Skyrmion Lattice Phases in Thin Film Multilayer

Abstract

Phases of matter are ubiquitous with everyday examples including solids and liquids. In reduced dimensions, particular phases, such as the 2D hexatic phase and corresponding phase transitions occur. A particularly exciting example of 2D ordered systems are skyrmion lattices, where in contrast to previously studied 2D colloid systems, the skyrmion size and density can be tuned by temperature and magnetic fields. This allows for the system to be driven from a liquid phase to the onset of a hexatic phase as deduced from the analysis of the hexagonal order. Using coarse‐grained molecular dynamics simulations of soft disks, the skyrmion interaction potentials are determined, and it is found that the simulations are able to reproduce the phase behavior. This shows that not only the static behavior of skyrmions is qualitatively well described in terms of a simple 2D model system but skyrmion lattices are versatile and tunable 2D model systems that allow for studying phases and phase transitions in reduced dimensions. Skyrmion lattices are shown to be promising playgrounds to study 2D phase transitions. While investigating phases with respect to skyrmion density, the hexatic phase is emerging, demonstrating the 2D nature of the multilayer system. Concluding from comparison with simulations, skyrmion lattices act as a 2D model systems with advantages in terms of tenability and speed over existing systems like colloids.

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Skyrmion Lattice Phases in Thin Film Multilayer
Jakub Zázvorka, Florian Dittrich, Yuqing Ge, Nico Kerber, Klaus Raab, Thomas Winkler,
Kai Litzius, Martin Veis, Peter Virnau,* and Mathias Kläui*
Phases of matter are ubiquitous with everyday examples including solids and
liquids. In reduced dimensions, particular phases, such as the 2D hexatic
phase and corresponding phase transitions occur. A particularly exciting
example of 2D ordered systems are skyrmion lattices, where in contrast to
previously studied 2D colloid systems, the skyrmion size and density can be
tuned by temperature and magnetic fields. This allows for the system to be
driven from a liquid phase to the onset of a hexatic phase as deduced from
the analysis of the hexagonal order. Using coarse-grained molecular dynamics
simulations of soft disks, the skyrmion interaction potentials are determined,
and it is found that the simulations are able to reproduce the phase behavior.
This shows that not only the static behavior of skyrmions is qualitatively well
described in terms of a simple 2D model system but skyrmion lattices are
versatile and tunable 2D model systems that allow for studying phases and
phase transitions in reduced dimensions.
DOI: 10.1002/adfm.202004037
Dr. J. Zázvorka, F. Dittrich, Y. Ge, N. Kerber, K. Raab, T. Winkler,
Dr. K. Litzius, Dr. P. Virnau, Prof. M. Kläui
Institute of Physics
Johannes Gutenberg-Universität Mainz
Staudingerweg 7, Mainz 55128, Germany
E-mail: virnau@uni-mainz.de; klaeui@uni-mainz.de
Dr. J. Zázvorka, Dr. M. Veis
Institute of Physics
Faculty of Mathematics and Physics
Charles University
Ke Karlovu 5, Prague 12116, Czech Republic
N. Kerber, Dr. K. Litzius, Dr. P. Virnau, Prof. M. Kläui
Graduate School of Excellence Materials Science in Mainz
Johannes Gutenberg-Universität Mainz
Staudingerweg 9, Mainz 55128, Germany
Dr. K. Litzius
Modern Magnetic Systems
Max Planck Institute for Intelligent Systems
Heisenbergstrasse 3, Stuttgart 70569, Germany
The ORCID identification number(s) for the author(s) of this article
can be found under https://doi.org/10.1002/adfm.202004037.
even be stabilized with no external mag-
netic field applied,[7–9] which makes them
potentially useful for memory and com-
puter logic devices.[3,10] In addition to such
devices based on the controlled operation
of single skyrmions, also thermally acti-
vated skyrmions and skyrmion ensem-
bles have been suggested for functional
devices for non-conventional computing
approaches: recently it was shown, that
skyrmions, including ensembles, can be
relevant for stochastic computing where a
functional skyrmion reshuer device was
implemented.[11] And in particular for res-
ervoir computing, we have suggested to
use ensembles of many skyrmions where
the skyrmion interaction and collective
behavior is of key importance.[12] Thus
advanced functionality in nanoscale devices
is enabled if the properties of ensembles of skyrmions can be
understood and controlled. Periodic ensembles called skyrmion
lattices have been found widely in bulk materials with B20 sym-
metry, where the topological structures are stabilized due to bulk
Dzyaloshinskii–Moriya interaction (DMI).[4,5,13,14] However, in
bulk systems the skyrmions are mostly not 2D like, as the “skyr-
mion tube” length can easily exceed the skyrmion diameter or
even the skyrmion-skyrmion distance. In advanced thin film
systems, skyrmions down to sub-nm thickness and diameters
in the range of micrometers are stabilized, making them prime
candidates for perfectly 2D systems. While skyrmion lattices have
been studied theoretically in such systems, only recently first
experimental reports of thin film lattices have been reported,
albeit with systems where the relatively large (≈100 nm) film
thickness is similar to the lateral skyrmion size making these
systems not necessarily 2D.[13,15–17] Thus to experimentally probe
the rich phase behavior of 2D systems[18–20] akin to colloids in
the past,[21–23] 2D skyrmion lattices occurring in ultra-thin film
stacks might be an ideal model system.[24] The nature of phase
transitions in 2D systems of hard and soft disks has been a grand
challenge in statistical physics, which has recently been numeri-
cally treated.[18] Apart from a liquid at low and a solid phase at
high density, a third intermediate phase may emerge: The hexatic
phase is characterized by short range translational and quasi-long
range orientational order, and there is a clear need for experi-
mental 2D systems to probe this unique phase behavior. This
calls for studying 2D skyrmion lattices and analysis of their phase
behavior with numerical simulations based on coarse-grained
models from Statistical Mechanics to identify possibly unique
2D properties as well as gauge the suitability of these systems to
study the exciting 2D phase behavior.
1. Introduction
Magnetic skyrmions, topologically stabilized whirls of magneti-
zation, are in the focus of the scientific community due to their
attractive properties for possible novel functional devices.[1–3]
Using spin-transfer torque and spin–orbit torque,[4–7] skyrmions
can be moved with high speeds at low current densities and can
© 2020 The Authors. Published by Wiley-VCH GmbH. This is an open
access article under the terms of the Creative Commons Attribution
License, which permits use, distribution and reproduction in any
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Thus, in this work, we use a sub-nm thick CoFeB-based
multi layer system to study the emergence of skyrmion lattices
as well as their response to tuning external parameters such as
temperature and field. Since the skyrmion diameter (Figure 1a)
is three orders of magnitudes larger than its thickness (0.9nm),
and as the thickness of the magnetic layer is much smaller than
the exchange length so that the magnetization texture is uni-
form along the z-direction, this system could be considered
to be inherently 2D. This is distinctly dierent from previous
reports on topologically trivial bubbles for instance in yttrium
iron garnet (YIG) films that are >µm thick and where no size-
able DMI is found. By experimentally ascertaining the phase
transitions, we demonstrate the 2D nature of the system as well
as its suitability as a model system to probe 2D phase behavior.
2. Results and Discussion
Using Kerr microscopy imaging we investigated a low-pinning
multilayer stack Ta(5)/Co20Fe60B20(0.9)/Ta(0.08)/MgO(2)/Ta(5)
similar to a material previously characterized in which the
skyrmions show thermally activated diusion at low skyrmion
densities.[11] The studied material exhibits perpendicular mag-
netic anisotropy (PMA) and interfacial DMI.[11] Using out-of-
plane magnetic field sweeps, stripe domains, and a low density
of skyrmions are present in the sample. Upon fixing the out-
of-plane field and a subsequent saturation of the sample using
an in-plane field in any direction, a high density of skyrmions
is nucleated in the sample when the in-plane field is reduced
back to zero abruptly by switching o the power supply to the
in-plane coil. Due to the interfacial DMI and concluded from
current induced motion experiments, the observed skyrmions
were topologically stabilized, with a fixed chirality, and rota-
tional symmetry and a topological charge Q= 1. This is desired
for our further investigation as there are no reports indicating
phase transitions with spatially ordered Q= 0 magnetic bubbles
that are not rotationally symmetric. The density and the mean
radius of the skyrmions is controlled by the values of the out-
of-plane magnetic field applied and the temperature. For details
on the MOKE hysteresis loops and skyrmion lattice nucleation,
see Supporting Information. By varying the out-of-plane field,
the size and as a result also the skyrmion lattice density and
Figure 1. Picture of skyrmion lattice and evolution of phase quantifiers. a) Kerr image of a skyrmion lattice at 335 K with µm sized skyrmions. b) The
evolution of 〈|ψ6|〉 averaged over all skyrmions in one frame in dependence of time after nucleation. The red, blue, and green backgrounds depict the
nucleation, stabilization, and a semi-steady skyrmion state, respectively. 〈|ψ6|〉 is dependent on the temperature and the applied out-of-plane field.
c)Pair correlation function g(r) right after nucleation, in the stabilization phase and in the semi-steady state for temperature 338 K and 20 µT applied
out-of-plane field. After switching o the in-plane field and the resulting skyrmion nucleation, the red pair correlation function (in (c)) emerges and
indicates typical nearest and next-nearest neighbor distances. Blue and green curves show the pair correlation function g(r) in the relaxation and semi-
steady state of the lattice, respectively. In the stabilization process, the correlation function is noisier, whereas in the semi-steady state the function
has a finer distribution.
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ordering is tuned, which is a unique handle compared to previ-
ously used systems, such as colloids with fixed sizes. Variations
in temperatures are found to tune the amount of thermally acti-
vated motion but also the average skyrmion radius, as well as
lattice density due to the changing magnetic properties.[11] To
evaluate phases in 2D systems such as the skyrmion lattice
phases (Figure1a), we employ two quantifiers:
The local orientational order parameter[18,25] (Figure1b)
kne
kl
ni
k
kl
∑
ψ
()
=
θ
=
1
6
1
6 (1)
and the pair correlation function (Figure1c)
gr Nrr r
k
N
kl
N
kl
∑∑
πρ δ
()
=−
−
=≠
1
2r
1
()
1
(2)
The local orientational order parameter is a standard measure
to quantify the emergence of local hexagonal order.[18,25] θkl
describes the angle of the connecting line between a (central)
skyrmion k and the lth of its nk nearest neighbors with respect
to a fixed axis (here the x-axis).[18] The cut-o distance to find
the nearest neighbors was selected to be the position of the first
minimum in the corresponding pair correlation function g(r).
A strict cut is implemented, so the number of neighbors nk will
usually but not necessarily be 6. For a perfect (periodic) trian-
gular lattice, the contribution of all six neighbors yields |ψ6| = 1.
The 1D pair correlation function g(r) (Equation (2)) contains
basic information such as typical nearest and next-nearest
neighbor distances and the general structure of a gas, liquid or
crystal. Particularly, it allows us to quantify the local structure of
a skyrmion lattice in area A by comparing it to a structureless,
homogeneous fluid of area density N
A
ρ
=. Essentially, we count
the number of particles located at a certain distance around
each particle and divide this number by the expected number
of particles in a fluid with no structure. In our modelling
approach we use this quantifier to reproduce the basic structure
of the system while keeping the fitting procedure manageable.
Equation (2) is, however, not suited to visualize the emergence
of hexagonal order like the 2D-pair correlation function, for
example, used in ref. [18].
To study the evolution of the phases of the system, we take a
video using the Kerr microscope after an in-plane magnetic field
is switched o. The observed skyrmions are tracked, their posi-
tions are evaluated, and quantifiers are calculated for each frame
in the video. Calculation of the correlation functions and indi-
vidual skyrmion position evaluation is described in Section 4. The
local orientational order parameter is calculated for every skyr-
mion in one frame except for those on the border of the frame.
Note that in this context the expression “order parameter” refers
to a parameter which quantifies the local orientational order of
a system and is not to be understood in the classical sense as a
parameter which characterizes second-order phase transitions. To
obtain a quick indication of the state and the phase of the system,
we introduce a heuristic parameter 〈|ψ6|〉, which averages the
absolute value of ψ6 over all skyrmions for which ψ6 was com-
puted.[25] From simulations of a soft disc system we find that the
liquid branch of the liquid to hexatic coexistence region is marked
by 〈|ψ6|〉≈ 0.69 irrespective of the exponent of the underlying
repulsive power-law potential used in the simulations. Larger
values correspond to hexatic or solid phases (respectively their
onsets), while smaller values are characteristic for liquid phases.
For details, we refer the reader to Supporting Information.
Figure1b shows 〈|ψ6|〉 of the skyrmion lattice at fixed out-of-
plane field and sample temperature as a function of time after
the initial lattice nucleation. As visible, the angular ordering
as well as translational ordering as quantified by the pair cor-
relation function (Figure 1c) is not constant instantly after
switching o the magnetic in-plane field: A local liquid-like
structure emerges and becomes more pronounced as relaxation
proceeds. Note that it is not possible to distinguish the 1D g(r)
of the hexatic phase from that of a dense liquid as pointed out
above. Immediately after switching o this field, the skyrmions
are nucleated on a timescale that is below the time resolution
of the measurement setup (ms). This is then followed by a
stabilization phase in the range of seconds to tens of seconds.
The stabilization time frame is influenced by the energy land-
scape of the multilayer material and the diusion parameter of
the skyrmions to form an ordered structure that we term here in
line with literature a skyrmion lattice. While the initial ordering
occurs rather quickly in all cases, 〈|ψ6|〉 is still increasing
slightly over the course of our measurement (60 s) consistent
with the expected prolonged equilibration times associated with
the emergence of hexagonal order. We refer to the last 30s of
the 〈|ψ6|〉(t) as the “semi-steady” state, which is a suciently
long period to robustly measure quantifiers. The initial 4 s after
switching o the in-plane field where the highest slope of 〈|ψ6|〉
(t) is found and where the nucleated skyrmions form a lattice is
referred to as the “nucleation” period. The “relaxation” period
covers the remaining part of the time evolution. These criteria
were chosen by comparing the obtained videos at every tem-
perature and out-of-plane field combination for a comparable
evaluation of the skyrmion lattice.
While as shown in Figure1b, at 338 K the system orders with
〈|ψ6|〉> 0.69 (indicating possibly a hexatic phase, see further
below for a detailed discussion), at 330 K 〈|ψ6|〉 only goes up
to the value of 0.55, consistent with the formation of a more
disordered dense liquid phase. Likewise, the pair correlation
function also changes in the course of equilibration (Figure1c).
Fluctuations are related to the thermally activated movement
of the skyrmions that occurs in the lattice. We observe that
skyrmions repel each other and we do not see any significant
skyrmion-skyrmion annihilation thus boding well to study
the phase transitions. Having established the time evolution
of 〈|ψ6|〉, we now systematically study the dependence of the
semi-steady state lattice properties on the external parameters,
temperature and magnetic field to explore the tunability. The
average t
ψ
||
6 is obtained from all frames after 30 s of equili-
bration and shown in Figure 2. With reducing temperature, the
range of out-of-plane field where skyrmions can be stabilized
becomes narrower and a monotonic trend of higher hexagonal
order with higher temperature is observed. The highest tem-
perature achievable was limited by the measurement tempera-
ture control as well as the spatial resolution since the skyrmion
diameter depends on temperature. At too low out-of-plane
fields, after in-plane field sweeps, not only skyrmions are sta-
bilized but also elongated chiral domains are present. These
eectively distort the lattice and hinder its higher ordering so
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that we have focused on parameter combinations where we
have only skyrmions. A decreasing tendency of angular order
is found at increased out-of-plane field values for every studied
temperature. This can result from higher skyrmion-skyrmion
distances, where the thermal movement of the magnetic tex-
tures is more pronounced and thus hinders the ordering of the
lattice. A maximum value of t
ψ
||
6 of around 0.73 is obtained
at the highest investigated temperature and 20 µT out-of-plane
field when also the highest observable skyrmion density is
reached.
As the onset of the hexatic (or even solid) phase is directly
visible in the spatially resolved map of the local orientational
order parameter, we study this at the maximum value of
t
ψ
||
6 (338 K, 20 µT): Figure 3a shows the hexatic skyrmion
domains with coincident orientation of ψ6 as measured by the
angle θ (Euler angle of the complex number ψ6 divided by 6 as
explained in Section 4). The average domain size is of the order
of 50 µm, corresponding to roughly 100 skyrmions. In par-
ticular we see a homogenous distribution of |ψ6| in Figure3c.
For comparison, we also show the corresponding liquid
phase results for T= 330 K and B = 40 µT in Figure3b,d). Note
that skyrmions are much larger under these conditions and
domains of similar orientation are of the order of 10 particles
or less. In this liquid phase, there is no homogeneous distribu-
tion of |ψ6| as shown by the irregular colors in Figure3d. To
understand our results and draw robust conclusions about the
phases and the 2D nature of the studied system, we support the
experimental results with numerical simulations using a model
of soft particles which interact with each other via a repulsive
power-law potential r−n. This choice is purely empirical but
allows us to describe the strong short range repulsive interac-
tion studied previously.[26] At the same time, the chosen poten-
tial benefits from the availability of exact phase diagrams for
a wide range of n.[18] For n≥ 6 (which includes the hard disk
scenario) the transition from the liquid to the hexatic phase was
shown to be of first order followed by a continuous transition to
Figure 2. Time averaged
||
6
ψ
t for dierent out-of-plane fields in the
temperature range 325–338 K. The field and temperature ranges are lim-
ited by the stability of a pure skyrmion lattice as well as the minimum
size of skyrmions that can be detected. The highest ordering achieved is
at 338 Kwith 20 µT.
||
6
ψ
t was calculated from the skyrmion position in
the sample in the semi-steady state part of the skyrmion lattice formation
(after 30 s since the skyrmion nucleation). Empty circles are simulation
results corresponding to T= 338 K and T= 330 K and were determined
after 106 simulation time steps. Dashed lines serve as guidelines between
points only.
Figure 3. Spatial distribution of the local orientational order parameter ψ6 of individual skyrmions. (a) and (c) were evaluated at 338 K and an out-of-
plane field value of 20 µT. This represents the state with the highest value of
||
6
ψ
t in Figure 2. (a) visualizes the orientation of ψ6, that is, the orienta-
tion angle θ, while (c) visualizes the absolute value of ψ6. (b) and (d) are corresponding figures for 330 K and 40 µT.
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the solid phase.[18] For smaller values of n, the transition from
the liquid to the hexatic phase becomes continuous and of the
Kosterlitz-Thouless–Halperin–Nelson–Young (KTHNY) type.[27]
In the following, we want to ascertain to which extent skyr-
mion lattices can be used as generic model systems to explore
the phase behavior of 2D systems akin to colloids.[21–23] At the
same time, we want to gauge if a coarse phenomenological
model from statistical physics is actually able to describe the
bulk macroscopic behavior of skyrmions accurately. Building
upon expansive numerical work, which has determined the
phase behavior of soft disks with great accuracy, Molecular
Dynamics simulations of this model were performed and
mapped onto our skyrmion system.[18,28–30] Parameters were
adjusted to match the pair correlation function of skyrmions
for a given density. Note that fixing n (to, e.g., 6 to represent
interactions between dipoles) will generally lead to a worse
agreement with the experimental g(r). For a detailed discus-
sion of the mapping procedure, see Section 4 and Supporting
Information.
Using this ansatz, we have reproduced the experimen-
tally observed behavior of t
ψ
||
6 for T= 338 K and T= 330K
(Figure 2). Qualitative agreement between simulations and
experiments is found. However, one should note that 〈|ψ6|〉 is
very sensitive to the details of the mapping (e.g., if all details
or only parts of g(r) are used). Another caveat for both simula-
tions and experiments at T= 338 K is the time after which 〈|ψ6|〉
is measured as it increases during the course of equilibration.
Nevertheless, considering that our mapping is purely based on
basic structural information (namely density and the 1D g(r)),
the qualitative agreement shows that static properties of skyr-
mion interactions can indeed be captured by a coarse-grained
phenomenological model.
A more quantitative approach relies on the decay of the spa-
tial correlation function G6 which can also be used to distin-
guish phases in 2D systems:[21,22]
Gr
n
rr
rrrr
kl
kl
∑
ψψ
=
−=
∗
() 1()
()
66
6 (3)
Here, we sum over all nr particle pairs whose distance is r.
In Figure 4, we compare the decay of G6 from experiment at
T= 338 K and B = 20 µT and simulation. While this correlation
function decays exponentially in the liquid phase, quasi-long
range orientational order is expected to emerge in the hexatic
phase.[27] Depending on the equilibration time after which the
correlation function is measured in the simulation, the enve-
lope of G6 increases toward an algebraic decay. We also observe
that the experimental data (black dashed curve) is still decaying
exponentially and is likely not fully equilibrated, yet, in line
with our observations in Figure3a.
The eect of equilibration can also be seen in simulation
snapshots. While after 104 equilibration steps the distribution of
θ in Figure 5a (as well as the decay of G6) is similar to the corre-
sponding experimental plot (Figure3a), the domains of similar
orientation continue to grow as indicated by a snapshot taken
after 108 equilibration steps (Figure5b).
3. Conclusions
Based on our numerical simulations, we conclude that the
observation of multiple domains in the experiment (Figure3)
is likely the result of an incomplete equilibration process as
equilibration times are notoriously large in an emergent hexatic
(or solid) phase. This is corroborated by the observation that
Figure 4. Decay of the spatial correlation function G6 for the experimental
system (338 K, 20 µT, averaged over frames 300–960) and matching simu-
lations after dierent runtimes (single snapshot of a quadratic simulation
box containing 40000 particles).
Figure 5. Spatial distribution of θ in a simulation corresponding to a sample temperature of 338 K and an out-of-plane field of 20 µT after a) 104 and
b) 108 equilibration steps. Only a small part of the simulation box is shown to make plots comparable to Figure 3a.
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the sizes of the experimental domains continue to grow up to
the maximum time which can be measured (that is limited by
the setup stability). Additionally, we occasionally see structural
defects that pin certain skyrmions that are thus not ordered
locally and remain unordered potentially leading to artificial
domain wall pinning.
In conclusion, we have analyzed the phases of skyrmion lat-
tices to identify the reduced dimensionality of this µm sized but
sub-nm thick system. We have shown that by using the pair
correlation function and local orientational order parameter we
can characterize the skyrmion lattice system, which allows us to
investigate 2D phase transitions. Temperature and out-of-plane
field impact density and mean skyrmion-skyrmion distance
and translate to dierent nucleation dynamics and hexagonal
ordering of the observed lattice. We find that the hexagonal
order increases with higher temperature and field values in the
range of 10–30 µT. Above 338 K, the skyrmion lattice cannot be
resolved with the optical microscope setup. For the majority of
the selected parameters, we observe behavior consistent with a
2D, dense liquid. However, we also find that for selected condi-
tions, our system is in an emergent hexatic (or even solid) phase
showing its 2D nature. As expected for the hexatic phase, we find
that the equilibration in this phase is very slow. By comparison
with theory, we were able to reproduce qualitatively the experi-
mentally observed phase behavior using computer simulations
with a simple phenomenological model based on soft disks by
matching density and the 1D pair correlation function. We thus
demonstrate that static behavior of skyrmion ensembles may be
described by a simple 2D model system highlighting that our
skyrmion lattices can indeed be used as 2D model systems with
major advantages in terms of tunability and speed compared to
conventionally used 2D model systems such as colloids.
4. Experimental Section
Sample Parameters: The sample was prepared using magnetron
sputtering in a Singulus Rotaris sputtering tool. The base pressure during
the growth process was less than 3 × 10-8 mbar. The composition of the
single stack was Ta(5)/Co20Fe60B20(0.9)/Ta(0.08)/MgO(2)/Ta(5), with
the thickness of individual layers given in nanometers in parentheses.
With the used deposition system, the thickness of the individual layers
can be tuned in the stack in a controlled manner with high accuracy. The
stack is similar to the one reported on previously where it was found that
the very thin Ta layer on top of the CoFeB plays a key role in setting the
eective anisotropy.[11] The sample was characterized using the magneto-
optical Kerr eect (MOKE) measurement. Single skyrmions can be
stabilized in the ferromagnetic layer using out-of-plane field sweeping,
meaning applying an oscillating out-of-plane field over several oscillation
periods. This procedure moves the domains and eventually they break
into smaller domains and in this case can form skyrmions. Elongated
domains and skyrmions exhibit thermally activated diusion. The
hysteresis loop in applied out-of-plane field shows an hour-glass shape,
typical for material with the presence of skyrmions (see Supporting
Information). With higher temperature, the hysteresis loop is tilted
toward larger applied fields. This indicates a change of the anisotropy
of the material with temperature. The lowest investigated temperature is
determined by the ability to stabilize the skyrmion lattice. Below 325K,
only stripe domains were nucleated by the saturation of the sample
with an in-plane field. The highest achievable temperature for the lattice
investigation was determined by the resolution of the microscope and
the thermally activated motion of skyrmions. Above 338 K, the size of the
skyrmions was comparable to the resolution of the Kerr microscope. The
skyrmions movement was also more rapid. Above this temperature, no
reliable skyrmion tracking in this material system could be performed.
The DMI was measured by investigating the domain periodicity at
zero magnetic field and by comparison with micromagnetic simulations.
Using the measured parameters of magnetic anisotropy and saturation
magnetization and an exchange parameter of A = 10 pJ m−1, it was found
that DMI is needed to stabilize skyrmion structures. The obtained DMI
from comparing the measurement with the simulations is comparable
to the value published previously for a similar stack (DI= 0.3 ±
0.1 mJ m−2).[11] The DMI in the material is suciently strong so that only
topologically non-trivial skyrmion spin structures are stable. The eect of
the topology on the skyrmion lattice phases and phase transitions could
be studied in a material where both topologically trivial bubbles and
non-trivial skyrmions are stable, which however goes beyond the scope
of the current work.
Measurement Setup: A commercial Evico GmbH MOKE microscope
was used. The optical spatial resolution is approximately 400 nm.
The temporal resolution remains the same at any temperature and
magnetic field and is 62.5 ms. At higher temperatures, the skyrmions
become much smaller and their diusion is enhanced and thus faster.
This hinders the reliable tracking to identify the position given the
time resolution. Therefore, the threshold where skyrmions can be
tracked is determined by both the temporal and spatial resolution of
the microscope setup. The in-plane field coil was supplied from the
microscope manufacturer. The highest achievable in-plane field was
300 mT. The coil for the out-of-plane field application was custom built
at the University of Mainz. The coil was designed to have negligible
coercivity and to be able to supply the sample with very small controlled
fields in orders of µT. A current versus magnetic field calibration was
performed using a Gaussmeter in the position of the sample and used
during the measurement along dierent directions. The calibration
for the Earth magnetic field was done using the hysteresis loop of
the material. The residual Earth field caused an oset in the x-axis
of the M-H loop. The authors compensated for this oset in the coil
calibration. The calibration of all stray-fields and resulting osets in the
field values was performed before every measurement and no changes
were observed during the timescale of the measurement. Adjustment
of the in-plane field coil was done the same way. When the coil was
tilted or set in a way that a cross field between the out-of-plane and
in-plane field was present, the M-H loop of the material was shifted. The
in-plane field coil was adjusted so that the hysteresis loop is the same as
without the coil, only with out-of-plane field. A stage with two HighTech
QuickCool QC-32-0.6.1.2 Peltier elements was used for the temperature
change of the sample in the range of 280–350 K. The temperature was
externally controlled by measurement of resistivity of a Pt100 resistor,
which was placed next to the multilayer sample. The stability of the set
temperature was measured to be within 0.3 K. The frame rate of the
microscope camera was 16 frames s−1; therefore, the time resolution of
the microscope measurement was 62.5ms.
Skyrmion Tracking: Skyrmion lattices are visualized using a Magneto-
Optical Kerr-Microscope. In the pictures the out-of-plane magnetization
is represented by a grey scale, so that the skyrmions appear as light blobs
on a dark(er) background. Videos recorded that way were consecutively
analyzed using the Trackpy[31] package. In a first stage, it locates the
skyrmions by detecting Gaussian-like blobs in the grey scale movies.
It was ensured that the software finds all skyrmions in the individual
video frames. Several parameters are set to optimize the recognition
for reliable results. Most importantly, the mask-parameter sets a rough
estimate for the pixel-diameter of the features to be found. During
the evaluation, it is set slightly above the average skyrmion diameter
determined by simple binarization of the frame. The separation-
parameter enforces a minimum separation between the recognized
features, this way over-recognition in defective areas is prevented. A
safe value for the recognition is several pixels lower than the average
skyrmion distance. The percentile-parameter depends on the contrast of
the video and indicates to which extend the features are expected to be
brighter than the surrounding area. The noise-parameter is a measure
for the “sharpness” of the features to be detected and can vary between
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2004037 (7 of 8) © 2020 The Authors. Published by Wiley-VCH GmbH
measurement videos with dierent external parameters. Most of the
skyrmion diameters are in a range from 7 to 13 pixels, corresponding
to 4.5–8.5 µm. For example, at the temperature of 338 K and the out-of-
plane field of 20 µT, we set the mask to 9, the separation to 4, and the
noise to 0.15.
Quantifiers for Phase Transitions in 2D Systems: The pair correlation
function (PCF) (Equation (2)) determines the probability of finding two
skyrmions at a distance r from each other. The position of the first peak
assesses the mean nearest neighbor distance and deep in the solid
phase characteristic sharp peaks resulting from the underlying lattice
appear. It is, however, impossible to distinguish g(r) of liquid, hexatic
and solid phases close to the phase transition and other identifiers
need to be considered. Since the disk-shaped skyrmions develop
hexagonal order as the bulk density increases, one can resort to the
local orientational order parameter ψ6 (Equation (1)).[27] This complex
parameter measures deviations from hexagonal order. The absolute
value |ψ6|= 1 for a perfect triangular lattice and decreases to 0 with
increasing disorder. In addition to the absolute value of ψ6 one can also
extract the local orientation angle of neighboring skyrmions, that is, the
Euler angle of ψ6 divided by 6. Note that the orientation of a hexagonally
ordered cluster consisting of the central particle and its six neighbors is
essentially determined by the angle between the x-axis and the vector of
the central particle and its neighbor in the range of 0–60°. The factor of
six in the definition of ψ6 projects all vectors between the central particle
and its neighbors on top of each other and ψ6 averages over these
projections. The orientation angle (ranging from 0° to 60°) is therefore a
gauge for the local orientation of the cluster with respect to the x-axis.
This parameter is well-suited to visualize clusters of equal orientation. In
simulations of soft disks, it was also noticed empirically that the mean
|ψ6| is roughly 〈|ψ6|〉≈ 0.69 at the liquid branch of the liquid to hexatic
phase transition (see Supporting Information), and this parameter was
used as an additional indicator for the transition. For computing g(r)
and ψ6, the MD analysis program FREUD was employed.[17]
Molecular Dynamics Simulations of Soft Disks: Molecular Dynamics
simulations of soft disks were performed using the model of Kapfer and
Krauth with the HOOMD Molecular Dynamics package and a Langevin
integrator:[18,32]
σ
()
=


Vr
r
n
(4)
In this coarse, phenomenological model for the bulk behavior of
skyrmions, σ roughly corresponds to the mean skyrmion distance, and
n denotes the steepness of the potential. By running MD simulations
at the experimentally determined skyrmion density, the authors were
able to adjust the simulation potential so that the pair correlation for
the simulated soft disks matches the pair correlation of the skyrmions.
In order not to overparameterize the mapping to the experimentally
measured PCFs, the authors only adjusted n and set σ constant. Even
though the position of the first peak of the PCFs is not necessarily
identical with the σ of the simulation potential, this approximation turns
out to be suciently accurate for the examined densities. Therefore,
σ was set in the simulations to be the position of the first peak of the
experimentally determined PCFs. Simulations were then run for varying
n in the range between 6 and 12 with 0.1 resolution. The matching of
the simulated and experimental PCFs is determined as mean squared
deviation measured up to the fourth maximum. This deviation shows a
smooth dependence of n and a clear minimum which were taken as best
match to the experiment. The optimal n is typically around 10 (for T= 338 K)
and somewhat lower for lower temperatures. The determined density,
σ and n allow running simulations mapped closely to the experiment
and the estimated underlying experimental potential. For these mapped
simulations, the mean absolute value 〈|ψ6|〉 were determined, which is
to some extent, dependent on the equilibration time of the simulations.
If not mentioned otherwise, the system was equilibrated for 106 time
steps before measurements were taken. It should also be noted that the
simulations employ a Langevin dynamics thermostat with a time step
of 10−3 and could be further improved by including additional specific
terms to account for gyrotropic dynamics.[33–35] It is not expected that
the current static equilibrium results are aected because such terms
do not contribute to the energy of the system and thus do not break
detailed balance.[35] However, to analyze the dynamics of the system
evolution in the future, such terms need to be considered.
Data Availability
The data that support the plots within this paper and other findings of
this study are available from the corresponding author upon reasonable
request.
Supporting Information
Supporting Information is available from the Wiley Online Library or
from the author.
Acknowledgements
The authors would like to thank Kurt Binder for insightful discussions
and acknowledge funding from TopDyn, SFB TRR 146, SPP 1726, and
SFB TRR 173 Spin+X (project A01). The experimental part of the project
was additionally funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) project No. 403502522 (SPP 2137
Skyrmionics) and the EU (3D MAGIC ERC-2019-SyG 856538, s-NEBULA
H2020-FETOPEN-2018-2020 863155).
Open access funding enabled and organized by Projekt DEAL.
Conflict of Interest
The authors declare no conflict of interest.
Author Contributions
J.Z. and F.D. contributed equally to this work. M.K., P.V., and J.Z.
proposed and supervised the study. J.Z. and N.K. fabricated and
characterized the multilayer samples. J.Z., N.K., and K.R. prepared the
measurement setup and conducted the experiments using the Kerr
microscope. F.D., J.Z., Y.G., T.W., and N.K. evaluated the experimental
data with the help of K.R., P.V., M.V., and K.L. F.D. and P.V. conducted
theoretical simulations and comparison with the experimental results.
J.Z. and F.D. prepared the manuscript with the help of M.V., P.V., and
M.K. All authors commented on the manuscript.
Keywords
2D phase transitions, hexatic phase, skyrmion lattice, skyrmion-skyrmion
interactions
Received: May 9, 2020
Revised: July 17, 2020
Published online: September 3, 2020
[1] A.Fert, N.Reyren, V.Cros, Nat. Rev. Mater. 2017, 2, 17031.
[2] G. Finocchio, F. Büttner, R. Tomasello, M. Carpentieri, M. Kläui,
J.Phys. D. Appl. Phys. 2016, 49, 423001.
[3] X.Zhang, M.Ezawa, Y.Zhou, Sci. Rep. 2015, 5, 9400.
Adv. Funct. Mater. 2020, 30, 2004037

www.afm-journal.dewww.advancedsciencenews.com
2004037 (8 of 8) © 2020 The Authors. Published by Wiley-VCH GmbH
[4] F. Jonietz, S. Mühlbauer, C. Pfleiderer, A. Neubauer, W. Münzer,
A.Bauer, T.Adams, R. Georgii, P.Böni, R. A. Duine, K. Everschor,
M.Garst, A.Rosch, Science 2010, 330, 1648.
[5] X. Z.Yu, N.Kanazawa, W. Z.Zhang, T.Nagai, T.Hara, K.Kimoto,
Y.Matsui, Y.Onose, Y.Tokura, Nat. Commun. 2012, 3, 988.
[6] W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch,
F. Y.Fradin, J. E.Pearson, Y.Tserkovnyak, K. L.Wang, O.Heinonen,
S. G. E. E.te Velthuis, A.Homann, Science 2015, 349, 283.
[7] S. Woo, K. Litzius, B. Krüger, M.-Y. Im, L. Caretta, K. Richter,
M. Mann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal,
I.Lemesh, M.-A.Mawass, P.Fischer, M.Kläui, G. S. D. D. Beach,
Nat. Mater. 2016, 15, 501.
[8] I. Lemesh, K. Litzius, M. Böttcher, P. Bassirian, N. Kerber,
D. Heinze, J. Zázvorka, F. Büttner, L. Caretta, M. Mann,
M. Weigand, S. Finizio, J. Raabe, M. Im, H. Stoll, G. Schütz,
B.Dupé, M.Kläui, G. S. D.Beach, Adv. Mater. 2018, 30, 1805461.
[9] F.Zheng, H.Li, S.Wang, D.Song, C.Jin, W.Wei, A.Kovács, J.Zang,
M. Tian, Y. Zhang, H. Du, R. E. Dunin-Borkowski, Phys. Rev. Lett.
2017, 119, 197205.
[10] K. Everschor-Sitte, J. Masell, R. M. Reeve, M. Kläui, J. Appl. Phys.
2018, 124, 240901.
[11] J. Zázvorka, F. Jakobs, D. Heinze, N. Keil, S. Kromin, S. Jaiswal,
K.Litzius, G.Jakob, P.Virnau, D.Pinna, K.Everschor-Sitte, L.Rózsa,
A.Donges, U.Nowak, M.Kläui, Nat. Nanotechnol. 2019, 14, 658.
[12] D. Prychynenko, M. Sitte, K. Litzius, B. Krüger, G. Bouriano,
M. Kläui, J. Sinova, K. Everschor-Sitte, Phys. Rev. Appl. 2018, 9,
014034.
[13] T. Nakajima, H. Oike, A. Kikkawa, E. P. Gilbert, N. Booth,
K.Kakurai, Y.Taguchi, Y.Tokura, F.Kagawa, T.Arima, Sci. Adv. 2017,
3, e1602562.
[14] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,
A.Neubauer, R.Georgii, P.Böni, Science 2009, 323, 915.
[15] P. Huang, T. Schönenberger, M. Cantoni, L. Heinen, A. Magrez,
A. Rosch, F. Carbone, H. M. Rønnow, Nat. Nanotechnol. 2020, 1,
https://doi.org/10.1038/s41565-020-0716-3.
[16] S. A. Díaz, C. J. O. Reichhardt, D. P. Arovas, A. Saxena,
C.Reichhardt, Phys. Rev. B 2017, 96, 085106.
[17] V. Ramasubramani, B. D. Dice, E. S. Harper, M. P. Spellings,
J. A. Anderson, S. C. Glotzer, Comput. Phys. Commun. 2020, 254,
107275.
[18] S. C.Kapfer, W.Krauth, Phys. Rev. Lett. 2015, 114, 035702.
[19] M. Engel, J. A. Anderson, S. C. Glotzer, M. Isobe, E. P. Bernard,
W.Krauth, Phys. Rev. E 2013, 87, 042134.
[20] E. P.Bernard, W.Krauth, Phys. Rev. Lett. 2011, 107, 155704.
[21] K.Zahn, R.Lenke, G.Maret, Phys. Rev. Lett. 1999, 82, 2721.
[22] K.Zahn, G.Maret, Phys. Rev. Lett. 2000, 85, 3656.
[23] P. Keim, G. Maret, H. H. Von Grünberg, Phys. Rev. E 2007, 75,
031402.
[24] Y. Nishikawa, K. Hukushima, W. Krauth, Phys. Rev. B 2019, 99,
064435.
[25] P. Dillmann, G. Maret, P.Keim, J. Phys. Condens. Matter 2012, 24,
464118.
[26] U. K.Rößler, A. A.Leonov, A. N.Bogdanov, J. Phys. Conf. Ser. 2011,
303, 012105.
[27] D. R.Nelson, B. I.Halperin, Phys. Rev. B 1979, 19, 2457.
[28] J.Zausch, P.Virnau, K.Binder, J.Horbach, R. L.Vink, J. Chem. Phys.
2009, 130, 064906.
[29] D.Winter, J.Horbach, P.Virnau, K.Binder, Phys. Rev. Lett. 2012, 108,
028303.
[30] S. K.Das, S. A.Egorov, B.Trefz, P.Virnau, K.Binder, Phys. Rev. Lett.
2014, 112, 198301.
[31] Trackpy, https://github.com/soft-matter/trackpy (accessed: July
2019).
[32] J. A. Anderson, C. D. Lorenz, A. Travesset, J. Comput. Phys. 2008,
227, 5342.
[33] A. A.Thiele, Phys. Rev. Lett. 1973, 30, 230.
[34] B. L. Brown, U. C. Täuber, M. Pleimling, Phys. Rev. B 2018, 97,
020405.
[35] B. L. Brown, U. C. Täuber, M. Pleimling, Phys. Rev. B 2019, 100,
024410.
Adv. Funct. Mater. 2020, 30, 2004037