# Thermoelectric properties of marcasite-type compounds MSb2 (M = Ta, Nb): A
combined experimental and computational study

Shamim Sk1 Naoki Sato1, Electronic mail<EMAIL_ADDRESS>Takao Mori1,
Electronic mail<EMAIL_ADDRESS>1Research Center for Materials
Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS),
1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan

###### Abstract

Here, we investigate the thermoelectric properties of the marcasite-type
compounds MSb2 (M = Ta, Nb) in the temperature range of 310–730 K. These
compounds were synthesized by a solid-state reaction followed by the spark
plasma sintering process. The Rietveld refinement method confirms the
monoclinic phase with space group C2/m for both compounds. The observed values
of Seebeck coefficients exhibit non-monotonic behaviour in the studied
temperature range, with the maximum magnitude of $-$14.4 and $-$22.7 $\micro$V
K-1 for TaSb2 and NbSb2, respectively at $\sim$444 K. The negative sign of S
in the full temperature window signifies the n-type behaviour of these
compounds. Both electrical and thermal conductivities show an increasing trend
with temperature. The experimentally observed thermoelectric properties are
understood through the first-principles DFT and Boltzmann transport equation.
A pseudogap in the density of states around the Fermi level characterizes the
semimetallic behaviour of these compounds. The multi-band electron and hole
pockets were found to be mainly responsible for the temperature dependence of
transport properties. The experimental power factors are found to be
$\sim$0.09 and $\sim$0.42 mW m-1 K-2 at 310 K for TaSb2 and NbSb2,
respectively. From the DFT-based calculations, the maximum possible power
factors for p-type conduction are predicted as $\sim$1.14 and $\sim$1.74 mW
m-1 K-2, while these values are found to be $\sim$1.16 and $\sim$1.80 mW m-1
K-2 for n-type TaSb2 and NbSb2, respectively at 300 K with the corresponding
doping concentrations. The present study suggests that the combined DFT and
Boltzmann transport theory are found to be reasonably good at explaining the
experimental transport properties, and moderate power factors are predicted.

Key words: Thermoelectric properties, density functional theory, electronic
structure, electron and hole pockets, semi- classical Boltzmann theory, power
factor.

##  I INTRODUCTION

The performance of thermoelectric (TE) materials is evaluated by the
dimensionless figure-of-merit, $ZT=\frac{S^{2}\sigma T}{\kappa}$, where S is
the Seebeck coefficient, $\sigma$ is the electrical conductivity, $\kappa$ is
the thermal conductivity (which is the sum of electronic, $\kappa_{e}$ and
lattice thermal conductivity, $\kappa_{l}$), and T is the absolute
temperature. The value of ZT determines the conversion efficiency; the higher
the ZT value, the better the TE performance. Finding materials having ZT
higher than unity is still a challenging task, though the research in TE has
made in progress for many decades [1, 2, 3, 4, 5, 6, 7]. Actually, efficient
TE materials have to pass through a typical tradeoff, which includes materials
that are good electrical conductors but have poor thermal conductivity. This
implies that the transport of charge carriers within the material should
experience weak electron scattering and strong phonon scattering.

Realizing high ZT has always been a challenging task due to the strong
correlation among the TE parameters through charge carriers [8, 9]. Till now,
the state-of-the-art TE materials are Bi2Te3 [10, 11], Sb2Te3 [12]; PbTe [13,
14], SnTe [15]; and SiGe [16, 17] based compounds. Recently, the antimonides
have gained increased interest with high performance discovered in Mg3Sb2-type
materials for example [18, 19, 20, 21]. The expression of ZT implies that
there are two ways for boosting the TE performance: either by decreasing the
lattice thermal conductivity without affecting or less affecting the
electronic properties [1, 22, 23, 24] or by improving the power factor
($S^{2}\sigma$) [25, 26, 27, 28, 29, 30].

In recent decades, the TE properties of FeSb2 were extensively studied because
of ultra-high S of up to $-$45000 $\micro$V K-1 at 12 K, resulting in the
highest power factor reported ever [31, 32]. But, due to the large $\kappa$ of
FeSb2, the ZT is diminished with the low value of 0.005 at 12 K [31, 32]. The
FeSb2 is a narrow-gap semiconductor with an orthorhombic marcasite structure
[31, 32]. The TE properties of other marcasite-type compounds including XTe2
(X = Fe, Co, Ni) [33], FeX2 (X = Se, Te) [34], FeS2 [35], FeAs2 [36], RuSb2
[36, 37], etc, have been studied in the last decades. In the same group of
marcasite-type compounds, the MSb2 (M = Ta, Nb) with monoclinic crystal
structure (Space group: C2/m, No. 12) have recently attracted remarkable
attention as Weyl semimetals having unexpected magneto-transport properties
[38, 39, 40, 41, 42]. Regrettably, the TE properties of these compounds are
rarely explored, especially at elevated temperatures. For instance, Masuda et
al. have studied the TE properties of TaSb2 compound in the temperature range
of 300–800 K [43]. They synthesized TaSb2 using solid-state reaction method
with the spark plasma-sintering process and explored the temperature dependent
S and $\sigma$ [43]. The experimental measurements of S and $\sigma$ of MSb2
(M = Ta, Nb) have been done by Failamani et al. [44] in the temperature range
of 300–800 K. However, there is a lack of theoretical understanding of the
experimental TE properties of MSb2 (M = Ta, Nb) at elevated temperatures. In
the present study, the TE properties of MSb2 (M = Ta, Nb) are investigated
using both experimental and computational tools at the high temperatures. The
experimental TE properties of said compounds are understood using density
functional theory (DFT)-based calculations, and we predict the possible
maximum power factors with a suitable amount of doping.

In this work, we have synthesized marcasite-type compounds MSb2 (M = Ta, Nb)
using the combined solid-state reaction and spark plasma sintering process.
All the TE properties are measured and analyzed. First-principle calculations
and Boltzmann transport equations are utilized to understand the experimental
results. The multi-band electron and hole pockets are found to reasonably
explain the experimental data. The maximum possible power factors for p-type
and n-type of MSb2 (M = Ta, Nb) are also predicted using the DFT-based
calculations.

##  II EXPERIMENTAL AND COMPUTATIONAL DETAILS

MSb2 (M = Ta, Nb) were synthesized using a combined solid state reaction (SSR)
and spark plasma sintering (SPS) process. The high purity powder of Ta
(99.99$\%$), Nb (99.99$\%$) and Sb (99.99$\%$) from Sigma-Aldrich were taken
as starting materials. The desired amount of metal powder were ground in
alumina mortar and cold pressed at 25 MPa to form pellets. Then the obtained
pellets were heated at 700 $\celsius$ for 3 days in vacuum-sealed quartz tube.
Although the target phase for both compounds were obtained in SSR, but the
pellets were too brittle for the transport measurements. In order to get dense
pellets, we ground the SSR pellets and did the SPS (SPS-1080, SPS Syntex Inc.)
at 750 $\celsius$ for 8 minutes. During the SPS, the constant uniaxial
pressure of 50 MPa was applied to a graphite punch of 10 mm diameter under a
partial argon atmosphere. The relative densities were calculated from the
sample densities determined by Archimedes principle, which are found to be
$\sim$95$\%$ and $\sim$94$\%$ for TaSb2 and NbSb2, respectively. The obtained
pellets were cut into the required dimensions for the transport measurements.

The simultaneous measurement of the electrical conductivity and Seebeck
coefficient were performed using ZEM-2 (ADVANCE RIKO) under partial helium
environment. The dimensions of TaSb2 and NbSb2 were taken as $4.11\times
1.92\times 7.57$ mm3 and $4.11\times 1.62\times 7.39$ mm3, respectively.

The thermal conductivity was obtained using the formula: $\kappa=D\times
C_{p}\times\rho$. The $\rho$ is the density of the sample. The D is the
thermal diffusivity, which is measured using the laser flash diffusivity
method implemented in LFA-467 Hyper flash (Netzsch) instrument. The heat
capacity, $C_{p}$ was estimated using a standard sample (pyroceram-9060) in
LFA-467. The circular pellets of TaSb2 and NbSb2 with diameter of 10 mm and
thickness of 1.3 mm and 1.6 mm, respectively were used for the measurements.

Figure 1: Measured room temperature X-ray diffraction patterns and calculated ones by Rietveld method of (a) TaSb2 and (b) NbSb2. The peak at 28.7$\degree$ indicated by an arrow in (a) is an impurity peak. Table 1: Fractional coordinates of atomic positions for relaxed structure of TaSb2 and NbSb2. The experimental values are also mentioned in the parentheses. TaSb2 | x (exp) | y (exp) | z (exp) | NbSb2 | x (exp) | y (exp) | z (exp)
---|---|---|---|---|---|---|---
Ta | 0.150 (0.152) | 0 (0) | 0.188 (0.193) | Nb | 0.151 (0.152) | 0 (0) | 0.190 (0.194)
Sb1 | 0.148 (0.143) | 0 (0) | 0.535 (0.532) | Sb1 | 0.149 (0.145) | 0 (0) | 0.536 (0.531)
Sb2 | 0.405 (0.401) | 0 (0) | 0.114 (0.111) | Sb2 | 0.405 (0.402) | 0 (0) | 0.115 (0.111)

X-ray diffraction (XRD) were taken (Smart Lab3, Rigaku) in 10–90$\degree$ as
shown in Fig. 1(a) and (b) for TaSb2 and NbSb2, respectively. Rietveld
refinement method confirms the monoclinic phase with space group C2/m (No. 12)
for both compounds. The refined lattice parameters are obtained as $a=10.22$ Å
, $b=3.64$ Å , $c=8.29$ Å and $\beta=120.39\degree$ for TaSb2, and $a=10.23$ Å
, $b=3.63$ Å , $c=8.33$ Å and $\beta=120.02\degree$ for NbSb2. The impurity
peak at 28.7$\degree$ (Fig. 1(a)) may arise from the surface oxide layer of
Sb2O3 [43].

In order to understand the experimental transport properties, we have carried
out the ground state electronic structure calculations within (DFT) [45]. The
projector augmented-wave method is used as implemented in Quantum Espresso
code [46]. The PERDEW-ZUNGER (LDA) [47] exchange-correlation (XC) functional
is used for the calculation. The calculations are done in relaxed structure
with optimized lattice parameters of $a=10.14$ Å , $b=3.62$ Å , $c=8.22$ Å and
$\beta=120.53\degree$ for TaSb2, and $a=10.16$ Å , $b=3.59$ Å , $c=8.25$ Å and
$\beta=120.01\degree$ for NbSb2. The force convergence criteria for structure
relaxation was set to be 10-4 Ry/Bohr. Table 1 shows the relaxed atomic
positions along with the experimental values for both compounds. The
structures relaxed using DFT are closely matched with the experimental ones.
The kinetic energy cut-off for wavefunctions is used as 60 Ry. The kinetic
energy cut-off for charge density is set to be 8 times the kinetic energy cut-
off for wavefunctions. The k-mesh grid was used as $7\times 15\times 7$ for
both compounds. The energy convergence criteria was set to be 10-8 Ry for the
self-consistent field calculation. The transport coefficients were calculated
using the BoltzTraP2 package [48] interfaced with Quantum Espresso code [46].

##  III RESULTS AND DISCUSSION

###  III.1 EXPERIMENTAL TRANSPORT PROPERTIES

Fig. 2(a) exhibits the experimentally measured Seebeck coefficients (S) of
MSb2 (M = Ta, Nb) in the temperature range of 310–730 K. The $|S|$ for both
compounds are found to increase up to $\sim$450 K, then decreases till the
highest temperature, consistent with the other reported data [43, 44]. The
highest magnitudes of S are found to be $-$14.4 and $-$22.7 $\micro$V K-1 for
TaSb2 and NbSb2, respectively at $\sim$444 K. The total S mainly comes from
the contributions of electrons and holes. Under the two-carrier conduction
model, the S can be expressed as [49]
$S=\frac{S_{h}\sigma_{h}+S_{e}\sigma_{e}}{\sigma_{h}+\sigma_{e}}$, where
$S_{h,e}$ is the Seebeck coefficients of holes and electrons and
$\sigma_{h,e}$ is the electrical conductivity of holes and electrons. The sign
of the S is determined by the major contribution coming from the electrons or
holes, because electrons generally yield the negative S, while holes give
positive S. Therefore, the negative sign of S in Fig. 2(a) signifies the
dominating n-type behaviour of these compounds. The magnitude of S for NbSb2
is larger than that of TaSb2 in the full temperature window. The non-linear
behaviour of S with temperature for both compounds can be explained by the
calculated band-structure, which is discussed later.

Figure 2: Temperature dependence of measured (a) Seebeck coefficient, S (b)
electrical conductivity, $\sigma$ and (c) thermal conductivity, $\kappa$ of
TaSb2 and NbSb2.

Figure 3: (a) Band-structure and (b) density of states (DOS) of TaSb2; (c)
band-structure and (d) DOS of NbSb2.

Fig. 2(b) shows the temperature dependence of electrical conductivities
($\sigma$) of TaSb2 and NbSb2. The $\sigma$ are found to be 0.88 $\times
10^{6}$ and 0.95 $\times 10^{6}$ $\ohm^{-1}$ m-1 for TaSb2 and NbSb2,
respectively at $\sim$310 K. Then as the temperature increases, the $\sigma$
decreases till the highest temperature ($\sim$730 K) with the corresponding
values of 0.39 $\times 10^{6}$ and 0.41 $\times 10^{6}$ $\ohm^{-1}$ m-1 for
TaSb2 and NbSb2, respectively. Our measured values of $\sigma$ are accordance
with the other experimentally reported data [43, 44]. The decrement behaviour
of $\sigma$ can be explained by the simple formula:
$\sigma=\frac{ne^{2}\tau}{m^{*}}$, where, $n$, $e$, $\tau$ and $m^{*}$ are the
carrier concentration, the electronic charge, the relaxation time and the
effective mass of charge carriers, respectively. Under the increment of
temperature, $n$ always increases, whereas $\tau$ decreases for any compound.
Among the opposite trend of $n$ and $\tau$ with temperature, the dominating
behaviour generally gives the temperature dependent trend of $\sigma$. Hence,
Fig. 2(b) shows that the temperature dependent $\tau$ is dominating over $n$
in $\sigma$. In general, with increase in temperature, the $\sigma$ of
semiconductors increases, while $\sigma$ of metals decreases. Hence, the
temperature dependent trend of $\sigma$ of present compounds show the metal-
like behaviour. However, the electronic structure calculations predict the
semimetallic (in between semiconductor and metals) behaviour of these
compounds, which is discussed later.

Using the measured S and $\sigma$, the power factors are calculated as
$\sim$0.91 $\times$ 10-1 and $\sim$4.17 $\times$ 10-1 mW m-1 K-2 at 310 K for
TaSb2 and NbSb2, respectively. At 730 K, the power factors are calculated as
$\sim$0.39 $\times$ 10-1 and $\sim$2.32 $\times$ 10-1 mW m-1 K-2,
respectively. Usually, semimetals possess low S with high $\sigma$. But recent
studies have shown that the S of the semimetals can be tuned by magnetic
field, strain, etc [50, 51, 52]. Semimetals having asymmetry in their electron
and hole pockets could have high S [53, 54].

The thermal conductivities ($\kappa$) of TaSb2 and NbSb2 are measured in the
temperature region 300–750 K as shown in Fig. 2(c). The $\kappa$ are gradually
decreasing with the increment of temperature in the whole temperature range.
At 300 K, the $\kappa$ are found to be 13.7 and 12.2 W m-1 K-1, while these
are decreased to 9.6 and 8.2 W m-1 K-1 at 750 K for TaSb2 and NbSb2,
respectively. In the full temperature window, the values of $\kappa$ for TaSb2
are higher than the NbSb2. The total $\kappa$ is a simple addition of
electronic thermal conductivity ($\kappa_{e}$) and lattice thermal
conductivity ($\kappa_{L}$), i.e., $\kappa=\kappa_{e}+\kappa_{L}$. In this
work, we have calculated the temperature dependence of $\kappa_{e}$ for both
compounds. The experimental $\kappa_{e}$ can be estimated using the
experimental $\sigma$ via Wiedemann-Franz law: $\kappa_{e}=L\sigma T$, L is
Lorenz number. Then it will be interesting to see how calculated $\kappa_{e}$
explains the experimental $\kappa_{e}$, which is described later. As can be
seen from the Fig. 2(b), the $\sigma$ of NbSb2 are higher than that of TaSb2.
Hence, larger values of $\kappa_{e}$ are expected for NbSb2 as compared to
TaSb2 according to the Wiedemann-Franz law.

###  III.2 ELECTRONIC STRUCTURE

To understand the experimentally measured transport coefficients, we have
calculated the electronic structure of MSb2 (M = Ta, Nb). The band-structures
for TaSb2 and NbSb2 are calculated along the lines between high symmetry
points ($\Gamma$–C–C2–$\Gamma$–M2–D2–A–L2–$\Gamma$–V2) in the first Brillouin
zone, which are shown in Fig. 3(a) and (c), respectively. The dashed red line
corresponding to zero energy defines the Fermi level, $E_{F}$ of the
compounds. It is clear that the two bands (which are indexed by 1 and 2)
around the $E_{F}$ are expected to contribute to the transport properties of
these compounds. In both Fig. 3(a) and (c), the occupied band 1 crosses the
$E_{F}$ at either side of the $\Gamma$ point and becomes unoccupied, while the
unoccupied band 2 crosses the $E_{F}$ in the M2–D2, A–L2 and $\Gamma$–V2
directions and becomes occupied. This type of mixing of occupied and
unoccupied bands around the $E_{F}$ predicts the semimetal-like character of
the compound, which is consistent with the other reported works [38, 39, 41,
55]. The electronic band-structure is the key input for calculating any
electronic transport properties. In the present study, we have calculated the
Seebeck coefficient, electrical conductivity and electronic part of thermal
conductivity for MSb2 (M = Ta, Nb). We will recall this part in the next sub-
section during the discussion of calculated transport properties.

Fig. 3(b) and (d) express the calculated total and partial density of states
(DOS) of TaSb2 and NbSb2, respectively. At $E_{F}$, the values of DOS are
calculated as $\sim$0.34 and $\sim$0.47 states/eV/f.u. for TaSb2 and NbSb2,
respectively. A pseudogap around the $E_{F}$ characterizes the semimetallic
behaviour of these compounds. In order to know the contributions in transport
properties from different atoms, we have calculated the partial DOS of Ta, Nb
and Sb as shown in the same figures. In the occupied band region, the
contribution in the DOS comes from Ta(Nb)-d and Sb-p orbitals almost equally.
In contrast, in the unoccupied band region, the dominant contribution in the
DOS comes from Ta(Nb)-d orbitals, with the small contribution from Sb-p
orbitals. In the energy range of $-1$ to 0 eV in the occupied band region of
Fig. 3(b), the contributions of Ta-d and Sb-p in the DOS are calculated as
$\sim$60$\%$ and $\sim$40$\%$, while these contributions are found to be
$\sim$80$\%$ and $\sim$20$\%$, respectively, in the energy range of 0 to 1 eV
in the unoccupied band region.

Figure 4: Calculated Seebeck coefficient, S at the function of chemical
potential, $\mu$ at different temperatures of TaSb2. The solid vertical black
line indicates the $\mu$ value, where transport properties are calculated.

###  III.3 CALCULATED TRANSPORT PROPERTIES

In this sub-section, we discuss the calculated transport properties, viz., the
temperature dependence of Seebeck coefficient (S), electrical conductivity
divided by relaxation time ($\sigma/\tau$) and electronic thermal conductivity
divided by relaxation time ($\kappa_{e}/\tau$) as calculated using the
BoltzTraP2 package [48] under semi-classical Boltzmann theory. The BoltzTrap2
is based on the combined constant relaxation time approximation (CRTA) and
rigid band approximation (RBA). In CRTA, relaxation time ($\tau$) is
considered as a constant, but in principle $\tau$ is dependent on both band
index and wave vector direction. The RBA means that the band-structure is
independent of temperature and doping.

At first, we have calculated the S at EF, which are $\sim$1 and $\sim$20
$\micro$V K-1 at 310 K for TaSb2 and NbSb2, respectively. These positive
values of S are far away from the experimental negative values of $\sim-10$
and $\sim-21$ $\micro$V K-1 for TaSb2 and NbSb2, respectively at the same
temperature. At this point, it is important to note that the calculations of S
have been done on single crystalline stoichiometric compounds. But, it is very
common to have off-stoichiometry in any synthesized polycrystalline samples.
This off-stoichiometry may come from many factors, including the purity of the
starting materials, inaccuracy in weighing the raw materials, inhomogeneous
mixing during the synthesis process, evaporation of low melting element during
the heat treatment etc. These factors are mainly responsible for the defects
and/or disorders in the samples which may affect the S of the sample. In
addition to this, the anisotropy often matters when comparing the calculation
of single crystal with the experimental polycrystalline one. All these factors
can be addressed in the calculation by shifting the chemical potential ($\mu$)
of the compound. But, quantifying the exact value of $\mu$ is a challenging
job for any compound. For doing this, we have calculated $\mu$ dependent S at
different temperatures as shown in Fig. 4. Then, the $\mu$ has been chosen at
300 K for a better representation of experimental S. We found that at $\mu$
$\approx$ 64 and 82 meV, the calculated T dependent S gives the best match
with the experimental T dependent S for TaSb2 and NbSb2 as shown in Fig. 5 (a)
and (b), respectively. This constant $\mu$ calculated at 300 K is used to
calculate the other transport properties in the full temperature range.

Figure 5: Comparison of experimental and calculated Seebeck coefficients, S of
(a) TaSb2 and (b) NbSb2.

The T dependent S of TaSb2 and NbSb2 at $\mu$ $\approx$ 64 and 82 meV,
respectively can be understood through electron/hole pockets formed with the
bands 1 and 2 in the band-structure of Fig. 3(a) and (c). The black horizontal
dotted lines above the $E_{F}$ indicate the $\mu$ values, where the
temperature dependent S are calculated. For both compounds, it is clear that
in S, the contributions of charge carriers mainly come from the hole pockets
formed with band 1 at the vicinity of $\Gamma$ and A points and in the
$\Gamma$–M2 direction, and electron pockets formed with the band 2 in the
C–C2, C2–$\Gamma$ M2–D2, A–L2, $\Gamma$–V2 and $\Gamma$–M2 (band 1) directions
at $\mu$ $\approx$ 64 (82) meV. The presence of dominating electron pockets
over hole pockets gives the negative S for both compounds. The size of the
hole pockets at $\Gamma$ point for TaSb2 (at 64 meV) is larger than the hole
pocket at $\Gamma$ point for NbSb2 (at 82 meV), which supports the less
magnitude of the S for TaSb2 as compared to that of NbSb2. However, in the
high temperature region, the calculated S are deviating from the experimental
S. At this conjuncture, it is important to note that the S is calculated using
the ground-state band-structures and constant $\mu$ of 300 K. But, the band-
structure and $\mu$ are the temperature dependent quantities. Hence, the
temperature dependence of these quantities are expected to improve the
accuracy of calculated S at higher temperature.

Figure 6: Calculated electrical conductivity divided by relaxation time,
$\sigma/\tau$ of (a) TaSb2 and (b) NbSb2. Inset shows $\tau$ as a function of
temperature, which is estimated by comparing the calculated $\sigma/\tau$ with
experimental $\sigma$.

Fig. 6 shows the calculated $\bm{\sigma}/\tau$ of TaSb2 and NbSb2 in the
temperature range of 300–750 K. The $\bm{\sigma}$ of $n^{th}$ band is defined
as [8]:

$\bm{\sigma}^{(n)}=e^{2}\int\frac{d\textbf{k}}{4\pi^{3}}\tau_{n}(\varepsilon_{n}(\textbf{k}))\textbf{v}_{n}(\textbf{k})\textbf{v}_{n}(\textbf{k})\bigg{(}-\frac{\partial
f}{\partial\varepsilon}\bigg{)}_{\varepsilon=\varepsilon_{n}(\textbf{k})},$
(1)

where $e$ is an electronic charge, $\tau_{n}(\varepsilon_{n}(\textbf{k}))$ is
the relaxation time of an electron of $n^{th}$ band with wave vector k and
$\varepsilon_{n}(\textbf{k})$ is an energy band. The
$\textbf{v}_{n}(\textbf{k})$ is the mean velocity of an electron of $n^{th}$
band with wave vector k. The f is the Fermi-Dirac distribution function, which
takes care of the temperature dependency of the compound. As we discussed
earlier, the BoltzTraP2 code [48] works under constant relaxation time
approximation, i.e., $\tau_{n}(\varepsilon_{n}(\textbf{k}))=\tau$. Hence, the
calculated $\bm{\sigma}/\tau$ depends on $\textbf{v}_{n}(\textbf{k})$ and
$\frac{\partial f}{\partial\varepsilon}$, and number of available states at a
given $\mu$. With an increase in temperature, the number of states always
increases. Hence, the increasing nature of $\bm{\sigma}/\tau$ with temperature
(Fig. 6) is directly related to the more number of available states at high
temperature. Initially, the charge carriers from band 1 at the vicinity of
$\Gamma$ point and in the $\Gamma$–M2 direction, and band 2 in the C–C2,
M2–D2, A–L2 and $\Gamma$–V2 directions are participated in the transport (Fig
3(a)). As the temperature increases, the more charge carriers from band 1 at
the vicinity of A point and in the M2–D2 and $\Gamma$–V2 directions; the band
2 in C2–$\Gamma$ direction are expected to contribute in the
$\bm{\sigma}/\tau$. At high temperature, more charge carriers from the
different electron/hole pockets participate in the conduction, and hence
$\bm{\sigma}/\tau$ increases. The calculated temperature dependent
$\bm{\sigma}/\tau$ is compared with the experimental $\sigma$ (Fig. 2(b)) to
extract the temperature dependent $\tau$. Insets of Fig. 6 shows the extracted
$\tau$ in the temperature range of 300–750 K. The values of $\tau$ are
calculated as $\sim$1.8 $\times$ 10-14 s and $\sim$2.5 $\times$ 10-14 s at 300
K for TaSb2 and NbSb2, respectively. With the increase in temperature, $\tau$
decreases monotonically and reaches $\sim$0.66 $\times$ 10-14 s ($\sim$0.82
$\times$ 10-14 s) at 750 K. The decreasing nature of $\tau$ is due to the
presence of more scattering centers at high temperature. For a real system,
the calculation of $\tau$ is a challenging task due to the involvement of many
scattering mechanisms, including electron-electron scattering, electron-phonon
scattering, electron-defect scattering etc [8]. This is the reason why we have
chosen the simple method to extract temperature dependent $\tau$. It will be
interesting to see how this temperature dependent $\tau$ explains the other
transport properties.

Figure 7: Comparison of experimental (estimated using Wiedemann-Franz law) and
calculated electronic part of thermal conductivity, $\kappa_{e}$ of (a) TaSb2
and (b) NbSb2.

Figure 8: Variation of power factor ($S^{2}\sigma$) with chemical potential at
different temperatures of (a) TaSb2 and (b) NbSb2.

We have calculated $\kappa_{e}/\tau$ for MSb2 (M = Ta, Nb) using the following
equation [8]

$\bm{\kappa_{e}}/\tau=\frac{\pi^{2}}{3}\bigg{(}\frac{k_{B}}{e}\bigg{)}^{2}T(\bm{\sigma/\tau}),$
(2)

where $k_{B}$ in the Boltzmann constant. Then the $\kappa_{e}$ is computed
using the temperature dependent $\tau$. Fig. 7 shows the $\kappa_{e}$ for
TaSb2 and NbSb2 in the temperature region 300–750 K. The calculated
$\kappa_{e}$ is compared with the experimental $\kappa_{e}$ in the same
figure. The experimental $\kappa_{e}$ is estimated using Wiedemann-Franz law:
$\kappa_{e}=L\sigma T$. Where the temperature dependent experimental $\sigma$
(Fig. 2(b)) and the constant value of L (2.45 $\times$ 10-8 W $\ohm$ K-2) are
taken to estimate the experimental $\kappa_{e}$. It is observed that the
calculated $\kappa_{e}$ gives quite good agreement with the experimental
$\kappa_{e}$. With an increase in temperature, $\kappa_{e}$ increases due to
contributions of more charge carriers from the different electron/hole
pockets. The figure shows that there is a small deviation at high temperature
region. At this point, it is important to note that the calculations have been
done using ground-state band-structure, constant $\mu$ and L values, as we
mentioned earlier. Considering temperature dependency of all these factors may
improve the reproducibility at high temperature, which demands extra
computational costs.

###  III.4 POWER FACTOR

Finally, the chemical potential dependence of the power factor divided by
relaxation time of MSb2 (M = Ta, Nb) is calculated in the temperature range of
300–1000 K as shown in Fig. 8. The red vertical dashed line at zero eV
indicates the Fermi level of the compound. The black vertical solid line at
$\sim$64 and $\sim$82 meV of TaSb2 and NbSb2, respectively indicates the $\mu$
value, at which the transport properties are calculated to explain the
experimental results. The maximum possible power factors are also calculated
for p-type and n-type of these compounds. The maximum power factors for p-type
conduction are calculated as $\sim$1.14 and $\sim$1.74 mW m-1 K-2 at
$\sim-$375 and $\sim-$320 meV, respectively, while these values are found to
be $\sim$1.16 and $\sim$1.80 mW m-1 K-2 at $\sim$215 and $\sim$215 meV for
n-type of TaSb2 and NbSb2, respectively at 300 K. The p-type and n-type are
confirmed from the sign of S at the corresponding $\mu$ values of Fig. 4. The
$\tau$ values are taken from the previous calculations (Inset of Fig. 6). The
carrier concentrations corresponding to the maximum power factors for p-type
of TaSb2 and NbSb2 are calculated as $\sim$2.56 $\times$ 1021 and $\sim$2.60
$\times$ 1021 cm-3, respectively, whereas for the n-type conduction of TaSb2
and NbSb2 these values are found to be $\sim$1.42 $\times$ 1021 and $\sim$1.58
$\times$ 1021 cm-3, respectively. The predicted power factors at 300 K in this
study are comparable with the power factors of $\sim$1–2 mW m-1 K-2 for Bi2Te3
parent compound [56, 57, 58], though the power factors of BiTe-based doped
compounds are reported as $\sim$4–7 mW m-1 K-2 in many literatures [59, 60,
61, 62] at/around the room temperature. However, a rigorous effort is required
to synthesize the suitable p and n-type doping of MSb2 (M = Ta, Nb) to
validate the computational prediction.

##  IV CONCLUSIONS

The thermoelectric properties of MSb2 (M = Ta, Nb) are studied in the present
study. These compounds were prepared by a combined solid-state reaction and a
spark plasma sintering process. The monoclinic phase with space group C2/m is
confirmed for both compounds through Rietveld refinement. The negative sign of
the Seebeck coefficient indicates the n-type behaviour of these compounds. The
DFT-based electronic structure calculations were carried out in order to
understand the experimentally observed thermoelectric properties. The
semimetallic behaviour of these compounds was confirmed from the band-
structure and density of states calculations. The multi-band electron and hole
pockets are found reasonably good in explaining the experimental results.
Further study of computational calculations gives the maximum possible power
factors of $\sim$1.14 and $\sim$1.74 mW m-1 K-2 for p-type conduction of TaSb2
and NbSb2, respectively at 300 K, while these values are found to be
$\sim$1.16 and $\sim$1.80 mW m-1 K-2 at 300 K for n-type conduction,
respectively. In conclusion, the combined DFT and Boltzmann transport theory
are found to be reasonably good at addressing the experimental transport
properties, and moderate power factors can be obtained if these compounds are
synthesized with the proper doping concentration.

##  V ACKNOWLEDGEMENTS

This work was supported by JST Mirai Program Grant Number JPMJMI19A1.

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