# Enhanced anomalous Nernst effects in ferromagnetic materials driven by Weyl
nodes

Ilias Samathrakis<EMAIL_ADDRESS>Teng Long  Zeying Zhang
Harish K. Singh  Hongbin Zhang  Institute of Materials Science, TU Darmstadt,
64287 Darmstadt, Germany

###### Abstract

Based on high-throughput first-principles calculations, we evaluated the
anomalous Hall and anomalous Nernst conductivities of 266 transition-metal-
based ferromagnetic compounds. Detailed analysis based on the symmetries and
Berry curvatures reveals that the origin of singular-like behaviour of
anomalous Hall/Nernst conductivities can be mostly attributed to the
appearance of Weyl nodes or nodal lines located in the proximity of the Fermi
energy, which can be further tailored by external stimuli such as biaxial
strains and magnetic fields. Moreover, such calculations are enabled by the
automated construction of Wannier functions with a success rate of 92%, which
paves the way to perform accurate high-throughput evaluation of the physical
properties such as the transport properties using the Wannier interpolation.

##  I Introduction

In the last decades, materials of the nontrivial topological nature have
attracted quite intensive attention of the community, leading to many
interesting physical properties such as ultra high mobility [1], protected
surface states [2], anomalous magnetoresistance [3, 1] and exotic optical
properties [4], promising for future applications. One particularly
interesting class of materials are those with long range magnetic orderings at
finite temperatures, including topological insulators, Dirac (Weyl) semimetals
(DSM,WSM) and nodal line semimetals. These materials are characterized by
gapless surface Dirac fermions with a spin-momentum-locked energy dispersion,
a spin-(non-)degenerate band touching point and lines respectively. [5, 6]
These special energy dispersions are responsible the generation of interesting
physical phenomena. For instance, for ferromagnetic materials, the anomalous
Hall effect (AHE), which describes the generation of a transversal current
perpendicular to the electric current in the absence of an external magnetic
field [7, 8], can be induced. The same can be achieved using the thermal
gradient as the external stimulus, resulting in the anomalous Nernst effect
(ANE) [9, 10]. Both AHE and ANE are caused by the nonzero Berry curvature of
the Fermi sea states, [8] which acts as a fictitious magnetic field [11]
giving rise to these effects that lead to promising applications such as data
storage and data transfer. [12, 13] Note that like the other linear response
properties, the anomalous Hall conductivities (AHC) can also be equally
evaluated by integrating the Fermi surfaces [14], arising one interesting
aspect whether AHC can be tuned by external stimuli.

Recently, motivated by the emergent antiferromagnetic spintronics [15], it is
demonstrated that significant AHCs and ANCs can be obtained in magnetic
materials with noncollinear antiferromagnetic ordering [16, 17]. The most
representative materials include the Mn3X with X=Ge,Ga,Sn,Rh,In,Pt family [18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28], the Mn3XN with X=Ga,Zn,Ag,Ni family
[29, 30, 31], Mn5Sn3 [32, 33] and Pr2Ir2O7 [34]. Besides AHC, the triangular
noncollinear antiferromagnetic structure of the Mn3X family is responsible for
the generation of a spin current in the absence of external field, dubbed Spin
Hall effect (SHE) [35] and the Magneto-optical Kerr effect (MOKE) [36] which
is the rotation of the plane of polarization of a light beam reflected from
the surface in the presence of a magnetic field. Such peculiar properties can
be traced back to the existence of Weyl nodes in the electronic structure.
[22, 37] WSM are topological materials where the electronic structure is
dominated by non-accidental linear touching and historically Y2Ir2O7 was the
first material to be realized as a WSM [38]. Recently, Co3Sn2S2 and several
magnetic Weyl Heusler compounds were classified as WSM which also exhibit
touching points with linear dispersion close to the Fermi level [39, 40],
giving rise to large AHC. Therefore, there is a strong impetus to search for
more magnetic WSM and to investigate the resulting physical properties.

In this work, after collecting the known ferromagnetic materials, we performed
high-throughput (HTP) calculations on AHC and ANC of 266 existing transition-
metal-based ferromagnetic compounds, where the magnetization directions and
biaxial strains are considered as perturbations to tune such effects. It is
observed that the appearance of linearly degenerate states such as Weyl nodes
and nodal lines will lead to singular-like behavior of AHC and hence enhanced
ANC. Nevertheless, there exist also materials with significant ANC but without
hot-spot contributions to AHC. Therefore, it is suspected that further HTP
calculations are required in order to identify promising candidates with
enhanced ANC for transversal thermoelectric applications, where automated
construction of Wannier functions can be valuable.

##  II Numerical details

Out of 5487 experimentally known ferromagnets, reported in AtomWork Adv
database [41], the crystal structure of 3956 was found and collected using
ICSD, [42] Materials Project, [43] and AtomWork databases. Compounds being
either non-stoichiometric or containing elements with partial occupation were
excluded, leading to 1827 compounds. Additionally, rare earths, oxides and
compounds including more than 30 atoms in their unit cell were screened out as
well since they increase the complexity and the time needed for the
calculation. These constraints further reduced the number of available
ferromagnetic compounds for calculations to 335.

The transport properties of the ferromagnetic compounds are computed
automatically using an in-house-developed scheme, written in Python, linking
VASP, Wannier90 and Wanniertools software. The first step involves the self
consistent first-principles calculation of each compound using the projected
augmented wave method (PAW), implemented in VASP. [44] The exchange
correlation functional is approximated in the general gradient approximation
(GGA), as parameterized by Perdew-Burke-Ernzerhof [45] and the spin orbit
coupling is included in all calculations. A $\Gamma$-centered kmesh of density
50, in respect to the lattice parameters, as well as an energy cutoff of
500$eV$ are selected. Subsequently, the Bloch wave functions are projected
onto maximally localised wannier functions (MLWF), using Wannier90, following
Ref. 46. The number of MLWF as well as the disentanglement and frozen windows
of each compound are automatically computed following Ref. 47. The AHC is then
evaluated by integrating the Berry curvature, according to the formula:

$\displaystyle\sigma_{\alpha\beta}=-\frac{e^{2}}{\hbar}\int\frac{d\mathbf{k}}{\left(2\pi\right)^{3}}\sum_{n}f\big{[}\epsilon\left(\mathbf{k}\right)-\mu\big{]}\Omega_{n,\alpha\beta}\left(\mathbf{k}\right)$
(1)
$\displaystyle\Omega_{n,\alpha\beta}\left(\mathbf{k}\right)=-2Im\sum_{m\neq
n}\frac{\braket{\mathbf{k}n}{v_{\alpha}}{\mathbf{k}m}\braket{\mathbf{k}m}{v_{\beta}}{\mathbf{k}n}}{\big{[}\epsilon_{m}\left(\mathbf{k}\right)-\epsilon_{n}\left(\mathbf{k}\right)\big{]}^{2}},$
(2)

where $f\big{[}\epsilon\left(\mathbf{k}\right)-\mu\big{]}$ denotes the Fermi
distribution function, $\mu$ the Fermi energy, n (m) the occupied (empty)
Bloch band,
$\epsilon_{n}\left(\mathbf{k}\right)\left(\epsilon_{m}\left(\mathbf{k}\right)\right)$
their energy eigenvalues and $v_{\alpha}$ $\left(v_{\beta}\right)$ the
velocity operator. The integration is performed on 400 dense kmesh in respect
to the lattice parameters, using Wanniertools. [48] The ANC is defined
according to the formula:

$a_{\alpha\beta}=-\frac{1}{e}\int d\epsilon\frac{\partial
f}{\partial\mu}\sigma_{\alpha\beta}\left(\epsilon\right)\frac{\epsilon-\mu}{T},$
(3)

where $\epsilon$ is the energy point of the energy grid, e the electron’s
charge, T the temperature and $\mu$ the Fermi level. Regarding the ANC, an in
house developed Python script in an energy grid of 1000 points at a range
$\big{[}-0.5,0.5\big{]}$ eV in respect to the Fermi level is used.

##  III Results and Discussion

As stated above, starting from 335 transition-metal-based ferromagnetic
materials, we managed to converge the DFT calculations for 289 compounds for
both magnetization directions, i.e., the [001] and [100]. MLWFs can be
successfully constructed for 266 of them, giving rise to a success rate of
92%. In comparison, a success rate of 93% and 97% is achieved in Ref. 49 and
Ref. 50 respectively. Our results provide an alternative to the automatic
construction of Wannier functions and enable us to perform further HTP
calculations to evaluate the desired physical properties, in particular making
use of the Wannier interpolation technique [51].

Figure. 1 shows the z-component ($\sigma_{z}=\sigma_{xy}$) of AHC and ANC for
266 FM intermetallic compounds with the magnetization aligned along the [001]
direction. Obviously, the magnitude of AHC ranges between -2051$S/cm$ for
Ni3Pt and 2040$S/cm$ for CrPt3, where there are 11 compounds with the absolute
value of AHC exceeding 1000$S/cm$. The largest magnitude of AHC is larger than
the already reported -1862$S/cm$ for Rh2MnGa and -1723$S/cm$ for Rh2MnAl, [52]
arising the question whether there is an upper limit and opens up the
possibility to engineer materials with more significant magnitudes.
Concomitant with AHC, the z-component ($\alpha_{z}=\alpha_{xy}$) of ANC varies
between -7.29$A/(m\cdot K)$ for Ni3Pt and 5.83$A/(m\cdot K)$ for BCo4Y, and
exceeds 3$A/(m\cdot K)$ in 16 compounds.

Figure 1: $\sigma_{z}$ (at T=0K) and $\alpha_{z}$ (at T=300K) for selected
ferromagnets with the magnetization direction parallel to the [001] axis. Our
results are indicated with black and compared to the ones available in
literature in red.

In general, our results match reasonably well with the existing literature for
the magnetization direction being parallel to z-axis. More specifically the
reported value of 1130$S/cm$ for Co3Sn2S2 [53] is comparable with the
988$S/cm$ obtained from our calculations. Moreover, good agreement is achieved
in Heusler compounds where our values of 140$S/cm$ and 200$S/cm$ for Co2VGa
and Co2MnSn agree with the reported cases of 137$S/cm$ in Ref. 54 and
118$S/cm$ in Ref. 55 respectively. Comparable results are also observed in the
XPt3 family with $X=\left(Cr,Mn\right)$ where the reported cases of 2040$S/cm$
and 1400$S/cm$ [56] are reproduced by our values of 2000$S/cm$ and 1471$S/cm$
respectively. Regarding ANC, the calculated value of 4.58$A/(m\cdot K)$ for
MnPt3 is consistent with the reported value of 4$A/(m\cdot K)$ in Ref. [56].
It is important to mention that the ANC is extremely sensitive to the density
of points used for the calculation as well as the temperature that might
explain several differences between the reported and the calculated values.

The shape of the AHC and ANC tensors is the same and can be determined by the
magnetic space group of the compound. For the AHC tensor given by Eq. 1, it
primarily depends on the Berry curvature which behaves as a pseudovector under
symmetry operations, yielding:

$s\mathbf{v\left(r\right)}=\pm
det\left(\mathbf{D}\left(R\right)\right)\mathbf{D}\left(R\right)\mathbf{v}\left(s^{-1}\mathbf{r}\right)$
(4)

where $\mathbf{v}\left(r\right)$ denotes the pseudovector Berry curvature,
$\mathbf{D}\left(R\right)$ the three-dimensional representation of a symmetry
operation without the translation part and $s$ an arbitrary symmetry
operation. Taking SiMnY as an example, we observe that its crystal structure
belongs to the space group $P4/nmm$ (129) and the magnetization direction
along the [100]-direction renders it to the magnetic space group is
$Pmm^{\prime}n^{\prime}$ (BNS: 59.410). Correspondingly, the Berry curvature
(in the Cartesian basis) behaves as an odd pseudovector under the application
of $m_{100}$ and obeys:

$\displaystyle\Omega_{x}\left(-k_{x},k_{y},k_{z}\right)=\Omega_{x}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{y}\left(-k_{x},k_{y},k_{z}\right)=-\Omega_{y}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{z}\left(-k_{x},k_{y},k_{z}\right)=-\Omega_{z}\left(k_{x},k_{y},k_{z}\right).$

Thus, the summation over the whole Brillouin zone forces
$\sigma_{y}=\sigma_{xz}$ and $\sigma_{z}=\sigma_{xy}$ to vanish. On the other
hand, the magnetic space group of SiMnY changes to $P4/nm^{\prime}m^{\prime}$
(BNS: 129.417) for the magnetization direction parallel to [001] axis. In this
case, the Berry curvature behaves as an odd pseudovector under the application
of $2_{100}$ and obeys

$\displaystyle\Omega_{x}\left(-k_{x},-k_{y},k_{z}\right)=-\Omega_{x}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{y}\left(-k_{x},-k_{y},k_{z}\right)=-\Omega_{y}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{z}\left(-k_{x},-k_{y},k_{z}\right)=\Omega_{z}\left(k_{x},k_{y},k_{z}\right).$

It is obvious that the summation over the whole Brillouin zone forces
$\sigma_{x}=\sigma_{yz}$ and $\sigma_{y}=\sigma_{xz}$ to vanish, leading to
the conclusion that for the high-symmetric magnetization directions, the
direction of the AHC tensor is always aligned with the magnetization
direction. It is further noted that the absence of symmetries dictating
specific components of the Berry curvature to be zero does not necessarily
guarantee finite AHC of the respective component, since AHC depends on the
distribution of the Berry curvature in the whole BZ. For instance, it is
calculated negligible AHC value for all components in Co2NbAl
($Im^{\prime}m^{\prime}m$, BNS:71.536) despite the absence of symmetry forcing
$\sigma_{x}=0$.

For magnetic materials without time-reversal symmetry, the actual symmetry and
the actual electronic structure depend on the magnetization directions after
considering SOC, leading to in general anisotropic responses. In order to
quantify the changes in AHC and ANC, we performed calculations on two
different magnetization directions, i.e., [100] and [001]. The resulting
anisotropy can be expressed as the ratio of the so obtained values for the two
magnetization directions, i.e.,
$\sigma_{x,M\parallel[100]}/\sigma_{z,M\parallel[001]}$ or
$\alpha_{x,M\parallel[100]}/\alpha_{z,M\parallel[001]}$. It is clear that
“large” values signify large changes in favor of the [100] direction, as it
happens for AHC and ANC in Rh2MnSb and Fe3Se4, where anisotropy of 53.32 and
-33.39 is calculated respectively. On the other hand, small values (excluding
the cases where both AHC values are lower than 10$S/cm$ and both ANC values
lower than 0.05$A/(m\cdot K)$ and therefore considered negligible) are in
favor of the [001] magnetization direction, such as -0.01 and 0.004, that are
present in PFe and Fe2Ge for AHC and ANC respectively. The full list of the
values obtained for AHC, ANC and anisotropy for all ferromagnetic compounds
are available in Table S1 of the supplementary[59]. Compounds with AHC larger
than or equal to 1000$S/cm$ (irrespective of the direction), ANC larger than
or equal to 3$A/(m\cdot K)$ and anisotropy larger than 4 or smaller than 0.25
are classified as extreme cases and they are highlighted.

Figure 2: (a) Crystal structure of MnZn. (b) The z-component of the AHC
($\sigma_{z}$) (blue) and the AHC contribution originating from the nodal
lines (red) as a function of energy for MnZn with magnetization direction
parallel to [001] axis. (c) The z-component of the ANC ($\alpha_{z}$)
evaluated at T=300K (blue) and the ANC contribution originating from the nodal
lines (red) as a function of energy for MnZn with magnetization direction
parallel to [001] axis. (d) Symmetry related nodal lines of MnZn, with
magnetiation direction parallel to [001] axis, contributing to the total AHC
value.

Recent works have shown that Weyl nodes and nodal lines, being bands touching
points, behave either as sinks or as sources of the Berry curvature [8, 57,
58] and hence they are expected to contribute to the total AHC, the origin of
which demonstrates the validity of the Mott relation. To further elucidate the
origin of singular-like AHC with resulting magnificent ANC, detailed analysis
is done on the energy-dependent AHC for MnZn (BNS: 123.345) (for crystal
structure see Fig. 2(a)), as shown in Fig. 2(b) with the magnetization
direction along the [001] axis. There exists a sharp peak of AHC about
1082$S/cm$ located about 4$meV$ below the Fermi energy. Explicit band
structure analysis reveals the presence of a circularly shaped nodal line at
$k_{y}=0.5$ plane (red part in Fig. 2(d)), which further appears at the planes
$k_{x}=\pm 0.5$, $k_{z}=\pm 0.5$ and $k_{y}=-0.5$ due to the presence of the
symmetries of the compound (black and blue parts). Importantly, we found that
about 73% of the total AHC (782$S/cm$ out of 1082$S/cm$) can be attributed to
the contribution of the nodal lines (Fig. 2(b)), similarly to the role of Weyl
nodes in Mn3PdN [57]. Exactly due to the presence of such nodal lines and
their contribution to the singular-like AHC, the behaviour of ANC around EF is
dominated by the contributions from the nodal lines (Fig. 2(c)). Therefore,
the presence of Weyl nodes and nodal lines close to the Fermi energy can lead
to an anomalous energy dependence of AHC and hence enhanced ANC.

A note about the role of symmetry of the nodal lines and their contributions
to AHC is in order. As discussed above, the symmetry of the Berry Curvature in
the Brillouin zone is essential to understand the origin of AHC. For MnZn with
the magnetization direction along the z-axis, the corresponding Magnetic Laue
group $4/mm^{\prime}m^{\prime}$ indicates the presence of the six closed-loop-
nodal lines of Fig. 2(d). However, not all of these symmetry equivalent nodal
lines contribute equally to the same component of AHC, though their geometry
are dictated by the energy eigenvalues and thus the symmetry. The underlined
Laue group for the compound includes the $m_{z}$ mirror plane symmetry that
flips the sign of $x$ and $y$ components of Berry curvature while it leaves
$z$ component unchanged, according to:

$\displaystyle\Omega_{x}\left(k_{x},k_{y},-k_{z}\right)=-\Omega_{x}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{y}\left(k_{x},k_{y},-k_{z}\right)=-\Omega_{y}\left(k_{x},k_{y},k_{z}\right)$
$\displaystyle\Omega_{z}\left(k_{x},k_{y},-k_{z}\right)=\Omega_{z}\left(k_{x},k_{y},k_{z}\right).$

Hence, the contribution from the top black nodal ring as well as from the
upper half of the red and the blue will exactly cancel out the one from the
bottom black and the lower half of the red and the blue respectively for $x$
and $y$ components. A complete list of Berry curvature transformations with
respect to each symmetry operation of the magnetic Laue group
$4/mm^{\prime}m^{\prime}$ is found in Table S2 of the supplementary [59].

Figure 3: (a) Band gap (in eV) at a slice of Brillouin Zone ($k_{z}=-0.479$)
for Ni3Pt. (b) The x-component of the AHC evaluated in 216 Brillouin Zone
cubes for Ni3Pt with magnetization direction parallel to [100] axis.

Enhanced ANC values may originate from AHC contributions other than isolated
Weyl nodes and nodal lines. The presence of nodal rings are responsible for
almost 3/4 of the total AHC of MnZn, as discussed above. However, there are
cases with more complicated behaviour, such as Ni3Pt that hosts a surprisingly
large AHC of -2051$S/cm$ that renders it as the largest calculated AHC. In
order to investigate its origin, we split the BZ
($k_{i}\in\left[-0.500,0.500\right)$) in $6\times 6\times 6=216$ cubes and
calculate the AHC within each of these, as illustrated in Fig. 3(b)). Our
results show that each part of the BZ contributes to the total AHC,
demonstrating the presence of numerous Weyl nodes, nodal lines as well as
extended small band gap areas. Such areas are illustrated in the region
$\left(k_{x},k_{y}\right)\in\left[0.33,0.50\right]$ of the band gap plot of
Fig. 3(a). Despite the absence of isolated Weyl nodes, there is a giant ANC of
-7.29$A/(m\cdot K)$ (at the Fermi energy) calculated, originating from the
whole BZ, showing at the same time that large ANC values are possible even in
the absence of isolated hot-spot contributions to the Berry curvature.

Figure 4: (a) Crystal structure of CoSe4. (b) The AHC components as a function
of energy for CoSe4 with magnetization direction parallel to [100] axis. (c)
The ANC components evaluated at T=300K as a function of energy for CoSe4 with
magnetization direction parallel to [100] axis. (d) The x-component of the AHC
of CoSe4 with magnetization direction parallel to [100] axis for different
values of applied strain

As the Fermi level can be tuned by several mechanisms including doping and
mechanical strain, it is fruitful to discuss the possibility of inducing
larger AHC and ANC values away from the charge neutral point. In order to
investigate the impact of such modifications to the transport properties, we
consider the hexagonal compound CoSe4 (BNS: 61.436) (for crystal structure see
Fig. 4(a). Interestingly, the compound exhibits an AHC of 202$S/cm$ at the
Fermi energy and moreover a sharp peak at almost 500$S/cm$, 5 meV above (see
Fig. 4(b)) that can be tuned by doping. Since, in general, compounds that
exhibit single-valued AHC peaks close to the Fermi energy offer great
opportunities for modifying AHC and and hence ANC values by means of doping,
some promising candidates include VAu4, CoS2 and Co2CrAl that exhibit AHC
peaks of 1955$S/cm$, -1048$S/cm$ and 1140$S/cm$ at 0.051$eV$, -0.024$eV$ and
0.027$eV$ respectively. AHC and ANC can also be modified by applying biaxial
strain. That is, the tensile strain of 0.5% reduces the size of the peak to
306$S/cm$ and moves it at 0.01$eV$ in respect to the Fermi energy. On the
other hand, the applied compressive strain of 1.0% has the opposite effect and
hence the size of the peak is increased to 683$S/cm$ and its position lies at
-0.001$eV$ (see Fig. 4(d)).

##  IV Conclusion

Based on high throughput first-principles calculations, we evaluated the
anomalous Hall conductivity as well as the anomalous Nernst conductivity of
266 transition-metal based ferromagnets. We report that the absolute value of
AHC (ANC) of 11 (16) compounds is larger than 1000$S/cm$ (3$A/(m\cdot K)$)
with the largest being equal to 2060$S/cm$ for CrPt3 (-7.24$A/(m\cdot K)$ for
Ni3Pt). Moreover, we find that the AHC and ANC values are by 3/4 originating
from linear degenerate states such as Weyl nodes and nodal lines in MnZn and
they can further be enhanced by a factor of 43% by applying external stimuli,
such as 1.5% compressive strain in CoSe4. However, large AHC and ANC values
can arise even without the presence of isolated singular hot spots in the
Berry curvature but instead from uniform extended small band gap areas from
the whole BZ in Ni3Pt.

ACKNOWLEDGMENTS

This work was financially supported by the Deutsche Forschungsgemeinschaft
(DFG) via the priority programme SPP 1666 and the calculations were conducted
on the Lichtenberg high performance computer of the TU Darmstadt.

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  * [59] See Supplemental Material for Table S1 summarizing all calculated AHC, ANC, anisotropy and reported literature values for all 266 compounds and for Table S2 illustrating the the Berry curvature transformation under all symmetry operations of Magnetic Laue group $4/mm^{\prime}m^{\prime}$..