Journal of Magnetism and Magnetic Materials 240 (2002) 469471
Simulation of domain formation and domain coarsening in
antiferromagnetic multilayers
M. Major*, L. Botty!an, D.L. Nagy
KFKI, Research Institute for Particle and Nuclear Physics, P.O.B. 49, H1525 Budapest, Hungary
Abstract
A Monte Carlo simulation of patch-domain formation and domain coarsening in antiferromagnetically coupled
compensated multilayers with fourfold in-plane anisotropy is presented. The simulation accounts for both the
emergence of small patch domains on unsaturation and the domain coarsening on spin flop. The autocorrelation
function of the simulated domain pattern is in good agreement with published Kerr microscopic images. r 2002
Elsevier Science B.V. All rights reserved.
Keywords: Domain growth; Domain pattern variations; Multilayers; Magnetic couplingFantiferromagnetic
Magnetic multilayers have been for a long time in the `unsaturation' region, when decreasing the field from
focus of interest due to their novel material properties saturation and (b) the spin-flop region when an
and diverse possible technological applications. The increasing field parallel/antiparallel to the layer magne-
antiferromagnetically (AF) coupled Fe/Cr system, for tisations results in a bulk spin-flop of the multilayer
example, shows the giant magnetoresistance (GMR) [4,6].
effect [1]. The GMR noise can be influenced by the In a strongly AF-coupled metallic multilayer, the
domain-size distribution of the multilayer [2]. The domain structure of the individual ferromagnetic layers
theoretically predicted [3] bulk spin-flop transition was is rather strictly correlated through the multilayer stack
first verified experimentally on a strongly AF-coupled from the substrate to the surface allowing for a two-
Fe/Cr epitaxial multilayer (superlattice, SL) with four- dimensional representation, e.g. by the domains of the
fold in-plane anisotropy by Temst et al. [4]. Recently, on topmost magnetic layer. In the cited SMR and PNR
an equivalent Fe/Cr system, a reproducible domain-size- experiments, the investigated multilayer (MgO(0 0 1)/
switching mechanism (domain-size coarsening at the [57Fe(26 (A)/Cr(13 (A)]20), had a saturation field of
spin-flop transition) was found by off-specular Synchro- approximately 700 kA m 1 [5,6]. In such a strongly
tron M.ossbauer Reflectometry (SMR) and was verified coupled multilayer, the unsaturation domain formation
by Polarised Neutron Reflectometry (PNR) [5]. In the is governed by the AF coupling and results in a
present paper, a simple phenomenological model and characteristic patch-domain pattern in remanence [7].
results of Monte Carlo simulations of the domain- This state will be called, henceforth, the primary domain
nucleation process and the domain coarsening in a state (PDS) of the multilayer. A so-called secondary
magnetic SL are presented. The studied AF-coupled SL domain state (SDS) is characteristic of a sample which
has even number of magnetic layers of equal thickness. has passed the spin-flop transition (in our particular case
The external field is applied along one of the two, at approximately 10 kA m 1 [6]) following a saturation.
mutually perpendicular easy axes. Two critical regions In the SDS, majority large and minority small domains
of the domain formation are investigated: (a) the were observed [5]. In our model [5], we associate the
domain formation and coarsening with the correlation
length of the saturation field and that of the spin-flop
*Corresponding author. Tel.: +36-1-392-2760; fax: +36-1-
395-9151. field, being related to the unavoidable variation of the
E-mail address: major@rmki.kfki.hu (M. Major). thickness of Cr spacer and the Fe layers, respectively.
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 9 0 2 - 7
470 M. Major et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 469471
We assume the absolute thickness variations of Cr and
Fe to be equivalent. Due to the strong oscillatory
thickness dependence of the coupling in the FeCr
system [8], the actual correlation length of the AF
coupling, consequently that of the `unsaturation' do-
mains is much smaller than the correlation length of the
Cr spacer thickness. Indeed, the Fe anisotropy energy is
linearly dependent on the iron layer thickness, thus the
correlation length of the anisotropy energy is equal to
the correlation length of the Cr spacer thickness. The
spin-flop field is, however, proportional to the geome-
trical mean of the AF coupling and the anisotropy
energy. Its autocorrelation function is, therefore, a linear
combination of the autocorrelation function of the
coupling and that of the spacer thickness thereby
resulting in both short-range and long-range correla-
tions [5]. Still, for clarity, we shall only use the dominant
(bigger) correlation length for the spin-flop field in the
present simulations.
The multilayer is modelled here by a two-dimensional
grid of pixels where the colour (grey scale gradation in
the figures) of each pixel represents the direction of the
magnetisation of the topmost layer in a given pixel area.
The mesh size of the grid (of the order of 0.1 mm) is
smaller than the actual domain size (see below) but in
our model, each pixel possesses a macroscopic `classical' Fig. 1. Primary (a, b) and secondary (c, d) domain formation.
magnetisation, saturation field and anisotropy energy. In primary domain formation black and white pixels represent
The primary and secondary domains are formed on top layer rotation to the left and right, respectively, grey pixels
this grid by first-neighbour rules explained later (see being still in saturation. The easy axes of the fourfold
Fig. 1). Domains are represented as contiguous sets of anisotropy are along [1 0 0] and [0 1 0]. The sample had been
pixels of the same colour. saturated in the [1 0 0] direction. The grid is 500 500 pixels.
The `unsaturation' or primary domain formation is The smoothing widths of the primary and secondary distribu-
governed by the distribution of the saturation field. The tions are w1 ž 10 and w2 ž 100; respectively. In (c) and (d), the
higher the saturation field of a given pixel is, the sooner system during and after the spin flop is shown. The spin flop is
the pixel unsaturates. First, we create a grid of induced by an increasing field along the [0 1 0] direction. Note
uncorrelated random numbers Uð~rrÞ of Gaussian dis- that the secondary large domains are perpendicular to the
primary small ones. Grey-scale scheme in (d): light grey pixels:
tribution, where ~rrž ðx; yÞ is the position vector [9]. The top layer up, dark grey pixels: top layer down. Note that the
saturation field distribution Dð~rrÞ is generated by scale was fitted to the Kerr image measured by R.uhrig et al. on
smoothing the grid by an empirical width w according to a thick Fe/Cr trilayer [7] and not to the domains observed by
X SMR on a Fe/Cr multilayer of 20 periods [5].
ð~rr ~rr0Þ2
Dð~rrÞ ž 1 Uð~rr0Þ: ð1Þ
w2
~rr ~rr0
j jow in saturation have no influence) or chooses by random if
Periodic boundary conditions are used. Decreasing no decision can be made using the previous rule.
the external magnetic field HextFalong an easy axisF According to this model, all domain walls are 1801
from saturation, the multilayer gradually unsaturates. walls in remanence. Indeed, assuming that the domains
When Hext matches the saturation field value Hsat of a do not change shape having been unsaturated, but only
given pixel, the pixel unsaturates. The pixel will choose rotate on further decrease of the field, the image is the
its sense of rotation (black: top layer left; white: top same in remanence as in complete unsaturation
layer right) according to so-called flipping rules (Fig. 1(b)), only the angles of the layer magnetisation
(Fig. 1(a)). When HextominðHsatÞ; the whole SL is directions change.
completely unsaturated (Fig. 1(b)).The set of first- By applying an in-plane external field along the
neighbour rules governs the `decision' of each pixel. In perpendicular easy axis, a bulk spin-flop transition can
our model, all eight first neighbours have equal weights. be induced [6]. At the transition, the pixel magnetisa-
To avoid creating domain walls, the pixel to decide tions rotate by 7901. The domain nucleation of the spin
chooses the colour of the majority (neighbour pixels still flop is now governed by the distribution of the spin-flop
M. Major et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 469471 471
field, which is much broader than that of the saturation 1
field. A pixel can choose its new direction if all pixels simulation
image
with lower spin-flop field have already decided. The rules 0.8
are now the same as those during unsaturation, but in
Fig. 1(c) the pixels may now flip from the left/right into 0.6
the up/down orientation. In a simple model of the
domain-wall energy, the energy penalty is proportional 0.4
to the sum of the square of the relative angle of the Correlation
neighbours. It can be shown that with these conditions, 0.2
the secondary domain formation is independent of the
primary structure and only depends on the lateral 0
distribution of the spin-flop field. In Fig. 1(c), the small
unflipped and the large flipped domains with perpendi- -40 -30 -20 -10 0 10 20 30 40
cular relative magnetisations coexist. This coexistence of x shift [pixel]
perpendicularly magnetised primary and secondary Fig. 2. The autocorrelation function (2) of the Kerr image of
domains has in fact been observed during spin flop by the AF patch-domains [7] compared with the autocorrelation
SMR and PNR [5]. In Fig. 1(d), the secondary domains function of the simulated pattern for the mesh size of 146 nm
are depicted after the spin flop has been completed. and w1 ž 10 pixels.
Note that the model introduced here applies to both
trilayers and multilayers as far as the two-dimensional presented for the first time to our knowledge. The
representation holds (i.e., the coupling is strong as generated patch-domain pattern is in quantitative
compared to the anisotropy energy). agreement with published Kerr-microscopic images of
To derive measurable quantities from the simulation a Fe/Cr/Fe trilayer. The simulation describes the
an appropriate transformation is needed. We may emergence of domains during unsaturation in accor-
choose the autocorrelation function Cð~rrÞ of the domain dance with observations and remarkably features the
image as an indicator of the correlation length. In the coexistence of the perpendicularly magnetised small and
Born approximation the off-specularly scattered inten- large domains observed by recent off-specular SMR and
sity is proportional to the Fourier transform of Cð~rrÞ: PNR during spin flop.
The autocorrelation function is defined by: This work was partly supported by the Hungarian
P Scientific Research Fund (OTKA) under Contract No.
Cð~rrÞ ž ~rr0ðmð~rr0Þ %mÞðmð~rr0 þ~rrÞ %mÞ
P ; ð2Þ T029409 and by the Hungarian Academy of Sciences,
~rr0 ðmð~
rr0Þ %mÞ2 Contract No. 2000103 2,3.
where mð~rrÞ is the chosen direction (colour) of the given
pixel (in the PDS white: +1, black: 1, in the secondary
domain state: light grey: +1, dark grey: 1), %m ž
P
ð1=L2Þ ~rrmð~rrÞ the average of mð~rrÞ and L is the width of
the image in pixels. The summation goes over the
complete grid. References
In order to test our model, the autocorrelation
function of the simulated primary domains and that of [1] M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F.
published Kerr images of R.uhrig et al. [7] were Petroff, Phys. Rev. Lett. 61 (1988) 2472.
compared. In Fig. 2, the autocorrelation function of [2] H.T. Hardner, M.B. Weissman, S.S.P. Parkin, Appl. Phys.
Lett. 67 (1995) 1938.
the model is adjusted to the autocorrelation function of [3] B. Dieny, J.P. Gavigan, J.P. Rebouillat, J. Phys. C. 2 (1990)
the Kerr image. The agreement is quite remarkable for 159.
the mesh size of 146 nm and w ž 10 pixels. The formal [4] K. Temst, E. Kunnen, V.V. Moshchalkov, H. Maletta, H.
relationship between the correlation lengths of the Fritzsche, Y. Bruynseraede, Physica B 276278 (2000) 684.
saturation as well as spin-flop fields and the correlation [5] D.L. Nagy, et al., Phys. Rev. Lett., submitted for
length of the domains obtained as a result of the publication.
described rules has not yet been elaborated. Further- [6] L. Botty!an, et al., J. Magn. Magn. Mater. 240 (2002), this
more, due to the inapplicability of the Born approxima- issue.
tion for the case of multiple scattering, the relation of [7] M. R.uhrig, et al., Phys. Status Solidi A 125 (1991) 635.
the domain correlation length to the off-specular SMR [8] J. Unguris, R.J. Celotta, D.T. Pierce, Phys. Rev. Lett. 67
(1991) 140.
scan shape is not a trivial task and is a subject of future [9] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flan-
study. nery, Numerical Recipes in C: the Art of Scientific
In summary, Monte Carlo simulations of patch- Computing, 2nd Edition, Cambridge University Press,
domain formation in antiferromagnetic multilayers were Cambridge, 1997, p. 288.