Lecture 3

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					3. Theoretical picture: magnetic impurities, Zener model,
mean-field theory

 DMS: Basic theoretical picture
  • Transition-metal ions in II-VI and III-V DMS

  • Higher concentrations of Mn in II-VI and III-V DMS
  • The “Standard Model” of DMS
  • DMS in weak doping limit
             DMS: Basic theoretical picture
We follow T. Dietl, Ferromagnetic semiconductors, Semicond. Sci.
Technol. 17, 377 (2002) and J. König et al., cond-mat/0111314

Consider the (by now) standard system:
Mn-doped III-V DMS (excluding wide-gap),
e.g., (Ga,Mn)As, (In,Mn)As, (In,Mn)Sb

 understand mechanism of ferromagnetic ordering
 learn where to look for desired properties:
 • high Tc
 • high mobility
 • strong coupling between carriers and spins
Transition-metal ions in II-VI and III-V DMS

        M2+ ! M1+
                                                M3+ ! M2+

                                            M3+ ! M4+
        M2+ ! M3+

    II-VI                                      III-V
                        Three cases (here for donors):
(a) level in the gap:    (b) level above CB bottom:         (c) level below VB top:
deep donor (d-like)      autoionization                     irrelevant for semi-
                         → hydrogenic donor (s-like)        conducting properties

 CB                           CB
Mn in II-VI semiconductors: no levels in gap, stable Mn2+ (half filled)
→ only introduces spin 5/2, no carriers

Mn in III-V semiconductors: acceptor level below VB top (hole picture!)
→ hydrogenic acceptor level
Mn3+ becomes Mn2+ (spin 5/2) + weakly bound hole
(experimental binding energy: 112 meV)

       Controversial in III-N and III-P, may be deep acceptor

Interaction between Mn2+ and holes consists of

 Coulomb attraction (accounts for ~ 86 meV)
 exchange interaction from canonical (Schrieffer-
  Wolff) transformation
antiferromagnetic, in agreement with experiment
Ab-initio calculations for Mn in DMS:
Density functional theory starts from Hohenberg-Kohn (1964) theorem:
For given electron-electron interaction (Coulomb) the potential V (due to
nuclei etc.) and thus the Hamiltonian and all properties of the system are
determined by the ground-state electronic density n0(r) alone.

Now write the energy E[n(r)] as a functional of density n(r) for given V.
Can show that E is minimized by n = n0.

E[n] is not known → approximations

Local density approximation (LDA):
Unknown (exchange-correlation) term in E[n] is written as

partially neglects correlations between electrons

Local spin density approximation (LSDA): keep full spin density s(r)
      (Ga,Mn)As with 3.125% Mn: typical results

d orbitals                              Mn d-orbital weight at EF, VB top,
                                        CB bottom: not seen in photoemission

                                       LDA+U: phenomenological incorporation
                                       of Hubbard U in d orbitals

                                        Mn d-orbital weight shifted away from
                                        EF, better agreement
                                        similar results from other methods
                                         going beyond LSDA: GGA, SIC-LDA

         Wierzbowska et al., PRB 70,
         235209 (2004)
Higher concentrations of Mn in II-VI and III-V DMS

 no carriers (II-VI): short-range antiferromagnetic superexchange
  → paramagnetic at low Mn concentration x, spin-glass at higher x
 with holes (not fully compensated III-V):

 low x          →          holes bound to acceptors, hopping
 intermediate x →          …overlap to form impurity band
 high x         →          …merges with valence band

 MBE growth also introduces compensating donors:
 antisites AsGa and interstitials Mni

                                                  Big question:
                                     What is “low”, “intermediate”, and “high”
                                                 for (Ga,Mn)As?

                                       Governed by Mn separation nMn–1/3
                                       vs. acceptor effective Bohr radius aB
Experimental evidence for holes with VB character in III-As and III-Sb:

 metallic conduction at low T, not thermally activated hopping
 high-field Hall effect
 Photoemission: anion p-orbital character
 Raman scattering
 very-high-field (500 T) cyclotron resonance of VB holes, not d-like
  Matsuda et al., PRB 70, 195211 (2004)

But does not fully rule out a separate impurity band of hydrogenic states

Experimental evidence that VB holes couple to impurity spins:
 large anomalous Hall effect
 spin-split VB, leading to large magnetoresistance effects

Consider the high-concentration case first
The “Standard Model” of DMS (T. Dietl, A.H. MacDonald et al.)
Step 1: Zener model [Zener, Phys. Rev. 83, 299 (1951)]
In terms of VB holes and impurity spins – here for single parabolic band:

                          hole spin 1/2    impurity spin 5/2

                                 hole position     impurity position

 canonical transformation really gives scattering form
 …and is not local
 no potential scattering – disorder only from exchange term
 (unrealistic band structure – can be improved)
The first (band) term can be improved to get a realistic band structure
Two main approaches:
(1) Kohn-Luttinger k ¢ p theory
(2) Slater-Koster tight-binding theory

(1) Kohn-Luttinger k ¢ p theory
Luttinger & Kohn, PR 97, 869 (1955)
Without spin-orbit coupling (now for single hole):

                                                           periodic part

Write wave function in Bloch form:
 treat k ¢ p term as small perturbation (valid if only small k are relevant)
 degenerate perturbation theory up to second order:
  if ground state is N-fold degenerate the Hamiltonian is, to 2nd order,

6-band Kohn-Luttinger Hamiltonian for VB top (still no spin-orbit):
3 periodic functions uk with p-orbital symmetry (one nodal plane per site)

Cannot calculate A, B, C precisely due to electron-electron interaction
→ treat as fitting parameter to actual band structure close to  (k = 0)
With spin-orbit coupling: treat

similarly. Obtain 6-band Hamiltonian:

                                               components are bilinear in ki
                                               Abolfath et al., PRB 63, 054418

 Fermi surface, Dietl et al. (2000)
 correctly gives heavy-hole, light-hole, split-off bands
 respects point-group of crystal
 only for region close to                                        
Spherical approximation for p-type semiconductors
(G. Zaránd, A.H. MacDonald etc.)
For light and heavy holes only: 4-band approximation

for heavy (–) and light (+) holes
average over all angles:

                 hole total angular momentum

   Spherical approximation
heavy holes:                    light holes:

Reasonable at small doping for some quantities
(2) Slater-Koster tight-binding theory
Slater & Koster, PR 94, 1498 (1954), for GaAs: Chadi, PRB 16, 790 (1977)
 tight-binding theory: consider atomic orbitals, express h1|H|2i, i.e.
  hopping matrix elements t, by 2- and 3-center integrals

 these integrals are not correct – no electron-electron interaction
 thus view them as fitting parameters: choose to fit the resulting band
  structure to known energies, usually at high-symmetry points in k space
                                                  respects full symmetry
                                                   (space group)

                                                 Chadi (1977): with only NN
                                                 hopping (few parameters) quite
                                                 good description of VB,
                                                 including spin-orbit coupling
Motivation for following steps: RKKY interaction
Idea: In the Zener model, impurity spins polarize the carriers by means of
the exchange interaction. Other impurity spins are aligned by this
polarization → interaction between impurity spins
         Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction

 localized impurity spin S → acts like magnetic field B(q) ~ S

 induces hole magnetization m(q) = (q) B(q)

 (q) from perturbation theory of 1st order for eigenstates (complicated
  integral over k vector of states |ki)
                              G(k + q,)

      (q) =       s’        s ’

                             G’(k,)          unperturbed Green function
 for single parabolic band:

                                 at 2kF

                               Anomaly at 2kF from scattering between locally
                               parallel portions of the Fermi surface
Hole magnetization in real space: Fourier transform



                                                        Friedel oscillations


Oscillating and decaying magnetization around impurity spin, leads to:
Interaction oscillates on length scale 1/2kF = F/2
What do we expect?
                                                           first zero
 If typical impurity separation ¿ 1/2kF:

Many neighboring impurity spins within first ferromagnetic maximum,
weaker alternating interaction at larger distances → ferromagnetism
 If typical separation > 1/2kF:


ferro-, antiferromagnetic interactions equally common → no long-range order
Step 2: Virtual crystal approximation
Replace impurity spins by smooth spin density

Ignores all disorder
 valid in stongly metallic regime (high x)
 …but not for all quantities (e.g., not for resistivity)
 requires impurity separation < 1/2kF (see RKKY interaction)
Step 3: Mean-field approximation
Hole spins only see averaged impurity spins and vice versa.
In homogeneous system: M(ri) = ni S

Selfconsistent solution:
Impurity spins:

Hole spins, assuming a parabolic band:
                                                spin- hole density:


Assuming weak effective field: EZ ¿ EF

Obtain Tc: linearize Brillouin function


                                          = 1 at Curie temperature
Gives mean-field Curie temperature

where N(0) is the density of states at the Fermi energy
(one spin direction)
For weak compensation nh ¼ ni, then Tc » ni4/3

Compare expriment:
Ohno, JMMM 200, 110 (1999)

Beyond simple parabolic band: result for Tc remains valid

 enhancement of Tc by ferromagnetic (Stoner) interactions of VB holes:
  Fermi liquid factor AF » 1.2 (from LSDA)

 reduction of Tc by short-range antiferromagnetic superexchange:
 correction term –TAFM (very small in III-V DMS, but not in II-VI)

 ni is the concentration of active magnetic impurities (not interstitials etc.)

Dietl et al., PRB 55, R3347 (1997); Science 287, 1019 (2000) etc.
but in our notation

Dietl et al. (1997) showed that this theory is equivalent to writing down a
Heisenberg-type model with interactions calculated from RKKY theory and
applying a mean-field approximation to that
Results for group-IV, III-V, and II-VI host semiconductors:

5% of cations replaced by Mn (2.5% of atoms for group-IV)
hole concentration nh = 3.5 £ 1020 cm-3

  experimentally confirmed

                                                         Mn replaces C2,
                              ☻                          low spin, deep level
                                                         → no DMS?
                              ☻             ?            Erwin et al. (2003)


Dietl, cond-mat/0408561 etc.
Magnetization: Numerical solution of equations for |hSi| and |hsi|,
parabolic band

                                                           Note that system
                                                           parameters only
                                                           enter through S
                                                           and Tc

                                                           All curves for Mn-
                                                           doped samples (S
                                                           = 5/2) should
                                                           collapse onto one
                                                           curve – but don‘t
Magnetization: numerical solution of equations for |hSi| and |hsi|

For k ¢ p Hamiltonian:

                                                         Curves become more
                                                         for increasing nh

Dietl et al., PRB 63, 195205 (2001)
Experiments well explained within k ¢ p/Zener/VCA/MF theory

 order of magnitude of Tc
 optical conductivity
 photoemission (partly)
 X-ray magnetic circular dichroism
 magnetic anisotropy & strain
 anomalous Hall effect – perhaps not for (In,Mn)Sb

Experiments that cannot be explained
 (change of) shape of magnetization curves → Lecture 5

 weak localization & metal-insulator transition → Lecture 4
 critical behavior of resistivity → Lecture 5
 photoemission: appearance of flat band
 giant magnetic moments in (Ga,Gd)N → Lecture 5
DMS in weak-doping limit (R. Bhatt et al.)
Step 1: Zener model for hopping between localized acceptor levels,
hole spin aligned (in antiparallel, Jpd<0) to impurity spin (bound magnetic
Valid if acceptor Bohr radius aB is small compared to typical separation

Bhatt, PRB 24, 3630 (1981):

Jij also decays exponentially on scale aB

Step 2: Mean-field approximation
Similar to band model but with position-dependent effective field

Step 3: Impurity average (or large system)
Advantage: takes disorder into account

 mean-field Tc determined by strongest coupling,
  real Tc determined by weak couping between clusters (percolation)
 only for very small concentrations x ¿ 1%
  [applied incorrectly by Berciu and Bhatt, PRL 87, 107203 (2001)]
 Upper limit for impurity concentration: Width of impurity band must be
 small compared to acceptor binding energy (band does not overlap VB)

                                                For x ~ few percent:
                                                exceedingly broad “IB”,
                                                merged with VB (and CB!)
                                                C.T. et al., PRL 90, 029701

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