Table of contents
1 Introduction
2
2 Micromagnetism
2
2.1 Domains in thin films with perpendicular anisotropy
2.2 RT films
2.3 Thin film stripe domains in magnetic fields
2.4 MFM and resonant x-ray scattering
3 X-ray scattering studies of stripe domains
3.1 Simulations and MFM studies
3.2 Skyrmion
4 OOMMF
4.1 Magnetic constants
4.2 Qualitative effect of parameters
5 Results
5.1 Comparison with 1-dimensional model
5.2 Magnetization loop and vortex structure
5.3 Magnetization loop: Vxx,Vyy and hysteresis
5.4 Skyrmion structure
5.5 Magnetization loop: contrast functions
5.6 Comparison magnetization curve with 1-d model
5.7 Energy
6 Discussion
6.1 First look
6.2 X-ray diffraction data
6.3 Nucleation and skyrmion collapse
6.4 The 1-dimensional model
3
4
4
5
5
6
7
8
8
8
9
9
10
11
13
15
17
19
23
23
24
25
27
7 Conclusion
28
8 Further research
28
9 References
29
Appendix
30
1 Introduction
Ferromagnetic objects consist of a great number of magnetic domains. These domains explain
why objects made from ferromagnetic materials do not always seem to be magnetic from the
outside. The study of magnetic domains has been important for electrical engineering, but
even more important for data recording. Better understanding of magnetic domains has
enabled magnetic storage technology to keep pace with semiconductor technology for the last
few decades.
The formation of magnetic domains is a complex process that depends on the properties of the
material but also on the size and shape of the object. In this thesis we studied thin-film
systems where the magnetic structure becomes a one-dimensional vortex lattice.
We used simulations to study the influence of an magnetic field applied to the film on this
structure, which transforms to a skyrmion structure at the coercive field. The skyrmion
topology has been observed before in condensed-matter systems, for instance in BoseEinstein condensates and 2-dimensional Quantum Hall systems and in thin-film chiral
magnets 5,6.
The systems we studied have been studied before with resonant x-ray scattering and with
magnetic force microscopy. In this thesis we compare the results with micromagnetic
simulations and with earlier a theoretical model. We will start with an introduction of the of
micromagnetism and of the films we studied. Then we discuss the results of the earlier
research and continue with our research; the simulations we have done and the results. In the
discussion we compare this to the resonant x-ray scattering studies and to the model. Where
we conclude that our simulations confirm both the model and the x-ray diffraction
experiments.
2 Micromagnetism
Magnetic domains are regions where the magnetization is uniform. Magnetic domain theory
explains why objects made of ferromagnetic materials do not always appear to be magnetic
from the outside. This is because the magnetization in the material has different directions in
each domain, which reduces the external magnetic field created. The regions in between
magnetic domains where the magnetization changes orientation are domain walls. When an
external magnetic field is applied the domain walls shift to increase the size of the domains
with magnetization parallel to the applied field.
Current domain theory is based on the work of Landau and Lifschitz which uses the
variational principle to find the configuration with the lowest energy.
Magnetic structures on the micro-scale are described by the micromagnetic equations.
The total energy of the system is given by:
1
Etot   [ A(m)2  Ku (1  mz 2 )  0 H  M  0 H d  M]V
2
v
The first term is the exchange energy density, given by the exchange constant times the
gradient of the local magnetization squared. The second term is the anisotropy energy,
associated with an easy axis (in this case the z-axis) to which the magnetic moment tends to
align. Ku is the anisotropy constant. The third term is the Zeeman energy originating from an
external field. The fourth term is the magnetostatic energy. This is the energy that comes from
the magnetic field an object with a certain magnetization would create in free space. This term
is dependant on the shape of the object and as Hd depends on M this term makes the problem
nonlinear.
2
Other terms are the magneto-elastic energies, which describe the influence of the magnetic
structure on the crystal lattice, but they are negligible in our case.
One of the characteristic lengths in a micromagnetic system is the Bloch wall width.
b  
A
Ku
This is the thickness of the area between two magnetic domains; regions with different
directions of magnetization. It ranges between a few nanometers and a hundred nanometers.
There are two ways to make the transition from one domain to the other, a Bloch wall and a
Neél wall. In a Neél wall the magnetization stays in the plane of the magnetization in the
domains and in a Bloch wall the magnetization goes out of this plane. The surface energy of a
180° Bloch wall is
  A Ku
Fig. 1. Schematic picture of a Bloch wall (top) and a Neél wall (bottom).
3
2.1 Domains in thin films with perpendicular anisotropy
In the thin film samples studied here, the thickness was reduced to the order of the domain
wall with, so there is only one domain in the direction perpendicular to the plane. This
reduces the domain structure to two dimensions, although the internal structure of the domain
walls is still three dimensional.
Thin films can be grown with perpendicular magnetocrystalline anisotropy. This anisotropy
aligns the magnetization in the direction perpendicular to the plane. However, if the
magnetization would be in this direction throughout the material there would be a large stray
field. To reduce this the magnetization must be either turned in the plane or break up in
domains. The combination of these effects gives rise to a configuration of alternating up and
down domains, as shown by Kittel 7. The size of these domains is influenced by the material
parameters, the exchange constant, the anisotropy constant and saturation magnetization, but
also by the field history. This phenomenon, where the current magnetic properties depend on
magnetic fields applied in the past is called hysteresis. A common source of hysteresis is the
fact that moving domain walls costs energy. In our system this effect is small since there are
no crystal grains to pin the domain walls. A high value of the exchange constant means that
rotating the magnetization is expensive, resulting in wider domains. Large saturation
magnetization produces high demagnetizing fields, leading to smaller domains. To reduce the
stray field energy, Neél closure domains can form at the surfaces (see fig. 2).
Fig. 2. Example of a Neél closure domain around a Bloch wall
4
2.2 RT films
Films made of rare earth transition metal alloys are particularly suited for the study of
magnetic domains due to the fact that they are amorphous. This means there are no crystal
boundaries on which domain walls can pin. They are ferrimagnetic and the material
parameters can be tuned by choosing the composition. Due to the amorphous structure, both
the Fe and the Gd sublattices have considerable spread in spin direction around the local
magnetization.
The materials we studied are GdFe9 and GdFe5. They are amorphous, ferrimagnetic materials
and the magnetization of the sublattices are antiparallel. The layers are about 40 nm thick.
During their growth, an out-of-plane magneto-crystalline anisotropy emerges due to the
anisotropy of the surface. This anisotropy aligns the magnetization in the direction
perpendicular to the plane. The value of the anisotropy constant depends on the growth
parameters.
Fig. 3. MFM image of the GdFe5 sample, image size is 5×5 . Red and yellow represent upand downward magnetization.
2.3 Thin film stripe domains in magnetic fields
The grown films display a magnetic stripe domain structure with a period that is the result
between the tendency to reduce the demagnetizing field, which favors small magnetic
domains, and the cost of forming domain walls.
The stripe domains respond to applied magnetic fields. Their size changes and they align with
the magnetic field. A strong magnetic field can magnetize the material uniformly, completely
destroying the domains. We studied the influence of an in-plane magnetic field on the
magnetic structure. The magnetic structure in the sample plane is hard to measure with most
techniques because many of them only measure the magnetization at the surface and cannot
be used in high magnetic fields. One cannot simply cut the sample and measure the
magnetization since cutting will influence the magnetization.
5
2.4 MFM and resonant x-ray scattering
A technique to study magnetic domain structures is magnetic force microscopy (MFM). It is a
scanning technique closely related to atomic force microscopy (AFM). One records the
atomic force between a sharp tip and a surface as the tip scans the surface. This returns a
topographic map of the surface. For magnetic force microscopy the tip has some magnetic
material ( usually some cobalt or nickel ) attached to it to measure the magnetic field
produced by the sample. This makes it hard to use MFM in high magnetic fields, since the
external magnetic field can dominate the interaction between the sample and the tip. First the
surface is scanned at a short distance with atomic force microscopy and then the process is
repeated with the needle at a larger distance from the surface to measure the magnetic field.
MFM is best suited for very flat surfaces but the magnetic structure must not be to fluxclosed; the field produced by the sample may not be too small. Another problem are tipdomain interactions where the magnetic tip influences the magnetic domains it is scanning.
The spatial resolution of MFM is in the order of a few tens of nanometers. MFM can only
measure the magnetization at the surface. This means that to determine the internal structure
you need to use other techniques or numerical methods.
The best technique to measure the internal structure is X-ray resonant scattering. It has a
spatial resolution in the order of ten nanometer and can be used in any external magnetic field.
A disadvantage of this technique, however, is that it cannot solve the magnetic structure
topographically like MFM. For a detailed explanation of this technique see Resonant soft xray scattering studies of the magnetic nanostructure of stripe domains, by Joost Frederik
Peters2.
6
3 X-ray scattering studies of the magnetic nanostructure of stripe domains
In 2003 J.F. Peters studied the evolution of the stripe domains in 40 nm thick amorphous
GdFe9 and GdFe5 films with resonant x-ray scattering 2. A magnetic field that saturated the
sample was applied in the in-plane direction (we call this the x-direction). The field was then
reduced and later applied in the minus x-direction. From the diffraction peaks the stripe
period, magnetization and the squared components of the magnetization were resolved. This
enabled Peters to deduce the magnetic structure of the domains and its evolution as the field
changed (see appendix). The results were also compared to a 1-dimensional model by Marty
et al4.
Fig. 4. Schematic view of the internal stripe structure. The direction of H is displayed on
the right of each picture.
It was found that when the in-plane magnetization changed from the positive to the negative
x-direction the large jumps in the magnetization at the coercive field is accompanied by a
huge internal reorganization of the magnetic structure while the domain period stayed the
same.. Two ways in which this could occur were proposed: either the magnetization would
start to turn at the edges, leaving the magnetization of the core in the original direction until a
sudden collapse E or the transition could be smooth and some domains go from C to F
directly.
Although the measurements were successful they only provided indirect evidence of the
magnetic structure and no conclusion on how the magnetization reversed. For a more detailed
understanding finite element numerical simulations were used.
7
3.1 Simulations and MFM studies
In 2009 we8 studied these systems with OOMMF9, a micromagnetic simulation program. We
also checked the domain period evolution found by Peters with magnetic force microscopy.
The simulations showed that Peters was largely correct in his depiction of the magnetic
structure. They showed that the domains have a vortex structure in alternating orientations
around the Bloch walls ( which are more like vortex cores than Bloch walls).
Fig. 5. Image of the simulated structure. The thickness is 42 nm. Red regions represent
magnetization pointing inward ( x ). The vertical direction is z and the y-axis lies
horizontal.
The simulations also showed the transition happens in the first way Peter proposed. When the
applied field already points in the negative x-direction, the magnetization of the vortex core
remains in the positive x-direction while the region on the outside twists in the negative xdirection. The mx magnetization is trapped by the circulating magnetization mz and my. This
state is stable until it collapses when the field becomes too large.
8
3.2 Skyrmion
During this phase the vortex has become a skyrmion. This skyrmion is a stable topological
defect; there is no continuous transition to a state where the magnetization is in the minus xdirection . Skyrmions were originally proposed by Peter Skyrme in a model for bosons in high
energy physics. Skyrmions also appear in Bose-Einstein condensates and in quantum hall
systems. In nanomagnetism they have been observed in thin film chiral ferromagnets5.
At low temperatures ( 0 to 50 Kelvin ) a ( meta- ) stable skyrmion lattice can form when a
magnetic field is applied perpendicular to the film.
Fig. 6. Example of a skyrmion in FeCoSi2 ( left ). The image on the right is the skyrmion
lattice that forms. Images taken from Yu5
The diameter of these skyrmions is in the same order of magnitude as the domain size in
GdFe5 and GdFe9. The length of the skyrmions in stripe domains could be a lot larger than in
these skrymions since in these systems the maximum length is approximately the thickness of
the film. A more important difference is the formation of the skyrmions. In the case of Yu and
Mühlbauer10 the skyrmion lattice is a static solution that exists in a particular space in the B-T
phase diagram. In our system the skyrmions are dynamically formed; by a changing magnetic
field.
In this study we further investigate the skyrmion and the transition that occurs when the
magnetization changes direction by using OOMMF. We compare our results to the research
of J.F. Peters and to the 1-dimensional model by Marty4
9
4 OOMMF
OOMMF is a finite element micromagnetic simulation program that solves the micromagnetic
equations iteratively. The physical input parameters are the exchange constant, the uniaxial
anisotropy constant and the saturation magnetization. OOMMF is extensible, meaning that
anyone can contribute by adding extensions for instance to calculate certain energy terms or
add periodic boundary conditions. We used an extension with periodic boundary conditions in
the x-direction, where we have translation symmetry.
Other input parameters are the domain sizes in x,y and z- directions and the cell size. The
number of cells has a strong influence on the simulation time and it is necessary to find a
balance between precision and simulation time.
The simulation starts with a randomly generated vectorfield. The vectors represent the local
magnetization. Each iteration the vectorfield changes to reduce the energy until a stable
solution has been found. Then the applied magnetic field changes and the simulation enters
the next stage, until the solution is stable again. The number of stages, the magnetic field
during each stage and the accuracy are also user controlled parameters. An example of an
input file can be found in the appendix.
4.1 Magnetic constants
The properties A, Ms and Ku were taken from Static and dynamic x-ray resonant scattering
studies on magnetic domains, by Jorge Miguel Soriano3. They were estimated from the
nucleation field and the stripe period, measured with vibrating sample magnetometry.
These parameters can only be measured indirectly so there is a lot of uncertainty in the value
of these parameters, especially in the value of the exchange constant A. We can even argue
that, if our simulations match the X-ray data perfectly the values we assumed for the
simulation are more accurate than the values obtained by Peters2 and Soriano3.
4.2 Qualitative effect of parameters
To qualitatively check the simulations and to find out how sensitive the results of the
simulations are to the input parameters we ran simulations with different values for A, Ms, Ku
and the thickness. First we did a reference simulation and we doubled each of the parameters
individually. In all simulations the sample was initially fully saturated with a field in the xdirection. That field was then reduced to a small value. The resulting domain structures are
displayed in the appendix.
They show that the effect of tweaking the parameters in OOMMF is qualitatively correct.
Reducing Ku means that there will be no domain formation. Increasing A results in larger
domains, reducing it results in smaller domains. They also show that the domain structure is
very sensitive to the values of the physical parameters. A factor of two can make the
difference between domain formation and no domain formation at all.
10
5 Results
5.1 Comparison with 1-dimensional model
J.F. Peters compared his results to a 1-dimensional model of the magnetization by A.Marty4
and so will we. This model assumes the magnetization to be in the xz plane and it varies along
the y-direction. The mz magnetization as function of y is given by the Jacobi sine function. Θ0
is the maximum angle between the magnetization and the plane, but my is ignored in this
model. Θ0, the reduced stripe width w and shape parameter σ parameterize mz(y) according
to mz  sin 0  sn( y ) , where sn(y) is the Jacobi sine function.
The magnetization is then found by minimizing the total energy.
The result is a Q-t diagram. The dashed ( vertical ) lines correspond to domain structures with
the same angle theta. The dotted ( horizontal ) lines are domain structures with the same
reduced width u. To the left of the solid line there is no domain nucleation and to the right
there are stripe domains. We checked these results by running our own simulations with
parameters from different points in the phase diagram. The results are displayed in the
following graph.
Fig. 7. Q-t diagram with the results from OOMMF. Red dots represent in-plane
magnetization, green dots stripe domain formation.
11
5.2 Magnetization loop and vortex structure
We ran our simulation with the following parameters:
A = 1.4 ∙ 10-12 J∙m-1
Ku = 2.1 ∙ 105 J∙m-3
Ms = 221 ∙ 103 A∙m-1
D = 44 nm
We did several simulations, with sample widths ranging from 550 to 1500 nm. The cell size
was 2 nm. A sample with a size of 574 nm was done both with a 2 nm and a 1 nm cell size for
more accuracy. All simulations showed similar results, both in the domain structure, as well
as their nucleation fields and their stripe evolution. The simulations with sample widths of
574 and 1500 nanometer are the ones we will discuss and look into mostly.
All simulations were carried out with an initial field of -200mT in the x-direction. The field
then increased linearly to 100 mT in the positive x-direction during 100 stages. Below are
images of the 574 nm simulation at different phases in this process.
This first graph shows the initial condition, a random vectorfield and the result after relaxation
at -200mT, when the sample is fully saturated.
The next graph shows the advent of domain nucleation in both the 574 nm and the 1500 nm
simulation. This occurs at a field of -77 mT. ( stage 42 ). It should be noted that nucleation
starts at the edges in this simulation.
This graph shows the 574 nm sample at a field of -56 mT ( stage 48). Nine domains have
formed. The x-component of the magnetization is still large. This configuration is stable and
the number of domains does not change now that they have formed.
12
The following graph is the magnetization at a field of just -2 mT (stage 66). Note the strong
closure component around the cores of the domain walls. The magnetization in these cores is
tied in the x-direction by the rotating magnetization around it. This is already a source of
hysteresis, the current state of the system is not only dependant on the current parameters, but
also on its history.
Fig. 8. Detailed image of two vortices at -2mT.
Fig. 9. Profile of mx trough two domains at -2mT( green line in figure 6). In the centre of
the vortex the magnetization is still fully in-plane
5.3 Magnetization loop: Vxx,Vyy and hysteresis
Vxx and Vyy are the integrals of the squared mx and my magnetization respectively,
normalized to a maximum value of 1.
2
Vxx  1  mx (r )r
VV
2
Vyy  1  my (r )r
VV
These quantities tell us how much magnetization is in the x and y directions. More
specifically, Vxx can tell us if there are regions where the magnetization is in the x-direction
with other regions where it is in the negative x-direction ( a skyrmion structure ). This would
be the case if Vxx is not equal to zero while mx is. The Vyy gives the strength of the closure
component. The bottom graph is a hysteresis loop, showing mx versus applied field.
13
Fig. 10. Vxx,Vyy and the hysteresis loop
When the sample is in a fully saturated state Vxx is equal to 1 and mx is equal to -1. At the
advent of domain formation Vxx decreases and mx increases, meaning the magnetization is
not fully in-plane anymore. The hysteresis curve progresses similarly to a hard magnet. The
magnetization goes to zero only when a field is applied in the other direction. But when mx is
equal to zero, Vxx is not. Vxx only reaches its minimum shortly afterwards.
At this point Vyy reaches it’s maximum, meaning the closure component is very strong. This
is exactly what Peters observed in the diffraction measurements. In the discussion we
compare our data with Peters’s.
Finally after a sudden jump in Vxx, Vyy and mx at about 62 mT the sample returns to an
almost fully saturated state. This means there has been a sudden change of the magnetic
structure.
The fact that Vxx is not equal to zero when mx is means that there must be regions where the
magnetization is in the x-direction and regions where the magnetization is in the –x direction.
The vortex cores are still in the –x direction while the domains have an x-component. We will
discuss the resulting structure in the next section.
14
5.4 Skyrmion structure
Fig. 11. Two domains in the skyrmion state. Red regions mean magnetization pointing
inward, blue means magnetization is pointing towards us.
The structure that forms has a skyrmion topology. It is a topological defect; there is no
smooth transition to a state where the magnetization in the core points in the positive xdirection like the magnetization in the domains. The magnetization has rotational symmetry
around the center. Any change in the direction of the magnetization of the center would
increase the exchange energy dramatically. This confirms the proposed structure to explain
the Vxx and mx data. It also confirms that the transition occurs in the first way proposed by
J.F. Peters: ‘ the vortex cores become highly frustrated and collapse catastrophically at the
magnetization reversal’, since there is no continuous transition from this structure to a vortex
lattice with magnetization in the x-direction.
The next graphs show the skyrmion just before collapse. The field is equal to 62 mT ( stage
88 ). We obtained the same result for the 1500 nm sample.
Fig. 12. The closure component is still present. Both the domains and the closure
components have a strong magnetization in the positive x-direction. Only the cores of the
skyrmions are still in the minus x-direction.
15
Fig. 13. Profile of the x-magnetization across the core ( green line in figure 10 ). In the
cores the magnetization is almost fully in the x-direction.
Fig. 14. Profile of the y-magnetization trough the core ( yellow line in figure 10). The
maxima correspond to the white areas around the core in figure 10, where the
magnetization makes a circle around the core.
16
Fig. 15. Surface plot of the mx magnetization in the two domains.
We will discuss the skyrmion collapse in section 6.3.
5.5 Magnetization loop: contrast functions
In x-ray scattering measurements of stripe structures one does not measure the magnetization
directly. The measured quantities are intensities of scattered x-rays with different
polarizations. They correspond with the Fourier transforms squared of contrast functions.
For the normal incidence geometry used by Peters et al contrast functions are integrals of the
magnetization components over the thickness of the sample defined as:
D
g z   mz ( y, z)z
0
D
and
g xx   mx ( y, z)2 z
0
For a more detailed explanation of the measured results see Resonant soft x-ray scattering
studies of the magnetic nanostructure of stripe domains, section 5.5.
The next graph shows the shapes of the fourier transforms of the contrast functions, Gxx and
Gyy at different fields. These results were obtained from the 1500nm simulation.
17
Fig. 16. Evolution of Gxx at different fields. As the structure becomes more complex higher
peaks emerge. The higher order peaks show the most hysteresis, similar to the x-ray
diffraction data. In our case the peaks do not move on the x-axis ( inverse length ) since
the domain size does not change.
Fig. 17. Evolution of Gyy at different fields. Again higher order peaks emerge as the
structure becomes more complex, especially in the skyrmion state ( blue and purple ). The
amplitude of the main first peak keeps increasing however, which is unlike the x-ray
diffraction data.
18
5.6 Comparison magnetization curve with 1-d model
We can compare gz and gxx, to the one-dimensional model and see if we can fit these with the
proposed curves. They are the shape of the z-component ( gz ) and the absolute value of the xcomponent ( gxx ) as a function of y.
g z  sin(0 )  sn( y) ,
where
sn( y) is the Jacobi sine function and 0 the canting angle.
gxx  sin(0 ) [1 sn( y)2 ]
Below is the curve for gz(y) obtained from the 1500 nm sample with a field of -8 mT; in the
normal vortex state.
Fig. 18. Shape of gz ( dots ) and a fit of the Jacobi sine function ( red). The shape proposed
by the 1-d model is correct.
The next graph shows gxx(y), again obtained from the 1500 nm simulation at the same field as
the previous graph. The second graph shows gxx(y) in the vortex state.
19
Fig. 19. gxx(y) and the fit. 1 sn( y)2 describes the shape of gxx.
Fig. 20. In the skyrmion state, however, 1 sn( y)2 is not a good fit of gxx. The model does
not incorporate domain reversal. The bumps correspond to the skyrmion cores and the
peaks are the regions where the magnetization has already switched directions.
20
5.7 Energy
With OOMMF we can analyze the different energy terms through the domain nucleation and
domain reversal processes. The following graph shows the different energy terms ( Zeeman,
demagnetization, exchange and anisotropy ) and the total energy versus the magnetic field for
the simulation with 574nm width and a 2nm cell size.
Fig. 21. All energy densities
The anisotropy and Zeeman energy terms dominate, the demagnetization energy is the
smallest. The exchange term is large in the skyrmion state, otherwise it is relatively small. We
will now look at the different energy terms separately. In the discussion we investigate two
critical points in more detail; domain nucleation and skyrmion collapse.
21
Fig. 22. Magnetization and Zeeman energy density
During the fist stage of the process, the Zeeman energy increases linearly with the field ( since
the sample remains fully saturated in the x-direction ). It then increases more rapidly when
domains form, going to zero as the field goes to zero. When the magnetic field changes
direction the Zeeman energy has a positive contribution. The net mx magnetization is in the
old direction while the magnetic field has already turned. When the domains reverse and mx
becomes positive the Zeeman energy is negative again. At 64 mT the energy decreases
rapidly; this is the point where the skyrmion collapses. Afterwards, the sample is fully
saturated and the energy decreases linearly with the field, like in the very beginning.
22
Fig. 23. Anisotropy, demagnetization and exchange energy densities. Vyy corresponds to the
amount of closure in the domains.
The anisotropy term tends to point the magnetization out of plane, so when the sample is fully
saturated the anisotropy energy is at its maximum. When domains form it decreases as the mz
component forms. When the field has reversed and mx squared increases again the anisotropy
energy also increases. At 64 mT, when the skyrmion collapses, the anisotropy energy rapidly
increases to its maximum value again.
There is no stray field produced by the sample in the fully saturated state ( the sample is
infinite in the x-direction ). When domains form the demagnetizing energy increases, reaching
its maximum right around when the applied field is zero. The stray field energy then
decreases again as the field points the magnetization in-plane and the closure component
keeps growing. The closure component emerges to reduce the demagnetization energy, at the
cost of exchange energy. Around the skyrmion collapse this breaks down.
The exchange term only comes into play when the magnetization twists between adjacent
cells. This starts at domain nucleation. The exchange energy keeps increasing as closure
domains form and becomes very large when the field reverses. In the skyrmion state the
exchange energy is huge; the magnetization rotates almost 360° along the z-axis through the
skyrmion core.
23
6 Discussion
6.1 First look
The simulations are qualitatively correct, the effect of modifying the exchange constant,
thickness, anisotropy constant and so on matches micromagnetic theory.. It is hard to judge
the simulations quantitatively, since there is a huge uncertainty in the value of several
material constants, most of all in the exchange constant and the anisotropy constant. The
saturation magnetization has been determined by vibrating sample magnetometry while the
exchange and anisotropy constants have been determined from the x-ray scattering
measurements and the nucleation field in both directions. The second order anisotropy has
also been neglected. Still we can say that the simulations are qualitatively correct and the
domain period and nucleation field match the experiments to a good degree.
6.2 Domain period
One important effect of the magnetic field is that the domain period varies with the magnetic
field. This has been observed in both GdFe9 and GdFe5. Below are graphs of the stripe period
versus applied field.
Fig. 24. Evolution of the domain period of GdFe5. The variation in the domain period
between nucleation and the maximum period is about 30 percent.
We have also measured the field evolution of the domain period of GdFe9 samples with
magnetic force microscopy (MFM) with similar results.
However, in our simulations the domain period does not change. Once a certain amount of
domains form, their number is stable. At the edges of the sample the magnetization is in the z
or minus-z direction, like in the middle of the domains. Any free pole on one side of the
sample must be compensated at the other side. Normally the domain size would increase,
reaching its maximum when the applied field has just changed direction as we can see in the
graph. So we would expect the number of domains to decrease during the simulation. This
could happen by the merging of two domains or a domain has to be squeezed out. But in the
simulations the vortex structure is too stable for either of these to occur.
A possible solution for this is to increase the sample size, since the relative impact of a
domain merger or a single domain being squeezed out is smaller. However, even in the 1500
nm simulation we do not observe such an event.
Alternatively, the experimentally observed periods could be taken as input for the simulation
size for each field. This would require considerably more effort and has not been attempted
yet.
24
6.3 X-ray diffraction data
Fig. 25. Diffraction data ( left ) versus data from the simulations (right)
The main goal of the simulations was to reproduce the x-ray diffraction data and investigate
the exact magnetic structure. Our data generally are in very good correspondence with the xray diffraction data. The main reason a structure was proposed with x-magnetization in both
directions was the fact that Vxx was not equal to zero while mx was. We obtain the same result
in the simulations.
Our Gxx and Gyy are also similar to the diffraction data. Higher order peaks emerge as the
structure becomes more complex and there is hysteresis, especially in the higher order peaks.
The largest difference between our data is the amount of hysteresis. In the simulations the
skyrmion collapse occurs at 62 mT, almost at the nucleation field of -77 mT while in the
diffraction data the collapse occurs earlier and is much smoother. After the collapse a domain
structure is still present while in our case it is destroyed. This could be because in the
experiment the transition is incomplete or because of edge effects in our simulation, removing
the possibility of a smooth transition. The true structure of the magnetic domains is threedimensional and there is a possibility that domains merge or a reverse domain nucleates in a
real sample. So the simulations cannot give a definitive answer to what happens when the
domains reverse, though they do strongly hint to the process proposed by Peters.
25
6.4 Nucleation and skyrmion collapse
Fig. 26. Energy densities around nucleation ( -77mT).
The most important energy terms are the Zeeman energy, which forces the magnetization in
the plane and the anisotropy energy, which points the magnetization out of plane. To first
order domain nucleation is the result of competition between these two processes. In the graph
we see that nucleation occurs when the Zeeman and anisotropy energies become equal.
Fig. 27. Energies around the skyrmion collapse (62 mT)
All the energy terms have discontinuous derivatives when the skyrmion collapses. When the
collapse is complete and the sample is fully saturated again the derivatives are continuous.
The discontinuity in the energies originate from the discontinuity in the evolution of the
magnetization. The skyrmion is a topological defect and its essence is that there is no
continuous transition to the relaxed state.
26
The skyrmion collapse is ultimately caused by the increase of the field. Increasing the field
turns the outer regions of the domains more and more in the x direction. This increases the
exchange energy density until the exchange energy becomes too large and it becomes
unfavourable to keep a domain structure. The following important things happen right before
collapse.
1. The difference between the Zeeman, exchange and anisotropy energies is small.
2. The Zeeman energy plus the anisotropy energy is almost equal to the demagnetizing
energy.
3. After the collapse the Zeeman energy plus the anisotropy energy has the same value as
at the advent of domain nucleation.
4. At the point of collapse the exchange energy has just become larger than the
difference between the anisotropy energy and its maximum value.
27
One would expect the difference between the energy densities to be small when the system is
in a critical stage. This also happens at nucleation, where the difference between the
anisotropy energy density and the Zeeman energy density is small
It is not strange either that the Zeeman plus the anisotropy energy has the same value as at the
beginning of domain nucleation. The systems makes a transition from a state with domains to
a saturated state, just the opposite of what happens at nucleation.
The fourth point is the most important and explains why the skyrmion collapses precisely
when it does. The energy cost of the skyrmion state has become larger than the gain; the cost
being the exchange energy and the gain is the lowered the anisotropy energy.
6.5 The 1-dimensional model
We checked the Q-t diagram from the model by Marty et al. Domains formation occurred at
slightly lower quality factors in the simulations compared to the model but the difference is
small. Also the shapes of gz and gxx match our fields perfectly when they are in a regular
vortex state. In the skyrmion state the model breaks down as it does not contains a description
of closure and core magnetization.
7 Conclusion
We have shown that 2-dimensional OOMMF simulations using periodic boundary conditions
give a good qualitative description of the experimental magnetization curve and internal
structure of stripe domains in in-plane magnetic fields. The main result is the confirmation of
the existence of a skyrmion structure prior to magnetization reversal. These skyrmions are
topologically stable and long, possibly in the order of hundreds of microns. The symmetry of
thestructure is the source of topological hysteresis in our system: there is hysteresis without
domain walls moving across grain boundaries like in normal ferromagnets. This leads to a
remarkable coupling of the nanometer structure to the macroscopically observed
magnetization.
The simulations confirmed the reversal process that was proposed by Peters. This makes a
strong case for the existence of the skyrmion structure but also confirms the usefulness of
resonant x-ray scattering as a method of stripe domain observation.
The simulations are also qualitatively correct and they match the 1-d model by Marty et al.
and confirm the shaped of the magnetization in that model.
8 Further research
Our data provide support for the existence of a skyrmion but cannot confirm it. More
experimental data would be needed to definitively prove this. This could be done with more
X-ray scattering experiments, illuminating the sample from different directions. This way the
entire structure can in principle be determined. The exact analysis of the diffraction data
would be extremely complicated however.
28
Another factor that may be important is the second term in the uniaxial anisotropy energy
density which goes like (sinθ)4 . This term becomes important when the effective anisotropy
given by the difference of the first order anisotropy energy and demagnetization energy
densities vanishes11. This will also be important in the vortex and skyrmion cores.
A shortcoming of the simulations is the lack of evolution of the domain size as a function of
the applied magnetic field. Simulations with periodic boundary conditions in the y-direction
or use of the periods obtained from the experiment could solve this problem.
9 References
1 Magnetic domains, the analysis of magnetic microstructures. Hubert & Schäfer.
Springer,1998
ISBN 3-540-64108-4
2 Resonant soft x-ray scattering studies of the magnetic nanostructure of stripe
domains. Joost Frederik Peters, PhD thesis, 2003.
ISBN 905776105X
3 Static and dynamic x-ray resonant scattering studies on magnetic domains. Jorge
Miguel Soriano, PhD thesis, 2005
ISBN 90-5776-142-4
4 Weak-stripe magnetic domain evolution with an in-plane field in epitaxial
FePd thin films: Model versus experimental results. A. Marty et al.
Journal of Applied Physics, Volume 87, Number 9. May 1st 2000
5 Real-space observation of a two-dimensional skyrmion crystal. Yu et al.
Nature Vol. 465. June 17th 2010
29
6 Observation of Unconventional Quantum Spin Textures in Topological Insulators
D. Hsieh, et al. Science 323, 919 (2009);
7 Physical theory of ferromagnetic domains. C. Kittel. Review of modern physics
21, 1949
8 Experimenteren, simuleren en analyseren van magnetische streepdomeinen van
.
GdFe5 en GdFe9 in een magneetveld W. Steggerda, D. Posthuma & T. van der
Vliet. 2009
9 OOMMF, micromagnetic simulation program by Mike Donahue and Don
Porter. See http://math.nist.gov/oommf/
10 Skyrmion Lattice in a Chiral Magnet. Mülbauer et al. Science Vol. 323 no. 5916,
Februari 13th 2009
11 Reorientation transitions in ultrathin ferromagnetic films by thickness- and
temperature-driven anisotropy flows. Y. Millev & J. Kirschner. Physical review
letters B, Vol. 54, Number 6. August 1st 1996
Appendix
Qualitative effect of parameters
Graph 1 shows the standard system, with A = 1.4E-12, Ku = 2.1E4 and Ms = 221E3.
Graph 2 shows the system with a Ku set to 4.2E4. The magnetization is stronger in the zdirection and the structure is not as flux closed anymore.
30
Graph 3 shows the system with Ku set to 1.05E4. The anisotropy is not strong enough and
there is no domain nucleation at all.
In graph 4 the exchange constant has been doubled to 2.8E-12. There are less domains than in
the original, with more magnetization remaining in the x-direction. This is explained by the
fact that with a higher exchange constant twisting the magnetization costs more energy.
The exchange constant has been halved to 0.7E-12 in graph 5. Domain walls cost less energy
now and the stray field energy is reduced by the great number of domains ( 12 compared to
the original 9 and 7 from the previous graph.
In graph 6 the saturation magnetization has been doubled to 442E3. The stray field energy and
the Zeeman energy are the terms most affected by this. They both tend to keep the
magnetization in-plane and because of this there is no domain nucleation.
In graph 7 the saturation
magnetization has been halved to
105E3. Anisotropy completely
dominates and there are only four
large domains, which already form at
a high magnetic field.
Magnetic structure by Peters
A, B and C are during the first
phase. D, E and F are during
domain reversal. See Resonant soft
x-ray scattering studies of the
magnetic nanostructure of stripe
domains, page 98
31
OOMMF input file
RandomSeed 1
Specify Oxs_BoxAtlas:atlas [subst {
xrange {0
574e-9}
yrange {0
44e-9}
zrange {0
2e-9}
}]
Specify Oxs_UniaxialAnisotropy {
32
K1 2.1e4
axis { 0 1 0 }
}
Specify Oxs_RectangularMesh:mesh [subst {
cellsize {2e-9 2e-9 2e-9}
atlas :atlas
}]
Specify Oxs_UZeeman [subst {
comment {Set units to mT}
multiplier 795.77472
Hrange {
{
0 0
-200
100}
0 0 100
}
}]
Specify Oxs_FixedZeeman {
comment {Set units to mT}
multiplier 795.77472
field { Oxs_RandomVectorField {
min_norm 1
max_norm 1
}}
}
Specify Klm_Demag_PBC [subst {
tensor_file_name
"[OOMMFRootDir]/tmp/demag_tensor/"
progress_script
{[OOMMFRootDir]/app/oxs/local/kl_progress.tcl}
}]
Specify Klm_UniformExchange {
A
1.4e-12
kernel
"6ngbrzperiod"
}
Specify Oxs_CGEvolve {
}
Specify Oxs_MinDriver [subst {
stopping_mxHxm 1
evolver Oxs_CGEvolve
mesh :mesh
normalize_aveM_output 1
Ms 221e3
m0 { Oxs_RandomVectorField {
min_norm 1
max_norm 1
}}
}]
33