Thermoelectric transport and phonon drag in Weyl semimetal monochalcogenides

The topological effect in thermoelectric transport is an important research aspect in semimetals, but whether an anomalous Nernst effect exists in nonmagnetic topological semimetals in a finite external field is still under debate. To demonstrate how to discern the topological effect in nonmagnetic topological semimetals, we present a comprehensive study on the magnetothermoelectric properties for four Weyl semimetals: TaAs, TaP, NbAs, and NbP. We observe large magneto-Seebeck and Nernst signals at intermediate temperatures, which are attributed to a multiband ordinary contribution and an inelastic, phonon-drag effect, while the latter is ignored in previous studies. This phonon-drag effect also induces an unusual, prominent temperature and field dependence of quantum oscillations in thermoelectric transport signals. On the other hand, only signatures of a relatively small anomalous thermoelectric effect are found in TaAs compared with the ordinary effect.

Figure 1

(a)–(d) Thermoelectrical signals $S_{ij}$ of TaAs, TaP, NbAs, and NbP at 2 K, respectively. Note the different scales. All these profiles feature strong QOs. Inset in (c): Zoom-in of $S_{xy}$ of NbAs under small magnetic field, showing a sharp steplike behavior and QOs with ultralow frequency.

Figure 2

(a) $S_{xx}$ and (b) $S_{xy}$ for the TaAs family at selected temperatures. The maximal values are in the order of $1\mathrm{mV}/\mathrm{K}$. Note the saturating profile of $S_{xx}$ and $S_{xy}$ under high magnetic field.

Figure 3

(a) $S_{xy}$ for the TaAs family at about 40 K. For NbAs there is 10 times magnification. (b) The off-diagonal component of the thermoelectric tensor ${\alpha }_{xy}$ for the TaAs family at 40 K. All curves approach zero in high magnetic field. Inset: corresponding ${\alpha }_{xy}B$. The horizontal dashed lines are a guide to the eye. (c) A simulation of $S_{xy}$ using the two-band model. Here $\mu$ is set to be 40 and 10 ${\mathrm{m}}^2/\mathrm{V}\mathrm{s}$ for electrons and holes, respectively. Total carrier concentration ($n_e+n_h$) is set to be ${10}^{19}{\mathrm{cm}}^{-3}$. ${\alpha }_0$ is set to be equal for electrons and holes. The dashed line shows the modified $S_{xy}$ in high field when a third minority carrier is considered. Here we assume the third carrier having ${10}^{18}{\mathrm{cm}}^{-3}$ carrier density and $0.03{\mathrm{m}}^2/\mathrm{V}\mathrm{s}$ mobility.

Figure 4

(a) ${\alpha }_{xy}B/T$ at selected temperatures for TaAs. Curves are shifted vertically for clarity. A linear slope with field as depicted by the dashed line is observed above 1 T, indicating the existence of a small but sizable anomalous contribution. (b) Extracted anomalous thermoelectric Hall conductivity ${\alpha }_{xy}^A/T$. The magenta line is the entropy density function $s_{\mathit{k}}$ at 1.5 meV, divided by temperature. We notice that both plots reach maximal in the vicinity of a characteristic temperature $0.55{\epsilon }_F/k_B\simeq 10$ K. Inset: $s_{\mathit{k},\epsilon =1.5\mathrm{meV}}$ at 2 and 15 K, and a sketch of $W_2$ nodes which are $1.5\mathrm{meV}$ above the Fermi level of TaAs. (c) Thermopower of TaAs when magnetic field is almost along the $a$ axis (about $2^{\circ }$) at 3.5 K. The longitudinal part $S_{xx}$ varies little with field, while the transverse component $S_{xy}$ has a large, plateaulike feature. The red dashed line shows the expected $S_{xy}$ inferred from $S_{xy}$ at $\theta ={90}^{\circ }$, assuming only the field component along $c$ contributes to transport signals. (d) Angle-dependent $S_{xy}$ at 3.5 K under a field of 0.5 and 1.5 T, respectively. Dashed lines are also the semiclassically expected $S_{xy}$.

Figure 5

(a) Zero-field Seebeck coefficient ($S_0$) for the TaAs family. Note the log temperature scale. Inset: zoom-in of $S_0$ for TaAs, showing a hump near 7 K. (b) Nernst coefficient divided by temperature ($\nu /T$). We adopt $\nu =S_{xy}/B$ at 0.5 T for TaAs and TaP while 0.1 T for Nb siblings. Vertical dashed lines indicate peak positions, being $\sim 5$, 15, 10, and 20 K for TaAs, TaP, NbAs, and NbP, respectively. Inset: Corresponding $\nu$ for TaAs and TaP, showing peaks at 25 and 35 K, respectively. (c) $-S_{xx}/T$ vs temperature, showing a similar profile to $\nu /T$ in (b). For comparison, we select the magnetic field at which the low-temperature $S_{xx}$ is saturated. The data for NbAs are magnified three times for clarity. Inset: a sketch for electron-phonon scattering N-process under a magnetic field. When colliding into a phonon with wave vector $\mathit{q}$, the Fermi vector $\mathit{k}$ changes its direction to ${\mathit{k}}^{\prime }$ while staying roughly on the Fermi sphere in the first Brillouin zone.

Figure 6

Low-temperature Nernst coefficients divided by temperature ($\nu /T$) in a variety of metals plotted vs the ratio of mobility to Fermi energy. The $\nu /T$ data for the TaAs family (magenta stars) in the low-temperature limit all lie close to the empirical line of $283\mu /E_F$. Black and blue data points are reproduced from Ref. [69].

Figure 7

(a) ${\tilde{\alpha }}_{xx}$ at 5 K for TaAs and corresponding ${\tilde{\sigma }}_{xx}\left|\beta \right|$ at 5 K (shifted for clarity). For brevity, the contribution from higher harmonics is not treated separately. Note the similar magnitudes and 1/4 phase shift. (b) ${\tilde{\alpha }}_{xy}$ at 5 K and ${\tilde{\sigma }}_{xy}\left|\beta \right|$ at 5 K for comparison. Unlike the case in (a), ${\tilde{\sigma }}_{xy}\left|\beta \right|$ is way smaller than ${\tilde{\alpha }}_{xy}$. Dashed line is $S_g^0\mu B{\tilde{\sigma }}_{xx}$ at 5 K, with $S_g^0$ taken to be 2.5 $\mu \mathrm{V}/\mathrm{K}$. (c) ${\alpha }_{xy}B/T$ at low temperatures. The height of the $N=1$ peak damps with decreasing temperature below 4 K. (d) Left scale: peak height for ${\alpha }_{xy}B/T$, plotted against temperature. The magenta curve is the peak height for ${\alpha }_{xx}B/T$ at 7.6 T, showing a monotonically decreasing feature.

Figure 8

${\alpha }_{xy}/T$ for TaAs at representative temperatures. The quantized value expected by Eq. (19) in EQL is by far smaller than the experimentally observed value.

Figure 9

$S_{xx}$ and $S_{xy}$ for TaP under high magnetic field. Dashed lines denote nonoscillatory backgrounds.