Transverse thermopower in Dirac and Weyl semimetals

Dirac semimetals (DSM) and Weyl semimetal (WSM) fall under the generic class of three-dimensional (3D) solids, which follow the relativistic energy-momentum relation ${\epsilon }_{\mathbf{k}}=\hslash v_F\left|\mathbf{k}\right|$ at low energies. Such a linear dispersion when regularized on a lattice can lead to remarkable properties such as the anomalous Hall effect, the presence of Fermi surface arcs, positive longitudinal magnetoconductance, and dynamic chiral magnetic effect. The last two properties arise due to the manifestation of chiral anomaly in these semimetals, which refers to the nonconservation of chiral charge in the presence of electromagnetic gauge fields. Here we propose the planar Nernst effect, or transverse thermopower, as another consequence of chiral anomaly, which should occur in both Dirac and Weyl semimetals. We analytically calculate the planar Nernst coefficient for DSMs (type I and type II) and also WSMs (type I and type II) using a quasiclassical Boltzmann formalism. The planar Nernst effect manifests in a configuration when the applied temperature gradient, magnetic field, and the measured voltage are all coplanar and is of distinct origin when compared to the anomalous and conventional Nernst effects. Our findings, specifically a 3D map of the planar Nernst coefficient in type-I Dirac semimetals (${\mathrm{Na}}_3\mathrm{Bi}$, ${\mathrm{Cd}}_3{\mathrm{As}}_2$, etc.) and type-II DSM (${\mathrm{PdTe}}_2$, VAI3, etc.), can be verified experimentally by an in situ 3D double-axis rotation extracting the full $4\pi$ solid angular dependence of the Nernst coefficient.

Figure 1

(a) Schematic illustration for the measurement of the planar Nernst coefficient in Dirac semimetals. A longitudinal temperature gradient $dT/dx$ produces a chiral anomaly-induced transverse electric field $E_y$ due to the coplanar component of the $\mathbf{B}$ field. As the magnetic field $\left(\mathbf{B}\right)$ is rotated along $\widehat{\theta }$ (polar angle) and $\widehat{\varphi }$ (azimuthal angle) directions, one can map the planar Nernst coefficient for a Dirac semimetal (note that the $B_z$ component in a DSM produces Weyl points, while the planar components of $\mathbf{B}$ produce the planar Nernst effect due to chiral anomaly). The 3D map of the planar Nernst coefficient can be verified experimentally by an in situ 3D double-axis rotation extracting the full $4\pi$ solid angular dependence [88]. (b) Mapping the planar Nernst coefficient ${\nu }^{\text{pl}}$ in a type-I Dirac semimetal, showing the angular dependence with respet to $\theta$ and $\varphi$, where $\theta$ is the polar angle and $\varphi$ is the azimuthal angle. The small plot on the left side shows the map of the anomalous Nernst coefficient for the same system.

Figure 2

(a) Density plot of the planar Nernst coefficient ${\nu }^{\text{pl}}$ for a type-I DSM. (b) Plot of ${\nu }^{\text{pl}}$ as a function of $\theta$ for various values of $\varphi$. We observe a peak in the Nernst coefficient away from $\theta =\pi /2$. The inset shows the plot of ${\nu }^{\text{pl}}$ as a function of $\varphi$ for various values of $\theta$, showing a $sin\varphi cos\varphi$ behavior. (c) Sketch of qualitative $\theta$ dependence of the anomalous and planar Nernst coefficient.

Figure 3

(a) Map of the full angular dependence of the planar Nernst coefficient ${\nu }^{\text{pl}}$ for a type-II DSM. (b) The planar Nernst coefficient ${\nu }^{\text{pl}}$ as a function of $\theta$ at various azimuthal angles, showing a double peaked behavior around $\theta =\pi /2$. The qualitative similarity with type-I DSM can be observed. The inset shows the $sin\varphi cos\varphi$ behavior with respect to the angle $\varphi$.

Figure 4

Top: Planar Nernst coefficient in inversion asymmetric type-II Weyl semimetal displaying the $cos\varphi sin\varphi$ feature. Bottom: Planar Nernst coefficient as a function of magnetic field. The linear in $B$ feature disappears here because as in the effects of tilts cancel out among all the nodes. We chose $t_x=t/2$, $m=2t$, $k_0=\pi /2$, $\gamma =2.5t$.