††thanks: These authors contributed equally to this work.

# Large anomalous Hall effect in single crystals of the kagome Weyl
ferromagnet Fe3Sn

Bishnu P. Belbase,1,7,† Linda Ye,2,† Bishnu Karki,1 Jorge I. Facio,3,4 Jhih-
Shih You,5 Joseph G. Checkelsky,2 Jeroen van den Brink,6 and Madhav Prasad
Ghimire,1,6<EMAIL_ADDRESS>1Central Department of Physics,
Tribhuvan University, Kirtipur 44613, Kathmandu, Nepal
2Department of Physics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA
3 Centro Atómico Bariloche and Instituto Balseiro, CNEA, 8400 Bariloche,
Argentina
4 Instituto de Nanociencia y Nanotecnología CNEA, 8400 Bariloche, Argentina
5Department of Physics, National Taiwan Normal University, Taipei 11677,
Taiwan
6Leibniz IFW Dresden, Helmholtzstr. 20, Dresden 01069, Germany
7Department of Physics and Astronomy, Purdue University, West Lafayette,
Indiana 47907, USA

###### Abstract

The material class of kagome metals has rapidly grown and has been established
as a field to explore the interplay between electronic topology and magnetism.
In this work, we report a combined theoretical and experimental study of the
anomalous Hall effect of the ferromagnetic kagome metal Fe3Sn. The compound
orders magnetically at 725 K and presents an easy-plane anisotropy. Hall
measurements in single crystals below room temperature yield an anomalous Hall
conductivity $\sigma_{xy}\sim 500\,(\Omega\textrm{cm})^{-1}$, which is found
to depend weakly on temperature. This value is in good agreement with the
band-intrinsic contribution obtained by density-functional calculations. Our
calculations also yield the correct magnetic anisotropy energy and predict the
existence of Weyl nodes near the Fermi energy.

##  I Introduction

In recent years, a number of materials with kagome layers as structural
building block has been investigated. Their electronic structures have been
found to host unconventional fermions, such as the Weyl fermions in Mn3(Ge,Si)
[1, 2] and Co3Sn2S2 [3, 4], quasi-2D Dirac fermions in Fe3Sn2 [5], and
quasi-2D Dirac fermions together with flat bands in FeSn [6].

The variety of kagome metals is being intensively studied including compounds
based on different transition metals, e.g, FeSn and CoSn [7, 8, 9], but also
different composition variants for a given transition metal, e.g. FeSn and
Fe3Sn2. The two-dimensional limit has been also addressed, e.g., in thin films
of Mn3Sn [10], FeSn [11] or Co3Sn2S2 [12] and in monolayers of Cu2Ge, Fe2Ge,
and Fe2Sn [13]. The latter are iso-structural monolayers combining the
honeycomb lattice structure of Cu or Fe with the triangular lattice of Ge or
Sn. Interesting electronic properties have been reported, such as massive Weyl
points in the FM state of Fe2Ge and massless Weyl points in the FM state of
Fe2Sn [13].

Large efforts have been made to correlate the geometrical aspects of the
electronic structure with the electronic response to external fields. In
particular, large anomalous Hall and Nernst conductivities have been reported
in several kagome metals, both in cases in which the ground state is
ferromagnetic [14, 4, 15, 16, 17] or antiferromagnetic [18, 1, 2, 10].

In this work, we study the compound Fe3Sn, one natural structural derivative
of FeSn and Fe3Sn2 [19, 6]. This material has recently attracted attention as
a possible permanent magnet due to its large ordering temperature (725 K) and
the possibility of changing its magnetic ground state from easy plane to easy
axis through alloying [20, 21]. Recently, large anomalous Nernst data in
polycrystals of Fe3Sn have been reported [22]. Here, we report the growth of
single crystals along with magnetic properties and measurements of the
anomalous Hall resistivity. In addition, we calculate the electronic structure
by means of density-functional theory (DFT) calculations. Our main result is
the experimental finding of a large anomalous Hall conductivity (AHC) of about
$500\,(\Omega\textrm{cm})^{-1}$, a value which is found to agree well with the
band-intrinsic contribution computed from DFT. The measured AHC is weakly
temperature dependent below room temperature. Further, our calculations
indicate the existence of Weyl nodes near the Fermi surface and suggest that
moderate hole doping could further increase the AHC up to values of $\sim
1250\,(\Omega\textrm{cm})^{-1}$.

This work is organized as follows. Section II describes the experimental and
computational methods used in our investigation. Section III presents
experimental results with focus on the anomalous transport properties. Section
IV contains our theoretical results, which include a characterization of the
bulk electronic and magnetic properties, as well as a characterization of the
Weyl nodes structure in the ferromagnetic phase. Last, Section V contains our
concluding remarks.

##  II Methods

###  A Experimental Methods

The bulk crystal structure of Fe3Sn shown in Fig. 1(a,b) consists of layers of
iron and tin piled in an A-B stacking. The Fe atoms form a kagome lattice with
Sn atoms sitting at the center of the hexagon. This structure belongs to space
group $P6_{3}/mmc$ (SG 194) with the experimental lattice constants $a$ = $b$
= 5.487 Å and $c$ = 4.31 Å, respectively [23, 24].

Figure 1: (a) Crystal structure of Fe3Sn. (b) Kagome lattice in the $ab-$plane
along the $a$-axis. (c) Photo of Fe3Sn single crystals where the hexagonal
facets are (001) planes. The vertical scale bar stands for 1 mm.

In view of the narrow stability regime of Fe3Sn in the iron-tin binary phase
diagram [25], we grew single crystals of Fe3Sn via a solid state reaction.
Iron powder (Alfa Aesar, 99.998%) and tin powder (Alfa Aesar, 99.995%) were
mixed in a ratio of $3.5\sim 3:1$ and loaded into an evacuated quartz tube.
After being heated up to 850 ∘C, kept for 120 hours and slowly cooled to 800
∘C over the course of 400 hours, the quartz tube was quenched to cold water
and we found crystals of Fe3Sn up to the lateral size of 500 $\mu$m as we show
in Fig. 1(c). We note that a small number of Fe5Sn3 crystals can also be found
in the quartz tubes. We confirmed with X-ray diffraction (XRD) that during the
growth process Fe3Sn and Fe5Sn3 tend to crystallize in distinct morphologies:
Fe3Sn forms hexagonal prisms (see crystals in Fig. 1(c)) and Fe5Sn3 tends to
form elongated hexagonal bipyramids. Phase purity of the sample under study is
confirmed by powder XRD (see supplemental material, Fig. S1) [26]. Electrical
transport measurements were performed using the standard five-probe methods in
a commercial superconducting magnet. The magnetic field is applied along the
$c$-axis and the current within the $ab$-plane.

###  B Computational details

DFT calculations were performed using the full potential local orbital (FPLO)
code version 18.00-52 [27, 28]. The Generalized Gradient Approximation with
the parametrization of Perdew, Burke, and Ernzerhof was used [29].
Relativistic effects were treated in the four component formalism as
implemented in FPLO. Brillouin zone (BZ) integrals were performed with the
tetrahedron method based on a $k$-mesh formed of $24\times 24\times 24$
subdivisions. The experimental structure was used as detailed above. To ensure
an accurate band structure close to the Fermi level, an enhanced basis set as
defined in the appendix to Ref. [30] was used.

Ferromagnetic (FM) configurations were considered with different quantization
axes. In particular, the magnetic anisotropy energy (MAE) was computed based
on calculations with the magnetic moments oriented within the kagome planes
([100] direction) or perpendicular to these layers ([001] direction).

The study of Weyl nodes, of the surface spectral properties and of the
anomalous Hall conductivities are based on a tight-binding model obtained by
constructing maximally projected Wannier functions. For this aim, we used the
PYFPLO [28] module of the FPLO code. The local Wannier basis set is formed of
Fe-3$d$, Fe-4$s$, Sn-5$s$ and Sn-5$p$ orbitals. The search for Weyl nodes was
done with a bisection method that detects sources of the Berry curvature
field. After a set of candidate nodes is found, the corresponding Chern number
associated to each point is computed by integrating the Berry curvature flux
through a small sphere centered at the node, as done in Ref. [31].

Finally, the band-intrinsic contribution to the anomalous Hall conductivity is
calculated using the formula

$\sigma_{\alpha\beta}^{A}=-\frac{\epsilon_{\alpha\beta\gamma}e^{2}}{\hbar}\sum_{n}\int_{BZ}\frac{dk}{(2\pi)^{3}}f_{n}(k)\Omega_{n,\gamma}(k)$
(1)

where $n$ is a band index, $\Omega_{n,\gamma}$ is the $\gamma$ component of
the Berry curvature, $f_{n}(k)$ is the occupation number of the Bloch state
and $\epsilon_{\alpha\beta\gamma}$ the Levi-Civita tensor [32]. A 300$\times$
300$\times$ 300 $k$-mesh was used for this calculation.

##  III Experimental results

Figure 2: (a) Resistivity $\rho$ of Fe3Sn as a function of $T$ with current
$I$ in the kagome lattice plane. (b) Hall resistivity $\rho_{yx}$ as a
function of $H$ at selected $T$. Here the magnetic field $H$ is applied along
the $c$-axis. (c) Anomalous Hall resistivity $\rho^{A}_{yx}$ and ordinary Hall
coefficient $R_{H}$ (inset) vs. $T$ in Fe3Sn extracted from $\rho_{yx}$. (d)
$T$-evolution of the anomalous Hall conductivity $\sigma^{A}_{xy}$ in Fe3Sn.

Fe3Sn is known to order ferromagnetically at $T_{C}=725$ K [20, 21]. The
resistivity $\rho$ of Fe3Sn as a function of temperature $T$ within the kagome
lattice plane is shown in Fig. 2(a), exhibiting a residual resistivity ratio
(RRR) of 2.9. We assume that the diffusive nature of the solid state reaction
process facilitates disorder formation throughout the synthesis process,
resulting in the high residual resistivity $\sim 35$ $\mathrm{\mu\Omega\cdot
cm}$ and low RRR in Fe3Sn as compared with it’s cousin compounds Fe3Sn2 [19]
and FeSn [6].

As expected in prototypical ferromagnetic metals, we observe in the Hall
traces of Fe3Sn a prominent anomalous component as shown in Fig. 2(b).
Saturation is observed at about 3 T external field, which we relate to the
complete re-orientation of the magnetization from its easy-plane ground state
to the c-axis. The Hall resistivity $\rho_{yx}$ can be decomposed into the
ordinary and anomalous contributions via $\rho_{yx}=R_{H}B_{z}+R_{S}M_{z}$,
where $R_{H}$ is the ordinary Hall coefficient, $B_{z}$ the magnetic induction
in $z$-direction, $R_{S}$ the anomalous Hall coefficient, and $M_{z}$ the
magnetization in $z$-direction. $R_{S}M_{z}$ gives the anomalous Hall
resistivity $\rho_{yx}^{A}$. We infer $\rho_{yx}^{A}$ and $R_{H}$ from the
high field intercept and slope (above 3 T) of $\rho_{yx}$, respectively. This
means, the extracted $\rho_{yx}^{A}$ data refers to saturated $M_{z}$. The
results are shown in Fig. 2(c). $\rho_{yx}^{A}$ grows monotonically with $T$
while $R_{H}$ exhibits a sign reversal; $R_{H}$ at $T=2$ K corresponds to a
carrier density of $1.2\times 10^{22}/\mathrm{cm}^{3}$. The observed $R_{H}$
suggests the coexistence of both large electron and hole pockets in the
system. The anomalous Hall conductivity $\sigma_{xy}^{A}$ estimated from
$\rho_{yx}^{A}$ via $\sigma_{xy}^{A}\sim{\rho_{yx}^{A}}/\rho^{2}$ is displayed
in Fig. 2(d). $\sigma_{xy}^{A}$ is found to depend weakly on $T$ and the
maximum is around 500 $\Omega^{-1}\mathrm{cm}^{-1}$. We note that the system
is characterized by longitudinal conductivity $\sigma_{xx}\approx 1/\rho$
between $1\sim 3\times 10^{4}$ $\mathrm{\Omega}^{-1}\mathrm{cm}^{-1}$, and
falls rather in the moderately dirty regime, such that the observed
$\sigma_{xy}^{A}$ is expected to reflect an underlying intrinsic Berry
curvature contribution [33, 34].

##  IV COMPUTATIONAL RESULTS

In this section, we first analyze the magnetic properties and electronic
structure of Fe3Sn, we report the existence of several Weyl nodes near the
Fermi surface and we present calculations of the band-structure-intrinsic
contribution to the anomalous Hall conductivity.

Figure 3: (a) Band structures in the scalar relativistic and fully-
relativistic approximations with the magnetic quantization axis along [100].
(b) Fully relativistic band structures with the quantization axis along [100]
and [001].

Table 1 shows the total energies and magnetic moments obtained for different
quantization axes. The FM state with quantization axis along the [100]
direction is found to be the ground state with a total magnetic moment of
7.22$\mu_{B}$ per Fe3Sn.

The magnetic moment is slightly reduced if the magnetization is oriented along
the c-axis. The MAE calculated from the energy difference between these two
states amounts to $\sim 0.64$ meV per Fe3Sn. Note that we refrain from an
analysis of in-plane magnetic anisotropy, which is very small in hexagonal
systems and not expected to affect the transport properties considered here.
Under the assumption of negligible higher-order terms, this translates into an
anisotropy constant $K_{1}$ = - 1.8 MJ$/$m3 and an anisotropy field of 3.1 T.
The former value agrees with earlier theoretical results of $\lvert
K_{1}\rvert$ = 1.59 MJ$/$m3 [20] (the authors of this reference use a sign
convention opposite to the usual one) and of - 1.5 MJ$/$m3 [35]. The
calculated anisotropy field agrees with the experimentally observed saturation
around 3 T, see Fig. 2 (b), and with the same result of 3 T obtained from
earlier magnetization data [20]. We tried the application of so-called orbital
polarization corrections (OPC) [36] which, however, yield a two-fold enhanced
anisotropy field. Thus, we conclude that Fe3Sn is better described by
conventional GGA than by GGA-OPC and use only GGA data for the further
investigation.

Table 1:  Total energy referred to the ground state energy and total magnetic moments (including spin and orbital contributions) for the two considered magnetic states. The reported data are calculated considering the extended basis set. State | Energy (meV / f.u.) | Total moment($\mu_{B}$ / f.u.)
---|---|---
FM [100] | 0 | 7.22
FM [001] | 0.64 | 7.20

Figure 3(a) presents the energy dispersion obtained in the scalar and in the
fully-relativistic calculations with [100] orientation of the magnetization.
Several bands, which mainly present Fe-$3d$ character, cross the Fermi energy.
Bands with relatively flat energy dispersion as a fingerprint of the kagome
lattice are present in the $\Gamma$-$M$-$K$ plane around -0.5 eV as well as in
the $A$-$L$-$H$ plane around +0.6 eV and around -0.8 eV. Without SOC, apparent
low-energy band crossings along the paths $\Gamma$-M and M-K-$\Gamma$ can be
observed, which appear to be robust to the SOC for this direction of the
magnetic moments. Fig. 3(b) compares the bands with SOC for the FM [001] and
FM [100] configurations and shows that these crossings become gapped when the
magnetization points along [001].
A search of Weyl nodes for the FM [100] ground state yields the existence of
several Weyl nodes, the closest to Fermi surface are found 8 meV below the
Fermi energy. Notice that in SG 194, for this magnetic configuration, there
can be up to eight symmetry-related Weyl nodes generated by the inversion
symmetry $I$, the two-fold rotations $C_{2}(y)\times\Theta$, $C_{2}(2x+y)$,
$C_{2}(z)\times\Theta$ and the reflection symmetries $m(y)\times\Theta$,
$m(2x+y)$ and $m(z)\times\Theta$, where $\Theta$ is the time-reversal
operator.
When the magnetization is rotated towards the hard axis, a situation
experimentally relevant in view of the MAE found above and the observed
saturation of $\rho_{yx}$, one can expect the Weyl nodes to move in momentum
and energy space as theoretically found in other magnetic compounds [37, 38].
A search of Weyl nodes for the FM [001] indeed leads to sizeable changes in
the number and energy of the Weyl nodes near the Fermi energy, as illustrated
in Fig. 4, which shows the found Weyl nodes ordered according to their energy,
relative to the Fermi energy, for both magnetic configurations. We also
present the distribution of these Weyl nodes in the Brillouin zone in the
supplemental material (see Fig. S2 and S3) [26].

Figure 4: Energy of Weyl nodes near the Fermi surface for the magnetization
along the [001] and [100] directions. Red, blue and orange dots correspond to
Weyl nodes in which the lower band has index $N-1$, $N$ and $N+1$,
respectively, with $N$ the number of valence electrons. Figure 5: Anomalous
Hall conductivity as a function of the chemical potential.

Fig. 5 presents the band-intrinsic contribution to the anomalous Hall
conductivity as a function of the chemical potential. For quantization axis
along [001] ([100]) only $\sigma_{xy}$ ($\sigma_{yz}$) is different from zero.
At the charge neutrality point ($\mu=0$), the related values are
$\sigma_{xy}^{A}=702\,(\Omega\textrm{cm})^{-1}$ and
$\sigma_{yz}^{A}=482\,(\Omega\textrm{cm})^{-1}$. The former value is in fairly
good agreement with the experimental result. In the supplemental material
[26], we analyze the contribution to the AHC from states at different parts of
the Brillouin zone and show that the most prominent hot spots acquire a loop
structure which reflects the importance of structures other than isolated Weyl
nodes for the AHC. The AHC obtained for Fe3Sn is significantly larger than
Fe3Sn2 with a value of 200 $(\Omega\textrm{cm})^{-1}$ [19], and is comparable
to Fe5Sn3 with a value of 507 $(\Omega\textrm{cm})^{-1}$ [39]. Interestingly,
according to the calculations a moderate amount of hole doping could increase
$\sigma_{xy}^{A}$ and $\sigma_{yz}^{A}$ up to maximum values of $\sim
950\,(\Omega\textrm{cm})^{-1}$ and $\sim 1250\,(\Omega\textrm{cm})^{-1}$,
respectively. However, this increase is contingent upon maintaining the
stability of the ground state property during the doping process [40, 41].
Recent values of AHC at $E_{\textrm{F}}$ [42] is found to be larger than our
reported value obtained from the enhanced basis calculations, but is well
supported with our earlier work [43].

##  V Conclusions

We have studied single crystalline kagome metal Fe3Sn by means of transport
experiments and on the basis of density-functional calculations. The system is
found to be ferromagnetic, with an easy-plane magnetic anisotropy. A moderate
magnetic field of about 3 T is sufficient to reorientate the magnetization
along the hexagonal axis. From transport measurements we have inferred the
anomalous Hall conductivity to be $\sim 500\,(\Omega\textrm{cm})^{-1}$. Our
DFT results reproduce the preference of the easy-plane anisotropy and our
calculation of the band-intrinsic contribution to the AHC agrees well with the
experimental results. Interestingly, our calculations indicate that moderate
hole doping could largely increase the AHC. All-together, our experimental and
theoretical results establish Fe3Sn as a kagome Weyl metal with large AHC.

###### Acknowledgements.

We thank Manuel Richter for fruitful discussions, suggestions and for reading
the manuscript. We also thank Ulrike Nitzsche for technical assistance. M.P.G.
thanks the Alexander von Humboldt-Stiftung for the financial support through
equipment grants and renewed research program. M.P.G. was also supported by a
grant from UNESCO-TWAS and the Swedish International Development Cooperation
Agency (SIDA). The views expressed herein do not necessarily represent those
of UNESCO-TWAS, SIDA or its Board of Governors. J.I.F. would like to
acknowledge the support from the Alexander von Humboldt-Stiftung and ANPCyT
grants PICT 2018/01509 and PICT 2019/00371. L.Y. acknowledges support by the
Tsinghua Education Foundation and STC Center for Integrated Quantum Materials,
NSF grant number DMR-1231319. This work was funded, in part, by the Gordon and
Betty Moore Foundation EPiQS Initiative, through Grants GBMF3848 and GBMF9070
to J.G.C. (material synthesis) and NSF grant DMR-2104964 (material analysis).
J.-S.Y. is supported by the National Science and Technology Council (Grant No.
MOST 110-2112-M-003-008-MY3) and National Center for Theoretical Sciences in
Taiwan. J.v.d.B. acknowledges financial support from the German Research
Foundation (Deutsche Forschungsgemeinschaft, DFG) via SFB 1143 Project No. A5
and under Germany’s Excellence Strategy through the Würzburg-Dresden Cluster
of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147,
Project No. 390858490).

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Supporting Information: Large anomalous Hall effect in single crystals of the
kagome Weyl ferromagnet Fe3Sn

## X-ray diffraction

This data confirms the phase purity of single crystals of Fe3Sn.

FIG. S1. Powder XRD pattern of Fe3Sn used in this study (the wavelength of
X-ray used is 1.54 Å).

## Weyl nodes distribution in Brillouin zone

In the main text, we have presented the energy of the Weyl nodes, as
illustrated in Fig. 4. Additionally, to provide further insight, we depict the
distribution of these Weyl nodes in the Brillouin zone in Fig. S2. Weyl nodes
closest to Fermi level are found 8 meV below the Fermi energy for the
magnetization along [100] direction. There are eight such Weyl nodes connected
by symmetry which we present in Fig. S3.

FIG. S2. Weyl nodes distribution in the three-dimensional (3D) and two-
dimensional (2D) Brillouin zone. (a) In 3D for Magnetization along [001]
direction (b) In 2D for Magnetization along [001] direction (c) In 3D for
Magnetization along [100] direction (d) In 2D for Magnetization along [100]
direction. Red, blue and orange dots correspond to Weyl nodes in which the
lower band has index $N-1$, $N$ and $N+1$, respectively, with $N$ the number
of valence electrons.

FIG. S3. Distribution of eight Weyl nodes located at 8 meV below the Fermi
level for magnetization along [100] direction (a) In 3D (b) In 2D. Red and
blue dots correspond to Weyl nodes with positive and negative chirality,
respectively.

## Anomalous Hall Conductivity (AHC) and Berry curvature

We have analyzed how states at different parts of the Brillouin zone
contribute to the AHC and found that the largest contributions evidence a
loop-like structure that reflects weakly gapped nodal lines rather than Weyl
nodes as the origin.

FIG. S4. Left: Contribution to the AHC from the different planes perpendicular
to kz. Right: $\Omega_{z}$ as a function of $k_{x}$ and $k_{y}$ for the plane
that contributes the most to the AHC, indicated by a vertical line in the left
panel. The loop-like structure of the hot spots indicates a significant
contribution from states other than isolated Weyl nodes. The data corresponds
to the Fermi energy and magnetization along [001].