# Large ordered moment with strong easy-plane anisotropy and vortex-domain
pattern in the kagome ferromagnet Fe3Sn

Lilian Prodan1,∗, Donald M. Evans1, Sinéad M. Griffin2,3, Andreas Östlin4,
Markus Altthaler1, Erik Lysne5, Irina G. Filippova6, Serghei Shova7, Liviu
Chioncel4, Vladimir Tsurkan1,6, and István Kézsmárki1 1Experimentalphysik V,
Center for Electronic Correlations and Magnetism, Institute for Physics,
Augsburg University, D-86135 Augsburg, Germany  2Materials Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  3Molecular
Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
4Theoretische Physik III, Center for Electronic Correlations and Magnetism,
Institute for Physics, Augsburg University, D-86135 Augsburg, Germany
Augsburg Center for Innovative Technologies (ACIT),D-86135 Augsburg, Germany
5Department of Materials Science and Engineering, Norwegian University of
Science and Technology (NTNU), 7043 Trondheim, Norway  6Institute of Applied
Physics, MD 2028, Chișinău, R. Moldova  7Department of Inorganic Polymers,
“Petru Poni” Institute of Macromolecular Chemistry, Romanian Academy, 700487
Iasi, Romania

###### Abstract

We report the structural and magnetic properties of high-quality bulk single
crystals of the kagome ferromagnet Fe3Sn. The dependence of magnetisation on
the magnitude and orientation of the external field reveals strong easy-plane
type uniaxial magnetic anisotropy, which shows a monotonous increase from
$K_{1}=-0.99\times 10^{6}J/m^{3}$ at 300 K to $-1.23\times 10^{6}J/m^{3}$ at 2
K. Our ab initio electronic structure calculations yield the value of total
magnetic moment of about 6.9 $\mu_{B}$/f.u. and a magnetocrystalline
anisotropy energy density of 0.406 meV/f.u. ($1.16\times 10^{6}J/m^{3}$) both
being in good agreement with the experimental values. The self-consistent DFT
computations for the components of the spin/orbital moments indicate that the
small difference between the saturation magnetisations measured along and
perpendicular to the kagome layers results from the subtle balance between the
Fe and Sn spin/orbital moments on the different sites. In zero field, magnetic
force microscopy reveals micrometer-scale magnetic vortices with weakly pinned
cores that vanish at $\sim$3 T applied perpendicular to the kagome plane. Our
micromagnetic simulations, using the experimentally determined value of
anisotropy, well reproduce the observed vortex-domain structure. The present
study, in comparison with the easy-axis ferromagnet Fe3Sn2, shows that varying
the stacking of kagome layers provides an efficient control over magnetic
anisotropy in this family of Fe-based kagome magnets.

Notes. Version 2. This manuscript is under review in Physical Review B.

##  I Introduction

Magnetic compounds with kagome-lattice arrangement of spins have recently
attracted much attention due to their unusual magnetic and electronic
properties related to the specific topology of their electronic band
structures [1, 2, 3, 4, 5]. Recent theoretical and experimental studies have
demonstrated that the existence of flat bands, nodal points and nodal lines
appearing close to the Fermi energy significantly affect magnetic, magneto-
transport and magneto-optical properties of kagome materials [6]. In
particular, fascinating physical phenomena like giant anomalous and
topological Hall effects, giant Nernst effect, topological superconductivity,
magnetic spin chirality and skyrmion bubbles have been reported for large
group of kagome materials like van der Waals $A$V3Sb5 ($A$ = K, Cs, Rb) [7,
8], rare earth based $ReT$6Sn6 ($Re$ = Gd, Tm, Tb, Y; $T$ = Mn,V) [9, 10],
magnetic Weyl-semimetal Co3Sn2S2 [11, 12], and binary metals $T$xSny ($T$ =
Fe, Mn, Co; $x$:$y$ = 1:1, 3:2, 3:1) [13, 14, 15, 16, 17, 18]. The coexistence
of these effects provide an exceptional platform to study electronic band
topology and its interplay with magnetic spin and orbital effects.

Here we focus on the magnetic properties of Fe3Sn quasi-2D kagome magnet,
investigated by anisotropy measurements, ab initio calculations and imaging of
the magnetic domain pattern. Fe3Sn is an itinerant ferromagnet, a member of a
family of iron stannides with the general formula FexSny. Our particular
interest is on compounds with the kagome-lattice arrangement, namely compounds
with $x:y$ = 1:1; 3:1; 3:2. Depending on the ratio of $x:y$, i.e. the stacking
of kagome layers, these compounds realize various hexagonal magnetic space
groups and different magnetic ground state. For example, FeSn crystallises in
the $P6/mmm$ symmetry and below 365 K it is an easy-plane antiferromagnet with
ferromagnetic arrangement of spins within each kagome layer [19, 20, 21].
Fe3Sn has a structure of $P6_{3}/mmc$ symmetry and exhibits ferromagnetic
order below 743 K [19, 22]. The crystal structure of Fe3Sn is shown in Figs.
1(a) and (b). The $ab$ layer consists of a breathing kagome lattice of two
sizes of equilateral Fe triangles with Sn atoms in the center of the kagome.
The unit cell of Fe3Sn contains two adjacent, laterally displaced kagome
layers, separated by half of the lattice constant along the $c$ axis. The
structure of Fe3Sn is much simpler than of Fe3Sn2, which realizes a third type
of stacking and crystallizes in the $R\overline{3}m$ symmetry. Namely, Fe3Sn2
contains two such basic blocks of Fe3Sn bilayers stacked along the $c$ axis
and separated by a honeycomb layer of Sn atoms [19, 22]. In contrast to Fe3Sn,
Fe3Sn2 is an easy-axis ferromagnet with a Curie temperature of 612 K [23],
which shows skyrmion bubbles at room temperature [18, 24]. However, this
compound shows a spin reorientation below room temperature, when the easy axis
of magnetisation gets tilted towards the kagome plane [22, 25, 26, 27].

Physical properties of Fe3Sn are investigated predomimantly on polycrystalline
samples synthesized by solid state reaction or by arc-melting of metalic Fe
and Sn in argon atmosphere. Early Mössbauer studies of Trumpy et al. [23] and
Djéga-Mariadassou et al. [28] revealed a ferromagnetic behavior of Fe3Sn and
concluded that the spin moment is oriented along the $c$ axis. In contrast,
recent analysis of magnetic properties of Fe3Sn by Sales et al. [29],
performed on field-oriented polycrystalline powder, implied predominantly in-
plane orientation of the magnetisation with the magnitude of easy-plane
anisotropy $K_{1}=-1.8MJ/m^{3}$ at 300 K. The easy-plane character of the
anisotropy was further supported by magnetisation studies of Fayyazi et al.
[30], that was performed on an oriented plaque of very small crystals with a
mass of 10 $\mu$g, obtained by reactive flux technique. However, this work
reported an unusual temperature dependence of the anisotropy field, namely its
decrease from 200 K to 10 K.

The precise quantification of magnetic anisotropy is the prerequisite to
understand the diversity of complex spin textures, for which kagome magnets
offer a fertile ground [31]. Moreover, the magnetic anisotropy in interplay
with the Dzyaloshinskii-Moriya interaction, competing and frustrated
exchanges, or dipole-dipole interactions, can affect the properties of
mesoscale spin textures, including conventional domain walls, vortices, and
skyrmions [32, 33, 34, 35, 36, 37]. As a prime example, the competition
between easy-axis magnetic anisotropy and dipole-dipole interaction can
stabilize magnetic bubbles and skyrmions in finite magnetic fields [33, 24,
18, 38]. On the other hand, the in-plane magnetized systems, e.g., magnets
with easy-plane anisotropy, can develop magnetic vortices and anti-vortices
[39, 40, 41, 42, 43].

Although FexSny compounds are known for long, the microscopic origin and the
precise value of exchange interactions and magnetic anisotropy constants is
still unsettled in these compounds, even the spin arrangement is under debate
in some of these materials. From the experimental point of view, the situation
is complicated by large variation of sample quality, leading to
inconsistencies of the data available in literature. A large piece of the data
was obtained on polycrystalline samples, while those reported for single
crystals show large scattering evidently related to different quality of
samples, e.g., homogeneity, deviation from the stoichiometry and impurity
content. Therefore, precisely quatifying the magnetic interactions in a single
bilayer Fe3Sn is a prerequisite for understanding magnetism on the microscopic
level in these kagome magnets.

Here we report detailed magnetometry studies performed on large stoichiometric
single crystals grown by the chemical transport reactions. The appropriate
size of the crystals (several mg) allowed us to accurately determine their
orientation prior to the magnetisation measurements along the different
crystallographic directions. Moreover, our angular-dependent magnetisation
measurements, performed in different fields and at various temperatures,
allowed a highly reliable quantification of the anisotropy constants of Fe3Sn.
The experimental studies were complemented by ab initio electronic structure
calculations to determine the magnetic properties in the ground state
(magnetic moments on Fe and Sn as well as anisotropy energies).

The spin/orbital moments were restricted to specific orientations and by
including the spin-orbit coupling, calculations revealed easy-plane magnetic
anisotropy. The computed uniaxial magnetocrystalline anisotropy energy, which
arises from the collective effect of the crystal structure and spin-orbit
coupling, is in excellent agreement with the experimentally determined value.
Furthermore, a slight departure from the collinear magnetic configuration is
predicted due to tilting of the orbital-magnetic moment of Fe with respect to
its spin moment. Finally, our MFM study reveals the formation of a complex
domain pattern built of in-plane flux enclosure domains, i.e. magnetic
vortices, on the micrometer scale. This vortex-domain pattern is well
reproduced by our micromagnetic simulations using the magnetic interaction
parameters determined experimentally and theoretically, further supporting the
high reliability of these parameters.

##  II METHODS

Crystal growth. Polycrystalline Fe3Sn was prepared by solid state reactions of
high purity elements of Fe (99.99$\%$) and Sn (99.995$\%$). A stoichiometric
amount of Fe and Sn has been mixed and sealed in quartz ampule evacuated to
$10^{-3}$ mbar. The mixture was annealed for four days at 780 ∘C followed by
quenching in ice water. In order to achieve the full reaction and homogeneity
two sintering cycles were performed. The phase purity after each sintering was
checked by x-ray powder diffraction. Single crystals of Fe3Sn have been grown
by the chemical transport reactions method using the polycrystalline powder as
starting material and I2 as the transport agent. The growth was performed in
exothermic conditions in a two-zone horizontal furnace with the temperature
gradient of 50 ∘C in a temperature range of 850-800 ∘C. Crystals up to 1.5 mm
size have been obtained after 6 weeks of transport. The chemical composition
of single crystals was analyzed by the ZEISS Crossbeam 550, using energy
dispersive x-ray spectroscopy (EDS) technique.

Single-crystal x-ray diffraction. The single-crystal x-ray diffraction
measurement was carried out with a Rigaku Oxford-Diffraction XCALIBUR E CCD
diffractometer equipped with graphite-monochromated MoK$\alpha$ radiation. The
data were corrected for the Lorentz and polarization effects and for the
absorption by multi-scan empirical absorption correction methods. The
structure was refined by the full matrix least-squares method based on $F^{2}$
with anisotropic displacement parameters. The unit cell determination and data
integration were carried out using the CrysAlis package of Oxford Diffraction
[44]. All calculations were carried out by the programs SHELXL2014 [45] and
the WinGX software [46].

Magnetic properties. Magnetic properties were measured by a SQUID magnetometer
(MPMS 3, Quantum Design) in the temperature range of 1.8–900 K and magnetic
fields up to 7 T. The angular-dependent magnetization measurements have been
performed both within $ab$ and $ac$ planes on cylindrical sample with the $ac$
plane as the basal plane of the cylinder. The measurements in fields up to 14
T were performed with a vibrating-sample magnetometer using a physical
properties measurement system (PPMS, Quantum Design).

Computational Details. Density Functional Theory (DFT) calculations were
performed with the Vienna Ab initio Simulation Package (vasp) [47, 48, 49]
using projector augmented wave (PAW) pseudo-potentials [50, 51]. We treated
the Fe(3d, 4s) and Sn(5s, 5p) electrons as valence. Plane waves were expanded
to an energy cutoff of 800 eV, and a Monkhorst-Pack Gamma-centered k-point
grid of $10\times 10\times 10$ for structural optimizations and $20\times
20\times 20$ for accurate total energy calculations to obtain the magnetic
energetics. A smearing value of 0.02 eV was used for all calculations. The
PBEsol functional was used throughout [52]. Spin-orbit coupling was included
self-consistently as implemented in VASP. The electronic convergence criteria
set to 10-8 eV and the force convergence criteria set to 0.002 eV/Å. Our
reference structure was taken from the Materials Project (mp-20883) [53, 54,
55, 56].

ELK [57] computation has been performed for the a equally large $k$-mesh
($21\times 21\times 21$), and the PBE-GGA functional was used with the
experimental lattice parameters given in Table I. For the FPLAPW calculations
the plane-wave cutoff $K_{max}$ was set so that $R_{mt}K_{max}=7.0$, where
$R_{mt}$ is the smallest muffin-tin radius.

MFM imaging.  The magnetic domain pattern was imaged using a low-temperature
attocube attoAFM I, in MFM mode, equipped with a superconducting magnet. The
magnetic texture on an as-grown ab surface was recorded via a phase-sensitive
feedback loop that records changes in the resonance frequency, which are
proportional to the gradient of the stray magnetic field of the sample along
the magnetisation of the MFM tip, that is normal to the surface. The magnetic
tips used were PPP-MFMR probes from NanoSensor with the magnetic moment of
$5.09\times{10^{-13}\,A/m^{2}}$. All data was collected at 100 K for
experimental reasons, such as the thermal stability of the equipment.

Micromagnetic simulations.  Simulations were carried out using MuMax3 v3.10
[58]. Here, the microstructure is found by minimizing a total energy made up
of terms representing, Heisenberg exchange, first-order uniaxial anisotropy,
Zeeman and demagnetisation energy. Practically this is done by solving the
equation

$\displaystyle\varepsilon=\int_{V_{S}}[A_{ex}(\Delta
m)^{2}-K_{u}m_{z}^{2}+M_{s}B_{ext}m-\frac{1}{2}M_{s}B_{dem}m]\textit{d}r$ (1)

Where the reduced magnetisation, m(x,y,z), is the magnetisation M(x,y,z)
divided by the saturation magnetization, $M_{s}$, within a sample volume
$V_{S}$, where the magnetization makes up a continuous vector field. The
exchange stiffness constant, the first order uniaxial anisotropy constant, the
external magnetic field and the demagnetisation field are represented by
$A_{ex}$, $K_{u}$, $B_{ext}$, $B_{dem}$, respectively. Periodic boundary
conditions were not necessary; the simulation was initiated from a random
state, and easy-plane anisotropy (0, 0, 1) was used. The mesh was $4\times
4\times 4$ nm cells, and the geometry was $2048\times 1024\times 256$ nm.
Note, that MuMax3 is able to calculate the MFM response as it knows the
orientation of the magnetisation, and therefore the stray fields, and the MFM
response comes from the interactions between the MFM tip and the gradient of
the stray fields.

##  III RESULTS AND DISCUSSIONS

###  III.1 Structure, magnetic properties and anisotropy

Typical as-grown Fe3Sn crystals used in our study are shown in the inset of
Fig. 2 (a). The energy-dispersive x-ray analysis found an excess of Fe
$\lesssim$1 $\%$, indicating that the stoichiometry Fe3Sn is close to ideal.
Single-crystal x-ray diffraction study revealed the hexagonal $P6_{3}/mmc$
($\\#194$) space group symmetry and showed no traces of impurity phases. The
calculated cell parameters $a$ and $c$ are close to those reported for
polycrystalline Fe3Sn [19, 29]. The structural parameters obtained from the
single-crystal refinement are summarised in Table [tab:table1].

The magnetisation measurements were carried out on a single crystal with a
size of $0.7\times 0.6\times 0.5\,mm^{3}$. The temperature-dependent
magnetisation in 1 T applied along the $a$ axis exhibits a steep increase
below 750 K indicating the onset of ferromagnetic order, as shown in Fig. 2
(a). The Curie temperature of $T_{C}=705$ K was obtained as the location of
the maximum in the temperature derivative of the magnetisation, and is close
to previously reported data for polycrystalline samples [19, 22].

Figure 1: (a) Schematic representation of the crystal structure of layered
Fe3Sn. Red and green spheres represent Fe and Sn ions, respectively.
Transparent and bold atoms are from different planes. (b) Breathing kagome
lattices of equilateral Fe triangles of two distinct sizes (marked by blue and
red lines) in the $ab$ plane, as seen when viewed along the $c$ axis. (c)/(d)
Magnetic structure with moment parallel/perpendicular to the $c$ axis. Green
and orange arrows show Fe and Sn spin moments, respectively. (e) Calculated
magnetocrystalline anisotropy energy for rotation of Fe spin within the $ab$
plane (purple) and the $ac$ plane (green), assuming a ferromagnetic ordering.
In the former no angular dependence is observed, while in the latter case the
energy has a maximum for spins pointing along the $c$ axis ($\phi=0$). (f)
Calculated orbital moment magnitude of Fe for in-plane ($l_{x}$) and out-of-
plane ($l_{z}$) projections as a function of Fe spin tilt angle.

Figure 2(b) shows the field-dependent magnetisation curves, $M(H)$, measured
at 2 K for magnetic fields applied along three orthogonal directions. Two
directions are in the basal $ab$ plane, $[\overline{1}2\overline{1}0]$ and
$[10\overline{1}0]$ being parallel and perpendicular to the $a$ axis,
respectively, and the hexagonal $c$ axis (equivalently $[0001]$) is chosen as
the third direction. The magnetisation within the $ab$ plane reaches the
saturation already at $\sim$1 T. We found that the saturation is reached in a
slightly higher field for $H||[10\overline{1}0]$ than for
$H||[\overline{1}2\overline{1}0]$, indicating that the $a$ axis is the easy
axis of the magnetisation. For fields along the $c$ axis, the saturation takes
place at higher fields, above 3 T. The uniaxial magnetocrystalline anisotropy
within the $ac$ plane was calculated from the hysteresis curves using an “area
method” [59], considering the difference of the integrals
$\int_{0}^{M_{s}}H_{i}dM$ for in-plane ($a$ axis) and out-of-plane ($c$ axis)
field orientations. Here $H_{i}$ is the internal field determined as the
difference between the applied magnetic field $H$ and the demagnetizing field
$DM$, where $D$ is the demagnetisation coefficient. We obtained the value of
$K_{u}=-1.27\times 10^{6}J/m^{3}$ at 2 K for the uniaxial anisotropy constant,
which decreases to $K_{u}$=-0.97$\times 10^{6}$ $J/m^{3}$ at room temperature.
A similar approach was used to calculate the sixth-order anisotropy in the
$ab$ plane, that led to the value of $2.3\times 10^{4}J/m^{3}$ at 2 K, which
decreases to $1.8\times 10^{4}J/m^{3}$ at 300 K. The anisotropy in the basal
plane is nearly two orders of magnitude weaker than uniaxial anisotropy.

The inset in Fig.2(b) shows the temperature dependence of the magnetisation
measured in 7 T. The value of the saturation moment is 2.27 $\mu_{B}$/Fe at 2
K for field in the $ab$ plane, which is $\sim$2 % larger than the saturation
value along the $c$ axis. Although this difference is rather small it cannot
originate from demagnetisation effects, since the sample has a circular shape
in the $ac$ plane. Additional measurements, not shown here, show this
difference is present up to 14 T.

Table 1:  Structural refinement details and crystal data for Fe3Sn determined
through single-crystal x-ray diffraction.

| Refined empirical formula |  | Fe3Sn
---|---|---|---
| Space group |  |  $P6_{3}/mmc$ (No. 194)
|  $a$ (Å) |  | 5.4604(3)
|  $c$ (Å) |  | 4.3458(3)
| Volume (Å3) |  | 112.215(15)
| Z, Dcalc (g/cm3) |  | 2, 8.472
|  $\mu$ (mm-1) |  | 29.551
|  $\Theta$ range (∘) |  | 4.310 - 29.014
| Reflections collected/unique |  | 577/68 [Rint = 0.0467)]
| No. variables |  | 8
| Goodness-of-fit on $F^{2}$ |  | 1.001
| Extinction coefficient |  | 0.174(9)
| R1, wR2 (all data) |  | 0.0143, 0.0352

Atom |  | Position |  | $x$ |  |  | $y$ |  |  | $z$ |  | Ueq
---|---|---|---|---|---|---|---|---|---|---|---|---
Fe |  | 6h |  | 0.1544(1) |  |  | 0.3088(2) |  |  | 0.2500 |  | 7(1)
Sn |  | 2d |  | 0.6667 |  |  | 0.3333 |  |  | 0.2500 |  | 5(1)

[t]

∗Ueq is defined as one third of the trace of the orthogonalized $U_{ij}$
tensor.

Figure 2:  (a) The temperature-dependent magnetisation of Fe3Sn single crystal
measured in 1 T applied along $a$ axis. The inset shows as-grown Fe3Sn single
crystals on a mm-scale paper.(b) Magnetisation curves measured at 2 K in
magnetic fields applied in two directions in the $ab$ plane—parallel to the
$a$ axis or $[\overline{1}2\overline{1}0]$ and perpendicular to the $a$ axis
or $[10\overline{1}0]$, and with fields along the $c$ axis or $[0001]$. The
inset shows the temperature dependence of the magnetisation in 7 T for these
three directions.

The angular dependence of the magnetisation $M(\phi)$ within the $ac$-plane,
measured in different fields at 300 K, is shown in Figure 3(b). Here $\phi$ is
the angle spanned by the magnetic field and the $c$ axis (see inset of Fig.
3(b)). As a general trend, the modulation of $M(\phi)$ is suppressed with
increasing magnetic field. In addition, there is a change in its functional
form. While $M(\phi)$ exhibits a minimum and a maximum in low fields parallel
and perpendicular to the $c$ axis, respectively, upon saturation a local
maximum develops also for fields applied along the $c$ axis. To avoid
complications due to the presence of multiple magnetic domains, we determined
the anisotropy based on $M(\phi)$ curves measured in the fully field polarized
state, as shown in Fig. 3(b) for the 4 T curve. The angular dependence of the
magnetisation was fitted based on commonly used phenomenological expression
for the magnetic anisotropy energy for a hexagonal ferromagnet [59]

$\displaystyle
E_{A}=K_{1}\sin^{2}\theta+K_{2}\sin^{4}\theta+K_{3}\sin^{6}\theta\cos 6\Psi.$
(2)

Here, $K_{1}$ and $K_{2}$ are the first- and second-order anisotropy
constants, respectively. $\theta$ is the angle spanned by the magnetisation
with the $c$ axis, while $\Psi$ denotes the angle between the projection of
the magnetisation to the $ab$ plane and the $a$ axis. In our fitting the
$K_{3}$ term is neglected, since our data show that the in-plane anisotropy is
less than 2 % of the total magnetic anisotropy, as already discussed above.

The fitting formula for calculation of the anisotropy constants $K_{1}$ and
$K_{2}$ for angular-dependent magnetisation measurements was derived by
minimising the total energy per unit volume described as
$E=E_{A}+E_{Z}+E_{D}$, where $E_{Z}=-M_{s}Hcos(\theta-\phi)$ corresponds to
the Zeeman energy, and $E_{D}=1/2(M_{||}^{2}\,D)$ is the magnetostatic energy
of a sample in its own field. The formula of fitting of the experimental data
is given by

$\displaystyle\sin
2\theta(K_{1}+2K_{2}\sin^{2}\theta)=-H_{i}\sqrt{M_{s}^{2}-M_{||}^{2}},$ (3)

where $M_{||}$ is the projection of $M_{s}$ on the applied field, which is
measured in experiment (see inset of Fig. 3(b)).

Figure 3:  (a) Angular dependence of the magnetisation measured in different
fields at 300 K. The magnetisation component parallel to the field is
detected. (b) Experimental data (open circles) and fit to the experimental
data (solid lines) in 4 T at 300 K.  Figure 4: (a) Temperature-dependent
anisotropy constants $K_{1}$ and $K_{2}$ as determined from the angular
dependence of the magnetisation (circles, marked as R) and by the S-T method
(triangles). (b) The uniaxial constant $K_{u}$ determined by the “area method”
(Ar) is plotted together with $K_{1}$ \+ $K_{2}$ taken from panel (a). Lines
are guides to the eye.

The fit to the angular-dependent magnetisation (marked as R method, fit 1 in
Fig. 3(b)) describes well the angular dependence of $M(\phi)$ over the whole
angular range except for the immediate vicinity of the $c$ axis, where it
overshoots the experimental data. The experimentally observed difference in
$M_{s}$ for fields parallel and perpendicular to the $c$ axis indicates an
anisotropy ($\sim 2\,{\%}$) of the $g$-tensor, i.e. imply peculiar orbital
contributions to the magnetisation. (Please notice, if the $g$-tensor
anisotropy is completely negligible, the magnetisation should point parallel
to the applied field in the high-field limit, hence $\theta$ in Eq. 2 and
$\phi$ in Fig. 3(b) would become identical.) From the fit to the experimental
data we derived the value of the first anisotropy constant $K_{1}=-1.32\times
10^{6}J/m^{3}$ at 2 K, which is in good agreement with the $K_{1}=-1.34\times
10^{6}J/m^{3}$ determined by Sucksmith-Thompson (S-T) method (see below) [60].

To account for the angular dependence of the saturation magnetisation
anisotropy we used the approach developed by J. Alameda ${et\,al.}$ and A. S.
Bolyachkin ${et\,al.}$ [61, 62]. This fit (fit 2 in Fig. 3(b)) improves the
description of the experimental data for angles close to the $c$ axis. Figure
4 (a) summarizes the temperature dependence of the anisotropy constants
$K_{1}$ and $K_{2}$, as determined by two independent methods: a) from the
fits to the angular dependence of the magnetisation measured in 4 T and b)
calculated from the magnetisation curves along the $c$ axis, following the
modified (S-T) method, which considers the anisotropy of the saturation
magnetisation [62]. At 300 K, both methods yield very close values
$K_{1}=-0.99\times 10^{6}J/m^{3}$ and $-1.01\times 10^{6}J/m^{3}$,
respectively. The magnitude of $K_{1}$ shows a monotonous increase with
decreasing temperature in contrast to non-monotonous behavior reported by
Fayyazi $et\,al.$ [30]. Moreover, the absolute values of $K_{1}$ reported in
[30] are by about 20% lower than our experimental values. These discrepancies
likely arise due to limitations of proper sample orientation in the former
study, which was not an issue in our case due to the larger single crystals.
$K_{2}$ has opposite sign and is lower by approximately one order magnitude
than $K_{1}$. At 300 K, we obtained $K_{2}=1.4\times 10^{5}J/m^{3}$ and
$2.7\times 10^{4}J/m^{3}$ using the R and S-T methods, respectively.

The reason of the difference in the values of $K_{1}$ and $K_{2}$ is unclear
and may be related to different data sets for fitting: high-field
magnetisation data in the saturated state in R method and low-field data in
S-T method. However, the value of $K_{2}$ derived from the R method should be
taken with precaution because of larger error in determination due to
nonlinear extension of the rotator spring and change of its elasticity with
lowering temperature.

To further validate our quantification of the uniaxial anisotropy in Fe3Sn, in
Fig. 4(b), we compare the sum of $K_{1}+K_{2}$, as determined using the
angular dependent magnetisation and the Sucksmith-Thompson methods, with the
overall $K_{u}$ anisotropy, as obtained using the “area method”. The values
obtained by the three methods agree within less than 10% at all temperatures.

###  III.2 Ab initio calculations of magnetic moments and anisotropy

The origins of magnetocrystalline anisotropy energy (MAE) is well-established
for $3d$-transition metals as arising from a combination of the spin-orbit
coupling and the crystal-field environment of the magnetic species. Being a
ground state property the MAE can in principle be calculated using DFT,
however, the small energy scales associated with MAE in $3d$ transition-metal
systems make this challenging owing to the tight convergence and sampling
needed. Because of this, we compare two different implementations of DFT - a
pseudo-potential code with a plane-wave basis set (VASP), and an all-electron
code with LAPW basis (ELK).

DFT computation of Fe3Sn binary system has been recently published [29, 30]
and in particular it was found that the easy magnetic axis lies in the
hexagonal plane. Here, we first present our calculations of the magnetic
properties using VASP. We find the calculated ground state to be ferromagnetic
order with a spin moment of 2.20 $\mu_{B}$ per Fe, an orbital moment of 0.06
$\mu_{B}$ per Fe, and a small antiparallel moment of 0.18 $\mu_{B}$ on the Sn
site, giving a total magnetisation of 6.4 $\mu_{B}$/f.u., consistent with our
experimental value of 6.8 $\mu_{B}$/f.u.. We find no off-plane canting of
moments aligned along the easy-plane direction in our VASP calculations, as
shown in Fig.1(f).

To calculate the magnetocrystalline anisotropy, we rotate the spin in the in-
plane and out-of-plane directions, finding the corresponding energy for
several angles, as depicted in Figure 1(e). We find the ground state magnetic
order to be ferromagnetic with an easy-plane anisotropy. In fact, we find the
in-plane $({ab})$ spin directionality to be fully degenerate (up to 2
$\mu$eV), in agreement with the negligible sixth-order $ab$-plane anisotropy
found experimentally is this study. Our calculated magnetocrystalline
anisotropy energy by comparing the energy of the in-plane and out-of-plane
spin axes is 0.406 meV/f.u. ($1.16\times 10^{6}J/m^{3}$), fully consistent
with our experiment. Note that this value is by ca. 20 % lower/higher than the
theoretical estimate reported by Sales $et\,al.$ [29]/Fayyazi $et\,al.$ [30].
In Fig.1(f) we also plot the variation of the calculated orbital moment on Fe
from varying the spin easy-axis in the $({ac})$ direction. For the ground-
state spin direction (fully in-plane), we find no out-of-plane contribution to
either the spin or orbital moment. However, as the spin axis is rotated in the
out-of-plane direction, we find a gradual increase in the out-of-plane orbital
moment which is compensated by a decrease in the in-plane orbital moment.

We now discuss our results using ELK. For systems containing $3d$-electrons,
the relatively weak SOC can be treated in a Russel-Sounders ($LS$)-coupling
scheme, with well defined spin and orbital quantum numbers. In this case it is
possible to define a quantization direction (direction of magnetisation). In
practice we have considered the quantization axis (spin moment $\vec{S}$)
aligned along the $a$\- and $c$\- crystallographic directions and performed
DFT self-consistent computations for the components of the spin/orbital
moments. The values of Fe spin/orbital moments are about 2.32/0.07 $\mu_{B}$
for the considered orientations. We have noticed, however, that in the self-
consistent computation with the moment along the crystallographic
$a$-direction the overall (unit cell) moment is slightly increased. The DFT
self-consistent configuration for spin moments of Fe and Sn atoms oriented
along the $c$/$a$-directions respectively are shown in Figs. 1 (c)/(d). In the
ferromagnetic ground state the induced spin moment on Sn-sites is of about
0.12 $\mu_{B}$. Orbital moments of Sn although existing, have no significant
magnitudes ($\leq 10^{-3}~{}\mu_{B}$). The total magnetic moment per formula
unit, as obtained from DFT is 6.9 $\mu_{B}$/f.u., in good agreement with the
experimental value of 6.8 $\mu_{B}$/f.u.

The DFT moments (on all atoms) computed for the configuration aligned along
the $a$-direction are somewhat smaller, in particular the decrease in the
magnitude of the Sn atoms spin-moments (pointing opposite to the Fe spins) are
larger than the corresponding decrease for Fe atoms. This is the cause for a
slightly larger total (unit cell) moments along the $a$-direction. Thus, the
anisotropy of the moments results from the subtle balance between spin and
orbital moments on the different sites of the unit cell.

Let us comment further upon the distinction between the anisotropy of magnetic
moments and the macroscopic anisotropy energies. In systems (containing
heavier elements) with lower than cubic symmetry the magnetic moment per atom
or the corresponding $g$-tensor is expected to be anisotropic and the spin-
orbit interaction might be important also in the absence of a long-range
order. Below the ordering temperature, the cooperative alignment of the
moments produces a net magnetisation. The energy cost to rotate the
magnetisation from the easy axis (lowest energy) into the hard direction is
higher at lower temperatures. In the framework of second-order perturbation
theory (at zero temperature) the anisotropy energy is proportional to the
anisotropy of the orbital magnetic moment. Note also that for Fe3Sn, for the
$c$ direction configuration, the spin and orbital moments align collinearly.
At the same time, we found that the orbital moments rotate slightly in the
$ab$ plane away from the initial $a$ direction (angular departure of about 7
degrees). Although the bulk measurements presented here are not able to
capture the small changes in the orientation and magnitude, recent progress in
XMCD allows for a precise determination of spin and orbital moments. Through
the XMCD sum rules also the magnitude of the orbital moment and its anisotropy
can be measured, note that the relative change in the orbital moment magnitude
and direction is important, not the absolute number, which is obviously small.

Figure 5: (a) Topographic image collected in contact mode on an as-grown Fe3Sn
$ab$ surface, also imaged in the MFM scans in panels (b), (c), and (d).
(b)-(d) MFM images, recorded at 100 K, showing the shift in resonance
frequency of the cantilever: Light/dark colors show an increase/decrease in
the resonance frequency. (b) Extended domain walls meeting at vortex cores in
the zero-field state. (c) Uniform MFM signal, observed except in the vicinity
of topographic features, indicate uniform magnetisation in 3 T. (d) The zero-
field state after field treatment hosts a high density of vortices, some of
which are much more angular in structure. The difference between the fully
relaxed vortex structure (b) and the metastable structure shortly after the
removal of a field (d) is highlighted by the dashed circle and square,
respectively. (e) Micromagnetic simulations of the local orientation of the
magnetisation (gray arrows), viewed down the $c$ axis. The false color code
(bottom right) indicates the in-plane orientation of the magnetisation vector.
(f) Magnified view of one of the closure structures (vortices) in (e). (g) MFM
response calculated for the domain pattern in panel (e).

###  III.3 Imaging and simulations of magnetic vortices

To move beyond the investigation of bulk properties, we use MFM to image the
local magnetic textures on an as-grown $ab$ plane. A contact-mode topography
image is given in Fig. 5(a), to show that the surface is smooth enough for the
magnetic signal to be collected in a single-pass, constant height mode. While
generally flat, there is a change in topography in the top-left of the image
and a few point defects which need to be considered when interpreting the MFM
data. The magnetic microstructure of the sample is represented by the relative
shift of the cantilever’s resonance frequency, at a lift height of 150 nm.
Figure 5(b) shows a representative MFM image of the virgin zero-field magnetic
state. Strikingly this evidences the formation of smooth vortices across the
sample. Such vortices are characteristic of systems where the magnetisation is
confined to a nearly isotropic plane, either via easy-plane anisotropy or
shape anisotropy, the latter being the consequence of dipole-dipole
interaction in thin samples.

Following the procedure used for the bulk measurements, we collect MFM images
in a 3 T magnetic field, perpendicular to the sample surface. As expected from
the bulk measurements, 2(a), Figure 5(c) shows a nearly mono-domain,
saturated, state. The remaing contrast predominantly originates at features
that correlate with the topography changes seen in Fig. 5 (a), suggesting that
they are not purely of magnetic origin. The field is removed and a subsequent
scan, Fig. 5(d), shows that the vortices have reformed but in different
positions and with different geometries. The different positions of the
vortices show that the vortex microstructure is not pinned. The more angular
geometry of vortex cores, see the high-lighted square in Figure 5(d), are
associated with larger stray fields, this likely indicates that the
microstructure, at 100 K, has not fully relaxed on the measurement time scale
of tens of minutes.

In order to directly connect our images of vortex-antivortex microstructure to
our bulk measurements, we use micromagnetic simulation. The bulk data are used
as input parameters to compute the expected orientation of the local magnetic
moments, as shown in Fig. 5(e). Here, the small arrows represent the nanoscale
orientation of the magnetisation vector and the rainbow false colors showing
areas of the same orientation. The pattern is typical for an easy-plane
magnet, with sets of vortex-antivortex pairs. Figure 5(f) shows one such
example of this, with a continuous in-plane rotation of magnetisation.

The connection between the local moment and the stray fields, that our MFM
senses, can be counterintuitive, and so we use MuMax3 to calculate the
expected MFM response of the microstructure (shown in Fig. 5(e)) which is
presented in Fig. 5(g). This is possible once the tip material and lift height
are given, as the force on the MFM tip goes as the gradient of the stray field
– which is known via the preceding magnetisation calculations. The calculated
MFM signal is in excellent qualitative agreement with the observed signal of
the zero-field state, e.g. Fig. 5(b) and (d), showing a series of vortex
points connected by domain walls. The simulations also show that the contrast
at the walls depends on the exact nature of the magnetisation reorientation:
some walls have very sharp bright-dark features, while others have smoother
transitions; again, as observed in the MFM data. Note, observing in-plane
closure domains with an MFM is long established for thin-films, for instance,
Ref. [63].

##  IV Conclusions

In conclusion, we have grown high-quality bulk single crystals of single
kagome bilayer ferromagnet Fe3Sn and performed magnetisation and magnetic-
force-microscopy studies. The studies revealed a strong uniaxial easy-plane
anisotropy characterized by the first-order anisotropy constant
$K_{1}=-0.99\times 10^{6}J/m^{3}$ at 300 K and $-1.23\times 10^{6}J/m^{3}$ at
2 K. The three independent methods, applied for the calculation of anisotropy
constants in our work, provide values that are in good agreement with each
other as well as the anisotropy energy obtained from our ab initio study.
Moreover, the used angular dependent measurements allowed to evidence the
anisotropy in the saturation magnetisation along the $a$ and the $c$ axes,
which we ascribe to orbital contributions. Our DFT calculations predict an
induced spin moment on the Sn sites of a similar magnitude to the Fe’s orbital
moment, the two pointing opposite to each other. All-electron DFT calculations
suggest that the orbital-magnetic moments of Fe tilt within the easy plane
with respect to the main crystallographic directions, thus a slight departure
from the collinear magnetic configuration is obtained. Although the current
experimental analysis of the bulk magnetisation data can not distinguish
between the orbital collinear and non-collinear configuration within the
$ab$-plane, the neutron diffraction measurements might confirm the existence
and the magnitude of the induce spin moment on Sn sites. Moreover, XMCD
studies have the potential to reveal the predicted non-collinearity of the
orbital moments.

Our MFM study further confirms the system is an easy-plane ferromagnet that
contains a rich microstructure of the magnetisation pattern, dominated by
(anti)vortices. The experimentally observed MFM contrast is well reproduced by
the micromagnetic simulations, using experimentally determined values of the
magnetic interaction parameters.

Overall, our bulk and microscopic experimental studies in combination with ab
initio and micromagnetic calculations provide a multi-scale, highly reliable
approach to quantify magnetic interactions, e.g., magnetic anisotropy and spin
arrangement, and to reveal their origins in the kagome bilayer model system
Fe3Sn. This is an important step towards understanding complex magnetism
emerging in kagome magnets in presence of remarkable spin-orbit effects.

## Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through
Transregional Research Collaboration TRR 80 (Augsburg, Munich, and Stuttgart),
and by the project ANCD 20.80009.5007.19 (Moldova). DME wishes to thank and
acknowledge funding by the DFG individual fellowship, number (EV 305/1-1). SMG
acknowledge funding by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences, Materials Sciences and Engineering Division
under Contract No. DEAC02-05-CH11231 (Nonequilibrium magnetic materials
program MSMAG). Computational resources were provided by the National Energy
Research Scientific Computing Center and the Molecular Foundry, DOE Office of
Science User Facilities supported under same contract.

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