Antisymmetric Seebeck Effect in a Tilted Weyl Semimetal

Tilting the Weyl cone breaks the Lorentz invariance and enriches the Weyl physics. Here, we report the observation of a magnetic-field-antisymmetric Seebeck effect in a tilted Weyl semimetal, ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. Moreover, it is found that the Seebeck effect and the Nernst effect are antisymmetric in both the in-plane magnetic field and the magnetization. We attribute these exotic effects to the one-dimensional chiral anomaly and phase space correction due to the Berry curvature. The observation is further reproduced by a theoretical calculation, taking into account the orbital magnetization.

Figure 1

Basic transport properties of ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ crystal and thermoelectric measurement setup. (a) Temperature dependence of longitudinal resistivity. A kink is observed at the Curie temperature $T_c=175\mathrm{K}$. The inset is an optical image of the thermoelectric measurement setup. The scale bar is 1 mm. (b) Normalized longitudinal resistivity ${\rho }_{xx}$ (black) and Seebeck coefficient $S_{xx}$ (red) as a function of the perpendicular magnetic field at $T=20\mathrm{K}$. The positive parabolic MR implies a dominant classic mechanism, while a negative and almost linear Seebeck may be attributed to the magnon-drag effect. (c) Anomalous Hall effect (black) and anomalous Nernst effect (red) at $T=20\mathrm{K}$. (d) Diagram of thermoelectric measurement setup. Two thermocouples are attached at the edge of the sample. (e) Longitudinal thermoelectric signal $V_{xx}$ as a function of the temperature difference $\mathrm{\Delta }T$. A linear dependence verifies the validation of our thermoelectric measurements.

Figure 2

$B$- and $M$-antisymmetric Seebeck coefficient when $\mathbf{B}\perp \widehat{x}$. (a) $S_{xx}$ as a function of the magnetic field at different angles. Data are shifted for clarity. Traces at $\pm \theta$ diverge with increasing field and angle. The inset shows the field dependence of $S_{xx}$ at $\theta =70°$ at the low field region. Below $B_c$, the slope of the linear dependence shows opposite sign for $\pm M$. (b) $S_{xx}^{\mathrm{O}}$ as a function of $B_z$ at different angles. The inset depicts the measurement setup and the angle definition. (c) $S_{xx}^{\chi }$ as a function of $B_{\Vert }$. For positive (solid line) and negative (dashed line) magnetization, the slopes display opposite signs. (d) Angular dependence of $S_{xx}$ at $B=8\mathrm{T}$. There exists a pronounced asymmetry with respect to $\theta =0$. Because of contact misalignment, there is a component of $S_{yx}$. Since $S_{yx}$ is nearly symmetric in $\theta$ when $B\perp x$, as seen in Fig. S7 [46], it does not affect the extracted $S_{xx}^{\chi }$. It is evident that $S_{xx}^{\chi }$ in (e) is very close to that in Fig. S7(e) [46], which is extracted after removing the $S_{yx}$. (e) $S_{xx}^{\chi }$ versus $\theta$. $S_{xx}^{\chi }$ is obtained by $\theta$ antisymmetrizing the data in (d). Open circles are experimental data and the solid lines are best fits to $\sin \theta$.

Figure 3

$B$- and $M$-antisymmetric Nernst effect when $\mathbf{B}\perp \widehat{y}$. (a) Field dependence of Nernst signals at different angles. Data are shifted for clarity. Traces at $\pm \theta$ diverge with increasing field and angle. (b) $S_{yx}^{\mathrm{O}}$ versus $B_z$ at different angles. The inset depicts the rotation angle. (c) $S_{yx}^{\chi }$ versus $B_{\Vert }$. The slope of the field dependence changes sign with $M$. (d) Angular dependence of $S_{yx}$ at $B=8\mathrm{T}$. There exists a pronounced asymmetry with respect to $\theta =0$. Because of the contact misalignment, there is a slight component of $S_{xx}$. Moreover, it is nearly symmetric in $\theta$ when $B\perp y$, as seen in Fig. S6 [46]. Therefore, it does not affect the extracted $S_{yx}^{\chi }$. (e) $S_{yx}^{\chi }$ versus $\theta$. $S_{yx}^{\chi }$ is obtained by $\theta$ antisymmetrizing the data in (d). Open circles are experimental data and the solid lines are best fits to $\sin \theta$.