta2 O5 are already discussed in
section 3.3.3. Bibes et al. (2003b) have investigated junctions with TiO2 barriers.
La2/3 Sr1/3 MnO3 -TiO2 -Co shows a negative TMR of around –3% at low temper-
ature. Regarding the positive spin polarization of La2/3 Sr1/3 MnO3 (though against
a SrTiO3 barrier by Worledge and Geballe (2000b)), the tunneling spin polariza-
tion of Co-TiO2 is negative, similar to the experiments by de Teresa et al. (1999a,
1999b) for interfaces of Co-SrTiO3 or Co-Ce0.69 La0.31 O1.845 . Also Co-Cr2 O3 and
Ni81 Fe19 -Cr2 O3 interfaces display a negative spin polarization as determined from
TMR in junctions with one half-metallic CrO2 electrode, the other electrode being
Co or NiFe (Gupta et al., 2001). A related experiment has been performed using the
ferromagnetic double perovskite Sr2 FeMoO6 with a TC of 415 K, and a predicted
half-metallicity (Kobayashi et al., 1998). Bibes et al. (2003a) have obtained a +50%
90                                                                       H.J.M. Swagten



TMR at low temperature in a junction consisting of SrTiO3 -Sr2 FeMoO6 -SrTiO3 -
Co. Using Julliere’s formula and the experimental fact that the epitaxial SrTiO3 -Co
interface carries a spin polarization at low bias voltage of about –25% (de Teresa et
al., 1999a), this yields a very strong tunneling spin polarization of –80%.
     It is important to be aware of the fact that only in a few systems the pres-
ence of negative tunneling spin polarization has been confirmed straightforwardly
by superconducting tunneling spectroscopy (section 3). In the case of Co-SrTiO3
interfaces, the expected negative tunneling spin polarization of around –25% (as
deduced from the low-bias TMR data of de Teresa et al. (1999a) using LSMO as
the second electrode) may be directly tested by STS on Co-SrTiO3 -Al supercon-
ducting junctions. However, the system of Co-SrTiO3 -Al is extremely difficult to
realize epitaxially and may suffer from oxidation of either Al or Co, both reducing
the spin polarization. STS data obtained by Thomas et al. (2005) using amorphous
SrTiO3 indeed did not yield the anticipated negative spin polarization as mentioned
earlier. A negative tunneling spin polarization of –9.5% is for the first time measured
by Worledge and Geballe (2000c) using ferromagnetic SrRuO3 in a supercon-
ducting junction consisting of SrTiO3 (100)-YBa2 Cu3 O7 -SrRuO3 -SrTiO3 -Al (see
Table 1.2). This negative sign is supported by theoretical calculations and emphasizes
the crucial role of weighting the density-of-states factors in Eq. (20) with transmis-
sion probabilities for the tunneling processes. Also in Co1–x Gdx ferrimagnetic alloys
for 0.2      x    0.75 a negative tunneling spin polarization has been observed di-
rectly from STS (Kaiser et al., 2005a). This is explained by the relative contribution
of independent spin-polarized tunneling currents from the two sublattice magneti-
zations (see section 3.3.2). Via the Julliere formula TMR = 2PL PR /[1 – PL PR ], the
negative tunneling spin polarization is in agreement with a negative magnetoresis-
tance in junctions with one electrode of Co1–x Gdx (P < 0) and a counter electrode
of Co70 Fe30 (P > 0).

4.6 Coherent tunneling in MgO junctions
In the previous sections, it is emphasized that TMR is certainly not determined by
the spin polarization of the individual ferromagnetic electrodes. Instead, it is sen-
sitively dependent on the full system of ferromagnetic electrodes and the adjacent
barrier, in which the electronic structure modifications at the barrier-electrode in-
terface and the symmetry and matching of the electron wave functions are playing
a crucial role. Based on this, it could be conceivable that certain electrode-barrier
material combinations would allow for a highly efficient polarization of the spin
currents, even with a bulk density-of-states displaying only a modest spin polar-
ization. In Fe-ZnSe-Fe(001) junctions (MacLaren et al., 1999), it is theoretically
shown that for thick enough barriers the conductance is dominated by slowly de-
caying s-states at k = 0 as provided by a 1 -band at the Fermi level of Fe(001).
Together with the absence of a minority 1 -band at EF , this leads to a very strong
asymmetry in the conductance and hence a large TMR.
     Experimentally, however, no such dramatic pseudo-half-metallic effects have
been observed for ZnSe barriers. Gustavsson et al. (2003) report on a low-
temperature TMR of only 16% in a Fe-ZnSe-Co0.15 Fe0.85 junction, disappearing
Spin-Dependent Tunneling in Magnetic Junctions                                     91


above T ≈ 50 K. Jiang et al. (2003b) have found magnetoresistance of less than
25% at low temperature and ≈ 10% at room temperature in ZnSe-based MTJ’s,
which, although potentially relevant for low RA product MTJs, is again not in
agreement with the promises given by theory. Although the interfaces between
ZnSe(001) and Fe are reported to be very sharp without magnetically dead or
modified interfacial regions even after annealing up to 300°C (Marangolo et al.,
2002), it could be that significant modifications of the Fe spin-polarized band
structure near EF as determined from spin-polarized inverse photoemission lead
to a suppression of TMR (Bertacco et al., 2004). Also the presence of mid-gap
localized states in the ZnSe barrier due to a small amount of disorder is shown
to significantly suppress or even change the sign of TMR in epitaxial Fe-ZnSe-Fe
junctions (Varalda et al., 2005). Similarly, the use of the II-VI compound ZnS has
yielded magnetoresistances of only 5% at room temperature (Guth et al., 2001b;
Guth et al., 2001a). In this case, it is suggested that the observation of an indirect
ferromagnetic interaction across the insulating ZnS is mediated by the tunneling
electrons (Dinia et al., 2003). Later on in this section, we will return to interlayer
coupling across insulating spacers.
    Now we will concentrate on the spin-dependent transport properties when
MgO barriers are employed. The experimental use of these barriers has also been
triggered by theoretical predictions of pseudo-half-metallic behavior in Fe-MgO-
Fe(001), and has resulted in a number of intriguing new observations, which will
be extensively discussed below.

4.6.1 TMR of MgO-based junctions
Using different theoretical approaches, both Butler et al. (2001b, 2005) and Mathon
and Umerski (2001) come basically to the same conclusion for coherent tunneling
in an Fe-MgO-Fe(001) magnetic tunnel junction, i.e. for electrons tunneling nor-
mal to the barrier in the [001] direction. For majority electrons, there are four
Bloch states of different symmetry present around the Fermi level for k = 0, viz.
a double-degenerate 5 state compatible with pd symmetry, 2 with d symme-
try, and a 1 state with spd symmetry. However, for the minority spins the 1
state is replaced by a d-type 2 state. Due to its s-type character, only the Bloch
states of 1 symmetry are able to effectively couple with the evanescent sp states in
the MgO barrier region, which, at the Fermi level, is only available for majority
electrons. This pseudo-half-metallicity of the band structure in the [001] direction
is schematically shown in Fig. 1.46 and Fig. 1.47a by the absence of 1 minority
states for tunneling electrons. For thick enough barriers, the majority conductance
in parallel alignment of magnetization becomes fully dominated by these 1 -band
contributions, and, correspondingly, extremely large TMR in these junctions (of
1000% and more) are expected to show up experimentally (Butler et al., 2001b;
Mathon and Umerski, 2001).
    Early experiments using MgO as a barrier have only been partially suc-
cessful. First of all, when the electrodes are polycrystalline and the MgO is
amorphous (Moodera and Kinder, 1996; Platt et al., 1997; Smith et al., 1998;
Kant et al., 2004c), only a modest TMR or tunneling spin polarization is found. In
fully epitaxial systems grown by molecular beam epitaxy combined with pulsed
92                                                                             H.J.M. Swagten




Figure 1.46 Layer-resolved tunneling density-of-states (DOS) for k = 0 in Fe(100)/ 8 mono-
layers MgO/Fe(100) for majority electrons (a) and minority electrons (b) when the magne-
tization of the Fe layers is parallel oriented. Each curve is labelled by the symmetry of the
incident Bloch state in the left Fe electrode, showing, for example, the absence of minority
states with 1 symmetry, whereas the majority 1 states decay only very slowly in the MgO
barrier. After Butler et al. (2001b).




Figure 1.47 (a) Calculated band dispersion of Fe in the [001] ( -H) direction. Solid and
dotted curves represent majority and minority-spin subbands, respectively; as indicated,
thicker lines are the 1 subbands. Adapted from Yuasa et al. (2004a). (b) Calculated lo-
cal spin-polarized density-of-states for Fe at the bottom interface with MgO in Fe/MgO/Fe
(grey) and Fe/FeO/MgO/Fe (black), the latter representing the presence of one complete O
layer between Fe and MgO. EF is the Fermi level. Top panel is for majority electrons, bottom
panel for minority electrons. After Tiusan et al. (2004).


laser deposition, the TMR is reported to be quenched by defects in the epi-
taxial MgO barrier (Klaua et al., 2001; Wulfhekel et al., 2001). Bowen et al.
(2001) have reported a TMR of 60% at 30 K and 27% at room temperature in
Fe-MgO-FeCo(001) by combining laser ablation and sputtering. In the case of
Spin-Dependent Tunneling in Magnetic Junctions                                        93




Figure 1.48 Inner-loop resistance when switching the free magnetic layer, expressed as
(R – RP )/RP , versus magnetic field strength H for MTJs with a crystalline MgO barrier.
(a) Junctions consisting of 100 Å TaN/250 Å IrMn/8 Å Co84 Fe16 /30 Å Co70 Fe30 /29 Å
MgO/150 Å Co84 Fe16 /100 Å Mg, annealed at TA = 120°C and 380°C (after Parkin
et al. (2004)). (b) Junctions of 100 Å Ta/150 Å PtMn/25 Å Co70 Fe30 /8.5 Å Ru/30 Å
Co60 Fe20 B20 /18 Å MgO/30 Å Co60 Fe20 B20 /100 Å Ta/70 Å Ru measured at T = 20 K
and T = 300 K, after an anneal at 360°C (after Djayaprawira et al. (2005)).


Fe-MgO-Fe-Co grown by molecular beam epitaxy (Faure-Vincent et al., 2003),
a TMR of 67% has been observed at room temperature, increasing up to around
100% at low T (see also the earlier work of Popova et al., 2002). Since the TMR
is still far from the existing theoretical predictions, the authors attribute this to the
growth-induced difference in topology of the two interfaces by which the required
symmetric matching of the wave functions is affected. By first-principle calculations
of the electronic structure of Fe-FeO-MgO-Fe, it is theoretically demonstrated that
the chemical bonding between Fe and O strongly reduces the conductance in par-
allel orientation (Zhang et al., 2003a). The corresponding reduction in TMR could
suggest that oxide formation at the barrier interfaces may be a common problem for
epitaxial MgO-based junctions; see also the surface X-ray diffraction experiments
by Meyerheim et al. (2001). On the other hand, Tusche et al. (2005) have shown
that oxygen at the barrier interfaces may promote a fully coherent growth of Fe
on top of the MgO spacer, leading to a coherent and symmetric MTJ structure
characterized by FeO layers at both Fe-MgO interfaces.
     A considerably improved room-temperature magnetoresistance in MgO junc-
tions has been found by Parkin et al. (2004). They have observed giant TMR
values up to ≈ 220%, whereas at low T it rises towards 300%. In their approach,
exchange-biased CoFe-MgO-CoFe(001) junctions are fabricated with regular sput-
tering deposition, the MgO being reactively magnetron-sputtered in an Ar-O2
mixture, and the full stack subsequently annealed at relatively high temperature (up
to 380°C). In Fig. 1.48a an example curve for these junctions is displayed. Obvi-
ously, these films are not epitaxial but polycrystalline and (001)-textured (including
the MgO barrier), which suggests that especially the well-defined crystalline orien-
tation of the barrier and electrodes is key to the strong tunneling spin polarization.
Separately, STS measurements on CoFe-MgO-Al junctions are used to directly
94                                                                      H.J.M. Swagten



measure the tunneling spin polarization. A positive P of 85% is found in optimized
junctions in accordance with the dominance of majority electrons with 1 sym-
metry as indicated above. Via the Julliere formula TMR = 2PL PR /(1 – PL PR ) this
relates to a magnetoresistance of ≈ 520% at low T , corresponding to a TMR effect
of around 260% at room temperature when correcting for the T dependence of
TMR, which is in close agreement with the magnetoresistance data (Parkin et al.,
2004).
    An even higher TMR at room temperature is found when MgO is sandwiched
between amorphous CoFeB ferromagnetic electrodes. Djayaprawira et al. (2005)
have compared the magnetoresistance of magnetron-sputtered structures containing
either Co70 Fe30 -MgO-Co70 Fe30 or CoFeB-MgO-CoFeB, where CoFeB is sput-
tered from a Co60 Fe20 B20 target. The barriers are deposited using rf sputtering
directly from a MgO target. All junctions are annealed at 360°C. As shown by
transmission electron microscopy, the structural quality of MgO in the CoFe junc-
tions is very poor and the interfaces are rough. In that case, a TMR of only 62% at
room temperature is observed. When growing MgO on top of the CoFeB, it shows
after the anneal a good crystallinity with a preferred (001) orientation, most proba-
bly due to the amorphous nature of the underlying CoFeB (although some parts are
crystallized). In the CoFeB-MgO-CoFeB junctions, the TMR ratio is now 230% at
room temperature increasing to 294% at T = 20 K; see Fig 1.48b. It seems that for
obtaining this very high TMR the correct structural symmetry of the MgO(001)
barrier is crucial, although it is presently not clear how an amorphous magnetic
electrode can give rise to giant TMR in view of the importance of the electrode
band structure along the k = 0 direction. In this respect, the authors do not ex-
clude the possibility that the annealed junctions show local (re)crystallization of a
few monolayers of CoFeB at the electrode-barrier interfaces, beyond the detection
limit of transmission electron microscopy. Indeed, Yuasa et al. (2005b) have shown
that a sputtered, amorphous Co60 Fe20 B20 layer grown on top of e-beam evapo-
rated Mg(001) crystallizes in a bcc structure with (001) orientation, after annealing
at temperatures of around 360°C. In this reflective high-energy electron diffrac-
tion study, it is also demonstrated that a MgO layer grown on amorphous CoFeB
initially has an amorphous structure as well, and begins to crystallize in the (001)
orientation only when tMgO exceeds 5 monolayers.
    Hayakawa et al. (2005b) have shown from transmission-electron-microscopy
images that annealing of Co40 Fe40 B20 -MgO-Co40 Fe40 B20 junctions at sufficiently
high temperature results in the formation of highly-oriented crystalline CoFeB
electrodes that were initially amorphous in the as-deposited state, a crystallization
process that is initiated at both the interfaces with MgO. When annealing a junction
with a MgO thickness of around 20 Å at 375°C, this yields an optimal TMR of
260% at room temperature and 403% at T = 5 K. By varying the Ar pressure for
sputter deposition of MgO (Ikeda et al., 2005), their optimized annealed junctions
exhibit a room-temperature TMR of 355%, and 578% at T = 5 K. An additional
improvement of the crystalline orientation of the Mg(001) layer may be achieved
by introducing an ultrathin ≈ 4 Å Mg layer between the bottom CoFeB electrode
and MgO (Tsunekawa et al., 2005). Especially for MgO(001) layers in the range
between 7 Å and 11 Å the addition of Mg during growth as suggested by Linn and
Spin-Dependent Tunneling in Magnetic Junctions                                        95




Figure 1.49 TMR at T = 20 K and T = 293 K at low bias voltage as a function of the
thickness of the MgO barrier tMgO . (b) Cross-sectional transmission electron microscopy
of an MTJ with tMgO = 18 Å, using two different magnifications showing the excellent
crystallinity of the layers. The junction stack consists of MgO(001)/MgO seed/1000 Å Fe/t
MgO/100 Å Fe/100 Å IrMn. After Yuasa et al. (2004b).


Mauri (2005) leads to a considerable enhancement of TMR and is typically well
above 100% (Tsunekawa et al., 2005). These huge TMR values are accompanied
by extremely small RA products of only a few µm2 , an attractive combination
never reached in alumina-based junctions (see section 2.2, Fig. 1.17).
    Comparable giant TMR values (180% at room temperature, 250% at T = 20 K)
have been demonstrated in single-crystalline (001)-oriented Fe-MgO-Fe-IrMn
junctions grown by molecular beam epitaxy (Yuasa et al., 2004a, 2004b). Apart
from the large magnetoresistances, the variation of TMR with the thickness of
the MgO barrier shows a number of interesting features, see Fig. 1.49. To start
with, on the average the TMR increases with the thickness of the MgO, and
saturates beyond tMgO ≈ 20 Å. This is in qualitative agreement with the ex-
periments by Hayakawa et al. (2005b) on sputtered junctions, and is also in line
with the aforementioned predictions (MacLaren et al., 1999; Butler et al., 2001b;
Mathon and Umerski, 2001). When the barrier is thick, the conductance is domi-
nated by electrons with the momentum vector normal to the barrier (k ≈ 0), by
which electrons in the highly polarized Fe- 1 band lead to giant TMR values. For
thinner barriers the TMR effect is suppressed by the increasing probability of elec-
trons tunneling with off-normal momentum vector. It is shown by Belashchenko
et al. (2005b) from first principles that for small MgO thickness the minority spin
bands at the interfaces make a significant contribution to the tunneling conduc-
tance in the antiparallel orientation of the Fe layers. In agreement with the data,
this efficiently reduces the TMR in Fe-MgO-Fe for small tMgO . Additionally, in the
experiments of Yuasa et al. (2004b) TMR versus tMgO is shown to clearly oscillate
with a period of 3.0 Å over the full spectrum of barrier thicknesses (Fig. 1.49),
not dependent on temperature or bias voltage. It is emphasized that the period is
not corresponding to the thickness of one monolayer of MgO(001), i.e. 2.2 Å.
96                                                                          H.J.M. Swagten




Figure 1.50 Resistance at T = 20 K and T = 300 K expressed as (R – RP )/RP versus mag-
netic field strength H for fully-epitaxial MgO-based MTJs consisting of (a) MgO(001)/200 Å
MgO(001)/1000 Å Fe(001)/23 Å MgO(001)/100 Å Fe(001)/100 Å IrMn/500 Å Au, and
(b) MgO(001)/200 Å MgO(001)/1000 Å Fe(001)/5.7 Å bcc Co(001)/21 Å MgO(001)/100 Å
Fe(001)/100 Å IrMn/500 Å Au. In (c) TMR is shown as a function of the composition x of
the Fe1–x Cox bottom electrode. For the junction with Fe50 Co50 the thickness of the MgO
                                                             .
barrier is 21 Å. In all experiments the bias voltage is 10 mV After Yuasa et al. (2005a).


The apparent intrinsic origin of the oscillation is not yet understood. It could
be related to quantum interferences in coherent tunneling processes across MgO,
e.g. due to the difference in complex wave vectors for the 1 and 5 evanes-
cent states in MgO (Butler et al., 2001b). On the other hand, this seems to be at
odds with hot spots in momentum space with extremely high tunneling prob-
ability, which would result in a single-period oscillation (Butler et al., 2001b;
Wunnicke et al., 2002).
    Using first-principles electronic-structure calculations, Zhang and Butler (2004)
have predicted that the magnetoresistance can be further enhanced by using bcc
Co and (B2-type) chemically ordered bcc Co50 Fe50 in MgO junctions. Again, for
these systems the large magnetoresistance can be understood from the slowly de-
caying 1 states available for majority electrons only. However, in the case of bcc
Co(Fe) the 1 band turns out to be the only majority band that crosses the Fermi
level, whereas for Fe there are others crossing EF for k = 0. Yuasa et al. (2005a)
have fabricated fully epitaxial bcc Fe1–x Cox (001)-MgO(001)-Fe(001) junctions to
test these predictions. In agreement with the calculations, the bcc Co electrodes
yield a higher TMR of 271% at room temperature and 353% at 20 K, as com-
pared to the Fe electrodes with 180% and 247%, respectively; see Figs. 1.50a and
1.50b. However, for bcc Co50 Fe50 TMR is of the same magnitude as for the Fe
bottom electrode (Fig. 1.50c). The authors suggest that the disordered character of
their Co50 Fe50 layer explains the discrepancy with the calculations assuming per-
fect B2-type chemical ordering. Finally, it is worth mentioning that apart from the
transport properties, the magnetic and electronic properties of thin 3d ferromagnetic
elements (such as bcc Co) in contact with MgO are being studied both theoreti-
cally and experimentally. As an example, Sicot et al. (2005, 2006) have used X-ray
absorption spectroscopy and X-ray photoemission spectroscopy to demonstrate a
weak hybridization between epitaxial MgO(001) and Co(001) or Fe(001), and to
Spin-Dependent Tunneling in Magnetic Junctions                                         97


rule out the formation of undesired magnetic oxides at the interfaces. Using X-ray
magnetic circular dichroism, it is observed that the magnetic moments of Fe and Co
are enhanced with respect to the bulk magnetic moments, ruling out the possibility
of magnetically quenched regions at the interfaces with epitaxial MgO layers (see
also Sicot et al., 2003, and the calculations for MgO-Fe by Li and Freeman, 1991).
Similar experiments have also been reported by Miyokawa et al. (2005), although in
this case exclusively focusing on the properties of 1 or 2 monolayers of bcc Fe(001)
embedded in a Co(001)-Fe(001)-MgO(001) structure.

4.6.2 Bias-voltage dependence of MgO-junctions
The bias-voltage dependence of the magnetoresistance of MgO-based junctions
needs special attention. The epitaxially grown junctions of Yuasa et al. (2004b)
show a remarkably small dependence on applied voltage, viz. V1/2 > 1.0 V for posi-
tive bias at room temperature (Fig. 1.51a). This is much better than usually reported
for alumina-based junctions where V1/2 ranges roughly between 0.3 and 0.6 V (see
also section 2.1.4). Interestingly, the bias voltage dependence is strongly asymmetric
with respect to the sign of the voltage, see again Fig. 1.51a for a junction with a bar-
rier of 20 Å MgO. However, the asymmetry becomes weaker upon an increase of
the barrier thickness (up to 32 Å). It is hypothesized that the asymmetry may be ex-
plained by structurally unequal MgO-electrode interfaces (Yuasa et al., 2004a). The
asymmetry is only slightly present in similarly grown Fe(001)-MgO(001)-Fe(001)
epitaxial junctions (Nozaki et al., 2005). In this case the room-temperature V1/2 is
around 0.7 V, combined with a TMR of almost 90%; see Fig. 1.51b. However, in
double barrier junctions of Fe-MgO-Fe-MgO-Fe grown by the same group (see
again the figure), the asymmetry with respect to the sign of the bias is also observed,
yielding V1/2 = +1.44 V, and V1/2 = –0.72 V for opposite bias direction. Tenta-




Figure 1.51 TMR of epitaxial MgO junctions as a function of applied bias voltage V at room
temperature. (a) MgO(001)/MgO seed/500 Å Fe/20 Å MgO/100 Å Fe/100 Å Co struc-
tures after Yuasa et al. (2004a). (b) Double-barrier junctions consisting of MgO(001)/MgO
seed/500 Å Fe/20 Å MgO/15 Å Fe/20 Å MgO/200 Å Fe (solid curve), and regular sin-
gle-barrier junctions of MgO(001)/MgO seed/500 Å Fe/20 Å MgO/15 Å Fe/100 Å Co
(symbols). After Nozaki et al. (2005). (c) Structures of MgO(001)/500 Å Fe/25 Å MgO/50 Å
Fe/100 Å Co junctions (open symbols) and Pd-seeded junctions of MgO(001)/400 Å Pd/20 Å
Fe/25 Å MgO/50 Å Fe/100 Å Co (closed). After Tiusan et al. (2004). In all cases positive
biasing (V > 0) corresponds to electron tunneling from the bottom into the top electrode.
98                                                                          H.J.M. Swagten



tively, this is ascribed to details of the band structure of the ultrathin 15 Å Fe layer
sandwiched between two barriers (Nozaki et al., 2005).
    To shed more light on this bias dependence and the role of the electronic struc-
ture of the Fe layers in MgO junctions, Tiusan et al. (2004) have shown that a
dramatic asymmetry in the bias dependence of TMR exists in their epitaxial Fe-
MgO-Fe junctions, even leading to a sign reversal at negative bias voltages. In
the case of V < 0, electrons are tunneling into the perfect bottom Fe(001) elec-
trode where a so-called interface resonance for minority electrons is believed to
strongly enhance the conductance in anti-parallel alignment; see Fig. 1.51c. The
impact of these interface resonances has been treated in several theoretical pa-
pers (MacLaren et al., 1999; Butler et al., 2001b; Mathon and Umerski, 2001;
Wunnicke et al., 2002) and is due to electron confinement between the bulk and
the barrier region. For thin enough barriers, at particular discrete values of k in the
two-dimensional Brillouin zone of the Fe minority electrons the tunneling trans-
mission can become very high, leading to huge conductance spikes. In Fig. 1.47b
the interface state is predicted from electronic-structure calculations on Fe-MgO-Fe
(Tiusan et al., 2004), and is still present when a full monolayer of O is introduced
between Fe and MgO, although shifted away with respect to the Fermi level. Note
that the effect of bias voltage and interface resonances on TMR in Fe-FeO-MgO-
Fe is treated from first principles by Zhang et al. (2004), although in that case the
predicted small or even negative TMR at low bias is strongly enhanced for higher
bias voltages (up to 0.5 V).
    Summarizing these remarks, it is suggested that for negative bias voltage across
such an epitaxial junction (Tiusan et al., 2004), electrons are tunneling into the
bottom epilayer at energies comparable to the location of the interface state lead-
ing to a strong enhancement of the antiparallel conductance. For large enough V
this finally reverses the sign of TMR. Interestingly, the effect of the interface res-
onances is almost completely destroyed when the bottom electrode is backed with
Pd (Tiusan et al., 2004). Mainly due to the absence of electron states starting 0.2 eV
above the Fermi level in Pd, the interfacial resonant states of the Fe are no longer
coupled to the electronic states of bulk Fe and drastically affect the propagation of
Bloch states. In the case of backing the Fe layer with Pd, TMR becomes almost
independent of bias voltage with V1/2 exceeding voltages of 1.5 V (Fig. 1.51c).
Surprisingly, the mechanisms involved in the reduction of TMR as described in
section 2.1.4 for Al2 O3 junctions are not active in these epitaxial junctions. As a
reminder, these mechanisms include quenching of TMR by magnon creation, by
intrinsic density-of-states effects, or by scattering at barrier imperfections. Note that
a similar insensitivity of bias voltage has been observed by vacuum tunneling be-
tween a magnetic CoFeSiB tip and a clean Co(0001) surface, suggesting that defect
scattering in the barrier of traditional alumina-based junctions is dominating the
observed bias dependencies (Ding et al., 2003). Finally, it should be mentioned that
the observation of the very strong bias dependence of TMR observed in Fig. 1.51c
for V < 0 (Tiusan et al., 2004) is probably related to the presence of a small amount
of carbon at the bottom interface between Fe and MgO incorporated during the
growth of the sample (including the required annealing steps). Due to this, the
conductance of propagating 1 states will be strongly suppressed and gets more
Spin-Dependent Tunneling in Magnetic Junctions                                              99




Figure 1.52 Bias-voltage and temperature dependence in junctions consisting of
100 Å Ta/150 Å PtMn/25 Å Co70 Fe30 /8.5 Å Ru/30 Å Co60 Fe20 B20 /0–10 Å Mg/18 Å
MgO/30 Å Co60 Fe20 B20 /100 Å Ta/40 Å Ru. (a) TMR and resistance V /I versus bias voltage
V at T = 77 K. (b) Conductance dI /dV versus bias voltage V (77 K). (c) TMR and resistance
V /I as a function of temperature. Resistances and conductances are shown for both parallel (P)
and antiparallel (AP) magnetization directions. After Miao et al. (2005).


sensitive to d-like features in the band structure including the interface resonances,
which otherwise would be negligible for the MgO thicknesses employed in these
experiments. Indeed, in C-free junctions, only a very small asymmetry in TMR(V )
is reported, probably due to a residual asymmetry in the Fe-MgO interfaces related
to roughness, defects, or lattice distortions (Tiusan et al., 2006).
     In the bias dependence of the conductance dI /dV and the derivative of the
conductance d2 I /dV 2 also a number of unique features are present in epitaxial
Fe(001)-MgO(001)-Fe(001) junctions, never observed in Al2 O3 -based magnetic
junctions (Ando et al., 2005). In the antiparallel orientation of the Fe layers at low
temperature (T = 6 K), a broad peak develops in d2 I /dV 2 at around ±1 V upon
an increase of the MgO barrier thickness (varied between 25.5 Å and 31.4 Å). Ten-
tatively, this is related to conduction channels between majority and minority spin
   1 bands that open up at sufficiently high bias voltages. For parallel orientation of
the Fe magnetizations, d2 I /dV 2 clearly oscillates with voltage and peaks at around
±0.1 V, ±0.35 V, and ±0.8 V, independent of the barrier thickness (again at 6 K).
This excludes the possibility of quantum-well formation in the barrier region, and,
moreover, these oscillations are not expected from tunneling dominated only by 1
bands at k = 0. It is suggested by Ando et al. (2005) that at higher bias voltage
tunneling may be governed by minority electrons tunneling via complex interface
resonant states at certain points in k-space with k = 0 (see Butler et al., 2001b and
Mathon and Umerski, 2001).
     Also for sputtered MgO-based junctions with high TMR, data are reported on
the bias dependence of the resistance, conductance dI /dV , and the derivative of
the conductance d2 I /dV 2 . As an example, Miao et al. (2005) report on a decrease
in conductance at a bias voltage of around 0.4 V in Co60 Fe20 B20 -MgO-Co60 Fe20 B20
junctions, indicating the emerging contribution from minority band states at these
energies (see Fig. 1.52b). At lower voltages, dI /dV and d2 I /dV 2 contain features
arising from magnon-assisted tunneling (see section 2.1.4) and phonon excitations
in MgO; see also the work of Ando et al. (2005). An intriguing result for these
junctions is seen in the bias voltage and temperature dependence of the resistance
100                                                                          H.J.M. Swagten



R = V /I when the magnetic layers have the same magnetization direction (paral-
lel). This is illustrated in Fig. 1.52a and Fig. 1.52c. Contrary to the typical signatures
for quantum-mechanical tunneling (section 2.1.4), there is hardly any variation of
R with temperature, nor is there a variation of R with applied bias voltage V ; see
also the I (V ) data obtained by Hayakawa et al. (2005b). This is probably related
to the decay rate in the MgO barrier for states of 1 symmetry. At energies up to
0.5 eV away from the Fermi level (covering the voltages used in the experiments),
the complex momentum vector varies extremely slowly with energy, typically less
than 1%. In accordance with the experimental data for parallel magnetized Fe lay-
ers, this leads to a surprisingly small resistance (or conductance) change with bias
voltage, typically less than 10% over this energy range (Miao et al., 2005).

4.6.3 Interlayer coupling across MgO-barriers
As an intriguing spinoff to the use of epitaxial MgO barriers, the observed superior
coherence of the tunneling electrons in these junctions could facilitate an interlayer
coupling between the two ferromagnetic electrodes across the barrier. Due to the
insulating character of the spacer, the well-known oscillatory Ruderman–Kittel–
Kasuya–Yosida interaction (Fert and Bruno, 1994; Bürgler et al., 1999) is excluded
to mediate the coupling. Slonczewski (1989) has derived an antiferromagnetic cou-
pling from the torque produced by rotation of the magnetization of one layer relative
to the other due to the tunneling electrons, which is further extended by Bruno
(1995) to predict the temperature dependence. At T = 0 K, the coupling strength
J across a barrier of thickness t and barrier height φ reads:
                                     φ
                         J =              e–2κt f (κ, kF ,maj , kF ,,min ),          (21)
                                 8π 2 t 2
where kF ,maj and kF ,min are the Fermi wave vectors of majority and minority elec-
trons, respectively, and κ the imaginary component of the wave vector of electrons
in the barrier with k = 0 at the Fermi level, corresponding to κ = (2me φ /h2 )1/2 .
                                                                                   ¯
Note that Eq. (11) in section 1 is based on a similar free-electron model calculation
by Slonczewski (1989), although in that case for the tunneling spin polarization.
The function f (κ, kF ,maj , kF ,min ) in Eq. (21) can be either positive or negative de-
pending on the Fermi wave vectors of the electrodes and the barrier height, and
therefore determines whether J is ferromagnetic (parallel magnetization of the two
layers) or antiferromagnetic (antiparallel magnetization).
    Faure-Vincent et al. (2002) have measured the magnetization loops of Fe-MgO-
Fe-Co multilayers with thin MgO spacers (down to 5 Å) to detect the presence of
coupling. As can be seen from the shift of the inner magnetization loop of the
magnetically softer bottom Fe layer, an antiferromagnetic coupling is present for
tMgO = 5.0 Å (see Fig. 1.53a). Upon an increase of the MgO spacer thickness the
shift of the hysteresis curve is drastically reduced as illustrated for tMgO = 6.3 Å. In
the right panel of the figure, the antiferromagnetic coupling is shown to be strongly
suppressed for thicker spacers and becomes ferromagnetic beyond 7 Å due to a
small ferromagnetic so-called orange-peel coupling (Néel, 1962). This is a dipolar
type of interlayer coupling when two magnetic layers have a correlated periodic
Spin-Dependent Tunneling in Magnetic Junctions                                          101




Figure 1.53 Interlayer coupling across MgO in layered structures composed of
                                                                   .
MgO(001)/500 Å Fe/4–25 Å MgO/50 Å Fe/500 Å Co/100 Å V (a) Inner-loop normal-
ized magnetization M/MSAT along the easy axis after a positive saturation magnetic field H ,
in a field range where the Fe/Co bilayer is magnetically rigid. The shift of the curve for
tMgO = 5.0 Å towards positive H is a signature of antiferromagnetic coupling. (b) Interlayer
coupling strength J versus thickness of the MgO spacer, tMgO , together with a fit using the
model of Slonczewski (1989), see Eq. (21). After Faure-Vincent et al. (2002, 2003).


modulation of interface roughness, frequently observed across metallic spacers (see,
e.g., Coehoorn, 2003) but also across insulating tunnel barriers; see section 2.1.2.
Note that the presence of a small amount of C at the bottom interface between
Fe and MgO as discussed earlier in the bias dependence of MgO-based junctions
(Tiusan et al., 2006) does not affect these data on interlayer coupling. Despite the
simplicity of the free-electron calculation of Slonczewski (1989), the authors show
that a perfect agreement can be obtained using Eq. (21) with realistic effective pa-
rameters, as can be seen from the fit in the right panel of the figure (Fig. 1.53b).
In conclusion, the results of Faure-Vincent et al. (2002) represent a clear signature
for the existence of an intrinsic interlayer coupling due to spin-polarized tunnel-
ing of electrons between ferromagnetic layers. In the case of ZnS barriers (Dinia
et al., 2003), qualitative fingerprints for interlayer coupling mediated by tunneling
electrons have been reported, viz. the coupling strength varies monotonically and
non-oscillatory with spacer thickness, and is increasing with temperature as pre-
dicted by Bruno (1995). However, since the sign of the coupling is positive in this
case, it could be partially obscured by ferromagnetic orange-peel coupling (Dinia
et al., 2003).
    In a calculation by Zhuravlev et al. (2005), an alternative explanation is presented
to describe the interlayer coupling data of Faure-Vincent et al. (2002). When it is
assumed that the barrier is not perfect but contains impurities or defects, the lo-
calized states within the gap of the insulator lead to a significant enhancement of
the coupling. Moreover, for certain impurity energies a crossover from antiferro-
magnetic to ferromagnetic coupling with spacer thickness is predicted. This is in
line with the experimental data shown in Fig. 1.53b, without the need to consider
dipolar orange-peel coupling. As to the temperature dependence of the impurity-
assisted interlayer coupling, it is shown that for impurity levels in the vicinity of the
102                                                                     H.J.M. Swagten



Fermi level the coupling strength is suppressed when raising the temperature. This
is opposite to the model of Bruno (1995) for perfect barriers, for which the thermal
population of the electronic states above the Fermi level leads to an increase of the
coupling strength with temperature.
    As a final remark to the interlayer coupling across MgO, it is suggested by
Tiusan et al. (2006) that interface resonances related to the minority channel in
Fe(001)-MgO(001)-Fe(001) could be crucial for the observed antiferromagnetic
coupling without the need to include impurity bands in the MgO. As a re-
minder, interface resonances can strongly enhance the tunnel conductance when
the Fe layers are aligned antiparallel (MacLaren et al., 1999; Butler et al., 2001b;
Mathon and Umerski, 2001; Wunnicke et al., 2002), and are believed to play
a crucial role in the bias dependence of MgO-based MTJs (Tiusan et al., 2004;
Zhang et al., 2004). To substantiate this hypothesis, it would be required to study
these electronic-structure effects by an ab-initio calculation of the coupling, which
is beyond the approach of Slonczewski (1989) and Zhuravlev et al. (2005) using
free-electron spin-split energy bands.



      5. Outlook
      In the foregoing sections, the physics of spin-dependent tunneling is addressed
focusing on a number of critical scientific breakthroughs in the field of MTJs. Since
the first discoveries in the mid-nineties, the field is rapidly expanding in many
other directions, related to various alternative hybrid material combinations often
motivated by new application potential for solid-state devices. A few prominent
examples are:
•   hybrid semiconductor magnetic tunnel junctions
•   tunnel barriers used for spin injection into semiconductors
•   magnetic tunnel transistors
•   magnetic semiconductor spin-filtering barriers
•   spin-torque effects in nanometer-scale ferromagnetic junctions
•   spin-logic devices using magnetic tunnel junctions.
    In the same order, these exciting research directions will be shortly introduced
below.
    One of the first observations of TMR in all-semiconductor magnetic tunnel junc-
tions is reported for Ga1–x Mnx As-AlAs-Ga1–x Mnx As (Tanaka and Higo, 2001).
Ga1–x Mnx As is a p-type ferromagnetic diluted magnetic semiconductor with a TC of
more than 100 K due to so-called carrier-induced ferromagnetism; see Story et al.
(1986) and Dietl (2002). For reviews on diluted magnetic semiconductors, see, e.g.,
de Jonge and Swagten (1991) and Dobrowolski et al. (2003). The magnetic elec-
trodes in the junctions of Tanaka and Higo (2001) are structurally well matched to
the AlAs spacer, the latter with a thickness typically between 13 Å and 28 Å. A large
TMR of more than 75% is observed for junctions with a thin (≤ 16 Å) AlAs bar-
rier when the magnetic field is applied along the [100] axis in the plane of the film,
Spin-Dependent Tunneling in Magnetic Junctions                                     103


see also Higo et al. (2001). Similarly, also in epitaxial ferromagnetic MnAs-AlAs-
MnAs junctions TMR has been observed, although in that case with a markedly
smaller magnitude of typically 1.4% at T = 10 K persisting up to room temperature
(Sugahara and Tanaka, 2002). Garcia et al. (2005) have improved this by measuring
a TMR of up to 12% at T = 4 K in MnAs-GaAs-AlAs-GaAs-MnAs incorporating
thin 10 Å GaAs layers to prevent Mn to diffuse towards the barrier region. Together
with data on junctions with single GaAs barriers, it is found that resonant tunneling
in these junctions may occur via a mid-gap defect band, which explains the rather
low TMR (less than 2% for the GaAs barriers). However, by using a resonant-
tunneling model a large tunneling spin polarization at the MnAs-GaAs interface is
deduced of around 60%. Ferromagnetic Cr1–δ with a TC in the range of 170-350 K
has been used in Cr1–δ -GaAs-AlAs-GaAs-Cr1–δ junctions with δ ≈ 0.33, where the
GaAs thin layers are used to prevent Mn, Cr, or Te atoms to diffuse into the barrier.
A magnetoresistance of up to 15% at T = 5 K is reported, though rapidly decreasing
with bias voltage and temperature (Saito et al., 2005a). Related to TMR in these
semiconductor tunnel junctions, Gould et al. (2004) have surprisingly found a new
magnetoresistance effect in GaAs(001)-Ga1–x Mnx As-Al2 O3 -Ti-Au using a single
ferromagnetic layer only. This so-called tunneling anisotropic magnetoresistance is due
to the large spin–orbit interaction in the valance band of Ga1–x Mnx As, which causes
the density-of-states at the Fermi level, and therefore the conductance, to depend
on the direction of magnetization; see also Brey et al. (2004). This observation may
have important consequences for the interpretation of the aforementioned results
in all-semiconductor MTJs, as shown by the extremely large tunneling anisotropic
magnetoresistance of more than 150,000% in Ga1–x Mnx As-GaAs-Ga1–x Mnx As. In
similar junctions with a ZnSe barrier, Saito et al. (2005b) have carefully disentangled
genuine TMR of up to 100% from a 10% tunneling anisotropic magnetoresistance
effect.
    To efficiently inject spin-polarized currents into semiconductor layers, one gen-
erally deals with the so-called conductivity mismatch between a ferromagnetic metal
and the poorly conducting semiconductor (Schmidt et al., 2000). One way to cir-
cumvent the mismatch is to separate the layers by a nonmagnetic tunneling barrier.
In this way, the spin-dependent resistance of the combined ferromagnetic-insulator
can be better matched to that of the semiconductor, leading to efficient spin injec-
tion in semiconductors (Fert and Jaffres, 2001). As an experimental example, it is
reported by Motsnyi et al. (2002) that electrons with more than 9% spin polarization
can be injected in GaAs-based light-emitting diodes at T = 80 K using a CoFe-
Al2 O3 tunneling system. A small magnetic field under 45 degrees with the film
normal is applied to manipulate the spins in the semiconductors via the so-called
oblique Hanle effect (Motsnyi et al., 2003). This is necessary to optically asses the
spin polarization in the light-emitting diode, for which a nonzero component of the
electron spin normal to the sample surface is required. Efficient spin injection from
Fe-Al2 O3 into a GaAs light-emitting diode is demonstrated by van ’t Erve et al.
(2004), with a spin polarization in the GaAs of up to 40% at T = 5 K, in their case
using large magnetic fields applied perpendicular to the film plane. A much higher
efficiency in this geometry is reported when exploiting the extremely large tunnel-
ing spin polarization for crystalline MgO barriers (see section 4). In the experiment
104                                                                          H.J.M. Swagten



of Jiang et al. (2005), the spin injection efficiency is estimated to be at least 52% at
T = 100 K and 32% at 290 K for a CoFe-MgO(100) tunnel injector. A fully elec-
trical demonstration of spin injection into semiconductors is reported by Mattana et
al. (2003). They use a Ga1–x Mnx As-AlAs-GaAs-AlAs-Ga1–x Mnx As double-barrier
junction to inject spins from the Ga1–x Mnx As-AlAs into the GaAs, which is subse-
quently measured by the AlAs-Ga1–x Mnx As detector. Similar to the aforementioned
all-semiconductor devices, thin GaAs films are incorporated between Ga1–x Mnx As
and AlAs to avoid diffusion. It is shown that the transport properties in these junc-
tions can be explained by sequential tunneling across the two barriers with a spin
relaxation small enough to efficiently transmit spins across GaAs (Mattana et al.,
2003).
    Al2 O3 tunnel barriers are also successfully employed in so-called magnetic tun-
nel transistors (Sato and Mizushima, 2001). Via an Al2 O3 layer, hot electrons are
injected from a nonmagnetic emitter layer into a metallic base layer consisting of
two ferromagnetic metals separated by a nonmagnetic metal. Subsequently, only
those hot electrons are collected in a n-type GaAs collector that have sufficiently
large energy to overcome a semiconductor Schottky barrier. The collector current
strongly depends on the relative orientation of the two ferromagnetic layers due to
spin-dependent filtering of hot electrons in the ferromagnetic layers. In a system of
GaAs(001)-CoFe-Cu-NiFe-Al2 O3 -Cu, van Dijken et al. (2003) have determined
that the relative change in the collector current is exceeding 3400% at T = 77 K
(see also the discussion of Jansen et al., 2003 and Jiang et al., 2003a). Also alternative
magnetic tunnel transistors with one ferromagnetic emitter and one ferromagnetic
base layer are feasible with relative collector current changes of 64% at room tem-
perature in GaAs(111)-CoFe-Al2 O3 -CoFe-IrMn-Ta (van Dijken et al., 2002). By
tuning the bias voltage across the tunneling barrier, it is demonstrated that magnetic
tunnel transistors can be used as a powerful tool to study hot-electron transport over
a wide range of energies. Using a transport model similar as for ballistic electron
emission microscopy (see section 2.1), spin-dependent inelastic electron scattering
in the ferromagnetic base layer and electron scattering at the base-collector inter-
face are included to describe the experimental data (Jiang et al., 2004b). As a final
promising direction, magnetic tunnel transistors can also be used for efficient injec-
tion of spin-polarized carriers in a GaAs-based light-emitting diode. In samples of
NiFe-CoFe-Al2 O3 -CoFe-Ta grown op top of (basically) a GaAs-InGaAs multiple
quantum well, a spin polarization of around 10% is determined from analyzing the
electroluminescence (Jiang et al., 2003b).
    As we have seen in section 4, the implementation of half-metallic electrodes
or the use of crystalline barriers have yielded very large magnetoresistance ratios,
corresponding to a tunneling spin polarization of up to 100%. Another route to
full spin polarization is the use of magnetic spin-filter tunnel barriers. Due to the spin
splitting of the conduction band of magnetic insulators such as EuS or EuO, their
barrier height becomes spin-dependent and can act as a very efficient spin filter.
This effect has been experimentally demonstrated by Moodera et al. (1988), Hao
et al. (1990), and Santos and Moodera (2004) in STS experiments using one su-
perconducting Al probe layer, the magnetic barrier, and a nonmagnetic layer as
the counter electrode. Inspired by the proposed electrical device by Worledge and
Spin-Dependent Tunneling in Magnetic Junctions                                    105


Geballe (2000d), a magnetoresistance effect can be obtained when combining a
magnetic spin-filter barrier with one magnetic and one nonmagnetic electrode. In
Al-EuS-Gd junctions, LeClair et al. (2002a) have reported TMR effects of more
than 100%, provided that the temperature is below TC of the ferromagnetic EuS
(≈ 17 K). However, the magnetic switching of the magnetic constituents is rather
poor, probably related to details of the EuS-Gd interface; see Smits et al. (2004).
Gajek et al. (2005) have used single-crystalline, insulating BiMnO3 as a spin filtering
barrier having a much higher Curie temperature of 105 K. In La2/3 Sr1/3 MnO3 -
SrTiO3 -BiMnO3 -Au junctions a thin 10 Å SrTiO3 layer is used to magnetically
separate LSMO from the spin filter. Although the observed TMR effects are only
50% at T = 3 K (corresponding to a filter efficiency of ≈ 22%), these experiments
show the potential of using insulating oxides for spin filtering and spin injection. In
this respect, NiFe2 O4 is considered as an extremely promising candidate for an in-
sulating spin-filtering barrier with TC above room temperature (850 K in the bulk),
although in thin films a metallic behavior is observed when growing it on single-
crystalline SrTiO3 (Lüders et al., 2005b). However, by tuning the conductivity
via different growth conditions, NiFe2 O4 has been used as an insulating spin-filter
in La2/3 Sr1/3 MnO3 -NiFe2 O4 -Au and La2/3 Sr1/3 MnO3 -SrTiO3 -NiFe2 O4 -Au junc-
tions where a thin SrTiO3 layer in the latter structures is again used for decoupling
the magnetic layers (Lüders et al., 2005a). Typically, a magnetoresistance of 52%
is observed at T = 4 K, corresponding to a NiFe2 O4 spin-filter efficiency of
around 23%. In a general perspective, calculations of Yin et al. (2005) have shown
that a further enhancement of the magnetoresistance for this class of devices is fea-
sible when a magnetic spin-filter barrier is combined with two instead of only one
magnetic electrode. However, no experimental evidence is yet available to confirm
this.
     When the current density within magnetic multilayers becomes sufficiently
high, it has been experimentally demonstrated that the magnetic moment of the
itinerant electrons may produce a so-called spin torque on the magnetization. Due
to the torque, a rotation or even switching of a magnetic layer is feasible (Katine et
al., 2000), as well as the possibility to create excitations of micro-wave frequencies
(Kiselev et al., 2003). Especially for advanced MRAM applications, it is envisioned
that such a novel switching scheme of the memory cell would no longer necessitate
the traditional use of separate word or bit lines. In MTJs based on Co88.2 Fe9.8 B2 -
Al2 O3 -Co88.2 Fe9.8 B2 and with a very low RA product of < 5 µm2 , Fuchs et al.
(2004) have demonstrated these effects when the junctions are laterally structured
down to sub-micrometer dimension. Higo et al. (2005) have shown reproducible
spin-torque switching of the free magnetic layer in 75 nm × 163 nm CoFe-Ru-
CoFeB-Al2 O3 -NiFe junctions at threshold current densities of around 106 A/cm2 .
A current density of 7 × 106 A/cm2 is reported for junctions based on a CoFeB free
layer in CoFe-Ru-CoFeB-Al2 O3 -CoFeB (Huai et al., 2005). A low current density
to switch the magnetization can be combined with the intrinsically large TMR in
MgO-based magnetic junctions (section 4.6). In CoFe-Ru-CoFeB-MgO-CoFeB
junctions of 100 nm×200 nm, threshold current densities of around 2×107 A/cm2
are reported Kubota et al. (2005a, 2005b). Hayakawa et al. (2005a) yield a current
106                                                                                      H.J.M. Swagten



density of only 2.5 × 106 A/cm2 in similar MgO junctions having a TMR well
above 100% at room temperature.
     Due to spin-torque effects, domain walls in magnetic materials are able to move
by a (spin-polarized) electrical current passing across the domain wall. In a so-called
shiftable magnetic shift register (Parkin, 2004), domain-wall movement is believed to
create a new storage solution with many advantages as compared to solid-state
memory and magnetic disks. An electric current is applied in order to move mag-
netic domains along a track, a strip of ferromagnetic material comprised of a large
number of magnetic domains with a magnetization direction representing the state
of the bit. Magnetic fields fringing from a domain wall in a strip-like writing device
are used to set the magnetization within the track of the shift register. By sending
a current through a magnetic tunnel junction that is part of the storage device, it is
possible to read the stored magnetization direction of the domains.
     Another new device using magnetic tunnel junctions is a so-called spin-torque
diode which may become relevant for applications in telecommunication circuits
(Tulapurkar et al., 2005). It is again based on the torque of a spin-polarized cur-
rent acting on small magnetic elements, leading, as mentioned before, to a high-
frequency rotation of the magnetization (Kiselev et al., 2003). In their experiment,
Tulapurkar et al. (2005) apply a radio-frequency alternating current to a nanometer-
scale magnetic junction, thereby generating a DC voltage across the device when
the frequency is resonant with the spin excitations arising from the spin-torque
effect.
     A final intriguing application of MTJs is emerging within the field of creating
logic devices for computing and programming. Although many variations are feasi-
ble for these novel magnetic devices, such as the implementation of thin metallic
layers at the barrier interface (You and Bader, 2000) or the combination of giant
magnetoresistance with resonant tunneling diodes (Hanbicki et al., 2001), a simple
concept is introduced by Hassoun et al. (1997) and later adapted specifically for
MTJs by Richter et al. (2002) and Ney et al. (2003). In the latter configuration, a
single MTJ cell offers the possibility to create nonvolatile output with basic logic
operations such as (N)AND and (N)OR by addressing a number of additional cur-
rent lines to predefine the magnetization directions; see also Moodera and LeClair
(2003).

ACKNOWLEDGEMENTS
Wim de Jonge, Corné Kant, Karel Knechten, Jürgen Kohlhepp, Bert Koopmans, Patrick LeClair, and
Paresh Paluskar are acknowledged for many useful discussions and for critically reading this manuscript.
Some of the results presented here are embedded in research programs of the Technology Founda-
tion STW and the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, both financially
supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”.


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         CHAPTER             TWO



         Magnetic Nanostructures: Currents and
         Dynamics
         Gerrit E.W. Bauer *,** , Yaroslav Tserkovnyak *,*** , Arne Brataas *,****
             and Paul J. Kelly *****




         Contents
         1. Introduction                                                                                       124
         2. Ferromagnets and Magnetization Dynamics                                                            125
         3. Magnetic Multilayers and Spin Valves                                                               127
            3.1 Non-local exchange coupling and giant magnetoresistance                                        127
            3.2 Non-equilibrium spin current and spin accumulation                                             129
            3.3 Spin-transfer torque                                                                           132
            3.4 Angular magnetoresistance of spin valves                                                       134
         4. Non-Local Magnetization Dynamics                                                                   135
            4.1 Current-induced magnetization dynamics                                                         136
            4.2 Spin pumping                                                                                   137
         5. The Standard Model                                                                                 139
            5.1 Enhanced Gilbert damping and spin battery                                                      139
            5.2 Current-induced magnetization reversal and high frequency generation                           140
            5.3 Dynamic exchange interaction                                                                   141
            5.4 Noise in magnetic heterostructures                                                             142
         6. Related Topics                                                                                     142
            6.1 Tunnel junctions                                                                               142
            6.2 Domain walls                                                                                   143
            6.3 Spin transport by thermal currents                                                             144
         7. Outlook                                                                                            144
    *   Centre for Advanced Study at the Norwegian Academy of Science and Letters, Drammensveien 78, NO-0271
        Oslo, Norway
   **   Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands
  ***   Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
 ****   Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
*****   Faculty of Science and Technology and Mesa+ Research Institute, University of Twente, P.O. Box 217, 7500 AE
        Enschede, The Netherlands

Handbook of Magnetic Materials, edited by K.H.J. Buschow                                       © 2008 Elsevier B.V.
Volume 17 ISSN 1567-2719 DOI 10.1016/S1567-2719(07)17002-5                                      All rights reserved.



                                                                                                               123
124                                                                           G.E.W. Bauer et al.



      Acknowledgements                                                                      144
      References                                                                            145

      Abstract
      The magnetization order parameter in magnetic nanostructures can be excited by torques
      from internal and external magnetic fields as well as electrically induced spin currents.
      Conversely, a time-varying magnetization emits spin currents that couple spatially sep-
      arated magnetic elements in circuits and devices. Here we review the principles of time-
      dependent magnetoelectronic circuit theory, a simple yet effective theoretical framework
      that captures the essential physics and in favourable limits allows quantitative descrip-
      tion of phenomena such as Gilbert damping enhancement, coupled magnetization dy-
      namics, and the spin battery effect.




      1. Introduction
      Since the discovery of giant magnetoresistance (GMR) in ferromagnetic mul-
tilayers (Grünberg, 2001; Levy, 1994), i.e. the modulation of electron transport by
magnetic-field-induced configuration changes of the magnetization profile, the use
of ferromagnetic elements in electronic circuits and devices has mushroomed. The
GMR effect is employed in read heads for mass data storage devices. Magnetic
random access memories (MRAMs) are based on the related effect of tunnelling
magnetoresistance (TMR) between two ferromagnets separated by a tunnel barrier.
MRAMs have the advantage of being non-volatile, which means that an applied
voltage is not needed to maintain a given memory state, and are therefore serious
competitors for flash memories in processors, reprogrammable logic applications
etc. These and other applications are reviewed by Parkin (2002) (see also the White-
book on Innovative Mass Storage Technology, http://www.ex.ac.uk/IMST2002).
    With decreasing feature size of magnetic elements in magnetic storage media,
magnetic read heads, and MRAM elements, the time and energy needed to read
and write a magnetic domain are crucial parameters studied intensively by industry
and academia. The magnetization dynamics of ferromagnetic films and particles
under the influence of a magnetic field are reviewed by Miltat et al. (2002).
    Basic research in magnetoelectronics is concentrated on small hybrid struc-
tures and novel materials. The unifying concept is that of spin-accumulation, i.e. a
non-equilibrium magnetization that can, e.g., be injected electrically into a non-
magnetic material from a ferromagnetic contact by applying a voltage (Johnson
and Silsbee, 1988a, 1988b). A breakthrough in magnetoelectronics was the obser-
vation of current-induced magnetization reversal in layered structures fabricated
into pillars with diameters down to about 50 nanometers (Kiselev et al., 2003;
Krivorotov et al., 2005; Myers et al., 1999). This effect was predicted a few years
earlier and arises from the transfer of spin angular momentum by the applied current
(Bazaliy et al., 1998; Berger, 1996; Slonczewski, 1996). Current-induced magneti-
zation switching has already been used to demonstrate a scalable MRAM concept,
the spin-RAM (Hosomi et al., 2005) based upon MgO magnetic tunnel junc-
tions (Parkin et al., 2004; Yuasa et al., 2004). Conversely, magnetization dynamics
Magnetic Nanostructures: Currents and Dynamics                                      125


induces spin currents in a conducting heterostructure (Tserkovnyak et al., 2002)
which in combination with the spin-transfer torque leads to crosstalk between dif-
ferent ferromagnets through conducting spacers: the dynamic exchange interaction
(Heinrich et al., 2003). The relation to earlier ideas of Berger (1996) is discussed by
Tserkovnyak et al. (2003).
     These novel effects arise from magnetization dynamics in hybrid devices
whereby magnetic elements are coupled by spin and charge currents induced by
either an applied bias or time-dependent magnetic fields. The dynamics therefore
becomes non-local, i.e. it is not a property of a single ferromagnetic element, but
depends on the whole magnetically active (or spin-coherent) region of the device.
We believe that the basic physics is well understood by now and undertake a brief
review in this contribution. For more technical expositions we refer to Brataas et
al. (2006a), Stiles and Miltat (2006), and Tserkovnyak et al. (2005).



      2. Ferromagnets and Magnetization Dynamics
       A ferromagnet has a broken symmetry ground state in which the spins of
a majority of the electrons point in a certain common direction below a critical
temperature which can be as high as 1388 K. The robustness of the magnetic or-
der (the insensitivity of its magnitude and direction to elevated temperatures and
external perturbations) is employed in applications as diverse as compass needles,
refrigerator-door stickers, and memory devices.
    In spite of its apparent stability, ferromagnetism is neither rigid nor static. As a
result of competition between exchange interactions, magnetocrystalline and shape
anisotropies, uniform magnetic order is often unstable with respect to domain
formation that lowers the magnetostatic energy. Thermal fluctuations reduce the
macroscopic moment until it vanishes at the critical temperature Tc . At tempera-
tures sufficiently below Tc , the internal dynamics of the ferromagnet are described
by low-energy transverse fluctuations of the magnetization (spin waves or magnons),
that are the magnetic equivalents of lattice vibrations (phonons). Classical coarse-
grained computer simulations of the detailed position- and time-dependent magne-
tization (“micromagnetism”) describe these phenomena well (Miltat et al., 2002).
    When magnetic grains become sufficiently small, the exchange stiffness ren-
ders domain structures energetically unfavorable and a single-domain picture is
adequate. At low temperatures, higher-energy spin waves freeze out and only the
lowest-energy, zero-wavevector spin wave is excited, which is nothing but a rigid
precession of the entire ferromagnetic order parameter. Restricting the ferromag-
netic degrees of freedom to this mode is often referred to as the “macrospin model”.
In thermodynamic equilibrium, the macrospin then points in a certain fixed di-
rection with small thermal fluctuations around it. It can still be forced to change
by applying external magnetic fields at an angle to the magnetization direction.
The system then moves in response to minimize its Zeeman energy. The com-
pass needle, a freely suspended single-domain ferromagnet with a sufficiently high
anisotropy (coercivity), does this by mechanical alignment. Here we are interested
126                                                                   G.E.W. Bauer et al.



in mechanically fixed magnets whose magnetic moments move in the presence of
external and internal (exchange and anisotropy) effective magnetic fields. Viscous
damping processes are required to achieve a reorientation (switching) of the mag-
netization under a suddenly applied magnetic field. Minimization of the switching
time by engineering magnetic anisotropies as well as magnetization damping rates is
an important goal in the design of fast magnetic memories. When the applied mag-
netic fields are large enough to surmount the anisotropies, the magnetization can
be reversed, often by large amplitude and complex trajectories, even in the simple
macrospin model. At finite temperatures, the magnetization reorientation becomes
probabilistic and is described by a Fokker–Planck equation on the unit sphere
(Brown, 1963).
    A traditional starting point in studying the transverse magnetization dynamics in
a ferromagnetic medium is based on the phenomenological Landau–Lifshitz (LL)
approach (Landau et al., 1980). The magnetization M(r, t) with direction m(r, t) =
M(r, t)/Ms and (constant) magnitude Ms is treated in this approach as a classical
position- and time-dependent variable, obeying equations of motion which are
determined by the free-energy functional F [M] for degrees of freedom coupled
to the magnetization distribution M (such as the electromagnetic field or itinerant
electrons experiencing a ferromagnetic exchange field):
                         ∂
                           m(r, t) = –γ m(r, t) × Heff (r),                  (1)
                        ∂t
where γ is (minus) the gyromagnetic ratio. For free electrons γ = 2μB /h > 0. In
                                                                       ¯
transition-metal ferromagnets, it is usually close to this value.
                                              ∂F [M]
                                Heff (r) = –                                     (2)
                                                ∂M
is the “effective” magnetic field. The magnetic free energy and effective field can be
decomposed into applied external, demagnetization, magnetocrystalline anisotropy,
and exchange fields.
     To lowest order in the frequency, dissipation can be described phenomenologi-
cally by an additional torque in Eq. (1) (Gilbert, 1955, 2004):
                        ∂                               ∂
                           m = –γ m × Heff + α0 m × m,                          (3)
                        ∂t                             ∂t
where α is the dimensionless Gilbert damping constant (it is sometimes conve-
nient to work with a different Gilbert parameter G = α0 γ Ms ). As required,
Eq. (3) preserves the local magnitude of the magnetization. For example, for a
constant Heff obeying Eq. (2) and α0 = 0, m precesses around the field vec-
tor with frequency ω = γ Heff . When damping is switched on, α0 > 0 (as-
suming positive γ , as in the case of free electrons), the precession spirals down
to a time-independent magnetization along the field direction, i.e., the lowest
free-energy state, on a time scale of (α0 ω)–1 . For an axially symmetric effective
field and close to equilibrium, the Landau–Lifshitz–Gilbert (LLG) equation (3)
is obeyed by a small-angle damped circular precession. Equation (3) very suc-
cessfully characterizes the dynamics of ultrathin ferromagnetic films as well as
Magnetic Nanostructures: Currents and Dynamics                                  127


bulk materials in terms of a few material-specific parameters that are accessi-
ble to ferromagnetic-resonance (FMR) experiments (Bhagat and Lubitz, 1974;
Heinrich and Cochran, 1993).
    In nanostructures of ferromagnets characterized by strong exchange interactions,
spatial magnetization gradients cost a great deal of energy and may be disregarded.
We then arrive at the “macrospin” model, in which (3) reduces to a non-linear
differential equation in the unit vector m(t).



      3. Magnetic Multilayers and Spin Valves

       The discovery that the energy of magnetic multilayers consisting of al-
ternating ferromagnetic (F) and normal (N) metal films depends on the rela-
tive direction of the individual magnetizations (Grünberg et al., 1986) is per-
haps the most important in magnetoelectronics. The existence of the antiparallel
(AP) ground-state configuration at certain spacer-layer thicknesses was essential
for the subsequent discovery of giant magnetoresistance (Baibich et al., 1988;
Binasch et al., 1989). Adjacent ferromagnetic layers in such structures are cou-
pled by nonlocal and, as a function of the thickness of the normal-metal layer,
oscillatory (Parkin et al., 1990) exchange interaction that can be qualitatively un-
derstood using perturbation theory in analogy to the RKKY exchange coupling
between magnetic impurities in a normal-metal host (Kittel, 2005). The differ-
ent oscillation periods that can be resolved when the magnetization configuration
(AP or P) is determined as a function of spacer thickness, are well explained in
terms of Fermi surface (FS) spanning vectors of the normal metal in the growth
direction. The magnetic ground-state configuration is, at least in principle, ac-
cessible to first-principles electronic-structure calculations in the spin-dependent
version of density-functional theory (DFT), and that is basically the end of the
story. However, in order to make a connection to the main topic of this review,
we briefly discuss the formulation of the equilibrium exchange coupling in terms
of scattering theory (Erickson et al., 1993; Slonczewski, 1989, 1993), that can
also be formulated from first-principles and calculated using DFT (Bruno, 1995;
Stiles, 2006). Another advantage of a scattering-theory formulation is that the ef-
fects of disorder can be understood employing the machinery of mesoscopic physics,
such as random-matrix (Beenakker, 1997) or diagrammatic perturbation theory.


3.1 Non-local exchange coupling and giant magnetoresistance
Let us consider a non-collinear N|F|N|F|N spin valve with angle θ between the
magnetizations and an N-spacer thickness L. Suppose we can view the F|N|F tri-
layer as some spin-dependent scatterer embedded in a normal-metal medium. The
total energy change induced by the scattering potential is given by an energy in-
tegral over the density of states that can be expressed by a standard formula as
128                                                                          G.E.W. Bauer et al.



(Akkermans et al., 1991)
                                        εF
                                  1              ∂
                    E(L, θ ) =               ε      ln det s(L, θ, ε)dε,                    (4)
                                 2πi   –∞        ∂ε
in terms of the energy-dependent scattering matrix s(L, θ , ε) of the trilayer F|N|F.
    Of special interest is the asymptotic dependence of energy E(L, θ ) for large L.
In this limit, the coupling is governed by the lowest term in the expansion of the
angular dependence in Legendre polynomials:
                                                      Jβ
                 E(L, θ )L → ∞ = cos θ                   sin(qβ⊥ L + φβ ),                  (5)
                                                  β
                                                      L2

which is a sum over all FS cross-sectional extremal vectors of the normal-metal
spacer labeled by β, and the parameters Jβ and φβ are model and material dependent
(Stiles, 1999); qβ⊥ is the distance between a pair of critical Fermi points in reciprocal
space in the layering direction.
    For configurations that are not in equilibrium, the derivative
                                           ∂
                                   τ =–      E(L, θ )                               (6)
                                          ∂θ
does not vanish. A finite τ is therefore interpreted as an exchange torque acting
on the magnetizations, pulling them into the energetically-favorable configuration.
Physically, this torque is a flow of angular momentum carried by the conduction
electrons in the normal metal spacer. A spin valve that is strained by a relative
misalignment of the magnetization directions from the lowest energy value therefore
supports dissipationless spin currents.
    Essential for the existence and the magnitude of the nonlocal exchange coupling
and the corresponding spontaneous persistent spin currents is the phase coherence
of the wave functions in the normal spacer. An incoming electron in the spacer
with information of the left magnetization direction has to be reflected at the right
interface and interfere with itself at the left interface in order to convey the cou-
pling information. This implies strong sensitivity to the effects of impurities, since
diffuse scattering destroys the regular interference pattern required for a sizeable
coupling. This qualitative notion has been formulated by Waintal et al. (2000) in
the scattering-theory formalism invoking the “isotropy” condition for validity of
the random-matrix theory. Isotropy requires diffuse transport, viz., that L is larger
than the mean free path due to bulk and interface scattering. It can then be shown
rigorously that the equilibrium spin currents vanish on average with fluctuations
that scale like N –1 , where N stands for the number of transverse transport channels
in the normal-metal spacer. In layered metallic structures, N is large and the static
exchange coupling and spin currents can safely be disregarded in the diffuse limit.
On top of the suppression by disorder, the absolute value of the coupling scales like
L–2 even in ballistic samples, see Eq. (5). Experimentally, even the best Co|Cu|Co
samples do not show any appreciable coupling beyond spacer-layer thicknesses of
20 atomic monolayers.
Magnetic Nanostructures: Currents and Dynamics                                     129


    The term giant magnetoresistance (GMR) stands for the reduction of the
resistance of multilayers when the magnetic configuration is forced by an ap-
plied magnetic field from an antiparallel configuration of neighboring layer mag-
netizations to a parallel one. GMR was originally discovered in a configura-
tion in which the current flow was in the plane of the film (CIP). More rel-
evant in the present context is the configuration in which the current flows
perpendicular to the planes (CPP) (Gijs and Bauer, 1997; Gijs et al., 1993;
Pratt et al., 1991). Assuming diffusive transport, the CPP GMR is easily under-
stood in terms of the two-spin-channel series-resistor model (Pratt et al., 1991;
Valet and Fert, 1993). In ferromagnets, the difference between electronic structures
of majority and minority spins at the Fermi energy gives rise to spin-dependent
scattering cross-sections at impurities resulting in spin-dependent mobilities. The
discontinuities of the electronic structure at a normal|ferromagnetic metal interface
can also be very different for both spin species, corresponding to a large spin de-
pendence of interface resistances. In the presence of applied electric fields and not
too strong spin-flip scattering processes, a two-channel resistor model is applica-
ble, according to which currents of two different spins flow in parallel. When the
magnetizations of the ferromagnetic layers are parallel the charge current is short
circuited by the low electrical resistance spin channel, explaining the reduction of
the resistance under the magnetic field-induced configuration change.

3.2 Non-equilibrium spin current and spin accumulation
The difference between spin-up and spin-down electric currents is called a spin-
current. It has a flow direction as well as a spin polarization, i.e. it is a tensor. In
ferromagnets, the electron spin angular momentum states are good quantum num-
bers in the directions of the magnetization, sz = ±h/2 for up and down spins. The
                                                      ¯
spin current is then polarized along the magnetization direction. In normal metals
there is no such preferred direction. Spin currents can flow without dissipation as
in the strained equilibrium configurations discussed in Section 3. A non-equilibrium
spin current excited by external perturbations is closely related to an imbalance in
the electrochemical potentials that is called spin-accumulation. This is again a vector
quantity that in a ferromagnet is parallel to the magnetization. Spin-related non-
equilibrium phenomena have lifetimes that are usually much longer than all other
relaxation time scales. Spin-flip scattering can originate from spin-orbit interaction
effects in the band structure plus potential disorder (“intrinsic”). “Extrinsic” spin
flips are caused by magnetic impurities or non-magnetic one with significant spin-
orbit interaction. Both depend strongly on the material, its chemical purity and
crystalline order and destroy a non-equilibrium spin-accumulation (we disregard
here the small spin accumulations that can be created by an electric bias in the pres-
ence of spin–orbit interaction, e.g. Edelstein, 1990). Spin-flip scattering can be very
weak in clean metals with simple band structures, but is often significantly larger
in 3d transition metals with a high density of states at the Fermi energy. In most
theoretical approaches to magnetoelectronics (and also here) spin-flip scattering is
treated phenomenologically, usually in terms of the spin-flip diffusion length, i.e.
the length scale over which an injected spin accumulation loses its polarization, that
130                                                                      G.E.W. Bauer et al.




Figure 2.1 The non-equilibrium spin accumulation injected from a ferromagnet (F) into a
normal metal (N), which decays over a length scale given by the spin-flip diffusion length
 N . The spin accumulation in the ferromagnet is not shown, being small compared to the
 sd
equilibrium magnetization.


is typically F ∼ 5 nm (Permalloy, Py)–50 nm (Co). In bulk ferromagnets the spin
             sd
accumulation vanishes at depths beyond F although spin currents persist.
                                              sd
    Much of magnetoelectronics is based on the notion that even a normal metal
such as copper can be magnetized. This magic is worked by applying a volt-
age over a normal|ferromagnetic metal contact (Fig. 2.1) so that a spin-polarized
current is injected from the ferromagnet into the normal metal (Aronov, 1976;
Johnson and Silsbee, 1987; van Son et al., 1987). The resulting spin accumulation is
equivalent to a net ferromagnetic moment in the non-magnetic metal. The decay
length of a spin accumulation injected into the normal metal is the spin-flip dif-
fusion length N that can be very large, e.g. about 1 µm in copper (Jedema et al.,
                sd
2001), much larger than the smallest feature size created by microelectronic fabrica-
tion technology. The spin accumulation is a vector that in Fig. 2.1 is collinear with
the ferromagnetic magnetization, i.e. parallel or antiparallel, depending on the spin-
dependent interface and bulk conductances. In non-collinear (i.e. neither parallel
nor antiparallel) spin valves, schematically F(↑)|N|F( ), or other devices with two
or more ferromagnetic contacts whose magnetizations are not parallel, the injected
spin currents are also non-collinear, and (when the spacer is sufficiently thin) the
resulting spin accumulation can be made to point in any direction. In contrast, in a
ferromagnet the spin accumulation direction is fixed along the magnetization vec-
tor. The manipulation of electronic properties via the long range spin-coherence
carried by the spin accumulation (Brataas et al., 2000) is the main challenge of
modern magnetoelectronics.
    Typical magnetoelectronic structures are made from ferromagnetic metals like
iron, cobalt or the magnetically soft permalloy (Py), a Ni/Fe alloy. The normal
metals are typically Al, Cu or Cr, where the spin-density wave in the latter is usu-
ally disregarded in transport studies. These metals cannot be grown as perfectly as
strongly bonded tetrahedral semiconductors; moreover, the Fermi wavelength is of
the order of the interatomic distances. These systems are said to be “dirty”, meaning
that size quantization effects on the transport properties may be disregarded (Waintal
et al., 2000). In this limit, semiclassical theories on the level of the Boltzmann or
diffusion equations most appropriately describe the physics. The spin accumulation
is then just the difference in the local chemical potentials for up and down spin
Magnetic Nanostructures: Currents and Dynamics                                       131


(Johnson and Silsbee, 1988a). Valet and Fert (1993) analyzed the GMR of magnetic
multilayers in the perpendicular (CPP) configuration with collinear magnetizations.
They used a spin-polarized linear Boltzmann equation to derive a diffusion equa-
tion including spin-flip scattering, that for vanishing spin-flip scattering reduces to
the two-channel series resistor model (Pratt et al., 1991). The total current can then
be interpreted as two parallel spin-up and spin-down electron currents, separately
limited by resistors in series that represent interfaces and bulk scattering.
    When magnetization vectors and spin-accumulations are not collinear with the
spin-quantization (z-)axis, the two-channel resistor model cannot be used anymore.
The concept of up and down spin states must be replaced by a representation in
terms of 2 × 2 matrices in Pauli spin space with non-diagonal terms that reflect the
spin-coherence, analogous to the anomalous Green functions in superconductiv-
ity that reflect the electron–hole coherence in the superconducting state. A circuit
theory for general non-collinear configurations can be derived from a given Stoner
Hamiltonian in terms of the Keldysh non-equilibrium Green function formalism
in spin space (Brataas et al., 2000, 2001) or simply by matching charge and spin
distribution functions at interfaces with transmission and reflection probabilities. It
reduces to a finite-element formulation of the diffusion equation with quantum
mechanical boundary conditions between distribution functions on both sides of
a resistor such as an interface. The initial step is an analysis of the circuit or de-
vice topology by dividing it into reservoirs, resistors and nodes that can be real or
fictitious. The expressions are greatly simplified by assuming that the electron dis-
tributions in the nodes are isotropic in momentum space. This implies the presence
of sufficient disorder (or chaotic scattering). The spin and charge currents through
a contact connecting two neighboring ferromagnetic and normal metal nodes can
then be calculated as a function of the distribution matrices on the adjacent nodes
and the 2 × 2 conductance tensor composed of the (diagonal) spin-dependent con-
ductances G↑ and G↓
                               e2                             e2
                       Gs =       M–             |rsnm |2 =             |tsnm |2 ,   (7)
                               h          nm
                                                              h    nm

and the (off-diagonal) spin-mixing conductance
                                       e2
                             Gs,–s =      M–             rsnm (r–s )∗ ,
                                                                nm
                                                                                     (8)
                                       h            nm

where rsnm , tsnm are the reflection and transmission coefficients in a spin-diagonal
reference frame with n, m the indices and M the total number of transport chan-
nels on the normal metal side of the contact. The expressions for the spin-up
and down conductances are the Landauer–Büttiker formula in a two-spin-channel
model (Bauer, 1992; Gijs and Bauer, 1997). Experimentally, these parameters
have been obtained by extensive measurements on multilayers in the CPP (cur-
rent perpendicular to the plane) configuration (Pratt et al., 1991). The complex
spin-mixing conductance parameterizes the spin currents that are perpendicu-
lar to the magnetization, as discussed in the next section. The resistance of spin
132                                                                   G.E.W. Bauer et al.



valves as a function of angle between the magnetizations is a sensitive mea-
sure of the spin-mixing conductance (Bauer et al., 2003b; Kovalev et al., 2006;
Urazhdin et al., 2005).
    A requirement for the validity of the circuit theory are nodes with charac-
teristic lengths smaller than the spin-flip diffusion length. When this criterion is
not fulfilled, the diffusion equation has to be solved, with boundary conditions
governed by the above conductance parameters (Huertas-Hernando et al., 2000).
When the resistances are small, the distributions are not isotropic in momen-
tum space anymore, reflecting the net electron flow. The assumption of isotropy
can be relaxed, leading to the conclusion that the diagonal (Bauer et al., 2002;
Schep et al., 1997) and the mixing conductances (Bauer et al., 2003b) in equations
7 and 8 contain spurious Sharvin resistances. A “drift” correction is essential for a
meaningful comparison of ab initio calculations with experiments.
    The scattering matrices for various clean and disordered interfaces have been
calculated from first-principles (Schep et al., 1997; Xia et al., 2001; 2006) and rep-
resentative results are listed in the tables. The computed conductance parameters in
general agree well with experiments (Bass and Pratt, 1999; Brataas et al., 2006a). In
the present formulation, circuit theory cannot describe spin-flip scattering processes
at interfaces addressed by Bass and Pratt (2007).

3.3 Spin-transfer torque
A spin accumulation with polarization normal to the magnetization direction can-
not penetrate deeply into the ferromagnet, but is instead absorbed at the inter-
face on an atomic length scale, thereby transferring angular momentum to the
ferromagnetic order parameter. A large enough torque overcomes the magnetic
anisotropy and damping to switch the direction of the magnetization (Berger, 1996;
Slonczewski, 1996).
    The spin transfer can be understood by analogy with the Andreev scattering
at normal|superconducting interfaces (Beenakker, 1997). This is illustrated by the
spin-transfer scattering process depicted in Fig. 2.2. Consider a spin-accumulation
in the normal metal (that was injected optically or electrically by other contacts),
polarized at right angles to the magnetization of the ferromagnet. No electric
voltage or charge current bias is required at this stage. In the idealized case of a
halfmetallic ferromagnet with an electronic structure that is perfectly matched to
that of the normal metal for one spin direction, a straightforward angular momen-
tum balance of incoming and scattering states shows that an incoming up-spin is
flipped during reflection. The electrons therefore lose not only linear momentum
(twice the electron wave vector normal to the interface when scattering at a specular
barrier) but also angular momentum of 2 × h . The (transverse) angular momen-
                                                ¯
                                                2
      ¯
tum h is conserved, however. Just as the linear momentum is absorbed by a wall
from which a soccer ball bounces, the electron angular momentum change is trans-
ferred to the ferromagnet. This spin-flip reflection process is equivalent to a spin
current that flows from the normal metal into the ferromagnet, where it is absorbed
and exerts a torque on the magnetization. The Andreev analogy becomes clear by
interpreting the ferromagnet as a condensate of angular momentum, just as a super-
Magnetic Nanostructures: Currents and Dynamics                                           133




Figure 2.2 Illustration of the magnetization torque exerted by a spin current. The magneti-
zation m of the ferromagnet is polarized at right angles to a spin accumulation in the normal
metal. For metallic interfaces, the “anomalous” spin-flip scattering illustrated in this figure
                                                                     ¯
dominates the spin-conserving scattering. The angular momentum h lost during this spin-flip
is transferred as a torque to the magnetization order parameter.


conductor is a condensate of charge. The magnitude of the spin transfer is obviously
proportional to the spin accumulation in the normal metal. Furthermore, for an
ideal interface between metals whose bands match perfectly for one spin direction,
each reflected electron is flipped with probability unity. The maximum spin current
absorbed (and thus the maximum magnetization torque) is then N(μ↑ – μ↑ )/4π,
proportional to the number of scattering channels in the normal metal.
    In more realistic interface models the qualitative picture remains the same, but
a few corrections should be taken into account. The transverse spin accumulation
is not absorbed on an exponential length scale, but by a destructive interference
process that leads to an algebraic decay (Stiles and Zangwill, 2002). The penetration
depth is still on an atomic length scale, viz. the magnetic coherence length
                                              π
                                    λc = F           ,                             (9)
                                          |k↑ – k↓ |
                                                  F

         F
where k↑(↓) are characteristic Fermi wave numbers for majority and minority spin
electrons. Spin conserving scattering at the interface, which becomes important in
the case of a potential barrier between N and F, reduces the spin transfer efficiency.
In the presence of conventional scattering processes, the number of channels N
must be replaced by the spin-mixing conductance (8) divided by e2 /h. Re G↑↓ can
therefore be interpreted as the material parameter governing spin transfer. First-
principles calculations predict (Xia et al., 2002; Zwierzycki et al., 2005) a small
imaginary contribution that corresponds to a spin transfer torque in the direction
normal to the plane in Fig. 2.2. It can be interpreted as an interface-generated
non-local exchange field.
    The general equations for transverse and longitudinal spin and charge currents
for arbitrary angles between spin accumulation and magnetization are a bit more
complicated (Brataas et al., 2000). A complete theory of the spin-accumulation
induced magnetization torque requires a quantitative treatment of the interface scat-
tering but also a description of the whole device that allows computation of the spin
accumulations. The magnetoelectronic circuit theory mentioned above (Brataas et
134                                                                             G.E.W. Bauer et al.



Table 2.1 Spin-dependent interface resistances ARs = A/Gs in units of (f m2 ), for a number
of commonly studied interfaces as calculated from first principles (Bauer et al., 2002). The drift
correction is included (Schep et al., 1997)

Interface                Roughness                        ARmaj                  ARmin
Au/Ag(111)               Clean                            0.094                  0.094
Au/Ag(111)               2 layers 50-50 alloy             0.118                  0.118
Au/Ag(111)               Exp.                             0.100 ± 0.008          0.100 ± 0.008
                         (Bass and Pratt, 1999)
Cu/Co(100)               Clean                            0.33                   1.79
Cu/Cohcp (111)           Clean                            0.60                   2.24
Cu/Co(111)               Clean                            0.39                   1.46
Cu/Co(111)               2 layers 50-50 alloy             0.41                   1.82 ± 0.03
Cu/Co(111)               Exp.                             0.26 ± 0.06            1.84 ± 0.14
                         (Bass and Pratt, 1999)
Cr/Fe(100)               Clean                            2.82                   0.50
Cr/Fe(100)               2 layers 50-50 alloy             0.99                   0.50
Cr/Fe(110)               Clean                            2.74                   1.05
Cr/Fe(110)               2 layers 50-50 alloy             2.05                   1.10
Cr/Fe(110)               Exp.                             2.7 ± 0.4              0.5 ± 0.2
                         (Zambano et al., 2002)



al., 2000) contains all of the necessary ingredients. The first microscopic treatment
explicitly addressing the spin torque in diffuse systems is the random matrix theory
by Waintal et al. (2000). We later showed that the theories are completely equivalent
for not too transparent interfaces (Bauer et al., 2003b), and can be generalized to
arbitrary circuits. An example of the application of this circuit theory to a complex
device is given by Bauer et al. (2003a).

3.4 Angular magnetoresistance of spin valves
The perpendicular F|N|F spin valve is the prototype magnetic structure in which
effects of the spin transfer torque can be observed. The (in theory) most simple
observable is the angular magnetoresistance, i.e. the linear electrical resistance as a
function of the angle θ between the magnetizations. For structurally symmetric spin
valves
                             G      p2      tan2 θ /2
                   G(θ ) =     1–                                 = G(–θ ),                  (10)
                             2    1 + gsf tan2 θ /2 + ζ
where
                                     (Re η + gsf )2 + (Im η)2
                               ζ =                            .                              (11)
                                      (1 + gsf )(Re η + gsf )
Magnetic Nanostructures: Currents and Dynamics                                           135


Table 2.2 Interface conductances in units of 1015   –1 m–2 (Brataas et al., 2006a). The drift
correction is included (Bauer et al., 2003b)

System                 Interface           G↑       G↓            Re G↑↓            Im G↑↓
Au|Fe (001)            clean               0.40     0.08          0.466               0.005
                       alloy               0.39     0.18          0.462               0.003
Cu|Co (111)            clean               0.42     0.38          0.546              0.015
                       alloy               0.42     0.33          0.564             –0.042
Cr|Fe (001)            clean               0.14     0.36          0.623               0.050
                       alloy               0.26     0.34          0.610               0.052



Here G = G↑ + G↓ is the total conductance of one contact, p = (G↑ – G↓ )/G
the polarization, η = 2G↑↓ /(G↑ + G↓ ) the relative mixing conductance and gsf =
2gsf /G the relative ‘spin-flip conductance’ gsf = De2 /(2τsf ) (Brataas et al., 1999)
(D is the energy density of states). Equation (10) deviates from a simple 1 – cos θ
dependence that reflects the projection of the two spin directions as expected for
a tunnel barrier (Slonczewski, 1989). The additional dissipation of the injected
spin accumulation at θ = 0 leads to an increase of the current for a voltage-based
structure relative to the cosine form. In asymmetric spin valves this may lead to
non-monotonic magnetoresistances and a sign change of the spin-transfer torque
(Manschot et al., 2004a). By engineering the asymmetry, an enhancement of the
spin-transfer torque can be achieved as well (Mancoff et al., 2004b). Using a single
parameter fit to the experimental angular magnetoresistance (Urazhdin et al., 2005),
Kovalev et al. (2006) found a mixing conductance for Cu|Py quite close to those in
Table 2.2.



      4. Non-Local Magnetization Dynamics
       In magnetic multilayer structures an applied current induces a spin transfer
torque on the ferromagnets that may excite the low-energy degrees of freedom
of a magnet and turn the static problem discussed above into a time-dependent
one (Bazaliy et al., 1998; Berger, 1996; Slonczewski, 1996). After being spin-
polarized by passing through a static ferromagnet a dc current can excite spin
waves, precessional and more complicated motions, and at a critical current, even
completely reverse the magnetization direction. The predictions have by now been
amply confirmed by many experiments (Katine et al., 2000; Myers et al., 1999;
Tsoi et al., 1998; Wegrowe et al., 1999) (see Stiles and Miltat, 2006). A reso-
nant finite-wavevector spin-wave excitation can be excited by an applied dc bias
even in a single ferromagnetic layer (Ji et al., 2003; Polianski and Brouwer, 2004;
Stiles et al., 2004; Özyilmaz et al., 2004). The current-induced magnetization dy-
namics are interesting from a fundamental-physics perspective, requiring a grasp of
136                                                                     G.E.W. Bauer et al.



the nontrivial coupling of non-equilibrium quasiparticles with the collective mag-
netization dynamics. It furthermore carries technological potential as an efficient
mechanism to write information into magnetic random-access memories and to
generate microwaves (Kiselev et al., 2003; Rippard et al., 2004).

4.1 Current-induced magnetization dynamics
Slonczewski’s spin transfer (Slonczewski, 1996) arises when a spin-current with a
polarization-component perpendicular to the magnetization is absorbed by the fer-
romagnet. Disregarding spin-orbit coupling (other than the crystal field anisotropy
in Heff ) and other spin-flip processes, the total spin angular momentum is conserved.
The angular momentum lost is transferred from the electron current to the coher-
ent magnetization of the ferromagnet contributing to the magnetization equation
of motion as a torque
                                τ torque = m × (Is × m),                             (12)
where Is is the spin current into the ferromagnet. To a very good approximation,
the parallel component of Is does not affect the magnetization dynamics in transi-
tion metals since changes in the modulus of the magnetization is energetically very
expensive. This is taken into account by the double outer vector product that only
projects out the component of Is normal to the magnetization. In Section 3.3 we
explained that the spin transfer torque can be written in terms of the spin accumu-
lation s next to the interface and the real part of the spin-mixing conductance
                                   h¯
                        τ torque =    Re G↑↓ m × (s × m).                    (13)
                                  2e2
The dynamics of a monodomain ferromagnet of volume V and magnetization Ms
that is subject to the torque (12) is modified by an additional source term on the
right-hand side of the LLG equation (3) (Slonczewski, 1996):
                          ∂m                  γ
                                         =        m × (Is × m).                      (14)
                           ∂t   torque       Ms V
For a fixed current density, Eq. (14) is proportional to the cross section of the
interface through the total spin current Is , and inversely proportional to the volume
V of the ferromagnet. In layered structures this term therefore scales with the inverse
of the ferromagnetic film thickness. The spin current Is is conveniently generated
in perpendicular spin valves, i.e., current or voltage-biased F|N|F structures. In
symmetric devices the spin transfer torques on both sides of the normal metal layer
are identical in size and direction. The simplest solution in the presence of a current
bias is then a constant rotation of both magnetizations. Usually one is interested
in a situation in which the magnetizations move relative to each other. This can
be achieved in asymmetric spin valves, in which the torque on one ferromagnet
is suppressed by making it much thicker (a larger V in Eq. (14)). This magnet
then behaves like a constant spin-current source, or polarizer, whereas the second,
thinner dynamic ferromagnet is a magnetically “soft” analyzer easily excitable by
the spin torque.
Magnetic Nanostructures: Currents and Dynamics                                        137




Figure 2.3 (a) The spin transfer torque is a spin-current that drives magnetization motion
somewhat like wind drives a windmill. (b) A moving magnetization excites a spin current
when in contact with a conductor, in analogy with a moving fan that causes a flow of air.


4.2 Spin pumping
When seeking a consistent theory of the magnetization dynamics in hybrid struc-
tures, the current-induced magnetization torque discussed above is only one side of
the coin: a time-varying magnetization of a ferromagnet that is in electrical contact
with normal metals emits (“pumps”) a spin current into its nonmagnetic environ-
ment (Tserkovnyak et al., 2002) as illustrated in Fig. 2.3b. Clearly, the magnetization
motion will be determined by the competition between the driving and the brak-
ing torques. The loss of angular momentum due to spin pumping enters as another
                   ˙
contribution to m, i.e. as an additional source term to the LLG equation. When
the magnetization dynamics are induced by external magnetic fields the process is
effective already in the absence of externally applied currents. When a bias is ap-
plied, the spin-pumping term is typically of the same order as the magnetization
torque (14) and should be treated on the same footing, although it appears to not
affect the current driven magnetization dynamics very strongly (Fuchs et al., 2007).
     Water streams through an elastic tube without external pressure by external
“peristaltic” modulation (the tube is periodically squeezed, out of phase, at two
different points). Similarly, electrons can be pumped through a scattering region
when externally applied gates are activated by time-dependent signals that are out of
phase. Such charge pumping is used in so-called single electron turnstiles. Formally,
the effect can be expressed in terms of time-dependent scattering matrices that
modulate the phases of transmitted and reflected electrons so that the net current
in the leads does not vanish (Brouwer, 1998; Büttiker et al., 1994). The parametric
spin-pumping formalism (Tserkovnyak et al., 2002), based on the low-frequency
limit of the scattering theory of transport is a general, versatile and practical method
to compute spin pumping and combine it with the spin-transfer torque as a source
term for the time-dependent circuit theory in the adiabatic limit.
     The reflection of an electron with given spin at a normal metal|ferromagnet in-
terface depends on the magnetization direction, implying that the scattering matrix
is time dependent when the magnetization varies. When the magnetization pre-
cesses with frequency ω, a spin-up electron incident on the interface has a chance
                                 ¯
to pick up an energy quantum hω by reflecting into an unoccupied spin-down state.
By this process the ferromagnet continuously loses angular momentum and energy
to the normal metal, thus “pumping” spins. A detailed analysis (Tserkovnyak et al.,
138                                                                                    G.E.W. Bauer et al.



2002, 2005) reveals that the pumped spin current obeys the following equation:
            ∂m                 γ
                         =–        Is,pump                                                          (15)
             ∂t   pump        Ms V
                              γ h  ¯            dm          dm
                         =           Re G↑↓ m ×    + Im G↑↓    .                                    (16)
                             Ms V 4π            dt          dt
Here we indicated that the (negative of the) pumped spin current is nothing but
a torque on the ferromagnet, i.e. an additional source term in the LLG equation
just as the spin-transfer torque, Eq. (14). The torque proportional to Re G↑↓ is
dissipative and has the same functional form as the Gilbert damping in Eq. (3). Its
effect can therefore be described by the modified Gilbert parameter:
                                                        γ h  ¯
                         α = α 0 + α = α0 +                    Re G↑↓ .                             (17)
                                                       Ms V 4π
Im G↑↓ acts like an effective magnetic field that can be taken into account by a
renormalized gyromagnetic ratio. For intermetallic interfaces (Zwierzycki et al.,
2005) this term can usually be disregarded (Table 2.2).
     When the pumped spin angular momentum is not dissipated sufficiently quickly
to the normal-metal lattice, a spin accumulation builds up. However, as discussed in
Section 3.3, the spin accumulation in proximity to a ferromagnet creates a reaction
spin-transfer torque in the form of a transverse-spin backflow into the ferromag-
net. In hybrid structures the magnetization dynamics and the non-equilibrium
spin-polarized currents are clearly mutually interdependent. The conversion of
magnetization movement into spin currents and vice versa at a possibly different
location characterizes the “nonlocality” of the magnetization dynamics.
     Spin pumping can be also understood in terms of the linear response of the
electron gas to a time-dependent magnetic perturbation (Barnes, 1974; Monod et
al., 1972; Šimánek and Heinrich, 2003). Assume that a magnetic impurity at the
origin in a normal metal perturbs the system with a localized and time-dependent
vector exchange potential Vx (t) = x m(t). The effective field due to the induced
non-equilibrium spin density is then given in terms of the linear-response magnetic
susceptibility of the metal
                                             i
                         χsa sa (r, t) =       (t) [sa (r, t), sa (0, 0)] ,                         (18)
                                             ¯
                                             h
where (t) is the Heaviside step function, [· · ·] the commutator and · · · the
ground state expectation value, as
                                                 ∞
               δHimp (r, t) =       x                   ˆ
                                                     dt aχsa sa (r, t – t )δma (t ),                (19)
                                        aa      –∞


                                                                   ˆ
where a, a ∈ {x, y, z} are the indices of the Cartesian axes and a stands for the
corresponding unit vectors. The leading terms for a slow and small-angle precession
Magnetic Nanostructures: Currents and Dynamics                                                139


in a uniaxial system are (Mills, 2003):
                                                              dm                       dm
                –γ m × δHeff (t) = γ Ms            2
                                                   x      1      +        2m       ×      ,   (20)
                                                              dt                       dt
where
                                                 dχsx sy (ω)
                                     1   = –i                        ,
                                                    dω         ω=0
                                                 dχsx sx (ω)
                                     2   = –i                                                 (21)
                                                    dω         ω=0

are real numbers. We see that            1   is an effective field term that modifies the gyro-
magnetic ratio γ in Eq. (3) and              2 is a Gilbert-like damping parameter


                                                       2 dχsx sx (ω)
                             αeff = –iγeff Ms          x                       .              (22)
                                                              dω         ω=0

When a precessing bulk ferromagnet is in contact with a normal metal, the spin
pumping can be explained in the same way (Mills, 2003; Šimánek, 2003). This
picture is physically appealing but cumbersome for material-specific calculations
including disorder as compared to the scattering theory approach (Zwierzycki et
al., 2005).



      5. The Standard Model
      Combining the charge and spin coupling mediated by the spin transfer torque
and spin pumping as explained above leads to “the standard model” for the charge
and magnetization dynamics in magnetic nanostructures in which many phenomena
can be discussed qualitatively and sometimes even quantitatively. The model is based
on the macrospin model dynamics (3) for each magnetic element with additional
surface torques due to spin pumping (15) and spin-transfer (14), where the latter
is governed by the spin accumulation close to the F|N interfaces. Circuit theory
(or any other semiclassical approach) can be used to compute the instantaneous
spin accumulation for a parametrically constant magnetization configuration that
changes slowly (adiabatically) compared to the electronic motion.
    In the following we briefly discuss some of the effects which can be understood
with this standard model without going into details or fully representing the quite
extensive literature on these topics, for which we refer to the technical reviews.

5.1 Enhanced Gilbert damping and spin battery
The loss of energy and momentum of a magnetization varying in time effectively
increases the damping with a functional dependence that is identical to that of the
Gilbert phenomenology as explained in Section 4.2. However, when the ejected
140                                                                    G.E.W. Bauer et al.



spins are not channelled away, a spin accumulation builds up close to the interface
that in the steady state exactly cancels the spin pumping term. The dissipation of
a spin accumulation in a normal metal layer is strongly material dependent; Pt is
a strong spin-flip scatterer and even a few monolayers act as an efficient spin sink,
whereas the spin lifetime in high-purity copper is very long. The magnetization
damping in thin magnetic films can therefore be engineered by covering them with
different normal metal films; α does not change when the ferromagnet is covered by
Cu, whereas a maximized enhancement of the damping Eq. (17) is achieved with
Pt. The excess enhancement due to spin pumping has been observed in several
FMR experiments, starting with (Mizukami et al., 2001, 2002) on N|F|N trilayers.
A second ferromagnet as in F|N|F spin valves can also be a good spin sink, thus
enhancing the damping, as discussed in Section 5.3.
    When the normal metal in an F|N bilayer is a weak spin-flip scatterer, a sig-
nificant spin accumulation suppresses the excess Gilbert damping. This spin ac-
cumulation can be considered as a source of spin motive force generated by the
magnetization dynamics (Brataas et al., 2002). The spin-battery spin-voltage can be
probed non-invasively by a magnetic tunnel junction attached to the normal metal
at a distance not exceeding the spin-flip diffusion length. In the presence of spin-
flip scattering in the ferromagnet, part of the spin accumulation can return into the
ferromagnet as a spin current polarized parallel to the magnetization. Due to differ-
ent resistances in both spin channels the spin current is transformed into a voltage
signal (Wang et al., 2006) that has recently been measured (Costache et al., 2006).

5.2 Current-induced magnetization reversal and high frequency
    generation
As indicated in the introduction, the most important consequence of the spin trans-
fer torque is the possibility to reverse the magnetization of a free layer relative to
that of a fixed layer (and back again) when the applied electrical current exceeds a
critical value. The experiments have by now become rather standard and the effect
has already been utilized in prototype memory devices (Hosomi et al., 2005). When
the magnetization becomes frustrated by conflicting spin-transfer and applied mag-
netic field torques, stable oscillations can be generated that, by means of the GMR,
cause periodic resistance oscillations in the GHz regime, that are tunable by e.g. the
applied current bias (Kiselev et al., 2003).
    The standard model appears to represent the experiment better than one might
have expected considering the very high current densities. A reliable quantitative
modelling of the effect is difficult and presumably requires a consistent treat-
ment of the spatially varying magnetization and (spin) current distributions. The
excess Gilbert damping is relevant since the free layer is usually very thin and be-
comes dependent on the instantaneous configuration during the reversal process
(Tserkovnyak et al., 2003). The competition between homogeneous (macrospin)
and spatially dependent (spin wave) excitations at criticality has been studied by
Brataas et al. (2006b). As in the case of current-induced excitation of a single
ferromagnetic layer (Polianski and Brouwer, 2004) it is essential to treat the in-
homogeneity of the magnetization damping selfconsistently with that of the spin
Magnetic Nanostructures: Currents and Dynamics                                              141




Figure 2.4 Single (left) and collective (right) ferromagnetic resonance in F1|NM|F2 spin valves
under a static applied field HDC and an oscillating field hRF . The collective resonance can
occur for different resonance frequencies of the isolated layers as a consequence of the dynamic
coupling, as explained in the text.


current dynamics. The challenge of integrating detailed micromagnetic modelling
with a realistic description of the electron transport for magnetic heterostructures
remains.

5.3 Dynamic exchange interaction
The combination of spin-pumping current and spin-transfer torque induces a dy-
namic crosstalk between moving magnetizations in magnetic heterostructures. This
has been studied in quite some detail for the FMR of planar F|N|F spin valves
(Tserkovnyak et al., 2005). The physics observed in experiments by Heinrich et al.
(2003) can be understood intuitively as follows.
     In Fig. 2.4(left), layer F1 is resonantly exited and its precessing magnetic moment
acts as a spin pump which creates a spin current propagating away from the F1|NM
interface in the direction of the NM|F2 interface (dotted blue arrow in NM). The
purple arrow indicates the instantaneous direction of the spin angular momentum
of the spin current. Layer F2 is assumed to be detuned from its FMR, and there-
fore its rf magnetization component is negligible compared to that in the layer F1.
A darker green area in F2 around the NM|F2 interface represents the region (of
thickness λc < 1 nm) in which the transverse spin momentum is absorbed by the F2
layer. F2 acts as a spin sink since the transverse momentum from the spin current
is transformed into a magnetization torque for the layer F2. The complete absorp-
tion of the spin current by F2 causes an additional damping of the magnetization
dynamics of F1 as discussed above.
     In the right drawing, F1 and F2 resonate at the same external field. In this
case both layers act as spin pumps and spin sinks. Consequently, the spin cur-
rents in the NM layer propagate towards both the NM|F2 and NM|F1 inter-
faces (see blue dotted arrows). An additional magnetic damping in F1 and F2
vanishes with the net spin flow through the interfaces. A dynamic locking be-
tween the magnetization dynamics has been observed and computed even when
the FMR resonance frequencies of F1 and F2 are somewhat detuned. This phe-
nomenon should be distinguished from the dynamic locking of magnetization
dynamics by spin waves in ferromagnets and AC charge currents (Kaka et al., 2005;
Mancoff et al., 2005).
     The mathematics of the coupled LLG equations for this system has been worked
out by Tserkovnyak et al. (2005), reproducing the measured line width narrowing
142                                                                     G.E.W. Bauer et al.



and frequency locking close to the common resonance for parameters determined
at off-resonance conditions.

5.4 Noise in magnetic heterostructures
Noise in magnetic devices can seriously limit their performance. The increased
noise in spin valve read heads was ascribed to the spin torque by Covington et al.
(2004). Indeed, both spin pumping and spin-transfer torque have strong effects on
the noise in magnetic nanostructures.
     Noise in ferromagnets such as the Barkhausen noise due to discrete domain
reorientation, has a long history. In nano-particles at finite temperature T the
magnetization fluctuates as a whole, due to thermal fluctuations in the effective
magnetic field h(t). These magnetization fluctuations are intimately related to the
magnetization damping by the fluctuation dissipation theorem (Brown, 1963):
                                                  α0
                      h(0) (t)h(0) (t ) = 2kB T
                       i       j                       δij δ(t – t ).              (23)
                                                γ Ms V
Foros et al. (2005) showed that the enhanced damping due to spin pumping is asso-
ciated with enhanced fluctuating magnetic fields caused by transverse spin current
fluctuations in the normal metal spin-sink that exert a stochastic spin-transfer torque
on the ferromagnet. The enhanced thermal magnetization noise has a white cor-
relation function identical to Eq. (23) with α0 replaced by the increased damping
α, see Eq. (17). In spin valves, the dynamic coupling by the exchange of fluctuat-
ing transverse spin currents depends on the configuration. The total magnetization
noise turns out to be larger in the antiparallel than parallel configuration (Foros et
al., 2007).



      6. Related Topics
      This article focuses on the spin and charge current and magnetization dynam-
ics of magnetic multilayers and spin valves. The general concepts are relevant for a
number of related phenomena, a few of which are very briefly introduced below.

6.1 Tunnel junctions
The spin-transfer torque through tunnel junctions has been addressed theoretically
by Slonczewski (2005). Especially since the discovery of single-crystalline MgO
as a superior tunneling barrier material (Parkin et al., 2004; Yuasa et al., 2004)
this is an important technological issue for MRAM applications (Hosomi et al.,
2005). The physics is quite similar to that of the spin-transfer torque in spin valves,
but without the complications of the spin accumulation in the normal metal. The
spin transfer torque and angular magnetoresistance are therefore well described by
simple geometric functions governed by the scalar product of the two magnetization
vectors (Slonczewski, 2005). A significant effective field contribution to the torque
Magnetic Nanostructures: Currents and Dynamics                                     143


has been found (Tulapurkar et al., 2005). Even relatively small currents induce large
voltage drops over tunnel junctions so “hot” electrons may become important (Levy
and Fert, 2006; Theodonis et al., 2006).

6.2 Domain walls
A magnetic domain wall is the region between two magnetic domains whose
magnetizations point in different directions. The study of current-driven domain
walls in thin film nanostructures is a very lively field in recent years. By apply-
ing a magnetic field, the energy of one domain can be lowered with respect to
the other, leading to a thermodynamic force that can accelerate the domain walls
up to considerable velocities. An electric current sent through a ferromagnetic
wire can also move the domain wall by the spin transfer torque (Berger, 1984;
Tatara and Kohno, 2004; Yamaguchi et al., 2004). Many experiments have since
confirmed this picture for ferromagnetic metal wires, but domain walls in di-
luted magnetic semiconductor (Ga,Mn)As can also be moved by electric currents
(Yamanouchi et al., 2006, 2004). Parkin (2004) has suggested a novel memory con-
cept in the form of ferromagnetic wire loops in which domain walls are collectively
moved by current pulses between pinning sites in a nanoscale shift register.
     According to the two-spin-channel model, an electric current in a metallic fer-
romagnet is polarized since the lower resistance spin channel carries a larger current
than the higher resistance one. When the current is passed through a domain wall,
the spin current has to accommodate its polarization to it. The angular momentum
lost by the electrons in this process is transferred to the magnetization in the domain
wall region. The strong exchange interaction in transition metals renders the mag-
netization very stiff so that domain walls can extend over hundreds of nanometers.
It is then safe to assume that the transfer of angular momentum occurs adiabatically,
which leads to the simple expression for the local transfer torque
                              ∂m(r,t)                 h¯
                                                  ≈      P (j · ∇r )m,            (24)
                                ∂t       torque       2e
where j is the charge current vector and P = (σ↑ – σ↓ )/(σ↑ + σ↓ ) its polarization
in terms of the conductivities σs . For sufficiently large currents the torque will
set the domain wall into motion. When the domain wall has very large gradients
(Tatara and Kohno, 2004) or in the presence of spin-flip scattering (Zhang and
Li, 2004), a “non-adiabatic” torque arises with a vector component normal to
Eq. (24). There is still some controversy concerning the correct equation of motion
for the magnetization in current-driven ferromagnets, in the presence of realistic
spin dephasing processes (due to spin-orbit interaction and/or magnetic disorder)
that correspond to a leakage of angular momentum into the lattice. In particular,
there are still some loose ends in understanding basic quantities like the critical
depinning current and domain-wall terminal velocity (Barnes and Maekawa, 2005).
Microscopic first-principles calculations should help to understand the origin of the
non-adiabatic torque and the role of Gilbert damping in current-induced domain
wall motion (Kohno et al., 2006; Skadsem et al., 2007; Tserkovnyak et al., 2006).
144                                                                            G.E.W. Bauer et al.



6.3 Spin transport by thermal currents
Increasing data storage density and access rate is a continuing challenge for the mag-
netic recording industry. The relatively high-current densities and voltages that are
required to operate magnetic random access memories will cause problems in fur-
ther downsizing device dimensions due to heating effects that complicate modeling
and deteriorate device stability and lifetime. Controlled heating can also be bene-
ficial as, for example, in the case of recording by thermally assisted magnetization
reversal. In magnetic nanostructures Peltier effects have been reported (Fukushima
et al., 2006; Gravier et al., 2006). Recently Hatami et al. (2007) predicted a
strong coupling of thermoelectric spin and charge transport with the magnetiza-
tion dynamics in nanoscale magnetic structures. The thermal spin-transfer torque is
believed large enough to realize a magnetization reversal by pure heat currents.



      7. Outlook

      The dynamics of magnetic nanostructures in the presence of currents is a fast-
moving field. It attracts many researchers since it combines interesting physics with
immediate practical relevance. We expect that the trends to ever small structures
will continue for some time. Topics to watch are
 (i) Semiconductor spintronics. This field has the promise of integrating the metal-
      based magnetoelectronics with semiconductor microelectronics. Very recently
      important progress has been made by demonstrating non-local electrical detec-
      tion of electrically injected spin currents in semiconductors (Lou et al., 2007).
      Unique to semiconductors are the optical creation and detection of spins.
      Current-induced spin accumulations can be created in the presence of spin-
      orbit interaction simply by an applied bias (Edelstein, 1990; Inoue et al., 2003;
      Kato et al., 2004).
(ii) Molecular spintronics. High effective spin-transfer fields have been observed
      in C60 (Pasupathy et al., 2004). The magnetoresistance observed in spin
      valves with single-wall carbon nanotubes (Sahoo et al., 2005) and single-sheet
      graphene (Hill et al., 2006) is very promising.
(iii) Magnetic nanoelectromechanical systems. The combination of small mechanical
      structures such as cantilevers with ferromagnetic particles have been employed
      to detect single spins by magnetic resonance microscopy (Rugar et al., 2004).
      Adding currents to such systems could lead to, e.g., current driven spin-transfer
      nanomotors (Kovalev et al., 2007).


ACKNOWLEDGEMENTS
We acknowledge the collaboration and helpful discussions with A. Kovalev, M. Zwiercycki, K. Xia,
M. Manschot, X. Wang, B. van Wees, J. Foros, H.-J. Skadsem, D. Huertas-Hernanod, Yu. Nazarov,
Magnetic Nanostructures: Currents and Dynamics                                                    145


B.I. Halperin, and A. Hoffmann. This work has been supported by NanoNed and the EC Con-
tracts IST-033749 “DynaMax” and NMP-505587-1 “SFINX”. G.E.W.B. and Y.T. are grateful for the
hospitality of the Centre of Advanced Study, Oslo.



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        CHAPTER             THREE



        Theory of Crystal-Field Effects in 3d-4f
        Intermetallic Compounds
        M.D. Kuz’min * and A.M. Tishin **




        Contents
        Foreword                                                                                               149
        1. Formal Description of the Crystal Field on Rare Earths                                              150
           1.1 The single-ion approximation                                                                    150
           1.2 Equivalent operator techniques for various subspaces: 4fN configuration,
                LS term, J multiplet, Kramers doublet                                                         155
           1.3 Local symmetry and the exact form of HCF ˆ                                                     161
        2. The Single-Ion Anisotropy Model for 3d-4f Intermetallic Compounds                                  166
           2.1 Macroscopic description of magnetic anisotropy                                                 166
           2.2 The notion of an exchange-dominated RE system                                                  170
           2.3 The single-ion model for 3d-4f intermetallics                                                  176
           2.4 The high-temperature approximation                                                              177
           2.5 The linear-in-CF approximation: main relations                                                 183
           2.6 Properties of generalised Brillouin functions                                                  188
           2.7 The linear-in-CF approximation (continued)                                                     193
           2.8 The low-temperature approximation                                                              199
           2.9 J -mixing made simple                                                                          203
        3. Spin Reorientation Transitions                                                                     210
           3.1 General remarks                                                                                210
           3.2 SRTs in uniaxial magnets                                                                       214
           3.3 Spontaneous SRTs in cubic magnets                                                              222
        4. Conclusion                                                                                         228
        References                                                                                            229




        Foreword
    Magnetic properties of 3d-4f intermetallic compounds have been reviewed
on numerous occasions in recent times (Kirchmayr and Burzo, 1990; Franse
   *   Leibniz-Institut für Festkörper- und Werkstofforschung, Postfach 270116, D-01171 Dresden, Germany
  **   Department of Physics, M.V. Lomonosov Moscow State University, 119992 Moscow, Russia

Handbook of Magnetic Materials, edited by K.H.J. Buschow                                       © 2008 Elsevier B.V.
Volume 17 ISSN 1567-2719 DOI 10.1016/S1567-2719(07)17003-7                                      All rights reserved.



                                                                                                               149
150                                                          M.D. Kuz’min and A.M. Tishin



              n
and Radwa´ ski, 1993; Liu et al., 1994; Andreev, 1995; Buschow and de Boer,
2003). Especially extensive is the literature on magnetically hard materials in
general (Buschow, 1991; Coey, 1996; Skomski and Coey, 1999) as well as on
specific classes of such materials (Buschow, 1988; Strnat, 1988; Herbst, 1991;
Li and Coey, 1991; Suski, 1996; Burzo, 1998), the magnetocrystalline anisotropy
playing invariably a central role. Somewhat apart stands the literature on experi-
mental (mainly by means of inelastic neutron scattering) studies of the crystal field
(CF) in intermetallics with ‘normal’ (Moze, 1998) and ‘anomalous’ (Loewenhaupt
and Fischer, 1993) rare earths. The separation between the subjects of magnetic
anisotropy and CF seems something artificial. Nowadays, when the single-ion
model has gained general recognition, little doubt remains about the indissol-
uble connection between the two phenomena. Perhaps the true cause of this
split is that theoretical activity in the area has been lagging the experiment ever
since the appearance of the last major review four decades ago (Callen and
Callen, 1966). Of course, theoretical advance on magnetic anisotropy and CF
did not cease in the meantime, it just took a different direction (Bruno, 1989;
Richter, 2001), stimulated by the advent of the density functional theory. As regards
the single-ion model proper, work on it proceeded at a rather slow pace. Nonethe-
less, a fair amount of new results has been published between the late 1960s (e.g.
Kazakov and Andreeva, 1970) and more recent times (Magnani et al., 2003).
    This Chapter is to review the progress in theory in the post-Callens era, filling
the gap in the literature between the anisotropy and the CF. We aim at reasserting
the statement that magnetocrystalline anisotropy is the most important manifesta-
tion of the CF.



      1. Formal Description of the Crystal Field on Rare
         Earths
      This section has an introductory character. We shall discuss the physical
foundations of the approach that enables quantitative treatment of CF effects in
RE-based hard magnetic materials—the single-ion approximation—and introduce
the basics of the mathematical apparatus required for that treatment. Admittedly,
this section contains mostly standard material, extensively covered in a number
of monographs published in the 1960–70s (Griffiths, 1961; Ballhausen, 1962;
Hutchings, 1964; Wybourne, 1965; Dieke, 1968; Abragam and Bleaney, 1970;
Al’tshuler and Kozyrev, 1974). Hence the brief style of our exposition. Like in
the rest of the Chapter, the approximate nature of the approach is emphasised and
the bounds of its validity are set out.

1.1 The single-ion approximation
The main experimental fact underlying the single-ion approach to 3d-4f inter-
metallics rich in 3d elements is the approximate non-interaction of the RE magnetic
moments therein. Of course, the single-ion model as such is not restricted to
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                        151




Figure 3.1 A two-dimensional view of a crystal of a 3d-4f intermetallic compound rich in the
3d element. The dark circles are the 3d atoms and the hatched areas are the RE ones.

intermetallic compounds (see e.g. Kolmakova et al., 1996), but it is in the 3d-4f in-
termetallics that it takes a particularly simple form. We shall therefore limit ourselves
to this special case. The non-interaction of RE moments should be understood as
relative weakness of the RE–RE exchange interaction in such compounds. That
is, if we consider a crystal consisting of atoms of 3d and 4f elements, as shown
schematically in Fig. 3.1, the 4f shell of each RE atom (hatched) interacts directly
(or rather, via the 5d states) only with the 3d electrons of the transition element, but
not with the 4f shells of neighbouring RE atoms. It is unimportant at this stage to
specify if the 3d electrons are regarded as localised, belonging to individual atoms,
or as itinerant electrons (dashed lines in Fig. 3.1).
     The weakness of the 4f-4f interaction should not be mistaken for its non-
existence or indetectability. Thus, the compound GdNi5 , where nickel is nonmag-
netic, orders ferromagnetically at TC = 32 K (Gignoux et al., 1976) entirely due to
the Gd–Gd exchange. The truth, however, is that in typical hard magnetic materials
the 4f-4f exchange coexists with much stronger 3d-3d and 3d-4f interactions and
that the former one is negligible in comparison with the latter two.
     This statement has been recently put to a direct test. The influence of the ex-
change on the 4f shell of Gd in GdCo5 was probed by two different techniques:
(i) by applying a very strong magnetic field that breaks the antiparallel orientation
of the Gd and Co sublattices (Kuz’min et al., 2004) and (ii) by inelastic neutron
scattering on the exchange-split 4f states (Loewenhaupt et al., 1994). In the first
case only the 3d-4f exchange is relevant, while in the second situation the 3d-4f
and the 4f-4f interactions produce a combined effect on the 4f shell. Now, the in-
tensity of the exchange interaction determined from these two kinds of experiment
turned out to be the same within the estimated uncertainty bounds. This is a direct
proof of the weakness of the Gd–Gd exchange in GdCo5 . In the other compounds
of the RECo5 series the 4f-4f interaction can be neglected with even more reason,
since its intensity decreases in proportion to the square of the total 4f spin.
     Another corner-stone of the single-ion approach is the weakness of the 3d-4f
interaction in comparison with the 3d-3d one. Hence, on the one hand, the 3d-
4f exchange is all-important for the 4f subsystem. On the other hand, its action
back on the 3d electrons, which are under the dominant influence of the intra-
sublattice 3d-3d exchange, is relatively insignificant. Once again, it does not mean
152                                                                 M.D. Kuz’min and A.M. Tishin



that the 3d electrons are completely unaware of the state of magnetisation of the
4f subsystem. The presence of a magnetic RE may result in a noticeable shift of
the Curie point in an intermetallic compound with a magnetic RE as compared to
its counterpart with Y or Lu. Insignificance in this instance means that the effect
of the 3d-4f exchange on the 3d sublattice can be reduced to a renormalisation
of the TC , without changing the dependence of the 3d sublattice magnetisation
M3d on reduced temperature, T /TC . The explicit form of M3d (T /TC ) is given by
Eq. (2.43) below.
    Thus, a peculiar hierarchy of exchange interactions takes place in 3d-4f hard
magnetic materials, which can be schematically expressed as (3d-3d)          (3d-4f)
(4f-4f) ≈ 0. This fact enables us to regard the 3d subsystem as something external
with respect to the RE, as something given, which orders magnetically largely
due to its internal forces. Then, from the viewpoint of the 4f electrons the 3d-
4f exchange can be presented as the action of an exchange field produced by the
ordered 3d subsystem. Therefore, the 4f subsystem can be regarded as a conjunction
of non-interacting RE atoms (ions) immersed in several fields: the 3d-4f exchange
field, applied magnetic field and the CF. The latter is not literally an electric field
in the crystal, so we shall avoid terming it “crystal electric field” as misleading.
Rather, CF is a combination of anisotropic time-even interactions involving the 4f
electrons, presented as a fictitious electrostatic potential seen by the 4f shell.
    The so formulated single-ion approximation is an enormous simplification: all
itinerant electron states have been eliminated and the attention has been concen-
trated on the 4f shell of one RE atom, which to a good approximation can be
considered localised. This explains the origin of the term “single-ion”, widely ap-
plied to non-ionic solids. Of course, there are no charged ions in metals. Just the
magnetic behaviour of RE’s in metallic systems is determined by the properties of
the ground configuration 4fN , which in most solids is the same as in trivalent RE
ions.
    Quantitatively this behaviour can be described by means of the following Hamil-
tonian:
                                                                           N
       ˆ     ˆ          ˆ               ˆ           ˆ    ˆ
      H4f = HCoulomb + Hs-o + 2μB Bex · S + μB B · (L + 2S) – e                  VCF (ri ).
                                                                           i=1
                                                                                 (1.1)
Here the first two terms describe the isotropic (Coulomb and spin-orbit) interac-
tions within the 4f shell; the third term presents the 3d-4f interaction by means of
                                               ˆ         ˆ
the exchange field Bex acting on the 4f spin, S = i=1 si ; the fourth one describes
                                                      N

the (Zeeman) effect of the applied magnetic field B. The last term in Eq. (1.1) con-
tains the CF potential VCF . Formally similar to an ordinary electrostatic potential,
VCF (r) is a function of coordinates in real space, which can be expanded over a
suitably chosen basis. It is convenient to use for the purpose spherical coordinates
                                                                       (n)
and to choose the so-called irreducible tensor operators (functions) Cm (θ , φ) as the
angular basis functions (Wybourne, 1965):
                                               n
                    VCF (r, θ , φ) =                          (n)
                                                      Vnm (r)Cm (θ , φ).                      (1.2)
                                       n=2,4,6 m=–n
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                              153


Following a long-standing tradition, one may assume in addition that this
anisotropic potential is created by electric charges situated entirely outside the re-
gion where the 4f shell is located. Then, within that region, VCF (r) must satisfy the
Laplace equation. The radial functions Vnm (r) would then turn into simple power
laws,
                                               Vnm (r) = Anm r n ,                                          (1.3)
where the coefficients are known as CF parameters. The approximation (1.3) is
neither physically justifiable nor really useful. However, it causes no formal difficul-
ties, as long as configuration mixing is neglected. Within the 4fN configuration the
radial functions reduce to their expectation values and Eq. (1.2) is equivalent to
                                           N             n
                                ˆ
                               HCF =                                  (n)
                                                                 Bnm Cm (θi , φi ).                         (1.4)
                                          i=1 n=2,4,6 m=–n

The quantities Bnm = –e Vnm (r) 4f = –eAnm r n 4f are also called CF parameters.
Generally speaking, Anm (as well as Bnm ) are complex numbers such that A* =      nm
                                                                          (n)
(–1)m An,–m . Indeed, the potential VCF must be a real quantity and Cm (θ , φ) are
complex-valued functions satisfying the condition [Cm (θ , φ)]* = (–1)m C–m (θ , φ)
                                                       (n)                    (n)

(Varshalovich et al., 1988). Terms with odd n’s have been omitted from the ex-
                                                              (n)
pansion (1.2), because for n odd all matrix elements of Cm (θ , φ) between any
                       1
two 4f orbitals are nil for parity reasons. Likewise, we have left out all terms with
n > 6, not complying with the triangle condition, n ≤ 2l. Finally, the isotropic
n = m = 0 term has been omitted as well; its only effect is shifting all energy levels
by the same amount, NB00 .
   For each n, the variable m may take 2n + 1 values, therefore the expansion (1.2)
may contain maximum 27 terms. Thus, there are 5 basis functions with n = 2:
                                           1
                               C0 (θ , φ) = (3 cos2 θ – 1),
                                (2)
                                           2
                                                             1
                                                  3          2
                                (2)
                               C±1 (θ , φ)     =∓                cos θ sin θ e±iφ ,                         (1.5)
                                                  2
                                                         1
                                                    3    2
                                (2)
                               C±2 (θ , φ)     =             sin2 θ e±2iφ ,
                                                    8
                            (n)
et cetera. Note that all C0 (θ , φ) possess cylindrical symmetry, in fact they are
just the nth -order Legendre polynomials of cos θ: C0 (θ , φ) = Pn (cos θ ). Another
                                                     (n)

simple particular case is C±n (θ , φ) = (∓1) [(2n – 1)!!/(2n)!!]1/2 sinn θ e±inφ .
                           (n)               n
                      (n)
    The choice of Cm (θ , φ) for the basis is by no means unique—any complete set
of functions can be used instead. For example, the spherical harmonics Ynm (θ , φ).
  1 This is not to say that A
                              nm must generally be nil for n odd. They genuinely vanish only when the RE occupies
a crystallographic site with a centre of inversion. But even when they are nonzero, these Anm have no effect on the
                                ˆ
eigenvalues or eigenvectors of H4f , as long as configuration interaction is neglected.
154                                                                                M.D. Kuz’min and A.M. Tishin



There is a simple connection between the two sets (Edmonds, 1957):
                                                                 1
                                             2n + 1              2
                           Ynm (θ , φ) =                              (n)
                                                                     Cm (θ , φ).                         (1.6)
                                               4π
So the five functions Y2,m (θ , φ) are obtained just by adding a prefactor (5/4π)1/2 to
Eqs. (1.5). Extensive collections of explicit expressions for the spherical harmonics
were compiled by Varshalovich et al. (1988; all Ynm with n ≤ 5) and by Görller-
Walrand and Binnemans (1996; n ≤ 7).
    An alternative basis set, favoured particularly by experimentalists, is the one
that uses the so-called Stevens normalisation (a term introduced by Newman and
                                                                           (n)
Ng, 1989). It is obtained by replacing the complex-valued functions Cm (θ , φ) by
their real combinations, ∝ Cm (θ , φ) ± C–m (θ , φ), and omitting all cumbersome
                                (n)           (n)

numerical prefactors. The result is simple-looking trigonometric expressions. Thus,
one gets instead of Eq. (1.5) the following 5 real functions:
                   O2 = 3 cos2 θ – 1,
                    0


                   O2 = cos θ sin θ cos φ,
                    1                                        1
                                                             2   = cos θ sin θ sin φ,                    (1.7)
                    2
                   O2   = sin θ cos 2φ,
                             2                               2
                                                             2   = sin θ sin 2φ.
                                                                         2

                                                         (n)
Also worth of mention are Racah’s unitary operators Um (Racah, 1942). The
                 (n)
connection to Cm (θ , φ) is provided by a factor which depends on n and also on
the orbital quantum number l:
                                                                             1
             .          [(n/2)!]2 (l – n/2)! (2l + n + 1)!                   2
       (n)
      Um     = (–1)n/2                                                            (n)
                                                                                 Cm (θ , φ),     n even.
                       n!(2l + 1)(l + n/2)!    (2l – n)!
                                                                                                         (1.8)
Within the 4f shell (l = 3) the corresponding relations are:
                                                     1
                                        .  15        2
                                  (2)
                                 Um     =–                (2)
                                                         Cm (θ , φ),
                                           28
                                                 1
                                        .   11   2
                                  (4)
                                 Um     =             (4)
                                                     Cm (θ , φ),                                         (1.9)
                                            14
                                                         1
                                        .  429           2
                                  (6)
                                 Um     =–                    (6)
                                                             Cm (θ , φ).
                                           700
    It should be emphasised that the introduced various sets of basis functions for
expanding VCF (r) differ only in normalisation. From the standpoint of the physical
approximations involved, all the above bases are absolutely equivalent and, if used
properly, must yield identical results. One should just be consistent with notation
and not allow indiscriminate use of CF parameters related to distinct normalisations.
To avoid confusion, it is advisable to write out expressions of type (1.2) explicitly
or at least to include references to such explicit expressions.
    The choice of a specific basis set is merely a matter of convenience. For ex-
                                                      (n)
ample, Wybourne’s irreducible tensor operators Cm (θ , φ) transform most simply
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                                      155


under rotations of the coordinate axes and are therefore particularly suited for seri-
ous theoretical work on magnetic anisotropy. The spherical harmonics Ynm (θ , φ) are
normalised to unity on a sphere, hence they are a natural choice for the angular part
of atomic wave functions (Condon and Shortley, 1935), being somewhat less con-
venient for expanding the CF potential. The ‘simple-looking’ basis functions in the
Stevens normalisation are useful only for very simple calculations in high-symmetry
cases. In particular, they are suitable for work in Cartesian coordinates (Hutchings,
                                                                        (n)
1964), which is seldom done nowadays. Finally, Racah’s operators Um possess the
important property of having unitary reduced matrix elements within any atomic
shell (hence their name). The significance of this can be learned e.g. from the book
of Wybourne (1965). In this Chapter we shall not deal with problems necessitating
              (n)                                                                  (n)
the use of Um or Ynm . So we shall use mostly the irreducible tensor operators Cm
                                                              m
(in various representations) and less often the operators On normalised according
to Stevens.

1.2 Equivalent operator techniques for various subspaces: 4fN
    configuration, LS term, J multiplet, Kramers doublet
Introducing equivalent operators in this section, we shall make a clear distinction
between the following two aspects of the method: (i) reduction of the dimen-
sionality of the space of available states at different stages of the method and the
underlying physical approximations; (ii) the (formally exact) algebraic techniques
facilitating the calculation of the matrix elements of VCF within those reduced sub-
                                                                             ˆ
spaces. The need and the possibility of the approximate treatment of H4f (1.1) are
due to the fact that its individual terms describe interactions of grossly disparate
intensities. On the other hand, to describe thermodynamic properties in a limited
                                                                           ˆ
temperature interval, we only require a few low-lying eigenstates of H4f (to about
                                                                        N
kT above the ground state), while the total number of states in a 4f configuration
can be a few thousand. From the point of view of physics, everything is determined
                                                                              ˆ
by the intensity relations between the individual terms entering in H4f . Several
distinct situations are possible, these are considered in subsections 1.2.1–1.2.5. Sub-
section 1.2.6 will then be devoted to the algebraic aspect of the method.

1.2.1 No restrictions2
                               ECoulomb ∼ Es-o ∼ Eex ∼ EZeeman ∼ ECF .                                            (1.10)
This is a trivial case included here for the sake of completeness. No new approxi-
mations are possible (apart from the already mentioned neglect of the configuration
                ˆ
interaction). H4f has to be diagonalised within the full 4fN configuration.

1.2.2 The Russel–Saunders approximation
                              ECoulomb          Es-o ∼ Eex ∼ EZeeman ∼ ECF .                                      (1.11)
 2  The energies in the symbolic relations (1.10)–(1.15) are of the order of the overall splitting of the relevant manifold
by the respective terms in Eq. (1.1). For example, Es-o    Eex can be understood as |λ|(2L + 1)         4μB Bex or even as
|λ|   μB Bex .
156                                                             M.D. Kuz’min and A.M. Tishin



The dominant Coulomb interaction is isotropic both in the coordinate and in the
spin spaces. Its eigenstates are (2L + 1)(2S + 1)-fold degenerate LS terms, i.e. sets
of states with given total orbital moment and total spin quantum numbers. The
construction of the |LML |SMS wave functions from one-electron orbitals |lm ↑,↓
is in principle a purely algebraic problem having an exact solution. The explicit ex-
pressions, however, may be extremely cumbersome and will not be required herein.
They have been described in full detail elsewhere (Sobel’man, 1972).
     The approximation here consists in restricting the space of states to those of the
ground LS term [subject to the usual Hund’s rules: S = 1 (2l + 1 – |2l + 1 – N|) and
                                                             2
                                            ˆ
L = S(2l +1–2S)]. Within that space HCoulomb reduces to a constant which will be
omitted. The remaining four terms of Eq. (1.1) are projected on the ground term
in the first approximation (i.e. their matrix elements on the |LML |SMS states are
computed) and diagonalised.
     Validity of the Russel–Saunders approximation is determined by the strong in-
equality ECoulomb     Es-o . Since it involves only intra-atomic interactions it has been
investigated in a rather exhaustive manner. Though generally inaccurate for the
RE’s, the Russel-Saunders approximation holds surprisingly well for their ground
LS terms, which are all more than 96% pure (Dieke, 1968). This suffices for our
purpose in this Chapter. In what follows we shall regard the Russel–Saunders ap-
proximation as valid in all cases.

1.2.3 The single-multiplet approximation (within the Russel–Saunders
      coupling scheme)
                      ECoulomb    Es-o     Eex ∼ EZeeman ∼ ECF .                    (1.12)
This is a particular case of the previous one. The added approximation here is the
one expressed by the inequality Es-o       max(Eex , EZeeman , ECF ). It is generally not
a very good approximation, particularly for the light RE’s and most notoriously
for samarium. On the other hand, this approximation is vital for the analytical
tractability of many important problems. Its validity is hard to estimate a priori,
since it depends on the characteristics of the solid, in particular on the relation
between Eex and ECF . We shall dedicate a special section (2.9) to the question of
validity of the single-J approximation in exchange-dominated systems (tentatively
defined by Eex       ECF ; see Section 2.2 for more a detailed definition).
                                                         ˆ              ˆ
    Technically the approximation is straightforward: HCoulomb + Hs-o reduces to a
                            ˆ     ˆ         ˆ
constant and is omitted; Hex + HZeeman + HCF is projected on the ground J manifold
comprising the 2J + 1 states of type |LSJ M , where J = L ± S (3rd Hund’s rule).
These are constructed from the states of the ground LS term according to the
following simple relation (Condon and Shortley, 1935):
                       |LSJ M =            CLM SM |LM |SM ,
                                            JM
                                                                                    (1.13)
                                    M ,M
        JM
where  CLM SM    are the so-called Clebsch–Gordan coefficients (CGC)—exactly
known functions of the quantum numbers J , M, L, M , S, M . The CGC have
been extensively tabulated (Varshalovich et al., 1988; Rotenberg et al., 1959). More
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                               157


information about the CGC can be found in Chapter 8 of Varshalovich et al. (1988),
                                    JM
including explicit expressions for CLM SM in the practically important particular
case of a ground multiplet obeying the 3rd Hund’s rule, J = L ± S.

1.2.4 CF-dominated RE systems

                            ECoulomb         Es-o       ECF        Eex ∼ EZeeman .                         (1.14)
A special case of the situation considered in the previous subsection. It has little
relevance to hard magnetic materials and is included here just for completeness.
The subspace of accessible states is shaped by the CF, so it depends essentially on
the symmetry of the crystallographic site occupied by the RE and also on the value
of J .
    The analysis is particularly simple at low temperatures, where the magnetic be-
haviour is determined by the ground CF level. This may typically be either a singlet
(J is an integer) or a Kramers doublet (J semi-integral). Within the ground CF
       ˆ              ˆ            ˆ
level HCF as well as HCoulomb + Hs-o reduce to irrelevant constants and can be omit-
ted. Of course, the CF still governs the properties dependent on the wave functions
of the ground state, e.g. the effective g-factor of a doublet.
    Often a pair of closely situated singlets make an accidental doublet, or a quasi-
doublet. The presence of the CF is then manifest in the zero-field splitting of such
a quasi-doublet, as well as determining the g-factor.

1.2.5 Exchange-dominated RE systems

                            ECoulomb         Es-o       Eex       ECF ∼ EZeeman .                          (1.15)
This special case is most closely relevant to RE-based hard magnetic materials,
it will be investigated in greater detail in subsequent sections. Either part of the
double inequality Es-o      Eex    ECF may fail under certain circumstances, which
imposes substantial limitations on the applicability of this approximation. These will
be considered separately for each half of the inequality (Sections 2.2 and 2.9).

1.2.6 Chains of equivalent operator representations
In the previous sections we have analysed several possible intensity relations between
                                      ˆ
the individual interactions within H4f . In each case the quantitative description
consisted in projecting some of the terms of Eq. (1.1) onto one of a hierarchically
organised chain of subspaces within the 4fN configuration: LS term, J multiplet,
Kramers doublet. . . Thus, the problem in each case reduces to computing the
matrix elements of VCF within one of those subspaces.3 The equivalent operator
technique makes this task easier. Let us consider the following chain of equivalence
 3 The rather straightforward handling of the remaining terms of Eq. (1.1) has been described in many standard texts
on quantum mechanics (e.g. Van Vleck, 1932; Condon and Shortley, 1935; Schiff, 1949). We shall make use of these
well-known results according as we need them.
158                                                                                            M.D. Kuz’min and A.M. Tishin



relations:
  N                                         N                                         N
                          .                                    .                             (2) ˆ
        ri2 Cm (θi , φi ) = r 2
             (2)
                                      4f         Cm (θi , φi ) = αl r 2
                                                  (2)
                                                                                 4f         Cm (li )
 i=1                                       i=1                                        i=1
         all space                   4fN    configuration                   4fN   configuration
                            .            (2) ˆ .            (2) ˆ .
                            = αL r 2 4f Cm (L) = αJ r 2 4f Cm (J ) =                              (±) const.
                                       LS   term                   J   multiplet              singlet, (quasi)doublet
                                                                                                                        (1.16)
      (2) ˆ
Here Cm (J ) denotes the following operator expressions4 :
                          (2) ˆ   1 ˆ
                         C0 (J ) = 3Jz2 – J (J + 1) ,
                                  2
                                                       1
                          (2) ˆ         3              2
                                                            ˆ ˆ       ˆ    ˆ      ˆ ˆ
                         C±1 (J )    =∓                    Jz Jx ± i Jy + Jx ± i Jy Jz ,                                (1.17)
                                        8
                                                   1
                          (2) ˆ              3     2
                                                        ˆ      ˆ       2
                         C±2 (J )    =                 Jx ± i Jy
                                             8
          (2) ˆ        (2) ˆ                                       ˆ   ˆ
while Cm (l) and Cm (L) are the same expressions, but with l or L substituted
for Jˆ. Each operator in the chain (1.16) is defined in a distinct space of states,
       (2) ˆ
e.g. Cm (L) operate within an LS term. Each subsequent space is a subspace of
the previous one. Any matrix element of an operator standing on the right of the
                     .
equivalence sign ‘=’ between any two states belonging to the space where that
operator is defined equal the corresponding matrix element of the operator on the
                   .
left-hand side of ‘=’. The opposite is not necessarily true. For example,
                             (2) ˆ                    (2) ˆ
                  LSJ M |αL Cm (L)|LSJ M = LSJ M |αJ Cm (J )|LSJ M .                                                    (1.18)
However, if J = J , then
                         (2) ˆ                        (2) ˆ
              LSJ M |αL Cm (L)|LSJ M = 0 = LSJ M |αJ Cm (J )|LSJ M . (1.19)
That is, the matrix elements of the two operators coincide only within a subspace
where both of them are defined. Hence the use of a special sign of equivalence in
Eq. (1.16) instead of the usual equality sign.
   The reason for having so many different representations for the CF is mere
convenience—each one is ideally suited for computing matrix elements within the
corresponding space of states. The choice of that space is not arbitrary; it constitutes
an approximation and is dictated by the physical situation under study, examples
 4    These can be obtained from Eqs. (1.5) using the following simple rules:
                   (n)
 (i) convert r n Cm (θ , φ) to Cartesian coordinates, replacing r 2 with x 2 + y 2 + z2 ;
(ii) symmetrise each monomial, e.g. xy = 1 (xy + yx);
                                                   2
                  ˆ ˆ ˆ
(iii) substitute Jx , Jy , Jz for x, y, z, respectively.

                                                        ˆ  (n)
The most complete list of explicit expressions for Cm (J ), with 0 ≤ n ≤ 8, was compiled by Lindgård and Danielsen
(1974).
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                159


being given in the preceding sections. For example, the J -representation is par-
ticularly convenient for a J multiplet, the matrix elements of the second-order
operators (1.17) are then given by
              (2) ˆ         1
      LSJ M |C0 (J )|LSJ M = [3M 2 – J (J + 1)]δM ,M ,
                            2
                                                                                   1
                  (2) ˆ                           3                                2
      LSJ M     |C±1 (J )|LSJ M        = ∓(M + M) (J ± M )(J ∓ M)                      δM ,M±1 ,
                                                  8
                                                                                             (1.20)
              (2) ˆ
      LSJ M |C±2 (J )|LSJ M
                                                                   1
          3                                                        2
        =   (J ± M )(J – 1 ± M )(J ∓ M)(J – 1 ∓ M)                     δM ,M±2 .
          8
Similar explicit expressions can be also written for the matrix elements of the
                            (4) ˆ    (6) ˆ
higher-order operators Cm (J ) and Cm (J ).
   The equivalence coefficients in Eq. (1.16), αl , αL , αJ , are known rational num-
bers. In general, αl is given by
                                                       2
                                      αl = –                                                 (1.21)
                                               (2l – 1)(2l + 3)
that is, αl = –2/45 for all RE’s.
    A general explicit expression for αL is quite complicated, but for a ground term
obeying Hund’s rules, S = 1 (2l + 1 – |2l + 1 – N|), L = S(2l + 1 – 2S), it simplifies
                             2
to (Bleaney and Stevens, 1953):
                                         2l + 1 – 4S
                                       αL = ±αl                                   (1.22)
                                             2L – 1
where the upper sign is required for a shell less than half full, and the lower sign for
a shell more than half full.
    Expressions for the Stevens coefficients αJ are generally extremely cumbersome
(Wybourne, 1965). Of interest to us here are only those for the ground multiplets
of the RE’s, J = L ± S. These are given by
                        ⎧
                        ⎪ (L + 1)(2L + 3) , if J = L – S (light RE)
                        ⎪
                        ⎨
                          (J + 1)(2J + 3)
           α J = αL ×                                                             (1.23)
                        ⎪ L(2L – 1)
                        ⎪
                        ⎩            ,          if J = L + S (heavy RE).
                          J (2J – 1)
The values computed using Eq. (1.23) are collected in Table 3.1.
   Chains of equivalence relations similar to (1.16) can be also written for higher-
order operators Cm and Cm , the coefficients therein being βl,L,J (n = 4) and γl,L,J
                  (4)      (6)

(n = 6). By analogy with the second-order case, βl and γl are given by general
expressions similar to Eq. (1.21); for all RE’s βl = 2/495 and γl = –4/3861.
Formulae similar to (1.22) exist for βL and γL in the case of a Hund’s ground term
(Bleaney and Stevens, 1953). The fourth- and sixth-order Stevens factors, βJ and
160                                                                         M.D. Kuz’min and A.M. Tishin


Table 3.1   The Stevens factors for the ground multiplets of the rare earths

N            Prototype        Ground           αJ                  βJ                      γJ
             RE ion           multiplet
                                               –2                      2
 1           Ce3+             2F
                                   5/2         5·7                  32 ·5·7
                                                                                           0
                                                –22 ·13                –22                      24 ·17
 2           Pr3+             3H
                                4              32 ·52 ·11           32 ·5·112              34 ·5·7·112 ·13
                                                 –7                  –23 ·17                 –5·17·19
 3           Nd3+             4I
                                9/2            32 ·112              33 ·113 ·13            33 ·7·113 ·132
                                                 2·7                  23 ·7·17               23 ·17·19
 4           Pm3+             5I
                                4              3·5·112              33 ·5·113 ·13          33 ·7·112 ·132
                                                 13                    2·13
 5           Sm3+             6H
                                  5/2          32 ·5·7              33 ·5·7·11
                                                                                           0
 6           Eu3+             7F
                                 0             –                   –                       –
 7           Gd3+             8S
                                7/2            –                   –                       –
                                                –1                       2                      –1
 8           Tb3+             7F
                                   6           32 ·11               33 ·5·112              34 ·7·112 ·13
                                                 –2                       –23                    22
 9           Dy3+             6H
                                15/2           32 ·5·7              33 ·5·7·11·13          33 ·7·112 ·132
                                                  –1                       –1                    –5
10           Ho3+             5I
                                8              2·32 ·52             2·3·5·7·11·13          33 ·7·112 ·132
                                                  22                       2                     23
11           Er3+             4I
                                15/2           32 ·52 ·7            32 ·5·7·11·13          33 ·7·112 ·132
                                                 1                      23                      –5
12           Tm3+             3H
                                6              32 ·11               34 ·5·112              34 ·7·112 ·13
                                                2                       –2                      22
13           Yb3+             2F
                                   7/2         32 ·7                3·5·7·11               33 ·7·11·13



γJ , for the ground multiplets of the RE’s are obtainable from equations similar
to (1.23) and are presented in Table 3.1.
    Equations (1.20) are generalised (Smith and Thornley, 1966) to

                      (n) ˆ          1                       (2J + n + 1)!
              LSJ M |Cm (J )|LSJ M = n                                        JM
                                                                             CJ Mnm .            (1.24)
                                    2                      (2J + 1)(2J – n)!
                            (n) ˆ                                             (n) ˆ
The matrix elements of Cm (L) between the states |LML or those of Cm (l) be-
tween |lm are obtained from Eq. (1.24) through the obvious substitution of L or l
for J etc.
    A less obvious fact with rather far-reaching consequences is that the matrix el-
ement of Cm in any representation between |LSJ M is proportional to the same
             (n)
         JM
CGC CJ Mnm , the proportionality factor being independent of the ‘projection’
quantum numbers m, M, M . This statement is known as the Wigner–Eckart the-
orem (Edmonds, 1957; Varshalovich et al., 1988). It is the foundation-stone of the
method of equivalent operators, since it directly follows that in order to compute
such matrix elements, one only needs (apart from the standard CGC) a set of coef-
ficients for n = 2, 4, 6, that is α, β and γ .
                                                  (n)
    Choosing one or another representation for Cm is thus a matter of convenience.
                                                                  ˆ
On the contrary, the choice of a correct set of basis states for H4f is very important,
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                                      161


it involves approximations and should be based upon intensity relations of type
(1.10)–(1.15)
                                                            (n)             (n)
    Likewise, it is a matter of convention whether to use Cm or Ynm or Um . The
Wigner–Eckart theorem is valid for all of them. The chains of equivalences for
            (n)                                      (n)
Ynm and Um look exactly the same as those for Cm , including the values of the
                                                                                   m
coefficients α, β and γ . Moreover, the same equivalences hold for the operators On
normalised according to Stevens, which do not obey the Wigner–Eckart theorem.
Indeed, there is a fairly straightforward connection between Eqs. (1.5) and (1.7):
                                                                 1
                                          1 2 (2)
                         =    0
                             O2       = (2)  1
                                      2C0 , O2           (2)
                                                C–1 – C1 , etc.                (1.25)
                                          6
These relations are valid in any representation: coordinate, l, L or J . Hence equa-
                                 m                             m
tions of type (1.16) hold for On as well. It was in fact for On that the principle of
equivalence was first stated (Stevens, 1952).
    Combining Eqs. (1.25) with the Wigner–Eckart theorem (1.24), one can express
                      m
matrix elements of On in terms of linear combinations of CGC. These expressions
are less convenient for analytical work and are seldom used because the coefficients
therein depend not just on n, but also on m (a high price to pay for the apparent
simplicity of the Stevens normalisation). More common are direct tabulations of
                           m
the matrix elements of On (Stevens, 1952; Hutchings, 1964; Abragam and Bleaney,
1970; Al’tshuler and Kozyrev, 1974).

                                          ˆ
1.3 Local symmetry and the exact form of HCF
Towards the conclusion of this introductory section, let us take a closer look at
what exactly determines the number of relevant terms in the expansion of the CF
potential (1.2). As long as configuration mixing is neglected, this number cannot
exceed 27, but is usually far less than that—just 2 in the highest-symmetry case
(cubic point groups Td , O or Oh ):5
                               ˆ
                              HCF = b4 O4 + 5O4 + b6 O6 – 21O6 .
                                        0     4       0      4
                                                                                                                  (1.26)
Not seldom one comes across an erroneous assertion of Eq. (1.26) being character-
istic of the ‘cubic symmetry’ in general (Lea et al., 1962, to give just one example).
In reality however, if the local symmetry of the RE site is described by either one of
                                                 ˆ
the cubic point groups T or Th (cf. Table 3.2), HCF must contain an extra sixth-order
term:
                   ˆ
                  HCF = b4 O4 + 5O4 + b6 O6 – 21O6 + b6 O6 – O6 .
                            0     4       0      4       2    6
                                                                                                                  (1.27)
This expression is clearly distinct from Eq. (1.26), so the need to take this matter
further should raise no doubts. (Many of such misstatements originate from the
  5 For compactness we use the Stevens normalisation for the CF operators, since we do not intend calculating their matrix
                                                             m
elements in this subsection. In this context the operators On should not be hastily identified with their J -representation,
            m ˆ                                                               ˆ
i.e. with On (J ). The symmetry considerations determining the form of HCF are quite independent of the chosen
representation. Therefore, depending on the situation, On in Eqs. (1.26)–(1.32) may be understood as N On (θi , φi )
                                                           m
                                                                                                            i=1
                                                                                                                  m
        m ˆ
or as On (L) etc. The loosely defined CF parameters are denoted with lower-case letters, to avoid confusion with properly
specified CF parameters.
162                                                                                     M.D. Kuz’min and A.M. Tishin


Table 3.2     The 32 point groups

No.                 Label                                           No.                  Label
Triclinic                                                           Trigonal
 1                  C1                   1                          16                   C3                   3
 2                  Ci                   ¯
                                         1                          17                   C3i                  ¯
                                                                                                              3
                                                                    18                   D3                   32
Monoclinic                                                          19                   C3v                  3m
 3                  C2                   2                          20                   D3d                  ¯
                                                                                                              3m
 4                  Cs                   m
 5                  C2h                  2/m                        Hexagonal
                                                                    21                   C6                   6
Orthorhombic                                                        22                   C3h                  ¯
                                                                                                              6
 6                  D2                   222                        23                   C6h                  6/m
 7                  C2v                  mm2                        24                   D6                   622
 8                  D2h                  mmm                        25                   C6v                  6mm
                                                                    26                   D3h                  ¯
                                                                                                              62m
Tetragonal                                                          27                   D6h                  6/mmm
 9                  C4                   4
10                  S4                   ¯
                                         4                          Cubic
11                  C4h                  4/m                        28                   T                    23
12                  D4                   422                        29                   Th                   m3
13                  C4v                  4mm                        30                   O                    432
14                  D2d                  ¯
                                         42m                        31                   Td                   ¯
                                                                                                              43m
15                  D4h                  4/mmm                      32                   Oh                   m3m



old CF theory, aimed exclusively at d-ions and therefore limited to fourth-order
terms.)
                                                          ˆ
    Now, with full rigour one can say that the form of HCF is uniquely determined
by the point group describing the local symmetry of the crystallographic site occu-
pied by the RE. The traditional combinations of point groups called syngonies, or
crystal systems (cubic, tetragonal etc., Table 3.2) provide no valid basis for judge-
ment in this question.6
                    ˆ
    The form of HCF also depends on the orientation of the coordinate system
in relation to the crystallographic directions. In the above example (1.26) all three
coordinate axes were set parallel to 4-fold symmetry axes. A rotation through ±π /4
around z would correspond to a simultaneous change of sign of the coefficients of
  4         4
O4 and O6 in Eq. (1.26). Setting the z axis along a 2- or a 3-fold crystal axis
would lead to Eqs. (6.14) or (6.15) of Hutchings (1964). Note that the number of
independent CF parameters (in this case, two) is independent of the choice of the
  6 Sometimes the name of a crystal system is used as a synonym for the most symmetric (holohedral) point group of that
crystal system (listed last under each of the headings in Table 3.2). Such liberty with the terms should be avoided, as it
only causes confusion.
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                        163




         Figure 3.2    A two-dimensional lattice with RE atoms shown as dark squares.




coordinate system. In the examples that follow the z axis will always be set along a
high-symmetry crystal direction.
    Local symmetry is the symmetry of the whole crystal seen from the standpoint
of the RE. (If RE atoms occupy several non-equivalent sites in a crystal, there are
generally as many distinct symmetries.) It should not be confounded with the ‘type
of coordination’, or the shape of the polyhedron made by the nearest neighbours of
the RE. Figure 3.2 illustrates this point in two dimensions: the nearest neighbours
of the central atom form a perfect square, described by the point group D4h . The
true local symmetry is however lower, C4h . The corresponding expressions for HCF   ˆ
are distinct, cf. Eqs. (1.31) and (1.32) below.
    On the other hand, the local symmetry of the RE site should be distinguished
from the crystallographic class, or the point group describing the crystal as a
whole, regardless of the viewpoint. Thus, the famous permanent magnet materials
                                          14
RE2 Fe14 B (space group P 42 /mnm—D4h ) belong to the tetragonal crystallographic
class D4h . The RE atoms occupy sites of two kinds, 4f and 4g, the symmetry
of both being described by the orthorhombic point group C2v . A definitive ref-
erence in this matter is the International Tables for Crystallography (Hahn, 1983).
For the space group P 42 /mnm one reads there, on page 459 of volume A, that
the 4f (xx0) and the 4g(xx0) sites both have the symmetry of m.2m, that is mm2,
or C2v . When comparing atomic positions described in the original literature and
in the International Tables, one should be aware of the possible multiple choices of
the origin and axes orientations.
    Having determined the local point group, we are ready to formulate the main
                                     ˆ                                   ˆ
principle governing the form of HCF . It sounds surprisingly simple: HCF may only
contain terms invariant under all symmetry operations of the local group. Alterna-
tively, in terms of the theory of representations: only the terms which belong to
the identity, or totally symmetric irreducible representation of the local point group
                 ˆ
may enter in HCF .
    Let us demonstrate this principle for the point group Cs (m). It is convenient to
rewrite Eqs. (1.7) in Cartesian coordinates:
164                                                           M.D. Kuz’min and A.M. Tishin



                                  3z2
                          O2 =
                           0
                                      – 1,
                                  r2
                                  xz              yz
                          O2 = 2 ,
                             1
                                              2 = 2,
                                              1
                                                                                (1.28)
                                  r               r
                                  x2 – y2              2xy
                          O2 =
                             2
                                          ,       2 =
                                                  2
                                                           .
                                    r2                  r2
    The group Cs contains a single non-trivial symmetry element—a mirror plane
perpendicular to the z axis. Being invariant in this case is equivalent to being even
                                                      0    2
in z. The allowed second-order terms thus are O2 , O2 and 2 . This analysis is
                                                                    2
                                                                                m
extended in a natural way to higher-order operators. The result is that all On and
  m                                         ˆ
  n with both n and m even enter in HCF (those with n > 6 do not affect the
4f shell and need not be included). A convenient collection of explicit expressions
      m
for On and m can be found in Appendix V of Al’tshuler and Kozyrev (1974).
                n
Alternatively, one may use the extensive list of tesseral harmonics, Znm ∝ On and
                                                                        c       m

Znm ∝ n , compiled by Görller-Walrand and Binnemans (1996, Appendix 2).
  s       m

    The orthorhombic point group D2h (mmm) contains three mutually perpendic-
ular mirror planes, as well as combinations thereof. The invariant terms are those
                                      0         2                                    m
even in all three coordinates, i.e. O2 and O2 from Eqs. (1.28), and generally On
with n and m even.
    One need not perform this analysis every time anew. Exhaustive results for all
32 point groups have been obtained and tabulated (Bradley and Cracknell, 1972;
Altmann and Herzig, 1994; Görller-Walrand and Binnemans, 1996). For example,
for the simplest hexagonal point group C6 one finds on page 64 of Bradley and
Cracknell (1972) the following table:
                          6 (C6 )                 m mod 6
                          A                       0
                          B                       3
                          1
                            E1                    4
                          2
                            E1                    2
                          1
                            E2                    1
                          2
                            E2                    5
The relevant information is in the first line, corresponding to the totally symmet-
ric irreducible representation A. It reads: allowed are all spherical harmonics Ynm
(or Cm , or Um ) with n arbitrary and m = 0 mod 6, i.e. 0, ±6, ±12 etc. (note
       (n)     (n)

that those with n odd are not forbidden, cf. footnote 1). Turning to the Stevens
convention and limiting ourselves to n = 2, 4, 6, we arrive at
                 ˆ
                HCF = b20 O2 + b40 O4 + b60 O6 + b66 O6 + b66
                           0        0        0        6             6
                                                                    6             (1.29)
where all the coefficients are real numbers. Equation (1.29) is related to the co-
ordinate system whose z axis is parallel to the 6-fold crystal axis [001] and whose
x axis is set along an elementary translation vector in the basal plane [100]. The
last term in (1.29) can be eliminated by rotating the coordinate system around the
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                  165


z axis through an (unknown a priori) angle φ0 , such that tan 6φ0 = b66 /b66 . Equa-
tion (1.29) is also valid for the hexagonal point groups C3h and C6h .
    Let us consider another hexagonal group, D6 . The corresponding table is on
page 65 of Bradley and Cracknell (1972). The top part of it looks as follows:
                   622 (D6 )      l       m mod (+6)        φ-dep
                   A1             0       0                 c
                                  7       6                 s
                   A2             6       6                 s
                   ...............................................

    Once again, what we need is in the first line of the table (the second line, though
related to the identity representation A1 , contains information on odd harmonics
of order seven or higher). According to the convention adopted by Bradley and
Cracknell (1972), l in the header is an abbreviation of l mod (+2). Thus, we read:
allowed are the harmonics with l, in our notation n = 0 mod (+2) and m =
0 mod (+6). The last column further specifies: when m = 0, only symmetric
combinations of type (Ynm + Yn–m ) ∝ cos mφ are admissible, i.e. On as opposed
                                                                         m

to m . One therefore has
     n


                           ˆ
                          HCF = b20 O2 + b40 O4 + b60 O6 + b66 O6 .
                                     0        0        0        6
                                                                                                (1.30)

The same expression is obtained for the hexagonal groups C6v , D3h and D6h .
  A further example—tetragonal point groups C4 , S4 and C4h :

         ˆ
        HCF = b20 O2 + b40 O4 + b44 O4 + b44
                   0        0        4                       4
                                                             4   + b60 O6 + b64 O6 + b64
                                                                        0        4         4
                                                                                           6.   (1.31)

Note that no rotation in the basal plane can reduce this expression to Eq. (1.32), i.e.
eliminate b44 and b64 simultaneously. (This was only possible in the old CF theory,
aimed mainly at spectroscopic properties of d-ions, hence the misconception of the
‘united tetragonal symmetry’.)
    The expression for the remaining tetragonal groups (D4 , C4v , D2d , D4h ) is as
follows:

                     ˆ
                    HCF = b20 O2 + b40 O4 + b44 O4 + b60 O6 + b64 O6 .
                               0        0        4        0        4
                                                                                                (1.32)

    Summarising this subsection, the precise form of the CF Hamiltonian cannot
be inferred from vague general categories like cubic (hexagonal, tetragonal etc.)
symmetry. Rather, it is uniquely determined by the local point group of the crys-
tallographic site occupied by the RE and can be inquired about in widely available
tables (Bradley and Cracknell, 1972; Altmann and Herzig, 1994). Finally, one can
consult the nearly complete list (missing are the cubic groups T , Th and O) of ex-
                        ˆ               (n)
plicit expressions for HCF in terms of Cm (Görller-Walrand and Binnemans, 1996,
Appendix 3).
166                                                              M.D. Kuz’min and A.M. Tishin




      2. The Single-Ion Anisotropy Model for 3d-4f
         Intermetallic Compounds
2.1 Macroscopic description of magnetic anisotropy
Let us consider a macroscopic system held at temperature T in an applied magnetic
field B. These external parameters may vary only quasi-statically, so that at all times
the system remains at thermal equilibrium. The standard thermodynamic descrip-
tion of such a system is afforded by specifying its free energy F (T , B, . . .), which
is a characteristic function of T , B and perhaps further external parameters. The equa-
tions of state are then obtained by taking partial derivatives of the free energy with
respect to its variables:
                                                    ∂F
                              S(T , B, . . .) = –            ,                         (2.1)
                                                    ∂T   B

                                                   ∂F
                             M(T , B, . . .) = –          .                        (2.2)
                                                   ∂B T
Here S, M, . . . are the system’s internal parameters: entropy, magnetisation, etc. Ex-
ternal and internal parameters make pairs of conjugate thermodynamic variables:
T –S, B–M, etc. In this Chapter we shall deal mainly with the magnetic equation
of state (2.2). This is not to say that the caloric equation of state (2.1) is less im-
portant. The entropy plays a central role in magneto-thermal properties, such as
specific heat and magnetocaloric effect (Tishin and Spichkin, 2003).
    Alternatively to using the equilibrium free energy F (T , B) one can take as a
starting point the non-equilibrium with respect to magnetisation thermodynamic
potential (T , B, M). Unlike the usual equilibrium potentials, depends on both
conjugate variables, B and M, and the corresponding equation of state is obtained
by minimising it with respect to the internal parameter M:
                                      ∂
                                                   =0                                  (2.3)
                                      ∂M    T ,B

and (∂ 2 /∂M 2 )T ,B > 0. The so minimised thermodynamic potential is the equi-
librium free energy, min (T , B, M) = F (T , B).
                       M
    Both approaches are of course equivalent, as long as they lead to the same mag-
netic equation of state, either in the form of Eq. (2.2) or of Eq. (2.3). The free
energy is the preferred route when the system’s partition function Z(T , B) can be
computed; then F is given by the well-known relation of the statistical mechanics,
F = –kT ln Z.
    In turn, is advantageous in phenomenological theories since its dependence
on M and B can be inferred from the rather general considerations of symme-
try. For example, a ferromagnet near its Curie point is described by (Landau and
Lifshitz, 1958):
                                      1        1
                  (T , B, M) = 0 + aM 2 + bM 4 + · · · – B · M                 (2.4)
                                      2        4
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                  167


where the quantities 0 , a, b, . . . depend on temperature and other external pa-
rameters, e.g. pressure, but not on magnetic field or magnetisation. The only
assumptions used in the construction of the expansion (2.4) are the time-evenness
of and the smallness of M. It is however sufficient to obtain a number of useful
predictions regarding the thermodynamic behaviour of ferromagnets near the Curie
point.
    Up to this point we have not paid much attention to the fact that both B and M
are vectors, which means that in effect one has to deal with three pairs of conjugate
magnetic variables. The system under consideration was tacitly assumed isotropic
(therefore at equilibrium M||B), so setting one of the coordinate axes along B
reduced the description to a single pair of scalar variables.
    Now we turn to the anisotropic case, which is both more general and more
interesting, since it demonstrates to full extent the advantages of the symme-
try approach based on the use of the non-equilibrium thermodynamic potential
   (T , B, M). The coefficients in the Landau expansion (2.4) now become tensor
quantities. For example, 1 aM 2 is replaced with 1 aαβ Mα Mβ , where aαβ is a ten-
                            2                      2
sor of rank two, the sum being taken over α, β = x, y, z. Many of the components
of aαβ , bαβγ δ etc. may be equal to each other or vanish for symmetry reasons. More
strictly, this depends on the point group describing the symmetry of the crystal
as a whole (since we are dealing with macroscopic properties), also known as the
crystallographic class. Thus, for any point group of the cubic crystal system (see
Table 3.2) the expansion (2.4) takes the following form:
                                   1         1
          (T , B, M) =        0+ aM 2 + bM 4 + b (Mx My + My Mz + Mz Mx )
                                                   2  2    2  2    2  2
                                   2         4
                            + · · · – B · M.                              (2.5)
In this case the anisotropy makes its first appearance in the terms of fourth order.
However, in lower-symmetry crystals it affects second-order terms as well.
     In certain classes of phenomena the magnitude of the magnetisation vector varies
little, while its direction may change significantly. In such a situation it is conve-
nient to use as internal thermodynamic parameter the direction of M, rather than
its Cartesian components. Direction cosines or spherical angles can be used for the
purpose. A specific example is the behaviour of a ferromagnet well below the Curie
point in a weak to moderate magnetic field. The energy associated with a noticeable
change of |M| is of the order of TC ∼ 103 K per atom (∼102 K for ferromagnets
less suitable for applications). The anisotropy energy is usually much smaller, reach-
ing ∼101 K per atom in YCo5 —one of the most strongly anisotropic 3d magnets
(Alameda et al., 1981). The RE contribution to the anisotropy energy may exceed
                                                      n
100 K per RE atom at low temperature (Radwa´ ski, 1986), which upon averag-
ing over all atoms in an iron- or cobalt-rich compound would yield ∼2 × 101 K
per atom. This is significantly less than the 3d-3d exchange energy. At any rate, it
should be kept in mind that the formalism of magnetic anisotropy is an approximate
one and that its validity is limited by the requirement that |M| should be essentially
constant. It does not apply near TC , in very strong magnetic fields or when |M|
is itself strongly anisotropic. The latter restriction concerns e.g. YCo5 , where |M|
changes by as much as 4% upon reorientation (Alameda et al., 1981). In general, the
168                                                                      M.D. Kuz’min and A.M. Tishin



approach might fail in RE compounds at low temperatures. From now on we limit
ourselves to such phenomena where |M(T , B)| = Ms (T ), i.e. the magnitude of the
magnetisation is practically independent of applied field and equals the spontaneous
magnetisation. The anisotropic version of Landau’s expansion then becomes
                     1                               1
           =    0   + Ms2 (T )            aαβ nα nβ + Ms4 (T )          bαβγ δ nα nβ nγ nδ
                     2              α,β
                                                     4         α,β,γ ,δ

               + · · · – Ms (T )n · B                                                          (2.6)
where n ≡ M /|M| is a unit vector in the direction of the magnetisation. The
fact that only quadratic and quartic terms have been written out explicitly does
not imply rapid convergence of the expansion (2.6) or that its truncation after the
quartic term is legitimised in any way. The situation here is radically different from
that in a ferromagnet near TC —the subject of Landau’s theory of second-order phase
transitions (Landau and Lifshitz, 1958). The truncation of Eq. (2.4) was based on
the obvious fact that Ms → 0 as T → TC . Conversely, Ms is not at all small in
Eq. (2.6). It is to be assumed that the series (2.6) diverges, unless proven otherwise.
    Equation (2.6) is too general to be useful. Let us rewrite it for some commonly
encountered special cases. The key point here is the invariance of each term of the
expansion under all symmetry operations of the crystallographic class. Thus, for the
cubic classes Td , O and Oh Eq. (2.6) becomes
                                    =      0   + Ea – Ms (T )n · B                             (2.7)
where
                                     1           1
                        0   =   0   + aMs2 (T ) + bMs4 (T ) + · · ·
                                     2           4
and
                Ea = K1 n2 n2 + n2 n2 + n2 n2 + K2 n2 n2 n2 + · · ·
                         x y     y z     z x        x y z                                      (2.8)
The quantity Ea —the anisotropic part of the thermodynamic potential in the ab-
sence of magnetic field—is known as anisotropy energy, while K1 , K2 etc. are called
anisotropy constants. The latter may depend on temperature and other external pa-
rameters, but not on magnetic field. The dependence of             on B is limited to
the last, Zeeman term of Eq. (2.7). Equation (2.8) was first obtained by Gans and
Czerlinsky (1932).
    For the cubic crystallographic classes T and Th the anisotropy energy contains an
extra sixth-order term in addition to that in Eq. (2.8):
               K2 n2 n2 n2 – n2 + n2 n2 n2 – n2 + n2 n2 n2 – n2 .
                   x y   x    y    y z   y    z    z x   z    x                                (2.9)
This term is invariant under the rotations around the 3-fold axes (equivalent to
cyclic permutations within the triplet n2 , n2 , n2 ), but is not invariant with re-
                                          x    y   z
spect to rotations through 90° about the 4-fold axes (pair permutations of the type
n2 ↔ n2 ). Therefore, it is allowed in the lower-symmetry cubic groups T and Th ,
 x      y
containing 3-fold axes only, and forbidden in the higher-symmetry cubic groups Td ,
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                                  169


O and Oh , which include 4-fold axes as well. A direct parallel can be drawn here
to Eqs. (1.26), (1.27) describing the CF for the same point groups.
    For non-cubic point groups it is customary to describe the direction of M by
means of the conventional spherical angles θ and φ. The polar axis is conveniently
set along a high-symmetry crystallographic direction, whose choice presents no dif-
ficulty except for the triclinic groups C1 and Ci . The anisotropy energy, Ea (θ , φ),
is then expanded over a suitably chosen basis, e.g. over the irreducible tensor oper-
ators:
                                                 ∞     n
                                      Ea =                       (n)
                                                            κnm Cm (θ , φ).                                   (2.10)
                                                n=2 m=–n
                                                even

Note that there is no generally valid reason to truncate this expansion after any
finite number of terms.
    An added advantage of presenting Ea as Eq. (2.10) is the possibility to exploit
the formal analogy with the CF potential (1.2). A particular point group dictates the
same form of both Ea (θ , φ) and VCF (r, θ , φ).7 Therefore, the expressions obtained
in Section 1.3 for specific point groups can be simply taken over. Alternatively,
one can consult the tables recommended therein (Bradley and Cracknell, 1972;
Altmann and Herzig, 1994; Görller-Walrand and Binnemans, 1996).
    Thus, by analogy with Eq. (1.30), we write for the hexagonal crystallographic
classes D6 , C6v , D3h and D6h :
                       Ea = κ20 P2 (cos θ ) + κ40 P4 (cos θ ) + κ60 P6 (cos θ )
                                 + κ66 C6 (θ , φ) + C–6 (θ , φ) + · · ·
                                        (6)          (6)
                                                                                                              (2.11)
It will be remembered that C0 (θ , φ) ≡ Pn (cos θ ), Pn (x) being the Legendre
                             (n)

polynomials.
   Another conventional form of this expression is due to Mason (1954):
             Ea = K1 sin2 θ + K2 sin4 θ + K3 sin6 θ + K3 sin6 θ cos 6φ + · · ·                                (2.12)
The fairly straightforward relations between the anisotropy constants Ki and κnm
                                                                     n
can be found e.g. in Appendix B to the review of Franse and Radwa´ ski (1993) in
Volume 7 of this Handbook.
   The corresponding expression for the hexagonal classes C6 , C3h and C6h contains
an extra sixth-order term, K3 sin6 θ sin 6φ, ∝ 6 in Eq. (1.29).
                                                  6
   Finally, the anisotropy energy of tetragonal crystals is given by
            Ea = K1 sin2 θ + K2 sin4 θ + K2 sin4 θ cos 4φ + K2 sin4 θ sin 4φ
                      + K3 sin6 θ + K3 sin6 θ cos 4φ + K3 sin6 θ sin 4φ + · · ·                               (2.13)
where the constants K2 and K3 are nonzero if the crystallographic class is C4 , S4
or C4h , but must vanish if it is D4 , C4v , D2d or D4h .
 7 This does not mean that the local symmetry of the RE site and the symmetry of the crystal as a whole are necessarily
described by the same point group. When these point groups are distinct, so are the expressions for Ea and VCF (cf. the
example of RE2 Fe14 B in Section 1.3).
170                                                               M.D. Kuz’min and A.M. Tishin



    Recapitulating, magnetic anisotropy energy Ea is that part of the non-
equilibrium with respect to M thermodynamic potential (T , B, M)|B=0 which
depends on the direction of M in a situation when |M| is known not to depend
on applied magnetic field B. The dependence Ea (M /|M|) is usually presented as a
series in powers of the direction cosines of M, whose form is dictated by the point
group describing the symmetry of the crystal as a whole—the crystallographic class.
As regards the convergence of the series, the formal theory is unable to make any
positive prediction in this respect. Such predictions, leading to truncated expan-
sions, can only be obtained in specific microscopic models, when the coefficients
(anisotropy constants) prove proportional to growing powers of a small parameter.
In the absence of a valid model, no custom or convention can justify the use of
expressions truncated after the terms of 2nd , 4th or 6th order.
    Throughout this subsection we have been dealing with standard thermodynamic
potentials, fit to describe macroscopic systems, containing very large numbers of
atoms. The introduced concepts of ‘anisotropy energy’ and ‘anisotropy constant’
are inapplicable to nanoscopic systems, just as does not apply to them the notion of
temperature.

2.2 The notion of an exchange-dominated RE system
Let us consider a RE–transition metal compound satisfying the validity conditions
for the single-multiplet approximation. We assume for simplicity that the applied
magnetic field is nil. Following Section 1.2.3, the properties of the RE subsystem
                                                               ˆ     ˆ
in this compound are described by a single-ion Hamiltonian, Hex + HCF , projected
on the ground J multiplet:
                                                      n
                 ˆ                                                (n) ˆ
                H4f = 2(gJ – 1)μB Bex · ˆ +
                                        J                    Bnm Cm (J ).             (2.14)
                                              n=2,4,6 m=–n

Here B ex is the exchange field on the RE produced by the ordered 3d sublattice,
Bnm are CF parameters incorporating the Stevens factors. Despite all simplifications,
the Hamiltonian (2.14) is still very complicated, since it contains in the general case
a large number of free parameters. Our consideration in this section will therefore
be limited to a special case of the so-called exchange-dominated RE system. On
the one hand, this will greatly simplify the calculations. On the other hand, the
approximation, if not taken too far, is likely to apply to real hard magnetic materials.
    It is clear from the outset that the CF can be regarded neither as infinitesimally
small, nor as negligible in comparison with the 3d-4f exchange. A strong CF is
indispensable to a good permanent magnet performance. At the same time, many
of these materials feature low-temperature RE magnetic moments close to the free-
ion value gJ J , as if the strong CF were not there at all. So, how exactly weak should
the CF be to account for this behaviour?
    To answer this question, let us first consider a fictitious ‘training’ RE whose
ground multiplet has J = 1. This is the simplest system displaying non-vanishing
CF effects. (Being time-even, the CF does not split the simpler J = 1/2 multi-
plets). When J = 1, the triangle rule dictates that only second-order CF terms are
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                              171




Figure 3.3 Normalised energy level pattern of a didactic RE with J = 1. The curved portions
of the middle level are hyperbolae described byE = 2/(1 – 3 η), η < – 2 , and E = 1 – 2/(1 + 3 η),
                                                           2          3                      2
η > 2 . The hatched interval around the origin is the location of broadly exchange-dominated
     3
systems.


relevant. Our hypothetical system is supposed to be a permanent magnet material,
i.e. a uniaxial crystal in which the magnetisation of the 3d sublattice is parallel to the
high-symmetry axis z (consequently, B ex is antiparallel to z). We assume in addition
that the local symmetry of the RE site is uniaxial as well, i.e. that it is described
by a point group belonging to either of the three syngonies: tetragonal, trigonal or
                                                             (2)
hexagonal. Then a single CF term is allowed—that in C0 —and Eq. (2.14) turns
into
                         ˆ                     ˆ        3 ˆ
                       H4f = 2(1 – gJ )μB Bex Jz + B20 Jz2 – 1 .                    (2.15)
                                                        2
The matrix of this Hamiltonian is obviously diagonal in the |J M basis, the three
eigenvalues being obtainable through the substitution of M = 0, ±1 for Jz in          ˆ
Eq. (2.15). Of interest to us here is the energy level pattern, rather than the eigen-
values as such. So we set the ground state energy to zero and normalise the overall
multiplet splitting to unity. The so defined level pattern (Fig. 3.3) is fully deter-
mined by a single dimensionless parameter—the CF-to-exchange ratio,
                                             B20
                                        η=                                          (2.16)
                                                         ex
where
                                      = 2|gJ – 1|μB Bex
                                          ex                                    (2.17)
is the exchange splitting between two adjacent levels of the multiplet.
    The converse is generally not true—knowing the pattern does not enable one
to establish the value of η even for such a simple system (unless the levels are un-
ambiguously labelled). For example, an equidistant spectrum could correspond to
either η = 0 or 2 or –2, the three situations being physically quite distinct.
    One finds by inspecting Fig. 3.3 that from the standpoint of the level sequence
the entire η axis is split into three domains separated by the points η = ± 2 , where
                                                                             3
two levels cross over.
172                                                             M.D. Kuz’min and A.M. Tishin



    We now define that a J = 1 RE system is exchange-dominated in the strict
sense if its locus in Fig. 3.3 lies near the origin, that is if |η| is much less than the
size of the central segment, |η|         2
                                         3
                                           . The same system will be called exchange-
dominated in the broad sense if its η lies somewhere within the central domain,
|η| < 2 , but not too close to its boundaries. (The latter clause is to exclude the
       3
nearly degenerate case, |η| = 2 – ε, 0 ≤ ε
                                  3
                                                     2
                                                     3
                                                       .) The domain corresponding to
broadly exchange-dominated systems is shown in Fig. 3.3 by hatching. Everywhere
within the hatched area (in fact, also everywhere left of it) the low-temperature
magnetic moment is constant and equals gJ μB , or rather gJ μB · sign(1 – gJ ).
    Before attempting a generalisation of these definitions for arbitrary J , let us con-
sider one more particular case. This time it is a ‘nearly real’ RE with J = 5/2 in a
hexagonal CF. Were it not for the extremely strong J -mixing that makes the single-
multiplet approximation fail, this example would be fully relevant to e.g. SmCo5 .
Our goal however is not so much to develop an accurate quantitative approach
to Sm-based magnets, as to demonstrate the concept of exchange-dominated RE
systems on something more realistic than the above example of plain J = 1.
    Thus, within the single-multiplet approximation, the Hamiltonian of a RE with
J = 5/2 in an exchange field B ex antiparallel to z and in a hexagonal CF has the
following form:

                     ˆ                      ˆ       3 ˆ 35
                    H4f = 2(1 – gJ )μB Bex Jz + B20 Jz2 –
                                                    2     8
                                   35 ˆ4 475 ˆ2 2835
                          + B40      J –        J +       .                         (2.18)
                                    8 z       16 z   128
The eigenvalues EM are obtained by substitution of M = ±1/2, ±3/2, ±5/2
     ˆ
for Jz :
                                   1
                         E±1/2 = ±    ex – 4B20 + 15B40 ,
                                   2
                                   3             45
                         E±3/2 = ±    ex – B20 –    B40 ,                    (2.19)
                                   2              2
                                   5                15
                         E±5/2 = ±    ex + 5B20 +      B40 ,
                                   2                2
where ex is the exchange splitting (2.17) and it has been assumed for definite-
ness that, like in Sm3+ , gJ < 1. The energy level pattern is determined by two
dimensionless parameters: η, defined according to Eq. (2.16), and ξ , given by
                                            B40
                                       ξ=          .                                (2.20)
                                              ex

The latter describes the strength of the 4th-order CF in relation to the 3d-4f
exchange. The ηξ plane can be divided into 53 domains, each one of them charac-
terised by a certain sequence of the energy levels (Fig. 3.4). Of primary interest to
us is the central domain, which contains the origin. There, the level sequence (i.e.
the dependence of EM on M) is monotonic, just as it would be without any CF.
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                         173




Figure 3.4 Character of the splitting of the ground sextet of a RE with J = 5/2, depending
on the intensities of second- and fourth-order CF in relation to the exchange. Within each of
the domains delimited by the oblique lines the eigenvalue sequence EM , arranged in ascending
order, defines a permutation of the six values of the quantum number M characteristic of
that domain. Level cross-over takes place at the domain boundaries. The domains within each
one of the three sectors separated by the bold lines have the same ground state: M = –5/2
(west), –3/2 (north), –1/2 (south-east). The hatched area near the origin marks the location
of broadly exchange-dominated systems. The dark diamond corresponds to SmCo5 .


    At the domain boundaries the levels cross over. Let us demonstrate this for the
levels with M = –5/2 and –3/2 assuming, by analogy with Sm3+ , that gJ < 1.
Then, in the absence of CF, the level with M = –5/2 is the ground state and the
one with M = –3/2 is the first excited state. The cross-over condition is obtained
by equating the last two of the equations (2.19), in which the lower signs must be
taken. Upon division by ex
                                            6η + 30ξ = 1.                             (2.21)
This equation corresponds to a straight line in the ηξ plane, namely, to that which
makes the north-east border of the central domain in Fig. 3.4. Taking the other
three combinations of upper/lower signs in the last two Eqs. (2.19) yields three
more equations similar to (2.21), but with different right-hand sides. The four equa-
tions describe the set of four negatively sloping parallel lines in Fig. 3.4. Two further
sets of four parallel lines are obtained in a similar fashion from the cross-over con-
ditions E±1/2 = E±3/2 and E±1/2 = E±5/2 . The four lines delimiting the central
parallelogram correspond to the crossings of the adjacent levels of the monotonic
spectrum, i.e. those with M = ±1. There are five pairs of adjacent levels in a
sextet, however no line is generated by the equation E1/2 = E–1/2 .
    Likewise, the conditions E3/2 = E–3/2 and E5/2 = E–5/2 produce no extra lines
in the ηξ plane. The physical reason of this is the time-even character of the CF, on
which grounds the latter does not affect the splitting of pairs of Kramers-conjugate
states. These are split by the exchange interaction alone, the level sequence within
each pair depending on the sign of the difference 1 – gJ . In the case of Sm3+ ,
174                                                                                         M.D. Kuz’min and A.M. Tishin



gJ = 5/7 < 1, the levels with negative M lie always below their positive coun-
terparts.8 Consequently, only these three levels, M = –1/2, –3/2, –5/2, can claim
the privilege of becoming ground state. On this principle, the entire ηξ plane can
be divided into three sectors. These are separated by the bold lines in Fig. 3.4.
We shall concentrate primarily on the west sector (containing the central parallel-
ogram), where the ground state is M = –5/2. The north sector corresponds to
M = –3/2 being the ground state, while in the south-east sector it is M = –1/2.
    A strictly exchange-dominated J = 5/2 system can now be readily defined as
such a system whose locus in the ηξ diagram is close to the origin, the proximity
being understood relative to the dimensions of the central domain. This definition
is equivalent to a pair of strong inequalities |η|  1/6, |ξ |    1/30.
    A J = 5/2 system is exchange-dominated in the broad sense if its locus lies
inside the central parallelogram of Fig. 3.4, excluding the regions close to its bound-
aries, as shown by the hatching. The hatched area belongs to the west sector of the
drawing, where the low-temperature magnetic moment is gJ J = gJ 5/2.
    The reason for the duplicate definition is that typical RE-based hard magnetic
materials are broadly exchange-dominated systems, without being such in the strict
sense. For example, the dark diamond, corresponding to the archetypal permanent
magnet material SmCo5 , is situated half way between the origin and the boundary
of the central domain of Fig. 3.4.9
    The above definitions can now be easily generalised for an arbitrary J > 5/2. In
addition to the parameters η and ξ , defined by Eqs. (2.16), (2.20), a third parameter
ζ needs to be introduced, to describe the relative intensity of the (axial) sixth-order
CF:
                                                                B60
                                                        ζ =             .                                           (2.22)
                                                                   ex

The ηξ ζ parameter space is divided by a number of planes into many domains, ac-
cording to the order of the eigenvalues EM . Within the central domain (containing
the origin) the eigenvalue sequence is monotonic: the ground state has the max-
imum (or negative maximum) possible M, the M of the first excited level differs
from the latter by 1, etc. The eigenvalue sequences in the other domains correspond
to permutations of the monotonic one.
    The boundaries of the central polyhedron are obtained from cross-over con-
ditions for the levels with M = ±1. The respective gaps can be presented as
  ex +    i , where   ex is the exchange contribution (2.17), common for all pairs
of adjacent levels, i are the CF contributions (numbered from bottom to top,
Kuz’min, 1995):
                            3             5
                     1   = – (2J – 1)B20 – (2J – 1)(2J – 2)(2J – 3)B40
                            2             4
                             21
                           – (2J – 1)(2J – 2)(2J – 3)(2J – 4)(2J – 5)B60 ,
                             32
 8   According to the adopted convention, the positive z direction is parallel to M 3d and antiparallel to B ex .
 9   The following values were used: Bex = 295 T, B20 = –2 meV, B40 = 0 (Kuz’min et al., 2002).
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                                     175


                           3            5
                   2   = – (2J – 3)B20 – (2J – 10)(2J – 2)(2J – 3)B40                                            (2.23)
                           2            4
                            21
                          – (2J – 21)(2J – 2)(2J – 3)(2J – 4)(2J – 5)B60 ,
                            32
                         etc.
Equating ex + i to zero, one obtains upon dividing by                                    ex   the equations of the
boundaries of the central domain, e.g.
              3             5
                (2J – 1)η + (2J – 1)(2J – 2)(2J – 3)ξ
              2             4
                    21
                 + (2J – 1)(2J – 2)(2J – 3)(2J – 4)(2J – 5)ζ = 1               (2.24)
                    32
and so on. In the special case of J = 5/2 Eq. (2.24) turns, as expected, into
Eq. (2.21).
    A RE system with an axial CF corresponds to a point in the ηξ ζ space. One
can demand that this point be close to the origin, the proximity being related to the
dimensions of the central polyhedron. The same condition can be expressed as a
strong inequality,
                                                   |   i|         ex .                                           (2.25)
A system satisfying (2.25) will be called exchange-dominated in the strict sense.
    Alternatively, the locus must lie inside the central polyhedron, not too close to
its boundaries:
                                     |   i|   <   ex (1   – δ),     0<δ          1.                              (2.26)
This is a broadly exchange-dominated RE system. Clearly, any strictly exchange-
dominated system is also broadly exchange-dominated. The converse is not true.
In order to ensure the free-ion value of the low-temperature magnetic moment,
gJ J μB , it suffices for an axially-symmetric RE system to be broadly exchange-
dominated.
    All the above-said can be applied to real materials—which do not possess the
axial symmetry—provided that Bnm with m = 0 are not too large as compared to
Bn0 (which appears to be fulfilled in most cases). Then, to first order in Bnm / ex ,
the presence of non-axial CF terms has no effect on the low-temperature magnetic
moment of the RE.10
    Summarising, we have formulated two different concepts of an exchange-
dominated RE system. When the first of them applies, it is strictly justified to
use the first-order perturbation theory in Bnm / ex (see subsection 2.511 ). Most real
RE-based magnets, however, fit the second (broad) but not the first definition. For
 10 Indeed, reduction of the ground-state magnetic moment is a time-even effect. The corresponding expression,

μCF /μfree-ion = 1 – const1 (Bnm / ex ) – const2 (Bnm / ex )2 – . . . , may not contain odd powers of ex ∝ Bex , therefore
const1 = 0.
 11 This does not include Sm-based magnets, on account of the failure of the single-multiplet approximation. Luckily, the
magnetic moment of Sm in intermetallics is so small—even its sign varies from compound to compound (Givord et al.,
1980)—that it can be safely neglected altogether.
176                                                                   M.D. Kuz’min and A.M. Tishin



such materials the equations of subsection 2.5 are still qualitatively correct, despite
some lack of rigorous foundation. These equations are strictly inapplicable when
neither of the above two definitions is satisfied.

2.3 The single-ion model for 3d-4f intermetallics
Our goal in this subsection is to formulate a general recipe for computing the RE
contribution to the magnetic anisotropy energy Ea proceeding from the parameters
                                               ˆ
entering in the single-ion RE Hamiltonian H4f . The latter will be treated in the
single-multiplet approximation, the only exception being Section 2.9, dedicated
specifically to J -mixing. As stated in Section 1.1, the single-ion approach to de-
scribing the properties of the RE subsystem in 3d-4f intermetallics relies on the
peculiar hierarchy of exchange interactions in these compounds. Thanks to it, the
3d subsystem can be regarded as something external, whose action on the RE is
described by means of an exchange field B ex . This enables one to treat the RE
subsystem as an ensemble of non-interacting ‘ions’, each one of which is described
by the following Hamiltonian:
              ˆ                      ˆ             ˆ                     (n) ˆ
             H4f = 2(gJ – 1)μB Bex · J + gJ μB B · J +              Bnm Cm (J ).          (2.27)
                                                              n,m

                                                                                       ˆ
As pointed out in Section 2.2, B ex is antiparallel to the 3d magnetisation M 3d . If H4f
is related to the crystallographic coordinate axes, the dependence on the orientation
of M 3d enters into the first term of Eq. (2.27). The angles θ and φ defining the
orientation of M 3d are external thermodynamic parameters in relation to the RE
subsystem. The latter is described by the usual equilibrium canonical distribution,
                                F4f = –kT ln Z4f (θ , φ)                                  (2.28)
where the RE partition function is
                                                         ˆ
                                                        H4f
                             Z4f (θ , φ) = tr exp –         .                             (2.29)
                                                        kT
   With respect to the 3d subsystem (and therefore to the combined 3d-4f system)
θ and φ are internal parameters, i.e. the former is described by means of a non-
equilibrium thermodynamic potential 3d (θ , φ). The equilibrium values of θ and
φ are determined through minimisation of the combined thermodynamic potential,
                             min    3d (θ , φ)   + F4f (θ , φ) .                          (2.30)
                              θ,φ

At this stage we do not specify the form of 3d (θ , φ). Suffice it to say that, like
Eq. (2.27), 3d contains anisotropy energy and a Zeeman term.
    Despite the fact that it only takes a few equations (2.27)–(2.30) to formulate
the single-ion model, it proves impossible to obtain a general explicit expression
for F4f (θ , φ). The main difficulty is taking the trace of the matrix exponential in
Eq. (2.29). One exception is the special case of J = 1, when such an expression
does exist (Kuz’min, 1995). This was used to demonstrate that F4f (θ , φ) cannot in
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                      177


general be presented as a truncated expansion of type (2.8)–(2.13). Such a presenta-
tion requires that the anisotropy constants Ki or κnm contain incrementing powers
of a small parameter, which is only true in some special cases when additional ap-
proximations can be made. If valid, the same approximations enable one to evaluate
the anisotropy constants.
    It is interesting to note that validity of the very same approximations allows to
settle the often raised question about the effect of non-collinearity of the 3d and
4f sublattices. The point is that the above-defined angles θ and φ determine the
orientation of just the 3d sublattice magnetisation M 3d . As regards the RE moment
μR , its orientation is described by the angles θR and φR , which are generally speak-
ing distinct from θ and φ. The RE magnetisation is an internal thermodynamic
parameter, so it does not figure explicitly in the above algorithm (2.8)–(2.13). For
any given θ and φ, the angles θR = arccos(μRz /|μR |) and φR = arctan(μRy /μRx )
can be found from the relation
                          ∂F4f                 gJ μ B                ˆ
                                                                    H4f
                 μR = –                 ˆ
                               = –gJ μB J = –               ˆ
                                                         tr J exp –            .   (2.31)
                          ∂B                  Z4f (θ ,φ)            kT
The problem posed by the non-collinearity is that the angles obtained through the
minimisation of the total thermodynamic potential of the system (2.30) do not
correspond to the orientation of the system’s total magnetisation. This difficulty
will be shown to disappear as soon as there is a valid reason to truncate the series
for Ea .
    To summarise, there are two possibilities. Either, one of the above-mentioned
approximations is valid and then one can (i) truncate the expansion for Ea , (ii) ob-
tain explicit expressions for the anisotropy constants entering in the truncated
expansion, and (iii) neglect the non-collinearity of the 3d and 4f sublattices. Or,
there is no such a valid approximation; then the standard description by means of
anisotropy constants is impossible, as none of the above three preconditions can be
secured. Of course, the general equations (2.27)–(2.31) are then still valid and can
be used to compute M(B) (not the anisotropy constants) numerically.
    Consequently, we shall no longer dwell on the sterile general formalism, but
rather proceed to the most important special cases.

2.4 The high-temperature approximation
Let us recast Eqs. (2.28), (2.29) in a different form,

                                            J       ex
                                                                          ˆ
                                                                         H4f
                          F4f (θ , φ) = –                ln tr exp –x              (2.32)
                                                x                       J ex
where x is a quantity equivalent to Langevin’s magneto-thermal ratio,
                                                         J ex
                                                x=                                 (2.33)
                                                          kT
and     ex   is the exchange splitting (2.17).
178                                                                    M.D. Kuz’min and A.M. Tishin



    Assuming that x is a small parameter, we can expand F4f in powers of x in the
spirit of Kramers-Opechowski:
                                    ln(2J + 1)
                                    J   ex
                                                              ˆ2
                                                          trH4f
                F4f (θ , φ) = –                  –                   x
                                       x            2J (2J + 1) ex
                                           ˆ3
                                       trH4f
                             + 2                   x2 + · · ·                  (2.34)
                                6J (2J + 1) 2   ex
                                         ˆ
It has been taken into account that tr H4f = 0.
    The first term in (2.34) is obviously isotropic, i.e. it does not depend on either
θ or φ. Let us demonstrate that the second term is isotropic, too. To this end it is
convenient to rewrite the Hamiltonian (2.27) as follows:
                          ˆ                         ˆ      ˆ
                         H4f = sign(1 – gJ ) ex n · J + HCF .                    (2.35)
Here n = M 3d /|M 3d | is a unit vector in the direction of the 3d magnetisation,
M 3d ↑↓ B ex , and it has been assumed that B         Bex , in order to ensure that μR
remains essentially independent of B. We shall make use of a helpful orthogonality
                            (n) ˆ
relation for the operators Cm (J ) (Kuz’min, 1995), which readily follows from the
well-known orthogonality of the CGC:
                (n) ˆ        ˆ               2–2n (2J + n + 1)!
            tr Cm (J )C–m) (J ) = (–1)m
                       (n
                                                                 δnn δmm       (2.36)
                                           2n + 1 (2J – n)!
where the trace is taken over the states of any J multiplet, such that 2J ≥ n.
                                                  ˆ      (1) ˆ
  Directing the z axis along n and noting that Jz ≡ C0 (J ), we write
             ˆ2
          trH4f =     2      ˆ2
                      ex tr(Jz )   + 2sign(1 – gJ )    ex tr
                                                                ˆ   (1) ˆ       ˆ2
                                                               HCF C0 (J ) + trHCF .       (2.37)
The first term is just  1
                       3
                         J (J + 1)(2J + 1)      2
                                              The second term vanishes by virtue of
                                                ex .
                                       ˆ                 (n) ˆ
the orthogonality relation (2.36), as HCF contains only Cm (J ) with n even. Finally,
the third term of Eq. (2.37) can be considered in the crystallographic coordinates,
then its independence of the orientation of M 3d or B ex becomes obvious.
    Going back to Eq. (2.34), the first non-vanishing contribution to the anisotropy
                                                         ˆ3
energy comes from the term in x 2 , which contains trH4f . We thus proceed to its
                                               ˆ
evaluation. We shall use the presentation of H4f as a binomial (2.35), just as we did
when computing trH     ˆ 4f .
                         2

    The invariance of the free energy with respect to time inversion means that
                   ˆ                        ˆ3
all terms odd in J must vanish, while trHCF obviously does not depend on the
                                                            ˆ3
orientation of M 3d . The only source of anisotropy in trH4f is the mixed product
             ˆ 2 ˆ
3 ex tr[(n · J) HCF ].
    2

    Let us write out this expression in the crystallographic coordinates, limiting
ourselves to tetragonal, trigonal and hexagonal point groups as most relevant to
permanent magnets:
                  3   2
                      ex tr
                               ˆ                ˆ                ˆ
                              Jx sin θ cos φ + Jy sin θ sin φ + Jz cos θ
                                                                              2

                           (2) ˆ
                    × B20 C0 (J ) + 4th - and 6th -order terms .                           (2.38)
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                              179


Note that the square of the parenthesis in this equation contains products of the
Cartesian components of the total angular momentum, which are linear combina-
          (n) ˆ
tions of Cm (J ) with n = 2 and 0, C0 ≡ 1:
                                    (0)


                                                                        1

                 ˆ   1           1 (2) ˆ                        1       2
                                                                             (2) ˆ     (2) ˆ
                Jx2 = J (J + 1) – C0 (J ) +                                 C2 (J ) + C–2 (J )
                     3           3                              6
                                                                    1

                 ˆ   1           1 (2) ˆ                       1    2
                                                                             (2) ˆ     (2) ˆ
                Jy2 = J (J + 1) – C0 (J ) –                                 C2 (J ) + C–2 (J )            (2.39)
                     3           3                             6

                 ˆ   1            2 (2) ˆ
                Jz2 = J (J + 1) + C0 (J )
                     3            3
                ....................................................

This is just a transformation inverse to Eqs. (1.17). There is no need to write out the
                    ˆ ˆ                                   (2) ˆ
mixed products, Jx Jy etc., since they do not contain C0 (J ). Due to the orthogo-
                                            (2) ˆ                                (2) ˆ
nality relation (2.36), only the terms in C0 (J ) survive in Eq. (2.38). Since C0 (J )
              ˆ 2     ˆ 2                           1                 2      2
enters into Jx and Jy with the same coefficient – 3 , the terms in sin θ cos φ and in
   2      2
sin θ sin φ will enter in the final expression for F4f (θ , φ) also with the same fac-
tor. Therefore, in this approximation F4f does not depend on the azimuthal angle
φ.
    Carrying out the calculations, we present the RE free energy as follows:

                                          F4f (θ , φ) = F0 + Ea ,
where F0 is the isotropic part and Ea is the anisotropy energy,

                       Ea = K1 sin2 θ + K2 sin4 θ + · · ·                                                 (2.40)
                                 (J + 1)(2J – 1)(2J + 3)
                       K1 = –                            B20 x 2 + O(x 3 ).                               (2.41)
                                          40J
    Thus, the leading term of the high-temperature expansion of the first anisotropy
constant is proportional to the second-order CF parameter B20 and to x 2 . Its
independence of the higher-order CF parameters arises from the orthogonality re-
lation (2.36).
    Quite similarly one arrives at the conclusion that K2 = const. × B40 x 4 + O(x 5 )
(Kuz’min, 1995). In general, the high-temperature series for an anisotropy constant
multiplying sinn θ begins with a term in x n , whose coefficient is a linear combi-
nation of nth -order CF parameters Bnm .12 This fact ensures the convergence of the
expansion (2.40) at small x (high T ).
12   In most high-symmetry cases these combinations contain a single CF parameter. For example, for the anisotropy
constants in Eq. (2.13) one gets K1 ∝ x 2 B20 , K2 ∝ x 4 B40 , K2 ∝ x4 ReB44 , K2 ∝ x 4 ImB44 , K3 ∝ x 6 B60 , K3 ∝
x 6 ReB64 , K3 ∝ x 6 ImB64 .
180                                                                 M.D. Kuz’min and A.M. Tishin




Figure 3.5 The unit cell of RE2 Fe14 B. The triangles indicate the positions of the 4f sites,
occupied by the RE. The other RE sites (4g, not shown) are situated on the vacant diagonals.


      Thus, to terms in x 2 , the anisotropy energy is simply K1 sin2 θ, where
                                                                                 2
                       1                                        μB Bex
             K1 = –      J (J + 1)(2J – 1)(2J + 3)(gJ – 1)2 B20                      .   (2.42)
                      10                                         kT
Note the very special role of the second-order CF parameter B20 . It and it alone
can guarantee the persistence of the anisotropy (and therefore of the coercivity) of
a permanent magnet material to high temperature, which is of vital importance for
most industrial applications.
    Even more important is to have a large exchange field Bex , since K1 is propor-
           2
tional to Bex . The value of Bex depends on temperature. To minimize its reduction
at elevated temperatures, one should not just seek to increase the TC , but also to
reduce the parameter s describing the shape of the dependence M3d (T ) (Kuz’min,
2005),
                                                 3                   5   1
                                      T          2             T     2   3
                    Bex ∝ M3d ∝ 1 – s                – (1 – s)               .           (2.43)
                                      TC                       TC
So far in this Section it has been assumed for simplicity that the local symmetry of
the RE site and the symmetry of the crystal as a whole are described by point groups
allowing just one second-order CF parameter B20 and accordingly, a single second-
order anisotropy term, K1 sin2 θ . However, we have already (Section 1.3) seen an
example of permanent magnet materials, RE2 Fe14 B, whose crystallographic class is
tetragonal, D4h , while the local symmetry is orthorhombic, C2v . The latter admits
an extra second-order CF parameter B22 (purely real in Wybourne’s notation). Con-
sequently, an extra term in sin2 θ should appear in Eq. (2.40), that describing the
anisotropy in the basal plane, K1 sin2 θ cos 2φ, with K1 = const.×B22 x 2 +O(x 3 ). It
turns out upon a closer look at the structure (Fig. 3.5) that four equivalent RE sites
split into two pairs with different orientations of the local symmetry axes. (Recall
that the simplest form of the CF—with two second-order CF parameters—refers
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                  181


to the local axes). Before the macroscopic anisotropy energy can be obtained by
summing up the single-ion contributions, these have to be related to the same co-
ordinates. The simplest way to do it is to rotate the local axes of one-half of the
RE sites (e.g. of those situated at z = 1/2, see Fig. 3.5) through 90° about [001].
Such a rotation is equivalent to a change of sign of B22 for those sites, while their
B20 remains unchanged. As the summation over all RE sites is performed, the terms
∝ B22 cos 2φ, incompatible with the crystallographic class, simply cancel out. Thus,
the presence of a nonzero B22 has no bearing on Eqs. (2.40)–(2.42) or on the ex-
ceptional role of B20 . This applies to most permanent magnet materials, insofar as
they belong to one of the medium-symmetry crystal systems: tetragonal, trigonal
or hexagonal.
    To conclude the discussion of RE2 Fe14 B, we note that there are two non-
equivalent RE sites, 4f and 4g. For simplicity, only the former are shown in Fig. 3.5.
The 4g sites are situated on the vacant diagonals and possess similar symmetry prop-
erties, to the extent that the above-mentioned cancellation of the terms in B22 takes
place for both kinds of RE sites independently. The final expression for K1 should
be a sum of two terms identical to (2.42) but with different B20 and Bex . There is,
however, direct experimental evidence that in Gd2 Fe14 B the exchange fields on the
two Gd sites are equal to within a few percent (Loewenhaupt et al., 1996). This fact
enables us to still use the simple expression (2.42) for the RE2 Fe14 B compounds,
provided that B20 is understood as 1 (B20 + B20 ). The averaging here is justified
                                          4f      4g
                                       2
by the fact that Bex = Bex and does not require that B20 ≈ B20 . A recent X-ray
                    4f      4g                              4f     4g

                                            4f       4g
diffraction experiment has revealed that B20 and B20 are essentially different (Haskel
et al., 2005).
    In order to set the lower bound to the domain of validity of the high-
temperature approximation, the expansion in powers of x should be continued.
It was established (Kuz’min, 1995) that the main contribution to K1 beyond x 2
comes from the term ∝ B20 x 4 (even though nonzero contributions from other CF
parameters may be present as well—not necessarily linear). In this approximation

                               (J + 1)(2J – 1)(2J + 3)
                     K1 = –                            B20 x 2 (1 – dJ x 2 )   (2.44)
                                        40J
where

                                               8J 2 + 8J + 5
                                       dJ =                  .                 (2.45)
                                                    84J 2
The quantity dJ is practically independent of J , varying between 0.11 for J = 8
and 0.14 for J = 5/2 (Kuz’min, 1995, Table I). For estimations one can take the
fractional error of the high-temperature approximation to be just 0.12x 2 for all
RE’s. This translates to 7% at T = 300 K for Tm2 Fe14 B. Though it may not always
be sufficient for accurate calculations in the room-temperature range, the simplicity
of Eq. (2.42) makes it nevertheless useful for analysing the behaviour of permanent
magnet materials at T > 300 K.
182                                                                                  M.D. Kuz’min and A.M. Tishin



  In the same approximation, to T –2 , Eq. (2.31) yields for the RE magnetic mo-
ment (Boutron, 1973):
                                    C       1                B20
                     μRx,y = Bx,y      1 + (2J – 1)(2J + 3)      + ···                                       (2.46)
                                    T       20               kT
                                  C       1                B20
                       μRz   = Bz     1 – (2J – 1)(2J + 3)     + ···                                         (2.47)
                                  T      10                kT
where B = 2(1 – gJ )Bex + B is the effective magnetic field on the RE, C =
                      –1

J (J +1)gJ μB /3k is the Curie constant. Within the same accuracy, the susceptibility
          2 2

can be recast in the Curie–Weiss form,
                                              C
                                   χ ,⊥ =                                     (2.48)
                                          T – θ ,⊥
where
                                   1
                          kθ = (2J – 1)(2J + 3)B20                            (2.49)
                                  10
                                     1
                         kθ⊥ = – (2J – 1)(2J + 3)B20 .                        (2.50)
                                   20
Note the validity of the Elliott formula (Elliott, 1965):
                                         3
                               k(θ – θ⊥ ) =(2J – 1)(2J + 3)B20 .                  (2.51)
                                        20
The first anisotropy constant K1 can be presented as
                                           χ – χ⊥
                                    K1 =          (B )2                           (2.52)
                                              2
which in the considered approximation is equivalent to the earlier obtained result
(2.42), provided that B        Bex . In other words, the RE subsystem behaves in this
approximation as an anisotropic paramagnet in an effective magnetic field. For this
reason at high T the susceptibility anisotropy, χ – χ⊥ , just like K1 , depends on a
single CF parameter B20 .
     To finalise this section, let us consider the effect of non-collinearity of the sub-
lattices and the possibility to allow for it by introducing two sets of orientation
angles and anisotropy constants, one for each sublattice. (Up no now we had to do
with a single set of angles, θ and φ, corresponding to the orientation of M3d , while
the respective anisotropy constants were mere sums, K3d + K4f ).
     Examination of Eqs. (2.46), (2.47) reveals two sources of non-collinearity: (i)
possible violation of the condition B        Bex while B Bex , and (ii) the terms in B20
in square brackets—a purely CF effect. As a result, M3d Bex B μR . We exclude
from the outset the possibility that the strong inequality B          Bex may fail—the
concept of anisotropy constants formulated in Section 2.1 requires that |μR | be
independent of B.13 Our consideration in this section will therefore be limited to
the CF-induced non-collinearity (ii).
 13 A situation when B ∼ B is neither unattainable experimentally (see e.g. Kostyuchenko et al., 2003) nor intractable
                            ex
theoretically—the formalism of Section 2.3 still applies.
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                   183


    Let us rewrite the square bracket of Eq. (2.47) as follows,
                                  (2J – 1)(2J + 3)
                                     1–              ηx                 (2.53)
                                         10J
where η is defined by Eq. (2.16). Assume that the system under consideration is
at least broadly exchange-dominated. Then its parameter η must not exceed the
η-axis intercept of the plane (2.24) delimiting the central polyhedron,
                                             2
                                          |η| <    .                           (2.54)
                                         3(2J – 1)
In a strictly exchange-dominated case this should become a strong inequality, cf.
(2.25). By virtue of (2.54), the square bracket of Eq. (2.47) can be recast as
                                                 1 + ax                         (2.55)
where |a| < (2J + 3)/15J ≈ 0.2. It will be recalled that the high-temperature
approximation is valid when the quantity dJ x 2 ≈ 0.12x 2 in Eq. (2.44) is small
as compared with unity. Fulfilment of this condition for a hard magnetic material
guarantees the smallness of the anisotropic terms in B20 in Eqs. (2.46), (2.47). Hence
the approximate collinearity of μR and B ≈ B ex M 3d , which becomes exact in
the limit x → 0, B /Bex → 0.
    One should take into account that permanent magnet materials are mostly light
RE-iron or cobalt intermetallics rich in the 3d element, so that the RE contributes
only a small part of the total magnetisation. Therefore, the apparent CF-induced
non-collinearity, that is the deviation of M 3d (its orientation defined by θ , φ) from
M 3d + M 4f , will be even less significant than suggested by the estimate (2.55).
    Conversely, the non-collinearity can play a very important role in heavy RE-
based ferrimagnets, especially when |M 4f | ≈ |M 3d |. We shall not consider such
systems any further since they were described in detail in Volume 9 of this Hand-
book (Zvezdin, 1995).
    Let us go back to the more important for applications class of phenomena
when the material is an iron- or cobalt-rich light RE-based intermetallic com-
pound and the applied magnetic field does not exceed what can be expected from
a permanent-magnet assembly, ∼2 T. In such a situation the high-temperature ap-
proximation (2.42) applies at and above room temperature. Its validity justifies the
truncation of the expansion (2.40). Furthermore, it guarantees the smallness of the
non-collinearity effects, so that the system behaves essentially as a single-sublattice
magnet.
    Validity of the high-temperature approximation has also interesting conse-
quences for spin reorientation transitions (SRT) in these materials. This question,
however, will be deferred until Section 3, devoted specially to the SRT.

2.5 The linear-in-CF approximation: main relations
In the preceding subsection we deduced a truncated quasi-single-sublattice expres-
sion for the anisotropy energy, Ea = K1 sin2 θ, and obtained an explicit formula
for the RE contribution to K1 . All this was achieved thanks to the presence of the
184                                                              M.D. Kuz’min and A.M. Tishin



small parameter x = J ex /kT . No specific restrictions had to be imposed on the
CF, as long as it remained not much stronger than the 3d-4f exchange, | i |     ex .
(Otherwise the high-temperature expansion should be in powers of i /kT rather
than x.)
   Let us now consider a special case when it is additionally known that the CF on
the RE is weak as compared with the 3d-4f exchange. The definition of a strictly
exchange-dominated RE system (2.25) can be rewritten as follows:
                                |Bnm |    (2J )1–n    ex .                           (2.56)
This condition applies, rigorously speaking, only to Bn0 . The extension to all
CF parameters is based upon a probably not unreasonable assumption that off-
diagonal CF parameters cannot be much greater than their diagonal counterparts:
|Bnm | ≤ |Bn0 |, m = 0. Anyhow, the smallness of the CF, justifying its treatment as
a perturbation with respect to the exchange, is the principal starting point of this
subsection.
    Our second assumption concerns the strength of applied magnetic field B.
Namely, we assume that B always remains much weaker than the 3d-4f exchange
field Bex , to make sure that the magnitude of the RE magnetic moment does
not depend on B. This is a necessary condition for the use of the formalism of
anisotropy constants (Section 2.1).
    Thus, in zeroth approximation only the first term of the RE Hamiltonian (2.14)
or (2.27) is taken into consideration. The result is an equidistant energy spectrum,
               EM = sign(1 – gJ )    ex M,     M = –J , –J + 1, . . . , J ,          (2.57)
leading to the well-known partition function (Smart, 1966),
                                          sinh( 2J +1 x)
                               ZJ (x) =          2J
                                                    1
                                                             .                       (2.58)
                                             sinh( 2J x)
Note that here, unlike in the preceding subsection, the Langevin ratio x (2.33) is
not necessarily small.
   The RE magnetic moment in this approximation is given by
                                ˆ
                   μR = –gJ μB Jz = sign(1 – gJ )gJ μB J BJ (x).                     (2.59)
The z axis here is directed along the 3d magnetisation vector M 3d , so that B ex points
in the negative z direction. There is of course no anisotropy or non-collinearity in
the zeroth approximation.
    Allowance for a nonzero applied magnetic field will to first approximation add
a Zeeman term –μR · B to the zeroth-order RE free energy, F4f = –kT ln ZJ (x),
whereas allowance for the CF will produce the anisotropy energy Ea . Thus, F4f
will acquire the structure of Eq. (2.7). Our primary objective in this subsection
is to compute the first-order, or linear in Bnm contribution to Ea . Note that the
absence of non-collinearity effects in this approximation is a natural feature of the
perturbation theory. The truncation of the expansion of Ea after the terms in sin6 θ
will occur automatically due to the presence of the small parameter.
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                    185


   The first-order anisotropy correction to F4f is merely a thermal average of the
              ˆ
perturbation HCF taken over the eigenstates of the unperturbed Hamiltonian:
                                     ˆ                             (n) ˆ
                               Ea = HCF =                     Bn0 C0 (J ) .      (2.60)
                                                    n=2,4,6

                                 (n) ˆ
This expression contains only Cm (J ) with m = 0, diagonal in the J M represen-
tation. We wish to emphasize that these operators are defined in the coordinate
system with z M 3d , the latter not necessarily parallel to any of the high-symmetry
crystal directions. To mark this fact, the CF parameters in Eq. (2.60) are primed.
They are related to the ‘usual’ non-primed CF parameters (defined with respect to
the crystallographic axes) by means of the following linear transformation:
                                               n
                                    Bn0 =                (n)
                                                    Bnm Cm (θ , φ).              (2.61)
                                             m=–n

The rotation angles θ and φ are just the angles determining the orientation of M 3d
in relation to the crystallographic axes.
                                                              (n) ˆ
    To find the average values in Eq. (2.60), we note that C0 (J ) are operator poly-
             ˆ
nomials in Jz , of order n and of the corresponding parity, cf. the explicit expressions
                                                                   ˆ
(Lindgård and Danielsen, 1974). The averages of powers of Jz can be computed
with the aid of the following identity (Kazakov and Andreeva, 1970):

                         ˆ       1                               xM
                        Jzn =             M n exp –sign(1 – gJ )
                               ZJ (x)                             J
                               [–J sign(1 – gJ )] d ZJ
                                                 n n
                             =                          .                        (2.62)
                                     ZJ (x)        dx n
                                                  ˆ
Apparently, for any given J and n, the quantity Jzn is up to a sign determined by
                                                        (n) ˆ
the Langevin ratio x alone. Therefore, the averages of C0 (J ) can be conveniently
presented as functions of x, the so-called generalised Brillouin functions (GBF)
  (n)
BJ (x):
                              (n) ˆ                   n
                             C0 (J ) = –sign(1 – gJ ) J n BJ (x).
                                                           (n)
                                                                                 (2.63)
Naturally, the sign only matters for n odd, whereas relevant to magnetic anisotropy
are n = 2, 4, 6. One could in principle omit the cumbersome square bracket from
Eq. (2.63), as it was done by Kuz’min (1992). However, we prefer to keep the sign
multiplyer, since we intend on using GBF with odd n later on in this section. In
fact, Eq. (2.59) above is nothing else but a special case of Eq. (2.63) with n = 1.
    By virtue of Eqs. (2.61), (2.63), the anisotropy energy (2.60) takes the standard
form (2.10), with the anisotropy constants given by
                                        κnm = Bnm J n BJ (x)
                                                       (n)
                                                                                 (2.64)
where n = 2, 4, 6. All higher-order anisotropy constants vanish in this approxima-
tion.
186                                                              M.D. Kuz’min and A.M. Tishin



     Equation (2.64) is the main result of the linear (in Bnm ) theory of magnetocrys-
talline anisotropy. Its convenience is the one-to-one correspondence between the
quantities κnm and Bnm . The temperature dependence of each κnm is given by a sin-
gle GBF. Expressions for the more conventional anisotropy constants Ki are readily
obtainable hence. For example, those entering in Eq. (2.12), relevant to the hexag-
onal point groups D6 , C6v , D3h and D6h , are given by:

                    3                                   21
             K1 = – B20 J 2 BJ (x) – 5B40 J 4 BJ (x) – B60 J 6 BJ (x)
                              (2)               (4)             (6)
                    2                                    2
                  35                   189
             K2 = B40 J 4 BJ (x) +
                              (4)                   (6)
                                           B60 J 6 BJ (x)
                   8                     8
                    231                                                              (2.65)
             K3 = –               (6)
                        B60 J 6 BJ (x)
                     16
                  √
                     231
             K3 =                  (6)
                         B66 J 6 BJ (x).
                    16
    The CF parameters in these relations are normalised according to Wybourne
(1965). The conversion to the Stevens convention is straightforward, an example
was given by Kuz’min et al. (1996). The only ‘advantage’ of the Stevens notation
is that the coefficients in Eqs. (2.65) become integers. Note the division of the
anisotropy constants and CF parameters in two groups: axial Ki ∝ linear combi-
nations of Bn0 , and basal-plane anisotropy constants Ki ∝ Bnm with m = 0. The
temperature dependence in all cases is described by three GBF, BJ (x), n = 2, 4, 6.
                                                                        (n)

    We wish to point out that while all anisotropy constants of order higher than six
are strictly nil in the considered approximation, there is no grounds whatsoever for
assuming hierarchical intensity relations of type |K1 |       |K2 |      |K3 | among those
which are nonzero. Such a situation may be realised at high temperature, x              1,
where BJ (x) ∝ x n and therefore BJ (x)
           (n)                            (2)         (4)
                                                    BJ (x)          (6)
                                                                 BJ (x). When it does
happen, it is only because x is small, irrespective of the strength of the CF in relation
to the 3d-4f exchange (see the preceding subsection). At low temperatures (large x)
all GBF are ∼1 and consequently all nonzero anisotropy constants are of the same
order of magnitude, cf. the coefficients of B60 in Eqs. (2.65). Neglecting K3 (or K3
and K2 ) in this situation is a serious mistake.
    It has been tacitly assumed that the local symmetry of the RE site and the
crystallographic class are described by the same point group. Another possibility is
that the local symmetry is lower than the crystallographic class, then some extra CF
parameters may be allowed. The contributions from the latter must however cancel
out upon summation over all RE atoms, just like it happened to B22 in RE2 Fe14 B
in the previous subsection. In the linear approximation such ‘latent’ CF parameters
do not affect the macroscopic magnetic anisotropy at all.
    Let us now turn to computing first-order CF corrections to the RE magnetic
moment (2.59). We should in principle repeat the calculation of the RE free en-
ergy with a nonzero applied magnetic field B and differentiate the former with
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                              187


respect to the latter. It is easier, however, to exploit the formal similarity of B
and Bex and to differentiate the already obtained expression for F4f with respect
to ex :

                                     ˆ                       ∂F4f
                        μR = –gJ μB Jz = –sign(1 – gJ )gJ μB                                (2.66)
                                                             ∂ ex
where

                         F4f = –kT ln ZJ (x) +                             (n)
                                                                  Bn0 J n BJ (x)            (2.67)
                                                        n=2,4,6

and x is related to ex through Eq. (2.33). The first term in Eq. (2.67) is
the isotropic part of the free energy while the sum represents the Ea . The an-
gles θ and φ are supposed to be constant and determine the easy magnetisa-
tion direction with respect to the crystallographic axes. (Typically in permanent-
magnet materials θ = 0, in which case the primes of the CF parameters in
Eq. (2.67) may be omitted.) Carrying out the differentiation in Eq. (2.66), we
arrive at
                                                                             Bn0
                 μR = sign(1 – gJ )gJ μB J BJ (x) 1 –                               (n)
                                                                                   DJ (x)   (2.68)
                                                                   n=2,4,6    ex


where
                                  (n)
                        J n–1 x dBJ (x)
         DJ (x) =
          (n)
                        BJ (x) dx
                                  n + 1 BJ (x)
                                         (n+1)
                    =J    n–1
                                x                  (n)
                                                – BJ (x)
                                  2n + 1 BJ (x)
                                        n (2J + n + 1)(2J – n + 1) BJ (x)
                                                                    (n–1)
                                  +                                        . (2.69)
                                      2n + 1        4J 2            BJ (x)
The structure of Eq. (2.68) is rather obvious: the prefactor of the square bracket is
the free-ion value (2.59), while the sum inside the square bracket is the CF correc-
tion, ∝ Bn0 / ex . The derivative has been taken with the aid of Eq. (2.78), proved
in the next subsection.
    Looking back at the main relations for Ea and μR obtained in the linear-in-
CF approximation, we note the central role played by the GBF, defined by means
of Eq. (2.63). In the case of Ea these are just three functions, with n = 2, 4, 6,
whereas the expression for the magnetic moment also contains GBF with n odd.
To understand the obtained results, we need to know some general properties of
the GBF. These will be formulated—and in most cases also proved—in the next
subsection. The discussion of the linear approximation will then be resumed in
Section 2.7.
188                                                               M.D. Kuz’min and A.M. Tishin



2.6 Properties of generalised Brillouin functions
2.6.1 Some elementary properties
2.6.1.1 Special values of n The following relations are obtainable directly from
                                                              ˆ
the definition (2.63), taking into account that C0 ≡ 1, C0 = Jz :
                                                (0)     (1)



         BJ (x) ≡ 1
          (0)
                                                                                      (2.70)
                               2J + 1      2J + 1      1       1
         BJ (x) = BJ (x) =
          (1)
                                      coth        x –    coth    x .                  (2.71)
                                 2J          2J       2J      2J
Thus, a GBF of order one is the usual Brillouin function. Hence the term ‘gener-
alised’ Brillouin functions.
2.6.1.2 Triangle inequality The GBF equal identically zero, unless n ≤ 2J . In-
deed, by virtue of the definition (2.63) and the Wigner–Eckart theorem (1.24),
                                          J
                             (n)
                            BJ (x)   ∝          e–xM/J CJ Mn0 .
                                                        JM
                                                                                      (2.72)
                                         M=–J

The CGC on the right vanish if the triangle inequality within the triplet (J , J , n)
is not satisfied (Varshalovich et al., 1988).
2.6.1.3 Parity   GBF of odd/even order n are, respectively, odd/even functions of
x:

                              BJ (–x) = (–1)n BJ (x).
                               (n)             (n)
                                                                                      (2.73)
This follows from Eq. (2.72) and the known parity property of the CGC
(Varshalovich et al., 1988):

                               CJ –Mn0 = (–1)n CJ Mn0 .
                                J –M            JM


2.6.1.4 Monotonicity     For any x > 0, 0 < n ≤ 2J ,
                                       (n)
                                     dBJ (x)
                                              > 0.                              (2.74)
                                       dx
The proof for arbitrary n is rather complicated and is not reproduced here. For
n = 1 it follows from the fact that the square bracket in Eq. (2.80) is the dispersion
    ˆ
of Jz , a positive-definite quantity.
2.6.1.5 The limit x → 0 (T → ∞)

                                              1, if n = 0
                              BJ (0) =
                               (n)
                                                                                      (2.75)
                                              0, if n > 0.
Taking the average in Eq. (2.63) is straightforward in this limit, since all Boltzmann’s
                                                                     (n) ˆ
exponentials equal unity and therefore BJ (0) ∝ (2J + 1)–1 trC0 (J ).
                                           (n)
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                   189


2.6.1.6 Asymptotic behaviour at low temperatures (x                J)
                           1      (2J )!     n(n + 1)          x
              BJ (x) ≈
               (n)
                              n (2J – n)!
                                          1–            exp –      + ··· .       (2.76)
                        (2J )                    2J           J
The prefactor is obtained in the limit T = 0 by averaging over the ground
state (M = J ) using Eq. (1.24) and the explicit expression for the CGC CJ J n0    JJ

(Varshalovich et al., 1988). Allowing for the (exponentially small) population of the
first excited level yields the correction term of Eq. (2.76). The special case of n = 1
describes the magnetic moment:
                              μR ∝ BJ (x) ≈ 1 – J –1 e–x/J .
Hence follows the approximate relation:
                                 BJ (x) ∝ (μR )n(n+1)/2
                                   (n)
                                                                                 (2.77)
known as Zener’s n(n + 1)/2 power law (Zener, 1954; Callen and Callen, 1966;
Goodings and Southern, 1971).

2.6.2 Differential relations and properties thence deduced
2.6.2.1 First derivative The first derivative of a GBF is given by
                (n)
            dBJ (x)           n + 1 (n+1)
                          =           BJ (x) – BJ (x)BJ (x)  (n)
                dx           2n + 1
                                    n (2J + n + 1)(2J – n + 1) (n–1)
                             +                                        BJ (x).  (2.78)
                                 2n + 1                4J 2
To prove this relation, differentiate the identity (2.62) with respect to x one more
time (the discrete parameters J and n = k being fixed):
                            ˆ
                         d Jzk        sign(1 – gJ )
                                 =–                        ˆ ˆ
                                                    cov Jz , Jzk
                          dx                J
                                      sign(1 – gJ ) ˆk+1          ˆ ˆ
                                 =–                   Jz     – Jz Jzk .        (2.79)
                                            J
Multiplying this relation by an appropriate coefficient and summing up over k of
                                                                (n) ˆ
the same parity so as to assemble the expression for C0 (J ), one arrives at
                     (n) ˆ
              d C0 (J )            sign(1 – gJ ) ˆ (n) ˆ            ˆ (n) ˆ
                              =–                   Jz C0 (J ) – Jz C0 (J )     (2.80)
                    dx                   J
or, on foot of the definition (2.63),
                 (n)
             dBJ (x)          [–sign(1 – gJ )]n+1 ˆ (n) ˆ
                          =             n+1
                                                                         (n)
                                                    Jz C0 (J ) – BJ (x)BJ (x). (2.81)
                dx                    J
Transforming the first term by means of the identity
             ˆ (n) ˆ           n + 1 (n+1) ˆ
            Jz C0 (J ) =              C      (J )
                              2n + 1 0
                                     n (2J + n + 1)(2J – n + 1) (n–1) ˆ
                              +                                        C0 (J ) (2.82)
                                 2n + 1                  4
one finally obtains Eq. (2.78).
190                                                             M.D. Kuz’min and A.M. Tishin



     It remains to prove Eq. (2.82). It is apparently an expansion of the product
   (1) (n)
C0 C0 in irreducible tensor operators of appropriate parity. The triangle rule ad-
                       (n±1)                                 (n+1)
mits only terms in C0 . The prefactor of the term in C0            is readily obtained by
equating the coefficients of J ˆzn+1 on both sides of Eq. (2.82). Note that the factor of
 ˆ        (n) ˆ
Jzn in C0 (J ) is the same as that of the leading term of the corresponding Legendre
polynomial, i.e. 2–n (2n)!/(n!)2 . Thus, we get

                 ˆ (n) ˆ     n + 1 (n+1) ˆ               (n–1) ˆ
                Jz C0 (J ) =       C    (J ) + f (n, J )C0 (J ).                    (2.83)
                             2n + 1 0
The remaining unknown factor f (n, J ) is evaluated by substituting Eq. (2.83) into
(2.81) and letting x go to infinity, whereas BJ (x) → (2J )–n (2J )!/(2J – n)! and
                                             (n)

dBJ (x)/dx → 0, cf. Eq. (2.76). This completes the proof of Eq. (2.82) and
    (n)

consequently of Eq. (2.78).
2.6.2.2 Power series expansion       At small x (high temperatures) the GBF can be
presented as follows:

                              1             (2J + n + 1)!
           BJ (x) =
            (n)
                       (2n + 1)!!(2J ) 2n (2J + 1)(2J – n)!


                                  J (J + 1) + 1 (n + 3) n+2
                       × xn – n                 8
                                                       x + O(x n+4 ) .              (2.84)
                                       3J 2 (2n + 3)
The leading coefficient of this expansion is readily obtained by applying the differ-
ential relation (2.78) k times, in order to compute the k th derivative of the GBF.
Note that only the lowest-order GBF needs to be followed. Thus we write:

   (n)
 dBJ (x)     n (2J + n + 1)(2J – n + 1) (n–1)
         =                             BJ (x) + higher-order GBF
   dx      2n + 1       (2J )2
     (n)
d 2 BJ (x)       n(n – 1)
           =
    dx 2     (2n + 1)(2n – 1)
                   (2J + n + 1)(2J + n)(2J – n + 1)(2J – n + 2) (n–2)
               ×                                               BJ (x)
                                      (2J )4
               + higher-order GBF
...........................................................................
     (n)
d k BJ (x)          n!/(n – k)!
           =
    dx k     (2n + 1)!!/(2n – 2k + 1)!!
                      (2J + n + 1)!(2J – n + k)!
               ×                                     B (n–k) (x) + higher-order GBF.
                   (2J )2k (2J + 1 + n – k)!(2J – n)! J
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                                  191


 By virtue of Eq. (2.75), one gets at x = 0:

                                      ⎧
           d   k    (n)
                   BJ (x)             ⎨0,                                                      if k < n
                                  =
                                 n!           (2J + n + 1)!               (2.85)
                   dx k     x=0
                                      ⎩                       , if k = n.
                           (2n + 1)!(2J )2n (2J + 1)(2J – n)!

Hence follows the leading coefficient of the expansion (2.84).
   The coefficient of the term in x n+2 is derived along the same lines, but this
requires more cumbersome algebra and will not be reproduced here.
2.6.2.3 Explicit expressions                Equation (2.78) can be recast as a recurrence formula
                         2n + 1 dBJ (x)
                                      (n)
                    BJ (x) =
                     (n+1)
                                             + BJ (x)BJ (x)
                                                      (n)
                          n+1        dx
                              n (2J + n + 1)(2J – n + 1) (n–1)
                          –                                  BJ (x).         (2.86)
                            n+1               4J 2
Starting from the already known GBF of zeroth and first orders (2.70), (2.71) and
working up, one can obtain in explicit form the GBF of an arbitrarily high order.
Note the cancellation of the terms in coth2 [x(2J + 1)/2J ] arising from the deriv-
ative and from the product of two GBF in the square bracket of Eq. (2.86). As a
consequence, GBF of any order n > 1 are linear in coth[x(2J +1)/2J ]. Conversely,
the highest power of coth[x /2J ] increments by one each time Eq. (2.86) is used.
Therefore, the GBF can be presented as follows (Kuz’min, 1992):
                                                                               2J + 1
                            BJ (x) = Pn (ξ , η) – Qn (ξ , η) coth
                             (n)
                                                                                      x
                                                                                 2J
                        1       x            1
                          cothξ=        η=     .                                                              (2.87)
                       2J      2J           2J
The polynomials Pn and Qn obey the following recurrence relations:
                         2n + 1                ∂Pn
                       Pn+1 =      (η2 – ξ 2 )     – (1 + η)Qn – ξ Pn
                         n+1                   ∂ξ
                              n
                         –          1 + 2η + (1 – n2 )η2 Pn–1
                           n+1                                                  (2.88)
                         2n + 1            2 ∂Qn
                Qn+1 =                2
                                   (η – ξ )        – (1 + η)Pn – ξ Qn
                         n+1                    ∂ξ
                              n
                         –          1 + 2η + (1 – n2 )η2 Qn–1 .
                           n+1
   Table 3.3 contains explicit expressions for Pn (ξ , η) and Qn (ξ , η) with n ≤ 7
obtained by means of Eqs. (2.88). Note that for n odd these functions differ in sign
from the analogous expressions of Magnani et al. (2003).14 The convention adopted
herein ensures that the plots of all GBF at x > 0 lie entirely in the first quadrant.
 14 We also note a misprint in their A (ξ , η): the inner-most parenthesis (–22 + 3η) should be multiplied by –η rather
                                      5
than by +η.
192                                                                      M.D. Kuz’min and A.M. Tishin


Table 3.3 Polynomials Pn (ξ , η) and Qn (ξ , η) entering in the explicit expression (2.87) for the
generalised Brillouin functions

P1 = –ξ
Q1 = –(1 + η)
P2 = 3ξ 2 + 1 + 2η
Q2 = 3ξ(1 + η)
P3 = –3ξ(5ξ 2 + 2 + 4η – η2 )
Q3 = –(1 + η)(15ξ 2 + 1 + 2η – 3η2 )
P4 = 105ξ 4 + 45ξ 2 (1 + 2η – η2 ) + 1 + 4η – 4η2 – 16η3
Q4 = 5ξ(1 + η)(21ξ 2 + 2 + 4η – 9η2 )
P5 = –15ξ [63ξ 4 + 14ξ 2 (2 + 4η – 3η2 ) + 1 + 4η – 7η2 – 22η3 + 3η4 ]
Q5 = –(1 + η)[945ξ 4 + 105ξ 2 (1 + 2η – 6η2 ) + 1 + 4η – 14η2 – 36η3 + 45η4 ]
P6 = 10395ξ 6 + 4725ξ 4 (1 + 2η – 2η2 ) + 105ξ 2 (2 + 8η
       – 20η2 – 56η3 + 15η4 ) + 1 + 6η – 20η2 – 120η3 + 64η4 + 384η5
Q6 = 21ξ(1 + η)[495ξ 4 + 30ξ 2 (2 + 4η – 15η2 ) + 1 + 4η – 19η2 – 46η3 + 75η4 ]
P7 = –7ξ [19305ξ 6 + 4455ξ 4 (2 + 4η – 5η2 ) + 225ξ 2 (2 + 8η – 26η2 – 68η3 + 27η4 ) + 4
       + 24η – 110η2 – 600η3 + 556η4 + 2376η5 – 225η6 ]
Q7 = –(1 + η)[135135ξ 6 + 17325ξ 4 (1 + 2η – 9η2 ) + 189ξ 2 (2 + 8η – 48η2 – 112η3
       + 225η4 ) + 1 + 6η – 41η2 – 204η3 + 463η4 + 1350η5 – 1575η6 ]



   The explicit expressions for the GBF were first obtained by Brillouin (1927,
n = 1), Yoshida (1951, n = 2), Kazakov and Andreeva (1970, n = 4, 6), Kuz’min
(2002, n = 3), Magnani et al. (2003, n = 5, 7).
2.6.2.4 The quasi-classical limit, J → ∞ In the limit of very large J the GBF
turn into the so-called reduced modified Bessel functions:
                                                 ˆ
                                   lim BJ (x) = In+ 1 (x).
                                        (n)
                                                                                             (2.89)
                                  J →∞                    2


The latter were introduced to the theory of magnetic anisotropy by Keffer (1955)
and are defined as follows:
                                  ˆ
                                 In+ 1 (x) = In+ 1 (x)/I 1 (x)                               (2.90)
                                     2            2           2

where Iν (x) are modified spherical Bessel functions of the first kind (Abramowitz
and Stegun, 1972, Chapter 10).
                                            ˆ
   To prove (2.89), note that by definition I 1 (x) ≡ 1 = BJ (x), while
                                                          (0)
                                                      2


                      ˆ
                     I3/2 (x) = coth x – 1/x = L(x) = lim BJ (x)                             (2.91)
                                                                  J →∞
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                     193


that is the well-known Langevin function. Finally, the recurrence formula (2.86)
goes over to
                              ˆ
                  2n + 1 d In+1/2 (x)                          n ˆ
      ˆ
     In+3/2 (x) =                         ˆ       ˆ
                                       + I3/2 (x)In+1/2 (x) –     In–1/2 (x) (2.92)
                   n+1           dx                           n+1
whose validity can be easily verified using the recurrence relation for the modified
spherical Bessel functions, Eq. (10.2.19) of Abramowitz and Stegun (1972). Thus,
                                            ˆ
by induction, for any n > 0 the function In+3/2 (x) obtained by means of Eq. (2.92)
                                         (n+1)
is the quasi-classical limit of the GBF BJ (x), q.e.d.
                                             (n)
2.6.3 Selected properties of the functions DJ (x)
                                                      (n)
Most of these readily follow from the definition of DJ (x), Eq. (2.69), and the
corresponding properties of the GBF. For example, the parity:
                                   DJ (–x) = (–1)n–1 DJ (x).
                                    (n)               (n)
                                                                                   (2.93)
It follows from the monotonicity of the GBF that
                                  (n)
                                 DJ (x) > 0,         if x > 0, n > 0.              (2.94)
The asymptotic behaviour at large x(x                  J ) is described by
                                            n(n + 1) (2J )!
                             DJ (x) ≈
                              (n)
                                                                x e–x/J .          (2.95)
                                             2n+1 J 3 (2J – n)!
Note that DJ (x) → 0 as x → ∞.
           (n)

  Alternatively, for x small, one has
                         3n               (2J + n + 1)!
    DJ (x) =
     (n)
                                                              x n–1 + O(x n+1 ).   (2.96)
                   (2n + 1)!!22n J n (J + 1)(2J + 1)(2J – n)!


Obviously, DJ (0) = 0 for all n > 1.
               (n)

   Vanishing at both ends of the semi-infinite interval 0 < x < ∞, positive and
                                                  (n)
continuous everywhere within, the functions DJ (x), n > 1, must have at least one
maximum at a certain point xmax > 0. In fact, there is exactly one maximum on the
positive semi-axis, see Fig. 3.6. For larger n the maxima are situated farther to the
right, their height scaling roughly as J n–1 /n.

2.7 The linear-in-CF approximation (continued)
Let us now return to the discussion of the main results of the linear theory,
Eqs. (2.64), (2.65), (2.68). We shall rely on our newly acquired knowledge of the
properties of the GBF. It is convenient to plot the GBF BJ (x), n = 2, 4, 6, against
                                                           (n)

inverse Langevin’s ratio 1/x, which is approximately proportional to absolute tem-
perature T , Fig. 3.7. A feature that immediately draws attention in this graph is
the presence of plateaus in the low-temperature region. On account of the expo-
nentially rapid approach to saturation characteristic of the GBF, cf. Eq. (2.76), the
plateaus in Fig. 3.7 have fairly sharply defined widths ∼1/5J .
194                                                                M.D. Kuz’min and A.M. Tishin




                                     (n)
Figure 3.6 Graphs of the functions DJ (x) (rescaled) with J = 6 and n = 2, 4, 6 (a); rescaled
                                                                       (n)
positions (b) and rescaled heights (c) of the maxima of the functions DJ (x), plotted vs J .
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                          195




           Figure 3.7   Generalised Brillouin functions for J = 8, plotted against 1/x.


    The plateaus disappear completely in the limit J → ∞, as the GBF go over to
the reduced Bessel functions (2.89), employed in the quasi-classical theory of mag-
netic anisotropy (Keffer, 1955; Callen and Callen, 1966). The awkward fact that this
constitutes a violation of the third law of thermodynamics was circumvented in the
classical theory by putting to the fore the dependence of the anisotropy constants on
magnetisation, rather than on temperature. This maneuvering has become obsolete
after the introduction of the GBF.
    Let us reiterate: below a certain point all anisotropy constants become inde-
pendent of temperature, or saturated. One of the consequences of this is that
spontaneous spin reorientation transitions (SRT) can take place only above a cer-
tain temperature. Quantitatively this temperature is determined by the exchange
field on the RE. For example, for HoFe2 , where the relation between x and T is
given by x ≈ 750/T (Kuz’min, 2001), the functions B8 (x) are saturated below
                                                           (n)

1/x ≈ 0.02 (Fig. 3.7), or T ≈ 15 K.
    The height of a low-temperature plateau is determined by the numerical factor
in front of the square bracket in Eq. (2.76); it tends to unity only when J → ∞.
For example, BJ (∞) = 1 – 1/2J . Accordingly, the low-temperature values of the
                 (2)

anisotropy constants are given by (Goodings and Southern, 1971):
                                            (2J )!
                                   κnm |T =0 =       Bnm .                     (2.97)
                                                  2n (2J
                                               – n)!
Here Bnm are the CF parameters normalised according to Wybourne. They include
the Stevens factors, therefore, their magnitudes decrease as n increases. As against
that, the coefficients of Bnm in Eq. (2.97) grow with n. As a result, at low tempera-
tures the anisotropy constants κ2m , κ4m and κ6m are of the same order of magnitude.
Similarly, there is no reason to presume that either K3 or K2 in Eq. (2.12) could be
neglected at low temperatures. (Terms of order higher than six in sin θ do vanish
though.)
196                                                              M.D. Kuz’min and A.M. Tishin




Figure 3.8   Three possible forms of temperature dependence of second anisotropy constant.


   The situation is quite different in the high-temperature case. According to
Eqs. (2.65), (2.84), Ki fall off with temperature as T –2i . Therefore, at around room
                                             (4)           (6)
temperature and above it one can neglect BJ (x) and BJ (x). In this approximation
K2 and K3 vanish, while K1 is given by
                                       3
                               K1 = – B20 J 2 BJ (x).
                                                 (2)
                                                                               (2.98)
                                       2
The quality of this approximation can be judged by the linearity of the magneti-
sation curves along the hard direction. Thus, the room-temperature magnetisation
curves of RE2 Fe14 B with the heaviest RE are practically linear (Yamada et al.,
1988). However, in the case of Nd2 Fe14 B one can still see some residual curvature
at T = 290 K, which disappears at higher temperatures. In such a situation it may
be sensible to leave K1 and K2 and to neglect K3 .
    From the fact that the GBF with n > 0 vanish at x = 0 and grow at any x > 0
it follows that these GBF are positive within the physically meaningful interval of
values of x, 0 < x < ∞. Therefore, the signs of the anisotropy constants κnm
entering in Eq. (2.10) are determined by the signs of the respective CF parameters
Bnm (these in turn depending on the signs of the Stevens factors) and cannot change
as temperature varies. The same is true in relation to the anisotropy constants K3
and K3 , cf. Eqs. (2.65). The latter quantity determines the orientation of the easy
magnetisation direction in the basal plane for most hexagonal crystals (point groups
D6 , C6v , D3h and D6h ).
    Equations (2.65) predict for the dependence K2 (T ) three possible shapes. These
are sketched in Fig. 3.8. Apparently, K2 can change sign no more than once. We
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                    197


shall come back to this argument in Section 3, in connection with the SRT in
TbCo5 .
    As regards K1 , it can in principle change sign twice, and even more times when
the 3d contribution is taken into account.
    Let us now consider the connection between the linear theory and the high-
temperature expansion of Section 2.4. That these two approximations are closely
related should not come as a surprise—we have already noted that the leading term
in the high-temperature expansion of anisotropy constants is always linear in CF.
For example, for the axial anisotropy constants one can write in general (Kuz’min,
1995):
                              Ki = const. × B2i,0 x 2i + O(x 2i+1 ).             (2.99)
                           2i+1
The coefficient of x         and of the subsequent terms may be non-linear in CF
parameters. (In fact, terms even in x must be odd in Bnm while those odd in x must
be even in Bnm ).
    The first term of Eq. (2.99) may be regarded as the leading term of an expansion
of Ki in powers of several variables: x, B20 , B40 etc. (or x and the dimensionless
quantities introduced in Section 2.2: η, ξ etc.). The same series can be obtained in a
different way: first Ki is expanded in powers of Bnm , then the coefficients of the ob-
tained expansion (first of all, those of the terms linear in Bnm ) are further expanded
in powers of x. In other words, Eqs. (2.65) of the linear theory are expanded in a
series of powers of x using Eq. (2.84):
                           (–1)n/2        (2J + n + 1)!
              Kn/2 =                                     B x n + O(x n+2 )
                       2n n!(2n + 1)J n (2J + 1)(2J – n)! n0
                      2
                 n = 2, 4, 6.                                                   (2.100)
Comparing this with the general expansion (2.99), one immediately gets an ex-
pression for the unknown constant therein. Equation (2.44) is not but a special case
of (2.100) with n = 2, where the expansion is taken to the next, quartic in x term.
Omitted from the approximate relation (2.44) are the term in x 3 (whose coefficient
is a homogeneous quadratic form in Bnm ) and parts of the term in x 4 , namely, one
which is linear in B40 and the other one cubic in Bnm . The omitted contributions
were evaluated for TbCo5 and found small (Kuz’min, 1995).
    Thus, the linear theory becomes asymptotically exact at high temperatures (x
small) because the leading terms of both expansions, (2.99) and (2.100) coincide. In
other words, non-linear terms die out more rapidly with temperature. For typical
permanent magnet materials the linear in Bnm contribution is also dominant at
moderately high temperatures, where terms in x 4 are no longer negligible, the term
in B20 x 4 prevailing over the one in B40 x 4 . Hence Eq. (2.44). One concludes that the
high-temperature version of the linear theory (2.98) should be more accurate than
Eq. (2.41) for exchange-dominated systems at intermediate temperatures, because
the former includes—albeit approximately—the higher-order terms: in x 4 , x 6 etc.
    An added advantage of Eq. (2.98) is its sensible behaviour in the limit of very
low temperatures, where it is generally speaking invalid. Thanks to the presence of
an unexpanded GBF, Eq. (2.98) inoffensively tends to a finite limit as x → ∞,
whereas (2.41) diverges.
198                                                                                   M.D. Kuz’min and A.M. Tishin



     As against that, Eq. (2.41) has the advantage of being very simple and is ideally
suited for discussing the high-temperature behaviour of permanent magnets. It can
also be easily solved for x.
     Before closing the subsection, let us touch upon the question of the influence of
CF on the RE magnetic moment, described by Eq. (2.68). According to the above-
                                        (n)
stated properties of the functions DJ (x), namely Eq. (2.95), this effect should be
negligibly small at the temperature of liquid He, that is exactly where experimen-
talists usually try to detect it. The reason why they prefer to compare the measured
RE moment with the free-ion expression (2.59) just at low temperatures is rather
mundane. The Brillouin function is saturated there, so one does not need to worry
about its unknown argument. Unfortunately, the saturation also kills off the sought
CF effect.
     This effect reaches its maximum at a finite temperature. For example, in
Nd2 Fe14 B the six-order CF contribution to μNd peaks at T ≈ 80 K (x ≈ 9),
where it amounts to –2% of the total Nd moment (at the same temperature).15 At
room temperature the sixth-order effect is a factor of 20 weaker, –0.1%, and may
be safely neglected. The fourth-order contribution is maximum at T = 120 K, or
x ≈ 6.2, where it reaches –1.6%. This reduces to about one-half of a per cent at
ambient temperature, negligible in most cases. Finally, the second-order CF effect
is maximum at room temperature, T = 290 K (x ≈ 2.3), where the function
   2
D9/2 (x) equals approximately 2.6. Accordingly, its relative contribution to μNd is
+7.7%. One should bear in mind however, that less than one-seventh of the total
magnetisation of Nd2 Fe14 B at ambient temperature comes from the Nd sublattice.
When related to the total magnetisation, the second-order CF effect reduces to a
mere 1%. Thus, the CF contribution to magnetisation hardly needs to be taken into
account in technical calculations of permanent-magnet devices. In any case, at or
above room temperature only second-order CF matters.
     The influence of the CF on the magnetisation is most noticeable near SRT,16
where rotation of the easy magnetisation direction leads to a rapid change of the
primed CF parameters (2.61) with temperature. Obviously, this effect is more pro-
nounced in ferrimagnetic intermetallic compounds with the heavy RE and at
first-order SRT. Further consideration of this phenomenon will be deferred till
Section 3.
     At very high temperatures (x         1) the influence of the CF on μR decreases.
The least rapidly falls off the second-order CF effect. According to Eq. (2.96), at
small x
                                         (2J – 1)(2J + 3)
                             DJ (x) ≈
                                (2)
                                                          x.                  (2.101)
                                                10J
Putting this into Eq. (2.68), we arrive at Eqs. (2.46), (2.47). Note that by virtue
of Eq. (2.61), B20 = B20 when B z (θ = 0), and B20 = – 1 B20 when B ⊥z
                                                                   2
 15 Our estimates are based on the CF parameters of Cadogan et al. (1988), converted to the Wybourne normalisation
and averaged over the two Nd sites: B20 = –4.4 K, B40 = 0.092 K, B60 = 0.02 K. For simplicity, no distinction was
made between primed and non-primed CF parameters below the SRT point, TSR = 135 K. The exchange splitting,
  ex = 168 K at T = 0, comes from the same source. At finite temperatures ex was scaled down in proportion to the
iron sublattice magnetisation. We used the scaling factors 0.987, 0.976 and 0.891 for T = 80, 120 and 290 K. These were
computed using Eq. (2.43) with s = 0.7 and TC = 592 K.
 16 This effect is not attributable to the RE sublattice alone.
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                     199


(θ = π /2). This manifests once again the intimate connection between the linear-
in-CF and the high-temperature approximations.

2.8 The low-temperature approximation
Following the introduction of the general single-ion model in Section 2.3, we
have considered two closely inter-related approximations: the high-temperature one
(Section 2.4) and the one linear in CF (Sections 2.5–2.7). These were ‘good’ ap-
proximations, in the sense that their validity in any particular case was easy to verify,
at least a posteriori.
     In this subsection we shall formulate an approximation that unfortunately
lacks the same quality. It has been already said that the standard theory of mag-
netic anisotropy is generally inapplicable to RE-transition metal magnets at low
temperatures—the expansion of the free energy in powers of sin θ cannot be trun-
cated and the very description in terms of a single angle θ is no longer meaningful
on account of significant non-collinearity of the sublattices. When an analytical
description of some sort at low T is possible, it should be necessarily limited to a
special case, i.e. it should involve extra assumptions apart from the smallness of T .
As such an additional condition, we shall now assume that the CF has a predomi-
nantly axial character, or that Bnm with m = 0 are small in comparison with Bn0 .
A rigorous verification of this hypothesis would require the knowledge of all CF
parameters, including those with m = 0, which in this approach we do not presume
to have. In that sense, this is an uncontrolled approximation.
     So let the CF be approximately axial and let the system under consideration
be broadly exchange-dominated. This implies that the ground state is |J M , with
M = – sign(1 – gJ )J , the first excited state has M = – sign(1 – gJ )(J – 1), etc.
The positive z direction is, as usual, that of the 3d magnetisation vector. The latter
coincides with the high-symmetry crystallographic axis [001], or at the most makes
with it an infinitesimal angle θ . Such a situation is characteristic of permanent
magnets—otherwise the material would simply lose its hard magnetic properties,
first of all the coercivity.
     The reason why it makes sense to limit the consideration to the vicinity of the
point θ = 0 is explained in Fig. 3.9. At T = 0 the free energy is the ground
state energy, shown as a solid line. It may not always be a smooth function of the
angle θ, because the RE energy levels may cross over at some finite values of θ.
Just such a case is shown in Fig. 3.9a. A more realistic example is REFe11 Ti (Hu et
al., 1990, Fig. 8 therein). It is clear that the displayed dependence F (θ) cannot be
approximated with a smooth function like K1 sin2 θ or K1 sin2 θ + K2 sin4 θ across
the entire interval from 0 to 90°. For θ small, such a presentation—in the spirit
of Landau’s theory (Landau and Lifshitz, 1958)—is still possible and even useful.
Indeed, when a second-order SRT occurs (Fig. 3.9b), the main events take place
near the point θ = 0 (or perhaps near θ = π /2, in which case the small angle π /2–
θ should be regarded as the order parameter). Of course, the benefit of presenting
the anisotropy energy as K1 sin2 θ in close vicinity to θ = 0 is not limited to second-
order SRT. A number of other phenomena—nucleation, transverse alternating-
current susceptibility, magnetic resonance etc.—require such a presentation.
200                                                                       M.D. Kuz’min and A.M. Tishin




Figure 3.9 Angular dependence of the free energy at T = 0 (solid line) when a level cross-over
takes place. The presence of a cusp in the solid curve does not interfere with the spin reorien-
tation transition—the displacement of the minimum from the origin to a nonzero angle.


   Thus, under the above assumptions, the RE Hamiltonian (2.35) falls into two
parts:
                ˆ                           ˆ
              H4f = sign(1 – gJ ) ex cos θ Jz +        (n) ˆ
                                                  Bn0 C0 (J )
                                                                n=2,4,6

                           + sign(1 – gJ )      ex
                                                            ˆ
                                                     sin θ Jx .                              (2.102)
The first one of them is diagonal in the J M representation, while the second one
can be treated as a perturbation since it contains an infinitesimal quantity sin θ .
    We restrict ourselves to the region of low temperatures, kT     2 ex + 1 + 2 ,
where we can neglect thermal population of all but the lowest two levels of the RE.
It will be recalled that ex + 1 is a gap separating the ground and the first excited
states when θ = 0. Its two parts, due to the exchange and the CF, are defined by
Eqs. (2.17) and (2.23), respectively. Similarly, ex + 2 stands for the gap between
the first and the second excited levels.
    Expanding the centre of gravity of the lowest two levels and the gap between
them in powers of the small parameter sin2 θ,
                        Ec.g. (θ ) = Ec.g. (0) + V sin2 θ + · · ·
                             (θ ) =    ex   +    1   + 2W sin2 θ + · · ·
we get
                                                   + 1     ex
                               K1 = V – W tanh            .                   (2.103)
                                                 2kT
The quantities V and W are evaluated using the standard second-order perturbation
theory:
                                                    2     2
                   2J – 1               2J – 1      ex sin θ
    Ec.g. = ECF +          ex cos θ –
                     2                     4    ex cos θ +   2
                                      2      2                   2     2      (2.104)
                                      ex sin θ       2J – 1      ex sin θ
     (θ ) = 1 + ex cos θ + J                      –                         .
                                  ex cos θ +   1        2    ex cos θ +   2
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                   201


Here ECF is the CF contribution to Ec.g. , independent of θ. Corrections of order
higher than 2 in perturbation theory contain only terms of order higher than 2
in sin θ and therefore need not be considered. Presenting cos θ as 1 – 1 sin2 θ + · · ·
                                                                       2
and collecting terms in sin2 θ in Eqs. (2.104), we arrive at
                                     2J – 1    ex 2
                                       V =                                    (2.105)
                                       4     ex +    2
                                         J    ex 1
                              W =V –                  .                       (2.106)
                                         2 ex + 1
The set of Eqs. (2.103), (2.105) and (2.106) gives the coefficient K1 of the Landau
expansion in powers of sin θ at low temperatures.
    The interplay of the CF and the exchange interactions can be best seen at T = 0,
as the hyperbolic tangent in Eq. (2.103) becomes unity and K1 reduces to
                                       J     ex 1
                                         K1 =      .                           (2.107)
                                       2 ex + 1
Insofar as their influence on K1 is concerned, the exchange and the CF act like two
resistors connected in parallel—additive are the reciprocal splittings rather than the
splittings themselves. In the extreme case of exchange domination, 1 / ex → 0,
Eq. (2.107) turns into
                                        J
                                            1. K1 =                       (2.108)
                                        2
Here one can recognise the first one of the equations (2.65) of the linear theory,
taken at T = 0, or x = ∞. Indeed,
                   1                                   n(n + 1) (2J )!         J
K1 = –               n(n + 1)J n BJ (∞)Bn0 = –
                                  (n)
                                                          n+1
                                                                         Bn0 =       1
           n=2,4,6
                   2                           n=2,4,6
                                                         2     (2J – n)!       2
where Eqs. (2.76) and (2.23) have been used.
   When 1 is small (as compared with ex ) but finite, Eq. (2.107) can be ex-
panded in powers of the ratio 1 / ex :
                                           J               1
                                  K1 =          1   1–          + ··· .        (2.109)
                                           2               ex

Thus, an added advantage of the low-temperature approximation is that it provides
a quantitative criterion of the performance of the linear theory at T = 0. Indeed,
according to Eq. (2.109), the fractional error of the ‘linear’ equation (2.108) can be
judged by the smallness of the ratio 1 / ex , or 2K1 /J ex . The latter combination
contains quantities more readily accessible to experiment. (We would like to remind
that K1 stands here for the RE contribution to the first anisotropy constant at
T = 0.) For example, at T = 4.2 K TbCo5 has K1Tb = –99 K (Ermolenko, 1980)
and Bex ≈ 220 T (Ballou et al., 1989), or ex ≈ 150 K. Hence 2K1Tb /6 ex =
–0.22, i.e. the linear approximation is accurate within 22%. This estimate, referred
to T = 0, is the upper bound of the inaccuracy of the linear-in-CF approximation.
202                                                                                      M.D. Kuz’min and A.M. Tishin



As stated in the previous subsection, the linear theory performs better at higher
temperatures and becomes exact in the limit T → ∞.
    Let us now demonstrate how the low-temperature approximation to K1 can be
used to evaluate the temperature of a second-order SRT of the type easy axis—easy
cone. Such transitions are not uncommon in hard magnetic materials, Nd2 Fe14 B
being the best-known example with TSR = 135 K (Deryagin et al., 1984; Givord
et al., 1984). A necessary condition for such a transition is that K1 = 0 at T = TSR ,
or17

                                                                    ex+       1
                                     K3d + V – W tanh                             =0
                                                                    2kTSR
whence

                                                           + 1 ex
                                    kTSR =                            .                                         (2.110)
                                                           V + K3d
                                                 ln 1 + 2
                                                          W – V – K3d
Taking for Nd2 Fe14 B the exchange and CF parameters of Cadogan et al. (1988)
averaged over the two Nd sites, one finds from Eqs. (2.23) 1 = –74.7 K and
   2 = 178 K, as well as      ex = 167 K. Hence V = 172 K and W = 476 K, by
way of Eqs. (2.105) and (2.106). The anisotropy constant of the iron sublattice K3d
is taken equal to that of Y2 Fe14 B, which at low temperatures was found to be 6.0
K/Y atom (Givord et al., 1984). Then Eq. (2.110) yields TSR = 117 K.
     This compares rather well with TSR = 122 K, obtained numerically by Piqué
et al. (1996) using the full algorithm of the single-ion model (Section 2.3) and the
same parameters as above. The discrepancy between the experimental transition
point, TSR = 135 K, and the calculated ones is inherent in the exchange and
CF parameters of Cadogan et al. (1988) rather than being a consequence of the
approximations introduced in this subsection—see the discussion by Piqué et al.
(1996).
     For Ho2 Fe14 B the situation is similar. The transition point found from
Eq. (2.110), TSR = 57 K (Kuz’min, 1995), agrees well with that obtained nu-
merically using the same parameters, TSR = 56 K (Piqué et al., 1996), both being
somewhat lower than the experimental value, TSR = 63 ± 2 K (Piqué et al., 1996).
     Let us recapitulate: at low temperatures the free energy of a RE-based hard
magnetic material generally cannot be presented as a truncated expansion in powers
of sin θ , Eq. (2.12) or similar. Such a presentation is however possible in a certain
neighbourhood of the point θ = 0, where Eq. (2.12) has the meaning of Landau’s
expansion, its convergence ensured by the smallness of sin θ . Then, upon some
additional assumptions, a useful analytical expression (2.103) can be obtained for
K1 (and in principle also for K2 etc.).

17   Here K1 is of course the total anisotropy constant, including the contributions from the 3d and the 4f subsystems.
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                  203


2.9 J -mixing made simple18
In this last part of Section 2 we shall finally get down to elucidating the importance
of the so far neglected effect of J -mixing on the single-ion magnetic anisotropy.
What most workers in the field know about J -mixing can be summarised as the
following ‘Three Myths’.
1. J -mixing is a complex phenomenon, not readily amenable to quantitative treat-
   ment. Allowing for it necessitates large-scale computer calculations. Too many
   factors play a role, so little can be demonstrated conclusively.
2. Quantitatively, J -mixing leads to serious consequences only in samarium com-
   pounds. For the other REs this effect is not very important, if not exactly
   unnoticeable. Given the formidable difficulties of its description (Myth 1), it
   is better to neglect it. The neglect is certainly justified for the heavy REs.
3. When it comes to allowing for J -mixing, the effect can be best visualised at
   low temperatures. There, both the calculations are more transparent and the
   anomalies are sharper. When proved unimportant at low T , J -mixing may be
   neglected at any temperature.
    The above three statements contain only one grain of truth: that the J -mixing
is most distinctly manifest in samarium compounds (and also in those of trivalent
europium, not often come across among hard magnetic materials). The rest is mis-
conceptions. In this subsection, according as the truth will gradually unfold, we
shall be coming back to the ‘Three Myths’ to point out the falsity of this or that
constituent statement.
    Since J -mixing in the light and the heavy REs is described by slightly different
equations, we shall first consider in some detail only the former. The main results
for the heavy RE case will be stated briefly towards the end of the subsection.
    We begin with writing down a model Hamiltonian for a single RE ion in the
absence of applied magnetic field. The Hamiltonian is defined on the ground LS
term, treated in the Russell–Saunders approximation, and contains terms describing
spin-orbit coupling, exchange interaction and the CF:
             ˆ     ˆ     ˆ     ˆ     ˆ ˆ             ˆ
            H4f = Hso + Hex + HCF = λL · S – 2μB Bex Sz +                               (n) ˆ
                                                                                   Anm Cm (L). (2.111)
                                                                             n,m

Here the z axis has been chosen to be parallel to the 3d sublattice magnetisation
M 3d (therefore antiparallel to the exchange field B ex ), which does not necessarily
coincide with any of the high-symmetry crystallographic directions. Accordingly,
the CF parameters in Eq. (2.111) are primed, to distinguish them from the usual,
non-primed CF parameters defined in the crystallographic coordinate system.
    We aim at describing the lower part of the energy spectrum of the RE, involved
in forming the thermodynamic properties of the solid. To this end we construct
                             ˜
an effective Hamiltonian H defined on the ground J manifold of the light RE,
J = L – S,
                                 ˜
                                 H=            ˆ +
                                            ex Jz
                                                                 (n) ˆ      ˜
                                                            Bnm Cm (J ) + δ V .                (2.112)
                                                      n,m
18   This subsection follows the work of Kuz’min (2002).
204                                                                      M.D. Kuz’min and A.M. Tishin



Here Bnm are CF parameters in the J representation, incorporating the Stevens
factors. They are related to the quantities Anm in Eq. (2.111) through known ra-
tional factors. E.g. for n = 2 this relation is given by Eq. (1.23): Bnm /Anm =
(L + 1)(2L + 3)/(J + 1)(2J + 3). Similar expressions can also be written for
n = 4 and 6.
                                   ˜
    The effective Hamiltonian H (2.112) differs from the usual single-multiplet
Hamiltonian (2.14) in a very important way: it incorporates an operator δ V con-˜
                                                  ˆ         ˆ
taining second-order corrections bilinear in Hex and HCF , the latter two regarded
as perturbations with respect to H  ˆ so . Although it operates within the ground mul-
          ˜                                                 ˆ
tiplet, δ V contains inter-multiplet matrix elements of Hex and HCF .ˆ
    Subsequently we intend treating the last two terms of Eq. (2.112) as pertur-
bations with respect to the first one, limiting ourselves to first-order corrections.
Therefore, all terms with m = 0 may be omitted from the sum in Eq. (2.112). As
                         ˜
regards the operator δ V , we only need to compute its matrix elements diagonal
in M:
                ˆ          2                 ˆ                    ˆ
              δ VMM = –      J + 1, M|Hex |J M J + 1, M|HCF |J M                (2.113)
                               so

where so = λ(J + 1) is the spin-orbit splitting between the centres of gravity of
the ground (J ) and the first excited (J + 1) multiplets.
                                               ˆ               ˆ
   The inter-multiplet matrix element of Hex = –2μB Bex Sz , required for
Eq. (2.113), is given by

                    ˆ           (L + 1)(J + 1)(2J + 1) J +1,M
          J + 1, M|Hex |J M = –                       CJ M10                           ex .   (2.114)
                                      S(2J + 3)
This expression has been obtained from the well-known formula (88) of Van Vleck
(1932) by setting J = L – S, as appropriate for the ground multiplet of a light RE,
and factoring out the CGC:

                                J +1,M                (J + 1)2 – M 2
                               CJ M10 =                              .
                                                     (2J + 1)(J + 1)
One of the advantages of Eq. (2.114) is that it depends on the exchange field Bex
through the quantity ex , which is the exchange splitting of the ground multiplet
defined by Eq. (2.17). This will enable us to reduce the effect of J -mixing to a
renormalisation of the standard (without J -mixing) expression for the anisotropy
constants (2.64), which depends on the characteristics of the ground multiplet only.
   For the inter-multiplet matrix element of the CF one can write
                    ˆ
          J + 1, M|HCF |J M
                               L        S       J +1
                               J        n         L    1      (2J + n + 1)!
           =             Bn0                                                 C J +1,M .       (2.115)
               n=2,4,6
                                    L       S    J     2n   (2J + 1)(2J – n)! J Mn0
                                    J       n    L
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                                     205


This is a generalisation of the intra-multiplet form (1.24) of the Wigner-Eckart
theorem. The change from Eq. (1.24) to Eq. (2.115) consists, apart from the obvious
modification of the CGC, in adding the ratio of the two 6j symbols, as follows from
Eqs. (6-4) and (6-5) of Wybourne (1965). The CF parameters Bn0 employed in
Eq. (2.115) include the Stevens factors. This is in line with our strategy of describing
the J -mixing in terms of the characteristics of the ground multiplet.
   As we further restrict ourselves to the Hund ground state of the light RE,
Eq. (2.115) simplifies significantly,19 to become
                       ˆ
             J + 1, M|HCF |J M
                                      1            n(n + 1)S(2J + n + 1)!
                =–              Bn0                                            C J +1,M . (2.116)
                      n=2,4,6
                                      2n       (L + 1)(2J + n + 2)(2J – n + 1)! J Mn0
Putting Eqs. (2.114) and (2.116) into Eq. (2.113) results in

         ˜                ex                    1        n(n + 1)(J + 1)(2J + 1)(2J + n + 1)!
       δ VMM = –                        Bn0
                           so n=2,4,6          2n–1        (2J + 3)(2J + n + 2)(2J – n + 1)!
                         J +1,M J +1,M
                      × CJ M10 CJ Mn0 .                                                                        (2.117)
Note the cancellation of explicit dependence on the quantum numbers L and S.
Of course, Eq. (2.117) still depends on L and S implicitly, through the relation
J = L – S, used in the derivation.
    Replacing the product of two CGCs in Eq. (2.117) by a linear combination
of two CGCs through the following identity [a particular case of Eq. (8.7.37) of
Varshalovich et al. (1988)],

                 J +1,M J +1,M                     1                       n(n + 1)(2J + 3)
                CJ M10 CJ Mn0 =
                                           2(2J + 1)(2n + 1)                    J +1
                                           ×            (2J + n + 1)(2J + n + 2)CJ M,n–1,0
                                                                                 JM



                                                    –    (2J – n)(2J – n + 1)CJ M,n+1,0
                                                                              JM


and comparing the result with Eq. (1.24), one concludes that the effective Hamil-
       ˜
tonian H, or rather its part diagonal in M, can be presented as follows:

  ˜              ˆ +                   (n) ˆ               exn(n + 1)
  H=         ex Jz                Bn0 C0 (J ) –
                        n=2,4,6                            so 2n + 1

                                                2J + n + 1 (n–1) ˆ      2               ˆ
                                           ×              C0 (J ) –           C (n+1) (J ) .
                                                    2               2J + n + 2 0
                                                                                       (2.118)
 19 To obtain Eq. (2.116) one should use the recurrence relation (9.6.5) of Varshalovich et al. (1988), with a = f = L =
J + S, b = S, c = d = J , and note that the last 6j symbol therein vanishes because its three upper indices do not satisfy
the triangle rule. One can then isolate the ratio of the remaining two 6j symbols and substitute it into Eq. (2.115).
206                                                           M.D. Kuz’min and A.M. Tishin



It is now apparent that the effect of J -mixing is a renormalisation of the CF, which
vanishes in the limit ex / so → 0.
     The remaining steps are quite similar to what was done in Section 2.5. The sum
                                                                               ˆ
in Eq. (2.118) is treated as a perturbation with respect to the first term, ex Jz . The
primed CF parameters are transformed to the crystallographic coordinates using
Eq. (2.61). The thermal averages of the irreducible tensor operators are replaced by
GBFs according to Eq. (2.63), recalling that sign(1 – gJ ) = 1 for the light REs.
The energy corrections of first order in Bnm take the form of the anisotropy energy
(2.10), in which
                                n(n + 1)
                                  ex
      κnm = Bnm J n BJ (x) +
                     (n)

                              so 2n + 1

                       2J + n + 1 (n–1)       2J
                     ×            BJ (x) –             (n+1)
                                                      BJ (x)                     (2.119)
                          2J               2J + n + 2
with x = J ex /kT .
    Thus, the effect of J -mixing on the nth -order anisotropy constant κnm consists
in renormalising its temperature dependence, which in the absence of the J -mixing
is described by a single GBF of the same order n, cf. Eq. (2.64). Now Eq. (2.119)
                                   (n±1)
contains two extra terms, with BJ (x). As one would expect, these corrections
vanish when ex / so → 0.
    Equation (2.119) enables us to reach a definite conclusion about the sense of
the effect. Let us consider the square bracket of Eq. (2.119) in the limit T → 0, or
x → ∞. Making use of Eq. (2.76), we get
                    2J + n + 1 (n–1)               2J
                                BJ (∞) –                    B (n+1) (∞)
                          2J                 2J + n + 2 J
                                  2J + n + 1         2J – n
                     = BJ (∞)
                           (n)
                                               –                    > 0.       (2.120)
                                   2J – n + 1 2J + n + 2
Obviously, the numerator of the first fraction in the parenthesis is greater than, and
its denominator is less than their respective counterparts in the second fraction.
It is easy to see that the square bracket of Eq. (2.119) will remain positive at any
                    (n+1)                                                     (n–1)
temperature, as BJ (x) decays with temperature more rapidly than BJ (x).
Therefore, J -mixing always enhances the intra-multiplet anisotropy, irrespective of
the sign of the latter.
    It follows from Eq. (2.120) that in the classical limit, J → ∞, the contribution
to κnm from the J -mixing in the light REs vanishes at T = 0—a first indication
that low temperatures may not be the best choice for appreciating the size of the
effect, contrary to the generally accepted view (Myth 3).
    Equation (2.119) can be put to a further good use: setting to zero all GBFs
of order higher than two, one arrives at a simple high-temperature version of the
formalism. Let us additionally limit ourselves to tetra-, hexa- or trigonal crystals.
Then the anisotropy energy is just K1 sin2 θ, where
                3       3                         6 2J + 3      ex
          K1 = – κ20 = – B20 J 2 BJ (x) +
                                  (2)
                                                                     BJ (x) .    (2.121)
                2       2                         5 2J          so
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                        207




Figure 3.10 Quantities relevant to Eq. (2.119) for RE = Nd (J = 9/2, n = 2), plotted against
                                      (3)
1/x: — y = 1.6B9/2 (x), - - - y = 54 B9/2 (x), · · · the difference of the previous two.
                                  65




Here the second term in the square brackets, representing the correction for J -
mixing, depends on temperature through the familiar Brillouin function—a sign
of simplicity yet to come (contrary to Myth 1). The correction term should be
small due to the ratio ex / so . However, at elevated temperatures it tends to zero
more slowly (as 1/T ) than the first, intra-multiplet term, BJ (J ex /kT ) ∝ 1/T 2 .
                                                               (2)

Therefore, the relative importance of the J -mixing effect must increase with tem-
perature. This constitutes a final departure from Myth 3.
    As has been demonstrated in Section 2.4, the leading term in the rigorous high-
temperature expansion for K1 is linear in the CF parameter B20 . In other words, all
other terms, including those nonlinear in Bnm , die out more rapidly as T → ∞.
Therefore, Eq. (2.121) becomes asymptotically accurate at elevated temperatures.
This happens irrespective of the strength of the CF in relation to the exchange, as
both become weak in comparison with the thermal energy kT .
    In practice, Eq. (2.121) applies to 3d-4f compounds upward of room tempera-
ture. Thus, according to Fig. 3.10, in Nd-based magnets the approximation breaks
down (the third-order term is no longer negligible) at x ∼ 3. This corresponds
to T ≈ 220 K, assuming for the exchange field on Nd a value typical for hard
magnetic materials, Bex = 400 T.
    The above argument is unaffected by the slow temperature variation of the
exchange splitting ex . Implicitly, we assume that, while in the high-temperature
regime, the system is still not close to the Curie point (the TC of a good permanent
magnet should exceed ambient temperature by a factor of at least 2).
    As a measure of relative importance of the J -mixing in the room-temperature
range one can use the ratio of the second term in the square brackets of Eq. (2.121)
208                                                                    M.D. Kuz’min and A.M. Tishin



to the first one,
                               6 2J + 3    ex   BJ (J   ex /kT )
                         ε=                    (2)
                                                                   .                      (2.122)
                               5 2J        so BJ (J     ex /kT )

Obviously, this quantity does not depend on the CF. In the high-temperature
regime, which is primarily of interest to us, ε is also independent of the exchange
field Bex , or ex . Indeed, expanding the Brillouin functions in Eq. (2.122) by means
of Eq. (2.84) and keeping just the leading terms, one gets
                                           12 kT
                                   ε=               .                        (2.123)
                                         2J – 1 so
     Thus, for a given RE, the fractional contribution of J -mixing to second-order
anisotropy is determined by temperature alone and does not depend on charac-
teristics of the solid, such as CF or exchange field. This conclusion is valid in the
room-temperature range as well as at higher temperatures. The importance of J -
mixing grows in direct proportion to absolute temperature.
     Calculations similar to the above can also be carried out for the second half of
the RE series, where the ground multiplets have J = L + S. Omitting the de-
tails, we only state the result. The heavy-RE counterparts of the general expression
(2.119), the high-temperature approximation (2.121) and the fractional contribu-
tion estimate (2.123) are, respectively, the following relations:
                              exn(n + 1) 2J – n + 1 (n–1)            2J
κnm = Bnm J n BJ (x) +
               (n)
                                                       BJ (x) –            B (n+1) (x)
                             so  2n + 1        2J                  2J – n J
                                                                                 (2.124)
         3                      6 2J – 1    ex
 K1 = – B20 J 2 BJ (x) +
                     (2)
                                               BJ (x)                            (2.125)
         2                      5 2J        so
          12 kT
   ε=               .                                                            (2.126)
        2J + 3 so
One peculiar feature of the J -mixing in the heavy REs is that the effect is strictly nil
at T = 0 [for verification put BJ (∞) = (2J )–n (2J )!/(2J – n)! into Eq. (2.124)].
                                  (n)

The physical reason is that the ground state of an exchange-dominated heavy RE
does not take part in the J -mixing (see Fig. 3.11) since it cannot find itself a partner
with the same magnetic quantum number, M = J , among the states of the first
excited multiplet, whose M do not exceed J = J – 1.
    To summarise, contrary to the common perception (Myth 3), the influence of
J -mixing on thermodynamic properties of RE magnets grows with temperature.
In the limit T → 0 the effect either vanishes completely (light RE with J → ∞,
heavy RE with arbitrary J ) or is very small. Its smallness at low temperatures is no
indication that it may be neglected in the room-temperature range.
    The insuperable complexity of the J -mixing has proved to be a yet another
myth (Myth 1). Where it matters most—at ambient temperature and above—this
effect can be accounted for by means of back-of-the-envelope calculations using
Eqs. (2.123), (2.126).
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                             209




Figure 3.11 Comparison of J -mixing in a light and a heavy RE. In both cases the mixing
states have the same M and different J . The ground state of a heavy RE is not involved in the
J -mixing.


Table 3.4 Fractional contribution of J -mixing to the second-order anisotropy constant K1 , as
given by Eq. (2.123) for the light REs (upper part of the table) and by Eq. (2.126) for the heavy
REs (lower part)

RE                 J                  so   (K)*            ε (T = 300 K)        ε (T = 400 K)
Pr                  4              3100                    0.166                0.221
Nd                  9/2            2740                    0.164                0.219
Sm                  5/2            1440                    0.625                0.833
Tb                  6              2880                    0.083                0.111
Dy                 15/2            4740                    0.042                0.056
Ho                  8              7480                    0.025                0.034

* Taken from Elliott (1972).



    Finally, the insignificance of the J -mixing in REs other than Sm (Myth 2) is
disproved by the data presented in Table 3.4. At T = 400 K even terbium—a
heavy RE—is subject to an 11% correction, while in Pr and Nd it is as high as
22%. In heavier REs the effect is noticeably smaller, for two reasons. Firstly, the
larger denominator in the prefactor of Eq. (2.126) as compared with Eq. (2.123).
210                                                             M.D. Kuz’min and A.M. Tishin



This alone makes the size of the effect in Ho a factor of 0.3 smaller than in Pr.
A reduction by a further factor 0.4 comes from the larger spin-orbit splitting so ,
especially high towards the end of the RE series.
    Sm-based magnets should be considered separately. There, the second-order
J -mixing correction to K1 , ∝ 1/ so , is the size of the main, intra-multiplet con-
tribution. It is therefore likely that corrections of higher orders in 1/ so , neglected
in Eqs. (2.119)–(2.126), are not small. The contribution of J -mixing to sixth-order
anisotropy constants (responsible e.g. for the anisotropy in the basal plane in the
hexagonal ferromagnets Sm2 Fe17 and Sm2 Co17 ) can hardly be called a correction,
since the intra-multiplet effect is strictly nil. Equation (2.115) for the matrix ele-
           ˆ
ment of HCF is invalid, because the 6j symbol in the denominator equals zero (the
numerator is also nil since B60 ∝ γJ = 0). Calculations in that case should be per-
formed using an alternative approach developed by Magnani et al. (2003) specially
for the purpose. The role of the Stevens coefficient γJ is then played by another
quantity, δ6 , which is negative for Sm. When the sign of δ6 is taken into consider-
ation, the anisotropic properties of Sm compounds—first of all the SRTs—are no
longer incomprehensible.
    Our review of the theoretical apparatus for the description of single-ion mag-
netocrystalline anisotropy has reached its close. All along we tried to illustrate the
relevance of the various approximations by performing simple calculations for well-
known hard magnetic materials. Our intention was to encourage experimentalists
to use, where possible, the approximate equations for do-it-yourself calculations.
Perhaps the best demonstration of the advantages of the single-ion model cast in
analytical form is still to come. We are just turning to the phenomena where mag-
netic anisotropy manifests itself most vividly—spin reorientation transitions (SRT).



      3. Spin Reorientation Transitions
3.1 General remarks
The third and last section of this Chapter is dedicated to the phenomenon of spin
reorientation transitions (SRT). Of interest to us here are not SRTs as such—a vast
subject covered in excellent reviews and monographs (Belov et al., 1976, 1979)—
but only some peculiar features of the SRTs viewed from the standpoint of the
single-ion anisotropy model. Our main goal is to demonstrate that the single-ion
model is more than an ad hoc theory explaining already known experimental facts.
Rather, it possesses a certain power of prediction. Where the underlying approxi-
mations are valid, the strength of the model is such that all experimental findings
not fitting in its framework eventually prove wrong. This point will be illustrated
with a number of examples.
    A spin reorientation transition (SRT) is a phase transition consisting in a change
of orientation of ordered magnetic moments—which can be distributed among
several sublattices—with respect to crystallographic axes. Obviously, such magnetic
transitions (called order-order transitions) are essentially distinct from the usual mag-
netic ordering of e.g. a ferromagnet at the Curie point. Even among order-order
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                                   211


transitions SRTs can be singled out in a separate class, on the grounds that they do
not involve any change of the mutual orientation of the sublattice moments, only
their orientation in relation to the crystal axes changes. Following this definition,
metamagnetic transitions or field-induced transitions of ferrimagnets into a non-
collinear state, are excluded from the scope of this section. Metamagnetism was
reviewed by Levitin and Markosyan (1988) and by Goto et al. (2001). A chapter
about the field-induced transitions in ferrimagnets appeared in Volume 9 of this
Handbook (Zvezdin, 1995).
    Just like phase transitions in general, SRTs can be of first or of second order.
In the former case the orientation angle experiences a discontinuity at the tran-
sition point, whereas in the latter case the angle itself varies continuously, but its
first derivative is discontinuous. For all that, the change of symmetry, essential in
second-order SRTs, is always abrupt. Therefore, no matter if an SRT is of first or of
second order—it takes place at a point, rather than within an interval. For instance,
the process of spontaneous20 spin reorientation shown in Fig. 3.12a comprises two
second-order SRTs as well as continuous rotation of the magnetisation vector in the
interval between the two transition points. This rotation is of course not a phase
transition. Likewise, a mere change of slope of the θ (T ) dependence (Fig. 3.12b)
does not amount to an SRT. The crucial difference is that at a transition point the
angle θ takes a special high-symmetry value, 0 or π /2. When this is the case, the
derivative of the orientation angle diverges on approach to the transition point from
the lower-symmetry phase (Landau and Lifshitz, 1958). In a first-order SRT there
is no restriction on the critical values of θ : in general both are distinct from 0 or
π /2 (Fig. 3.12c), but either one of them or both may also take higher-symmetry
values.
    Second-order SRTs need not always come in pairs as shown in Fig. 3.12a. It is
not inconceivable that the process of spin reorientation starting at the point T2 may
not reach completion before the temperature reaches 0 K. Well-known examples of
single second-order SRTs are those taking place in Gd metal (Corner et al., 1962)
and in Nd2 Fe14 B (Deryagin et al., 1984; Givord et al., 1984).
    All the above applies, practically without change, also to magnetic field-induced
transitions. Interestingly, an infinitesimal magnetic field applied at an angle to the
easy magnetisation direction is sufficient to provoke a second-order SRT. Satura-
tion in a finite field, characteristic of magnetisation along a hard direction, is also
a second-order SRT (by contrast, under general orientation of the field the ap-
proach to saturation is asymptotic, without an SRT). Apart from such ubiquitous
and trivial second-order SRTs, there are also field-induced SRTs of first order. In
this case the upper critical value of the angle θ may correspond either to low or
to high symmetry (Figs. 3.13d, 3.13f), however, the lower critical value is always
low-symmetry. The reason is the afore-mentioned transition to a low-symmetry
phase induced by an infinitesimal magnetic field. Hereafter we shall concentrate
on spontaneous SRTs. Magnetic field-induced SRTs of first order (FOMPs) were
described in detail in Volume 5 of this Handbook (Asti, 1990).
20 According to the generally accepted definition, spontaneous SRTs take place as a result of temperature change, at zero
magnetic field and ambient pressure.
212                                                              M.D. Kuz’min and A.M. Tishin




Figure 3.12 Temperature dependence of orientation angle θ: (a) 2 second-order SRTs, (b) no
SRT, (c) a first-order SRT.


    It is an interesting peculiarity of second-order SRTs that Landau’s theory of
second-order phase transitions (Landau and Lifshitz, 1958) applies to them practi-
cally without restrictions. In this sense SRTs differ significantly from order-disorder
transitions. Estimations show that the interval where Landau’s theory fails due to
critical fluctuations is very narrow in the case of SRTs, 10–7 . . . 10–4 K (Belov et al.,
1976). Physically, this is because the fluctuations arising near an SRT have a very
large correlation length.
                                                                                                                                           Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds
Figure 3.13 Examples of magnetic field-induced SRTs: (a, b) 2 second-order SRTs, (c, d) 2 second-order SRTs and 1 first-order SRT (type II
FOMP), (e, f) 1 second-order SRT and 1 first-order SRT (type I FOMP). Note the ubiquity of the trivial second-order SRT at B → 0.




                                                                                                                                           213
214                                                            M.D. Kuz’min and A.M. Tishin



3.2 SRTs in uniaxial magnets
3.2.1 Graphic representation
In this subsection we shall introduce the concept of phase diagrams and relate the
diagrams of different levels within the hierarchy of approximations to the anisotropy
energy.
    We begin with the simplest expression,
                                   Ea = K1 sin2 θ.                                   (3.1)
According to Section 2.4, this formula is relevant to a uniaxial magnet at high
temperatures (T       300 K). Obviously, a system described by Eq. (3.1) may have
just two stable states:
                                     0,      if K1 > 0
                               θ=                                                    (3.2)
                                     π /2,   if K1 < 0.
These are traditionally called ‘easy axis’ and ‘easy plane’. A first-order phase transi-
tion takes place at the point K1 = 0. This information is summarised in Fig. 3.14.
    Let us now turn to a more complicated example. Consider a system whose
anisotropy energy is given by
                            Ea = K1 sin2 θ + K2 sin4 θ.                              (3.3)
Minimisation with respect to θ yields the following equilibrium phases (Casimir et
al., 1959):
                  ⎧
                  ⎨0,                      if K1 > max(0, –K2 )
             θ = arcsin –K1 /2K2 , if – 2K2 < K1 < 0                         (3.4)
                  ⎩
                    π /2,                  if K1 < min(–2K2 , –K2 ).
This rather complex combination of if-statements can be visualised with the aid
of a simple diagram in the K1 –K2 plane, Fig. 3.15. Note the presence of a new
phase ‘easy cone’, with intermediate values of θ between 0 and π /2. The bold lines
separating the domains of different phases are phase transition lines of first (dashed)
or second (solid) order. For simplicity we shall not go into the difference between
the existence of a phase and its stability.
    Introducing a dimensionless ratio K1 /|K2 |, one can display the same informa-
tion in quasi-one-dimensional diagrams, Fig. 3.16. Obviously, two such graphs are
needed to show the qualitatively distinct cases of K2 > 0 and K2 < 0, Figs. 3.16a
and 3.16b, respectively.
    What happens if K2 = 0? In other words, how can one graphically go over to
the limit K2 → 0? The above-considered Eq. (3.1) is not but a particular case of
the more general Eq. (3.3). Therefore, there must be a way to obtain Fig. 3.14 from
Figs. 3.15 and/or 3.16.
    The graphic operation turning Fig. 3.16 into Fig. 3.14 is zooming out. Indeed,
letting K2 go to zero means scaling Fig. 3.16 down, reducing it. Looking at Fig. 3.16
on an ever decreasing scale, one gradually ceases to distinguish the details. On a very
small scale the easy-cone domain shrinks to non-existence and the transition occurs
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                         215




                   Figure 3.14 Phase diagram of the simplest uniaxial magnet.




Figure 3.15 Phase diagram of a uniaxial magnet with two anisotropy constants (Casimir et al.,
1959).




Figure 3.16 The information of Fig. 3.15 presented as two one-dimensional diagrams: (a)
K2 > 0, (b) K2 < 0.



at the origin. Figures 3.16a and 3.16b are no longer different from each other, both
becoming identical to Fig. 3.14.
    Moving in the opposite direction, one can regard Fig. 3.16 as a refinement of
Fig. 3.14. According as one zooms in, it becomes apparent that the transition occurs
not quite at the origin and that the sign of K2 does matter.
216                                                                                      M.D. Kuz’min and A.M. Tishin



    Let us apply these ideas to the analysis of a more complete expression for the
anisotropy energy:
                               Ea = K1 sin2 θ + K2 sin4 θ + K3 sin6 θ.                                              (3.5)
                                                                                              21
This expression is relevant to exchange-dominated RE magnets at arbitrary tem-
perature. The properties of such a system cannot be concisely formulated either as
inequalities of type (3.2) or (3.4) (the latter is already too cumbersome and far too
involved), nor as a phase diagram similar to Fig. 3.15 in the K1 K2 K3 parameter
space—comprehensibility does not belong to the virtues of 3-dimensional draw-
ings. We are left with the only acceptable choice—a quasi-2-dimensional diagram
in reduced coordinates. (We say ‘quasi’, because one always needs two drawings to
show all possible cases, cf. Figs. 3.16a and 3.16b.) Choosing the reduced variables,
we follow certain guidelines. The diagrams should be easy to relate to those of the
cruder approximations (3.1) and (3.3). Quantities prone to changing sign are un-
suitable candidates for the denominators. For instance, Asti’s choice of K2 /K1 and
K3 /K1 (Asti, 1990) is rather inconvenient: every time K1 (T ) changes sign, the locus
of the system goes to infinity, to reappear on the other sheet of the diagram. Clearly,
K1 /K3 and K2 /K3 would be a better choice because, as stated in Section 2.7, K3
as a function of temperature is always sign-definite.
    We find the variables K1 /|K3 | and K2 /|K3 | even more suitable, in accordance
with the requirement that the shape of the diagram should depend possibly little on
the sign of K3 and that |K3 | should act as a scaling factor when going over to the
limit K3 → 0. The phase diagram in such coordinates is displayed in Figs. 3.17a
(K3 > 0) and 3.17b (K3 < 0). The notation for the phases is the same as in Figs. 3.15
and 3.16, however, now the angle θ in the easy-cone phase is determined from the
condition (Asti, 1990):
                                          2
                                       K2 – 3K1 K3 – K2
                                       sin2 θ =           .                      (3.6)
                                             3K3
    SRTs of five different kinds are possible: there can be a first- and a second-order
SRT between every two phases out of the three present in Fig. 3.17, except the pair
easy axis—easy plane, where only a first-order transition can take place. A second-
order transition easy axis—easy plane is impossible in principle, because none of the
two phases is more symmetric than the other (in other words, neither of the two
symmetry groups is a subgroup of the other one).
    The domain boundaries in Fig. 3.17 are mainly straight lines, the curved por-
tions AO and BC being parabolic arcs. The equations describing these boundaries—
the necessary conditions of the SRTs—are collected in Table 3.5. A general neces-
sary condition of a second-order SRT is that the second derivative of the anisotropy
energy with respect to the angle must vanish at the point corresponding to the
higher-symmetry phase. E.g. for the transition easy axis—easy cone this condition
is ∂ 2 Ea /∂θ 2 |θ=0 = 0, whence K1 = 0. In the case of first-order SRTs, a general
21 In real crystals the anisotropy energy may also depend on the angle φ, cf. Eqs. (2.12), (2.13). Our simplified analysis
makes use of the well-known fact that in the vast majority of SRTs only the angle θ changes, while φ remains constant,
fixed by the symmetry. For instance, in the case of the hexagonal crystallographic classes D6 , C6v , D3h and D6h this means
φ = 0 or π /6, which reduces Eq. (2.12) to (3.5) with K3 → K3 ± K3 .
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                       217




Figure 3.17 Phase diagram of a uniaxial magnet with three anisotropy constants: (a) K3 > 0,
(b) K3 < 0.




Table 3.5   Necessary conditions of the SRTs in Fig. 3.17

Phases involved                    1st -order SRT                   2nd -order SRT
                                             √
Easy axis–easy cone                K2 = –2 K1 K3                    K1 = 0
Easy plane–easy cone               K2 /K3 = 1 – 2 1 + K1 /K3        K1 + 2K2 + 3K3 = 0
Easy axis–easy plane               K1 + K2 + K3 = 0                 no SRT
218                                                                                    M.D. Kuz’min and A.M. Tishin



necessary condition is that the function Ea (θ ) must take equal values at the points
corresponding to the two phases in question. Thus, for the first-order transition easy
axis—easy cone this condition is as follows: Ea (θcone ) = Ea (0) = 0.√  Substituting
Eq. (3.6) into Eq. (3.5) and equating the latter to zero yields K2 = –2 K1 K3 .
    Finally, neglecting graphically the third anisotropy constant (K3 → 0) consists
in zooming out of Fig. 3.17. Then the small details are gradually lost, the points
A, B and C merge with the origin and both sheets of the phase diagram turn into
Fig. 3.15. Thus, Fig. 3.15 can be regarded as a cruder, lower-resolution version of
Fig. 3.17, and vice-versa Fig. 3.17 is a more precisely defined version of Fig. 3.15,
taking account of the sign of K3 .

3.2.2 Peculiarities following from the single-ion model
In this subsection some specific predictions of the single-ion model regarding spon-
taneous SRTs in uniaxial magnets will be considered. Admittedly, these predictions
can only be formulated as a number of separate statements, or ‘rules’, unlike in the
case of cubic magnets, where a fully-fledged coherent theory can be developed (see
Section 3.3 below). Nevertheless, these ‘rules’ deserve some respect. Experience
shows that ignoring them may lead to easily avoidable mistakes.
    In the crudest approximation, zooming out of all phase diagrams, an SRT
takes place in a uniaxial magnet when its first anisotropy constant changes sign.
This statement is certainly true for the room-temperature range and might be
somewhat qualified for low temperatures. Staying for the moment near room tem-
perature, there is one natural reason for K1 to change sign in a 3d-4f intermetallic
compound—the competition of the 3d and the 4f contributions. Indeed, since the
high-temperature approximation applies to the RE,
                                       3
                          K1 = K3d – αJ B20 J 2 BJ (x).
                                                    (2)
                                                                               (3.7)
                                       2
Here αJ is the first Stevens factor and B20 is the leading CF parameter in the co-
ordinate representation, Eq. (1.4). Note that the product αJ B20 in Eq. (3.7) equals
the quantity B20 in Eq. (2.98). On account of the known properties of the func-
       (2)
tion BJ (x), Section 2.6, the second term in Eq. (3.7) is sign-definite and falls off
monotonically with temperature. The same is true in respect of K3d ;22 it is smaller
in magnitude but also decreases more slowly with temperature than the RE con-
tribution. Therefore, in order for K1 to become zero near ambient temperature,
the two terms in Eq. (3.7) must have opposite signs. Applied to a particular family
of RE-iron or RE-cobalt compounds, where K3d and B20 are both sign-definite,
this means that, depending on the combination of signs of K3d and B20 , spon-
taneous SRTs will occur either in the compounds of the REs with αJ positive,
or on the contrary, only in those where αJ is negative. This ‘Stevens αJ rule’ is
followed by the vast majority of uniaxial RE-iron and RE-cobalt intermetallics
(Buschow, 1988, 1991; Kirchmayr and Burzo, 1990; Li and Coey, 1991; Franse
            n
and Radwa´ ski, 1993). Exceptions do happen, however, for the obvious reason
that higher-order CF terms may interfere in the anisotropy energy balance, as the
high-T approximation gradually breaks down below room temperature.
22   A rare exception is the compound Y2 Fe14 B, where K1 (T ) is non-monotonic (Bartashevich et al., 1990).
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                    219


     We wish to emphasise that the single-ion theory is not concerned with the
task of evaluating ab initio the CF parameters Bnm , nor with predicting their signs.
Rather, it regards them as phenomenological parameters. It should not be con-
founded with the point-charge model. The well-known fact that the latter generally
fails in metallic systems leaves the single-ion theory undefeated.
     Let us refine our analysis and take into consideration the second anisotropy con-
stant K2 [in the case of tetragonal magnets, Eq. (2.13), this should be understood as
the combination K2 – (K2 )2 + (K2 )2 ]. As we still remain in the room-temperature
range, we may neglect all anisotropy constants but K1 and K2 , taking for the latter
just the fourth-order CF term in Eq. (2.65):
                                      35
                                    K2 =             (4)
                                         βJ B40 J 4 BJ (x).                        (3.8)
                                       8
Here βJ is the second Stevens factor and B40 is the CF parameter in the coordinate
representation, Eq. (1.4). The product βJ B40 in Eq. (3.8) is equivalent to the quan-
tity B40 in Eq. (2.65). In tetragonal magnets the role of B40 in Eq. (3.8) is played by
the combination B40 – ( 35 )1/2 sign βJ |B44 |.
                            2

    The occurrence of a spontaneous spin reorientation is still subject to the ‘Stevens
αJ rule’. However, the presence of a nonzero K2 brings about some variety: the
reorientation may proceed as a single first-order SRT or by way of two second-
order SRTs (Fig. 3.12a displays the latter possibility). Which of the two scenarios
will take place is decided by the sign of K2 (Horner and Varma, 1968), which is in
turn determined by the sign of the second Stevens factor βJ (the 3d contribution
to K2 is negligible).
    As an illustration, let us consider the well-studied archetypal permanent magnet
materials RECo5 . In accordance with the ‘Stevens αJ rule’, spontaneous SRTs are
observed in all RECo5 where αJ < 0, that is with RE = Pr (Yermolenko, 1983),
Nd, Tb (Lemaire, 1966), Dy (Ohkoshi et al., 1977) and Ho (Lemaire, 1966; Chuev
et al., 1981a). Moreover, for RE = Pr, Nd, Dy and Ho the SRTs are distinctly of
second order. This is in perfect agreement with the ‘Stevens βJ rule’, because all
four above-mentioned REs have βJ < 0 (Table 3.1).
    In contrast, Tb has βJ > 0, therefore, the SRT in TbCo5 must be of first
order. This definite prediction of the single-ion theory is apparently at variance with
experiment, which interprets the reorientation process in TbCo5 as two closely
situated second-order SRTs. The situation is aggravated further by the fact that it is
not just from bulk magnetic measurements (Ermolenko, 1980) that this conclusion
was made. Also neutron diffraction experiments (Lemaire and Schweizer, 1967;
Kelarev et al., 1980) reportedly detected in TbCo5 , in a narrow interval just above
400 K, the presence of an easy-cone phase with intermediate values of θ between
0 and π /2.
    It should be noted that these data are open to another interpretation. Namely,
that two phases—easy axis and easy plane—coexist in the vicinity of the SRT
(which is of first order, as predicted by the single-ion model). The relative con-
tent of the two phases varies gradually with temperature, from the pure easy plane
below ∼400 K to the pure easy axis above ∼425 K. This interpretation is corrobo-
rated by the peculiar shape of the temperature dependence of the angle θ observed
220                                                                                   M.D. Kuz’min and A.M. Tishin




Figure 3.18 Orientation angle of the easy magnetisation direction versus temperature. Two
paradigms of continuous spin reorientation in a uniaxial magnet: (i) two second-order SRTs at
T1 and T2 , solid line, (ii) one first-order SRT broadened by normally distributed inhomogeneity,
dashed line (after Kuz’min, 2000).


in TbCo5 . It resembles the dashed curve in Fig. 3.18. Such a shape of the de-
pendence θ (T ), described by the complementary error function, is characteristic
of ‘smeared-out’ first-order SRTs (Kuz’min, 2000). It is clearly distinct from the
arcsine-type dependence (solid line) with two sharp second-order transition points
T1 and T2 . The inhomogeneity of composition, necessary for smearing out of a
first-order SRT, is present in TbCo5 , whose real stoichiometry is TbCo5+δ , with
δ ≈ 0.1.
    According as the experimental arguments in favour of two second-order SRTs
in TbCo5 become less unambiguous, the single-ion theory, on the contrary,
strengthens its insistence on a single first-order SRT. Beyond all doubt is the general
analysis of Horner and Varma (1968) demonstrating that an SRT must be of first
order if K2 < 0 and of second order if K2 > 0. On the other hand, the shape of
the dependence θ (T ) in the other RECo5 undergoing spontaneous spin reorien-
tation (RE = Pr, Nd, Dy, Ho) bears close resemblance to the continuous curve of
Fig. 3.18, with two square-root-type anomalies characteristic of Landau’s theory.23
The SRTs are clearly of second order, and therefore K2 > 0, in the aforementioned
RECo5 with βJ < 0. By virtue of Eq. (3.8), K2 must be negative in TbCo5 , where
βJ > 0.
    The only (unlikely) loophole in our logic might be that T = 400 K is not high
enough a temperature and that K2 in TbCo5 , negative at the highest temperatures
as it should be, becomes positive somewhere above T = 425 K due to an extraor-
dinarily large sixth-order CF term, cf. the second one of Eqs. (2.65). This remote
possibility can be ruled out completely. Indeed, as stated in Section 2.7, K2 cannot
change sign more than once. And in TbCo5 K2 < 0 at T = 4.2 K, as found experi-
23 In the case of PrCo , where the reorientation process is incomplete, only the higher-temperature anomaly is observed,
                        5
T1 being effectively negative.
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                     221


mentally by Ermolenko (1980). Therefore, K2 is negative at any temperature. Thus,
the single-ion theory still insists that the SRT in TbCo5 must be of first order.
    The true situation in TbCo5 has been finally established in recent scanning dif-
ferential calorimetry experiments (Tereshina et al., 2007), which revealed a nonzero
latent heat of the SRT—an ultimate proof that it is of first order. The single-ion
model made no mistake.
    Let us formulate a yet another specific prediction of the model. When the easy
direction lies in the basal plane, its orientation within the plane for a certain family
of tetragonal compounds is determined by the sign of the second Stevens coeffi-
cient βJ , whereas in hexagonal compounds it follows the third Stevens factor γJ .
Moreover, in the latter case the orientation within the basal plane cannot change
as temperature is lowered. In tetragonal compounds such a reorientation may take
place no more than once, but in reality SRTs of this kind are extremely rare.
    For definiteness, we limit ourselves to the hexagonal crystallographic classes D6 ,
C6v , D3h and D6h , where the anisotropy energy is presentable as Eq. (2.12) with the
anisotropy constants given by Eqs. (2.65). By assumption, θ ≡ π /2. Therefore, the
equilibrium value of the angle φ, either 0 or π /6, is determined by the sign of the
quantity K3 ,
                                     √
                                        231
                             K3 =                       (6)
                                            γJ B66 J 6 BJ (x).                      (3.9)
                                       16
Again, we have factored out the Stevens coefficient γJ , so that the product γJ B66
in Eq. (3.9) is equivalent to the quantity B66 in the last one of Eqs. (2.65).
    Thus, if the product γJ B66 is negative, the easy magnetisation direction within
the basal plane is the a axis, or [100], corresponding to φ = 0. If γJ B66 > 0, then
the easy axis is b, or [120], φ = π /6. For a given family of RE-iron or RE-cobalt
compounds, all having B66 of the same sign, the easy direction is determined by the
sign of the third Stevens factor γJ .
    This ‘Stevens γJ rule’ can be best illustrated by our favourite example—the
RECo5 compounds—whose symmetry is described by the holohedral hexagonal
group D6h . Back in the 1960’s it was found from neutron diffraction on NdCo5
and TbCo5 (Lemaire, 1966; Lemaire and Schweizer, 1967) and also from magnetic
measurements on a single crystal of HoCo5 (Katsuraki and Yoshii, 1968) that the
easy direction within the basal plane in all those compounds is the a axis. This
is not illogical, since Nd, Tb and Ho have γJ < 0, see Table 3.1. Following the
same logic, the easy direction in PrCo5 and in DyCo5 must be the b axis, because
γJ > 0 for both Pr and Dy. Indeed, magnetisation data obtained on a PrCo5 single
crystal (Yermolenko, 1983) confirmed that the easy direction there rotates from the
c towards the b axis (even though the reorientation process is not completed down
to T = 0).
    Unexpectedly, DyCo5 falls out of line. A magnetisation study of a single crystal
(Ohkoshi et al., 1977) concluded that the easy direction at and below room tem-
perature is the a rather than the b axis. This statement was reiterated in the work
of Berezin et al. (1980) and even in the neutron diffraction paper of Chuyev et al.
(1981b). The single-ion model seems to have received a fatal blow and will never
recover.
222                                                            M.D. Kuz’min and A.M. Tishin



    However, a more careful perusal of the above articles reveals a number of dis-
crepancies. Thus, according to Ohkoshi et al. (1977), the a axis is the same as the
[110] direction and the b axis is [100], which is rather unusual. A clue to their
unconventional notation may be found in their previous paper (Ohkoshi et al.,
1976), stating that in NdCo5 the easy direction below T = 245 K is the b axis, or
[100] (while in reality it is the a axis, or [100]). Apparently, the authors use a non-
standard set of Bravais vectors where two of them, lying in the basal plane, make
an angle of 60°, rather than 120° as in the standard hexagonal set. Moreover, the
direction of those Bravais vectors is called the b axis, while their bisector is called
the a axis. In short, Ohkoshi et al. (1976, 1977) seem to have swapped the a and b
axes, as compared with the standard notation. Of course, this is not but a plausible
conjecture.
    Furthermore, the text of the paper of Berezin et al. (1980) speaks of an easy axis
a and a hard axis b. However, from the experimental magnetisation curves in Fig. 1
thereof one concludes that the opposite is true. Namely, that the easy magnetisation
direction is the b axis.
    Finally, in their neutron diffraction experiments the unsuspecting Chuyev et al.
(1981b) did not at all pose the problem of checking the orientation of the easy
direction within the basal plane of DyCo5 , having taken for granted that is was
along the a axis.
    At our instance, Skokov (2007) have recently conducted a series of purposeful
tests on a single crystal of DyCo5 , which have established that the easy magnetisation
direction at and below room temperature lies along the crystallographic axis b, i.e.
[120], exactly as predicted by the single-ion model, or by the ‘Stevens γJ rule’.
    The above examples demonstrate—quite convincingly in our view—that the
single-ion theory of magnetocrystalline anisotropy has the power of prediction. To
the extent that it enables an armchair theoretician to find mistakes in experimental
papers. In this connection, the recent attempts to question the validity of the single-
ion approach and even to supplant it with a new mechanism (Irkhin, 2002) can only
arouse bewilderment. We reiterate, however: in order for the strength of the single-
ion model to be fully appreciated, it has to be kept strictly apart from the task of
computing the CF parameters ab initio. Certain progress has been achieved in the
latter field, too (Hummler and Fähnle, 1996; Novák, 1996), based on the density
functional theory rather than on the naive point-charge model.

3.3 Spontaneous SRTs in cubic magnets
In the crudest approximation, the anisotropy energy of any cubic crystal is given by

                          Ea = K1 n2 n2 + n2 n2 + n2 n2 .
                                   x y     y z     z x                             (3.10)
The equilibrium phases are the 6-fold degenerate [100] and 8-fold degenerate
[111]. The former is energetically favourable at K1 > 0, the latter at K1 < 0.
A first-order SRT takes place at the point K1 = 0. This information is presented
graphically in a one-dimensional phase diagram, Fig. 3.19.
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                                 223




Figure 3.19 Phase diagram of a cubic magnet described by Eq. (3.10). The positive semi-axis
is the domain of the [100] phase, the negative semi-axis (hatched) is the domain of the [111]
phase, the origin being the transition point.


    Let us consider a more realistic expression for the anisotropy energy (2.8), valid
for the cubic crystallographic classes possessing 4-fold symmetry axes,24 O, Td and
Oh :
                          Ea = K1 n2 n2 + n2 n2 + n2 n2 + K2 n2 n2 n2 .
                                   x y     y z     z x        x y z                                          (3.11)
This expression is relevant to cubic exchange-dominated 3d-4f compounds, such
as e.g. Laves phases REFe2 and RECo2 . In that case the omission of terms of order
higher than six is a valid approximation. It is also justified to neglect the 3d con-
tribution to the fourth- and sixth-order anisotropy constants. This contribution,
originating from the relatively weak spin-orbit coupling, decreases rapidly as the
order of the anisotropy constants n increases (∝ λn ). In practice, it can play a role
only in second-order anisotropy constants, here forbidden by the symmetry. Thus,
not unreasonably we expect that the linear theory of Section 2.5 should apply to
the systems under consideration to full extent.
                                                              m
    Proceeding from the CF Hamiltonian (1.26), where On are meant to be the
usual Stevens operators in the J representation, and repeating the manipulations of
Section 2.5, we get for the anisotropy constants the following expressions:
                              K1 = –40J 4 b4 BJ (x) – 168J 6 b6 BJ (x)
                                              (4)                (6)
                                                                                                             (3.12)
                              K2 = 1848J 6 b6 BJ (x).
                                               (6)

         (4,6)
Here BJ (x) are the fourth- and sixth-order generalised Brillouin functions (GBF,
Section 2.6) and x is the magneto-thermal ratio (2.33).
    Figure 3.20 displays the phase diagram of a cubic magnet described by Eq. (3.11)
(Smit and Wijn, 1959). We observe the presence of an additional phase—the 12-
fold degenerate [110]. In the considered approximation all the transitions are of first
order.
    It is convenient to present the information contained in the two-dimensional
phase diagram (Fig. 3.20) as quasi-one-dimensional diagrams, Fig. 3.21. There, the
role of the coordinate is played by the ratio K1 /|K2 |. The need to distinguish two
essentially distinct cases according to the sign of K2 brings about the two sheets of
the diagram, Figs. 3.21a and 3.21b. This insignificant complication is outweighed
by the advantages of Fig. 3.21. The latter is related in a rather straightforward way
to Fig. 3.19, this relation being a graphic realisation of neglecting K2 , or taking
the limit K2 → 0. Zooming out of Fig. 3.21, one gradually loses out of sight the
details like the intermediate [110] domain or the deviation of the transition point
 24 It will be recalled that in the case of the cubic classes T and T an extra six-order term (2.9) must be taken into
                                                                     h
account.
224                                                             M.D. Kuz’min and A.M. Tishin




Figure 3.20 Phase diagram of a cubic magnet with two anisotropy constants (Smit and Wijn,
1959). The oblique phase boundaries are 9K1 + 4K2 = 0 (second quadrant) and 9K1 + K2 = 0
(fourth quadrant).




Figure 3.21 A quasi-one-dimensional presentation of the phase diagram of Fig. 3.19: (a)
K2 > 0, (b) K2 < 0.



from the origin. What eventually remains of either part of Fig. 3.21 is two semi-
axes, the blank positive [100] and the hatched negative [111], that is just Fig. 3.19.
    The more important advantage of Fig. 3.21 is its one-dimensionality. Thanks
to it, the SRT conditions can be presented simply as taking on of certain universal
values, –4/9, 0 and 1/9, by the variable K1 /|K2 |. By means of Eqs. (3.12) these
conditions can be readily expressed in terms of the dimensionless CF ratio b4 /|b6 |.
It is still necessary to distinguish two particular cases according to the sign of b6 .
Namely:
A. b6 > 0 (K2 > 0). There are two transitions:
   1. [111]–[110] at K1 /|K2 | = –4/9. By virtue of Eqs. (3.12), this is equivalent
      to
                                     b4    49 B (6) (x)
                                          = J2 J        .                           (3.13)
                                    |b6 |       (4)
                                           3 BJ (x)
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                     225




Figure 3.22 Phase diagrams of a cubic RE magnet in the dimensionless coordinates ‘temper-
ature–crystal field’.


    2. [110]–[100] at K1 /|K2 | = 0, whence
                                            b4      21 B (6) (x)
                                                 = – J2 J        .                (3.14)
                                           |b6 |         (4)
                                                     5 BJ (x)
B. b6 < 0 (K2 < 0). One further transition is possible:
   3. [111]–[100] at K1 /|K2 | = 1/9, which yields
                                            b4      14 B (6) (x)
                                                 = – J2 J        .                (3.15)
                                           |b6 |         (4)
                                                    15 BJ (x)
It is time to take advantage of the one-dimensionality of the chosen representation.
Having saved a dimension in Fig. 3.21, we now have the option of adding a new
second dimension to our diagrams. We choose the quantity 1/x for this role, which
will enable us to include temperature evolution of the system into the picture. The
variable 1/x is more convenient for the purpose than x, because at low temperatures
1/x is directly proportional to T ; in any case its dependence on T is monotonic.
     Now Eqs. (3.13)–(3.15) describe curves in the plane 1/x – b4 /|b6 |, Fig. 3.22. It
is interesting to note that one and the same special function is involved in all three
cases—the ratio of the sixth- to the fourth-order GBF—only the prefactors differ.
The advantage of the coordinates employed in Fig. 3.22 is that the phase bound-
aries therein are universal (apart from their dependence on the quantum number
J , Fig. 3.22 corresponds to J = 8). Anyhow, the topology of the phase diagrams
does not depend on J , while the ordinates of the points A, B and C are given by
                                                                 7
simple formulae: 49 (J – 2)(2J – 5), – 21 (J – 2)(2J – 5) and – 15 (J – 2)(2J – 5), re-
                    6                   10
spectively. Therefore, for any other J diagrams similar to Fig. 3.22 can be sketched
rather straightforwardly. Accurate drawings should present no major difficulties ei-
ther, since all GBF have been tabulated (Kuz’min, 1992).
     Temperature evolution of a specific compound can be depicted in Fig. 3.22 by a
horizontal line, because the quantity b4 /|b6 | proper of the system remains constant
226                                                            M.D. Kuz’min and A.M. Tishin



as ‘temperature’ 1/x varies. If this horizontal line crosses one of the curves, a spon-
taneous SRT takes place at the temperature T corresponding to the abscissa of the
crossing-point. Thus, spontaneous SRTs follow a scenario fully determined by the
ratio b4 /|b6 | and independent of the strength of the exchange interaction (as long
as the system is exchange-dominated, see Section 2.2). Knowledge of the exchange
field on the RE and of its temperature dependence is only needed for establishing
the quantitative relation between x and T . The sign of b6 is very important, since
it decides to which sheet of the phase diagram, Figs. 3.22a or 3.22b, the system
belongs. To determine sign(b6 ), it suffices to find sign(K2 ) at any temperature.
    A number of more specific conclusions can be drawn.
1. No more than one spontaneous SRT can take place in any one system.
2. If the low-temperature phase is [100], a spontaneous SRT is in principle impos-
   sible.
3. If the low-temperature phase is [110], a spontaneous SRT will take place in-
   evitably.
4. If the low-temperature phase is [111] and b4 > 0, no spontaneous SRT is possi-
   ble.
5. If the low-temperature phase is [111] and b4 < 0, an SRT is inevitable, [100]
   being the high-temperature phase.
     Table 3.6 summarises the predictions of the single-ion theory and the experi-
mental information on the easy magnetisation directions and spontaneous SRTs in
the cubic Laves phases REFe2 and RECo2 . Full-potential density-functional calcu-
lations (Diviš et al., 1995) yielded a positive sign for the fourth-order CF parameter
in the coordinate representation (i.e. for the quantity b4 /βJ ) and a negative sign for
b6 /γJ . Accordingly, in Table 3.6, sign(b4 ) = sign(βJ ) and sign(b6 ) = – sign(γJ ),
cf. Table 3.1. For RE = Sm γJ is undefined, its role being played by the quantity
δ6 < 0 (Magnani et al., 2003). Therefore, b6 > 0 for SmFe2 and SmCo2 .
     Examining Table 3.6 one observes that the single-ion model agrees with ex-
periment in all cases without exception. It should be noted that in each case the
model makes a binding prediction of the high-T phase as well as predicting an op-
tional low-T phase. In order for the low-T option to be realised, i.e. in order for
the spontaneous SRT to actually happen, the ratio b4 /|b6 | must be within certain
bounds, also predicted by the theory. In some exceptional cases, e.g. in ErFe2 and
in ErCo2 , the model can rule out the possibility of an SRT altogether. However, in
general the single-ion theory is not concerned with calculating CF parameters ab
initio, therefore it cannot be held responsible for wrongly predicted SRTs in specific
compounds. Allegations of failure of the single-ion model sometimes found in the
literature are in fact reports of failures of various modifications of the point-charge
model (confused with the single-ion one).
     In those cases when the CF parameters are considered known, their non-
compliance with the single-ion model is a sure sign of mistake. Thus, according to
Gignoux et al. (1975, last line of Table I) b4 /|b6 | in HoCo2 is about –204. However,
the fact that HoCo2 undergoes a spontaneous SRT [110]–[100] means that this
ratio must be between 0 and –138.6, cf. Fig. 3.22a. Further checks unearth an ap-
parent inconsistency between the values b6 = 2.3 × 10–5 K and K2 = 109 erg/cm3 ,
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                                                          227


Table 3.6 Easy directions of magnetisation in the cubic Laves phases REFe2 and RECo2 . In
each case the model makes a binding prediction of the high-temperature orientation (right-
hand symbol), the predicted low-temperature phase being optional

RE           Single-ion model                                                Experiment
             Sign b4     Sign b6                Easy direction               REFe2                       RECo2

Pr          –1               –1                  [111]/[100]                      –/[100]a                    –/[100]b
Nd          –1               +1                  [110]/[100]                  [110]/[100]a                [110]/[100]c
Sm          +1               +1                  [110]/[111]                  [110]/[111]d                    –/[111]e
Tb          +1               +1                  [110]/[111]                      –/[111]f                    –/[111]b
Dy          –1               –1                  [111]/[100]                      –/[100]f                    –/[100]b
Ho          –1               +1                  [110]/[100]                      –/[100]f                [110]/[100]g
Er          +1               –1                      –/[111]                      –/[111]f                    –/[111]b
Tm          +1               +1                  [110]/[111]                      –/[111]f                    –/[111]h
Yb          –1               –1                  [111]/[100]                      –/[100]a               paramagnetism
a Meyer et al. (1981). b Atzmony and Dublon (1977). c Atzmony et al. (1976). d Van Diepen et al. (1973). e Gratz et al. (1994).
   f Taylor (1971). g Gignoux et al. (1975). h Déportes et al. (1974).




reported by Gignoux et al. (1975). According to the second one of Eqs. (3.12),
at T = 0 and J = 8, K2 must equal 166,486,320 b6 = 3.8 × 103 K/f.u., or
1.15 × 1010 erg/cm3 , which is an order of magnitude too high. Given that K2 was
determined experimentally, one is left to conclude that the reported value of b6
(and most likely of b4 as well) is mistaken.
    There are also examples of false SRTs in the literature, later proved to be arte-
facts. For instance, an impossible transition [100]–[111] ‘discovered’ by Shimotomai
et al. (1980) in PrFe2 (a quick glance at Fig. 3.22 is sufficient to conclude that [100]
cannot be the low-T phase in a spontaneous SRT). Or, e.g. an incomplete transi-
tion [100]–[110] in YbFe2 (Meyer et al., 1981), where the easy direction ‘slightly
deviates’ from [100] above T ≈ 50 K. Apart from the afore-mentioned fact that
[110] can only be the low-T phase in a spontaneous SRT and [100] can only be the
high-T one, the general theory (Section 3.1) states that the orientation angle always
changes sharply near an SRT, even if the latter is of second order (Fig. 3.12a). It is
like an airplane, which cannot take off or land ‘slightly’.
    Most tortuous was the way to the truth in the case of HoFe2 . An early review by
Taylor (1971, Table 7 thereof) gave the correct easy magnetisation direction, [100],
without any SRT. This view was soon reiterated by Atzmony et al. (1972), who
interpreted the Mössbauer spectrum of HoFe2 at T = 4.2 K as being characteristic
of the [100] phase. Had the authors seen the above Conclusion 2, they would have
put a full stop at this point. The [100] phase has the simplest, most reliably identified
Mössbauer spectrum, our Conclusion 2 is therefore quite useful.
    Anyhow, this was not to be, and during the decade of the 1970’s articles of
the same authors were coming out (Atzmony and Dublon, 1977 and references
therein), claiming having observed an SRT in HoFe2 at around 14 K. The disproval
finally came in the form of a direct specific heat measurement (Germano et al.,
228                                                           M.D. Kuz’min and A.M. Tishin



1979), which found in that temperature range no anomaly characteristic of a phase
transition.
    To be fair to Atzmony et al., their version of the events was not impossible, but
rather improbable. Between T = 0 and 14 K the ratio B8 (x)/B8 (x) in Eq. (3.14)
                                                            (6)     (4)

changes by as little as a quarter of a percent [for HoFe2 , x ≈ 750 K/T (Kuz’min,
2001)]. In order for the SRT at about 14 K to become reality, nature would have
to set the ratio b4 /|b6 | within a very narrow interval immediately above –138.6. In
reality, b4 /|b6 | ≈ –184 (Germano et al., 1979).
    The controversy around the SRT in HoFe2 is typical of the state of the theory
in the 1970’s. The basics of setting and diagonalising the RE Hamiltonian were
well known by then, whereas the approximations introduced in this Chapter were
not. Thus, it was unknown that the sequence of phases is determined by a sole
quantity—the ratio b4 /|b6 |—and does not depend on the exchange field at all. That
is, out of the three disposable model parameters only one is relevant to deciding if
a spontaneous SRT is to take place. The presence of irrelevant parameters in the
early calculations could not bring about but confusion.
    Also unsound were the attempts to ‘explain’ the complex Mössbauer spectra
observed in some REFe2 by way of intermediate low-symmetry orientations of
the easy magnetisation direction (Atzmony and Dariel, 1974). Such an explanation
involves necessarily a large anisotropy constant of eighth order. The linear theory—
whereby this and all higher-order anisotropy constants are strictly nil—is admittedly
an approximation. It is, however, a well-founded approximation, so what is forbid-
den by it can only be small. A rather more plausible but prosaic explanation could be
that the samples investigated by Atzmony and Dariel (1974) were not single-phase.
    The theory developed in this subsection and expressed graphically in Fig. 3.22
is not limited to the cubic Laves phases. Without major modifications it applies e.g.
to the RE6 Fe23 compounds. One subtlety needs to be taken into consideration,
however: the local symmetry of the 24e sites occupied by the RE in RE6 Fe23 is
not cubic, but rather tetragonal, C4v . Therefore, five nonzero CF parameters are
allowed, cf. Eq. (1.32). Equations (3.12) are still valid, provided that linear combi-
nations of 4th- and 6th-order CF parameters, 12 (7B40 + B44 ) and 24 (3B60 – B64 ),
                                                   1                    1

are substituted for b4 and b6 , respectively. These combinations arise in the process
of averaging over the 24e sites with differently oriented local 4-fold axes (paral-
lel to [100], [010], and [001]). The second-order CF parameter B20 is averaged
out completely. Figure 3.22 is then valid too, provided the ordinate is defined as
(14B40 + 2B44 )/|3B60 – B64 |.



      4. Conclusion
      We are about to close this Chapter about crystal-field effects in 3d-4f in-
termetallics. From the subject of CF on REs we moved on to magnetocrystalline
anisotropy and further on to SRTs. En route we touched upon the influence of the
CF on the magnetic moment of the RE. Of course, the narrow path we took does
Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds                              229


not cover the whole area of CF-related phenomena. For example, magnetostric-
tion can be described—similarly to anisotropy constants—by expressions involving
generalised Brillouin functions (Kuz’min, 1992). However, in this Chapter we de-
liberately did not enter into the topic of magnetostriction. It will be exhaustively
covered in an extensive monograph by Professor A. del Moral, due to appear shortly.
Likewise, the route towards SRTs is not but one of many ways to proceed from
the subject of magnetic anisotropy. Other possible connections include micromag-
netism, ac susceptibility, coercivity, magnetic resonance etc.
    This Chapter is addressed primarily to experimentalists. At the intuitive level,
most of them would be well familiar with the physics of the phenomena discussed
above. They might be less forthcoming when it comes to committing themselves
to a quantitative estimate, for the obvious reason that a computer program able
to ‘take account of everything’ is hard to come by. A message we tried to get
across is that taking everything into account is not always necessary. When valid, a
suitable approximation may offer the invaluable advantages of a concise analytical
expression—greater transparency and simplicity of calculations.
    This turned out particularly well in the case of the J -mixing effect, Section 2.9,
where expressions suitable for back-of-the-envelope calculations (2.123, 2.126)
were obtained. Another example worthy of mention is the newly developed in
Section 3.3 theory of spontaneous SRTs in exchange-dominated cubic magnets. Its
main statement is ultimately simple: as temperature varies, the system goes through
a sequence of (at the most two) phases which is unambiguously determined by a
single quantity—a ratio of fourth- and sixth-order CF parameters. As regards uni-
axial magnets, there the main results can be formulated as the ‘Stevens αJ , βJ and
γJ rules’. Even though they do not constitute an accomplished theory, these rules
are nonetheless binding necessary conditions, to the extent that their violation is a
nearly certain sign of a mistake. It was also graphically shown how the classical phase
diagram of a uniaxial magnet (Fig. 3.15) is modified when a third anisotropy con-
stant is allowed for, establishing a simple visual relation between Figs. 3.15 and 3.17.
Last but not least, the interplay of the 3d-4f exchange and the CF in the expression
for the leading anisotropy constant K1 , was shown to take a particularly transparent
form at high T (2.42) and also when T → 0 (2.107). Our goal was to bring all
these simple findings to the notice of workers in the field of magnetic materials.


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        CHAPTER              FOUR



        Magnetocaloric Refrigeration at Ambient
        Temperature
        Ekkes Brück *



        Contents
        List of Symbols and Abbreviations                                                                         237
        1. Brief Review of Current Refrigeration Technology                                                       237
        2. Introduction to Magnetic Refrigeration                                                                 239
        3. Thermodynamics                                                                                         241
        4. Materials                                                                                              247
            4.1 Gd metal and alloys                                                                               247
            4.2 Gd5 (Ge,Si)4 and related compounds                                                                248
            4.3 La(Fe,Si)13 and related compounds                                                                 253
            4.4 MnAs based compounds                                                                              256
            4.5 Heusler alloys                                                                                    259
            4.6 Fe2 P based compounds                                                                             262
            4.7 Other Mn intermetallic compounds                                                                  265
            4.8 Amorphous materials                                                                               267
            4.9 Manganites                                                                                        269
        5. Comparison of Different Materials and Miscellaneous Measurements                                       270
        6. Demonstrators and Prototypes                                                                           274
        7. Outlook                                                                                                280
        Acknowledgements                                                                                          281
        References                                                                                                281

        Abstract
        Modern society relies on readily available refrigeration. Magnetic refrigeration has three
        prominent advantages compared to the most commonly used compressor-based refrig-
        eration. First there are no harmful gasses involved, second it may be built more compact
        as the working material is a solid and third magnetic refrigerators generate much less
        noise. Recently a new class of magnetic refrigerant-materials for room-temperature ap-
        plications was discovered. These new materials have important advantages over existing
        magnetic coolants: They exhibit a large magnetocaloric effect (MCE) in conjunction with a
        magnetic phase-transition of first order. This MCE is, larger than that of Gd metal, which
   *   Department of Mechanical Engineering, Federal University of Santa Catarina, Florianopolis SC, Brazil and
       Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam,
       The Netherlands
       E-mail: bruck@science.uva.nl

Handbook of Magnetic Materials, edited by K.H.J. Buschow                                          © 2008 Elsevier B.V.
Volume 17 ISSN 1567-2719 DOI 10.1016/S1567-2719(07)17004-9                                         All rights reserved.



                                                                                                                  235
236                                                                                     E. Brück



      is used in the demonstration refrigerators built to explore the potential of this evolving
      technology. In the present review we compare different materials, however, concentrat-
      ing on transition metal containing compounds, as we expect that the limited availability
      of Rare-earth elements will hamper the industrial applicability. Recently also more and
      more demonstrators and prototypes are being developed. We compare the different con-
      cepts and discuss some important design issues.



      Key Words: magnetic refrigeration, transition metal compounds, magnetic entropy,
      magnetocaloric effects
Magnetocaloric Refrigeration at Ambient Temperature                            237




      List of Symbols and Abbreviations
      AMR        Active magnetic regenerator
      A          Helmholtz free energy
      α          thermal expansion coefficient
      BJ         Brillouin function
      Bmax       maximal applied field
      C          Curie constant
      C*         effective Curie constant
      CH         heat capacity at iso-magnetic intensity
      Cm         magnetic contribution to specific heat
      COP        coefficient of performance
      χ          magnetic susceptibility
      DSC        differential scanning calorimetry
        B        magnetic field change from 0 to Bmax
        Sm       magnetic entropy change
        Tad      adiabatic temperature change
      g          Lande’s-factor
      H          magnetic intensity
      H0         externally applied field
      J          quantum number of angular momentum
      k          Boltzmann constant
      μB         Bohr magneton
      M          magnetization
      Ms         saturation magnetization
      MCE        Magnetocaloric effect
      N          numbers of magnetic moments
      NW         Weiss-field constant
      S          entropy
      T          absolute temperature
      TC         Curie temperature
      T0*        effective Curie temperature
      T0         Curie temperature if the lattice was not compressible
      U          internal energy
      Um         magnetic contribution to internal energy
      V          volume
      V0         volume in absence of exchange interaction
      W          magnetic work
      Z          partition function


      1. Brief Review of Current Refrigeration Technology
     The art and science of refrigeration is concerned with the cooling of matter
to temperatures lower than those available in the surroundings at a particular time
and place.
238                                                                                E. Brück



    Refrigeration can be achieved in many different ways. The principal methods
of refrigeration, which have found practical use in industry and consumer products,
are:
(a)   vapor compression,
(b)   vapor absorption,
(c)   air cycle,
(d)   thermo-electric.
    Vapor-compression refrigeration systems are the most common refrigeration systems
in use today. Like in vapor absorption systems, the refrigerating effect is produced
by making a volatile fluid boil at a suitably low temperature. The vast majority of
cooling devices of all sizes from domestic refrigerators to large industrial systems
use the vapor compression principle and, basically, this is owed to the better cost-
to-benefit ratio attainable from a vapor compression installation in comparison with
other systems.
    Absorption refrigeration systems are used in large chemical plants, in air-conditioning
and in some domestic refrigerators. Because it needs an input of heat at a moder-
ately high temperature to drive it, it finds applications where such heat is readily
available or where mechanical power is not available. With the advent of cogener-
ation plants, absorption refrigeration is experiencing a renewal but limited to this
field, for a review see (Srikhirin et al., 2001). Efficiency attainable from absorption
systems is not competitive to that one attained from a vapor compression system.
    Air cycle refrigeration, in which the temperature of air is reduced by an expansion
process in which work is done by the air, was used for many years as the principal
method of refrigeration at sea, chiefly on account of the inherent safety of the
method. The open cycle cold air machine or heat pump may seem very attractive
by its simplicity and environmental advantage, and numerous attempts have been
made over the years to revive the idea, eliminating some of its drawbacks by using
turbo or other high speed rotary machinery, for a review see (Gigiel, 1996). The
problem of excessive power consumption remains, however. It is clear that the open
cold air system has little chance of gaining any importance for refrigeration or heat
pumps in the normal temperature range unless a significant break-through should
occur and this does not seem very likely at the present time.
    Thermo-electric refrigeration works on the principle of the Peltier effect; i.e. the
cooling effect produced when an electric current is passed through a junction of two
dissimilar metals. With the materials so far available its efficiency is rather low, but it
has many uses in circumstances where efficiency does not matter much, as in very
small refrigerators for specimens on microscope stages, instruments for measuring
dew point, food and liquid storage for camping and a few others. Thermoelectrics,
despite the billion dollars in research spent to date, have not cracked 10% in effi-
ciency though they may achieve as high as 20% to 30% if recent reports are true,
for a review see (Riffat and Ma, 2003). Still, the technology is far less efficient than
vapor compression systems.
    The design of a refrigeration system is a type of problem of which the solution
involves many considerations. Design invariably requires the critical evaluation of
the solutions by considering factors such as economics, safety, reliability, and envi-
Magnetocaloric Refrigeration at Ambient Temperature                              239


ronmental impact, before choosing a course of action. The vapor compression cycle
has dominated the refrigeration market to date because of its advantages: high effi-
ciency, low toxicity, low cost, and simple mechanical embodiments. Perhaps this is
because as much as 90% of the worlds heat pumping power; i.e. refrigeration, water
chilling, air conditioning, various industrial heating and cooling processes among
others, is based on the vapor compression cycle principle.
    In recent years environmental aspects have been becoming an increasingly im-
portant issue in the design and development of refrigeration systems. Especially in
vapor compression systems, the banning of CFCs and HCFCs because of their en-
vironmental disadvantages has made way for other refrigeration technologies which
until now have been largely ignored by the refrigeration market. As environmental
concerns grow, alternative technologies which use either inert gasses or no fluid at
all become attractive solutions to the environment problem. A significant part of
the refrigeration industry R&D expenditures worldwide is now oriented towards
the development of such alternative technologies to the replacement of vapor com-
pression systems in a mid- to long-term perspective. One of these alternatives is
magnetic refrigeration.



      2. Introduction to Magnetic Refrigeration
       Magnetic refrigeration, based on the magnetocaloric effect (MCE), has
recently received increased attention as an alternative to the well-established
compression-evaporation cycle for room-temperature applications. Magnetic ma-
terials contain two energy reservoirs; the usual phonon excitations connected to
lattice degrees of freedom and magnetic excitations connected to spin degrees
of freedom. These two reservoirs are generally well coupled by the spin lattice
coupling that ensures loss-free energy transfer within millisecond time scales. An
externally applied magnetic field can strongly affect the spin degree of freedom
that results in the MCE. In the magnetic refrigeration cycle, depicted in Fig. 4.1,
initially randomly oriented magnetic moments are aligned by a magnetic field, re-
sulting in heating of the magnetic material. This heat is removed from the material
to the ambient by heat transfer. On removing the field, the magnetic moments ran-
domise, which leads to cooling of the material below ambient temperature. Heat
from the system to be cooled can then be extracted using a heat-transfer medium.
Depending on the operating temperature, the heat-transfer medium may be water
(with antifreeze) or air, and for very low temperatures helium. The cycle described
here is very similar to the vapour compression refrigeration cycle: on compression
the temperature of a gas increases, in the condenser this heat is expelled to the
environment and on expansion the gas cools below ambient temperature and can
take up heat from the environment. In contrast to a compression cycle the magnetic
refrigeration cycle can be performed quasi static which results in the possibility to
operate close to Carnot efficiency.
     Therefore, magnetic refrigeration is an environmentally friendly cooling tech-
nology. It does not use ozone depleting chemicals (CFCs), hazardous chemicals
240                                                                                  E. Brück




Figure 4.1 Schematic representation of a magnetic-refrigeration cycle, which transports heat
from the heat load to the ambient. Left and right depict material in low and high magnetic
field, respectively.


(NH3 ), or greenhouse gases (HCFCs and HFCs). The difference between vapour-
cycle refrigerators and magnetic refrigerators manifests itself also is the amount
of energy loss incurred during the refrigeration cycle. From thermodynamics it
appears feasible to construct magnetic refrigerators that have very high Carnot
efficiency compared to conventional vapour pressure refrigerators (Brown, 1976;
Steyert, 1978). This higher energy efficiency will also result in a reduced CO2
release. Current research aims at new magnetic materials displaying larger magne-
tocaloric effects, which then can be operated in fields of about 2 T or less, that can
be generated by permanent magnets.
    The heating and cooling described above is proportional to the change of mag-
netization and the applied magnetic field. This is the reason that, until recently,
research in magnetic refrigeration was almost exclusively conducted on super-
paramagnetic materials and on rare-earth compounds, see the earlier review in
this Handbook (Tishin, 1999). For room-temperature applications like refrigera-
tors and air-conditioners, compounds containing manganese or iron should be a
good alternative. Manganese and iron are transition metals with high abundance.
Also, there exist in contrast to rare-earth compounds, an almost unlimited number
of manganese and iron compounds with critical temperatures near room tempera-
ture. However, the magnetic moment of manganese generally is only about half the
size of heavy rare-earth elements and the magnetic moment of iron is even less. En-
hancement of the caloric effects associated with magnetic moment alignment may
be achieved through the induction of a first order phase-transition or better a very
rapid change of magnetisation at the critical temperature, which will bring along
a much higher efficiency of the magnetic refrigerator. In combination with cur-
rently available permanent magnets (Dai et al., 2000a) based on modern rare-earth
transition-metal compounds (Kirchmayr, 1996), this opens the path to the devel-
opment of small-scale magnetic refrigerators, which no more rely on rather costly
and service-intensive superconducting magnets. Another prominent advantage of
magnetocaloric refrigerators is that the cooling power can be varied by scaling from
Magnetocaloric Refrigeration at Ambient Temperature                                     241


milliwatt to a few hundred watts or even kilowatts. To increases the temperature
span of the refrigerator, in comparison with the temperature change in a single
cycle, all demonstrators or prototypes nowadays are based on the active magnetic
regenerator design (Barclay and Steyert, 1981).
    In the first section we will discuss the thermodynamics of magnetic refrigera-
tion and some numerical models that are developed to simulate the performance
characteristics of magnetocaloric devices. Then we discuss recent developments in
magnetocaloric materials and finally we shall discuss several recent demonstrators
and prototypes.



      3. Thermodynamics
       Recently a description of the thermodynamics of a refrigeration cycle was
given starting with the first law of thermodynamics (Kitanovski and Egolf, 2006).
Alternatively one may start from a microscopic view on the atomic magnetic mo-
ments in a solid in combination with textbook statistical mechanics e.g. (Pathria,
1972). With J the quantum number of angular momentum, a magnetic field will
lift the (2J + 1) degeneracy of the eigenstates. At finite temperatures, thermal
agitation prevents that only the eigenstates with lowest energy are occupied. For
noninteracting magnetic moments like in a simple paramagnetic salt it has been
shown that the partition function of the system is given by

                                           2J + 1
                                                                    N
                                                            1
                       Z = sinh                   x / sinh    x         ,               (1)
                                             2J            2J
where x = gμB J H /(kT ), g is known as Lande’s-factor, μB is the Bohr magneton,
k is the Boltzmann constant, H is the magnetic intensity, T is the absolute tempera-
ture, and N is the numbers of the magnetic moments. Using Eq. (1) and Helmholtz
free energy A = –kT ln Z, we can calculate the entropy S, internal energy U ,
magnetization M and heat capacity CH at iso-magnetic intensity as,
              ∂A                   2J + 1              x
         S=–         = Nk ln sinh         x – ln sinh                       – xBJ (x)   (2)
              ∂T H                   2J               2J
        U = A + T S = –NkT xBJ (x)                                                      (3)
                  ∂A
       M=–                    = NgμB J BJ (x)                                           (4)
                  ∂H      T

and
                 ∂U
       CH =
                 ∂T       H

                              2J + 1              2J + 1
                                       2                            2
                                                                1                1
            = –Nkx    2                       2
                                           csch          x –            csch2      x    (5)
                                2J                  2J         2J               2J
242                                                                        E. Brück



where
                         2J + 1      2J + 1      1       1
              BJ (x) =          coth        x –    coth    x
                           2J          2J       2J      2J
is the Brillouin function that varies between 0 and 1 for x = 0 and x = ∞, respec-
tively. When x      1, Eq. (4) becomes the Curie law (Zemansky, 1968; Vonsovskii,
1974; Buschow and de Boer, 2003),
                                        CH
                                       M=    ,                                 (6)
                                         T
where C = Ng 2 μ2 J (J + 1)/3k is the Curie constant. This equation gives the well
                 B
known inverse proportionality of the magnetic susceptibility χ = M /H .
   When the argument of the Brillouin function is very large, thus either at low
temperatures and or high magnetic field, the magnetization will saturate to the
maximal value MS
                                 MS = NgμB J .                                 (7)
As initially stated the above is valid for noninteracting magnetic moments. When
the distance between magnetic moments is small, the Pauli exclusion principle,
which states that two identical fermions may not have the same quantum states, re-
sults in interaction between magnetic moments. Heisenberg introduced a model to
describe this exchange interaction on microscopic scale. The Heisenberg exchange
Hamiltonian may be written in the form
                             Hexch = –         2Jij Si · Sj                    (8)
                                         i<j

the summation extends over all magnetic moment pairs in the crystal lattice. For
positive values of the exchange constant Jij one finds parallel alignment else an-
tiparallel. Ferromagnetism is observed for positive exchange interactions below a
critical temperature.
    The exchange interaction can be regarded as effective field acting on the mo-
ments. This field is produced by the surrounding magnetic moments and called
molecular field. As the size of the surrounding moments is proportional to the
magnetization, the molecular field Hm is written as
                                   Hm = NW M                                   (9)
with NW the Weiss-field constant, that was already introduced in the early 20th
century long before the development of quantum physics. The total magnetic field
experienced by a magnetic material is thus the sum of the externally applied field
H0 and the internal field
                                  H = H0 + Hm                                 (10)
and Eq. (6) needs to be rewritten as
                                       C
                              M=         (H0 + NW M)                          (11)
                                       T
Magnetocaloric Refrigeration at Ambient Temperature                            243


thus in the presence of ferromagnetic interaction much lower fields are sufficient to
saturate the magnetization. The magnetic susceptibility is given by
                                            C         C
                                  χ=              =                            (12)
                                         T – NW C   T – TC
this is the Curie–Weiss law, where TC is the Curie temperature. Below the Curie
temperature spontaneous magnetization is observed. For most materials, the phase
transition from the paramagnetic state to the ferromagnetic state is found to be of
second order. This means that the temperature dependence of the first derivative
of the free energy (S, M, V ) is continuous and the second derivative of the free
energy (CH , χ , thermal expansion α) is discontinuous. Some materials like MnAs
however, show a magnetic phase transition of first order, thus the volume changes
abruptly at the critical temperature. To account for this, Bean and Rodbell (1962)
assumed a strong distance dependence of the exchange energy. Within a molecular
field model they introduced an additional parameter the volume dependent TC .
                                 TC = T0 1 + β(V – V0 )/V0                     (13)
here TC is the Curie temperature, T0 would be the Curie temperature if the lattice
was not compressible, ß is the slope of the dependence of TC , V is the volume and
V0 would be the volume in absence of exchange interaction. As pointed out by
de Blois and Rodbell (1963), the Curie temperature will depend on the thermal
expansion and the Curie–Weiss law will be modified
                               C              C             C*
                       χ=           =                   =                      (14)
                             T – TC   T – T0 (1 + αβT )   T – T0*
where the effective Curie constant C * and the effective Curie temperatures T0* are
given by
                                     C                                T
                         C* =                   and      T0* =             .   (15)
                                 1 – αβT0                         1 – αβT0
Starting from experimentally determined parameters, the temperature and field de-
pendence of the magnetization of MnAs were calculated using this model (Bean
and Rodbell, 1962; de Blois and Rodbell, 1963).
    Another ansatz that accounts for magnetic ordering with a first order phase
transition is the model of itinerant metamagnetism (Moriya and Usami, 1977). In
this model spin fluctuations are treated in a Landau–Ginzburg type of approach,
including higher order terms. For certain ranges of parameters a first order phase
transition is observed. This situation may be considered as competing ferro- and
antiferro-magnetic interactions. This model also produces a strong volume depen-
dence of TC (Yamada et al., 2002a, 2002b).
    From Eq. (4) it is obvious, that the work necessary to magnetize a unit volume
of a material is
                                                    M
                                     W =–               μ0 H dM                (16)
                                                0
244                                                                                     E. Brück




Figure 4.2 Schematic representation of the effect of a magnetic field on the entropy of a
material. The response in a cooling cycle will lie in between the isothermal (A–C) and adiabatic
(D–C) process.


in the same way one finds within the molecular field model for a ferromagnetic
material a contribution to the internal energy per unit volume.
                                                 M
                                  Um = –             μ0 Hm dM                              (17)
                                             0
with Hm = NW M the molecular field generated by the exchange interaction. The
specific heat due to the spontaneous magnetization of a ferromagnet is thus
                                 dUm                  dM 2
                             Cm =       = –1/2μ0 NW        .                      (18)
                                  dT                   dT
At low temperatures the magnetization M is usually almost temperature indepen-
dent, therefore the magnetic contribution to the specific heat is very small. The
largest magnetic contribution to the specific heat occurs close to TC where M
strongly varies. Near TC also an applied magnetic field has an intense effect on M
and on the temperature dependence of the magnetization in constant field. This
then will change the magnetic contribution to the internal energy U . Under
adiabatic conditions this will result in a change of temperature T (line D–C in
Fig. 4.2). These changes in internal energy or temperature due to a magnetic field
change are called magnetocaloric effect.
    The magnetocaloric effects are not restricted to the magnetically ordered state
but also in the paramagnetic state they are observed. Equations (4) and (16) are valid
in any magnetic state and thus magnetic work will change the total energy of the
system. The temperature change due to the application of a magnetic field may be
recorded directly.
    There exist nowadays a few experimental setups to do this direct measurement.
Either an adiabatically mounted sample is exposed to a varying magnetic field or the
Magnetocaloric Refrigeration at Ambient Temperature                               245


sample is quickly inserted into a high field region (Dankov et al., 1997; Giguere et
al., 1999b; Tishin, 1999; Hu et al., 2003). Alternatively one may perform specific
heat and magnetization measurements in different applied magnetic fields.
     For magnetization measurements made at discrete temperature intervals, Sm
can be calculated by means of
                                           Mi+1 (Ti+1 , B) – Mi (Ti , B)
                      Sm (T , B) =                                       B,       (19)
                                       i
                                                    Ti+1 – Ti
where Mi+1 (Ti+1 , B) and Mi (Ti , B) represent the values of the magnetization in
a magnetic field B at the temperatures Ti+1 and Ti , respectively. On the other
hand, the field-induced magnetic entropy change can be obtained more directly
from a calorimetric measurement of the field dependence of the heat capacity and
subsequent integration:
                                            T
                                         C(T , B) – C(T , 0)
                          Sm (T , B) =                       dT ,                (20)
                                      0          T
where C(T , B) and C(T , 0) are the values of the heat capacity measured in a field B
and in zero field, respectively. It has been confirmed that the values of Sm (T , B)
derived from the magnetization measurement coincide with the values from calori-
metric measurement (Gschneidner et al., 1999; Tegus et al., 2002d).
   The adiabatic temperature change can be integrated numerically using the ex-
perimentally measured or theoretically predicted magnetization and heat capacity
                                                    B
                                                           T      ∂M
                          Tad (T , B) = –                                  dB .   (21)
                                                0       C(T , B ) ∂T   B
In practice the adiabatic temperature change derived from direct measurements is
generally somewhat smaller than the one derived from Eq. (21). The reason for this
is probably because the specific heat near the phase transition is a function which
is heavily dependent on temperature and field and some interpolation is inevitable
but difficult.
     Recently, magnetic entropy changes are also calculated theoretically either based
on local moment descriptions in the Bean Rodbell model or on itinerant electron
descriptions (von Ranke et al., 2000, 2001, 2004; Yamada and Goto, 2003, 2004;
de Oliveira and von Ranke, 2005). For the degree of agreement between exper-
iment and calculations, the starting point does not matter too much. The mean
field approaches all give reasonable agreement though the shape of the tempera-
ture dependence of the magnetic entropy change is generally not well reproduced.
However, this shortcoming of the mean field approach is well known, as it does not
account for so-called short-range ordering phenomena and local fluctuations.
     To account for the colossal magnetocaloric effect, von Ranke et al. added a
lattice contribution to the magnetic entropy change (von Ranke et al., 2005, 2006).
This contribution is found to be due to a change in Debye temperature when
the crystal structure changes. The model put forward parameterizes this via the
Grüneisen parameter that connects the thermal expansion and the lattice specific
heat. Also for this model description, the magnitude of the effect is well reproduced
246                                                                                  E. Brück




      Figure 4.3 T –S diagram representing the Brayton and Ericsson cycles (see text).



but the experimentally determined shape of the temperature dependence of the
magnetic entropy change is quite different.
    Magnetic refrigeration near room temperature will always involve a cycle process
in which the material will be magnetized and demagnetized. The cycles which
are most discussed in literature either consist of two isomagnetic field and two
isothermal processes (Ericsson) or two isomagnetic field and two adiabatic processes
(Brayton) or two isothermal processes and two iso-magnetization processes (Stir-
ling). The former two cycles are depicted in the T –S diagram see Fig. 4.3 where
H1 < H2 . In the Brayton cycle the processes 5–1 and 3–4 are the isofield processes
and 1–3 and 4–5 are the adiabatic processes. For the Ericsson cycle 6–2 and 3–4
are the isofield processes and 2–3 and 4–6 the isothermal processes. Regeneration
takes place during the isofield processes. Today the literature theoretically discussing
various aspects of these cycles is quite numerous see (Steyert, 1978; Huang and
Teng, 2004; Lin et al., 2004; Allab et al., 2005; Kitanovski and Egolf, 2006; Xia
et al., 2006; Yu et al., 2006; Zhang et al., 2006c). The performance of Gd in a
Stirling cycle was discussed already 30 years ago by Steyert (Steyert, 1978). Huang
et al. compare the Brayton and Ericson cycle in a numerical model based on the
Brillouin function description of a magnetic material (Huang and Teng, 2004). Lin
et al. consider the effect of irreversibility in the Stirling cycle with regeneration
(Lin et al., 2004). Allab et al. implement the effect of finite heat transfer in their
numerical model (Allab et al., 2005). Kitanovski et al. discuss various aspects of the
thermodynamics involved in magnetic cooling (Kitanovski and Egolf, 2006). Xia et
al. discuss irreversibility in the Ericsson cycle with regeneration (Xia et al., 2006).
Yu et al. consider a paramagnetic material in an Ericsson cycle and study the effect
of increasing the applied field on the temperature span and the performance of the
refrigerator (Yu et al., 2006). Zhang et al. discuss the effect of irreversibility on a
two stage Brayton cycle (Zhang et al., 2006c).
    Next to specific cycles also more general effects like demagnetizing effects
(Peksoy and Rowe, 2005) and hysteresis (Basso et al., 2005, 2006) are considered
theoretically.
Magnetocaloric Refrigeration at Ambient Temperature                                  247




      4. Materials

      In the following section we shall discuss various materials that have been put
forward to be used in magnetic refrigerators. From Eq. (21) it is obvious that a
strong temperature dependence of the magnetization will result in a large mag-
netocaloric effect. This is true for a paramagnet near zero temperature and for a
ferromagnet near the Curie temperature. However, we will see that there are still
some other possibilities like a transition from antiferromagnetic to ferromagnetic.
What was realized only quite recently, is that a material with a combined mag-
netic and structural transition, often exhibits a very large magnetocaloric effect. We
will start with Gd as this is nowadays considered as a reference material, though it
is rather unlikely that this will be employed in future refrigerators operating near
room temperature.

4.1 Gd metal and alloys
The only elemental ferromagnet with a Curie temperature near room tempera-
ture is Gd. The ordered magnetic moment of Gd at low temperatures is quite
high 7.63 μB and magnetic ordering occurs below 293 K with a second order
phase-transition. This metal is currently used as magneto refrigerant in various mag-
netocaloric refrigeration demonstrator prototypes (see section 6). As pointed out by
Dan’kov et al. (Dan’kov et al., 1998), the Curie temperature and the magnetocaloric
properties of Gd strongly depend on the impurity content. As the impurity content
is generally quoted as weight percent, different batches with nominal the same im-
purity content may yet have quite different properties. It appears, that especially the
interstitial impurities H, C, N, O and F, which are rather light and therefore hardly
count in the weight percentage, weaken the exchange interaction, which leads to
an reduction in Curie temperature and enhanced spin fluctuations below and above
the Curie temperature. The importance of knowing the type of impurities is further
demonstrated by the same authors when studying the time dependence of the mag-
netocaloric effect. For high purity samples they show that the adiabatic temperature
change is the same when measured in quasi-static fields or in pulsed fields with full
loop duration of 0.2 s, the latter simulates a refrigerator operating at 5 Hz. However,
commercial Gd with a high C content exhibits in pulsed field measurement only
about 60% of the maximal adiabatic temperature change compared to high purity
materials. Additionally to the reduction in adiabatic temperature change the whole
effect is shifted a few K to lower temperatures. On the other hand, pulsed field
measurements, on a sample with a higher overall impurity content that contains
mainly oxides, almost reproduce the results found for high purity materials.
    The thermal conductivity of Gd also depends on the impurity content (Glorieux
et al., 1995). This should be especially taken into account when the performance
of a refrigerator is simulated. It is not always possible to just plug in some literature
data and start simulating. The high reactivity of Gd can be accounted for by adding
some NaOH into the water in the heat exchanger (Zhang et al., 2005b).
248                                                                              E. Brück



    As the magnetocaloric effect of a material is large only in the vicinity of the
Curie temperature, one needs more than one material if one wants to increase
the temperature span of operation. Other rare-earth metals have lower ordering
temperatures, however the magnetic ordering is often rather complicated and the
associated magnetocaloric effects are often not very large. A way out from this is
Gd that is doped with heavy rare-earth, these alloys show a lower ferromagnetic
ordering-temperature with yet large ordered magnetic moments. A Gd alloy with
26% Tb orders at 280 K and shows a somewhat larger MCE compared to pure Gd
(Gschneidner et al., 2000). A Gd alloy with 50% Dy orders at 230 K and the MCE
reported for this alloy is about the same as for pure Gd (Dai et al., 2000b). Dai et al.
also studied further addition of Nd to reach even lower temperatures, however the
Nd doped alloys show strong time dependence of the magnetic response which may
obstruct application. There also exist several compounds of Gd with nonmagnetic
elements that have magnetic ordering temperatures above room temperature. The
structural and magnetocaloric data of these are included in Table 4.1. In the case
of Gd7 Pd3 also the eutectic composition Gd76 Pd24 has been proposed as composite
magnetic refrigerant (Canepa et al., 2002, 2005).

4.2 Gd5 (Ge,Si)4 and related compounds
The search for alternatives to rare-earth elements and their alloys lead to the
discovery of a sub-room temperature giant-MCE in the ternary compound
Gd5 (Ge1–x Six )4 (0.3 ≤ x ≤ 0.5) (Pecharsky and Gschneidner, 1997b). Since
then there is a strongly increased interest from both fundamental and practi-
cal points of view to study the MCE in these materials (Choe et al., 2000;
Morellon et al., 2000). The most prominent feature of these compounds is that
they undergo a first-order structural and magnetic phase transition, which leads to
a giant magnetic field-induced entropy change, across their ordering temperature.
We here therefore will discuss to some extend the structural properties of these
compounds.
     At low temperatures for all x Gd5 (Ge1–x Six )4 adopts an orthorhombic Gd5 Si4 -
type structure (Pnma) and the ground state is ferromagnetic (Pecharsky et al., 2002).
However, at room temperature depending on x three different crystallographic
phases are observed. For x > 0.55 the aforementioned Gd5 Si4 structure is stable,
for x < 0.3 the materials adopt the Sm5 Ge4 -type structure with the same space
group (Pnma) but a different atomic arrangement and a somewhat larger volume,
finally in between these two structure types the monoclinic Gd5 Si2 Ge2 type with
space group (P1121 /a) is formed, which has an intermediate volume. The latter
structure type is stable only below about 570 K where again the orthorhombic
Gd5 Si4 -type structure is formed in a first-order phase transition (Mozharivskyj et
al., 2005). The thermal evolution of the structural change is depicted in Fig. 4.5. As
one may guess, the three structure types are closely related (see Fig. 4.4) (Pecharsky
and Gschneidner, 1997c); the unit cells contain four formula units and essentially
only differ in the mutual arrangement of identical building blocks which are either
connected by two, one or no covalent-like Si–Ge bonds, resulting in successively
increasing unit-cell volumes. The giant magnetocaloric effect is observed for the
                                                                                                                                               Magnetocaloric Refrigeration at Ambient Temperature
Table 4.1   Structural, magnetic and magnetocaloric data of Gd based intermetallic compounds. | S| at   B = 2 T if not indicated differently

Compound                   Structure         Space           Tc           | S|          Comment           References
                           type              group           (K)          (J/kg·K)

Gd                         HCP               P63/mmc         294           5                              Dan’kov et al. (1998)
Gd0.74 Tb0.26              HCP               P63/mmc         280           6                              Gschneidner and Pecharsky (2000)
Gd0.5 Dy0.5                HCP               P63/mmc         230           5                              Dai et al. (2000b)
Gd7 Pd3                    Th7 Fe3           P63mc           334           2.5                            Canepa et al. (2002)
Gd4 Bi3                    Th3 P4            I43d            332           1.5                            Niu et al. (2001)
Gd5 Si4                    Gd5 Si4           Pnma            346           4.2                            Spichkin et al. (2001)
Gd5 (Si0.5 Ge0.5 )4        Gd5 (Si2 Ge2 )    P1121 /a        272          27            Optimal           Pecharsky et al. (2003a)
Gd5 (Si0.5 Ge0.5 )4        Gd5 (Si2 Ge2 )    P1121 /a        275          14            First             Pecharsky et al. (2003b)
Gd5 (Si0.5 Ge0.5 )4        Gd5 Si4           Pnma            305          20(5T)        Fe doped          Provenzano et al. (2004)
Gd5 (Si0.425 Ge0.575 )4    Gd5 (Si2 Ge2 )    P1121 /a        246          43            2 T//a-axis       Tegus et al. (2002b)
Gd5 (Si0.425 Ge0.575 )4    Gd5 (Si2 Ge2 )    P1121 /a        246          47            2 T//b-axis       Tegus et al. (2002b)
Gd5 (Si0.425 Ge0.575 )4    Gd5 (Si2 Ge2 )    P1121 /a        246          38            2 T//c-axis       Tegus et al. (2002b)
Gd5 (Si0.5 Ge0.5 )4        Gd5 (Si2 Ge2 )    P1121 /a        278          14                              Casanova (2004)
Gd5 (Si0.45 Ge0.55 )4      Gd5 (Si2 Ge2 )    P1121 /a        257          17                              Casanova (2004)
Gd5 (Si0.365 Ge0.635 )4    Gd5 (Si2 Ge2 )    P1121 /a        215          28                              Casanova (2004)
Gd5 (Si0.3 Ge0.7 )4        Gd5 (Si2 Ge2 )    P1121 /a        182          31                              Casanova (2004)
Gd5 (Si0.25 Ge0.75 )4      Gd5 (Si2 Ge2 )    P1121 /a        155          38                              Casanova (2004)




                                                                                                                                               249
250                                                                                    E. Brück




Figure 4.4 Crystal structure adopted at room temperature in the pseudo-binary system
Gd5 Si4 –Gd5 Ge4 (Pecharsky and Gschneidner, 1997c).




Figure 4.5 X-ray diffraction spectra of Gd5 Ge2.4 Si1.6 taken at different temperatures (Duijn,
2000).
Magnetocaloric Refrigeration at Ambient Temperature                                  251




Figure 4.6 Temperature dependence of the electrical resistivity of an annealed crystal of
Gd5 Ge2.4 Si1.6 (Duijn, 2000).


compounds that exhibit a simultaneous paramagnetic to ferromagnetic and struc-
tural phase-transition that can be either induced by a change in temperature, applied
magnetic field or applied pressure (Morellon et al., 1998a, 2004a). In contrast to
most magnetic systems the ferromagnetic phase has a 0.4% smaller volume than the
paramagnetic phase which results in an increase of TC on application of pressure
with about 3 K/kbar.
     The structural change at the phase transition brings along also a very large
magneto-elastic effect and the electrical resistivity behaves anomalous see Fig. 4.6.
The strong coupling between lattice degrees of freedom and magnetic and elec-
tronic properties is rather unexpected, because the magnetic moment in Gd origi-
nates from spherical symmetric S-states that in contrast to other rare-earth elements
hardly couple with the lattice. First principle electronic structure calculations in
atomic sphere and local-density approximation with spin-orbit coupling added vari-
ationally, could reproduce some distinct features of the phase transition (Pecharsky et
al., 2003b). Total energy calculations for the two phases show different temperature
dependences and the structural change occurs at the temperature where the ener-
gies are equal. There appears a distinct difference in effective exchange-coupling
parameter for the monoclinic and orthorhombic phase, respectively. This difference
could directly be related to the change of the Fermi-level in the structural transi-
tion. Thus the fact that the structural and magnetic transitions are simultaneous is
somewhat accidental as the exchange energy is of the same order of magnitude as
252                                                                                E. Brück



the thermal energy at the structural phase-transition. The electrical resistivity and
magneto resistance of Gd5 Ge2 Si2 also shows unusual behaviour, indicating a strong
coupling between electronic structure and lattice. For several compounds of the se-
ries, next to a cusp like anomaly in the temperature dependence of the resistivity,
a very large magnetoresistance effect is reported (Morellon et al., 1998b, 2001a;
Levin et al., 2001; Tang et al., 2004).
    In view of building a refrigerator based on Gd5 (Ge1–x Six )4 , there are a few points
to consider. The largest magnetocaloric effect is observed considerably below room
temperature (see Table 4.1), while a real refrigerator should expel heat at least at
about 320 K. Because the structural transition is connected with sliding of building
blocks, impurities especially at the sliding interface can play an important role. The
thermal hysteresis and the size of the magnetocaloric effect connected with the first-
order phase transition strongly depend on the quality of the starting materials and
the sample preparation (Pecharsky et al., 2003a). For the compounds Gd5 (Ge1–x Six )4
with x around 0.5 small amounts of impurities like Al, Bi, C, Co, Cu, Ga, Mn or
O may suppress the formation of the monoclinic structure near room tempera-
ture. These alloys then show only a phase transition of second order at somewhat
higher temperature but with a lower magnetocaloric effect (Pecharsky and Gschnei-
dner, 1997a; Provenzano et al., 2004; Mozharivskyj et al., 2005; Shull et al., 2006).
The only impurity that appears to enhance the magnetocaloric effect and in-
creases the magnetic ordering temperature are so far Pb and Sn (Li et al., 2006b;
Zhuang et al., 2006). This sensitivity to impurities like carbon, oxygen and iron
strongly influences the production costs of the materials which may hamper broad-
scale application. Next to the thermal and field hysteresis the magneto-structural
transition in Gd5 (Ge1–x Six )4 appears to be rather sluggish (Giguere et al., 1999b;
Gschneidner et al., 2000). This will also influence the optimal operation-frequency
of a magnetic refrigerator and the efficiency. An aspect which is hardly ever taken
into consideration is the availability of the components. For this compound Ge will
be the limiting ingredient as the worldwide yearly production of Ge amounts to
only about 90 metric tons. This Ge is mainly consumed for electronic and optical
devices so about 10 t may be available for the production of magnetic refriger-
ants. Which, limits the yearly production of Gd5 (Ge1–x Six )4 to about 160 t (see
Table 4.9).
    Other R5 (Ge,Si)4 compounds are also found to form in the monoclinic
Gd5 Ge2 Si2 type structure and when the structural transformation coincides with the
magnetic ordering transition a large magnetocaloric effect is observed (Table 4.2).
This is most strikingly evidenced in the experiments of Morellon et al. (2004b)
on Tb5 Ge2 Si2 where the two transitions were forced to coincide by application of
hydrostatic pressure, which results in a strong enhancement of the magnetic entropy
change at the ordering temperature. The magnetic ordering temperatures of other
R5 (Ge,Si)4 compounds are all lower than for the Gd compound as expected. For
cooling applications below liquid nitrogen temperatures some of these compounds
may be interesting.
Magnetocaloric Refrigeration at Ambient Temperature                                     253


Table 4.2 Structural, magnetic and magnetocaloric data of several heavy rare-earth based in-
termetallic compounds. | S| at B = 2 T if not indicated differently

Compound           Structure           Space          TC       | S|          References
                   type                group          (K)      (J/kg·K)

Tb5 Si4            Gd5 Si4             Pnma           225       5.2          Morellon et
                                                                             al. (2001b)
Tb5 Si4            Gd5 Si4             Pnma           223       4.5          Spichkin et al.
                                                                             (2001)
Tb5 (Si3 Ge)       Gd5 (Si2 Ge2 )      P1121 /a       215       4.5(3T)      Thuy et al.
                                                                             (2002)
Tb5 (Si2 Ge2 )     Gd5 (Si2 Ge2 )      P1121 /a       110      10.4          Morellon et
                                                                             al. (2001b)
Tb5 (Si2 Ge2 )     Gd5 (Si2 Ge2 )      P1121 /a       116       7.8(3T)      Thuy et al.
                                                                             (2002)
Tb5 (Si2 Ge2 )     Gd5 (Si2 Ge2 )      P1121 /a        76      12            Tegus et al.
                                                                             (2002c)
Tb5 Ge4            Sm5 Ge4             Pnma            91a      1.0          Morellon et
                                                                             al. (2001b)
DyTiGe             CeFeSi              P4/nmm         165       0.5          Tegus et al.
                                                                             (2002b)
HoTiGe             CeFeSi              P4/nmm          90       1.0          Tegus et al.
                                                                             (2001)
HoTiGe             CeFeSi              P4/nmm          90       2.5//c       Tegus et al.
                                                                             (2002b)
TmTiGe             CeFeSi              P4/nmm          15       4.2          Tegus et al.
                                                                             (2002b)



4.3 La(Fe,Si)13 and related compounds
Another interesting type of materials are rare-earth–transition-metal compounds
crystallizing in the cubic NaZn13 type of structure. LaCo13 is the only binary com-
pound, from the 45 possible combinations of a rare-earth and iron, cobalt or nickel,
that exists in this structure. It has been shown that with an addition of at least 10%
Si or Al this structure can also be stabilized with iron and nickel (Kripyakevich et
al., 1968). The NaZn13 structure contains two different Zn sites. The Na atoms
at 8a and ZnI atoms at 8b form a simple CsCl type of structure. Each ZnI atom is
surrounded by an icosahedron of 12 ZnII atoms at the 96i site. In La(Fe,Si)13 La goes
on the 8a site, the 8b site is fully occupied by Fe and the 96i site is shared by Fe and
Si. The iron rich compounds La(Fe,Si)13 show typical invar behavior, with mag-
netic ordering temperatures around 200 K that increase to 262 K with lower iron
content (Palstra et al., 1983). Thus, though the magnetic moment is diluted and
also decreases per Fe atom, the magnetic ordering temperature increases. Around
200 K the magnetic-ordering transition is found to be distinctly visible also in the
electrical resistivity, where a chromium-like cusp in the temperature dependence is
254                                                                              E. Brück



observed. In contrast to Gd5 Ge2 Si2 this phase transition is not accompanied by a
structural change, thus above and below TC the material is cubic. Recently, because
of the extremely sharp magnetic ordering transition, the (La,Fe,Si,Al) system was
reinvestigated by several research groups and a large magnetocaloric effect was re-
ported (Hu et al., 2000, 2001b; Fujieda et al., 2002). The largest effects are observed
for the compounds that show a field- or temperature-induced phase-transition of
first order. Unfortunately, these large effects only occur up to about 210 K as the
magnetic sublattice becomes more and more diluted. When using standard melting
techniques, preparation of homogeneous single-phase samples appears to be rather
difficult especially for alloys with high transition metal content. Almost single phase
samples are reported when, instead of normal arc melting, rapid quenching by melt
spinning and subsequent annealing is employed (Liu et al. 2004, 2005; Gutfleisch
et al., 2005). Samples prepared in this way also show a very large magnetocaloric
effect. To increase the total magnetic moment partial substitution of Ce for La has
been successful (Fujieda et al., 2006a, 2006b). This substitution however, leads next
to an enhanced magnetocaloric effect, to a lower magnetic ordering temperature
and a broader thermal hysteresis. Small additions of Nd were also studied and were
found to increase the ordering temperature, however at 50% Nd the phase tran-
sition becomes of second order and the entropy change steeply drops (Zhu et al.,
2005). To increase the magnetic ordering temperature without loosing too much
magnetic moment, one may replace some Fe by other magnetic transition-metals.
Because the isostructural compound LaCo13 has a very high critical temperature
substitution of Co for Fe is widely studied. The compounds La(Fe,Co)13–x Alx and
La(Fe,Co)13–x Six with x ≈ 1.1 and thus a very high transition-metal content, show a
considerable magnetocaloric effect near room temperature (Hu et al., 2001a, 2005;
Shen et al., 2004; Proveti et al., 2005). This is achieved with only a few percent
of Co and the Co content can easily be varied to tune the critical temperature to
the desired value. It should be mentioned however that near room temperature the
values for the entropy change steeply drop.
    The fact that the alloys with the highest Fe content have an antiferromagnetic
ground-state indicates that antiferromagnetic direct exchange-interaction plays an
important role in these compounds. Taking into account that this occurs at a very
high Fe density, one may expect that expansion of the lattice will lead to an increase
in ferromagnetic exchange. Hydrogen is the most promising interstitial element. In
contrast to other interstitial atoms, interstitial hydrogen not only increases the crit-
ical temperature but also leads to an increase in magnetic moment (Irisawa et al.,
2001; Fujieda et al., 2002, 2004a; Fujita et al., 2003; Nikitin et al., 2004; Mandal
et al., 2005). The lattice expansion due to the addition of three hydrogen atoms
per formula unit is about 4.5%. The critical temperature can be increased to up
to 450 K, the average magnetic moment per Fe increases from 2.0 μB to up to
2.2 μB and the field- or temperature-induced phase-transition is found to be of
first-order for all hydrogen concentrations. This all results for a certain Si percent-
age in an almost constant value of the magnetic entropy change per mass unit over
a broad temperature span see Table 4.3 and Fig. 4.7. Heat capacity measurements
on La(Fe,Si) and La(Fe,Si)H in applied magnetic field, confirm a large adiabatic
temperature change (Podgornykh and Shcherbakova, 2006). The reduction of the
Magnetocaloric Refrigeration at Ambient Temperature                                   255


Table 4.3 Magnetic and magnetocaloric data of several NaZn13 type intermetallic compounds.
| S| at B = 2 T if not indicated differently

Material                         Remark          Tmax     Smax     Ref.
                                                 (K)    (J/kg·K)

La(Fe0.90 Si0.10 )13                             184    28         Fujita et al. (2003)
La(Fe0.89 Si0.11 )13                             188    24         Fujita et al. (2003)
La(Fe0.880 Si0.120 )13                           195    20         Fujita et al. (2003)
La(Fe0.877 Si0.123 )13                           208    14         Fujita et al. (2003)
LaFe11.8 Si1.2                   Melt spun       195    25         Gutfleisch et al. (2005)
La(Fe0.88 Si0.12 )13 H0.5                        233    20         Fujita et al. (2003)
La(Fe0.88 Si0.12 )13 H1.0                        274    19         Fujita et al. (2003)
LaFe11.7 Si1.3 H1.1                              287    28         Fujita et al. (2003)
LaFe11.57 Si1.43 H1.3                            291    24         Fujita et al. (2003)
La(Fe0.88 Si0.12 )H1.5                           323    19         Fujita et al. (2003)
LaFe11.2 Co0.7 Si1.1                             274    12         Hu et al. (2002)
LaFe11.5 Al1.5 C0.1                              185     5.5       Wang et al. (2004)
LaFe11.5 Al1.5 C0.2                              210     5.7       Wang et al. (2004)
LaFe11.5 Al1.5 C0.4                              238     5.5       Wang et al. (2004)
LaFe11.5 Al1.5 C0.5                              255     5         Wang et al. (2004)
La(Fe0.94 Co0.06 )11.83 Al1.17                   273     4.8       Hu et al. (2001a)
La(Fe0.92 Co0.08 )11.83 Al1.17                   303     4.5       Hu et al. (2001a)




Figure 4.7 Magnetic entropy change for different LaFe13 based alloys at a field change from
0 to 2 T (Fujita et al., 2003; Wang et al., 2004; Hu et al., 2005).

electronic contribution to the heat capacity observed for the hydrogenated sam-
ple however, is in conflict with the model of itinerant metamagnetism for these
materials.
256                                                                            E. Brück



    From the materials cost point of view the La(Fe,Si)13 type of alloys appear to
be very attractive. La is the cheapest from the rare-earth series and both Fe and
Si are available in large amounts (see Table 4.9). The processing will be a little
more elaborate than for a simple metal alloy but this can be optimized. For the use
in a magnetic refrigerator next to the magnetocaloric properties also mechanical
properties and chemical stability may be of importance. The hydrogenation process
of rare-earth transition-metal compounds produces always granular material due
to the strong lattice expansion. In the case of the cubic NaZn13 type of structure
this does not seem to be the case. At the phase transition in La(Fe,Si)13 type of
alloys also a volume change of 1.5% is observed (Wang et al., 2003). If this volume
change is performed very frequently the material will definitely become very brittle
and probably break into even smaller grains. This can have distinct influence on the
corrosion resistance of the material and thus on the lifetime of a refrigerator. The
suitability of this material definitely needs to be tested.

4.4 MnAs based compounds
MnAs exist similar to Gd5 Ge2 Si2 in two distinct crystallographic structures (Pytlik
and Zieba, 1985). At low and high temperature the hexagonal NiAs structure is
found and for a narrow temperature range 307 K to 393 K the orthorhombic MnP
structure exists. The high temperature transition in the paramagnetic region is of
second order. The low temperature transition is a combined structural and ferro-
paramagnetic transition of first order with large thermal hysteresis. The change in
volume at this transition amounts to 2.2% (Fjellvag et al., 1984). The transition from
paramagnetic to ferromagnetic occurs at 307 K, the reverse transition from ferro-
magnetic to paramagnetic occurs at 317 K. Very large magnetic entropy changes are
observed in this transition (Kuhrt et al., 1985; Wada and Tanabe, 2001). Similar to
the application of pressure (Menyuk et al., 1969; Yamada et al., 2002b) substitution
of Sb for As leads to lowering of TC (Wada et al., 2002, 2003), 25% of Sb gives
an transition temperature of 225 K (Table 4.4). However, the thermal hysteresis is
affected quite differently by hydrostatic pressure or Sb substitution. In Mn(As,Sb)
the hysteresis is strongly reduced and at 5% Sb it is reduced to about 1 K. In the
concentration range 5 to 40% of Sb TC can be tuned between 220 and 320 K with-
out loosing much of the magnetic entropy change (see Table 4.4) (Morikawa et
al., 2004; Wada and Asano, 2005). Direct measurements of the temperature change
confirm a T of 2 K/T (Wada et al., 2005b). On the other hand MnAs under
pressure shows an extremely large magnetic entropy change (Gama et al., 2004)
in conjunction with large hysteresis. A similar effect can be produced at ambient
pressure when part of the Mn is substituted by Fe see Fig. 4.8 (De Campos et
al., 2006). This effect is larger than what may be expected from aligning the Mn
magnetic moments, to account for this excess magnetic entropy change von Ranke
et al. introduced a model which involves the difference in phonon spectra for the
different crystal structures (von Ranke et al., 2005, 2006).
     The materials costs of MnAs are quite low, processing of As containing alloys
is however complicated due to the biological activity of As. In the MnAs alloy
Table 4.4 Structural, magnetic and magnetocaloric data of Mn based intermetallic compounds. | S| at    B = 2 T if not indicated differently.




                                                                                                                                               Magnetocaloric Refrigeration at Ambient Temperature
M3d is the magnetic moment at low temperature per 3d atom

Compound                   Structure         Space              Tc              M3d              | S|                References
                           type              group              (K)             (μB /3d)         (J/kg·K)

MnFeGe                     Ni2 In            P63 /mmc           159             0.97                  1.6a           Lin et al. (2006)
MnFe0.9 Co0.1 Ge           Ni2 In            P63 /mmc           173             0.97                  1.8a           Lin et al. (2006)
MnFe0.8 Co0.2 Ge           Ni2 In            P63 /mmc           209             1.13                  2.5a           Lin et al. (2006)
MnFe0.7 Co0.3 Ge           Ni2 In            P63 /mmc           220             1.13                  2.9a           Lin et al. (2006)
MnFe0.6 Co0.4 Ge           Ni2 In            P63 /mmc           223             1.20                  3.2a           Lin et al. (2006)
MnFe0.5 Co0.5 Ge           Ni2 In            P63 /mmc           228             1.30                  3.5a           Lin et al. (2006)
MnFe0.4 Co0.6 Ge           Ni2 In            P63 /mmc           242             1.51                  2.9a           Lin et al. (2006)
MnFe0.3 Co0.7 Ge           Ni2 In            P63 /mmc           249             1.55                  4.0a           Lin et al. (2006)
MnFe0.2 Co0.8 Ge           Ni2 In            P63 /mmc           289             2.34                  9.0a           Lin et al. (2006)
MnFe0.15 Co0.85 Ge         TiNiSi            Pnma               306             1.97                  5.3a           Lin et al. (2006)
MnFe0.1 Co0.9 Ge           TiNiSi            Pnma               340             2.05                  5.7a           Lin et al. (2006)
MnCoGe                     TiNiSi            Pnma               345             2.06                  6.1a           Lin et al. (2006)
Mn5 Ge2.5 Si0.5            Mn5 Si3           P63 /mcm           299             2.56                  7.8            Zhao et al. (2006)
Mn5 Ge2 Si                 Mn5 Si3           P63 /mcm           283             2.52                  7.6            Zhao et al. (2006)
Mn5 Ge1.5 Si1.5            Mn5 Si3           P63 /mcm           258             2.46                  6.9            Zhao et al. (2006)
Mn5 GeSi2                  Mn5 Si3           P63 /mcm           198             2.36                  6.8            Zhao et al. (2006)
Mn5 Ge3                    Mn5 Si3           P63 /mcm           298             2.64                  9.3            Songlin et al. (2002a)
Mn5 Ge2.9 Sb0.1            Mn5 Si3           P63 /mcm           304             2.63                  6.6            Songlin et al. (2002a)
Mn5 Ge2.8 Sb0.2            Mn5 Si3           P63 /mcm           307             2.60                  6.2            Songlin et al. (2002a)
Mn5 Ge2.7 Sb0.3            Mn5 Si3           P63 /mcm           312             2.40                  5.6            Songlin et al. (2002a)
LaMn1.9 Fe0.1 Ge2          ThCr2 Si2         I4/mmm             310             1.2                   1.02b          Zhang et al. (2006b)
LaMn1.85 Fe0.15 Ge2        ThCr2 Si2         I4/mmm             295             1.2                   0.93b          Zhang et al. (2006b)
LaMn1.8 Fe0.2 Ge2          ThCr2 Si2         I4/mmm             275             1.2                   0.88b          Zhang et al. (2006b)




                                                                                                                                               257
                                                                                                                           258
                                Table 4.4     (Continued.)

Compound                           Structure            Space      Tc     M3d        | S|       References
                                   type                 group      (K)    (μB /3d)   (J/kg·K)

(Fe0.9 Mn0.1 )3 C                  Fe3 C                Pnma       305    1.39         3.4      Brück et al. (2007)
(Fe0.8 Mn0.2 )3 C                  Fe3 C                Pnma       109    0.62         1.8      Brück et al. (2007)
(Fe0.7 Mn0.3 )3 C                  Fe3 C                Pnma        31    0.22         1.3      Brück et al. (2007)
Mn3 GaC                            CaTiO3                 ¯
                                                        Pm3m       160*   1.2/1.8     15        Tohei et al. (2003)
MnAs                               NiAs                 P63 /mmc   317    3.4         43        Nascimento et al. (2006)
(Mn,Fe)As                          NiAs                 P63 /mmc   310    –          320        De Campos et al. (2006)
Mn1+δ As0.8 Sb0.2                  NiAs                 P63 /mmc   250    3.7         26        Wada and Asano (2005)
MnAs0.75 Sb0.25                    NiAs                 P63 /mmc   232    3.7         14c       Morikawa et al. (2004)
Mn1.1 As0.75 Sb0.25                NiAs                 P63 /mmc   227    3.3         17c       Morikawa et al. (2004)
Mn1.5 As0.75 Sb0.25                NiAs                 P63 /mmc   204    3.2         14c       Morikawa et al. (2004)
a   B = 5 T; b   B = 1.8 T; c   B = 1.0 T; * Tt .




                                                                                                                           E. Brück
Magnetocaloric Refrigeration at Ambient Temperature                                259




Figure 4.8 Magnetic entropy changes of MnAs alloys at a field change from 0 to 5 T (Wada
and Asano, 2005; De Campos et al., 2006; Nascimento et al., 2006).


the As is covalently bound to the Mn and would not be easily released into the
environment. However, this should be experimentally verified, especially because
in an alloy frequently second phases form that may be less stable. The change in
volume in Mn(As,Sb) is still 0.7% which may result in aging after frequent cycling
of the material.

4.5 Heusler alloys
Heusler alloys frequently undergo a martensitic transition between the martensitic
and the austenitic phase which is generally temperature induced and of first order.
Ni2 MnGa orders ferromagnetic with a Curie temperature of 376 K, and a mag-
netic moment of 4.17 μB , which is largely confined to the Mn atoms and with
a small moment of about 0.3 μB associated with the Ni atoms (Webster et al.,
1984). As may be expected from its cubic structure, the parent phase has a low
magneto-crystalline anisotropy energy (Ha = 0.15 T). However, in its martensitic
phase the compound is exhibiting a much larger anisotropy (Ha = 0.8 T). The
martensitic-transformation temperature is near 220 K. This martensitic transforma-
tion temperature can be easily varied to around room temperature by modifying
the composition of the alloy from the stoichiometric one.
    The low-temperature phase evolves from the parent phase by a diffusionless,
displacive transformation leading to a tetragonal structure, a = b = 5.90 Å,
c = 5.44 Å. A martensitic phase generally accommodates the strain associated with
the transformation (this is 6.56% along c for Ni2 MnGa) by the formation of twin
variants. This means that a cubic crystallite splits up in two tetragonal crystallites
sharing one contact plane. These twins pack together in compatible orientations to
260                                                                           E. Brück



minimize the strain energy (much the same as the magnetization of a ferromagnet
may take on different orientations by breaking up into domains to minimize the
magneto-static energy). Alignment of these twin variants by the motion of twin
boundaries can result in large macroscopic strains. In the tetragonal phase with its
much higher magnetic anisotropy, an applied magnetic field can induce a change in
strain. This is the reason why these materials may be used as actuators. Next to this
ferromagnetic shape memory effect, very close to the martensitic transition temper-
ature, one observes a large change in magnetization for low applied magnetic fields.
This change in magnetization is also related to the magnetocrystalline anisotropy.
This change in magnetization is resulting in a moderate magnetic entropy change
of a few J/mol · K, which is enhanced when measured on a single crystal (Hu et al.,
2001c; Marcos et al., 2002). When the composition in this material is tuned in a way
that the magnetic and structural transformation occurs at the same temperature, the
largest magnetic entropy changes are observed (Kuo et al., 2005; Long et al., 2005;
Zhou et al., 2005b). Recently in the Heusler alloy NiMnSn a large inverse mag-
netocaloric effect was reported (Krenke et al., 2005b), this effect is related to the
increase of magnetization with increasing temperature over the martensitic transi-
tion temperature. Substitution of Co for Ni leads to an increase of the transition
temperature close to room temperature (see Fig. 4.9).
    CoMnSb is a half Heusler alloy with a rather high ordering temperature when
one considers magnetocaloric applications, the effect of Nb addition on the mag-
netic properties and magnetocaloric effect (MCE) of CoNbx Mn1–x Sb alloys was
investigated recently (Li et al. 2006c, 2007). The Curie temperature of these com-
pounds slightly decreases with Nb substitution. As seen in Table 4.5, Nb substitution
strongly lowers the magnetic moment and the MCE of CoMnSb alloy. With in-
creasing Nb content, the magnetic moment decreases linearly and the magnetic
phase transition is smeared out over a larger temperature interval. These facts re-
sult in a reduction of the magnetic entropy change, but lead to a broader working
temperature span.
    The Fe rich Heusler alloys generally show different behavior. A series of
Fe2 MnSi1–x Gex compounds (x = 0–1) was prepared by Zhang et al. (2003) using
a mechanically activated solid-state diffusion method. Both X-ray diffraction and
differential scanning calorimetry evidenced the presence of an amorphous phase
after 10 h of milling. The X-ray data reveal that in the high-temperature annealing
the single D03 -type phase can be retained up to 50% substitution of Ge for Si in
Fe2 MnSi. A metastable D03 phase is obtained after crystallization of the as-milled
amorphous compounds with x > 0.5. High-temperature annealing transforms the
low-temperature D03 phase into a single D019 phases (x = 1) or a mixture of
D03 and D019 phase (x = 0.6 and 0.8). Low-field thermomagnetic measurements
show a moderately sharp ferromagnetic-paramagnetic transition, which becomes
enormously broad in higher magnetic fields. The Curie temperature is significantly
enhanced when going from the D03 phase to the D019 phase. Neither a magnetic-
field-induced transition nor a reversible structural transition is observed throughout
this compound series. The magnetocaloric effect associated with the magnetic tran-
sition is small. This may be illustrated for the compound Fe2 MnSi0.5 Ge0.5 listed in
Table 4.5.
                                                                                                                                                Magnetocaloric Refrigeration at Ambient Temperature
Table 4.5   Structural, magnetic and magnetocaloric properties of Mn based Heusler alloys and intermetallic compounds with Fe2 P structure

Compound                       Structure          Space            Tc             Ms (μB /3d)          | S|             Ref.
                               type               group            (K)            at 5 K               (J/kg·K)

Fe2 MnSi0.5 Ge0.5              BiF3               Fm3m             260             0.93                  0.8            Zhang et al. (2003)
Ni52.9 Mn22.4 Ga24.7           BiF3               Fm3m             305           ∼1.3                    8.65T          Zhou et al. (2005a)
Ni50.9 Mn24.7 Ga24.4           BiF3               Fm3m             272           ∼1.3                   3.55T           Zhou et al. (2005a)
Ni55.2 Mn18.6 Ga26.2           BiF3               Fm3m             315           ∼1.3                  20.45T           Zhou et al. (2005a)
Ni51.6 Mn24.7 Ga23.8           BiF3               Fm3m             296           ∼1.3                   7.05T           Zhou et al. (2005a)
Ni52.7 Mn23.9 Ga23.4           BiF3               Fm3m             338           ∼1.3                  15.65T           Zhou et al. (2005a)
CoMnSb                         MgAgAs             F43m             472             2.00                  2.10.9T        Li et al. (2007)
CoNb0.2 Mn0.8 Sb               MgAgAs             F43m             470             3.2                   1.40.9T        Li et al. (2007)
CoNb0.4 Mn0.6 Sb               MgAgAs             F43m             465             2.3                   1.20.9T        Li et al. (2007)
CoNb0.6 Mn0.4 Sb               MgAgAs             F43m             463             1.9                   0.60.9T        Li et al. (2007)
Ni50 Mn35 Sn15                 Cu2 MnAl              ¯
                                                  Fm3m             187                                   5.8            Krenke et al. (2005b)
Ni50 Mn37 Sn13                 10M                Pnnm             303                                   6.8            Krenke et al. (2005a)
MnFeP0.45 As0.55               Fe2 P              P62m             300             2.0                 15               Brück et al. (2005)
MnFeP0.47 As0.53               Fe2 P              P62m             293             2.0                 15               Brück et al. (2005)
Mn1.1 Fe0.9 P0.47 As0.53       Fe2 P              P62m             298             2.1                 21               Brück et al. (2005)
MnFeP0.89–x Six Ge0.11
x = 0.22                       Fe2 P              P62m             270             2.1                 38               Brück et al. (2007)
x = 0.26                       Fe2 P              P62m             292             2.1                 41               Brück et al. (2007)
x = 0.30                       Fe2 P              P62m             288             2.1                 39               Brück et al. (2007)
x = 0.33                       Fe2 P              P62m             260             2.1                 36               Brück et al. (2007)




                                                                                                                                                261
262                                                                                  E. Brück




Figure 4.9 Magnetic entropy change for a field change from 0 to 2 T for different Heusler
alloys. Note that NiMnSn shows the inverse effect (Krenke et al., 2005b; Long et al., 2005).


    For magnetocaloric applications the extremely large length changes in the
martensitic transition will definitely result in aging effects. It is well known for
the magnetic shape-memory alloys that only single crystals can be frequently cycled
while polycrystalline materials spontaneously crack and convert to powder after
several cycles. For the Ga containing alloys similar to Ge there is only a very limited
supply of Ga metal as the worldwide production is of the order of 90 t. As most of
the Ga is consumed for GaAs wafers and as dopand in semiconductor industries the
yearly production of NiMnGa for transducer or magnetocaloric applications would
be limited to about 140 t.

4.6 Fe2 P based compounds
The magnetic phase diagram for the system MnFeP-MnFeAs (Beckman and Lund-
gren, 1991) shows a rich variety of crystallographic and magnetic phases. The most
striking feature is the fact that for As concentrations between 30 and 65% the hexag-
onal Fe2 P type of structure is stable and the ferromagnetic order is accompanied by a
discontinuous change of c/a ratio. While the total magnetic moment is not affected
by changes of the composition, the Curie temperature increases from about 150 K
to well above room temperature. We reinvestigated this part of the phase diagram
(Tegus et al., 2002b; Brück et al., 2003) and investigated possibilities to partially
replace the As (Tegus et al., 2003, 2005; Dagula et al., 2005, 2006; Zhang et al.,
2005a; Ou et al., 2006; Thanh et al., 2006).
    Polycrystalline samples can be synthesised starting from the binary Fe2 P and
FeAs2 compounds, Mn chips and P powder (red) mixed in the appropriate propor-
tions by ball milling under a protective atmosphere. After this mechanical alloying
Magnetocaloric Refrigeration at Ambient Temperature                                        263




Figure 4.10 Temperature dependence of the magnetization of MnFeP0.45 As0.55 measured in
an applied field of 50 mT. T1 and T2 indicate the onset and finalization of the phase transition.
  Th indicates the thermal hysteresis.


process one obtains amorphous powder. To obtain dense material of the crystalline
phase, the powders are pressed to pellets wrapt in Mo foil and sealed in quartz tubes
under an argon atmosphere. These are heated at 1273 K for 1 hour, followed by a
homogenisation process at 923 K for 50 hours and finally by slow cooling to am-
bient conditions. The powder X-ray diffraction patterns show that the compound
crystallises in the hexagonal Fe2 P type structure. In this structure the Mn atoms
occupy the 3(g) sites, the Fe atoms occupy the 3(f) sites and the P and the As atoms
occupy 2(c) and 1(b) sites statistically (Bacmann et al., 1994). From the broadening
of the X-ray diffraction reflections, the average grain size is estimated to be about
100 nm (Tegus et al., 2002a).
     Figure 4.10 shows the temperature dependence of the magnetisation measured
with increasing and decreasing temperature in an applied field of 50 mT. The ther-
mal hysteresis is signature of a first-order phase transition. Because of the small size
of the thermal hysteresis (less than 1 K), the magnetisation process can be con-
sidered as being reversible in temperature. From the magnetisation curve at 5 K,
the saturation magnetisation was determined as 3.9 μB /f.u. This high magnetisa-
tion originates from the parallel alignment of the Mn and Fe moments, though
the moments of Mn are much larger than those of Fe (Beckman and Lundgren,
1991). Variation of the Mn/Fe ratio may also be used to further improve the
magnetocaloric effect. Recently, we have observed a surprisingly large magne-
tocaloric effect in the compound MnFeP0.5 As0.3 Si0.2 at room temperature (Dagula
et al., 2006). After replacing all As a considerable large magnetocaloric effect is
still observed for MnFe(P,Si,Ge) (Thanh et al., 2006). A Mössbauer study on the
MnFe(P,As) series evidences the importance of competing AF and F interactions
that depend on the local environment (Hermann et al., 2004).
264                                                                              E. Brück




Figure 4.11 Magnetic entropy change of various MnFe(P,As,Si,Ge) alloys for a field change
from 0 to 2 T.



    The magnetic-entropy change of different MnFe(P,As,Si,Ge) alloys is shown in
Fig. 4.11. The origin of the large magnetic-entropy change should be attributed to
the comparatively high 3d moments and the rapid change of the magnetisation in
the field-induced magnetic phase transition. In rare-earth materials, the magnetic
moment fully develops only at low temperatures and therefore the entropy change
near room temperature is only a fraction of their potential. In 3d compounds, the
strong magneto-crystalline coupling results in competing intra- and inter-atomic
interactions and leads to a modification of metal-metal distances which may change
the iron and manganese magnetic moment and favours the spin ordering.
    The large MCE observed in Fe2 P based compounds originates from a field-
induced first-order magnetic phase transition. The magnetisation and structural
change is reversible in temperature and in alternating magnetic field as was evi-
denced also in X-ray diffraction experiments in applied magnetic field (Koyama
et al., 2005; Yabuta et al., 2006). The magnetic ordering temperature of these
compounds is tuneable over a wide temperature interval (200 K to 450 K). The
excellent magnetocaloric features of the compounds of the type MnFe(P,Si,Ge,As),
rather simple sample preparation (Kim and Cho, 2005; Yan, 2006) in addition to the
very low material costs, make it an attractive candidate material for a commercial
magnetic refrigerator. However, similar to MnAs alloys, it should be verified that
materials containing As do not release this to the environment. The fact that the
magneto-elastic phase-transition is rather a change of c/a than a change of volume,
makes it feasible that this alloy even in polycrystalline form will not experience
severe aging effects after frequent magnetic cycling.
Magnetocaloric Refrigeration at Ambient Temperature                               265


4.7 Other Mn intermetallic compounds
MnFe1–x Cox Ge
The intermetallic compound MnFeGe crystallizes in the hexagonal Ni2 In-type
structure. In this structure Mn atoms occupy 2a sites with a moment of 2.3 μB /Mn,
Fe atoms are at 2d sites with 1.1 μB /Fe, and Ge at 2c sites (Beckman and Lund-
gren, 1991). The Curie temperature of MnFeGe is 228 K. On the other hand, the
compound MnCoGe crystallizes in the orthorhombic TiNiSi-type structure with a
Curie temperature of 337 K. In this structure Mn has a moment of about 3 μB /Mn
and Co has a moment of 0.78 μB /Co. When replacing Fe by Co, it is expected
that both magnetic moment and Curie temperature should increase and a structural
transformation from the hexagonal Ni2 In-type to the orthorhombic TiNiSi-type
occurs.
     It turns out, that the samples have the Ni2 In-type structure (hexagonal, space
group P63/mmc) for x < 0.8 and the TiNiSi-type structure (orthorhombic, space
group Pnma) for x ≥ 0.85 (Lin et al., 2006). The MnFe0.2 Co0.8 Ge compound crys-
tallizes mainly in Ni2 In-type, but a small amount of orthorhombic phase traces were
present. The lattice parameter a decreases and c increases, but the unit cell volume
becomes smaller with increasing of Co contents. Figure 4.10 depicts the concentra-
tion dependence of the structure and the Curie temperatures, which also are listed
in Table 4.4. MnFeGe has a Curie temperature of 159 K, which is much lower than
the value of 228 K reported earlier (Beckman and Lundgren, 1991). MnCoGe has
a Curie temperature of 345 K, being close to the earlier reported values of 337 K.
The data listed in the table show that the Curie temperature increases with increas-
ing Co contents. The spontaneous moment of the MnFe1–x Cox Ge compounds at
5 K, derived from extrapolation to zero field of the high-field magnetization, has
also been listed in Table 4.4. The magnetic moments increase with increasing Co
content in the Ni2 In-type structure, reaching a maximum value of 2.34 μB /3d
atom. for x = 0.8. In the NiTiSi-type structure, the magnetic moments almost sat-
urate at a value of 2.06 μB /3d atom. When the symmetry changes from hexagonal
to orthorhombic, TC and magnetic moment increase abruptly see Fig. 4.12.
     The magnetic-entropy change is derived from the magnetization data by using
Eq. (19). Table 4.4 shows the magnetic-entropy change of MnFe1–x Cox Ge com-
pounds in a field change from 0 to 2 and 5 T, respectively. The magnetic-entropy
change in the compounds, which crystallized in the Ni2 In-type structure, increases
with increasing Co content. A comparatively large magnetic-entropy change, which
reaches 9 J/kg · K, is observed for x = 0.8 in a field change of 5 T.

Mn5–x Fex Si3
Because field induced transitions can produce large magnetocaloric effects the ma-
terial Mn5 Si3 crystallizing in the hexagonal Mn5 Si3 -type structure with space group
P63 /mcm attracted some attention (Songlin et al., 2002b). The compound Mn5 Si3
is an antiferromagnet with a field-induced transition ( Sm = 2.9 J/kg · K at 58 K
and B = 5 T). On the other hand the isostructural compound Fe5 Si3 is a fer-
romagnet with a Curie temperature of 363 K. The magnetic phase transitions and
the magnetocaloric properties have been investigated in the pseudo binary system
266                                                                                E. Brück




      Figure 4.12 Magnetic and crystal structure of the MnFe1–x Cox Ge compounds.


Mn5–x Fex Si3 for x = 0, 1, 2, 3, 4, 5. With increasing Fe content, the anti-
ferromagnetic ordering temperature shifts to higher temperatures. At 4.2 K, the
Mn5–x Fex Si3 compounds with x = 1 and 2 display antiferromagnetic behavior up
to 38 T. The compounds with x = 4 and 5 show ferromagnetic order. The largest
value for the magnetic-entropy change is observed for the MnFe4 Si3 compound
( SM = –4.0 J/kg · K at 310 K and B = 5 T).

Mn5 Ge3–x Six
One of the Mn rich alloys with group four elements that orders near room temper-
ature is Mn5 Ge3 , the magnetocaloric effect of this alloy is fairly large but yet smaller
than for Gd metal (Hashimotoa et al., 1981). The magnetic properties and the mag-
netocaloric effect of Mn5 Ge3–x Six alloys were investigated by (Zhao et al., 2006) for
x = 0.1, 0.3, 0.5, 1.0, 1.5 and 2.0. All Mn5 Ge3–x Six compounds crystallize in the
Mn5 Si3 -type hexagonal structure with space group P63 /mcm. The lattice parame-
ters and the Curie temperature of Mn5 Ge3–x Six alloys decrease with increasing x. As
can be seen in the table, a fairly large magnetic-entropy change has been observed
in these alloys near room temperature. The average Mn magnetic moment decreases
with increasing Si content. The substitution of Si in Mn5 Ge3 does not result in a
change of the crystal structure. But the Si substitution has two kinds of effects on
the magnetocaloric effects. One is that the magnetic-entropy change decreases with
increasing Si content, the other one is that the magnetocaloric effect peak becomes
broadened.

Mn5 Ge3–x Sbx
Compared to the Mn5 Ge3–x Six series, substitution of Ge by Sb should give the op-
posite effect on unit-cell volume and Curie temperature In this series the magnetic
Magnetocaloric Refrigeration at Ambient Temperature                                267


and magnetocaloric properties were investigated by (Songlin et al., 2002a) for com-
pounds with x = 0, 0.1, 0.2 and 0.3 (see Table 4.1) The compounds crystallize in
the hexagonal Mn5 Si3 -type structure with space group P63 /mcm. The Sb substitu-
tion leads to slightly enhanced Curie temperatures but decreasing average magnetic
Mn moments with increasing Sb content. The Sb substitution has two kinds of
effects on the magnetocaloric effect (MCE) of Mn5 Ge3–x Sbx . One is the magnetic
entropy change decreases with increasing Sb content, the other is that the MCE
peak becomes broadened.

LaMn2–x Fex Ge2
These compounds crystallize in the tetragonal ThCr2 Si2 -type structure and the
Curie temperature gradually decreases with increasing Fe concentration from
310.7 K at x = 0.10 to 274.5 K at x = 0.20 (Zhang et al., 2003). As can be
seen in Table 4.4, the magnetic entropy change in this series of compounds, mea-
sured with a field change of 1.8 T, also decreases with Fe content.

(Fe1–x Mnx )3 C
The (Fe1–x Mnx )3 C compounds crystallize in the orthorhombic Fe3 C structure. The
Curie temperature can be adjusted very well from 31 to 483 K. However, there is a
large loss of magnetization with the addition of manganese by changing the Fe/Mn
ratio. The magnetocaloric effects remain relatively low see Table 4.4.

Mn3–x Cox GaC
The magnetocaloric effect in the Mn3–x Cox GaC compounds has been investigated
by Tohei et al. (2003). Mn3 GaC shows a first-order antiferromagnetic to ferromag-
netic transition at Tt = 160 K. Large magnetocaloric effects of Smag = 15 J/kg · K,
were observed at this transition. The substitution of Co for Mn lowers Tt without
significant loss of magnetocaloric effects (Table 4.4). It was indicated that the system
could cover a wide temperature range of 50–160 K by combining the compounds
with various compositions from x = 0 to 0.05.

4.8 Amorphous materials
Super paramagnetic or very soft magnetic materials change their magnetization re-
versibly in very low magnetic fields. This is the motivation that a few amorphous
materials are being studied for possible magnetocaloric applications. On the one
hand the Fe rich alloys have good mechanical and corrosion properties but rather
high magnetic ordering temperatures (Franco et al., 2006a, 2006b). Another prob-
lem with these materials is the rather low adiabatic temperature change, for effective
heat transfer in a short time interval this temperature change should be a few degrees
and not a few tenth of a degree.
    The flatter top of the temperature dependence of the magnetic entropy change,
is another motivation for the use of amorphous materials in AMRs. It is expected
that the performance of these materials is better as this reduces the cycle losses
in e.g. an Ericsson cycle. With this motivation mainly rare-earth based materials
are studied for rather low temperature applications (Foldeaki et al., 1997a, 1998;
268                                                                                       E. Brück


Table 4.6 Magnetocaloric properties of a few amorphous or nanocrystalline transition metal
based materials. | S| at B = 2 T if not indicated differently

Material                 Remark              Bmax     Tmax        Smax      Ref.
                                             (T)      (K)       (J/kg·K)

Fe90 Zr10                                    7        237       6.52        Maeda et al. (1983)
Fe82 Mn8 Zr10            Melt spun           5        210       2.8         Min et al. (2005)
Fe80 Mn10 Zr10           Melt spun           5        195       2.3         Min et al. (2005)
Co66 Nb9 Cu1 Si12 B12    Melt spun           0.15     175       ?           Didukh and
                                                                            Slawska-Waniewska
                                                                            (2003)
Co66 Nb9 Cu1 Si12 B12    Partly              0.15     120        ?          Didukh and
                         recrystallize                                      Slawska-Waniewska
                                                                            (2003)
Co66 Nb9 Cu1 Si12 B12    Partly              0.15         80     ?          Didukh and
                         recrystallize                                      Slawska-Waniewska
                                                                            (2003)
Pd40 Ni22.5 Fe17.5 P20   Bulk                5            94    0.58        Shen et al. (2002)
                         amorphous
FeMoSiBCuNb              Finemet             1.5      480       1.1         Franco et al. (2006a)



Table 4.7 Magnetocaloric properties of a few amorphous or nanocrystalline rare-earth metal
based materials

Material       Remark                    Bmax       Tmax         Smax      Ref.
                                         (T)        (K)        (J/kg·K)

Gd70 Ni30      Melt spun                 1          126         2.5        Foldeaki et al. (1997a)
Gd70 Ni30      Melt spun                 7          130        11.5        Foldeaki et al. (1997a)
Gd70 Ni30      Melt spun ground          7           95         7.5        Foldeaki et al. (1998)
Gd70 Fe30      Melt spun                 1          288         1.5        Foldeaki et al. (1997a)
GdNiAl         Melt spun                 2           40         6.7        Si et al. (2002)
GdNiAl         Ball milled               1           35         1.6        Chevalier et al. (2005)
NdFe12 B6      Melt spun                 1          218         8.4        Zhang et al. (2006a)
               recrystallized
GdMn2          Ball milled               9          130         2.2        Marcos et al. (2004)
GdMn2          Poly                      9           50         4.9        Marcos et al. (2004)



Giguere et al., 1999a, Si et al., 2001a, 2001b, 2002). The key data of a few amor-
phous or nanocrystalline alloys are summarized in Table 4.6 for the transition metal
based materials and Table 4.7 for rare-earth based materials.
   In general the magnetic entropy change in amorphous materials is smeared out
over a wider temperature interval than what is observed in crystalline materials.
This behavior is beneficial for the efficiency of a regenerator. However, if the mag-
Magnetocaloric Refrigeration at Ambient Temperature                                     269


netocaloric effect results in less than 1 degree temperature change, the driving force
for heat transfer becomes quite low and very low frequency operation will be re-
quired. Therefore evaluation of the cooling capacity of a material as proposed a
few years ago (Wood and Potter, 1985) should not be done by just integrating over
the width at half maximum. Instead, the region of the curve that results in a too
low temperature change should be truncated. Some reports of very large cooling
capacity of a certain material should therefore be taken with some caution as the
temperature span used for the calculation is far too wide.

4.9 Manganites
Field or temperature induced first order phase transitions are also a common feature
of a large group of colossal magneto resistance CMR materials. In recent years
a few of these rare-earth manganese oxide materials that crystallize in Perovskite
type structure, were studied with respect to their magnetocaloric properties. Very
recently a review on the magnetocaloric properties of these materials was published
(Phan and Yu, 2007). Indeed a few materials produce decent magnetic entropy
changes near or below room temperature see Fig. 4.13. Some key parameters of
this type of materials are summarized in Table 4.8.
    Concerning the perspective of these materials for room temperature magnetic
refrigeration, a few aspects need to be considered. The materials definitely have
excellent corrosion stability and generally are quite cheap. However, though the
magnetic entropy change is quite decent, the magnetic field induced change in
temperature is often quite low in these materials. The reason for this is the low




Figure 4.13 Magnetic entropy change of several manganites (Zhong et al., 1999; Hueso et al.,
2002; Phan et al., 2005).
270                                                                                E. Brück


Table 4.8 Magnetocaloric properties of a few perovskite type manganites. FO, SO stand for
first and second order transitions, respectively

Material                         Remark    Bmax   Tmax      Smax Ref.
                                           (T)    (K)     (J/kg·K)

La0.6 Ca0.4 MnO3                 SO?       3      263     5         Bohigas et al. (1998)
La0.67 Ca0.33 MnO3               Sol gel   1      263     5         Hueso et al. (2002)
                                 process
La0.8 Ca0.2 MnO3                 Sol gel   1.5    230     5.5       Guo et al. (1997)
                                 process
La0.7 Ca0.3 MnO3                 FO        1      216     6.3       Ulyanov et al. (2007)
La0.958 Li0.025 Ti0.1 Mn0.9 O3   SO        3       90     2         Bohigas et al. (1998)
La0.65 Ca0.35 Ti0.1 Mn0.9 O3     SO        3      103     1.3       Bohigas et al. (1998)
La0.799 Na0.199 MnO2.97          SO        1      334     2         Zhong et al. (1998)
La0.88 Na0.099 Mn0.977 O3        SO        1      220     1.5       Zhong et al. (1998)
La0.877 K0.096 Mn0.974 O3        SO        1      283     1.1       Zhong et al. (1998)
La0.65 Sr0.35 Mn0.95 Cu0.05 O3   FO        1      345     3.1       Phan et al. (2005)
La0.7 Nd0.1 Na0.2 MnO3           Sol gel   1      312     2.1       Hou et al. (2006)
                                 process
La0.5 Ca0.3 Sr0.2 MnO3           Hydro-    2      317     1.5       Li et al. (2006a)
                                 thermal


number of magnetic ions per formula unit, thus the lattice specific heat is quite
high compared to most other magnetocaloric materials. From Eq. (21) it is obvious
that this will lead to a low T . The low electrical conductivity in these materials
could be an advantage as the generation of eddy currents at high cycle frequencies is
reduced. However this low electrical conductivity comes along with a low thermal
conductivity which limits the cycle frequency.



      5. Comparison of Different Materials and
         Miscellaneous Measurements
        The MCEs for field changes of 2 T (if available) are summarized in the ta-
bles. It is obvious that above room temperature a few transition-metal-based alloys
perform the best. If one takes into account the fact that T also depends on the
specific heat of the compound (Pecharsky and Gschneidner, 2001) these alloys are
still favorable not only from the cost point of view. This makes them likely candi-
dates for use as magnetic refrigerant materials above room temperature. However,
below room temperature a number of rare-earth compounds perform better and
for these materials a thorough cost vs performance analysis will be needed.
     The main parameters of the most important magnetocaloric materials are sum-
marized in Table 4.9 which allows a fast comparison. Because of the limited
availability, the Gd, Ge and Ga containing materials will be restricted to niche
                                                                                                                                                      Magnetocaloric Refrigeration at Ambient Temperature
Table 4.9 Availability of different types of magnetocaloric materials, possible range of use, thermal hysteresis and temperature change in 2 T. The
worldwide production is estimated from data of US geological service, hysteresis is strongly sample dependent, Gd has a second order transition,
thus no hysteresis, below 2 K is 0, above 2 K is –

Material                     Limiting        Estimated                     Total availability       Temperature         Thermal          Tmax for
                             ingredient      availability                  of MC material           range (°C)          hysteresis       B2T

Gd metal                     Gd              1000                          1000 t                   0–20                +              6K
Gadolinium silicon           Ge              WW prod. = 90 t,              140 t                    –100 to 0           –              7K
alloys                                       avail. 10 t?
Gd5 (Si1–x Gex )4
Manganese alloys             None            None                          No limitation            –50 to 50           0              6K
Mn(As1–x Sbx )                                                             for an industrial        –100 to 120         0              8K
MnFe(P1–x Asx )                                                            production


NaZn13 type alloys           La              4000                          22000 t                  –80 to 50           –              6K
La(Fe13–x Mx )

Manganites                   La              4000                          7000 t                   –100 to 50          –              3K
LaMnO3
Heusler alloys               Ga              WW prod. = 90 t,              60 t?                    –50 to 50           –              3K
Ni0.501 Mn0.227 Ga0.258                      avail. 10 t?




                                                                                                                                                      271
272                                                                                 E. Brück



markets. At present it is not clear which material will really get to the stage of real
life applications and one may expect that still other materials will be developed.
Though it is already feasible that for applications with limited temperature span and
a cooling power in the kW range like air conditioning, commercial competitive
magnetic refrigerators are quite possible, it is not yet obvious, which of the above
mentioned materials shall be employed. Currently most attention is paid to the pure
magnetocaloric properties which are derived from magnetic measurements as de-
scribed in section 3. In the last few years more specific experimental equipment has
been developed to characterize magnetocaloric materials.
    At Moscow State University two setups exist to determine the adiabatic tem-
perature change. One setup utilizes a rather simple electromagnet (B < 2 T) and a
nitrogen cryostat. The other setup is a liquid nitrogen cooled pulse magnet in com-
bination with a He flow cryostat. In the former equipment the variation of sample
temperature during the sweeping of the field is monitored with a thermocouple.
Sweeping to the maximum field takes approximately 3 s (Tishin, 1999). Data of the
most important magnetocaloric materials exist from this equipment (Chernyshov
et al., 2002; Hu et al., 2003, 2005; Brück et al., 2005; Ilyn et al., 2005; Wada et al.,
2005b). The last few years however the maximal field employed is only 1.45 T. The
reported temperature changes are 4.3 K at 292 K for Mn1.1 Fe0.9 P0.47 As0.53 , 4 K at
312 K for MnAs, 4 K at 188 K for LaFe11.7 Si1.3 , 3.2 K at 274 K for LaFe11.2 Co0.7 Si1.1 ,
and 3 K at 271 K for Gd5 Ge2.05 Si1.95 . The pulse field setup which provides up to
8 T fields with a pulse length of 0.2 s is up to now only used to study the classical
materials (Dankov et al., 1997).
    At the University of Quebec a special sample holder for a Quantum Design
PPMS system is constructed, which enables the insertion of the sample from a low
field position into the 9 T maximum field within 1 s (Gopal et al., 1997). The
sample temperature is monitored with a Cernox resistance thermometer and can
be varied between 2 and 400 K. However, only the temperature at the high field
position is controlled so that the measurements need to be performed rather fast to
avoid excessive drift of the temperature. Eddy current heating of the sample may
occur under these conditions. Next to classical magnetocaloric materials like (Gd,Y)
alloys (Foldeaki et al., 1997b), also measurements on Gd5 Ge2 Si2 are reported, it
should be noted that the latter results were quite controversial (Giguere et al., 1999b;
Gschneidner et al., 2000; Sun et al., 2000).
    Direct measurements of the adiabatic temperature-change of La0.6 Ca0.4 MnO4
were reported by a Danish group. They utilize a nitrogen cryostat that can be in-
serted in the field of an electro magnet. For a field change of 0.7 T at 270 K a
maximal adiabatic temperature change of 0.5 K is observed (Dinesen et al., 2002).
Recently, the same group reports on the extension of the temperature range to well
above room temperature for La0.67 Ca0.33–x Srx MnO3+δ (Dinesen et al., 2005). The
largest effect measured directly in again 0.7 T is for the compound without Sr at
267 K a temperature change of 1.5 K. For 22% Sr at 344 K a temperature change
of 0.5 K is observed.
    A pulsed magnet setup with rather long pulse duration of more than a second
has been developed at Sichuan University (Tang et al., 2003) there are however little
results published yet.
Magnetocaloric Refrigeration at Ambient Temperature                                    273




Figure 4.14 Photographs of the DSC in field insert open (left) and closed (right) the sample
is mounted on the thermo batteries attached to the Cu block a reference sample is mounted
on the backside (Marcos et al., 2003).


     At Tohoku University also the mechanical insertion of the thermally insulated
sample into a 2 T field region is employed for direct measurements (Fujieda et al.,
2004b). The results on LaFe11.9 Si1.1 T = 5.9 K at 188 K are in good agreement
with earlier published data (Hu et al., 2003), the temperature change observed for
LaFe11.9 Si1.1 H1.6 T = 4.0 K at 319 K is rather unexpected.
     A group at the University of Genoa reports on a fast direct measurement device
that employs an electromagnet with maximum field of 1 T and can be utilized
from 100 K up to 340 K (Canepa et al., 2005). The adiabatic temperature change
of several Gd eutectic alloys are reported, the authors also observe a rather high
sensitivity of the MCE on impurities in Gd.
     A commercial adiabatic calorimeter from Thermis Ltd placed in a 6 T super-
conducting magnet is employed in Barcelona (Tocado et al., 2005). The calorimeter
works in the temperature range 4–370 K and magnetic fields may be applied as long
as the vacuum grease used to attach the sample to the sample holder keeps it at its
place, the latter may become problematic near room temperature for magnetic sam-
ples. For the direct measurements performed in a field sweep to 5 T field a maximal
temperature change T = 6.5 K at 109 K is reported for Tb5 Ge2 Si2 .
     With the aim to study the heat effects occurring in magnetostructural transi-
tions, Marcos et al. developed a differential scanning calorimeter (DSC), which is
depicted in Fig. 4.14, capable to work in magnetic fields up to 5 T (Marcos et
al., 2003). In contrast to most other calorimeters a DSC can measure in a heating
and a cooling mode so the thermal hysteresis is easily monitored. Also the en-
tropy change in a first order phase-transition is normally difficult to determine as
274                                                                              E. Brück




Figure 4.15 Thermal conductivity of various magnetocaloric materials (Battabyal and Dey,
2004; Fujieda et al., 2004c; Fukamichi et al., 2006).


accurately as with a DSC. In combination with applied magnetic field the phase
transition of giant magnetocaloric materials thus can be studied very accurately. In-
teresting cycle time dependent effects in the magnetic field induced entropy change
in Gd5 Ge3.8 Si0.2 have been observed with this equipment (Casanova et al., 2005).
    Another important property of magnetic refrigerants is the thermal conduc-
tivity. At Tohoku University the temperature dependence of the thermal con-
ductivity was studied for several magnetic refrigerants (Fujieda et al., 2004c;
Fukamichi et al., 2006). At the Indian Institute for Technology, Karagpur the ther-
mal conductivity of (La,Sr,Ag)MnO3 perovskites was studied (Battabyal and Dey,
2004). As one may expect the perovskites have a much lower thermal conductivity
but astonishingly the conductivity of MnAs is not much higher and Gd5 Ge2 Si2 is
found to be intermediate, the results are summarized in Fig. 4.15.
    In the near future also other properties like corrosion resistance, mechanical
properties, heat conductivity, electrical resistivity and environmental impact should
be addressed more.



      6. Demonstrators and Prototypes
     The growing interest in magnetic refrigeration near room temperature is also
reflected in a growing number of projects that do not study materials properties but
Magnetocaloric Refrigeration at Ambient Temperature                              275


study the performance of certain refrigerator designs. Next to theoretical papers
that discuss for example different thermodynamic cycles, in the last few years sev-
eral demonstrators and prototypes were built. We shall here discuss a few of them
in more or less chronological order, some key aspects of these prototypes are sum-
marized in Table 4.10. A few years ago already reviews on magnetic refrigerators
were published but the development is rather fast so it is worthwhile to discuss it
here again (Yu et al., 2003; Gschneidner et al., 2005).
     Already more than 30 years ago, the advantages of a regenerator process for
magnetic refrigeration near room temperature were pointed out in a paper that
discusses Gd as a possible magnetic refrigerant near room temperature in combina-
tion with a 7 T superconducting magnet (Brown, 1976). The paper predicts near
Carnot efficiency and a maximal temperature span of 46 K at a sink temperature of
340 K. Shortly after this a rotary design of the magnetic Stirling cycle was proposed
(Steyert, 1978). Here also a 7 T magnetic field is used to magnetize a porous Gd
disk that rotates in and out of the high field region and heats or cools a counter-
flowing heat-transfer fluid. This machine was predicted to reach a cooling power
of 32 kW/l Gd at an operating frequency of 1 Hz. Note that the rule of thumb,
low fluid thermal capacity is needed for high efficiency, which holds for a passive
regenerator, does not hold in the case of an active regenerator.
     The first realization of a reciprocating magnetic refrigerator that very much
resembles the design proposed by Brown was reported 20 years later (Zimm et
al., 1998). This magnetic refrigeration demonstrator built in collaboration of Ames
Laboratory and Astronautics in Madison WI, utilized a 5 T superconducting magnet
and 3 kg of Gd spheres. Extremely good performance parameters were reported on
this machine. A cooling power of 600 W at an operating frequency of 0.17 Hz
and a temperature span of 10 K was realized. The COP reached a value of 10 or
in other words a Carnot efficiency of 75% was reported, the authors mention that
they neglected a few things, but in the light of more recent publications this high
Carnot efficiency must be taken with caution. Other authors report for very similar
heat-exchanger designs rather high losses due to the high flow resistance of the
heat-exchanger (see below).
     Having in mind to design a very simple demonstrator a group in Barcelona uti-
lizes the rotary device proposed by Steyert in combination with permanent magnets
(Bohigas et al., 2000). A thin ribbon of Gd metal mounted on a plastic wheel rotates
in and out of the field of 0.3 T. Commercial olive oil is used as heat transfer medium
and regenerator. The maximal temperature difference achieved in steady state op-
eration was 1.6 K which clearly demonstrates that regeneration worked even in this
simple device. When the device is operated at 1/3 Hz steady state is reached after
about 1000 s. The main losses in this device are probably due to flow of transfer
medium between different sections. The authors do not comment on the efficiency
of the device and the numbers quoted are not sufficient to determine it.
     To test the possibility of increasing the temperature span of an active magnetic
regenerator a team at the University of Victoria BC designed a compact reciprocat-
ing device that accommodates AMR pucks of 2.5 cm diameter and 2.5 cm length
(Richard et al., 2004). The field source for this device is a 2 T superconducting
magnet and the operating frequency could be varied between 0.2 and 1 Hz. Two
                                                                                                                                              276
Table 4.10 Magnetic refrigeration demonstrators, the magnetic field source is (S) superconducting magnet, (P) permanent magnet, (E) electro-
magnet. When authors quote the performance there exist COPT only taking into account the cooling power and the power dissipated at the hot
heat exchanger and COPR cooling power divided by total electrical power input

Name                   AMR type             AMR material             Magnetic field           Remarks              Ref.
                                                                     (T)

Ames                   Reciprocating        Gd spheres               5 (S)                   COPT 10              Zimm et al. (1998)
Laboratory/
Astronautics
Barcelona              Rotary               Gd foil                  0.3 (P)                 Olive oil            Bohigas et al. (2000)
University of          Reciprocating        Gd, Gd0.74 Tb0.26        2 (S)                   Epoxy                Richard et al. (2004)
Victoria                                                                                     bonded
                                                                                             pucks
Lab. Electric          Reciprocating        Gd foil                  0.8 (P)                 COPR 2.2             Clot et al. (2003)
Grenoble
Astronautics           Rotary               Gd, Gd-Er,               1.5 (P)                 4 Hz                 Zimm et al. (2006)
                                            spheres LaFeSiH
                                            particles
Tokyo Inst.            Rotary               Gd-Dy,                   0.7 (P)                 Torque 52 Nm         Okamura et al. (2006)
Techno./Chubu                               Gd-Y spheres                                     COPR 0.2
Natl Inst. Appl.       Rotary               Gd plates                1 (P)                   Torque 10 Nm         Vasile and Muller (2006)
Sci./Cooltech
Xian Jiaotong          Reciprocating        Gd spheres;              2.18 (E)                COPT 25              Gao et al. (2006)
Univ.                                       Gd5 (Si,Ge)4
                                            pwdr.
University of          Reciprocating        Gd, Gd0.74 Tb0.26        2.0 (S)                   T 50 K             Rowe and Tura (2006)
Victoria                                    Gd0.85 Er0.15




                                                                                                                                              E. Brück
Magnetocaloric Refrigeration at Ambient Temperature                                 277


AMR’s were alternating in and out of the high field region. The heat transfer fluid
used in this experiment was He gas at up to 10 atm. pressure and a maximal mass
flow of 0.4 g per half cycle. Two types of regenerators were tested one with two
pucks of Gd with a total mass of 90 g, and one consisting of two materials with
different TC , a puck of Gd at the hot side and a puck of Gd0.74 Tb0.26 at the cold side
with a total mass of 85 g. The tests clearly show that for reaching larger temperature
spans the multimaterial heat exchanger is more suited. Another interesting point
is that the low heat capacity of the helium gas in combination with the low mass
flow rate seemingly limited the performance of the device. The cooling power of
this device is depending on the temperature span only a few W. This results in a
very long startup period of about 1.5 h before steady state is reached. The authors
do not quote a COP or other figures of performance. Just recently the same group
reported on the use of three different materials, Gd, Gd0.74 Tb0.26 and Gd0.85 Er0.15
with a total mass of about 135 g for each AMR and a field of 2 T (Rowe and
Tura, 2006). In this configuration a maximal no-load temperature span of 50 K was
realized. The authors however comment that this span quickly decreases when the
hot reservoir is at temperatures above 307 K.
    At the Laboratoir Electric de Grenoble a reciprocating magnetic refrigerator was
designed that utilizes a Halbach magnet generating 0.8 T transverse field and Gd
foil as magnetic refrigerant (Clot et al., 2003). The device is only poorly isolated
which results in considerable thermal losses, therefore the authors only quote the
COP for a temperature span of 4 K. For this condition the cooling power is found
to be 8.8 W and the electrical power needed to operate the device is 4 W thus a
COP of 2.2 is derived. This COP is a really measured value and not just estimated
after neglecting parasitic influences.
    The second AMR constructed at Astronautics, Madison WI is a rotary device
that utilizes a permanent magnet as field source (Zimm et al., 2006). In this machine
the magnetic field is about 1.5 T and the regenerator rotates with up to 4 Hz in
and out of the high field region while the heat-exchange fluid (water) is pumped in
the opposite direction. This device has been tested with several different materials.
Pure Gd, Gd0.94 Er0.06 , a combination of these two and the giant MCE material
La(Fe0.88 Si0.12 )13 H. The performance of the different beds strongly depends on the
temperature interval that is studied. The authors also mention that the performance
of the device is strongly influenced by the fluid-flow resistance of the AMR matrices
which depends on the material. The Gd and Gd-Er particles are spherical with a
rather narrow size distribution but in the case of LaFeSiH the particles are with
irregular shape and a large spread of sizes. The latter limited the operation frequency
of the LaFeSiH AMR to 1 Hz. Interestingly, for low temperature spans the authors
find the cooling power to be higher at lower cycle frequencies while the opposite is
true for larger temperature spans. This effect is attributed to the dominance of valve
friction and dynamic flow losses, which are lower at low frequency. This finding
may be of interest for the optimization of startup procedures.
    Two other rotary devices have been presented recently which instead of moving
the regenerator move the magnet (Okamura et al., 2006; Vasile and Muller, 2006).
The obvious advantage of moving the magnet is that one avoids sliding seals that
may deteriorate after extended period of use.
278                                                                                 E. Brück



     The device developed at the Tokyo Institute of Technology and Chubu Electric
power employs a 0.77 T permanent magnet and four AMRs that consist each of
four different Gd alloys Gd0.92 Y0.08 , Gd0.84 Dy0.16 , Gd0.87 Dy0.13 and Gd0.89 Dy0.11 with
TC ranging from about 5°C to 10°C (Okamura et al., 2006). The AMR’s of 1 kg
each are packed beads of spheres with 0.6 mm diameter with a filling factor of 63%.
In the cooling cycle the magnet is rotated in 90 degree steps switching from one
AMR to the next. In the stopping period water is pumped through the AMR.
The flow direction depends on the field if the field is high the water flows from
the cold side to the hot side and vice versa. Actually the heat exchange period
can extend into the rotating period. Typical rotation periods of 0.5 s and cycle
times of 2.4 s were used. The authors realize a maximal cooling power of 60 W at
10°C with zero temperature span, however the coefficient of performance of this
device is disappointing low (below 0.2). This is due to the high torque (52 Nm)
needed to switch the magnetic field and the high flow resistance of the water in
the AMR resulting in a very high power consumption. The authors conclude that
new arrangement and configuration of the AMR beds are inevitable to improve the
design.
     These two points are especially addressed in the design presented by the French
National Institute of Applied Sciences and Cooltech Applications (Vasile and
Muller, 2006). They employ a heat exchanger with micro channels and separated
circuits for hot and cold flow. This heat exchanger consists of quadratic plates of
pure Gd with a width of 45 mm and thickness 0.65 mm (see Fig. 4.16). The fluid
channels are 0.2 mm wide. The filling factor is 77.7%. The magnets are rotating
NdFeB based permanent magnets that generate a 1 T magnetic field. In the same
publication a modified Hallbach magnet design is presented that generates 1.9 T.
This magnet is open on one side and can thus be rotated over the heat exchangers.
The regenerator with straight micro channels has a very low flow resistance and
yet a good heat transfer rate. Dividing hot and cold flow in separate channels re-
duces the dead volume. Additionally the torque needed to rotate the magnet array
is rather low as the AMRs are arranged almost continuous. This torque of 10 Nm
is more than 5 times less compared to the Japanese design with yet a higher applied
field (Muller, 2006). A near industrial design prototype is depicted in Fig. 4.17.
     At Xian Jiaotong University a study of different materials in a given mag-
netic refrigerator is performed (Gao et al., 2006). On the one hand Gd spheres
with different diameters 0.3 and 0.55 mm and 0.3–0.75 mm particles of the al-
loy Gd5 Si2 Ge2 . An electromagnet with 160 mm diameter pole pieces and an air
gap of 60 mm generates the magnetic field of 2.18 T. The AMR with dimension
140 × 76 × 36 mm3 is made of stainless steel and isolated by 2 mm thick layer
of isolation material. The Gd particles are prepared by milling the Gd hydride as
the pure Gd is too ductile to be easily milled. After the required size is achieved
the material is dehydrided. However, the authors mention that the magnetocaloric
effect of the powder is 10–25% lower than what was measured on the ingot which
was used as starting material. The Gd5 Ge2 Si2 prepared from 2N5 commercial grade
Gd which was not heat treated after preparation does not exhibit a first order phase
transition but shows similar properties as the material reported by (Provenzano et
al., 2004). The alloy is milled and particles with sizes between 0.15 and 0.3 mm
Magnetocaloric Refrigeration at Ambient Temperature                                279




Figure 4.16 Front view of a rotary prototype the black squares are the Gd plates of the
micro-channel heat exchangers (Vasile and Muller, 2006).


are selected by sieving. The refrigeration cycle is of the type move, flush, move,
flush where the periods and the flow rates can be varied and the flows and tem-
perature variations are recorded automatically. Similar as Zimm et al. (1998) the
power consumption of the magnet and the pumps are neglected for the calculation
of the COP. Actually only the refrigeration capacity and the heat released at the
hot reservoir are considered. Thus also the power consumption of the step motor
driving the magnet is neglected. The results for the different materials are rather
puzzling. 930 g of 0.3 mm diameter Gd produce a higher cooling power and a
better performance than 1109 g of 0.55 mm diameter Gd. The worst performance
comes from the 1213 g of 0.15–0.3 mm Gd5 Ge2 Si2 . As the external parameters
like force and flow resistance are neglected, these differences must originate from
heat transfer, regenerator and demagnetization parameters. The poor performance
of Gd5 Ge2 Si2 may be explained by the rather low thermal conductivity and the fact
that the very irregular shape of the particles can lead to enhanced demagnetization
effects. On the other hand the higher specific heat of Gd5 Ge2 Si2 should improve
the performance of the regenerator. All this can not hold for the two Gd batches
as different cycle frequencies and fluid-flow rates were tested, the effect of differ-
280                                                                                   E. Brück




      Figure 4.17   Photograph of most recent prototype of Cooltech (Muller, 2007).


ent sizes should have been compensated. This makes me suspect that parasitic heat
sources were not detected in the system. One possible heat source is the pump in
the primary circuit that probably will consume much more power when the pack-
ing factor and thus the pressure drop in the AMR is increased. Leakage of the heat
released in the motor into the water circuit is quite feasible.
    The design of a permanent magnet field source is also an important issue for
cost efficient magnetic refrigeration. The very simple bar magnets used in some
prototypes are limiting the field to below 1 T. However, nowadays closed and open
Halbach magnet arrays are built that can produce fields exceeding 2 T (Lee and Jiles,
2000; Lee et al., 2002a, 2002b; Xu et al., 2004, 2006a, 2006b). These arrays have in
common that several magnetized bar magnets are combined and the magnetization
is then concentrated in a soft magnetic pole piece. A soft magnetic shell that acts
as a flux return path further enhances the performance. The art of getting maximal
performance out of a minimal number of segments will determine the price of
these advanced field sources (Russek and Zimm, 2006).



      7. Outlook
      From the above it is obvious that the field of magnetic refrigeration is very
fast developing, both in research on new materials, modeling and prototype design.
The ideal magnetocaloric material is yet to be developed and the quest for higher
efficiency refrigerators is becoming more and more important as the reduction of
global CO2 production becomes a high priority issue. After the compressor being
the technology of the 20th century, magnetic refrigeration has all potential to be-
come the technology of the 21st century. However, to bring this into reality a broad
research effort is needed on all three fields of research. The THERMAG confer-
ence series started in 2005 in Lausanne and continued in 2007 in Portoroz brings
together researchers from industry and academia which is necessary to accelerate
Magnetocaloric Refrigeration at Ambient Temperature                                               281


the development. Having the next conference held in Asia would also give credit
to the vast amount of research performed on this subject on that continent.


ACKNOWLEDGEMENTS
This work is supported by the CNPQ process number 304385/2006-9-PV and the Dutch Technology
Foundation STW, applied science division of NWO and the Technology Program of the Ministry of
Economic Affairs. We want to especially thank A. Planes of University of Barcelona and C. Muller of
Cooltech Applications for supplying photographs of the DSC in field and of prototypes, respectively.


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        CHAPTER             FIVE



        Magnetism of Hydrides
        Günter Wiesinger * and Gerfried Hilscher *



        Contents
        1. Introduction                                                                                      293
        2. Formation of Stable Hydrides                                                                      295
        3. Electronic Properties                                                                             296
        4. Basic Aspects of Magnetism                                                                        300
        5. Review of Experimental and Theoretical Results                                                    304
           5.1 Binary rare-earth hydrides                                                                    304
           5.2 Binary actinide hydrides                                                                      329
           5.3 Binary transition metal hydrides                                                              332
           5.4 Ternary rare-earth–transition-metal hydrides                                                  335
           5.5 Hydrides of amorphous alloys                                                                  417
        Acknowledgement                                                                                      422
        References                                                                                           422




        1. Introduction
      The present review is based upon our article (Wiesinger and Hilscher, 1991)
which covered the literature until 1990. Despite the article of Vajda (1995a), deal-
ing exclusively with binary RHx systems, no comprehensive review about metal-
hydrogen-systems was published since then. Thus, when studying the literature, it
seems worthwhile to update the review to the articles published to date, particularly,
since ten metal-hydrogen conferences took place in the last 15 years (MH 1988: Z.
Phys. Chem. NF 163 (1989), MH 1990: J. Less-Comm. Met. 172–174 (1991), MH
1992: Z. Phys. Chem. NF 179 (1993), MH 1994: J. Alloys Comp. 231 (1995), MH
1996: J. Alloys Comp. 253–254, MH 1998: J. Alloys Comp. 293–295, MH 2000:
J. Alloys Comp. 330–332 (2002), MH 2002: J. Alloys Comp. 356–357 (2003), MH
2004: J. Alloys Comp. 404–406 (2005), MH 2006: J. Alloys Comp. (2007). In or-
der to avoid redundancy we have tried to condense the text of the previous article
under the consideration of improvements in theory and experiment happening in
   *   Institute for Solid State Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Wien,
       Austria

Handbook of Magnetic Materials, edited by K.H.J. Buschow                                      © 2008 Elsevier B.V.
Volume 17 ISSN 1567-2719 DOI 10.1016/S1567-2719(07)17005-0                                     All rights reserved.



                                                                                                              293
294                                                            G. Wiesinger and G. Hilscher



the last one and a half decades. This is visible particularly in the tables which have
been comprehensively updated and completed.
    Intermetallic compounds of 3d metals (particularly Mn, Fe, Co and Ni) with
rare earth elements exhibit a large variety of interesting physical properties. The
magnetic properties of these intermetallics (for reviews see e.g. Wallace, 1973;
Buschow, 1977a, 1980a and Kirchmayr and Poldy, 1979) are a matter of interest
for two main reasons: Firstly their study helps to elucidate some of the funda-
mental principles of magnetism (RKKY interaction, crystal field effects, valence
instabilities, magnetoelastic properties, coexistence of superconductivity and mag-
netic order). Secondly they are of technical interest, because several compounds
(RCo5 , R2 Co17 , Nd2 Fe14 B, RFe11 T) were found to be a suitable basis for high per-
formance permanent magnets. More recently the unique soft magnetic properties
made amorphous metal-metalloid alloys to a further class of materials which has
attained considerable importance with regard to industrial application.
    Since the discovery of LaNi5 as a hydrogen storage material roughly four decades
ago, a vast number of intermetallic compounds and alloys has been involved in stud-
ies of the hydrogen induced changes of their physical properties. A large variety of
techniques has been applied in order to elucidate the mechanism of hydrogen uptake
which is particularly complex in intermetallic compounds. They can roughly be
divided into surface sensitive methods (photo emission and related spectroscopies,
X-ray absorption (XANES, EXAFS), X-ray magnetic circular dichroism (XMCD),
transmission electron microscopy, conversion electron Mössbauer spectroscopy and
to some extent susceptibility measurements, NMR and ESR) and surface insensitive
experiments, where only the bulk properties can be studied (calorimetric and trans-
port studies, magnetic measurements, neutron and X-ray diffraction, transmission
Mössbauer spectroscopy).
    Despite the complex hydrogen absorption mechanism, some general statements
concerning the influence of hydrogen upon the physical properties can be made.
Hydrogen uptake commonly leads to a considerable lattice expansion. Although
the absorption of hydrogen can lead to a volume increase of up to 30%, the overall
crystal structure frequently is retained. The hydrogen induced rise in volume is
to a large extent the essential reason for the altered magnetic properties in the
hydrides. A larger volume implies narrower bands which, on the other hand, may
reduce a hybridization having perhaps been present in the host compound. When
a transition metal (TM) is alloyed to a rare earth or a related metal (R), the R-3d
exchange interaction (3d-5d overlap) leads to a significant reduction of the TM
moment. The strong hydrogen affinity of the R metals brings about a decrease of
the 3d-5d overlap in the hydrides. Thus the absorption of hydrogen commonly
cancels this moment depression to a certain degree. This partial restoration of the
3d moment is interpreted as an hydrogen induced screening effect.
    The predominant part of published results connected with magnetism considers
binary R hydrides (R = rare earth element) and hydrides of binary compounds of
the general formula Ry TMz , R being a rare earth element, which may be replaced
by elements such as Sc,Y, Zr, Ti, and TM standing for a transition metal. Particu-
larly Mn, Fe, Co and Ni compounds have been examined with regard to hydrogen
absorption properties. Consequently, after some theoretical considerations the re-
Magnetism of Hydrides                                                               295


view will deal with the experimental results regarding binary rare earth hydrides,
followed by a short treatment of transition metal hydrides. The main part covers
ternary hydrides along the element order mentioned above, the final part contain-
ing hydrides of less common compounds and alloys (e.g. GdRh2 , oxygen stabilized
TiTM compounds, several ternary R-compounds, amorphous alloys). Experimen-
tal data are only to some extent mentioned in the text. They have been summarized
in several tables according to the transition element present in the compound.
     In order to limit the number of references to a reasonable number, initially at-
tention was focused to the literature cited subsequently to 1980, except those papers
which contain physical quantities given in the tables. For the remaining former lit-
erature the reader is referred to the comprehensive review articles of Buschow et
al. (1982a), Buschow (1984a), Burger (1987) and Wiesinger and Hilscher (1988a).
Since we suggest that a reference list, complete as much as possible, is desired by the
reader, the articles published after 1991 are merely added to the initial reference list
in Wiesinger and Hilscher (1991).



      2. Formation of Stable Hydrides
      In order to predict the formation of metal hydrogen systems, the heat of for-
mation has to be evaluated. Up to now, only a few first principle calculations have
been performed. However, empirical and semi-empirical models have been pro-
posed for the heat of formation and heat of solution of metal hydrides. For a recent
review we refer to chapter 6 of Hydrogen in Intermetallic Compounds (Griessen
and Riesterer, 1988). The cellular model of Miedema et al. (1976) and, more re-
cently, the band structure model of Griessen and Driessen (Griessen and Driessen,
1984a, 1984b; Griessen et al., 1984) have successfully been applied in metal hy-
dride research. While the former model is already known fairly well and thus needs
not to be introduced separately, the latter one shall be described briefly, particu-
larly because the electronic band structure is involved and thus the connection with
magnetism is obvious.
    Empirical linear relations are proposed between the standard heat of formation
   H and characteristic band structure energy parameters of the parent elements in
order to predict H of the ternary hydrides. In the case of binary metal hydrides
the standard heat of formation is correlated with the difference between the Fermi
energy and the energy of the centre of the lowest s-like conduction band of the host
metal. In the case of ternary metal hydrides the energy difference for intermetallics
of two d-band metals has been evaluated using the model of Cyrot and Cyrot-
Lackmann (1976). The exact density of states (DOS) of an alloy is approximated by
a “simple” DOS, where the individual contributions of the elements are acting in an
additive way (coherent potential approximation). There are various steps involved in
the scaling of the DOS function of each metal. In the first step the widths of the d
bands of both metals are set equal to their weighted average and the DOS curves are
brought to a common width. In the second step the Fermi energies are equilibrated.
The agreement of the calculated heat of formation values with the experiment was
296                                                              G. Wiesinger and G. Hilscher



found to be remarkably good. In most of the cases the band structure model yields
better results than the Miedema-model, which furthermore has the disadvantage of
involving more fit parameters.
    The development of reasonable computational methods in the last decade led
to a step forward compared to the semi-empirical models mentioned above. In
particular, we want to mention the theoretical study of Gupta (1999) applying an ab
initio self-consistent linear muffin-tin orbitals method to study the stability of several
Zr- and La-3d ternary hydrides. As an example the results of ZrNiHx (x = 0, 3)
are given.
    The total density of states (DOS) of both, parent compound and hydride is dis-
played in (Fig. 5.1). The comparison with the pure intermetallic reveals the several
modifications:
 (i) On the low energy side a new structure associated with metal–hydrogen bond-
      ing and H–H interactions appears in the hydride between 4 and 13 eV below
      EF ;
(ii) The energy separation between the Ni 3d and the Zr 4d main peaks decreases
      upon hydrogen uptake as a consequence of the lattice expansion, the Zr 4d
      DOS having been considerably modified by the presence of hydrogen;
(iii) The Fermi energy of the hydride lies closer to the main Ni 3d peak. The ob-
      served reduction of EF is associated predominantly with the lattice expansion.
      Furthermore, the Zr–H interaction leads to a substantial lowering of the Zr 4d
      states located above the Fermi level in the parent intermetallic.
     This factor is of essential importance for the stability of the hydride. The con-
tribution of the Ni 3d states to the metal–hydrogen bonding is sizable although it
is lower than that of the Zr 4d states. This has to be attributed to the larger coor-
dination number of H with Zr in both, the pyramidal (Zr3 Ni2 ) and the tetrahedral
(Zr3 Ni) sites. The most important difference observed in the bonding of H with
Ni and Zr lies in the fact that the Ni-d states are already occupied in the parent
compound, while a majority of the Zr-d states are located above EF . This feature
has important consequences on the position of the Fermi energy in the hydride.
     The effect of chemical substitution plays an important role in the reduced stabil-
ity of a compound (e.g. LaNi4 M compared to LaNi5 ). The reason of the decreased
stability of LaNi4 M compared to LaNi5 is found in the lattice expansion which
accounts for about 50% of the decohesion of the compound. On the other hand,
the Fermi energy is always found to rise upon hydrogenation, a factor which affects
adversely the stability (Gupta, 2002).



      3. Electronic Properties
     The knowledge of the electronic properties (band structure, DOS) consider-
ably helps in understanding a material’s magnetic properties. In most of the stable
metal hydrides, a simple interpretation of the band structure can be obtained, as
Magnetism of Hydrides                                                                    297




                                            (a)




                                            (b)
Figure 5.1 Total density of states of ZrNi (a) and ZrNiH3 (b) and number of electrons (dotted
line). EF is chosen as origin of the energy axis (Gupta, 1999).



follows: since the hydrogen potential is more attractive than the metal atom poten-
tial, the lowest energy bands result from the hydrogen-metal bonding and the H–H
antibonding interactions. The number of corresponding bands is usually equal to
the number of hydrogen atoms in the unit cell (Gupta, 2002).
     Starting from the pioneering work of Switendick (1978, 1979), in the last
years the number of papers dealing with band structure calculations has increased
considerably (see e.g. Gupta, 1989, 1999; Singh and Papaconstantopoulos, 1994;
298                                                                 G. Wiesinger and G. Hilscher




Figure 5.2 Electronic density of states for majority spin and minority spin electrons for both
(a) YCo3 H2 and (b) parent YCo3 . Zero of the energy axis is the Fermi energy (Cui et al.,
2005).


Gupta and Rodriguez, 1995; Orgaz and Gupta, 1995, 2002; Elsässer et al., 1998a,
1998b, 1998c; Michalowicz et al., 2002; Crivello and Gupta, 2003; Wu et al., 2004;
Jezierski et al., 2005; Orgaz et al., 2005). Moreover, the accuracy of the DOS and
the Fermi energy calculations has grown substantially. Decomposition of the DOS
into site and angular momentum components are now available for many metal
hydrides. Even charge transfer calculations and hydrogen induced changes of the
magnetic properties can now be explained in satisfactorily agreement with experi-
mental data (Cr-H and YFe2 H4 , Crivello and Gupta, 2005). By using an ab initio
density functional theory, Cui et al. (2005) succeeded in predicting the structure
and the electronic structure of YCo3 H2 (Fig. 5.2). A somewhat different approach
to study the electronic structure of LaNi5 Hx was performed by Monma et al. (2006)
applying the DV-Xα method.
    The kind of the electronic charge transfer upon hydrogenation is an essential
point for interpreting the hydrogen-induced change of the magnetic properties. In
Magnetism of Hydrides                                                               299


order to explain the magnetization data of rare-earth-transition-metal hydrides, a
few earlier works favored a hydrogen-transition-metal charge transfer in connection
with the rigid-band model (see e.g. Wallace, 1978, 1982). Later on, a similar inter-
pretation has been given with regard to Mn, Fe and Ni hydrides (see e.g. Antonov et
al., 1989). However, theory (energy band and DOS calculations, see e.g. Vargas and
Christensen, 1987; Gupta, 1982, 1987, 1989, 1999) and experiment (Mössbauer
studies performed on R nuclei and X(U)PS investigations, see e.g. Cohen et al.,
1980; Schlapbach, 1982; Schlapbach et al., 1984; Höchst et al., 1985; Osterwalder
et al., 1985) proved the indefensibility of this position. Details will be found below.
     There are a number of experimental methods in order to compare theory and
experiment in the field of the electronic properties. The Pauli contribution of the
magnetic susceptibility and the electronic specific heat coefficient γ are propor-
tional to N(EF ). Resistivity measurements yield valuable results for binary hydrides
(see section 5.1), for hydrides of intermetallic compounds this method is rarely
applied because of experimental difficulties (contacting brittle samples or disinte-
gration of the specimens into powder). Spectroscopic techniques such as electron
and X-ray photo emission belong to the most powerful methods to study the elec-
tronic structure. A valence-band photoelectron spectrum resembles a one-electron
DOS curve. Within some approximations, photoelectron spectra yield directly po-
sition and width of the occupied bands, charge transfer is indicated by XPS core
level and Mössbauer isomer shifts. In valence fluctuation systems X-ray absorption
experiments are particularly valuable. The X-ray absorption near-edge structure
(XANES) contains information about the partial DOS and has become an increas-
ingly important technique. Compton scattering, in particular, when associated with
band structure calculations, was proved to be a powerful tool in studying the elec-
tronic structure of metal hydrogen systems (Mizusaki et al., 2003, 2005; Yamaguchi
et al., 2007).
     Even complicated ternary hydrides are subject to theoretical studies nowadays.
As an example, hydrides of the iron rich rare earth intermetallics, R2 Fe17 , are given,
where a significant enhancement of the Curie temperature TC is observed which
was claimed to correlate with the rate of the increase in the lattice constant a upon
the introduction of hydrogen (Fujii et al., 1995). This effect is further discussed on
the basis of the electronic band structure. According to the spin fluctuation theory
of Moriya (1987), Lonzarich (1987), Mohn and Wohlfarth (1987), TC is propor-
tional to the inverse of the density of states in the spin-up and spin-down bands at
the Fermi level, N↑ (EF ) and N↓ (EF ). Several experimental results carried out on
R2 Fe17 compounds suggest that the a axis expansion due to the introduction of
interstitial hydrogen into the 9e sites within the dense (001) plane mainly brings
about the reduction of the hybridization between Fe 3d and R 5d states, leading to
a decrease in both N↑ (EF ) and N↓ (EF ). This hybridization reduction might play an
important role in the suppression of the spin fluctuations yielding a rise in TC . Later
on, Beurle and Fähnle (1992) presented a study on hexagonal Y2 Fe17 H3 (Th2 Ni17
type of structure) containing calculations within the framework of the local-spin-
density approximation (LSDA) and the linear-muffin-tin-orbital (LMTO) method
in atomic sphere approximation (ASA). The authors were able to show that there
are two counteracting effects of the interstitial hydrogen: a geometrical effect (vol-
300                                                            G. Wiesinger and G. Hilscher



ume expansion and local relaxation), increasing Fe moment and hyperfine field and
a hybridization effect of the hydrogen atom with the neighboring Fe atoms, re-
ducing these values. Concerning the magnetic moments, the results were found to
be in good agreement with self-consistent augmented spherical wave (ASW) cal-
culations for the rhombohedral Th2 Zn17 phase (Coehoorn and Daalderop, 1992).
Band structure calculations with the LAPW method and the LMTO-ASA method
were used to study the magnetic properties of YFe2 and its hydrides (Singh and
Gupta, 2004 and Crivello and Gupta, 2005, respectively). The magnetic properties
were explained by the two competing effects, mentioned above. In the hydride, the
majority spin states of Fe were found to be fully occupied, the Fermi energy falling
in a peak of the minority spin density of states. The increase in magnetization, ob-
served for limited hydrogen concentration is attributed almost entirely to the lattice
expansion.
    For a comprehensive description of the electronic properties of metal–hydrogen
systems the reader is referred to the reviews of Switendick (1978), Gupta and
Schlapbach (1988), Gupta (2002) and, published quite recently, to chapter 7 in
Fukai’s book on basic bulk properties of metal–hydrogen systems (Fukai, 2005).



      4. Basic Aspects of Magnetism
       Metallic magnetism covers a wide range of phenomena, which are intimately
correlated with both the electronic structure and the metallurgy of a given metal or
compound. Particularly the latter appears to be an important factor when consider-
ing the formation and properties of intermetallic compounds and binary (ternary)
hydrides. Frequently the studies of hydrides of intermetallic compounds have led to
a deeper insight into the fundamental properties of the parent system.
    For quite a long time 3d magnetism has been a controversial topic Wohlfarth,
1980, where still some problems are not completely settled. The reason for this con-
troversy is the absence of a general agreement upon the microscopic nature of the
magnetic state above and below the Curie temperature. Two opposite standpoints
have so far been used to explain the magnetic order as a function of temperature. In
the Heisenberg model magnetism is described in terms of localized moments and
the magnetization vanishes at TC because of disorder in the local moments due to
thermal fluctuations. Nevertheless, their absolute value remains almost independent
of temperature.
    In the Stoner–Wohlfarth itinerant-electron model the magnetic moment is de-
termined by the number of unpaired electrons in the exchange-split spin-up and
spin-down bands. Within this model the thermal excitations of electron-hole pairs
(single particle excitations of the Fermi–Dirac distribution) reduce the exchange
splitting and thus favor the paramagnetic state. Consequently, the magnetization
disappears only if the absolute value of the magnetic moment goes to zero, which
only happens if the exchange splitting is zero, too. This model sufficiently describes
magnetism in metals at 0 K, provided that the band structure and density of states
is known to a sufficient accuracy and electron correlations are not too strong as in
Magnetism of Hydrides                                                                301


heavy fermion materials. For an analysis of experimental data obtained from ternary
hydrides in terms of the Stoner–Wohlfarth theory, see section 5.4.2.4 (Fruchart et
al., 1995).
     Parallel to the development of band structure theory in terms of the density
functional formalism and the local spin density approximation (LSDA) there was a
search for “simple toy-models” as e.g. the Hubbard model to reproduce solid state
magnetism. Thus calculations of the ground state properties with high accuracy and
reliability are now available. Accordingly, the understanding of complex mechanisms
in the solid improved significantly. Among the remaining problems the temperature
dependence of the magnetization was one of the most crucial ones. While for the
localized moment models finite temperature effects became reasonably clear at a
rather early stage this development, however, took some time for itinerant electrons
in most solids since the calculated exchange splitting of the spin up and down bands
was too large. Thus, the corresponding Curie temperatures were also too large by
a factor of 4–8 and the inverse susceptibility is expected to show a T 2 rather than
a linear temperature dependence as frequently observed. It become clear that the
single particle excitations in the Stoner model are not—or only to a small extent—
responsible for the finite temperature behavior of metallic magnetism.
     These results suggest that one has to consider two extreme limits: (i) the localized
limit for which the magnetic moments and their fluctuations are localized in real
space (delocalized in reciprocal space), with their amplitudes being large and fixed;
(ii) the itinerant limit for which the moments and their fluctuations are localized in
reciprocal space (delocalized in real space), with their amplitudes being temperature
dependent. A Curie–Weiss law is observed in both cases, however, its physical origin
and the corresponding Curie constant are different.
     To solve these inconsistencies Moriya (1987) included thermally induced col-
lective excitations of the spin system—as they were already known for localized
spins—in order to formulate an unified picture of magnetism. A similar approach
was introduced by Murata and Doniach (1972) who introduced local and random
classical fluctuations of the spin density (spin fluctuations) which should be excited
thermally. The latter two models become equivalent at high temperatures and lead
to a Curie–Weiss law. Thus, about hundred years after Langevin, there exists a fairly
good knowledge about the basic mechanisms of localized and itinerant electron
magnetism but many open questions still remain and a practical unified picture of
magnetism is still not at hand.
     Magnetism of strongly correlated electron systems (in particular with unstable
4f and 5f moments as e.g. in Ce-, Yb- and U-intermetallics) is still not well un-
derstood but an actual research topic, see e.g. the conference series on Strongly
Correlated Electron Systems (SCES) Physica B 378–380 (2006): One of the goals
of modern condensed matter research is to couple magnetic and electronic prop-
erties to develop new classes of material behavior, such as high temperature super-
conductivity or colossal magneto-resistance, spintronics and the newly discovered
multi-ferroic materials. Strong correlations between electrons lead to a renormal-
ization of the electron mass by a factor of 1000 or more, which are therefore called
heavy Fermions and are usually well described in terms of the Fermi liquid the-
ory. Heavy electron materials lie frequently at the verge of a magnetic instability,
302                                                            G. Wiesinger and G. Hilscher



in a regime where quantum fluctuations of the magnetic and electronic degrees are
strongly coupled and significant deviations from Fermi liquid behavior may occur.
Thus, these materials appear to be an important test-bed for development of the
understanding about the interactions between magnetic and electronic quantum
fluctuations, see e.g.: Coleman (2006) and v. Löhneysen et al. (2006).
    Contrary to the magnetism of the 3d-metals, the magnetic properties of the
“stable” trivalent rare earth (R) elements are unambiguously described in terms of
the RKKY theory; because of the localized nature of the 4f electrons no overlap
exists between 4f wave functions on different lattice sites. Thus, the magnetic
coupling can only proceed indirectly via the spatially non-uniform polarization of
the conduction electrons.
    The pure 4f -4f interaction and its behavior upon the absorption of hydrogen
can be studied directly not only in binary rare-earth hydrides, but also in ternary
hydrides with a zero transition metal moment. As a first approximation one would
expect that hydrogen induced changes in the magnetic properties of the latter can be
explained in analogy with the binary hydrides, i.e. in terms of the anionic model.
There, the conduction electron concentration is lowered after hydrogen uptake
which in turn reduces the RKKY interaction.
    The rare earths form binary hydrides with the stoichiometries x = 2 and x = 3.
In the case of x approaching 3, metallic conductivity disappears, which has been
attributed by Switendick (1978) to the formation of a low-lying s-band with the
capacity to hold six valence electrons. This in fact equals the number of electrons
supplied to the conduction band by one R and three H atoms. Since this low lying
bonding band is completely filled up with electrons, in the RH3 conduction elec-
trons are no longer present, prohibiting the transmission of the RKKY interaction.
This accounts for the suppression of the magnetic interactions, which indeed is gen-
erally observed experimentally. However, as will be seen later, details the physical
properties of the rare-earth hydrides in the α-phase and of the broad homogeneity
range of the R-dihydrides are only partly solved and several exceptions from the
simple approach can be found.
    In R-3d intermetallics and their hydrides, where both the R and the 3d ele-
ment carry a magnetic moment we can distinguish between three main types of
magnetic interactions which are quite different in nature: that (i) between the local-
ized 4f moments; (ii) between the more itinerant 3d moments; and (iii) between
3d and 4f moments. Generally, it is observed that these interactions decrease in the
following sequence: 3d-3d > 4f -3d > 4f -4f . Actually, it is the combined effect
of itinerant 3d electrons (providing a large Curie temperature), and localized 4f
states (providing the magnetocrystalline anisotropy), which frequently make these
compounds suitable for permanent magnet application.
    In contrast to the binary 4f hydrides, for ternary R-3d hydrides no similar
straightforward arguments can be used about the hydrogen induced change of the
magnetic order. The only statement being generally valid is that upon hydrogen
absorption the magnetic order of Co and Ni compounds is considerably weakened
which is not observed in the case of Fe compounds.
    As will be described in detail below, hydrogen absorption usually weakens the
magnetic coupling between the 4f and the 3d moments and can lead to substantial
Magnetism of Hydrides                                                            303


changes of the 3d transition-metal moment in either way. As mentioned earlier,
hydrogen in the lattice reduces the 4f -3d exchange interaction. This is explained
by a reduced overlap of the 3d-electron wave functions with the 5d-like ones due
the narrower bands as a consequence of the hydrogen induced increase in volume.
Furthermore concentration fluctuations of H atoms over a few atomic distances
may frequently occur, leading to a difference in electron concentration between
one site and an other and, therefore, to a varying coupling strength. Additionally, a
disturbance of the lattice periodicity takes place in the hydrides, reducing the mean
free path of the conduction electrons (see section 5.1). This leads to a damping of
the RKKY conduction electron polarization which in turn decreases the magnetic
coupling strength.
    If the magnetic order in R-intermetallics is dominated by the 4f moments, the
concept of an R-H charge transfer in analogy with the binary rare earth hydrides
has proved to be a reasonable explanation for the hydrogen induced changes in
magnetism (see isomer shift data obtained from Mössbauer studies on rare earth
nuclei).
    In the case where 3d magnetism is dominant in the R-3d compounds, no gen-
eral rule can be given. Commonly, hydrogen absorption leads to a loss in the 3d
moment in Ni- and Co-based intermetallics, but to an enhancement of the Fe
moment. As an example, the hydrogen induced change of the magnetic moment
in GdCo2 H4 has been computed by using self-consistent electronic structure cal-
culations, where further the significant drop in TC found experimentally could be
confirmed (Severin et al., 1993). For Mn-intermetallics both changes from para- to
ferromagnetism and vice versa are obtained. In Fe-containing intermetallic hydrides
the 3d states are localized to a greater extent compared to the parent compound.
This leads to an enhancement of the molecular field which, on the other hand, is
opposed by the influence of the grown Fe–Fe distance, tending to reduce it. As it is
observed experimentally, the former is apparently the dominating one, yielding an
increased or at least an unchanged molecular field constant nRFe upon hydrogena-
tion.
    When discussing the hydrogen induced change of the magnetic properties one
is, among other things, faced with the problem of finding confidential moment
data. Frequently, one has to rely on magnetization measurements, which may lead
to wrong results in those cases, where from experimental reasons (lack of a high
field facility) only incomplete saturation has been achieved. As will be seen below,
particularly in the case of ternary hydrides, magnetic saturation is difficult to ob-
tain. The situation, however, has been improved in the last decade, since in rare
cases, ultra high magnetic fields, exceeding 100 T are available. An alternative way
is offered by Mössbauer measurements carried out in zero applied field. However,
the problem of correlating the hyperfine field unambiguously with the magnetic
moment (particularly in the case of the hydrides) still remains. Fortunately, how-
ever, an increasing number of neutron diffraction results have been achieved more
recently, showing that the assumption of a unique hyperfine coupling constant is
an oversimplification. For a detailed summary of neutron diffraction data we refer
to chapter 4 in Hydrogen in Intermetallic Compounds I (Yvon and Fischer, 1988)
and the proceedings of the more recent metal hydrogen conferences (see section 1).
304                                                              G. Wiesinger and G. Hilscher




      5. Review of Experimental and Theoretical Results
5.1 Binary rare-earth hydrides
5.1.1 α-, α * -RHx solid solutions
Hydrogen is readily absorbed by the rare earths (R) and forms solid solutions (α-
phases) at high temperatures. The solubility limits at a certain temperature generally
increase with the atomic number. An extensive review of the situation has been
given by Vajda (1995a). Thus, here we shall only recall the most essential facts with
emphasis put on the magnetic properties.
    The rare-earth–hydrogen (R-H) phase diagram as presented in Fig. 5.3 is gen-
erally valid for the hcp heavy lanthanides Gd through Lu (with some limitations for
Yb) and for Y. It is characterized by relatively broad existence ranges around the
stoichiometric compositions, both in the α-phase solid solutions and for the cubic
β-phase dihydrides and h.c.p. γ -phase trihydrides.
    As concerns the magnetic properties of the rare earths with incomplete 4f -
shells, their interaction with hydrogen is favored by the stability of the single-phase
regions displayed in Fig. 5.3 at low temperatures, where these metals are magneti-
cally ordered. In certain cases a metastable low-T α * -phase occurs (see below). The
(upper) phase boundaries lie in the range between x = 0.03 (Ho) up to x = 0.35
(Sc), those for the β-phase RH(D)2+x between x = 0.03 (Lu) and x = 0.3 (Gd)
(Vajda, 1995a, 2005; Udovic et al., 1999). The lower limits of the β-phase are pu-
rity dependent and reach ideally values close to 2.00; the width of the γ -phase has
not been established definitely in most cases but is of the order of 0.1 H atoms/R.
An important fact following from the particular shape of the phase diagrams is the
tendency of the excess hydrogens, x, in each phase to form sub-lattices at higher
x- and lower T -values, with a strong influence upon the electronic and thus, upon
the magnetic properties.
    As mentioned above, the unusual fact of an existing H solid solution phase down
to the lowest temperatures, without precipitation of the dihydride, has permitted
to study its interaction with magnetism in Ho, Er, and Tm. Thus, it was found
(for details and references, see Vajda, 1995a) that the transition temperatures to
sinusoidal or helical antiferromagnetism (AFM), TN or TH , decreased in all three
metals as well as the TC towards ferrimagnetism in Tm, while TC towards conical
ferromagnetism (FM) in Er increased strongly upon hydrogenation. At the same
time, a kind of magnetic hardening took place manifesting itself e.g. by a decrease
of the critical field needed for the ferri-to-ferromagnetic spin-flip transition in Tm
(Fig. 5.4) or by the increasing spiral period of the helical phase in Ho (Fig. 5.5).
The observations were explained by the competition of several processes: on the
one hand, a diminishing role of the RKKY exchange mechanism due to a decrease
of the carrier density with increasing H concentration and, on the other, by a
simultaneously growing influence of magneto-elastic and anisotropy effects.
    It should be pointed out that the special configuration of the α * -phase, in fact, is
not a real solid solution but-as determined by neutron scattering-consists of H-R-H
pairs on second-neighbor T-sites aligned in modulated quasi-unidimensional chains
along the c-axis (Vajda, 1995a).
Magnetism of Hydrides                                                                    305




Figure 5.3 Generic phase diagram for R-H solubility, valid for bulk specimens, the hydrogen
solubility is usually larger for thin films (Vajda, 2005).




Figure 5.4   Magnetization as a function of applied field for several α * -Tm hydrides (Vajda,
1995a).


    This reminds one of the equally modulated AFM of those rare earths, where
the α * -phase is present. In Fig. 5.5 this parallelism between the spin-density waves
(SDW) in the concerned metals and the charge-density wave (CDW) formed by
the zig-zagging structure of the α * -phase is demonstrated. The occurrence of the
CDW of the α * -phase is related to an electronic topological transition on the Fermi
surface, in particular to its webbing features. The α * -phase was found to form in
systems where a situation is present similar to a Peierls transition. The three metals
306                                                                 G. Wiesinger and G. Hilscher




Figure 5.5 c-axis modulated magnetic configurations (SDW) in Ho, Er and Tm (to the left)
and the modulated H-R-H chain structure (CDW) of the α * -phase (to the right) (Vajda, 2005).


with modulated configurations exhibiting a suitable turning angle ωi are just those
forming an α * -phase, while the equally modulated AFM of Dy and Tb with a
lower ωi do not (Vajda, 2005 and references therein). Theoretical efforts to study
the eventual evolution of the Fermi surface upon introduction of hydrogen have
been undertaken (to begin with) on the non-magnetic YHx system (Garcés et al.,
2005).
     In the heaviest lanthanides (and Sc and Y) which retain hydrogen in solution
(metastable α * phase) down to 0 K no evidence is found of an α–β phase transition,
however, a resistivity anomaly in the range between 150 and 170 K is observed.
This anomaly was attributed to short range ordering of the interstitial H atoms. In
the case of α-LuHx it has been identified by neutron scattering as creation of linear
chains of H–H pairs on tetrahedral sites along the 3c-axis surrounding a metal atom
(Blaschko et al., 1985). Contrary to the heaviest rare earths, hydrogen in solution
appears to be unstable in the lighter R elements below a certain temperature (de-
creasing from ≈700 K to ≈400 K for La to Dy, respectively) and precipitates into
the β-phase (dihydride). α-HoHx , α-ErHx and α-TmHx yield a hydrogen intersti-
tial solubility limit of 3, 7 and 11 at.%, respectively (Daou and Vajda, 1988), which
was previously believed to be lower.
     Hydrogen in solution reduces the antiferromagnetic ordering temperature TN
of Ho (133 K) at a rate of about 2 K/at.% H(D) which is in agreement with the
results obtained for the two other magnetic R-hydrogen systems, α-ErHx and α-
TmHx (Daou et al., 1987). Details of this phenomenon have been described in the
review of Vajda (1995a). These results were confirmed by resonant magnetic X-ray
scattering at the Ho LIII -edge on hydrogenated thin Ho films (Sutter et al., 2001),
where furthermore a hydrogen induced increase of the length of the magnetic spiral
could be observed.
Magnetism of Hydrides                                                                    307


     The effect of hydrogen absorption upon the magnetic properties in α-ErHx has
been studied by resistivity (Daou et al., 1980; Vajda et al., 1987b; Daou and Vajda,
1992), magnetic (Vajda et al., 1983; Ito et al., 1984; Vajda and Daou, 1984; Burger et
al., 1986b; Boukraa et al., 1993a, 1993b) and specific heat measurements (Schmitzer
et al., 1987). Er is well known to exhibit three different magnetic structures: below
the Néel temperature TN of 85 K there is a sinusoidally modulated magnetization
along the c axis, while the basal plane magnetization remains zero. The basal plane
component starts to order in a helicoidal structure at TH = 51 K and spiral (conical)
ferromagnetism is stable below TC = 19.5 K (for a review see Coqblin, 1977). The
above mentioned measurements show that TN and TH decrease, while TC and TC1 (at
which a transition to an incommensurate structure of 15 layers occurs) increase with
H or D content (see Fig. 5.6). The rise of TC is interpreted by Burger et al. (1986b,
1987) in terms of a hydrogen induced dilatation of the c axis as a consequence of
the interplay between the uniaxial anisotropy and the magnetoelastic energy and
the coupling between the axial and basal plane magnetization. The increase in the
electronic specific heat with rising H content (up to 1.5 at.%) indicates a growing
density of states at EF for Er in the α-phase (at least at low concentrations), which
leads to the suggestion that the decrease of TN , TH and the spin disorder resistivity
ρspd is due to a reduction in the exchange interaction in agreement with magnetic
measurements. The thermal variation of the resistivity and the specific heat in the
ferromagnetic cone structure regime (below TC ) gives evidence for the existence of
a gap like behavior in the spin wave spectrum, which is reduced with the addition
of hydrogen in solution. From this variation of the exchange anisotropy gap and the
nuclear specific heat Schmitzer et al. (1987) and Vajda et al. (1987b) draw the same
conclusion as above, namely that the exchange field is reduced with rising hydrogen
content in α-ErHx .
     Contrary to α-ErHx , for α-TmHx both TN (57.5 K) and the order-order
transition at TC (39.5 K) decrease with rising H content down to 45.5 K and
29 K respectively for x = 0.1 (Daou et al., 1981; Vajda and Daou, 1984;
Vajda et al., 1989c). This different behavior is suggested to arise from the specific
magnetic structures, which is in the case of Tm sinusoidally modulated antiferro-
magnetic below TN and gradually squares up to yield below TC = 39 K an antiphase
ferromagnet with three spins up and four spins down and has therefore no basal
plane component in contrast to Er below TC . The fall of TN and TC in α-TmHx is
governed also by the reduction of the indirect exchange interaction. In Tm, there
is no interaction present between the short range ordered H–H pairs and the uni-
axially aligned moments.
     On the other hand, in α-ErHx a strong interaction between the H–H pairs
and the conical structure appears to change the magnetoelastic energy giving rise
to a TC enhancement in Er with H in solution. In Fig. 5.7 a global view of the
temperature dependence of the electrical resistivity of an α-TmHx single crystal
parallel to the b and c axis is presented, showing a significantly different ρ(T )
behavior between the two crystal orientations with x. While the b axis oriented
crystal exhibits a nearly linear increase of the residual resistivity ρrb with x (Fig. 5.7a),
the apparent increase of ρrc (ρ parallel to the c axis) is much larger and nonlinear.
In fact the resistivity decrease due to ferrimagnetic ordering is strongly suppressed
308                                                                G. Wiesinger and G. Hilscher




Figure 5.6 (a) Variation of the magnetization with temperature for pure Er (x = 0) and
ErH0.035 . A field of H = 0.02 T is applied parallel to the c axis [(o) x = 0, (+) x = 0.035]
(Burger et al., 1986b). (b) Heat capacity of α-ErHx with various hydrogen concentrations:
(!) x = 0; (E) x = 0.01; (1) x = 0.03 and (2) ErD0.03 (Schmitzer et al., 1987).
Magnetism of Hydrides                                                                      309




Figure 5.7 Temperature dependence of the resistivity parallel to the b axis for various α-TmHx
crystals with the x values labeled; the insert shows the magnified low-temperature region. The
arrows indicate the high-temperature anomaly above 150 K as well as the variation of the Néel
temperature TN . (b) The same as (a) for single crystals with a c axis orientation. Both figures
after Daou et al. (1988b).



by hydrogen in solution, disappearing completely for x > 0.05, which is attributed
to the evolution of magnetic superzones, a phenomenon already observed to a
smaller degree in single crystals of α-ErHx (Vajda et al., 1987b). The insert of
Fig. 5.7a shows a substantial rise of the magnetic contribution to the resistivity
310                                                            G. Wiesinger and G. Hilscher



ρmag (T ) with x. The analysis of these data in terms of a sum of a power function
and an exponential expression for the anisotropy gap

                       ρmag (T ) = AT n + BT 2 exp(– /kT)
                        b


shows that both n and decrease with rising x. Specific heat measurements show a
similar trend (Daou et al., 1988b), namely that an anisotropy gap occurs in the spin
wave spectrum which is reduced with growing amount of hydrogen and goes hand
in hand with the development of a complex magnetic structure below 4 K.
     Of the magnetically disordered elements Sc, Y and Lu, the H-system of the latter
has been investigated thoroughly by specific heat and susceptibility measurements
(Stierman and Gschneidner, 1984). These authors state that Lu is a spin fluctuation
system, where the fluctuations are quickly degraded by impurities and by hydrogen
in solid solution. Both the susceptibility χ and the electronic specific heat coeffi-
cient γ show a similar variation as a function of H composition yielding a peak at 3
at.% and 1.5 at.% H, respectively. This difference is attributed to hydrogen tunneling
giving rise to a linear contribution to the heat capacity. Correcting for this brings
into good agreement the concentration dependence of both γ and χ with a peak at
about 3% H. In this context it is worth to note that also in α-ErHx and presumably
in α-TmHx an increase of the electronic specific heat with x is observed. However,
in α-TmHx the increase of γ with growing x is not established for x > 0.02, since
in this regime the magnetic contribution to the heat capacity could not unambigu-
ously be resolved, a not yet determined magnetic structure occurring below 4 K
(Daou et al., 1990).
     Susceptibility measurements of α-ScHx by Volkenshtein et al. (1983) indicate
that the spin paramagnetism is reduced by a factor of two for x = 0.36. This can be
associated (neglecting a possible change of the Stoner enhancement factor) with the
decrease of the density of states at EF , whereby EF passes through a maximum of
the N(E) curve down to lower energies. This trend of the α-phase is also observed
in the dihydride. According to band structure calculations, susceptibility and spin-
lattice relaxation time measurements the DOS at EF in comparison with that of
parent Sc is reduced by a factor of 3.5, 3 and 4.5, respectively.

5.1.2 Rare-earth dihydrides
The rare earths commonly form dihydrides and trihydrides. The dihydrides ex-
hibit a broad homogeneity range and crystallize except for Eu and Yb in the CaF2
structure. The CaF2 structure forms a f.c.c. unit cell, where in the ideal case all
tetrahedral (T) sites are occupied. On increasing the hydrogen content, the octa-
hedral (O) sites become gradually filled up with hydrogen atoms to form the BiF3
structure. Dihydrides of Eu and Yb are of the orthorhombic (Pnma) structure.
    In reality, the “pure” dihydride is frequently substoichiometric. The occupation
of the octahedral (O) sites already starts sometimes for x = 1.8, depending on
the material purity. In particular the oxygen content is of importance and also
the sample shape (foils, powder etc.) used for hydrogen loading. It seems that the
larger the purity of the parent material, the closer is the approach to the ideal
stoichiometry of the dihydride. In the case of heavy rare earth with a purity of
Magnetism of Hydrides                                                              311


99.9% and 99.99%, the stoichiometry of the dihydride is usually 1.90 < x < 1.95
and 1.96 < x < 2.0 (Vajda, 2000, 2004), respectively.
     The absorption of hydrogen affects the magnetic properties of the rare earths
indirectly via a reduction of the number of conduction electrons and a volume
expansion. Both effects lead to a drastic decrease of the RKKY indirect exchange
interaction between the localized 4f electrons mediated by the conduction elec-
trons. Consequently, the magnetic ordering temperatures are much lower than in
the parent metals (e.g. TC = 291 K for Gd, TN = 20 K for GdH1.93 ). On raising the
hydrogen content above the pure dihydride, it is obvious that a random and/or or-
dered occupation of the octahedral sites with hydrogen significantly influences the
crystalline electric field (CEF). This plays an important role in the change of the
magnetic properties of those compounds located in the intermediate range between
the di- and trihydrides.
     The rare earth dihydride acts as a monovalent metal (one conduction electron
per atom), the trihydride as an insulator or a semiconductor. Whereas the electronic
structure and the magnetic properties of stoichiometric dihydrides and trihydrides
seem to be rather well understood, considerable confusion exists in the intermediate
composition range (–0.15 to –0.05 < x < 1 of REH2+x ). To elucidate the transi-
tion between these two extreme situations this regime became therefore of growing
interest about three decades ago, the research activities still ongoing. Simplifying
the matter, one may expect a continuous decrease of the conduction electron den-
sity with rising x, implying that each H atom depopulates the conduction band
by one electron through the formation of a low-energy metal–H band. However,
this simple model is complicated by several structural transitions: attractive H–H in-
teractions lead to a phase segregation with the formation of a dilute metallic phase
(x = 0.1–0.2) and a concentrated nearly insulating phase or γ phase (x = 0.8–0.9).
This seems to be the case in the heavy rare earths (R = Gd