# Coexistence of phononic Weyl, nodal line, and threefold excitations
in chalcopyrite CdGeAs2 and associated thermoelectric properties

Vikas Saini<EMAIL_ADDRESS>Department of Condensed Matter Physics
and Materials Science,Tata Institute of Fundamental Research, Mumbai 400005,
India  Bikash Patra<EMAIL_ADDRESS>Department of Condensed Matter
Physics and Materials Science,Tata Institute of Fundamental Research, Mumbai
400005, India  Bahadur Singh  Department of Condensed Matter Physics and
Materials Science,Tata Institute of Fundamental Research, Mumbai 400005, India
A. Thamizhavel<EMAIL_ADDRESS>Department of Condensed Matter Physics and
Materials Science,Tata Institute of Fundamental Research, Mumbai 400005, India

###### Abstract

Realization of topologically protected quantum states leads to unprecedented
opportunities for fundamental science and device applications. Here, we
demonstrate the coexistence of multiple topological phononic states and
calculate the associated thermoelectric properties of a chalcopyrite material
CdGeAs2 using first-principles theoretical modeling. CdGeAs2 is a direct
bandgap semiconductor with a bandgap of $0.65$ eV. By analysing the phonon
spectrum and associated symmetries, we show the presence of nearly isolated
Weyl, nodal line, and threefold band crossings in CdGeAs2. Specifically, the
two triply degenerate points (TDPs) identified on the $k_{z}$ axis are formed
by the optical phonons bands 7, 8, and 9 with type-II energy dispersion. These
TDPs form a time reversal pair and are connected by a straight nodal line with
zero Berry phase. The TDPs formed between bands 14, 15, and 16 exhibit type-I
crossings and are connected through the open straight nodal line. Our
transport calculations show a large thermopower exceeding $\sim$500 and $200$
$\rm\mu V/K$ for the hole and electron carriers, respectively, above 500 K
with a carrier doping of 1018 cm-3. The large thermopower in $p$-type CdGeAs2
is a consequence of the sharp density of states appear from the presence of a
heavy hole band at the $\Gamma$ point. We argue that the presence of
topological states in the phonon bands could lead to low lattice thermal
conductivity and drive a high figure-of-merit in CdGeAs2.

##  I Introduction

The existence of topologically protected nontrivial states has been
demonstrated in wide classes of crystalline materials [1, 2, 3]. For example,
Dirac semimetals [4, 5, 6, 7] are formed by the crossings of spin degenerate
valence and conduction bands in the momentum space. The low-energy excitations
around the crossing points mimic the linear energy-momentum relation of Dirac
fermions. These Dirac points are invariant under perturbations preserving
parity and time-reversal symmetries. Breaking either of these symmetries
splits a Dirac point into two Weyl points with opposite chirality [8, 9, 10,
11]. The band crossings in presence of non-symmorphic crystallographic
symmetries can lead to higher fold fermionic excitations and have been
realized in numerous materials [12, 13, 14, 15]. The triply degenerate point
(TDP) semimetal states have been established in MoP, WC, ZrSe, etc. materials
where the presence of three-fold crossings are shown to exist between a double
degenerate and single degenerate bands [16, 17, 18, 19, 20, 21]. Importantly,
the TDP semimetal phase conceptually lies between the Dirac and Weyl phases.
Topological band crossings can also be classified based on the dimensionality
of the band touching points. They can form nodal points with zero-dimensional
point-like crossings or nodal lines with one-dimensional band crossings. Such
one-dimensional nodal lines can constitute nodal lines, nodal links, nodal
knots, nodal rings, and nodal chains, etc. [22, 23, 24, 25, 26]. Besides the
degeneracies and dimensionalities of the crossing points, chiral charges of
the nodal points, energy dispersion and slope of the bands are essential to
distinguish among a variety of topological states. In particular, the type-II
band crossings are identified at the touching point of the hole and electron
pockets and break the Lorentz symmetry [27, 28, 29, 30, 31]. The topology in
the electronic structure has been explored over the last several years, and
many novel phenomena have been proposed and verified in experiments. Common to
the topology of electronic structure is that they are constrained by the Pauli
exclusion principle.

Topology of the bosonic states especially topological states in the phonon
spectrum is an emerging research field. Phonons obey the Bose-Einstein
statistics and are not constrained by the Fermi energy, giving access to the
whole spectrum of phonon energy range from the THz and IR to probe the
topological states [32, 33, 34]. Recent studies on graphene, FeSi, and other
materials uncover the presence of topological excitations such as Weyl, double
Weyl, and multifold Weyl in the phonon spectrum. The higher fold degenerate
phonons have been proposed for several materials with specific space groups
which show the topological phonon states protected by various crystalline
symmetries [35, 36, 37, 38, 39, 40, 41]. Motivated by the studies of
topological phonons in materials and their possible effects on thermal
transport, we explore the topological phonons and transport properties of
chalcopyrite material CdGeAs2. Our phonon calculations reveal topological Weyl
phonons, triply-degenerate nodal points, and nodal lines, among other phases
in various phonon bands. Specifically, we find Weyl points with chiral charge
$\pm 1$ in multiple phonons branches. Moreover, multiple TDPs protected by
$C_{3v}$ symmetry are found on the $k_{z}$ axis with both the type I and type
II energy dispersions.

We also investigate the electronic and thermoelectric properties of CdGeAs2.
Electronic structure and transport calculations are carried with both the
generalized gradient approximation (GGA) [42] and mBJ [43] exchange-
correlation (XC) functionals and with the inclusion of spin-orbit coupling
(SOC). The results obtained with mBJ functional shows a band gap of 0.65 eV,
close to experimentally observed values of 0.57 eV at room temperature [44,
45, 46]. The calculated thermopower ($S$) of $p$-type carriers is found to be
more than the $n$-type carriers in our considered carrier density range of
1018 \- 1021 cm-3. This large thermopower for $p$-type carriers is attributed
to the steepness in the density of states (DOS) below the Fermi energy level
due to the presence of heavy hole-type bands. We also discuss the effect of
carrier effective mass in generating the large thermopower and figure of merit
$ZT{{}_{e}}$ $=$ $\frac{S^{2}\sigma/\tau}{k_{e}/\tau}$. Specifically, the
upper limit of $ZT{{}_{e}}$ without considering the lattice thermal
conductivity, is remarkably large with a value of $\sim 11.5$ at 600 K and
carrier density $\sim$ 1.1$\times$ 1018 cm-3 for $p$-type carriers. In this
way, our work identifies CdGeAs2 as a potential chalcopyrite material for
exploring topological phonons and thermoelectric properties.

Figure 1: Crystal structure and Brillouin zone of CdGeAs2. (a), (b)
Conventional tetragonal unit cell of CdGeAs2. (c) Primitive unit cell and (d)
associated Brillouin zone structure of CdGeAs2. High-symmetry points are
marked in red.

##  II Methods

Electronic structure calculations were performed within the framework of
density functional theory (DFT) using the full-potential linearized augmented
plane wave (FP-LAPW) method as implemented in WIEN2k package [47, 48, 49].
Self-consistent calculations were performed on a dense $k$-mesh with $10000$
$k$ points. Plane-wave cut-off for $\rm R_{MT}K_{MAX}$ was set to be 9 (RMT is
the muffin tin radius and KMAX is the maximum value of reciprocal lattice
vector). Thermoelectric properties were calculated by solving the Boltzmann
transport equations under the constant scattering time approximation (CSTA) as
implemented in the BoltzTraP2 [50, 51]. We used $149784$ irreducible $k$
points to accurately model transport properties. The phonon spectrum was
obtained using the finite displacement method by considering a
2$\times$2$\times$2 supercell employing the VASP [52, 53, 54] and Phonopy
codes [55]. The lattice parameters for the first-principles and thermoelectric
calculations were obtained from the powder x-ray diffraction experiments using
our synthesized CdGeAs2 polycrystalline sample. The refined parameters were
obtained as $a$ = 5.94(4) Å, $c$ = 11.21(6) Å and the Wyckoff positions
(0.22009, 0.25, 0.125), (0.0, 0.0, 0.5), and (0.0, 0.0, 0.0) for As, Cd, and
Ge atoms, respectively.

##  III Results and Discussion

Figure 2: (a) Calculated phonon dispersion of CdGeAs2 along high symmetry
Brillouin zone directions. The topological crossings are highlighted. (b)
Partial and total phonon density of states (DOS) of CdGeAs2. Figure 3:
Topological phonon structure for bands 8 and 9 (see Fig. 2 for band index).
(a) Nodal crossing points formed by the bands $8$ and $9$ in 3D momentum
space. (b) Triple degenerate point and nodal line along $\Gamma$-$Z$
direction. (c-d) 3D dispersion around the TDP in $k_{x}-k_{y}$ and
$k_{x}-k_{z}$ planes. (e) Phonon dispersion for a nodal line point along in-
plane axis. (f) Phonon dispersion for a point present at nodal ring along the
diagonal axis. (g) Topological fermi arc surface states connecting the
projected phonon Weyl points on the (001) surface. (h) Non-zero Berry
curvature around the positive Weyl point. (i) Schematic diagram of the Weyl
points on the (001) surface. Blue and yellow circles denote the positive and
negative charges. Figure 4: Topological phonon structure for bands 14 and 15
(see Fig. 2 for band index). (a) Nodal crossings of bands 14 and 15 inside the
BZ. (b) Isolated part for the TDP and NL along $\Gamma$-$Z$ direction. (c-d)
3D phonon dispersion in $k_{x}-k_{y}$ and $k_{x}-k_{z}$ planes around the TDP.
NL is marked along the $k_{z}$ axis in (d). (e-f) Dispersion along the $k_{y}$
and $k_{z}$ directions around a point present at the nodal line. (g) Weyl
points distribution on the (001) surface. The WPs are drawn as circles. (h-i)
Wannier center evolution on a sphere enclosing the positive and negative
chiral charge WPs. (d) Topological fermi arc surface states connecting
opposite chiral WPs on the (001) surface. (k-l) Chiral topological surface
states connecting the projected bulk WPs.  Figure 5: (a) Electronic band
structure of CdGeAs2 obtained with GGA XC functional. (b) Orbital resolved
band structure obtained with GGA XC functional. (c) Same as (a) but obtained
with the mBJ XC functional. (d) Calculated total DOS for the GGA and mBJ XC
functionals. (e) Band dispersion of valence and conduction bands for
$\Gamma$-$X$ and $\Gamma$-$Z$ directions. (f) Calculated carrier effective
masses for in-plane and out-of-plane directions for $p$-type carriers at 300
K. (g) Topologically nontrivial surface states on crystallographic (001)
surface obtained for GGA XC functional without SOC. (h) Anisotropic nature of
topmost hole band below 20 meV energy from the Fermi level for mBJ XC
functional calculation. (i) Electron pocket of the lowest conduction band at
an energy around 1 eV upward from the Fermi level for mBJ XC functional.

The CdGeAs2 crystallizes in a non-centrosymmetric tetragonal crystal lattice
with space group $I$$\bar{4}2d$ (No. 122). The conventional unit cell of
CdGeAs2 is shown in Fig. 1(a). The As atoms in the top layer are shifted with
respect to the As atoms in the bottom layer, as shown in Fig. 1(b). This
suggests the absence of inversion symmetry in the crystal. Figure 1(c) shows
the primitive unit cell and Fig. 1(d) illustrates the associated Brillouin
zone (BZ) with high-symmetry points marked. The phonon dispersion of CdGeAs2
along various high symmetry paths is shown in Fig. 2(a). The absence of
negative phonon frequencies suggests the dynamical stability of the crystal.
From the phonon calculations, we found that the acoustic phonons are mainly
contributed by the heavy Cd atoms, as depicted from the atom projected density
of states in Fig. 2(b).

The topological phonon analysis of CdGeAs2 is shown in Figs. 2, 3, and 4. The
acoustic phonon modes along the in-plane $\Gamma$-$X$ direction are non-
degenerate whereas, the two lowest energy acoustic modes are degenerate along
the $\Gamma$ \- $Z$ direction. Top transverse acoustic (TTA) mode possesses
linear phonon dispersion along the in- and out-of-plane directions.

For the topological analysis, a few of the phonon bands are selected, which
are highlighted in Fig. 2(a) from their bulk band crossings. Bands $8$ and $9$
show band crossings leading to the symmetry-protected nodal rings, nodal line,
Weyl, and triply degenerate points inside the BZ. All the nodal phases formed
by the crossings of these bands are summarized in Fig. 3(a). The Weyl points
are located parallel to $\Gamma$ \- $X$ directions but not at the $k_{z}$ = 0
plane. The nodal line formed from these bands is represented by the green
color in Fig. 2(a), at the endpoint of the nodal ring band 7 appears to form
the triply degenerate point along the $\Gamma$ \- $Z$ direction. The triply
degenerate points of the 7, 8, and 9 bands crossings are protected by the
$C_{3z}$ rotational symmetry. The dispersion along the $k_{z}$ direction for
these three bands are shown separately in Fig. 3(b). The triply degenerate
points are obtained from the crossings of doubly degenerate and non-degenerate
bands. The 3D visualization of these three bands in $k_{x}-k_{y}$ and
$k_{x}-k_{z}$ planes can be observed from the Fig. 3(c,d). Fig. 3(c) shows
that the TDP point is formed by the crossings of two nearly flat and one
dispersive band. Since the slope of the crossing bands appears to be the same,
therefore, type-II TDP is resolved in the $k_{x}-k_{y}$ plane. The two TDP
points along the $k_{z}$ axis are connected through a straight doubly
degenerated Weyl nodal line. The calculated Berry phase for this nodal line is
0 suggesting a topologically trivial character and that is followed by the in-
plane quadratic dispersion as shown in Fig. 3(e). The nodal line is formed by
the bands 8 and 9 as depicted in Fig. 3(a). Below the TDP along $k_{z}$ axis
two nodal rings intersect which are lying on the diagonal mirror planes and
the Berry phase of both rings is 0, which again reveals the topologically
trivial character of these rings and the band dispersion along the diagonal
axis is depicted in Fig. 3(f). The crossing point is formed by one nearly flat
and one dispersive band. For the Berry phase calculation, a circle contour is
defined perpendicular to the nodal plane with a small finite radius to avoid
inclosing other band crossings.

Moreover, eight Weyl points are observed parallel to $k_{x}$ and $k_{y}$ axes
at $k_{z}$ = $\pm$ 0.21 Å-1. These eight Weyl points formed by these bands
possess a magnitude of charge $1$ and all the Weyl points are at the same
frequency 2.41 THz as a result of mirror inversion in the $m_{xy}$ diagonal
planes. For same $k_{x}$ and $k_{y}$ coordinates opposite $k_{z}$ WPs have the
same chiral charge owing to the absence of the $m_{z}$ symmetry thus the
projected WPs on the (001) surface possess a chiral charge of $\pm$ 2 and are
at the $k_{x}$ and $k_{y}$ axes as illustrated in the Fig. 3(i). The Berry
curvature around a positive chiral WP is non-zero as depicted in Fig. 3(h).
Opposite chiral charge Weyl nodes connect by the topologically nontrivial arcs
on (001) surface as depicted in Fig. 3(g) for phonon energy 2.44 THz.

Similar to the preceding analysis, we also analyzed topological features of
the crossings of $14$ and $15$ phonon bands as identified in Fig. 4(a). The
nodal line is guided in green color and a TDP is present inside the shown box.
Unlike bands 8 and 9, these phonon bands have two pairs of Weyl points along
the principle momentum axes at $k_{z}$ = 0 plane. Notably, the magnitude of
the chiral charges remains the same as before. The nodal points of these bands
are summarized in Fig. 4(a). Along $k_{z}$ axis a pair of TDP is observed at
$k_{z}$= $\pm 0.37$ Å-1. These points are connected with the open straight
line that possesses zero Berry phase and obeys the non-linear band characters
along the in-plane direction. Whereas along the crossing line it shows the
linear band crossings as represented in Fig. 4(e,f). The TDP is formed by the
crossings of 14, 15, and 16 bands, whose 3D dispersion along $k_{x}k_{y}$ is
shown in Fig. 4(c). Bands 15 and 16 are doubly degenerate along $k_{z}$ axis
before the TDP crossings and after the crossing bands, 14 and 15 become
degenerate for a range of $k$-values as shown in Fig. 4(b,d).

We also discuss the Weyl phonons which are present on the $k_{z}$ = 0 plane
along the $k_{x}$ and $k_{y}$ axes at $|k_{x}|=|k_{y}|=0.31~{}$ Å-1. The
energy-momentum relations and locations of WPs in the 3D are shown in Fig.
4(g). The Berry phases around the opposite chiral WPs are calculated using the
Wilson loop method from the Wannier charges. Positive and negative chiral WPs
possess $\pi$ and $-\pi$ Berry phases that can be observed from Fig. 4(h-i).
WPs on $k_{x}$ axis and $k_{y}$ axis have opposite chiral charges. The
opposite chiral Weyl points are connected through the arcs as depicted in Fig.
4(j) and the corresponding surface states of the opposite WPs on to the (001)
surface are shown in Fig. 4(k,l).

The topology in the phonon bands leads to the symmetry protected states which
stays robust against symmetry respected perturbations. There are many other
bands in the phonon spectrum featuring topological states, for example
acoustic and optical bands possess several straight nodal lines and TDP along
$k_{z}$ direction protected by the crystalline symmetries. Topological
protection of phonon states lead to many novel phenomena and increase in the
phonon scatterings result in the reduction of the lattice thermal conductivity
that leads to the thermoelectric properties. It has been established for many
of the materials such as TaSb, TaBi, NbSb etc. that having the topological
phonons lead to the lower lattice thermal conductivity than the materials
which do not have the topological characters in the phonons [56, 57, 58]. In
parallel to topological phonons, many electronic properties such as high band
gap value between valence and conduction bands that suppress the bipolar
effect prevents the reduction of thermopower below a certain temperature.
Heavy hole band for $p$-type doping gives rise to the large thermopower, and
the anisotropic nature of hole band supports to amplify the thermoelectric
power factor and $ZT$ by increasing the mobility of charge carriers along
highly dispersive band directions.

We present the calculated bulk band structure obtained with both the GGA and
mBJ XC functionals in Figs. 5(a-c) in the presence of spin-orbit coupling
(SOC). A negative bandgap of $\sim 0.08$ eV is obtained with GGA XC functional
with a clear band inversion at $\Gamma$ point [Figs. 5(a) and Fig. 5(b)]. The
orbital resolved band structure is shown in Fig. 5(b) where As $p$ (blue
color) and Ge $s$ (green color) states contribute dominantly near the Fermi
level. A band inversion between the As $p$ and Ge $s$ states is evident at the
$\Gamma$-point. We also examine the electronic structure in the absence of
SOC.

Table 1: Location of Weyl nodes on the $k_{z}=0$ plane without SOC obtained with GGA XC functional. Strain |  $\rm b^{{}^{\prime}}$$=$b | 1.01b | 1.02b | 1.03b
---|---|---|---|---
Weyl position |  |  |  |
$k_{x}$ $=$ $k_{y}$ ($\rm\AA^{-1}$) | 0.028 | 0.022 | 0.016 | 0.008

Weyl points of positive chiral charges are located at $k_{x}$ axis and
negative charges are found at $k_{y}$ axis. For surface states, four Weyl
points in the bulk are projected onto (001) surface, and two opposite chiral
Weyl points are connected with the arcs as represented in Fig. 5(g). To check
the effect of strain on the location of Weyl points in momentum space we
summarized a few of the strains in Table 1. Keeping the unit cell volume
constant, compression on the small $a$-axis of the tetragonal structure is
applied in the form of the tensile strain on the longest $b$-axis by 1 to
3$\%$. In absence of SOC, the Weyl phase is robust with four Weyl nodes on the
$k_{z}$ $=$ 0 plane. The separation between the Weyl nodes points continuously
decreases as the strain is increased from 1$\%$ to 3$\%$ as tabulated in Table
1.

GGA XC functional underestimates band gap in the materials therefore we have
carried out thermoelectric calculations in presence of the mBJ XC functional.

Figure 5(c) shows the band structure obtained with mBJ XC functional. The band
inversion is now disappeared and the system becomes trivial with a direct
bandgap of 0.65 eV. This is close to the experimentally reported value of 0.57
eV at room temperature [44, 45, 46]. Fig. 5(d) represents the density of
states (DOS) for GGA and mBJ XC functionals in red and blue colors,
respectively. The DOS below the Fermi energy shows a rapid increase with
energy which leads to the large thermopower for $p$-type doping as discussed
below.

Figure 6: Calculated thermoelectric properties of CdGeAs2 for $p$-type and
$n$-type doping. (a) Thermopower plot with varying carrier density at constant
temperatures. (b) Electrical conductivity divided by relaxation time as a
function of carrier density in the regime of $10^{18}$-$10^{21}$ cm-3. (c) The
power factor divided by relaxation time with carrier density at constant
temperatures. (d) Electrical thermal conductivity over relaxation time plotted
against carrier density for various temperatures. Figure 7: (a) Anisotropic
parameters $\sigma_{zz}/\sigma_{xx}$ and $S_{zz}/S_{xx}$ plotted as a function
of hole carrier density at $T$= 300 K. (b) The upper limit of the figure-of-
merit ($ZT_{e}$) as a function of carrier density at constant temperatures.
Figure 8: Calculated figure-of-merit $ZT$ for CdGeAs2. (a-b) $ZT$ values as
the functions of hole and electron carrier densities at a constant scattering
time $\tau$ $=$ 10-13. (c-d) Estimation of $ZT$ values at $\tau$ $=$ 10-14 s,
for hole and electron type of carriers, respectively.

Our first-principles results show heavy hole bands along $\Gamma$-$X$
direction whereas the bands are highly dispersive along $\Gamma$-$Z$ direction
[Fig. 5(e)]. This suggests that the charge carriers of in-plane direction
($\Gamma$-$X$) possesses high effective mass compared to the out-of-plane
direction ($\Gamma$-$Z$) which possesses light effective mass and high
mobility. To see the anisotropic effects, the weighted mobility can be defined
as $\mu_{w}$ = $\mu$($\frac{m^{*}}{m_{e}})^{\frac{3}{2}}$, where $\mu$ is the
mobility of charge carriers, $m^{*}$ = $N_{v}^{\frac{2}{3}}$
${(m_{x}m_{y}m_{z})}^{\frac{1}{3}}$; $N_{v}$ valley degeneracy, and $m_{e}$ is
the rest mass of the electron. The highly dispersive band along out-of-plane
($\Gamma$-$Z$) direction gives rise to the higher weighted mobility $\mu_{w}$
because of high mobility as compared to the in-plane $\Gamma$-$X$ direction.
The high value of the $\mu_{w}$ lead to the higher power factor and $ZT$
parameters. The anisotropy in effective masses is calculated from transport
parameters at room temperature $\frac{1}{m^{*}_{\alpha\beta}}$ $=$
$\frac{\sigma_{\alpha\beta}/\tau}{e^{2}n}$; where $\sigma_{\alpha\beta}/\tau$
is electrical conductivity tensor divided by relaxation time, $n$ is the
carrier density. Further, the anisotropic band structure is consistent with
the calculated effective masses for $ab$-plane ($m_{xx}$) and $c$-direction
($m_{zz}$) for $p$-type CdGeAs2 as shown in Fig. 5(f). In the low carrier
density $p$ $\sim$ 1.27 $\times$ $10^{18}$ cm-3 the anisotropy in the masses
is maximum $m_{xx}$ $\sim$ 1.9 $m_{zz}$, and as the hole carrier density
increases the anisotropy decreases and the effective masses for in- and out-
of-plane bands are around $m_{xx}$ = 0.73 $m_{e}$ and $m_{zz}$ = 0.63 $m_{e}$
for $p$ $\sim$ $10^{21}$ cm-3.

Figures 6, 7, and 8 show the calculated thermoelectric properties of CdGeAs2
within constant relaxation time approximation [59, 60, 61, 62, 63]. The
thermopower ($S$) of $p$-type carriers (solid curves) is greater than $n$-type
carriers (dotted curves) in the entire carrier density range 1018 \- 1021 cm-3
at a constant temperature as shown in Fig. 6(a). The thermopower increases
with increasing temperature and reaches a value of 567 $\mu$V/K at 700 K for
$p$-type doping and 270 $\mu$V/K at 800 K for $n$-type at carrier
concentration 1018 cm-3.

To understand the large thermopower of $p$-type carriers, we have used the
Mott relation in constant relaxation time approximation which is given by [64,
65].

$S(n,T)=\frac{\pi^{2}k_{B}^{2}T}{3q}\left[\frac{1}{n}\frac{dn(E)}{dE}+\frac{1}{\mu}\frac{d\mu(E)}{dE}\right]_{E_{F}}$
(1)

Here, $q$ denotes the charge, $n(E)$ represents the density of states, $\mu$
is the mobility, and $T$ denotes the absolute temperature. The equation is
composed of the addition of two derivative terms. It can be simply related
from the first term of Eqn. 1 that steepness in the DOS enhances thermopower.
The mBJ XC calculated DOS (Fig 5(d)) for $p$-type doping shows a steep rise
than the $n$-type doping in CdGeAs2 resulting large thermopower for the $p$
doped system. The second term of the Mott relation exhibits that if the
mobility of charge carriers increases with energy as a consequence of the
critical scatterings near the Fermi level then the thermopower can be boosted
further.

Thermopower of both types of carriers gradually reduce with increasing carrier
density except for the $p$-type 800 K plot. For $p$-type carriers, owing to
electronic thermal excitations, the thermopower reduces with decreasing
carrier density at high temperatures as observed in Fig. 6(a) for 800 K.

The bipolar conduction effect in the low carrier density regime for the narrow
gap semiconductors is kind of normal, as studies suggest, to suppress this
effect the band gap energy should be more than around 8 $k_{\rm B}$$T$ [66].

The band gap of CdGeAs2 (0.65 eV) is reasonably large which ensures that the
bipolar effect is not seen at temperatures below 700 K. The band gap of
CdGeAs2 is higher than the calculated values for the other TE materials such
as (PbSe ($E_{g}$ = 0.28) and PbTe ($E_{g}$ = 0.36) from ref. [67]). Also,
thermopower at 300 K for the $p$-type CdGeAs2 is more than the PbSe, SnTe and
PbTe compounds [59, 68, 60].

The electrical conductivity divided by the relaxation time
$\frac{\sigma}{\tau}$ is represented in Fig. 6(b) on a logarithmic scale for
both types of charge carriers for different temperatures from 200 to 800 K.
The $\frac{\sigma}{\tau}$ of the $n$-type charge carriers is higher than the
$p$-type carriers that is consistent from the highly dispersive nature of the
$n$-type band compared to the $p$-type band that results to have relatively
small effective mass of $n$-type band which essentially increases the
electrical conductivity. The increase in temperature reduces the electrical
conductivity in the entire range of carrier density 1018 \- 1021 cm-3.
However, the magnitude of change with the temperature at a given carrier
concentration is not much for both types of carriers. For $p$-type charge
carriers $\frac{\sigma}{\tau}$ is (0.1 and 0.06) $\times$ 1018 $\Omega^{-1}$
$m^{-1}$ s-1 at carrier concentration 1018 cm-3 and for temperatures 300 and
800 K.

The power factor divided by the relaxation time $\frac{S^{2}\sigma}{\tau}$ is
shown in Fig. 6(c) for the both types of charge carriers in the range of
carrier density 1018 \- 1021 cm-3. It is obvious from the figure that the
$p$-type power factor is dominated over the $n$-type power factor in
intermediate carrier density regime before crossover appears around
1.07$\times$1020 cm-3 and after the crossover $n$-type power factor is
dominated over $p$-type until carrier density reaches up to 1021 cm-3 for each
temperature ranging from 200 to 800 K.

The power factor is one of the considerable parameters to increase the
thermoelectric performance of materials. However, for the rough estimation of
power factor if empirically relaxation time is considered in the order of
10-14 s then it governs reasonably good values of the power factor in the
range of mWK-2m-1 to $\mu$WK-2m-1 for the $p$-type carriers from high to low
density regime, attributed to the optimized band structure of CdGeAs2.

The anisotropic ratio of out-of- and in-plane thermopower $S_{zz}/S_{xx}$ and
electrical conductivity $\sigma_{zz}/\sigma_{xx}$ are plotted against hole
carrier density at $T$= 300 K in the Fig. 7(a). The thermopower along the out-
of-plane direction is lower than the in-plane direction as a consequence of
light band along $\Gamma$-$Z$ direction which leads to the small effective
mass. The $S_{zz}/S_{xx}$ is weakly changing in the low carrier density
regime. Conversely, the electrical conductivity for out-of-plane direction is
$\sigma_{zz}$ $\sim$ 1.9 $\sigma_{xx}$ at carrier density $p$ $\sim$
1.27$\times$1018 cm-3. Beyond this hole density, the anisotropy in
conductivity decreases up to 1.9$\times$1020 cm-3 and later this increases
very gradually which attains $\sigma_{zz}$ $\sim$ 1.16 $\sigma_{xx}$ at
carrier density $p$ $\sim$ 1021 cm-3. The anisotropy in the calculated
transport parameters is consistent with the anisotropic effective mass as
discussed in Fig. 5(f).

Finally, the anisotropic nature of the hole band at $\Gamma$ point is shown in
Fig. 5 (h) where the Fermi level is shifted below $0.2$ eV from its pristine
value which drives the anisotropy in the electrical conductivity that helps to
increase the weighted mobility $\mu_{w}$ for out-of-plane ($\Gamma$-$Z$)
direction thus resulting in the high power factor and $ZT$ for out-of-plane
direction compared to in-plane direction. For comparison, we also have the
lower conduction band visualization while the Fermi level is around 1 eV up
from the maxima of the valence band that shows the isotropic nature of
electron band. Therefore, anisotropic advantage of the hole bands may enhance
the thermoelectric properties of $p$-type carriers compared to the $n$-type
carriers.

The electronic thermal conductivity ($k_{e}$) can be defined as $k_{e}$$=$
L$\sigma$T. The calculated $k_{e}$ is shown in Fig. 6(d) for both carriers in
the temperature range from 200 to 800 K. The electronic thermal conductivity
divided by relaxation time $\frac{k_{e}}{\tau}$ for $p$-type doping is smaller
than the $n$-type doping in the entire carrier density regime
$10^{18}-10^{21}$ cm-3 at the constant temperatures which supports the higher
$ZT{{}_{e}}$ values compared to $n$-type doping as shown in Fig. 7(b).

For $n$-type doping, $\frac{k_{e}}{\tau}$ increases gradually with increasing
temperature in the entire carrier density range $10^{18}-10^{21}$ cm-3 and for
$p$-type doping $\frac{k_{e}}{\tau}$ increases gradually with increasing
temperature in the entire carrier density regime below 700 K. At 700 K and
above in low density regime 1018 \- 1019 cm-3 electronic thermal conductivity
increases much more rapidly with lowering the carrier density as shown in Fig.
6(d) which results in the reduction of $ZT_{e}$ values for $T\gtrsim 700$ K
(Fig. 7(b)).

The upper limit of figure-of-merit $ZT{{}_{e}}$ $=$
$\frac{S^{2}\sigma/\tau}{k_{e}/\tau}$ in the low-carrier carrier density
regime are exceptionally high for $p$-type carriers (11.5 at $T=600$ K and $n$
$\sim$ 1.1$\times$ 1018 cm-3) than the $n$-type carriers (3.1 at $T=600$ K and
$n$ $\sim$ 1.1$\times$ 1018 cm-3) as a consequence of optimized band structure
as depicted in Fig. 7(b).

As the phonon dispersion of CdGeAs2 shown in Fig. 2(a) uncover that the small
frequencies of acoustic phonons attribute to the low group velocity which can
lead to low lattice thermal conductivity ($k_{l}$ $=$
$\frac{1}{3}C_{V}v_{g}l$). Moreover, as we discussed above the mixing of
acoustic and optical modes along with the topological protection give rise the
enhanced phonon scatterings resulting into small mean free path and thus
indicate a small lattice thermal conductivity in CdGeAs2.

However, the experimentally observed lattice thermal conductivity of CdGeAs2
at 300 K is 4 Wm-1K-1 [69]. As we discussed the electronic part of the figure-
of-merit $ZT_{e}$ is remarkably high for $p$-type CdGeAs2 and notably this
does not depend on the relaxation time $\tau$. The thermoelectric figure-of-
merit can be written in terms of electronic part of figure-of-merit as $ZT$
$=$ $\frac{S^{2}\sigma}{k_{e}+k_{l}}$= $\frac{ZT_{e}}{(1+k_{l}/k_{e})}$. To
estimate $ZT$ we need to have the electronic thermal conductivity $k_{e}$.
Since from the calculations, we obtain $\frac{k_{e}}{\tau}$ therefore to get
$k_{e}$ the knowledge of relaxation time $\tau$ is required. For the
materials, $\tau$ can be a function of temperature and carrier density, at the
simplest electron-phonon scatterings show the inverse temperature behavior
$\tau$ $\propto$ $T$-1. To get the idea of $ZT$ values in $p$\- and $n$-type
CdGeAs2, we approximate the relaxation time as 10-13 and 10-14 s independent
of temperature and carrier density and later we will add the effect of these
parameters on $\tau$ for $ZT$ values.

Figure 8 depicts the calculated $ZT$ for $\tau$ = 10-13 and 10-14 s. For
$\tau$ $=$ 10-13 s $ZT$ reaches from 0.25 at 300 K to 1.16 at 600 K for
$p$-type dopings as shown in Fig. 8(a). Whereas it goes up to 0.37 at 600 K
from 0.12 at 300 K, there is one thing to note that we have used the
temperature independent lattice thermal conductivity $k_{l}$ as 4 W/m-K but
conventionally $k_{l}$ decreases with increasing temperature, therefore, we
may expect further increase in the $ZT$ values.

For the $\tau$ $=$ 10-14 s maximum values of $ZT$ are around 0.047 and 0.049
obtained at 300 K, respectively for $p$\- and $n$-type dopings, and with
increasing temperature $ZT$ increases and attains 0.26 and 0.22 at the 600 K
for $p$\- and $n$-type dopings, respectively. Similar to the previous
discussion, drops in the $k_{l}$ with increasing temperature may be expected
to enhance the $ZT$ values further. Of course with increasing temperature
$\tau$ reduces and without experimental data the scaling of carrier density is
challenging but as the previous studies suggest this makes $\tau$ to be
suppressed [62]. Thus incorporating both of the these effects into the $\tau$
would lead to the decrease in the estimated values of $ZT$.

These analytical results indicate towards a reasonably good $ZT$ values at
high temperatures $T$ $\gtrsim$ 300 K. Experimental engineering of the
reinforcing phonon scatterings from defects and grain boundaries will further
reduce the $k_{l}$ and enhancement of the $k_{e}$ would increase the $ZT$
values for the better performance of thermoelectric devices. In this work, we
have covered the complete comprehensive study of the CdGeAs2 from the
calculated results these will be surely helpful for the further leads to
explore the experimental aspects.

## Summary and Conclusion

We have studied the topological states in the phonon spectrum and resolved the
triply degenerate phonons on the $k_{z}$ axis. Multiple pairs of Weyl points
in the bulk structure, and their topological surface states are observed on
the (001) surface. The topological features in the phonon spectra could
suppress lattice thermal conductivity enhancing the thermoelectric figure of
merit. We discuss electronic properties of CdGeAs2 that exhibit many
thermoelectric supportive features and result in high thermopower for $p$-type
dopings. The anisotropic hole bands lead to the high-weighted mobility that
supports the high power factor and $ZT$ values. The calculated value of
electronic figure-of-merit $ZT_{e}$ is high (more than 2 for temperatures 300
K and above) in carrier density regime 1018 \- 1019 cm-3 for $p$-type CdGeAs2
that result in the noble response to the thermoelectric performance. Our study
unfold that CdGeAs2 has topologically nontrivial phonon states and exhibits
very good theromelectric properties that are surely useful for the potential
applications.

## Acknowledgement

We thank Prof. Kalobaran Maiti for providing the computational resources and
acknowledge the TIFR computing resources. This work was supported by the
Department of Atomic Energy of the government of India under Project No.
12-R$\&$D-TFR-5.10-0100.

## Data Availability

The data supporting the findings of this study are available within the
article. More data can be provided on a reasonable request to the
corresponding author.

## References

  * [1] B. Singh, H. Lin, and A. Bansil, Adv. Mater. , 2201058.
  * Bansil  _et al._ [2016] A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. 88, 021004 (2016).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Liu  _et al._ [2014] Z. Liu, B. Zhou, Y. Zhang, Z. Wang, H. Weng, D. Prabhakaran, S.-K. Mo, Z. Shen, Z. Fang, X. Dai, _et al._ , Science 343, 864 (2014).
  * Jenkins  _et al._ [2016] G. Jenkins, C. Lane, B. Barbiellini, A. Sushkov, R. Carey, F. Liu, J. Krizan, S. Kushwaha, Q. Gibson, T.-R. Chang, _et al._ , Phys. Rev. B 94, 085121 (2016).
  * Nie  _et al._ [2018] T. Nie, L. Meng, Y. Li, Y. Luan, and J. Yu, J. Condens. Matter Phys. 30, 125502 (2018).
  * Wang  _et al._ [2012] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012).
  * Sun  _et al._ [2015a] Y. Sun, S.-C. Wu, and B. Yan, Phys. Rev. B 92, 115428 (2015a).
  * Yan and Felser [2017] B. Yan and C. Felser, Annu. Rev. Condens. 8, 337 (2017).
  * Xu  _et al._ [2015] S.-Y. Xu, N. Alidoust, I. Belopolski, C. Zhang, G. Bian, T.-R. Chang, H. Zheng, V. Strokov, D. S. Sanchez, G. Chang, _et al._ , arXiv preprint arXiv:1504.01350 (2015).
  * Lv  _et al._ [2015] B. Lv, S. Muff, T. Qian, Z. Song, S. Nie, N. Xu, P. Richard, C. E. Matt, N. C. Plumb, L. Zhao, _et al._ , Phys. Rev. Lett. 115, 217601 (2015).
  * Yáng  _et al._ [2020] X. Yáng, T. A. Cochran, R. Chapai, D. Tristant, J.-X. Yin, I. Belopolski, Z. b. u. b. a. Chéng, D. Multer, S. S. Zhang, N. Shumiya, M. Litskevich, Y. Jiang, G. Chang, Q. Zhang, I. Vekhter, W. A. Shelton, R. Jin, S.-Y. Xu, and M. Z. Hasan, Phys. Rev. B 101, 201105 (2020).
  * Thirupathaiah  _et al._ [2021] S. Thirupathaiah, Y. Kushnirenk, K. Koepernik, B. R. Piening, B. Buechner, S. Aswartham, J. van den Brink, S. Borisenko, and I. C. Fulga, SciPost Phys. 10, 004 (2021).
  * Yang  _et al._ [2019] Y. Yang, H.-x. Sun, J.-p. Xia, H. Xue, Z. Gao, Y. Ge, D. Jia, S.-q. Yuan, Y. Chong, and B. Zhang, Nat. Phys. 15, 645 (2019).
  * Hasan  _et al._ [2021] M. Z. Hasan, G. Chang, I. Belopolski, G. Bian, S.-Y. Xu, and J.-X. Yin, Nat. Rev. Mater. 6, 784 (2021).
  * Lv  _et al._ [2017] B. Lv, Z.-L. Feng, Q.-N. Xu, X. Gao, J.-Z. Ma, L.-Y. Kong, P. Richard, Y.-B. Huang, V. Strocov, C. Fang, _et al._ , Nature 546, 627 (2017).
  * Kumar  _et al._ [2019] N. Kumar, Y. Sun, M. Nicklas, S. J. Watzman, O. Young, I. Leermakers, J. Hornung, J. Klotz, J. Gooth, K. Manna, _et al._ , Nat. Commun. 10, 1 (2019).
  * Zhu  _et al._ [2016] Z. Zhu, G. W. Winkler, Q. Wu, J. Li, and A. A. Soluyanov, Phys. Rev. X 6, 031003 (2016).
  * Ma  _et al._ [2018] J.-Z. Ma, J.-B. He, Y.-F. Xu, B. Lv, D. Chen, W.-L. Zhu, S. Zhang, L.-Y. Kong, X. Gao, L.-Y. Rong, _et al._ , Nat. Phys. 14, 349 (2018).
  * Weng  _et al._ [2016] H. Weng, C. Fang, Z. Fang, and X. Dai, Phys. Rev. B 93, 241202 (2016).
  * Mardanya  _et al._ [2019] S. Mardanya, B. Singh, S.-M. Huang, T.-R. Chang, C. Su, H. Lin, A. Agarwal, and A. Bansil, Phys. Rev. Mater. 3, 071201 (2019).
  * Yan  _et al._ [2017] Z. Yan, R. Bi, H. Shen, L. Lu, S.-C. Zhang, and Z. Wang, Phys. Rev. B 96, 041103 (2017).
  * Bi  _et al._ [2017] R. Bi, Z. Yan, L. Lu, and Z. Wang, Phys. Rev. B 96, 201305 (2017).
  * Fang  _et al._ [2016] C. Fang, H. Weng, X. Dai, and Z. Fang, Chin. Phys. B 25, 117106 (2016).
  * Rui  _et al._ [2018] W. Rui, Y. Zhao, and A. P. Schnyder, Phys. Rev. B 97, 161113 (2018).
  * Zhu  _et al._ [2022] J. Zhu, W. Wu, J. Zhao, H. Chen, L. Zhang, and S. A. Yang, npj Quantum Mater. 7, 1 (2022).
  * Zhang  _et al._ [2017] T.-T. Zhang, Z.-M. Yu, W. Guo, D. Shi, G. Zhang, and Y. Yao, J. Phys. Chem. Lett. 8, 5792 (2017).
  * Soluyanov [2017] A. A. Soluyanov, Phys. 10, 74 (2017).
  * Sun  _et al._ [2015b] Y. Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan, Phys. Rev. B 92, 161107 (2015b).
  * Xia  _et al._ [2019] B. Xia, R. Wang, Z. Chen, Y. Zhao, and H. Xu, Phys. Rev. Lett. 123, 065501 (2019).
  * Xu  _et al._ [2017] S.-Y. Xu, N. Alidoust, G. Chang, H. Lu, B. Singh, I. Belopolski, D. S. Sanchez, X. Zhang, G. Bian, H. Zheng, M.-A. Husanu, Y. Bian, S.-M. Huang, C.-H. Hsu, T.-R. Chang, H.-T. Jeng, A. Bansil, T. Neupert, V. N. Strocov, H. Lin, S. Jia, and M. Z. Hasan, Sci. Adv. 3, e1603266 (2017).
  * Zhu  _et al._ [2018] H. Zhu, J. Yi, M.-Y. Li, J. Xiao, L. Zhang, C.-W. Yang, R. A. Kaindl, L.-J. Li, Y. Wang, and X. Zhang, Science 359, 579 (2018).
  * Wang  _et al._ [2021a] J. Wang, H. Yuan, M. Kuang, T. Yang, Z.-M. Yu, Z. Zhang, and X. Wang, Phys. Rev. B 104, L041107 (2021a).
  * Litvinchuk and Valakh [2020] A. P. Litvinchuk and M. Y. Valakh, J. Condens. Matter Phys. 32, 445401 (2020).
  * Li  _et al._ [2019] J. Li, L. Wang, J. Liu, R. Li, Z. Zhang, and X.-Q. Chen, arXiv preprint arXiv:1907.08547 (2019).
  * Miao  _et al._ [2018] H. Miao, T. Zhang, L. Wang, D. Meyers, A. Said, Y. Wang, Y. Shi, H. Weng, Z. Fang, and M. Dean, Phys. Rev. Lett. 121, 035302 (2018).
  * Liu  _et al._ [2021] Q.-B. Liu, Z. Wang, and H.-H. Fu, Phys. Rev. B 103, L161303 (2021).
  * Wang  _et al._ [2021b] X. Wang, F. Zhou, T. Yang, M. Kuang, Z.-M. Yu, and G. Zhang, Phys. Rev. B 104, L041104 (2021b).
  * Xie  _et al._ [2021] C. Xie, Y. Liu, Z. Zhang, F. Zhou, T. Yang, M. Kuang, X. Wang, and G. Zhang, Phys. Rev. B 104, 045148 (2021).
  * Liu  _et al._ [2020] Y. Liu, X. Chen, and Y. Xu, Adv. Funct. Mater. 30, 1904784 (2020).
  * Zhong  _et al._ [2021] M. Zhong, Y. Liu, F. Zhou, M. Kuang, T. Yang, X. Wang, and G. Zhang, Phys. Rev. B 104, 085118 (2021).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Tran and Blaha [2009] F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009).
  * McCrae  _et al._ [1997] J. McCrae, R. Hengehold, Y. Yeo, M. Ohmer, and P. Schunemann, Appl. Phys. Lett. 70, 455 (1997).
  * Akimchenko  _et al._ [1973] I. Akimchenko, V. Ivanov, and A. Borshchevsky, Sov. Phys. Semiconduct. 7, 309 (1973).
  * Bai  _et al._ [2005] L. Bai, C. Xu, P. Schunemann, K. Nagashio, R. Feigelson, and N. Giles, J. Phys. Condens. Matter 17, 549 (2005).
  * Blaha  _et al._ [2001] P. Blaha, K. Schwarz, G. K. Madsen, D. Kvasnicka, J. Luitz, _et al._ , An augmented plane wave+ local orbitals program for calculating crystal properties 60 (2001).
  * Schwarz and Blaha [2003] K. Schwarz and P. Blaha, Comput. Mater. Sci. 28, 259 (2003).
  * Schwarz [2003] K. Schwarz, J. Solid State Chem. 176, 319 (2003).
  * Madsen and Singh [2006] G. K. Madsen and D. J. Singh, Comput. Phys. Commun. 175, 67 (2006).
  * Madsen  _et al._ [2018] G. K. Madsen, J. Carrete, and M. J. Verstraete, Comput. Phys. Commun. 231, 140 (2018).
  * Kresse and Hafner [1993] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Kresse and Joubert [1999] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Togo and Tanaka [2015] A. Togo and I. Tanaka, Scr. Mater. 108, 1 (2015).
  * Singh  _et al._ [2018] S. Singh, Q. Wu, C. Yue, A. H. Romero, and A. A. Soluyanov, Phys. Rev. Mater. 2, 114204 (2018).
  * Li  _et al._ [2016] W. Li, J. Carrete, G. K. Madsen, and N. Mingo, Phys. Rev. B 93, 205203 (2016).
  * Yu and Hong [2021] X. Yu and J. Hong, J. Mater. Chem. 9, 12420 (2021).
  * Singh [2010a] D. J. Singh, Phys. Rev. B 81, 195217 (2010a).
  * Parker and Singh [2010] D. Parker and D. J. Singh, Phys. Rev. B 82, 035204 (2010).
  * Wang  _et al._ [2011] H. Wang, Y. Pei, A. D. LaLonde, and G. J. Snyder, Adv. Mater. 23, 1366 (2011).
  * Sun and Singh [2016a] J. Sun and D. J. Singh, Phys. Rev. Appl. 5, 024006 (2016a).
  * Sun and Singh [2016b] J. Sun and D. J. Singh, APL Mater. 4, 104803 (2016b).
  * He and Tritt [2017] J. He and T. M. Tritt, Science 357, eaak9997 (2017).
  * Irkhin and Irkhin [2007] V. Y. Irkhin and Y. P. Irkhin, (Cambridge Int Science Publishing, 2007).
  * Dehkordi  _et al._ [2015] A. M. Dehkordi, M. Zebarjadi, J. He, and T. M. Tritt, Mater. Sci. Eng. R Rep. 97, 1 (2015).
  * Ekuma  _et al._ [2012] C. E. Ekuma, D. J. Singh, J. Moreno, and M. Jarrell, Phys. Rev. B 85, 085205 (2012).
  * Singh [2010b] D. J. Singh, Funct. Mater. Lett. 3, 223 (2010b).
  * Spitzer [1970] D. Spitzer, J. Phys. Chem. Solids 31, 19 (1970).