Domain Nucleation and Wall Motion in Magnetically Frustrated Bilayers
Introduction
Professor Robert Stamps (RLS) is a leading theoretician, currently based at the University of
Western Australia. He has made substantial contributions to our understanding of magnetisation
reversal in magnetic thin films. Much of his recent work has been in connection with exchange-
biased bilayers. The purpose of EPSRC grant GR/T07053/01 was to allow RLS to spend time at
the University of Glasgow to:
further develop his existing model of exchange bias;
develop a model for magnetisation reversal in ferromagnetic/antiferromagnetic systems
that took account of the phenomenon of magnetisation ripple;
compare his predictions with experimental results obtained using Lorentz microscopy at
the University of Glasgow.
During his period in the UK he split his time equally between the Universities of Glasgow and
Leeds, hosted at the latter by Professor Denis Greig. His time at Leeds was supported by a
parallel grant from EPSRC. The report that follows focuses on the research undertaken in
Glasgow.
RLS was in Glasgow during May and June 2004. He revisited Glasgow in December 2004, whilst
a further interchange of results and ideas took place at an EU summer school on magnetism in
Anglet during September 2004.
Background
It has been well known for 40 years that ferromagnetic ordering deforms into ‘magnetisation
ripple’ in response to locally random fields [1]. Spatially random fields arise naturally, for
example, in polycrystalline films for which the orientation of magnetic anisotropy axes changes
from one region to another according to the local orientation of crystallites. The experimental
study of how ripple evolves during the magnetisation process thereby provides unique
information on how ferromagnetic order develops in the presence of random fields. Ripple then is
an important topic for applications in which hysteresis needs to be understood and controlled.
Experimentally our knowledge of magnetisation ripple is based on observations made by Lorentz
microscopy. Ripple strength and coherence are two features that can be identified in the ripple
spectrum, and there is a need to explore how these features can be interpreted in terms of
different magnetisation processes determined by intergranular exchange. In particular,
experiment shows that the ripple spectrum of a ferromagnet is strongly modified by interlayer
coupling to an antiferromagnet with random anisotropy axes [2]. In this case the ripple patterns
can be different for the outward and return branches of a hysteresis loop in a manner that can
depend strongly on the magnitude of intergranular exchange within the antiferromagnet. Indeed,
the ripple spectrum strongly influences the domains that develop and hence the magnetisation
reversal mechanism itself. The new model developed over the summer by RLS allowed us to
explore the relationship between the various phenomena described briefly in this Background
section.
Outcomes of the Research
(i) The model developed
A numerically efficient model for simulating essential features of magnetisation ripple in a thin
film during field cycling was constructed in the following way. The film was represented by a
two dimensional array of elements, each with a vector mn representing the orientation of the
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element’s magnetisation. Each element was identified by a position vector n. For simplicity, a
square array was assumed, and the magnetisation and volume of each element in the array
assumed to be independent of position.
The magnetisation of each element was required to align parallel to a local effective field Hn. The
effective field was the sum of an applied magnetic field Ho, a local uniaxial anisotropy field Hn(2),
a field representing intergranular exchange coupling to neighbouring elements with magnitude
determined by F, and a dipolar field produced by the magnetisation of all other ferromagnetic
elements in the film. The magnetisation of each element was approximated as a single point
dipole at the location specified by the position vector. Defining the position vector between
elements n and m as dnm, the total local effective field is:
M
m dnm m
H n xH o H(2) F mn d m 5 dnm m 3
ˆ n 3
mn
dnm dnm .
Periodic boundary conditions were used. A simplifying approximation for the dipolar fields was
made by restricting the sum over m to the nearest 5 neighbours. This reduced overall computation
time greatly and simplified application of periodic boundary conditions. The effect of this
approximation was to include only contributions of the dipolar field that had short scale spatial
variations. As will be discussed below, the short scale structure of the ripple determines
coherence in the ripple pattern. The use of a truncated dipole sum was able to reproduce this
feature. A disadvantage to this approximation was that it was not possible to give an accurate
estimate of the magnitude of d in terms of shape demagnetizing factors and the saturation
magnetisation.
A minimum energy configuration of magnetic element orientations was obtained by rotating each
mn in the array into the direction of Hn and iterating until a steady state configuration emerged. A
system size of 720 elements was used for the results presented later in this report and was
sufficiently large to mimic ripple contrast and coherence for domain wall formation in exchange
coupled grains.
As noted in the Background, ripple appears when long range order, controlled by exchange and
dipolar interactions, competes with short range disorder in the local effective fields. Disorder was
introduced by assuming a distribution of anisotropy axis orientations. Each element was assigned
an anisotropy axis direction, and the local field was rotated into a common film coordinate frame
of reference. The rotation was performed such that the anisotropy energy Ea was independent of
coordinate system.
Contrast in the Fresnel mode of Lorentz microscopy appears because of the way the local
magnetisation deflects the electron beam as it passes through the sample [3]. The result at the
image plane was estimated from the above model by forming a finite difference between
neighbouring elements in the following way:
Cn x ny mn y y mn x mnx
ˆ m ˆ
.
The quantity Cn is interpreted as the amplitude of the signal at location n due to electrons passing
through the four elements in the film neighbouring element n. In our notation, these elements
were identified as the n+x, nx, ny and ny elements.
Using the model we explored how ripple contrast is influenced by interactions between grains in
the case of thin single layer films and exchange-coupled bilayers. The average magnetisation m
2
was taken as the average component of mn along the applied field direction, normalised to MS,
whilst contrast is presented as c, the average of Cn, again normalised to MS.
(ii) Magnetisation reversal and dispersion in single layer films
Figure 1 shows how the contrast predicted by the model varies spatially with applied field
through one half of a magnetisation cycle for a single layer film; figure 1a shows the ripple
patterns for zero dipolar and exchange interactions and figure 1b shows the ripple patterns for a
non-zero interaction strength. The effect of the interactions is to align the contrast in the ripple
pattern leading to significant coherence. We associate coherence in the pattern with this
alignment. A stronger interaction creates closer alignment of the oscillations in contrast across
greater distances. Figure 2 shows a sequence of Fresnel images that show qualitatively similar
features to those observed in figure 1b. At this stage we have not progressed sufficiently far to
undertake a quantitative comparison but we are able to explore the effect of varying some of the
key parameters in the model.
To illustrate this, figure 3a shows the effect of varying the random anisotropy on the form taken
by the hysteresis loop whilst figure 3b plots the corresponding variation of c. Figure 3a is
symmetrical, as would be expected for a single layer film, but we draw attention to this point as
the same is not true for the exchange-biased bilayers discussed in the following sub-section. An
increase in is associated with increasing randomness. That the coercivity decreases smoothly
with increasing randomness is expected because large magnetisation dispersion facilitates
reversal at smaller applied field values.
Experimentally determined values of c, obtained by processing the Fresnel image sequence
shown in figure 2 are displayed in figure 4. It can be seen once again that the principal features
are well reproduced by the model. Note the greatly enhanced dispersion that occurs just before
the magnetisation reverses abruptly.
(iii) Magnetisation reversal and dispersion in exchange-biased bilayers
Exchange coupling between grains in the ferromagnetic film to grains in an adjacent
antiferromagnetic film can lead to induced anisotropies in the ferromagnet. One manifestation of
this is the shifted hysteresis loop. The model described above has been extended to include a
description of exchange coupling between ferro- and antiferromagnetic grains. In essence
expressions have been derived for the effective fields acting on the ferromagnetic and the
antiferromagnetic elements in the model. Steady state configurations of the mn were found in the
same manner as before by rotating each element of the ferromagnet and antiferromagnet into the
direction of their respective local fields. The entire array of coupled elements was then iterated
until a convergent configuration was found. In the examples that follow, anisotropies in the
ferromagnet were set to zero. The anisotropy axes of the antiferromagnet were chosen randomly
from a uniform distribution across all possible angles. The orientations of the antiferromagnetic
vectors were initially chosen at random according to a uniform distribution across all possible in-
plane orientations. Field cooling was next simulated by allowing the system to relax into a steady
state configuration at a field large enough to saturate the ferromagnet. Hysteresis loops were
calculated after this ‘cooling’ procedure.
An example of how the magnetisation reversal takes place over half a magnetisation cycle and
the corresponding hysteresis loop are shown in figures 5 and 6 respectively. Whilst ripple
contrast is once again visible, the ripple pattern shows a marked lack of coherence when
compared to the equivalent single layer film. Moreover, the domain structure that develops is on
a much smaller scale, comparable to the ripple wavelength. Both predictions are in excellent
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qualitative agreement with experiment [2] and are quite unlike what is calculated or observed
when the antiferromagnet is not present. Typical experimental results are shown in figure 7. As
well as the model showing a shift to the hysteresis loop, the loop itself is asymmetric. That the
reversal path on the outward and return parts of the hystesis loop differ is again in accord with
experimental observation [4].
Project plan review
Despite the comparatively short time that RLS was able to spend in Glasgow, the majority of the
objectives of this small project were met or surpassed. We anticipate that the collaboration will
continue with effort being devoted to refining the model, especially the magnetostatic term, and
to examining more closely the extent to which the model succeeds in reproducing quantitatively
what is observed experimentally. The research conducted at Leeds has also been extremely
successful and steps are now underway to use the analytical TEM facilities at Glasgow to study
the uniformity of the spin-glasses deposited in Leeds (for details of the spin-glasses, see the
corresponding final report from Leeds). If clustering of Mn atoms is found, this will be an
important step in validating a further model developed by RLS.
Research impact, benefits to society and dissemination
The recent discovery that use of very high applied fields during film deposition can induce
significant anisotropy in soft thin films has re-awakened an interest in magnetisation ripple and
the role it may play in contributing to noise in, for example, magnetic logic devices [5]. In
addition, exchange-biased bilayers, though still very imperfectly understood, continue to be used
in an ever-increasing range of magnetic devices making studies such as this that link exchange-
bias and ripple potentially of great scientific importance. That excellent qualitative agreement
with experiment has been demonstrated provides incentive for further development of the model
to assess its potential for making quantitative predictions. Immediate beneficiaries are those with
an interest in understanding how the properties of thin films can be controlled and, as such, span
academia and industry. The industrial sectors to whom this work is most relevant are those
concerned with magnetic information storage, though other sectors employing high performance
sensors (aerospace, security, automotive) are also likely to be interested. At the time of writing, a
paper is under preparation, some of the results have been presented at an EU School (Anglet –
09/04) - at which exchange-biasing was a major topic), and in seminars; moreover the results
were discussed briefly with colleagues from Seagate during a visit in 01/05.
Expenditure
Professor Stamps did not require any additional salary so the financial requirements were modest.
A contribution was made to his internal and overseas travel, his accommodation and living
expenses whilst he was based in Glasgow and to the costs of preparing and characterising the
samples used to validate the predictions of the model he developed. This was in accord with the
proposal.
References:
[1] K J Harte, J. Appl. Phys. 39, 1503, (1968).
[2] P Gogol, J N Chapman, M F Gillies, F W M Vanhelmont, J. Appl. Phys. 92, 1458, (2002).
[3] J N Chapman and M R Scheinfein, J Magn. Magn. Mat. 200, 729, (1999).
[4] J P King, J N Chapman, M F Gillies, J C S Kools, J. Phys. D: Appl. Phys. 34, 528, (2001).
[5] D. A. Allwood, Gang Xiong, M. D. Cooke, C. C. Faulkner, D. Atkinson, N. Vernier,
R. P. Cowburn, Science, 296 2003, (2002)
4
1 2
-10.6 Oe -1.2 Oe 2.4 Oe
H
4 µm
4.7 Oe 5.9 Oe 44.6 Oe
Figure 1. Modelled ripple contrast during reversal of magnetisation in a single
layer film with zero and weak coupling.
Figure 2. Experimental Fresnel images of reversal in a single layer NiFe film.
3(a) (b)
Figure 3a. Modelled magnetszation as a
function of field for different distributions of
uniaxial anisotropy axes (variance ).
Figure 3b. Modelled Lorentz contrast c as a
function of applied field for different .
4
0.16
0.14 Figure 4. Lorentz contrast c as a
function of applied field for the
c
experimental images in fig 2.
0.12
0.1
0.08
-25 -15 -5 5 15 25
H applied (Oe) 5
5 6
Figure 5. Modelled ripple contrast for an exchange coupled
ferromagnet/antiferromagnet bilayer during reversal of
magnetisation in the ferromagnet. Results are shown for two
interlayer coupling JI strengths.
Figure 6. Modelled magnetisation as a function of field for
different interlayer coupling strengths
7 39 Oe 241 Oe 320 Oe
H
Figure 7. Experimental Fresnel
15 µm images of reversal in a CoFe
layer exchange biased by an
358 Oe 453 Oe 544 Oe antiferromagnetic IrMn layer.
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