††thanks: The last three authors contributed equally.

# Anomalous transport in itinerant van der Waals ferromagnets FenGeTe2 (_n_
=3, 4, 5)

Jyotirmoy Sau  Department of Condensed Matter Physics and Material Science, S.
N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake,
Kolkata 700106, India  S. R. Hassan<EMAIL_ADDRESS>The institute of
mathematical sciences, CIT Campus, Tharamani, Chennai, Tamil Nadu 600113,
India  Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India  Nitesh Kumar<EMAIL_ADDRESS>Department
of Condensed Matter Physics and Material Science, S. N. Bose National Centre
for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India
Manoranjan Kumar<EMAIL_ADDRESS>Department of Condensed Matter
Physics and Material Science, S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata 700106, India

###### Abstract

Ferromagnetic (FM) semimetals FenGeTe2(n=3, 4, 5), exhibit several symmetry-
protected band-crossing points or lines near the Fermi energy (EF) and these
topological properties of energy bands lead to interesting transport
properties. We study these materials employing the first-principle
calculations and the tight-binding Hamiltonian constructed by fitting the
parameters of the first principles calculation. In the presence of spin-orbit
coupling (SOC) for n=3,5 a large Berry curvature (BC) concentrated on the
nodal lines is observed. The consequence of the correlation of the topological
nodal line and magnetic moments on anomalous Hall conductivity (AHC)
$\sigma_{xy}$ and anomalous Nernst conductivity (ANC) $\alpha_{xy}$ have been
investigated. We find $\sigma_{xy}=150$ S/cm for n=3, 295 S/cm for n=4, and 90
S/cm for n=5 at 0 K, while the ANC is observed as $\alpha_{xy}=0.55$ A/Km for
n=3, 0.10 A/Km for n=5, and 0.80 A/Km for n=4, at the EF at room temperature.
Our calculated AHC values at 0 K, i.e., 150 S/cm for Fe3GeTe2 and 90 S/cm
Fe5GeTe2, are consistent with the experimentally reported values. Also the
experimentally reported value of ANC for Fe5GeTe2 is close to our calculated
value at room temperature, i.e., 0.10 A/Km.

##  I INTRODUCTION

Recent experimental discovery of two and three dimensional non-spatial
symmetry protected topological insulating materials and their prospective
applications inspired further prediction of a group of novel states protected
by different symmetries, i.e., crystalline symmetry over the last few yearsQi
and Zhang (2011); Kitaev (2009); Ryu  _et al._ (2010). Furthermore, the
concept of topological band insulator has been extended to topological
semimetals (TSMs)Koshino  _et al._ (2014); Hořava (2005); Volovik (2013); Lau
and Timm (2013); Ran  _et al._ (2009); Hou (2013); Kobayashi  _et al._ (2014).
TSMs are described by crossing valance and conduction bands in the Brillouin
zone (BZ); for example, the nodes formed by touching the conduction and
valance bands at a discrete point are the Weyl/Dirac semimetals since the
Dirac/Weyl equations govern their low excitation behaviorHou (2013); Kobayashi
_et al._ (2014); Burkov and Balents (2011). The recent addition is topological
nodal line semimetals (TNLSMs), where bands cross each other along a line or
closed-loop in BZ and, in principle, it can exist in quasi-2DLin  _et al._
(2017) and 3D systemsChiu  _et al._ (2013).

In TNSLMs, the degenerate crossing point of the conduction and the valence
bands near EF are protected by the crystal and time-reversal symmetries (TRS).
Perturbations on the Hamiltonian cannot lift the degeneracy without breaking
any of its symmetries. By breaking time-reversal or spin parity symmetry,
these systems may have fully gapped nodal lines or gapped into nodal points.
For example, the first principles calculations demonstrate that the electronic
band structure of TaAs Weng  _et al._ (2015); Huang  _et al._ (2015) in the
absence of SOC exhibits two nodal lines, which are protected by mirror
reflection and spin-rotation symmetries. Each nodal line gaps into three pairs
of Weyl nodes in the presence of SOC. A double nodal line in SrIrO3 is another
example of nodal lines gapping into a pair of Dirac nodes when a mirror
reflection symmetry is broken Carter  _et al._ (2012); Fang  _et al._ (2016).
These nontrivial topological energy bands lead to many exotic phenomena, such
as the AHE and ANE. ANE is a thermoelectric counterpart of the AHE, both of
which are associated with the BCXiao  _et al._ (2006). Moreover, compared to
AHE, which investigates the Berry curvature of the whole Fermi sea, ANE is
exposed to the BC near the Fermi Energy(EF). Consequently, with an enhanced
Berry curvature at the nodal line near the EF , ANE evolves as the general
term of the total Nernst signal in TNLSMs.

Recently, the van der Waals (vdW) ferromagnet Fe3GeTe2 has been demonstrated
to be a magnetic variant of TNLSM, wherein the large BC induced AHC has been
attributed to the presence of a gapped nodal line near EFSeo  _et al._ (2020).
Later related iron rich compounds in the series Fe4GeTe2 and Fe5GeTe2 were
preparedSeo  _et al._ (2020). Interestingly, both are vdW compounds and
exhibit ferromagnetic behavior. This motivated us to investigate the interplay
between magnetism and topology in the magnetic vdW materials series, FenGeTe2
(n=3, 4, 5) in order to understand their potential anomalous transport
properties.

Figure 1: Three stable vdW structures in the series of Fe3GeTe2 for n=3, 4, 5.
The upper panels (a), (b), and (c) are side views of the structures Fe3GeTe2,
Fe4GeTe2, and Fe5GeTe2 respectively. While the lower panels (d), (e), and (f)
are corresponding top views and d represents the inter-layer distance.

Owing to the dearth of vdW ferromagnets, FenGeTe2 have recently attracted a
lot of attention. This series of compounds are also technologically important
due to the large Curie temperature (TC); TC = 230 K,270 K and 293 K
forFe3GeTe2 Deng  _et al._ (2018); Seo  _et al._ (2020), Fe4GeTe2Mondal  _et
al._ (2021) and Fe5GeTe2Ribeiro  _et al._ (2022), respectively. Chen et al.
showed that TC of Fe5GeTe2 could be further enhanced to 478 K upon Ni-
dopingChen  _et al._ (2015). The monolayer of these systems shows interesting
magnetic behavior, and the thin atomic layer can be tuned using the
femtosecond laserLiu  _et al._ (2020). Fe3GeTe2 also indicated towards high
electronic correlation in terms of the emergence of Kondo behaviorZhang  _et
al._ (2018). The Density functional theory study of Fe3GeTe2 and Fe4GeTe2
monolayers show that these systems have large magnetic anisotropy under an
electric fieldKim  _et al._ (2021). Fe5GeTe2 shows butterfly type of magneto-
resistanceAli  _et al._ (2016).

The structural characteristics of FenGeTe2 could be understood in terms of the
triangular Fe-layers and Fe-Fe dumbbells. In Fe3GeTe2, Fe1 makes a triangular
lattice when considered together with Ge,which can be viewed as honeycomb
lattice. Fe2-Fe2 dumbbells align perpendicular to the layer, centered at each
hexagon of the honeycomb, with one Fe2 above the hexagon and another Fe2 below
the hexagon. Fe3GeTe2 belongs to space group P63/mmc (No.194). Fe4GeTe2
consists of two types of dumbbells each displaced with respect to the other
along the direction perpendicular to the vdW layer. Fe4GeTe2 has a
rhombohedral structure with space group R$\bar{3}$m (No.166). The structure of
Fe5GeTe2 also consists of two types of dumbbells as in the case of Fe4GeTe2,
in addition to a triangular layer of Fe. The Space group of Fe5GeTe2 is P3m1
(No.156)Seo  _et al._ (2020). These materials also have C3y symmetry in common
in addition to a mirror or C2 symmetry. The top and side views of FenGeTe2
structures have been depicted in Fig.1. where different Fe atoms are
distinguished with separate colors.

In this manuscript, we perform a comparative study of the electronic
properties of FenGeTe2 (n=3, 4, 5) to explore the origin of the different
topologies in FenGeTe2. We also study AHE and ANE using a tight-binding
Hamiltonian constructed by fitting parameters with density functional theory
(DFT). The magnetic moments of all these materials are close to each other;
therefore, our main focus is to highlight the correlation between topology and
magnetism.

The remainder of this paper is organized as follows. In Sec. II we briefly
discuss the computational details, followed by a discussion on the magnetism
of these three materials in Sec. III and Sec. IV and Sec. V summarise our
results on the nontrivial band topology, nodal lines, and corresponding Berry
curvature distribution.We present our paper’s primary focus, anomalous
transport, in Sec. VI and conclude with a summary and conclusion in Sec. VII

Figure 2:  (a) The total DOS for Fe3GeTe2 as a function of (E-EF) is plotted
where the blue and red curves represent the total DOS for the majority spin
carrier (solid) and minority spin carrier(dotted)of Fe1 and Fe2, respectively.
The magnetic moments, MFe1 MFe2, are shown with the black (green) curve, the
main contribution from the majority spins. (b) Similarly the total DOS for
Fe4GeTe2 as a function of (E-EF) is plotted where the blue and red curves
represent the total DOS for the majority spin carrier (solid) and minority
spin carrier(dotted)of Fe1 and Fe2, respectively. The magnetic moments, MFe1
MFe2, are shown with the black (green) curve. (c)the total DOS for Fe5GeTe2 as
a function of (E-EF) is plotted where the red, green, blue, deep green, orange
curves represent the total DOS for the majority spin carrier (solid) and
minority spin carrier (dotted) of Fe1 and Fe2, Fe3,Fe4, Fe5 respectively.

##  II Theory and Computational Details

We examine the electronic structures and transport properties of FenGeTe2 in
the scheme of DFT. First principles calculations are performed using the VASP
package Blöchl (1994) based on the generalized gradient approximation (GGA) of
the Perdew-Burke-Ernzerhof (PBE) for the exchange-correlation functional. A
plane-wave basis set with a kinetic energy cutoff of 600 eV is considered
while performing first-principles calculations. Furthermore, we have used the
BZ sampling, a 6$\times$6$\times$1 k-point mesh forFe3GeTe2 and Fe5GeTe2, and
a 6$\times$6$\times$6 k-point mesh for Fe4GeTe2. The Gaussian smearing method
is acquired for broadening the Fermi surface with a width of 0.05 eV. Both
cell parameters and internal atomic positions were fully relaxed until the
forces on all atoms were smaller than 0.01 eV/A. To explore the nontrivial
band topology and the intrinsic AHE, the tight-binding Hamiltonian was
constructed with the maximally localized Wannier functions Pizzi  _et al._
(2020); Marzari and Vanderbilt (1997). The intrinsic AHC is computed using the
linear-response Kubo formula approach in the clean limit, and a
500$\times$500$\times$500 k-grid in the BZ was used for the integral of the
AHC.

##  III Magnetic Properties

In this section, the magnetic properties of all three vdW compounds Fe3GeTe2,
Fe4GeTe2 and Fe5GeTe2 are studied and we calculate spin resolved total density
of states (DOS) to understand the magnetic moments. We notice that the
d-orbitals of Fe atoms dictate the magnetism; therefore, we show only the DOS
of the d-orbitals where down DOS is shown as negative of its value. We
calculate the total magnetic moment of Fe atoms as a difference of total up
and down spin electrons filled up to EF and density of up or down electron at
any energy is proportional to the DOS of up and down spins at that energy.

Fe3GeTe2: In this compound, there are two in-equivalent Fe atomic sites
denoted as Fe1 and Fe2, as shown in Fig.1(a), Fe1 lies in a-b plane and
arranges on hexagonal structure, whereas Fe2 lies perpendicular to this plane.
Energy dependence spin up (solid line) and down ( dashed line) DOS are shown
in Fig.2(a): red lines represent the DOS of Fe1 atoms, whereas the blue line
represents DOS of Fe2 atoms. The partial DOS calculations suggest that the DOS
of all d-sub-orbitals are spread near the EF, therefore, electrons of all the
sub-orbitals are delocalised. The DOS of Fe1 compared to Fe2 has a higher
magnitude at -2.5 eV below the EF, whereas the DOS of Fe2 has a marginally
larger value compared to Fe2 between the energy range 2 eV to the EF. We note
that the difference between up and down DOS of Fe2 atoms have higher a spread
than that of Fe1 atoms; therefore, we expect a higher magnetic moment for Fe1
than Fe2. The magnetic moments MFe1 and MFe2 for Fe1 and Fe2 atoms are shown
as solid black and green lines for Fe1 and Fe2 atoms in Fig.2(a). The magnetic
moment contribution of Fe1 is higher than that of Fe2 and these are MFe1=2.5
$\mu_{B}$/f.u. and MFe2=1.5 $\mu_{B}$/f.u. respectively at zero temperature.
In Fe3GeTe2, the average contribution magnetic moment of Fe is MFe=2.12
$\mu_{B}$/f.u and it matches very well with experimentally reported valueSeo
_et al._ (2020).

Figure 3: The band structure of FenGeTe2 without and with SOC. After opening a
gap, there is a Berry curvature along symmetry lines. (a) For Fe3GeTe2, the
crossing point at K, 0.03 eV above the Fermi energy. The gap is opened below
the Fermi energy due to SOC. (b) The crossing point is between K and M points
at the Fermi energy for Fe4GeTe2. (c) The crossing point at K point below the
Fermi energy in Fe5GeTe2. With SOC, the gap opens : (d) at the K point below
the Fermi energy for Fe3GeTe2, (e) between K and M point at the Fermi energy
for Fe4GeTe2, and (f) at the K point below the Fermi energy. A large negative
Berry curvature: (g) at the K point for Fe3GeTe2, (h) between K and M point
for Fe4GeTe2, and (i) at the K point for Fe5GeTe2.

Fe4GeTe2: This system also has two in-equivalent Fe sites, labeled as Fe1 and
Fe2 in Fig.1(b). Fe2 atoms lying below the Te atom of hexagonal plaquettes and
Fe1 atom is present in hexagonal plaquettes. The up and down spin DOS are
plotted as a function of energy with solid and dotted line. Red and blue
colour represent DOS of Fe1 and Fe2 d-orbitals. The Fe2 $d$-bands are more
dispersed compared to Fe1. The DOS of Fe1 has a larger amplitude at -2.8 eV
below EF for spin up electron than Fe2 DOS, as shown in Fig.2(b). The magnetic
moments MFe1 and MFe2 for atoms Fe1 and atoms Fe2 are shown as solid black and
green lines for Fe1 and Fe2 atoms in Fig.2(b). Magnetic moments of Fe1 and Fe2
are 1.7 and 2.47$\mu_{B}$/f.u and the average magnetic moment of Fe is
MFe=2.084 $\mu_{B}$/f.u, which agrees well with the experimental valueSeo  _et
al._ (2020) .

Figure 4: For Fe3GeTe2 with SOC: (a) 2D Fermi surface using spectral function
A(kx,ky, kz=0, Ef). (b) Energy gap, $\Delta
E(k_{x},k_{y},k_{z}=0)=(E_{a}(k_{x},k_{y},k_{z}=0)-E_{b}(k_{x},k_{y},k_{z}=0))$,
between two crossing bands (a(red) and b(blue)) and the high symmetry points
are also shown. (c) The difference in Berry curvature distribution,
$\Delta\Omega_{z}(k_{x},k_{y},k_{z}=0)=(\Omega_{z}^{a}(k_{x},k_{y},k_{z}=0)-\Omega_{z}^{b}(k_{x},k_{y},k_{z}=0))$
of two crossing bands, has a large negative value at the K point (red spot).
(d) Energy gap, $\Delta E(k_{y},k_{z},k_{x}=0)$, and gapped nodal at
${k}_{y}$=1.2. (e) the difference of the Berry curvature distribution between
two bands, $\Delta\Omega_{z}(k_{y},k_{z},k_{x}=0)$.

Fe5GeTe2: It contains five in-equivalent Fe sites which are shown in Fig.1(c)
and two atoms Fe1 and Fe4 are in the same a-b plane. Fe2 and Fe3 are also in
same plane and have a similar chemical environments, but have different
bonding along the c-axis. d-orbitals of these atoms are also highly
hybridized. The DOS of all five Fe d-orbitals are shown in Fig.2(c) and DOS of
Fe1 and Fe4 looks similar, whereas it is distinct for other three Fe atoms.
The Green curve represents the large peak in up DOS of Fe2 at -3.0 eV below
$E_{F}$ shows localization magnetic moment and it is contributed from
d${}_{z^{2}}$ sub-orbitals. Fe2 atoms have the highest magnetic moments and
Fe5 and Fe3 have the lowest contribution. Fe5 and Fe3 d-orbitals have higher
DOS for both up and down spins near the $E_{F}$. The colored thick lines
represent the magnetic moments for all five Fe atoms and the largest magnetic
moment is MFe2=2.55$\mu_{B}$/f.u. for Fe2. Magnetic moment of Fe1 and Fe4 are
nearly the same MFe1=2.4$\mu_{B}$/f.u. and MFe4=2.3$\mu_{B}$/f.u.
respectively, whereas the magnetic moment of Fe3 and Fe5 are
MFe3=1.29$\mu_{B}$/f.u. and MFe5=1.84$\mu_{B}$/f.u. The average magnetic
moment per Fe atom is MFe=2.064$\mu_{B}$/f.u., which agrees with the
experimental reported value.Li  _et al._ (2020).

##  IV Nontrivial band topology

We study the topology of energy bands of these three materials and analyze the
energy band crossovers. The contributions of various sub-orbitals of Fe
d-orbitals to bands near EF are also investigated.

Fe3GeTe2: The energy dispersion curve near EF is shown in Fig.3(a) without SOC
and in Fig.3(b) with SOC for this material. As shown in Fig.3(a), in the
absence of SOC, two bands, represented by ’a’ (thick red line) and ’b’ ( thick
black line), cross at point K, 0.03 eV above EF. These two bands are solely
contributed from d orbitals of Fe1 and Fe2 atoms. These two bands are solely
contributed from d orbitals of Fe1 and Fe2 atoms. We observe that the bands
crossing resemble Mexican-hat shapes induced by the Rashba effect with
quenched spins, as reported in the literatureSeo  _et al._ (2020). The
degeneracy at the K point arises due to C2y symmetry in the crystal and on the
application of SOC along 001 direction, lifts this degeneracy due to breaking
of the time-reversal symmetry. We also find some other band crossings which
contribute large BC contribution which lie along the high symmetry direction,
M-$\Gamma$ and $K$-$\Gamma$, as shown in Fig.3(a) . Non-trivial crossover
points open a gap in the presence of the SOC as shown in Fig.3(b). Therefore,
we investigate the BC (It is defined in Eqn (1) in the Sec. VI.) of these
bands in the above mentioned high symmetry directions. The total BC has a
large magnitude corresponding to these at nontrivial band crossing close to
the EF, as shown in Fig.3(c). It shows the largest negative BC at the K point
and relatively smaller values along the M-$\Gamma$, and $K$-$\Gamma$.

Fe4GeTe2:It has a rhombohedral structure, and the space group is R$\bar{3}$m.
This material also has three-fold rotational symmetry about the z-axis
(c-axis) and two-fold rotation about the y-axis. All the energy bands close to
the EF are contributed by Fe atoms, Many energy bands cross close to EF, as
shown in Fig.3(d), and these bands are formed from the dxz ,dyz, and
d${}_{x^{2}-y^{2}}$ sub-orbitals of Fe atoms. The nontrivial degeneracies near
the EF are along the K-M and K-$\Gamma$ symmetry points in the absence of the
SOC. However, the presence of the SOC along 001 breaks the TR symmetry and
lifts the degeneracy, as shown in Fig.3(e). The BC in this system shows a
large negative value along the K-$\Gamma$ symmetry point, as shown in
Fig.3(f).

Fe5GeTe2: This material belongs to P3m1 symmetry group and has three-fold
rotational symmetry along the z-axis (c-axis) and mirror symmetry along the
y-axis. Most energy bands are contributed from the dxz,dyz and
d${}_{x_{2}-y_{2}}$ sub-orbitals of Fe atoms. A prominent band crossing -0.11
eV below EF is detected at the K point, and the participating bands are
occupied by minority spins, as shown in Fig.3(g). As was mentioned earlier,
the crossing point is encircled with a red dotted circle in the absence of the
SOC. There are two other crossings close to K points along the K-M and
K-$\Gamma$, 0.1 eV below the EF. But in the presence of the SOC, gap is opened
at these points as shown in Fig.3(h). The Berry curvature along the K-M and
K-$\Gamma$ symmetry points are shown in Fig.3(i). We notice a large negative
berry curvature at the K point and two other crossings around, the K point as
mentioned earlier.

Figure 5: For $Fe_{5}GeTe2$ with SOC: 2D Fermi surface, the energy gap,
$\Delta E(k_{y},k_{z},k_{x}=0)$, and the difference of Berry curvature
distribution are plotted in (a),(b), and (c), respectively.

##  V Nodal line and Berry curvature distribution

This section highlights the band crossing feature by the 2D Fermi surface
using the spectral function A(${\bf k}$, $E_{f}$) computed from wanniertoolsWu
_et al._ (2018), the energy gap, $\Delta E$(${\bf k}$), between the band
crossing, and the difference of the z component of the Berry curvature,
$\Delta\Omega_{z}({\bf k})$, at each point of the BZ. We usually observe large
BC at the crossing points or the nodal lines. We show our results in the
$k_{z}$=0, $k_{x}$=0 and $k_{y}$=0 planes.

Fe3GeTe2: Fig.4(a) shows the FS plot in $k_{z}=0$ plane. The calculated FS
consists of a circular-shaped pocket centered at the six K-point and a
hexagonal-shaped pocket centered at $\Gamma$. The red region shows a high
spectral function value, and the blue has values close to zero, i.e., a small
spectral function. The spectral function’s intense value reveals the band’s
degeneracy and crossing between different bands. Most of the region is blue,
i.e., low, low spectral function regime, except the red line on $\Gamma$
points and six k points exhibit high spectral function. The yellow color
represents finite gaps, whereas the white and red regions show tiny gaps
opened at the high symmetry points due to the SOC. In Fig.4(b), gap
distribution , $\Delta E(k_{x},k_{y},k_{z}=0)$, is shown, and a gap smaller
than 10-5 eV is considered zero. Mainly the white region is at K points, and
corresponding to this gap, there is a finite BC, as shown in Fig.4(c) with the
red spot. We also studied the energy gap in $k_{x}=0$ plane and observed a
gapped line, as shown in Fig.4(d) by a white line (at Ky=1.2 Å-1) called a
gapped nodal line. The BC is large at this nodal line, and we show it in
Fig.4(e) by a red line(at Ky=1.2 Å-1).

Figure 6:  For Fe5 with SOC : (a) 2D Fermi surface, (b) $\Delta
E(k_{x},k_{y},k_{z}=0)$, (c)$\Delta\Omega^{z}(k_{x},k_{y},k_{z}=0)$, (d)
$\Delta E(k_{y},k_{z},k_{x}=0)$ and (e)
$\Delta\Omega^{z}(k_{y},k_{z},k_{x}=0)$

Fe4GeTe2: In Fig.5(a), 2D FS using the spectral function is plotted in the
Kx=0 plane, and there is a $\Gamma$-centered hexagonal-shaped and six K and M
point-centred circular-shaped FS. Like the previous system, the red region
shows a high spectral function value indicating band degeneracy and the
crossing point. The blue has values close to zero, i.e., a small spectral
function. The intense value of the spectral function at the red dots around
the $\Gamma$ points and red lines around K and M points indicate band crossing
and band degeneracy. The higher symmetry around M points has consequences on
the transport properties. The SOC opens tiny gaps around the degenerate points
at $\Gamma$, along the K-M direction, as shown in Fig.5(b). White spots
between the K-M line and $\Gamma$ points correspond to gapped regions, and BC
at these points are high, as shown in Fig.5(c) by red spots. The gaps and BC
are also calculated in the K${}_{y}=0$ plane, but there is not much feature
associated with it. Gaps and finite BC are scattered in the first BZ;
therefore, we safely conclude the absence of a nodal line.

Fe5GeTe2: $Fe_{3}GeTe_{2}$ and $Fe_{5}GeTe_{2}$ have similar symmetries;
therefore, the spectral function behavior and its symmetry are expected to be
similar, as can be seen in Fig.6(a) of the contour plot of FS. Most of the
region is blue with a low spectral function value, except the red line with a
high spectral function value in the neighborhood of $\Gamma$ points and six K
points, reflecting the band degeneracy and the band crossing. In Fig.6(b), gap
distribution, $\Delta E(k_{x},k_{y})$, is shown in the first BZ, and blue and
white regions show tiny gaps opened at the high symmetry points due to the
SOC. Mainly the white region is around the K points, and corresponding to this
gap, there is a finite BC, as shown in Fig.6(c) with the red spot. In Kx=0
plane, white stripes represent the gaped nodal line as shown in Fig.6(d), and
there is a large BC associated with it as shown in Fig.6(e) by a red and blue
line(at Ky=1.2 Å-1).

##  VI Anomalous transport properties

We also study anomalous transport properties described by a tight-binding
Hamiltonian, $H$, constructed by fitting its parameters with DFT results. As
discussed in the previous section, these materials exhibit a large BC on the
nodal lines in the presence of SOC. The BC (${\bf\Omega}$) acts as a magnetic
monopole in the momentum space, giving rise to anomalous Hall and heat
conductivities. The computation of the BC requires the energy eigenvalues
$\epsilon_{n{\bf k}}$ and the energy eigenstates $\;|n({\bf k)}\rangle$ where
${\bf k}$ and n are the wave vector and band index respectively.
${{\bf\Omega}({\bf k})}$ is given as:

$\displaystyle{\bf\Omega^{n}({\bf k})}=$ (1) $\displaystyle i\sum_{n\neq
m}\frac{\langle n({\bf k})|\nabla_{\bf k}H({\bf k})|m({\bf
k})\rangle\times\langle m({\bf k})|\nabla_{\bf k}H({\bf k})|n({\bf
k})\rangle}{(\epsilon_{n{\bf k}}-\epsilon_{m\bf k})^{2}}.$

The sum of the BC defines the anomalous Hall conductivity ($\sigma_{xy}$) for
all the occupied bands up to the Fermi energy EF. $\sigma_{xy}$ is defined as:

$\sigma_{xy}=-{\frac{e^{2}}{\hbar}}\sum_{n}\int\frac{d^{3}k}{(2\pi)^{3}}\Omega^{n}_{z}({\bf
k})f_{n}({\bf k})$ (2)

where f${}_{n}(\vec{k})$ is the Fermi distribution function corresponding to
the nth eigenstate and $\Omega_{z}^{n}({\bf k})$ is a z-component of
$\bf{\Omega}^{n}({\bf k}).$

The BC-effect also manifests in thermoelectric transport is driven by a
statistical force, for example, temperature gradient. In the presence of a
temperature gradient, the local current of carriers acquires an additional
term from the carrier’s magnetic moment in the presence of a non-uniform
distribution. The extra term, an extrinsic Hall current ${\bf j}_{in}$, can be
written in terms of the BC as,

$\displaystyle{\bf
j}_{in}=-\frac{{\bf\nabla}T}{T}\times\frac{e}{h}\sum_{n}\int{}{d{\bf
k}}\Omega^{n}({\bf k})[(\epsilon_{n\bf k}-\mu)f_{n}({\bf k})$
$\displaystyle+k_{B}Tlog(1+e^{\beta(\epsilon_{n\bf k}-\mu)}]$ (3)

the Nernst conductivity $\alpha_{xy}$ can be extracted using
$j_{x}=\alpha_{xy}(-\Delta T_{y})$ and the $\alpha_{xy}$ can be written as

$\alpha_{xy}=-\frac{1}{e}\int
d\epsilon\frac{-df(\epsilon)}{d\mu}\sigma_{xy}(\epsilon)\frac{(\epsilon-\mu)}{T}$
(4)

where intrinsic anomalous Hall conductivity $\sigma_{xy}(E_{f})$ at zero
temperature with Fermi-energy $E_{f}$.

Figure 7: (a) Red, green, blue curve represent Energy ($E-E_{F}$) dependence
of the AHC For Fe3GeTe2, Fe4GeTe2 and Fe5GeTe2 respectively. (b) Red, green,
blue curve represent Energy ($E-E_{F}$) dependence of the ANE For Fe3GeTe2,
Fe4GeTe2 and Fe5GeTe2 respectively.

All three materials show the variation in AHC as (E-EF) varies, which is
prominent in n = 3, 5 compared to n=4 and for n = 3 as the energy (E-EF)
varies, AHC first decreases from 300 S/cm to -400 S/cm at (E-EF) = 0.1 eV and
then increases as (E-EF) approaches zero and acquires a value of 150 S/cm.
This material reveals the large BC at the nodal line, as seen in Fig.3(c),
which causes a high value of AHC. The SOC opens the gap in the energy bands
below the Fermi energy, $E_{F}$, which contributes to a large variation in BC.
Further, we can tune its values by doping, for example, up to 600 S/cm by
electron doping and 450 S/cm by hole doping. Our result is consistent with the
experimental value of AHC, 150 S/cm, at a low temperature. In n=4 the six-fold
degeneracy around K and M points lying close to EF contributes to the large
value of AHC, 295 S/cm at (E-EF) = 0.4 eV decrease to -100S/cm at (E-EF) = 0.3
eV and at EF its value reaches to 300 S/cm. In the case of $Fe_{5}GeTe_{2}$,
the various bands contribute to BC, which is responsible for fluctuating AHC
with (E-EF). The calculated AHC is 90 S/cm and most of the contribution comes
from the nodal line below the EF.

We now discuss ANE, which is intimately related to AHE via anomalous
thermoelectric response tensor and emerges when charge carriers acquire an
anomalous transverse velocity in a longitudinal temperature gradient and a
finite BC. AHE probes the BC of the whole Fermi sea, while ANE is sensitive to
the BC of the near Fermi energy. As a result, ANE may become the dominant term
of the total Nernst signal in this series of materials with an enhanced BC
near the Fermi energy. Fig.7(b) shows ANE at room temperature and as a
function of (E-EF) for all three materials. The behavior of Fe3GeTe2 and
Fe5GeTe2 are similar but opposite to Fe4GeTe2. Near Fermi energy E-EF = 0,
Fe4GeTe2 has a large value while Fe3GeTe2 and Fe5GeTe2 have the small value
and are close to each other. This suggests that band crossing near the Fermi
energy in all these materials is different, as evident in the band structure
calculation discussed in the previous section. At room temperature, the
anomalous Nernst conductivity is observed 0.55 A/Km for Fe3GeTe2 and 0.10 A/Km
for Fe5GeTe2. The observed Nernst conductivities at room temperature are 0.55
A/Km for Fe3GeTe2, 0.10 A/Km for Fe5GeTe2, and 0.8 A/Km for Fe5GeTe2. The
experimentally observed value of ANE for Fe5GeTe2, 0.15 A/Km, is close to our
calculated value.

##  VII Summary and Conclusions

In this manuscript, we compare the electronic properties of FenGeTe2 (n=3, 4,
5) using the ab-initio theory and showed these materials are semimetal and
have interesting topological electronic energy bands and finite magnetization.
Fe3GeTe2 and Fe5GeTe2 have a nodal line in the electronic band, whereas the
Fe4GeTe2 no-trivial energy band crossing in the neighborhood of the K and M
high symmetry points. All these three systems have three-fold rotational
symmetry about the c-axis and either two-fold mirror symmetry or rotational
symmetry perpendicular to this axis and these symmetries get reflected in
spectral densities at the Fermi-surface as shown in Fig.4(a), Fig.5(b) and
Fig.6(c). We also notice that Fe3GeTe2 system shows extra non-trivial
degenerate points along $K-\Gamma$ and $M-\Gamma$ high symmetry line other
than the bands crossing at the $K$ points found in ref. Seo  _et al._ (2020).

We also show that in the presence of the SOC the time-reversal symmetry breaks
and the energy degeneracies at high symmetry points get lifted. We observe the
finite magnetic moments of Fe atoms which are consistent with the experimental
results in the literature Seo  _et al._ (2020). The finite magnetic moments
and non-zero BC in momentum space at these non-degenerate points lead to the
anomalous Hall and Nernst conductivity. The experimentally reported AHC Seo
_et al._ (2020) at the low temperature is consistent with our predicted value,
150 S/cm and 90 S/cm, for Fe3GeTe2 and Fe5GeTe2, respectively. Among the three
Fe4GeTe2 has the largest AHC, 295 S/cm, due to the higher contribution of the
BC stemming from the near degenerate energy bands around K and M symmetry
points. We obtain the ANC at the room temperature, 0.55 A/Km, 0.80 A/Km, and
0.10 A/Km, for Fe3GeTe2, Fe4GeTe2, and Fe5GeTe2, respectively. The theoretical
value of ANC, 0.10 A/Km, for Fe5GeTe2 is close to the experimentally reported
value. The large ANC in these materials offers promising applications as the
new generation of thermoelectric energy conversion devices.

In summary, we have studied the electronic band structure of FenGeTe2 (n=3, 4,
5) materials and explored the topology in energy bands. The magnetic moments
of all three materials are close but our results showed that anomalous
transport properties are not the same since the nontrivial topological points
in the energy bands having large BC appearing at the different symmetry points
in the BZ in each system.

## Acknowledgement

M.K. thanks Professor Atindra Nath Pal, Professor Prabhat Mandal and Professor
Thirupathaiah Setti for fruitful discussion. J.S. thanks U.G.C for financial
support. M.K thanks SERB for financial support through Grant Sanction
No.CRG/2020/000754. N.K. thanks SERB for financial support through Grant
Sanction No.CRG/2021/002747.

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