AIP ADVANCES 7, 056001 (2017)
Vortex motion in amorphous ferrimagnetic thin
film elements
Harald Oezelt,
1,
a
Eugenie Kirk,
2,3,
b
Phillip Wohlh
¨
uter,
2,3
Elisabeth M
¨
uller,
4
Laura Jane Heyderman,
2,3
Alexander Kovacs,
1
and Thomas Schrefl
1
1
Center for Integrated Sensor Systems, Danube University Krems, 2700 Wiener Neustadt,
Austria
2
Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich,
Switzerland
3
Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute, 5232 Villigen PSI,
Switzerland
4
Laboratory of Biomolecular Research, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
(Presented 3 November 2016; received 23 September 2016; accepted 14 October 2016;
published online 27 December 2016)
Amorphous Fe
64
Gd
36
thin film square elements are investigated by imaging in the
Fresnel mode of a transmission electron microscope (TEM). The equilibrium state
without an applied field shows the well-known four-domain flux closure pattern
with in-plane magnetization. However, the vortex is displaced from the center of
the square element and the domain walls are curved. In a reference measurement of
a thin Ni
81
Fe
19
element, the vortex core is perfectly centered and the domain walls
straight. When an increasing external field is applied in-plane, the vortex core can
be moved. While this motion of the vortex core is linear in NiFe elements, in the
ferrimagnetic FeGd squares the vortex core moves by sudden jumps. Micromagnetic
simulations show that the asymmetry of the domain patterns as well as the vortex
core pinning and depinning can be attributed to random anisotropy and a patchy
microstructure in amorphous films. © 2016 Author(s). All article content, except where
otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(
http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4973295]
I. INTRODUCTION
In patterned ferromagnetic thin film elements multidomain states form if the elements are suf-
ficiently large. Most commonly the ground state is a symmetric flux closure pattern. In medium
sized square elements the remanent state is a four-domain pattern with straight domain walls and
a center vortex.
1
Dietrich and co-workers
2
showed that in permalloy (Ni
81
Fe
19
) squares a distorted
flux closure pattern may arise from the substrate curvature which may lead to curved domain walls.
Similarly, the magnetic field created by the tip of a magnetic force microscope can induce twisted
flux closure patterns.
3
A distortion of the equilibrium domain patterns caused by magnetic fields was
also reported by Hertel and co-workers.
4
They showed that the demagnetizing field of nano-islands
with inclined surface leads to asymmetric magnetic states.
Whereas the above mentioned distortions of the domain patterns are a result of local magnetostatic
fields, Heyderman and co-workers
5
reported asymmetric domain configurations in Ni
83
Fe
17
and Co
elements when decreasing the thickness to below 17 nm. They attributed the observed patterns to
material defects such as edge and surface roughness, but also local magnetocrystalline anisotropy
variations, which lead to pinning of domain walls or vortices. For thicker elements they observed a
symmetric domain pattern.
a
harald.oezelt@donau-uni.ac.at.
b
eugenie.kirk@psi.ch
2158-3226/2017/7(5)/056001/5 7, 056001-1 © Author(s) 2016
056001-2 Oezelt et al. AIP Advances 7, 056001 (2017)
In this work, we investigate the domain structures in amorphous Fe
64
Gd
36
elements. Due to
shape anisotropy the magnetization lies in the plane of the film. Magnetic images are obtained by
transmission electron microscopy imaging in the Fresnel mode. In most samples an asymmetric flux
closure pattern was observed in zero applied field. Micromagnetic simulations that take into account
the random anisotropy distribution
6
of the amorphous film show the same key features of the domain
pattern: 1) an off-centered vortex state with 2) curved domain walls at equilibrium, 3) increasing the
element thickness leads to a reduced vortex offset, and 4) pinning of the vortex core during application
of magnetic fields in the plane of the sample.
II. EXPERIMENTAL METHOD
Amorphous Fe
64
Gd
36
thin film square elements of edge lengths ranging from 3 µm to 10 µm
were fabricated by electron beam lithography and subsequent lift-off. The magnetic material with a
thickness of 50 nm was deposited by ultra high vacuum (UHV) magnetron sputtering onto silicon
nitride membranes and covered by 5 nm Pt to prevent rapid oxidation. For reference measurements
permalloy (Py, Ni
81
Fe
19
) squares of edge length 10 µm and thickness 15 nm were fabricated.
The vortex pinning is imaged with transmission electron microscopy (TEM) in the Fresnel mode.
This technique is based on the interaction between an electron beam and the magnetic flux of the
sample. Deflection of the electron beam due to the Lorentz force occurs if the magnetization changes
in the plane of the sample. In the Fresnel imaging mode, the sample is defocused revealing magnetic
contrast. The vortex position is tracked as a function of applied in-plane field, which is created by the
remanent field of the TEM objective lens. Tilting the sample inside the objective lens field results in
increased in-plane fields. At zero tilt, the in-plane component of the magnetic field is minimal, which
is reflected by a center position of the vortex in NiFe. On increasing the tilt, and thus the applied
magnetic field, the vortex is displaced from the center.
III. MICROMAGNETIC SIMULATIONS
The micromagnetic treatment of amorphous ferrimagnets such as Fe
64
Gd
36
was pioneered by
Mansuripur and co-workers.
7
Introducing patches that resemble local disorder of the amorphous
film, they were able to explain domain nucleation, domain wall pinning, and coercivity in amorphous
rare-earth transition metal films used for magneto-optic recording.
8
By means of the patchy structure,
they introduced a structural correlation length into their micromagnetic model. Within a patch, which
can have arbitrary shape, the direction of the local anisotropy was assumed constant. Fu and co-
workers
9
showed that the critical field for the nucleation of reversed domains strongly depends on
the patch size, whereas the pinning field of domain walls depends on the patch-to-patch easy axis
orientation. They emphasize that the nanoscale patches are magnetic entities and not microstructural
features, for example columnar structures or poly-crystalline grains. Mansuripur
10
derived an equation
for magnetization dynamics in ferrimagnetic rare earth transition metal alloys. Assuming a strong
coupling between the rare earth and transition metal spins, he deduced an equation for the effective
magnetization.
FIG. 1. Schematic of the micromagnetic model used to simulate the FeGd element: It is divided into patches which in turn are
subdivided into tetrahedral finite elements. Random magnetization across the nodes of the mesh are used as the initial state.
056001-3 Oezelt et al. AIP Advances 7, 056001 (2017)
We adapt this approach for our micromagnetic model as we described in previously published
work.
11
Patches are created using Voronoi tessellation
12
of the thin film element. The patches are
further discretized using a tetrahedral finite element mesh. In order to compute the remanent domain
configuration, the sample is initialized with a random magnetization (Fig. 1). Then the system is
relaxed by solving the effective Landau-Lifshitz equation with infinite damping.
13
Fast convergence
of the algorithm is achieved by using the steepest descent method with a modified Barzilai-Borwein
step size selection.
14
The magneto-static field is computed from a magnetic scalar potential. The
parallelepipedic shell transformation
15
is applied to treat the boundary conditions at infinity.
IV. RESULTS AND DISCUSSION
The remanent Fresnel images of the specimens show the well-known four-domain flux closure
pattern. However, in thin ferrimagnetic specimens, for example the 20 nm Fe
64
Gd
36
square in Fig. 2c
it can be seen that the vortex is displaced from the center of the square element and the domain walls
are curved. In the reference measurement on 15 nm thick Ni
81
Fe
19
, the vortex core is centered as
expected and the domain walls are almost straight (Fig. 2a). Further measurements revealed that with
increased thickness of the FeGd squares the vortex core moves towards the center of the square and
therefore closer to the ideal symmetric pattern.
In the micromagnetic simulations we compare a continuous element to a patchy element. The
continuous element has no structural features. For both elements the saturation polarization was
set to J
s
= 1 T and the exchange stiffness constant to A
x
= 10 pJ/m. The patchy element was given
a mean magnetocrystalline anisotropy of
¯
K
u
= 0.1 MJ/m
3
with standard deviation σ
K
= 20% across
the patches, but with random anisotropic easy axis for each patch. The patchy structure was deeply
investigated by Mansuripur for amorphous films in magneto optical recording.
10
Nucleation fields and
domain wall pinning fields computed with the assumption of random anisotropy fluctuation compare
well with experimental data. The mean anisotropy field assigned to the patches of our micromagnetic
model corresponds to the experimentally measured one in Fe
64
Gd
36
.
16
For comparison we modelled
FIG. 2. The remanent domain pattern of a permalloy square with edge length 10 µm shows an almost ideal flux-closure pattern
and straight domain walls in the TEM image a) and also in the simulation b). The ferrimagnetic FeGd squares with edge lengths
10 µm clearly show a displaced vortex core and curved domain walls in the measurement c) as well as in the simulation with
a patchy microstructure d). When the thickness of the simulated element is doubled e) the vortex core is centered again, but
the randomness can still be recognized by the twist in the domain walls. The white dotted lines are a guide to the eye to mark
the ideal four-domain flux closure pattern. The domain walls in the simulations are shown as gray ribbons and the magnetic
moments as gray arrows. The in-plane angle of the magnetic moments is represented according to the color-map.
056001-4 Oezelt et al. AIP Advances 7, 056001 (2017)
FIG. 3. Measured vortex core displacement in a 3 µm × 3 µm, 50 nm thick FeGd element a) and the simulated magnetization
of a 0.3 µm × 0.3 µm, 20 nm thick FeGd element b). Both show nonlinear vortex motion when magnetizing the elements (red
solid curves) and decreasing the external field back to zero (blue dashed curves).
a continuous film with zero magneto-crystalline anisotropy. The exchange constant and the magneti-
zation of this film match the values for NiFe. The uniaxial anisotropy of NiFe is essentially zero unless
the film shows a stress induced magneto-elastic anisotropy. By this comparison we want to show that
the presence of randomness in the magneto-crystalline anisotropy creates pinning sites for domain
walls and vortex cores. In GdFe structural randomness is more important than in permalloy because
the magneto-crystalline anisotropy in GdFe is several orders of magnitude higher than in NiFe. For
the ease of computation the lateral extension of the square elements was reduced to 300 nm× 300 nm.
In Fig.
2 the simulated remanent states are compared to the measurement.
In the continuous case the domain pattern is perfectly symmetric, thus in Fig.
2b the gray
ribbon representing the domain wall is perpendicular and can not be distinguished from the dotted
white line. In the patchy sample the random anisotropy introduced through the patches leads to an
asymmetric closure domain pattern. The vortex is shifted and the domain walls are curved. This
is in agreement with the experimental results. When we double the film thickness from 20 nm to
40 nm of the patchy element the vortex moves towards the center. This is shown in Fig.
2e where the
vortex is located close to the center. However, the randomness still can be observed by the twist in the
domain wall. The now symmetric pattern may be attributed either to the more dominant magnetostatic
energy in the thicker film or the averaging of the anisotropy fluctuations throughout the thickness of
the film.
Starting from the remanent state, we now apply an increasing in-plane field. In TEM this is done
by tilting the sample inside the objective lens field. This allows us to observe how the vortex core gets
pushed away from its original position due to the growth of domains parallel to the applied field. In
a material with little pinning such as NiFe, the vortex is displaced reversibly with the applied field,
which can be also observed in the simulated continuous model. In case of ferrimagnetic Fe
64
Gd
36
,
the behavior is more complex: at low fields, the vortex displacement is exponential; at intermediate
fields, the displacement is hysteretic; and at high fields, the displacement is linear (Fig.
3a). In
the simulations with the patchy model we can reproduce the hysteresis in the magnetization curve
(Fig.
3b).
V. CONCLUSION
Asymmetric four-domain patterns in remanent state were observed in thin ferrimagnetic FeGd
square elements by Fresnel imaging in a transmission electron microscope. When an increasing in-
plane field is applied, nonlinear vortex movements are detected. These findings were investigated
by micromagnetic simulations. By dividing the amorphous ferrimagnet into patches with varying
anisotropic properties structural inhomogeneities are introduced. While possible sources such as
magnetostatic fields can not be ruled out, the introduced randomness is sufficient to explain several
features observed in the measurements: 1) off-center vortex core and 2) curved domain walls in
remanent state, 3) increasing thickness of the square decreases the vortex offset, and 4) pinning and
056001-5 Oezelt et al. AIP Advances 7, 056001 (2017)
depinning of the vortex core during magnetization by an external in-plane field. All these features
could not be seen in the reference measurements and simulations of ferromagnetic NiFe squares.
ACKNOWLEDGMENTS
The authors would like to acknowledge the financial support provided by the Austrian Science
Fund (FWF Grand No. I821), the Vienna Science and Technology Fund (WWTF Grant No. MA14-
44), and the Swiss National Science Foundation (SNF Grant No. 200021L
137509).
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