# Skyrmionic Spin Structures in Layered Fe5GeTe2 Up To Room Temperature

Maurice Schmitt  Johannes Gutenberg Universität Mainz, Institut für Physik,
Staudingerweg 7, 55128 Mainz, Germany  Thibaud Denneulin  Ernst Ruska Centre
for Microscopy and Spectroscopy with Electrons and Peter Grünberg Institute,
Forschungszentrum Jülich, 52425 Jülich, Germany  András Kovács  Ernst Ruska
Centre for Microscopy and Spectroscopy with Electrons and Peter Grünberg
Institute, Forschungszentrum Jülich, 52425 Jülich, Germany  Tom G. Saunderson
Johannes Gutenberg Universität Mainz, Institut für Physik, Staudingerweg 7,
55128 Mainz, Germany  Philipp Rüßmann  Institute of Theoretical Physics and
Astrophysics, University of Würzburg, Am Hubland, 97074 Würzburg, Germany
Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich, 52425 Jülich, Germany  Aga Shahee  Johannes
Gutenberg Universität Mainz, Institut für Physik, Staudingerweg 7, 55128
Mainz, Germany  Tanja Scholz  Max Planck Institute for Solid State Research,
Heisenbergstraße 1, 70569 Stuttgart, Germany  Amir Tavabi  Ernst Ruska Centre
for Microscopy and Spectroscopy with Electrons and Peter Grünberg Institute,
Forschungszentrum Jülich, 52425 Jülich, Germany  Martin Gradhand  University
of Bristol, School of Physics, HH Wills Physics Laboratory, Tyndall Avenue,
Bristol BS8 1TL, England  Phivos Mavropoulos  National and Kapodistrian
University of Athens, Department of Physics, University Campus, GR-157 84
Zografou, Athens  Bettina Lotsch  Max Planck Institute for Solid State
Research, Heisenbergstraße 1, 70569 Stuttgart, Germany  Rafal Dunin-Borkowski
Ernst Ruska Centre for Microscopy and Spectroscopy with Electrons and Peter
Grünberg Institute, Forschungszentrum Jülich, 52425 Jülich, Germany  Yuriy
Mokrousov  Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich, 52425 Jülich, Germany  Stefan Blügel  Peter Grünberg
Institut and Institute for Advanced Simulation, Forschungszentrum Jülich,
52425 Jülich, Germany  Mathias Kläui  Johannes Gutenberg Universität Mainz,
Institut für Physik, Staudingerweg 7, 55128 Mainz, Germany  QuSpin, Norwegian
University of Science and Technology, Department of Physics, NTNU NO-7491,
Trondheim, Norway

###### Abstract

The role of the crystal lattice, temperature and magnetic field for the spin
structure formation in the 2D van der Waals magnet Fe5GeTe2 is a key open
question. Using Lorentz transmission electron microscopy, we experimentally
observe topological spin structures up to room temperature in the metastable
pre-cooling and stable post-cooling phase of Fe5GeTe2. Over wide temperature
and field ranges, skyrmionic magnetic bubbles form without preferred
chirality, which is indicative of a centrosymmetric crystal structure. In the
pre-cooling phase, these bubbles are observable even without the application
of an external field, while in the post-cooling phase, a transformation from
bubble domains to stripe domains is seen. To understand the magnetic order in
Fe5GeTe2 we compare macroscopic magnetometry characterization results with
microscopic density functional theory calculation. Our results show that even
up to room temperature, topological spin structures can be stabilized in
centrosymmetric van der Waals magnets.

##  1 Introduction

Two-dimensional (2D) magnetic van der Waals (vdW) materials have seen rising
attention since their recent introduction into the field of spintronics and
beyond [1]. Many of these materials show promising properties for future
applications, such as the semiconducting 2D ferromagnet CrI3 [2], the metallic
2D ferromagnet Fe3GeTe2 [3] and the very recently discovered metallic 2D
ferromagnet Fe5GeTe2 that orders even at relatively high temperatures [4]. The
operation of a spintronic device requires ferromagnetic materials with room-
temperature magnetic ordering, but most of the 2D vdW magnetic materials are
either not ferromagnetic or do not possess a high ordering temperature. Here,
Fe5GeTe2 is unique in that it has attracted specific scientific attention due
to its room-temperature ferromagnetism with an ordering temperature of
$T_{\text{C}}\approx 310\,\text{K}$. To make use of a magnetic system for
applications, spin structures are often required as information carriers.
Interest in the dynamics of spin structures accordingly motivated recent
studies on domain walls in vdW magnets [5, 6]. A particularly promising and
exciting spin structure is the magnetic skyrmion [7, 8, 9]. So for 2D magnet
applications, the required next step is the observation of topological
magnetic spin structures near room temperature in Fe5GeTe2. Fe5GeTe2 is
additionally intriguing as it exhibits a phase transition at $\approx
100\,\text{K}$ when it is first synthesized in a metastable phase [10]. So
there is particular interest in the spin structures in a Fe5GeTe2 crystal,
before and after going through the irreversible first-order phase transition
upon cooling below $T\approx 100\,\text{K}$. These phases are termed the
metastable pre-cooling and the stable post-cooling phases. Real space probing
of the crystal lattice and spin structure in both of these phases is still a
major open question.

In the present work, bulk magnetic properties in the phases of 2D Fe5GeTe2 are
characterized using a superconducting quantum interference device (SQUID), in
combination with magnetic domain images using Lorentz transmission electron
microscopy (L-TEM), which reveals stable topological spin structures up to
room temperature. Further, the crystallographic space groups previously
suggested for Fe5GeTe2 are investigated and we resolve open questions about
their symmetry by presenting a spin chirality analysis. The spin chirality
analysis involves careful considerations of the chirality of observed spin
structures in Fe5GeTe2, in order to deduce if the chiral Dzyaloshinskii–Moriya
interaction (DMI) is present in the sample. Finally, the magnetic ordering of
Fe5GeTe2 is investigated closely and we determine whether or not the system is
in a spin glassy state. This is done via a careful comparison of the effective
anisotropy obtained on a macroscopic scale with the effective anisotropy
obtained on a microscopic scale.

##  2 Results and Discussion

###  2.1 Structural and composition analysis

Fig. (1)(a) is a cross-section atomic resolution HAADF STEM image of the
single-crystal Fe5GeTe2 used in the present work. Fig. (1)(b) is a color-coded
composition map obtained using EDX, which shows the periodic distribution of
iron, germanium and tellurium. 2D layers consist of a sandwich of two planes
of Te (brightest columns) with Ge and Fe planes located in-between. The layers
are stacked along the c-axis and are approximately 1 nm thick including the
vdW gap (dark area between two neighboring Te planes). A schematic model of
the atomic structure is overlaid in the middle of Fig. (1)(a) with the same
color code as in Fig. (1)(b). Following the notation used in Ref. [4], the
Fe(2) and Fe(3) columns can be identified but the Fe(1) columns are not
visible. Since STEM images show a projection of the crystal structure over the
thickness of the lamella, this iron site is not easily visible because of its
split-site nature [4]. Fig. (1)(c) is another HAADF STEM that shows the
presence of a stacking fault as indicated by red lines and an arrow, which has
been previously reported in Fe5GeTe2 [11]. Here, stacking faults were observed
primarily near the surface of the crystal (down to 200 nm below the surface).
They are scarce deeper in the bulk of the sample, which has essentially a
regular crystal structure, as shown in Fig. (1)(a). Fig. (1)(d) is a plan-view
image where the c-axis is perpendicular to the image, which shows the
hexagonal shape of the lattice.

Figure 1: High resolution analyses of the crystal structure of Fe5GeTe2. (a)
High resolution HAADF STEM image of the structure with the c-axis along the
vertical direction in the image plane. (b) Color-coded EDX composition map
showing the distribution of iron, germanium and tellurium. A schematic model
of the atomic structure is overlaid on image (a) with the same color code as
in panel (b). (c) HAADF STEM image showing a stacking fault as indicated by
red lines and an arrow. (d) HAADF STEM image of the structure with the c-axis
perpendicular to the image plane.

###  2.2 Magnetic imaging of skyrmionic spin structures

We first observe the magnetization configuration at room temperature in our
sample, as the type of spin structure also entails information about the
underlying crystal structure of Fe5GeTe2. The space group of Fe5GeTe2 has not
so far been identified unambiguously. Li et al. and May et al. report the
centrosymmetric space group $R\bar{3}m$ (No. 166) [12, 4], whereas Stahl et
al. reported the non-centrosymmetric space group $R3m$ (No. 160) [11]. Here we
consider the skyrmionic spin structures in Fe5GeTe2 to resolve this space
group issue. A prerequisite for chiral spin structures, such as skyrmions and
chiral domain walls, is the DMI, which occurs in systems with broken inversion
symmetry [13]. In this regard, chiral spin structures would not be stabilized
in bulk Fe5GeTe2 if its space group is $R\bar{3}m$, whereas they could be
stabilized in bulk Fe5GeTe2 with the non-centrosymmetric space group $R3m$.

Fig. (2)(a, b) show an in-focus phase shift image and the corresponding
magnetic induction map reconstructed from an off-axis electron hologram
obtained in a lamella cut perpendicular to the c-axis. The sample is
magnetized along the crystallographic c-axis, which can also be deduced from
SQUID measurements, as discussed in section S1 of the supplementary
information. Two different types of circular spin structures can be observed.
These bubbles with a black or white electron phase contrast correspond to
type-I bubbles with a clockwise or counter-clockwise field rotation [14].
These two possible chiralities were found to occur with equal probabilities.
Bubbles that show a black to white phase gradient correspond to topologically
trivial type-II bubbles in which the rotation of the field changes [14]. Fig.
(2)(c) shows schematically the magnetic field in type-I and type-II bubbles.
The presence of both type-II and type-I bubbles with two possible chiralities
indicate that they are not stabilized by DMI, but by dipolar interactions
[15]. These observations are indicative of a centrosymmetric crystal structure
and thus support the results of Li et al. and May et al. regarding the
centrosymmetric space group $R\bar{3}m$ for Fe5GeTe2 [12, 4].

Figure 2: Magnetic imaging of Fe5GeTe2, revealing skyrmionic spin structures.
(a) phase shift image obtained using off-axis electron holography at $T$ = 200
K and $B=0\,\text{mT}$ in Fe5GeTe2 with the c-axis perpendicular to the image
plane after previously applying out-of-plane fields up to $B=566\,\text{mT}$.
(b) Corresponding color-coded magnetic induction map where the direction of
the magnetic field is given by the color wheel in the bottom right corner.
Type-II as well as Type-I bubbles with opposite winding numbers are present.
(c) Schematic illustration of type-I and type-II bubbles. (d) Series of
Fresnel images taken at various temperatures and external out-of-plane fields
indicated in the figure, with a defocus of -1 mm, in a plan-view lamella
(c-axis is perpendicular to the image plane). Bubbles form at higher external
fields for lower temperatures.

A series of Fresnel defocus images taken at different temperatures and
external fields, as shown in Fig. (2)(d), reveals that the formation of
bubbles is favorable over labyrinth domains at larger applied fields, and that
the bubbles form at fields slightly smaller than the saturating fields at any
temperature. This is in line with previous results on other systems, where it
was shown that skyrmion formation can be favorable at sufficiently strong
external fields [16]. Furthermore, we observe that the sizes of the magnetic
bubbles shrink as the opposing net magnetization grows with external field,
which is also in line with previous reports for chiral skyrmions and non-
chiral skyrmions [17, 14].

###  2.3 Pre-cooling and post-cooling magnetic properties

In order to explore the bulk magnetic properties of Fe5GeTe2 in both the
metastable pre-cooling and stable post-cooling phases, we have taken SQUID
measurements of our sample while it is going through the phase transition. The
temperature dependence of the magnetization for two magnetic field directions,
$B$ parallel to the ab-plane ($B_{\text{ab-plane}}$) and $B$ parallel to the
c-axis ($B_{\text{c-axis}}$) in an external field of 0.1 T for Fe5GeTe2 is
shown in Fig. (3)(a) and (b), respectively. For $B_{\text{ab-plane}}$, the
first Field Cooled Cooling (FCC1) curve reveals three magnetic anomalies. The
anomaly at 271 K corresponds to the magnetic ordering temperature
($T_{\text{C}}$). The anomaly related to decreasing magnetization at 136 K
($T_{C2}$) may correspond to a transition to a spin glassy phase [18]. Another
possible explanation is that Fe5GeTe2 may be in a ferrimagnetic order under
these conditions, as previously suggested for a similar temperature range by
Ohta et al. and Alahmed et al., respectively [19, 20]. However, as soon as our
sample reaches the stable post-cooling phase, it it found to be in a
ferromagnetic state. Finally, the anomaly of a sudden increase in
magnetization at $T=87\,\text{K}$ corresponds to an irreversible magneto-
structural phase transition ($T_{S}$) from a metastable pre-cooling phase to a
stable post-cooling phase, consistent with previous results [4, 11]. In the
Field Cooled Warming (FCW) curve and the second Field Cooled Cooling (FCC2)
curve, no such anomaly is evident at $\approx 100\,\text{K}$, thus confirming
the 87 K anomaly of FCC1 data was due to the non-reversible nature of this
magneto-structural phase transition. While a weak anomaly with an increase in
magnetization is evident at 118 K in FCW followed by a narrow hysteresis
between FCW and FCC2, it may be due to some spin reoriented phase transition.
Further, the critical temperature $T_{\text{C}}$ has shifted from 270 K to 310
K. Such a shift in $T_{\text{C}}$ is consistent with previous reports [10].
The difference in the net magnetization between 100 K to 300 K for FCW and
FCC1 curves may be partially caused by the two different spin textures that
Fe5GeTe2 exhibits between the stable post-cooling and the metastable pre-
cooling phases, which will be discussed later in more detail.

Figure 3: Magnetization vs temperature of Fe5GeTe2, where the external field
$B$ is applied along the c-axis and in the ab-plane with a field strength of
0.1 T. Zero field cooled (ZFC) indicates that the sample was cooled without
application of an external field and then the field of 0.1 T was applied at
the lowest temperature followed by measurements performed during warming.
FCC/FCW indicates measurements performed subsequently during cooling/warming
with the field kept on, respectively. The numbers 1, 2, 3 defines the
measurement sequence. When the external magnetic field is in the ab-pane, the
sequence is FCC1, followed by FCW & finally FCC2, while for $B$ along the
c-axis, the sequence is FCC followed by FCW and finally ZFC.

Due to the irreversibility of the phase transition that occurs at 87 K, newly
synthesized crystal pieces were chosen for the $B_{\text{c-axis}}$
measurements. The $B_{\text{c-axis}}$ data curves show similar magnetic
behavior to that of the $B_{\text{ab-plane}}$ configuration. The slightly
different values of $T_{C2}$ and $T_{S}$ for the $B_{\text{ab-plane}}$ and
$B_{\text{c-axis}}$ FCC curves may be due to the different magneto-crystalline
anisotropy. That $T_{\text{C}}$ is almost the same value in both
configurations further indicates that the magneto-crystalline anisotropy is a
possible origin of the observed difference in $T_{C2}$ and $T_{S}$ for the two
measurement configurations. The third and final curve is a zero-field cooled
(ZFC) curve, which also shows a $T_{\text{C}}$ of 310 K with low net
magnetization. The large difference between the ZFC and FCW curves may have
resulted from the occurrence of different domain configurations due to the
external magnetic field-induced reorientation of ferromagnetic components.

In Fig. (4), one can quantitatively see an increase in magnetization in the
post-cooling phase compared to the pre-cooling phase under otherwise
equivalent conditions. This is true for the wide range of measured
temperatures and external fields, in both the $B_{\text{c-axis}}$ and
$B_{\text{ab-plane}}$ configurations. e.g. at 150 K and 100 mT in the
$B_{\text{ab-plane}}$ configuration:

$M_{\text{pre-cooling}}^{\text{150K}}(B=100\,\text{mT})=(5.75\pm 0.12)\cdot
10^{5}\text{A m}^{-1}\;,$ (1) $M_{\text{post-
cooling}}^{\text{150K}}(B=100\,\text{mT})=(6.54\pm 0.13)\cdot 10^{5}\text{A
m}^{-1}\;.$ (2)

The SQUID measurements yielded a saturation magnetization of

$M_{\text{s}}=(8.12\pm 0.16)\cdot 10^{5}\text{A m}^{-1}\;.$ (3)

Figure 4: The magnetic field dependent magnetization curves with the external
field along the c-axis and in the ab-plane at different temperatures. (a, b)
for the pre-cooling and (c, d) for the post-cooling phase.

To confirm the validity of this saturation magnetization value, an independent
method to determine the saturation magnetization via analysis of the stripe
domain patterns is used [21]. The resulting value is $M_{\text{s}}=(2.56\pm
1.40)\cdot 10^{6}\,\text{A m}^{-1}$. This value has a relatively large
uncertainty, and the method neglects e.g. shape anisotropy and surface
defects. However, it is sufficiently close to the saturation magnetization
measured with the SQUID and thus confirms its validity. The details of the
method and calculation can be found in section S2 of the supplementary
information.

Finally, we have taken a series of Fresnel defocus images of our sample at
temperatures ranging from $95\,\text{K}$ to $290\,\text{K}$ and at a constant
external field of $B=100\,\text{mT}$, both in the pre-cooling and post-cooling
phases. The phase transition from the metastable pre-cooling to the stable
post-cooling phase in the TEM lamellas was achieved by cooling the sample down
to $\approx 77\,\text{K}$, leaving it in liquid nitrogen for one hour. The
images for these temperature sweeps are shown in Fig. (5). A striking
difference between the two sets of images is that skyrmions are formed in the
pre-cooling phase, but stripe domains form in the post-cooling phase, under
otherwise equivalent circumstances. Since magnetic skyrmions predominantly
form when an external field close to the saturation field is applied, this
suggests that the saturation field of our sample has increased during the
phase transition to the post-cooling phase. This could be explained by an
increase in the magnetization in the post-cooling state, as this would mean
that the dipolar energy is still large relative to the external field
contribution. Magnetometry measurements of a bulk sample using a SQUID support
this idea, and yield larger magnetizations at moderate fields in the post-
cooling phase compared to the pre-cooling phase, as discussed earlier in this
subsection.

Figure 5: Fresnel images of Fe5GeTe2 at various temperatures indicated in the
figure and a constant external field of $B=100\,\text{mT}$. A defocus of -1 mm
was used in a plan-view lamella (c-axis is perpendicular to the image plane),
recorded before (a) and after (b) leaving the sample in liquid nitrogen for
one hour.

###  2.4 Magnetic ordering

It was shown in the previous subsection that Fe5GeTe2 shows a decrease in
magnetization upon cooling from $T_{C2}$ down to $T_{S}$ while the sample is
in the metastable pre-cooling phase (see Fig. (3)), which indicates that
Fe5GeTe2 might exhibit ferrimagnetic order or is in a spin glassy state. To
distinguish these effects, we study next whether or not Fe5GeTe2 is in a
glassy spin state. To this end, the micromagnetic anisotropy constant $K$ is
experimentally determined in two independent ways and compared. One method is
to consider the hard-axis saturation field $B_{\text{sat}}^{\text{IP}}$, which
is simply related to the exchange constant via the Stoner-Wohlfarth model
[22]. By finding the total energy minimum with a given field applied along the
hard-axis, and including Zeeman and anisotropy interactions, one finds that
the expression for in-plane saturation is:

$K_{\text{IP}}=B_{\text{sat}}^{\text{IP}}\frac{M_{\text{s}}}{2}\;.$ (4)

A complementary approach to determine $K$ is via the domain wall width
$\delta$ [23]:

$\delta=\pi\Delta=\pi\sqrt{\frac{A}{K_{\delta}}}\;,$ (5)

where $A$ is the micromagnetic exchange constant and $\Delta$ is a fit
parameter used in the determination of the domain wall width $\delta$ from the
measured domain wall profile. Note that when measuring $K_{\text{IP}}$, the
quantities involved are determined on a macroscopic scale, whereas the
determination of $K_{\delta}$ will require knowledge of the exchange
interaction, which is short-ranged and dominates on the nanoscale. Thus, local
disorder present in spin glasses [24] would manifest itself by affecting
$K_{\delta}$ and $K_{\text{IP}}$ differently.

To obtain $K_{\delta}$, the micromagnetic exchange coupling constants were
calculated using the JuKKR density functional theory (DFT) code [25] (see
method section (4.4)). This code employs the method of infinitesimal rotations
[26, 27] to map the interactions among spins $\vec{S}_{i}=\vec{M}_{i}/\mu_{i}$
($\mu_{i}=|\vec{M}_{i}|$) at lattice sites $i$ onto the extended classical
Heisenberg Hamiltonian

$H=-\sum_{i\neq j}J_{ij}\big{(}\vec{S}_{i}\cdot\vec{S}_{j}\big{)}-\sum_{i\neq
j}\vec{D}_{ij}\cdot\big{(}\vec{S}_{i}\times\vec{S}_{j}\big{)}-\sum_{i}K(\vec{S}_{i}\cdot\hat{e}_{z})^{2}\,,$
(6)

where the first term describes the exchange interaction, the second term
represents DMI, and the final term describes the uniaxial out-of-plane
anisotropy. The micromagnetic spin stiffness is

$A=\frac{1}{2V}\sum_{i\neq j}J_{ij}\vec{r}_{ij}^{2}\,,$ (7)

where $V$ is the volume per Fe atom, which can be calculated from the exchange
coupling constants [27]. These parameters define the micromagnetic energy
functional

$E[\vec{m}]=\int d\vec{r}\left[A\dot{\vec{m}}^{2}+Km^{2}_{z}\right],$ (8)

where the contribution of the spiralization that is related to the DMI vectors
is neglected because it is found to vanish. The micromagnetic exchange
constant obtained from this method is

$A_{\text{DFT}}=0.1508\;\text{eV nm}^{-1}\;.$ (9)

An alternative, independent method to approximate the micromagnetic exchange
constant $A$ by considering the magnetic ordering temperature and crystal
structure, is presented and executed in section S3 of the supplementary
information. The result is $A_{\text{H}}=(0.060\pm 0.007)\,\text{eV nm}^{-1}$
and it is fairly close to the DFT value considering that many assumptions are
made in this alternative method. Further calculations and comparisons to
experiment, which support the DFT models and calculations are presented in the
supplementary sections S4 and S5.

To determine the effective anisotropy $K_{\delta}$ using the spin stiffness
$A$, we next need to determine the domain wall width $\delta$. The domain wall
parameter $\Delta$ can in principle be determined from phase shift images
obtained using off-axis electron holography by fitting the magnetic phase
gradient profile perpendicular to a domain wall and using the well-known
theoretical profile $M_{\text{z}}^{\text{theo}}(x)=M_{0}\cdot\tanh(x/\Delta)$.
In reality the profiles will not be perfectly centered, so we use a fit
function which allows for shifts:

$M_{\text{z}}^{\text{fit}}(x)=M_{0}\cdot\tanh((x-x_{\text{shift}})/\Delta)+M_{\text{shift}}\;,$
(10)

where $M_{0}$ is the amplitude of the magnetization, $x$ is the position
perpendicular to the domain wall, and $x_{\text{shift}}$ and
$M_{\text{shift}}$ are the free fit parameters accounting for imperfect
centering. To obtain reliable results, each line drawn perpendicular to the
domain walls has a width of 277.5 nm (300 pixels) to average across. An
example of such a line is shown in Fig. (6)(a), and the corresponding magnetic
profile and fit is shown in panel (b). Overall, the domain wall width has been
determined via four independent lines at $95\,\text{K}$ and five different
lines at $200\,\text{K}$. The resulting domain wall widths are
$\delta^{95\text{K}}=(24.2\pm 5.2)\,\text{nm}$ and
$\delta^{200\text{K}}=(36.3\pm 3.5)\,\text{nm}$. The uncertainties are
estimated via the standard deviation of the domain wall widths obtained from
all lines used at each temperature. Accordingly, the fit parameters are
$\Delta=\delta/\pi$. Thus, all required parameters to find $K_{\delta}$ via
equation (5) are known, and we find using $A_{\text{DFT}}$ and the parameters
$\Delta$:

$K_{\delta}^{\text{95K}}=(2.55\pm 0.55)\cdot 10^{-3}\,\text{eV nm}^{-3}\;,$
(11) $K_{\delta}^{\text{200K}}=(1.13\pm 0.11)\cdot 10^{-3}\,\text{eV
nm}^{-3}\;.$ (12)

Figure 6:  Analysis of the magnetic contrast of a magnetic domain wall, with
the aim to determine the domain wall width. (a) Magnetic phase gradient image
obtained using off-axis electron holography. A profile was extracted in the
direction perpendicular to a 180° domain wall along the wide line. (b) Phase
gradient profile (blue) and corresponding fit (orange). This example shows one
magnetic profile across a domain wall at 95 K.

For the alternative determination of $K$ via the saturation field along the
hard-axis, images of the cross-section lamella sample are recorded at various
field strengths at $95\,\text{K}$ and $200\,\text{K}$ (examples of images at
$95\,\text{K}$ are shown in Fig. (7)(c-e)). The point at which the magnetic
contrast completely vanishes as a function of magnetic field is approximated
by splining the data for which there is magnetic contrast, and interpolating
to the point where the minority domain has a width of zero. It should be noted
that, since the field may not be applied perfectly perpendicular to the
c-axis, this field is the monodomainization field, and serves as a lower bound
for the hard-axis saturation field. However, it can be used as an estimation
for the hard-axis saturation field. In this case, the saturation fields are
estimated to be $B_{\text{sat}}^{\text{IP, 95K}}=(285\pm 85)\,\text{mT}$ and
$B_{\text{sat}}^{\text{IP, 200K}}=(143\pm 43)\,\text{mT}$. Accordingly, in
conjunction with the saturation magnetization $M_{\text{s}}$ from the SQUID
measurements and inserting into equation (4), the micromagnetic anisotropy
constants from this method are evaluated to be

$K_{\text{IP}}^{\text{95K}}=(0.72\pm 0.22)\cdot 10^{-3}\,\text{eV nm}^{-3}\;,$
(13) $K_{\text{IP}}^{\text{200K}}=(0.36\pm 0.11)\cdot 10^{-3}\,\text{eV
nm}^{-3}\;.$ (14)

Finally, the anisotropy constants obtained via both the domain wall and hard-
axis saturation method can be compared. At 95 K the ratio between the
effective anisotropy constants is
$K_{\delta}^{\text{95K}}/K_{\text{IP}}^{\text{95K}}=3.53\pm 1.30$, and at 200
K it is $K_{\delta}^{\text{200K}}/K_{\text{IP}}^{\text{200K}}=3.11\pm 0.98$.
Thus, the two methods to determine $K$ used in this work, although they are
completely independent and cover vastly different length scales, yield results
which are on the same order of magnitude. The fact that the ratio
$K_{\delta}/K_{\text{IP}}$ may systematically be larger than $1$ could be
attributed to the fundamental approximations used. Namely, since the hard-axis
saturation field will be slightly larger than the monodomainization field,
which we used to estimate the saturation field, $K_{\text{IP}}$ should in
reality be systematically slightly larger than the values we obtained. If the
obtained anisotropy constants would differ by a large margin one could suggest
that the material might be in a spin glassy state. But since they do not
differ greatly, this analysis implies that the material is not in a spin
glassy state. As such, this analysis in conjunction with the SQUID results
indicates that Fe5GeTe2 might not be in a spin glassy state, but in a
ferrimagnetic state while it is in the pre-cooling phase, as previously
claimed [19, 20].

##  3 Conclusions

In conclusion, we have presented experimental observations of spin structures
and magnetic properties in the promising 2D vdW magnet Fe5GeTe2, to understand
the magnetic ordering of this material. Notably, non-chiral magnetic skyrmions
form in the material near room temperature when magnetic fields close to the
saturation field are applied. The achirality of these magnetic bubbles
suggests that there is no considerable DMI present in Fe5GeTe2, which in turn
supports the conclusion that Fe5GeTe2 has the centrosymmetric $R\bar{3}m$
crystal structure, rather than the otherwise proposed non-centrosymmetric
$R3m$ crystal structure. Since Fe5GeTe2 exhibits a lower magnetization in the
pre-cooling phase than in the post-cooling phase, we were able to observe
bubbles at lower fields in the pre-cooling phase than in the post-cooling
phase. Furthermore, by considering the anisotropy constants obtained via two
independent methods, namely via the hard-axis saturation field and via the
domain wall profile and predicted exchange constants, we conclude that
Fe5GeTe2 is not in a spin glassy state. However, a notable drop of the
magnetization with decreasing temperature observed in SQUID measurements
suggests that Fe5GeTe2 is a ferrimagnet instead of a simple ferromagnet whilst
in the pre-cooling phase. Future neutron diffraction experiments are required
to fully ascertain the spin structure of Fe5GeTe2. We have thus completed the
necessary steps toward room temperature spintronics devices based on Skyrmion
type spin structures, which are of significant interest for novel information
storage and non-conventional computing concepts using 2D magnets.

##  4 Methods

###  4.1 A. Crystal synthesis

Single crystals of Fe5GeTe2 with an average size of $2\times 2\times
0.5\,\text{mm}^{3}$ were synthesized in quartz glass ampoules from the
elements Fe, Ge, and Te in a 6 : 1 : 2 ratio in the presence of iodine as
mineralizer. Similar to reference [4], Ge was supplied as a powder on the
bottom of the quartz glass tube. Fe and Te were pressed to separate pellets of
$1\,\text{mm}$ diameter and positioned on top of the Ge powder without direct
contact between the pellets. The reaction thus predominantly occurs through
the gas phase and increases the single crystal size to several millimeters.
The vacuum-sealed ampules were heated up to 750 °C with 120 K/h and kept at
this temperature for about 2 weeks before quenching them in room temperature
water. The composition of the single crystals was confirmed by energy
dispersive X-ray spectroscopy (Tescan SEM Vega TS 5130 MM equipped with a
silicon drift detector, Oxford) with a ratio of Fe : Ge : Te of 4.72(5) : 1 :
1.94(4).

###  4.2 B. Lamella preparation

Electron transparent cross-section and plan-view lamellas were prepared using
a 30 kV Ga+ focused ion beam and scanning electron microscope (FIB-SEM) FEI
Helios platform. The energy of the ion beam was decreased to 5 kV in the last
steps of the thinning to minimize surface damage.

###  4.3 C. Transmission electron microscopy

Scanning transmission electron microscopy (STEM) and energy dispersive X-ray
(EDX) spectroscopy were carried out using an FEI Titan TEM equipped with a
Schottky ﬁeld emission gun operated at 200 kV, a CEOS probe aberration
corrector, a high angle annular dark-ﬁeld detector (HAADF) and a Super-X EDX
detection system [28]. Composition maps were obtained using the Thermo
Scientific Velox software.

Magnetic imaging was carried out using Fresnel defocus and off-axis electron
holography in an FEI Titan TEM equipped with a Schottky field emission gun, a
CEOS image aberration corrector, a post-specimen electron biprism and a
4k$\times$4k Gatan K2-IS direct detection camera [29]. A liquid-nitrogen-
cooled specimen holder (Gatan model 636) was used to vary the sample
temperature. The microscope was operated at 300 kV in magnetic field-free
conditions (Lorentz mode) by using the first transfer lens of the aberration
corrector as the primary imaging lens. The conventional objective lens was
used to apply the chosen magnetic fields to the sample, which were pre-
calibrated using a Hall probe. The field is applied along the electron beam
direction i.e. perpendicular to the sample plane. For off-axis electron
holography, the electron biprism was used to overlap a reference wave
travelling in vacuum with an object wave passing through the sample. An
elliptical illumination was used to optimize the coherence of the beam in the
direction perpendicular to the biprism. The hologram width was 2.7 µm and the
fringe spacing was 2.5 nm. Magnetic induction maps were reconstructed using
Fourier transforms with the Holoworks plugin in the Digital Micrograph
software (Gatan) [30]. The contribution of the mean inner potential to the
phase was removed by subtracting the phase of an hologram acquired above the
Curie temperature.

###  4.4 D. DFT atomistic exchange tensor calculations

We performed density functional theory (DFT) calculations for bulk Fe5GeTe2
using the experimental lattice constants, $a=4.04(2)$Å, $c=29.19(3)$Å, from
Ref. [4]. In order to account for the two Fe atoms at 50% occupancy we
employed the use of the coherent potential approximation (CPA) [31]. We
perform our calculations with the JuKKR code [25] which implements the full
potential Korringa Kohn Rostoker (KKR) Green’s Function method [31] with an
exact description of the shape of the atomic cell [32, 33]. We use an angular
momentum cutoff $l_{max}=3$ and a generalised gradient approximation (GGA) to
obtain the exchange-correlation potential [34]. Our calculations of the
exchange coupling constants for the determination of the atomistic spin
stiffness are based on the model of infinitesimal rotations [26, 27]. The
domain wall widths calculated in the supplementary S5 are determined using the
Spirit code [35].

##  5 Data Availability

All data used in the present work are available upon reasonable request.

##  6 Code Availability

All code used in the present work is available upon reasonable request.

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##  7 Acknowledgements

The work at JGU Mainz was funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) - TRR 173 - 268565370 (projects A01 and B02),
project 403502522 (SPP 2137 skyrmionics) the EU (FET-Open grant agreement no.
863155 (s-Nebula)), and the Research Council of Norway (QuSpin Center 262633).
The work at JGU Mainz and FZJ was funded by ERC Synergy grant agreement no.
856538 (3D MAGiC), T.S., P.R., S.B. and Y.M. gratefully acknowledge the Jülich
Supercomputing Centre for providing computational resources and Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - TRR 173 - 268565370
(project A11), TRR 288 - 422213477 (project B06). T.G.S. Would like to thank
Dr. Sarah Jenkins and Dr. Fabian Lux for fruitful discussions.

##  8 Author Information

###  8.1 Affiliations

Johannes Gutenberg Universität Mainz, Institut für Physik, Staudingerweg 7,
55128 Mainz, Germany
Maurice Schmitt, Tom G. Saunderson, Aga Shahee & Mathias Kläui

Ernst Ruska Centre for Microscopy and Spectroscopy with Electrons and Peter
Grünberg Institute, Forschungszentrum Jülich, 52425 Jülich, Germany
Thibaud Denneulin, András Kovács, Amir Tavabi & Rafal Dunin-Borkowski

Forschungszentrum Jülich GmbH, Wilhelm-Johnen-Straße, 52428 Jülich, Germany
Philipp Rüßmann, Yuriy Mokrousov & Stefan Blügel

Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569
Stuttgart, Germany
Tanja Scholz, Bettina Lotsch

University of Bristol, School of Physics, HH Wills Physics Laboratory, Tyndall
Avenue, Bristol BS8 1TL
Martin Gradhand

National and Kapodistrian University of Athens, Department of Physics,
University Campus, GR-157 84 Zografou, Athens
Phivos Mavropoulos

###  8.2 Contributions

Tanja Scholz synthesized all Fe5GeTe2 samples used in the present work.
Thibaud Denneulin prepared the TEM lamellas and conducted the LTEM experiments
with the help of Amir Tavabi. András Kovács carried out the STEM and EDX
measurements. Aga Shahee conducted the SQUID measurements. Tom G. Saunderson
and Philipp Rüßmann contributed equally to the DFT results. Martin Gradhand
and Phivos Mavropolous helped with the determination of the spin stiffness via
$T_{C}$. Phivos Mavropoulos provided the theoretical $T_{C}$ of Fe5GeTe2 based
on DFT results. Maurice Schmitt analyzed the experimental and theoretical
results to yield insights into the spin ordering of Fe5GeTe2, and wrote the
paper with Mathias Kläui. Bettina Lotsch supervised the synthesization of the
Fe5GeTe2 crystals. Yuriy Mokrousov and Stefan Blügel discussed and supervised
the DFT work. Rafal Dunin-Borkowski supervised the microscopy work. Mathias
Kläui supervised the experimental SQUID measurements and devised the study.
All authors commented on the results and contributed to the manuscript.

##  9 Ethics declarations

###  9.1 Conflict of Interest

The Authors declare that there is no conflict of interest.

##  10 Supplementary information

###  10.1 S1 Magnetic anisotropy in Fe5GeTe2

Isothermal magnetization vs. magnetic field (M vs. H) curves measured using a
SQUID, under the applied field either parallel to the c-axis
($B_{\text{c-axis}}$) or the ab plane ($B_{\text{ab-plane}}$) are shown in
Fig. (4) of the main text. Fig. (4)(a) and (b) are for the pre-cooling phase
and Fig. (4)(c) and (d) for the post-cooling phase respectively. These
magnetization curves are presented without taking into consideration the
demagnetization factor, which plays a great role in defining the internal
field in 2D flake-like and 1D needle-like crystals. These M vs. H curves
indicate a soft ferromagnetic nature of both the pre-cooling and post-cooling
phases with a weak magneto-crystalline anisotropy. The raw data indicates a
smaller external field is needed to saturate magnetization in the ab-plane
than along the c-axis, which may at first indicate that Fe5GeTe2 is an easy
plane ferromagnet. However, consideration of the large contribution to
internal field from the demagnetization factor for the c-axis configuration
indicates an opposite behavior, where spins prefer to align along the c-axis,
as has been reported by May et al. [4].

Figure 7: L-TEM images of a Fe5GeTe2 lamella with the crystallographic c-axis
in the image plane. (a) phase shift image obtained using off-axis electron
holography at 95 K and 0 mT in a cross-section lamella (c-axis is vertical in
the image plane). (b) Corresponding color-coded magnetic induction map where
the direction of the magnetic field is given by the color wheel in the bottom
right corner. (c-e) Fresnel images obtained at 95 K and different external
fields indicated on the images and with a defocus of -1 mm. (f-h) Fresnel
images obtained at 0 mT and different temperatures indicated on the images.

Further, we investigated magnetic domains in lamellas cut from Fe5GeTe2
crystals parallel to the c-axis. Fig. (7)(a,b) shows a phase shift image and
the corresponding color-coded magnetic induction map obtained at 95 K and 0
mT. The presence of 180° stripe domains with domain walls (DWs) orientated
along the [001] direction confirm the easy-axis magnetic anisotropy of
Fe5GeTe2 with the c-axis (001) as the easy-axis. Fig. (7)(c-h) shows Fresnel
defocus images obtained in the presence of different external fields applied
nearly perpendicular to the c-axis (0 to 162 mT) and at different temperatures
(95 to 275 K). Even though the domain walls move in the presence of external
fields and the width of the domains changes as a function of temperature, it
can be observed that the DWs remain parallel to the c-axis even close to
$T_{\text{C}}$ at 250 K for instance, which confirms the easy-axis magnetic
anisotropy of Fe5GeTe2 over a wide range of temperatures, which as shown in
the main text, we have observed directly using L-TEM.

###  10.2 S2 Determination of the saturation magnetization from the stripe
domain pattern

An independent way to determine the saturation magnetization is to closely
examine the stripe domain patterns seen in the L-TEM images. This way, the
validity of the SQUID results can be checked. According to theoretical
considerations [21], the saturation magnetization of a film of magnetic
material is given by

$M_{\text{s}}=2\cdot\frac{a^{2}}{t}\cdot\frac{(B_{\text{c}}-B)/\mu_{0}}{\pi
r_{\downarrow\text{c}}}\;,$ (15)

where $\mu_{0}$ is the magnetic vacuum permeability, $M_{\text{s}}$ is the
saturation magnetization, $a$ is the overall domain periodicity, $t$ is the
film thickness, $r_{\downarrow\text{c}}$ is the width of the minority spin
domains, and $B$ and $B_{\text{c}}$ are an arbitrarily chosen field and the
saturating field, respectively. This theory assumes the considered sample is a
magnetic film with uniaxial anisotropy and a large domain period, and
considers exchange, Zeeman, anisotropy, and stray field energy contributions.
The film thickness in our sample is not uniform, as scanning electron
microscopy (SEM) measurements on various spots of the sample reveal. However,
the measured local thicknesses all range from $90\,\text{nm}$ to
$210\,\text{nm}$, so the film thickness can be estimated to be $t=(150\pm
60)\,\text{nm}$. All other parameters on the right hand side of equation (15)
can be estimated from the recorded TEM images. The domain periodicity $a$ at
the chosen field $B=0\,\text{mT}$ can be estimated by adding the length of
both blue lines in Fig. (8) and dividing by eight, since both blue lines
together span a total of eight domain periods. The average length of the red
lines is used to estimate $r_{\downarrow\text{c}}$. Since no magnetic contrast
is visible at $655\,\text{mT}$ anymore, the saturating field is estimated to
be $B_{\text{c}}=(550\pm 100)\,\text{mT}$ in this case. This method leads to a
relatively large uncertainty in the minority domain width, since it is not
only a relatively small distance to measure, but also because the measured
widths cannot be measured exactly at the critical field. Therefore, each
individual red line can be estimated to have an error of $50\,\%$, which
propagates to yield the average length of all lines of
$r_{\downarrow\text{c}}=(53.6\pm 10.2)\,\text{nm}$. Via the same method, the
domain periodicity is estimated to be $a=(272.1\pm 96.5)\,\text{nm}$.

Figure 8: L-TEM images of Fe5GeTe2 at $95\,\text{K}$. The red lines are used
to estimate the minority stripe domain width, whereas the blue lines are used
to determine the overall domain periodicity. The green line is used to convert
lengths in pixels to real lengths. The c-axis is perpendicular to the image
plane.

Accordingly, using error propagation, the obtained saturation magnetization is
$M_{\text{s}}=(2.56\pm 1.52)\cdot 10^{6}\,\text{A m}^{-1}$. Although this way
the saturation magnetization has a larger relative uncertainty, its order of
magnitude agrees with the values obtained from the SQUID measurements, thus
confirming the validity of the SQUID measurements.

###  10.3 S3 Determination of the micromagnetic exchange constant from the
Curie temperature

To confirm the results obtained from DFT, the micromagnetic exchange constant
(or spin stiffness) will be approximated in the following. To this end, one
can invoke a simple Heisenberg model, which predicts in a mean-field approach
[36]:

$J_{\text{H}}=\frac{3kT_{\text{C}}}{2\epsilon z}$ (16)

where $k$ is the Boltzmann constant, $T_{\text{C}}$ is the Curie temperature,
$z$ is the number of nearest neighbors in a unit cell and $J_{\text{H}}$ is
the Heisenberg exchange constant between magnetic sites. The parameter
$\epsilon$ is a correction factor accounting for spin waves, which is slightly
smaller than one and is well-known for common crystal structures [37]. Here,
we will simply estimate $\epsilon=0.8\pm 0.1$. One can then find the
micromagnetic exchange constant $A_{\text{H}}^{\text{IP}}$ according to
equation (7) by assuming $J_{ij}=J_{H}$ for nearest neighbors only and
averaging the sum over nearest neighbors as well as the nearest neighbor
distance squared $\overline{r_{ij}^{2}}$:

$A_{\text{H}}^{\text{IP}}=J_{H}\cdot\frac{z\cdot\overline{r_{ij}^{2}}}{2V}\;,$
(17)

with the volume per iron atom $V$. Combining (16) and (17) yields

$A_{\text{H}}^{\text{IP}}=\frac{3kT_{\text{C}}}{4\epsilon}\cdot\frac{\overline{r_{ij}^{2}}}{V}\;,$
(18)

The Curie temperature $T_{\text{C}}=(310\pm 10)\,\text{K}$ can be estimated
from the SQUID data. Note that in Fe5GeTe2 the Curie temperature changes
depending on the thermal phase it is in. The larger value of
$T_{\text{C}}=(310\pm 10)\,\text{K}$ is valid for the post-cooling phase. The
volume per magnetic site is $V=0.0276\,\text{nm}^{3}$ according to the DFT
results. For the mean squared neighbor distance, all neighbors in table (1)
are considered and averaged over. Here, the split site nature of the Fe1 site
has to be considered, by weighting them only $50\%$. The result is
$\overline{r_{ij}^{2}}\approx 0.06594\,\text{nm}^{2}$. Inserting these values
into equation (18) yields

$A_{\text{H}}^{\text{IP}}=(0.060\pm 0.007)\,\text{eV nm}^{-1}\;.$ (19)

The value of the micromagnetic exchange constant obtained this way
$A_{\text{H}}$ is close to the value obtained via DFT $A_{\text{DFT}}$.
Considering that many assumptions are made when calculating the micromagnetic
exchange constant via the critical temperature due to the complex crystal
structure of Fe5GeTe2, the obtained value $A_{\text{H}}$ turns out to be
compatible with the more robust DFT result $A_{\text{DFT}}$ and thus supports
it.

type | Neighbor | Number of neighbors | Bond length
---|---|---|---
Fe1 | Fe3 | 3 | 2.35464 Å
Fe1 | Fe2 | 3 | 2.73786 Å
Fe2 | Fe3 | 3 | 2.59186 Å
Fe2 | Fe3 | 1 | 2.53864 Å
Fe2 | Fe1 | 3 | 2.73786 Å
Fe3 | Fe2 | 1 | 2.53864 Å
Fe3 | Fe2 | 3 | 2.59186 Å
Fe3 | Fe1 | 3 | 2.35464 Å
Table 1: The distances between individual iron sites in Fe5GeTe2 assuming the
space group $R\bar{3}m$.

###  10.4 S4 Determination of T${}_{\text{C}}$ using the $J_{ij}$ elements
from DFT

In order to further confirm the validity of our DFT models and calculations,
the critical temperature of Fe5GeTe2 has been determined by using both mean-
field [36] and a Monte Carlo [38] methods.

In the mean-field method [36] we calculate two separate values for the Curie
temperature. The first is an effective 2-dimensional Curie temperature,
$T^{2D}_{\text{C}}$, which we obtain by extracting purely the intra-layer
$J_{ij}$ terms which yields $T_{\text{C}}^{\text{2D}}=551\,\text{K}$. The
second is a bulk Curie temperature, $T_{\text{C}}^{\text{3D}}$, which
additionally contains the inter-layer $J_{ij}$ terms, yielding
$T_{\text{C}}^{\text{3D}}=871\,\text{K}$. While the experimental samples used
in the present work are more than $100\,\text{nm}$ thick,
$T_{\text{C}}^{\text{2D}}$ might be more appropriate than $T_{C}^{\text{3D}}$,
because ferromagnetic inter-layer coupling could be mediated by the long-range
dipolar interaction. The Monte Carlo method, detailed in Ref. [38] yields a
Curie temperature of $T^{\text{MC}}_{\text{C}}=500\,\text{K}$. This implies
that the bulk Curie temperature is, as previously suggested, suppressed.

To conclude, using mean-field and Monte-Carlo methods we have determined three
values for the $T_{\text{C}}$ using the $J_{ij}$ terms determined from ab
initio methods. We conclude that it is more reasonable to assume the 2D mean-
field value describes the system the best because inter-layer coupling could
be mediated by the long-range dipolar interaction. The $T_{\text{C}}$
determined from the Monte Carlo method is within a factor of 2 of the measured
experimental $T_{\text{C}}$. As the independent determination of
$T_{\text{C}}$ through the $J_{ij}$ terms is comparable to the experiment, it
further justifies their use to determine $K_{\delta}$ in equations ((11)) and
((12)).

###  10.5 S5 Domain wall widths using the $J_{ij}$ terms from DFT

To further analyse the determination of the exchange parameter
$A^{\text{DFT}}$, we turn to atomistic simulations using the Spirit code to
use the $J_{ij}$ terms to accurately determine a domain wall width. By
simulating a domain wall we can further justify our use of $A^{\text{DFT}}$ in
the determination of the experimental value of $K^{\delta}$. Firstly, we
additionally determine $K^{\text{DFT}}$ directly from the juKKR package giving
a value of $K^{\text{DFT}}=24\cdot 10^{-3}\,\text{eV nm}^{-3}\;$. This value
is approximately a factor of 30 too large compared to experiment, however this
quantity is determined from the bulk when in reality the domain wall is on the
surface. This quantity can therefore be considered unknown and hence free to
be determined. For our simulations we chose a $100\times 20\times 10$
supercell consisting of 200,000 spins in the simulation where periodic
boundary conditions are chosen in $\vec{b}$ and $\vec{c}$ directions whereas
open boundary conditions are used in the $\vec{a}$ direction. The first and
last plane of spins in the $\vec{a}$ direction are then fixed in opposite
($\pm z$) directions which enforces a domain wall to form in the center of the
simulation box. We solve the Landau-Lifshitz-Gilbert equation with the Spirit
code [35] where the Depondt solver is applied for a maximum of 100,000 time
steps with a maximal simulation time of $1,000\,\text{ps}$ to relax the
magnetic texture. These simulations include the effect of temperature noise
which is varied between 0 and 300 K. Fig. (9)(a) is a plot of the domain wall
width at $95\,\text{K}$ for different choices of $K^{\text{DFT}}$ within the
range of experimental values of $K$. For large $K$ t domain wall is shortened
significantly compared to the experimental values and is hence not necessary
to go beyond 0.7 meV/Fe. As the anisotropy constant decreases we see a rather
flat distribution until 0.2 meV/Fe at which point the domain wall diverges for
vanishing $K$. Choosing a particular value for $K=0.04\,\text{meV/Fe}$ to
match with the experimental value, Fig. (9)(b) shows how the domain wall
evolves as a function of temperature. We observe a steadily increasing value
for the domain wall width with temperature, crossing $\delta=23.2\,\text{nm}$
at $100\text{K}$ and $\delta=27.7\,\text{nm}$ at $200\text{K}$. This is in
good agreement with experimentally determined domain walls when using the
experimental anisotropy instead of the value determined by DFT which is more
specific to the bulk. Three exemplary domain wall fit at temperatures of 20 K,
100 K and 300 K are shown in Fig. (10)(a-c).

Figure 9: Domain Wall widths using ab initio $J_{ij}$’s as a function of
anisotropy and temperature. (a) The domain wall width at $95\,\text{K}$ as a
function of anisotropy constant $K$. (b) The domain wall width with increasing
temperature in the simulation. Green and blue lines are indicate the maximal
and minimal determined values from the different lines of the Fe atom
positions in the unit cell and among the multiple unit cell in the $200\times
20\times 10$ cell large supercell. The orange line gives the average over the
$20\times 10$ unit cell cross section parallel to the domain wall. Figure 10:
Simulated domain Wall profiles. (a-c) Profiles of the domain wall across all
lines of Fe atoms in the $20\times 10$ cell cross section of the Spirit
supercell for increasing temperature of 20 K, 100 K and 300 K. The faint lines
in the background are the inividual line profiles where the temperature
fluctations are visible. The thick red lines show the average over the lines
and the black line the fitted domain wall profile. The simulation uses the ab
initio $J_{ij}$’s and the optimized value of the anisotropy of
$K=0.04\,\text{eV/Fe}$. For large temperatures the fit starts to deviate from
the average line (most visible in c) which is attributed to finite size
effects in the simulation.