An ab-initio study of nodal-arcs, axial strain’s effect on nodal-lines & Weyl nodes and Weyl-contributed Seebeck coefficient in TaAs class of Weyl semimetals.

Inspired by our previous studies on TaAs & TaPPandey and Pandey (2023a), there is a strong interest in investigating the presence of nodal-arcs in NbAs & NbP. Our focus is on analyzing the band characteristics of the topmost valence band (TVB) and bottommost conduction band (BCB) of NbAs & NbP along the directions $\Gamma$-N-Z-P-X-$\Gamma$-P-N in their electronic structure. Our analysis reveals that these states are primarily influenced by Nb-4d and As-4p/P-3p orbitals. At the atomic level, Nb-4d orbital possess higher energy when compared with the energy of As-4p/P-3p. Thus, it is typically expected that the BCB (TVB) must have higher (lower) contributions from Nb-4d than that from As-4p/P-3p across the BZ. However, deviations from this behaviour are observed in certain regions of the BZ, where BCB (TVB) is found to have higher (lower) contributions from As-4p/P-3p than that from Nb-4d orbital. This leads to what is termed as intraband dp inversion (IDPI)Pandey and Pandey (2023a). The IDPI observed along the $N-Z$ high symmetric directions of NbAs & NbP is shown in Fig. S1 of supplementary materials. It is well known that such bands-inversion generally leads to non-trivial bands-touchings. Further analysis in this direction reveal that NbAs & NbP possess a pair of nodal-lines on each of its mirror planes ($k_{x}$=0 & $k_{y}$=0 planes). Moreover, examination shows that IDPI predominantly occurs within these nodal-lines when compared with the regions outside these rings, highlighting its crucial role in their formation.

Our previous workPandey and Pandey (2023a) reveal that the four nodal-lines in TaAs & TaP can be generated from a single nodal-arc by applying on it the crystal symmetry operations associated with these materials. Based on this result, the nodal-arc degeneracy was reported to be associated with the irreducible parts of the BZ of these WSMs and thus, considered as fundamental degeneracy. As NbAs & NbP share similarities with TaAs & TaP in terms of crystal symmetriesArmitage et al. (2018), it is anticipated that each nodal-line of NbAs & NbP can be further devided into nodal-arcs. These arcs are supposed to be related to each other through crystal symmetries including the time-reversal symmetry. In that case, one of the arcs can be considered as arising from fundamental degeneracy phenomena, while the others can be termed as derived arcs. This expectation is supported by energy analysis of nodal-line in NbAs & NbP as shown in Fig. 2(a), revealing symmetric energy distribution about $k_{z}$=0 plane. Thus, the periphery of the shaded region can be considered as nodal-arc. This arc can generate all the four nodal-lines upon the application of crystal symmetries as discussed in our previous workPandey and Pandey (2023b). Moving further, the comparison of energy spread between NbAs & NbP indicates a higher spread in NbP ($\sim$215 meV) as compared to NbAs ($\sim$157 meV). The relevant graphs are shown in Fig. 2(b). The nodal-arcs in TaAs & TaP were reported to be extended beyond the boundaries of first BZPandey and Pandey (2023a). In this direction, investigation in NbAs & NbP reveals a similar results. The extended nodal-arcs of NbAs & NbP beyond the boundaries of the first BZ is shown in Fig. S2 of supplementary materials. The figure also shows the degeneracy created in the first BZ when the extended regions of the nodal-arcs are mapped inside the first BZ.

Another well-established aspect about TaAs & TaP is that the strength of hybridization among the atomic orbitals play an important role in the occurrence of IDPI and the formation of nodal-linesPandey and Pandey (2023b). As the hybridization-strength depends on the inter-atomic distances, changing the lattice parameters was used to probe the hybridization-strength among these materials. In this direction, the study has been carried out to explore the sensitivity of IDPI in TVB & BCB of NbAs & NbP to the variation in lattice parameters. The obtained results are shown in Fig. 1(a) (Fig. 1(b)) corresponding to NbAs (NbP). It is found that spatial-extent of IDPI in the bands increases with the decrease in magnitude of lattice parameters. The size of nodal-lines also shows exactly similar variations with the change in lattice parameters of these WSMs. Corresponding plots are shown in Fig. 1(c) (Fig. 1(d)) for NbAs (NbP). A comparative analysis of change in the area of nodal-lines with the variation in primitive unit cell volume of NbAs & NbP is presented in Fig. 1(e). These observations mimics to the one observed in TaAs & TaPPandey and Pandey (2023b). This establishes the crucial role of hybridization of atomic orbitals in the formation of nodal-lines in NbAs & NbP.

In the presence of SOC, the nodal-arcs degeneracy in NbAs & NbP gets destroyed. The magnitude of energy-gap created along these arcs varies along the $k_{y}$ direction, showing a different trend when compared to the case in TaAs & TaP. Unlike TaAs & TaPPandey and Pandey (2023a), where the energy-gap initially decreases, reaching a minimum non-zero value before increasing again to a maximum value as $k_{y}$ increasesPandey and Pandey (2023b), in NbAs & NbP, it first increases to a local-maximum value (NbAs: 15.2 meV & NbP: 24.2 meV) and then decreases to a non-zero minimum. Subsequently, it rises to a maximum value (NbAs: 46.9 meV & NbP: 38.9 meV) at the end of the nodal-arc along the $k_{y}$ direction. Comparing these results with TaAs & TaP, it is found that the value of maximum energy gap created due to SOC follows the trend: TaAs (162.0 meV)Pandey and Pandey (2023a)$>$TaP (143.7 meV)Pandey and Pandey (2023a)$>$NbAs (46.9 meV)$>$NbP (38.9 meV). It is important to note that this trend is in accordance to the order of SOC-strength associated with these WSMs. The quatitative comparision of this feature in NbAs & NbP is shown in Fig. 3. Before moving further, it is important to highlight that the nodal-arcs in NbAs & NbP, as already mentioned, are extended beyond the boundaries of the first BZ. Thus, in further discussion, the term nodal-arc will refer to the one obtained when the extended portion is also mapped into the first BZ. Moving further, although SOC leads to the vanishing of nodal-arc, it causes the formation of two pairs of Weyl nodes close to the plane containig the nodal-arc. The two nodes forming each Weyl pair are found to be situated on either sides of the plane containing the nodal-arc and also perpendicular to it. Out of the two pairs, one is positioned on either side of the middle portion of the nodal-arc (called as W2 points), while the other pair is situated at the junction point of the two arcs forming a nodal-line (referred as W1 points). This second pair shares contributions from both arcs, resulting in a total of three pairs of Weyl nodes per nodal-line and 12 pairs of Weyl nodes in the entire BZ. These findings closely parallel to those observed in TaAs & TaPPandey and Pandey (2023a). Detailed figures illustrating these features for NbAs & NbP can be found in supplementary materials (Fig. S3). Furthermore, the Fermi-level-scaled energy of representative W1 & W2 points in NbAs & NbP are mentioned in table 1. Grassano et.al. have previously investigated the energy of these nodes in TaAs class of WSMsGrassano et al. (2020). For NbAs (NbP) they reported the energy of W1 point as -33 (-56) meV whereas for W2, it is 4 (26) meV with respect to the Fermi energy. However, in present analysis, the energy of W1 points is found to be $\sim$-40 ($\sim$-68) meV while for W2 it is obtained as $\sim$13 ($\sim$8) meV with respect to Fermi energy in NbAs (NbP). The differences in the results of present work with that of Grassano et.al. is probably because they have used k-mesh gridding method to search the Weyl nodes while in the present analysis, the highly efficient PY-Nodes code is employed for searching these bands-touching points. On the top of that, Grassano et.al. have used the pseudopotential methods as implemented in the Quantum expressoGiannozzi et al. (2009) package while in the present analysis, full-potential Linearized augmented-plane-wave method (FP-LAPW) method as implemented in WIEN2k package is used. This FP-LAPW method is considered to be the most accurate one in the research community. Thus, the results of the present analysis can be considered as more reliable and accurate.

The next point of discussion is whether the strength of SOC affects the number of Weyl-nodes into which these arcs evolve. As the SOC Hamiltonian ($H_{so}$) is inversely proportional to the $c^{2}$Blaha et al. (2001), value of $c$ is decreased in our calculations to increase the strength of SOC. The method is similar to the one as followed in our previous workPandey and Pandey (2023a). The SOC-strength per Nb atom is calculated by finding the difference between the ground state energy obtained before and after adding the SOC-effect in our calculations. The obtained results are shown in table 1. It is seen that as the value of $c$ decreases from $\sim$137 to $\sim$123, the strength of SOC in NbAs (NbP) increases from $\sim$0.08 ($\sim$0.06) eV to $\sim$0.11 ($\sim$0.09) eV. This increase is found to be extremely small when compared with increase in SOC strength in TaAs ($\sim$0.7 eV) & TaP ($\sim$3.0 eV) for the same change in value of $c$Pandey and Pandey (2023a). Similar to the case of TaAs & TaP, the evolution of nodal-arc into two pairs of Weyl-nodes in NbAs & NbP is found to be robust against the given increase in the magnitude of SOC-strength. That is, the nodes forming a Weyl pair are still found to be situated on either sides of the plane containing the nodal-arc and also at equidistant from the plane. The coordinates and energy of one node from each pair is mentioned in table 1. The coordinated of obtained nodes are also found to be robust to the given increase in SOC strength of NbAs & NbP, which is similar to the case as observed in TaAs & TaPPandey and Pandey (2023a). As the increase in the SOC strength in NbAs & NbP is extremenly small, it is expected that the energy of Weyl nodes in these semimetals will be less sensitive when compared to that of TaAs & TaP. The obtained results as mentioned in table 1 reflects these aspects.

Understanding the effects of strain along different crystallographic directions is crucial for tailoring the electronic properties of TaAs class of WSMs. The study of impact of strain on nodal lines is carried-out under two different cases: (i) strain created along $a$ direction and (ii) strain created along the $c$ direction. Fig. S7 illustrates the results for nodal lines situated in the $k_{x}$=0 plane of NbAs. For clarity, we divide a nodal line into three parts: a larger portion, a smaller portion, and an extended portion forming beyond the boundaries of the first BZ, as indicated in the figure. It is found that strain (either tensile or compressive) along the $a$ or $c$ direction has no effect on the smaller portion of the rings. However, it significantly impacts the larger and extended portions. Increasing strain magnitude along the $a$ direction enlarges these portions, while the opposite trend occurs with strain along the $c$ direction. Notably, a 3% tensile strain along $a$ (or $\geq$2% compressive strain along $c$) leads to the partial merging of nodal lines 1 and 2 in the extended BZ. This merging is schematically depicted in Fig. S4 of supplementary materials. Similar observations hold for nodal lines in the $k_{y}$=0 plane. Also, nodal-lines in the $k_{x}$=0 plane of NbP, TaAs & TaP are also found to be sensitive to the creation of such strain (see Figs. S5, S6 & S7 of supplementary materials). It is important to note that partial merging of rings at specific strain values (in between -3% to 3%) is observed only in NbAs & NbP but not in TaAs & TaP. This difference may stem from the larger area covered by nodal lines in NbAs & NbP at experimental lattice parameters when compared with that of TaAs & TaP. Consequently, further ring enlargement upon creating a strain in NbAs & NbP brings the rings closer in the extended BZ, resulting in their partial merging. This discussion establishes the effect of lattice strain on nodal lines in TaAs class of WSMs (under no SOC). Further discussion will investigate the impact of SOC on these strained nodal lines.

Fig. S10(a) (S10(b)) depict the influence of SOC on nodal lines in the $k_{x}$=0 plane of NbAs under compressive (tensile) strain along the $a$ direction. In Fig. S10(c) (S10(d)), the effect of SOC on these rings is shown for compressive (tensile) strain along the $c$ direction. It is seen in these figures that some of the nodes (clustered at discrete places of the BZ) are encircled in blue dashed-circles. These nodes are forming at the points in the $k_{x}$=0 plane, where end portions of nodal-arcs existed in the absence of SOC. One must not confuse these nodes (encircled in blue-dashed circles) with the nodal-lines, because nodal-lines generally form due to the crossing of two bands to form a closed loop-like structure. However, these nodes (encircled in blue-dashed circles) do not form a closed loop. Instead, they are discrete bands-degenerate points forming in the $k_{x}$=0 plane. Apart from these nodes, there are also other node-points that are created away from the $k_{x}$=0 plane as shown in Fig. S10. One must note that each red dots that are not encircled in a dashed-circle denotes two nodes, which are getting projected at the same point when shown in 2D plots. Next, we shall discuss how the number of nodes get affected on creating strain along $a$ direction. One must keep in mind that for the experimental lattice parameter, nodal-lines of NbAs in $k_{x}$=0 plane evolve into 6 pairs of nodes off to the $k_{x}$=0 plane, as discussed before.

Fig. S10 (a) shows that compressive strain of 1% or 2% or 3% along $a$ direction has no effect on the number of nodes created in $k_{x}\neq$0 plane. However, with increasing compressive strain, new nodes are found to get created within the $k_{x}$=0 plane, particularly where the ends of nodal-arcs existed under no-SOC case. Now moving to the effect of tensile strain along $a$ direction (shown in Fig. S10 (b)), it is observed that the 4 pairs of nodes that used to be formed in the $k_{z}\neq$0 plane under no-strain, get vanished upon the application of 1% tensile strain. However, the two pairs of nodes are still generated in the $k_{z}$=0 plane as used to be in the no-strain case. With further increase in the strain, the number of nodes in $k_{z}\neq$0 plane starts increasing along with maintaining the number of nodes in $k_{z}$=0 plane as was in no-strain case. The total number of nodes generated from the nodal-lines in the $k_{x}$=0 plane under SOC-effect along with 2% (3%) strain in $a$ direction is found to be 20 (28). Regarding the impact of compressive strain on node energy, it is observed in Fig. S10 (a) that nodes in the $k_{z}$=0 plane become increasingly negative in energy relative to the Fermi energy as the magnitude of compressive strain along the $a$ direction increases. Similar behavior is observed for nodes situated within the $k_{x}$=0 plane (encircled in blue-dashed circles). Conversely, nodes not encircled and located in $k_{z}\neq$0 planes approach the Fermi energy from the positive energy direction with increasing compressive strain magnitude. Moreover, the energy of nodes with respect to the Fermi-energy, obtained after tensile strain are also found to be sensitive to the magnitude of strain. Nextly, the effect of strain along $c$ direction on the number and energy of nodes is discussed.

Fig. S10(c) illustrates the impact of compressive strain (1%, 2%, and 3%) along the $c$ direction on the number of nodes in NbAs. The strain of 1% have no effect on the number of nodes generated from nodal-lines in the $k_{x}$=0 plane. However, at 2% strain, the number increases to 28. Notably, 2% compressive strain along $c$ has no effect on the number of nodes in the $k_{z}$=0 plane. At 3% strain, the extra nodes vanish, reverting to the 1% compressive strain scenario. Additionally, the effect of tensile strain along the $c$ direction is discussed in Fig. S10(d). Strain percentages of 1%, 2%, and 3% have no effect on the number of nodes generated off the $k_{x}$=0 plane. However, the number of nodes in the $k_{x}$=0 plane increases with higher tensile strain percentages, clustering at the regions where the ends of nodal-arcs existed when SOC was ignored. Similar to strain along the $a$ direction, increasing compressive or tensile strain along $c$ direction leads to nodes in NbAs spreading further apart in energy and with respect to the Fermi level, as depicted in Figs. S10(c) and S10(d).

It is important to note here that as the nodal-lines situated in $k_{y}$=0 plane are related with crystal-symmetry to those situated in the $k_{x}$=0 plane, similar observations as discussed above will be obtained for the effect of SOC on the nodal-lines in $k_{y}$=0 plane of NbAs. Moreover, similar effect of lattice strain along $a$ and $c$ direction of unit-cell is obtained on the nodal-lines and Weyl nodes of other family of TaAs class of WSMs (i.e., for TaAs, TaP & NbP). Corresponding results are shown in supplementary materials (Figs. S8, S9 & S10). However, as already mentioned, for strain of -3% to 3% along $a$ or $c$ direction do not lead to the partial merging of two nodal-lines in a given plane for TaAs & TaP.

For studying the Seebeck coefficient ($S$) behaviour of any given material, it is important to know the physical parameters upon which it depends. Within the Kubo linear-response formalism, expression for $S$ is defined asTomczak et al. (2010); Oudovenko et al. (2006)-

$$S^{\alpha\alpha\textquoteright}=-\frac{k_{B}}{|e|}\frac{K_{1}^{\alpha\alpha\textquoteright}}{K_{0}^{\alpha\alpha\textquoteright}}$$

(1)

where $K_{n}^{\alpha\alpha\textquoteright}$ are called the kinetic coefficients. Their expression is given bySihi and Pandey (2023)-

$$K_{n}^{\alpha\alpha\textquoteright}=N_{sp}\pi\hbar\int d\omega(\beta\omega)^{n}f(\omega)f(-\omega)\Gamma^{\alpha\alpha\textquoteright}(\omega,\omega)$$

(2)

where, $N_{sp}$ and $f(\omega)$ are the spin factor and Fermi function, respectively. $\hbar$ denotes the reduced Planck’s constantSihi and Pandey (2023). Here, $\Gamma^{\alpha\alpha\textquoteright}(\omega_{1},\omega_{2})$ is known as transport distribution function and is defined as-

$$\Gamma^{\alpha\alpha\textquoteright}(\omega_{1},\omega_{2})=\frac{1}{V}\sum_{k}Tr\{v_{n}^{\alpha}(\textbf{{k}})A(\textbf{{k}},\omega_{1})v_{n}^{\alpha\textquoteright}(\textbf{{k}})A(\textbf{{k}},\omega_{2})\}$$

(3)

The term $A(\textbf{{k}},\omega)$, known as spectral function in real $\omega$, is expressed asImada et al. (1998)-

$$A(\textbf{{k}},\omega)=-\frac{1}{\pi}\frac{Im\Sigma(\omega)}{[\omega-\varepsilon_{\textbf{{k}}}^{0}-Re\Sigma(\omega)]^{2}+[Im\Sigma(\omega)]^{2}}$$

(4)

Here, $\omega$ is the energy with respect to the chemical potential ($\mu$) where the transport is supposed to be calculated. In the present work, the $S$ is calculated by the setting $\mu$ at the energy of Weyl node. Nextly, as the topological and transport analysis in the present work is carried out at DFT level, a slight modification in the above formalism is required to use it for studying the behaviour of $S$ obtained from Weyl cones of TaAs class of WSMs. In the Eq. 4, the $Re\Sigma(\omega)$ must be taken as 0 while the value of $Im\Sigma(\omega)$ should be taken to be extremely small. As TRACK codeSihi and Pandey (2023) provides such options, we have used it in our present study. The value of $Im\Sigma(\omega)$ is taken as 0.5 meV in our calculations. Before discussing the behaviour of $S$ obtained from Weyl cones, we are motivated in exploring that under what conditions will a Weyl cone contribute to a non-zero value of $S^{\alpha\alpha\textquoteright}$ to a given WSM. This aspect is discussed further.

To understand the contribution of Weyl cone to the $S$ of a given materials, it becomes extremely important to analyse the minute aspects of the Weyl cone. In this regard, the two most important aspects of the cone are (i) tilt of the Weyl cone and (ii) asymmetry of the Weyl cone. The term ‘tilt of the cone’  means that how much angle does the axis of the Weyl cone makes with the energy axis. Thus, if the axis of the Weyl cone is parallel with the energy axis, the Weyl cone is said to be not at all tilted (see Figs. 6 (b) and (c)). Moving further, in this work the term ‘symmetric cone’  stands for the one which possess either the inversion symmetry with respect to the Weyl node or mirror symmetry with respect to plane perpendular to energy axis and passing through the Weyl node. To make it more clear, let us explain the meaning of asymmetry of the Weyl cone using Fig. 6. If the cone is symmetric, then on applying inversion symmetry operation (with respect to Weyl node) or mirror symmetry operation (with respect to plane perpendular to energy axis and passing through Weyl node) on the red cone, full blue cone can be obtained. In the other case, the cone is said to be asymmetric. In addition to this, in the present work the term ideal cone stands for the case in which the Weyl cone is symmetric and not tilted. In further discussions, the condition under which the Weyl cone will have non-zero contribution to $S$ is explained.

Firstly, consider the case of ideal cone as shown in Fig. 6 (b), the non-tilted cone. It is well known that the velocity $v_{n}^{\alpha}(\textbf{{k}})$ is defined as (1/$\hbar$)($\partial\varepsilon_{n}(\textbf{{k}})/\partial k_{\alpha}$)Madsen and Singh (2006). As the red & blue cones are symmetric, corresponding to magnitude of velocity of the red cone at each k-point, there will be a k-point where the blue cone will have the same magnitude of velocity. If the Weyl cone is inversion symmetric about the Weyl node, these two k-points will be different. However, if the Weyl cone possess mirror symmetry about the energy plane passing through the energy of Weyl nodes, the red & blue cone will have same magnitude of velocity at the same k-point. Moreover, corresponding to these k-points, the value of $A_{n}(\textbf{{k}},\omega)$ for both the bands forming the Weyl cone will also be same. This is because, as the cone is symmetric at these k-points, they will possess same magnitude of $[\omega-\varepsilon_{\textbf{{k}}}^{0}]^{2}$. Moving further, due to such behaviour of $v_{n}^{\alpha}(\textbf{{k}})$ & $A_{n}(\textbf{{k}},\omega)$, the quantity $\Gamma^{\alpha\alpha\textquoteright}(\omega,\omega)$ will also become symmetric with respect to the energy of the Weyl node. As can be seen in Fig. 6 (a), at any temperature, the term $\beta\omega f(\omega)f(-\omega)$ is an odd function of $\omega$. Due to this, the value of $K_{1}^{\alpha\alpha\textquoteright}$, as mentioned in equation 1, will vanish out at any given temperature ($T$). Hence, an ideal Weyl cone will have zero contribution to the $S$ of a material possessing it. Now moving to the case of symmetric Weyl cone having non-zero tilt as shown in Fig. 6 (b). The discussion for such cone will be exactly similar to case of an ideal Weyl cone as discussed above. The only difference is that it will have different magnitude of velocity at a given k-point when compared to case when the cone was not tilted. This is because the tilt in the cone with respect to the energy axis modifies the value of velocity at each k-points. Thus, in this case also there will be a compensating effect from the red & the blue cones resulting into zero contribution to the value of $S$ at any given temperature. Now we are left with the case of asymmetric Weyl cone which is discussed further.

For the asymmetric Weyl cone with no tilt, there will be some k-points where corresponding to the magnitude of $v_{n}^{\alpha}(\textbf{{k}})$ of red cone, there will not be a k-point where the magnitude of $v_{n}^{\alpha}(\textbf{{k}})$ of blue cone will be same with same magnitude of $A_{n}(\textbf{{k}},\omega)$ for both the cones. Thus, for such a asymmetric Weyl cone there will be no compensating effect from the red & the blue cone. This will result in a non-zero contribution to the value of $S$ at any given temperature. Moving further, for such an asymmetric Weyl cone, tilting of the cone will change the profile of temperature dependent value of $S^{\alpha\alpha}$. The reason behind this is that tilting of the cone will modify the value of $v_{n}^{\alpha}(\textbf{{k}})$ and due to the non-compensating effect from red & blue cones, the value of $S^{\alpha\alpha}$ will get affected. From these discussions, a general conclusion that can be drawn is that the basic requirements for a Weyl cone to have finite contribution to the $S$ of WSMs is its asymmetric nature. At the top of that, tilting of the cone may change the magnitude of contribution to the $S$ at any given T. In the further discussion, this conclusion is validated over the TaAs class of WSMs by analysing their Weyl cones and calculating the contribution of such cones to $S$ of these semimetals.

It is well-known that TaAs class of WSMs contain 24 Weyl nodes in their full BZ. These nodes are arranged in a particular fashion: 8 Weyl nodes in the $k_{z}$=0 plane and 8-8 nodes are also situated in a similar fashion on the $k_{z}\sim$0.58 & $k_{z}\sim$-0.58 planes of these WSMs. In the present analysis of $S$, we have separately calculated $S$ for 3 different nodes for each semimetals, taking one from each plane. Also we have examined the features of each of these Weyl cones. The results corresponding to Weyl node (0.283, 0.020, 0.590) in case of TaAs is shown in Fig. 7. Let us first discuss the features of Weyl cone around this node (shown in Fig. 7(a)). The cone is drawn by taking the k-points sampled in $k_{x}$-$k_{y}$ plane. In the preliminary analysis of the cone, it is seen that the axis of the cone is tilted with respect to the energy axis. Further analysis reveal that the cone is assymmetric for symmetry operations: (i) inversion symmetry with respect to Weyl node and (ii) mirror symmetry with respect to energy plane corresponding to Weyl energy. Corresponding plots are shown in Figs. 7(b) and (c), respectively. Thus, according to our previous discussion, the cone must have non-zero contribution to temperature dependent $S$. It is important to note here that such an asymmetry is found corresponding to all the three cones (in the above mentioned three planes) and also in such cones for other members of TaAs class of WSMs. The relevant plots are shown in supplementary materials (Figs. S11-S21). Now, we are interested in studying the magnitude of $S$ obtained from the above mentioned Weyl cone in TaAs, which is discussed below.

Generally, a Weyl cone is visualised as a 3-dimensional figure in the reciprocal space where the x-y plane consists of states and the z-axis contains the energy of the two bands forming the cone. For studying the Weyl dominated transport, it is needed to consider the excitations only from the states close to the Weyl nodes. However, one must keep in mind that passing through a Weyl-node, infinite number of 2-dimensional plane can be constructed which will produce the Weyl cones (as all these planes will contain the Weyl node). Thus, for calculating the Weyl-contributed transport coefficients such as $S$, it is necessary to consider the states contained in all these planes. Hence, in the present work, a 3-dimensional k-points sampling around the Weyl nodes is performed to calculate the $S$. The results obtained for Weyl-contributed $S$ to the TaAs form the Weyl node (0.283, 0.020, 0.590) is shown in Fig. 7(d). $S$ obtained corresponding to the Weyl cone is found to be anisotropic. That is, the $S^{xx}$, $S^{yy}$ and $S^{zz}$ components are all differently contributed from the Weyl cone. This occurs because in general, the $v_{n}^{x}$$\neq$$v_{n}^{y}$$\neq$$v_{n}^{z}$ for the states comprising the two bands forming Weyl cones in these WSMs. Another important feature that can be seen is that the magnitude of each components of $S$ (i.e., $S^{xx}$, $S^{yy}$ & $S^{zz}$) changes in a non-monotonous fashion with the rise in the value of $T$. It is found that with the rise in temperature, each components attains a peak at some value of $T$. With the further increase in the magnitude of temperature, the value of each components of $S$ is found to decrease. The observation of such behavious of each components of $S$ is found to be common for the $S$ contributed from the Weyl cones corresponding to the nodes in $k_{z}$=-0.590 and $k_{z}$=0 planes in TaAs (see Figs. S11 & S12). Furthermore, similar observations have been found for the Weyl cones in case of other members of TaAs class of WSMs, i.e., NbAs, NbP & TaP. The relevant plots are shown in supplementary material (Figs. S13-S21). With this analysis, the fact has been successfully validated that the assymmetric nature of the Weyl cone is necessary for it to affect the $S$ of any WSM.