Berry curvature unravelled by the anomalous Nernst effect in ${\mathrm{Mn}}_3\mathrm{Ge}$

The discovery of topological quantum materials represents a striking innovation in modern condensed matter physics with remarkable fundamental and technological implications. Their classification has been recently extended to topological Weyl semimetals, i.e., solid-state systems which exhibit the elusive Weyl fermions as low-energy excitations. Here we show that the Nernst effect can be exploited as a sensitive probe for determining key parameters of the Weyl physics, applying it to the noncollinear antiferromagnet ${\mathrm{Mn}}_3\mathrm{Ge}$. This compound exhibits anomalous thermoelectric transport due to enhanced Berry curvature from Weyl points located extremely close to the Fermi level. We establish from our data a direct measure of the Berry curvature at the Fermi level and, using a minimal model of a Weyl semimetal, extract the Weyl point energy and their distance in momentum space.

Figure 1

(a) Mn atoms and magnetic structure of ${\mathrm{Mn}}_3\mathrm{Ge}$. Ge atoms are in the center of each Mn hexagon (not sketched here). By mirroring ($xz$-mirror plane) and translating by $c/2$ along the $z$ axis the different layers are transformed into each other. (b) Schematic Nernst setup for measuring $S_{ij}$. On the upper side a resistive chip heater creates the thermal gradient. The bottom of the sample is coupled to a thermal bath of controlled temperature, therefore the temperature gradient arises along the $j$ direction. The magnetic field is applied in the $k$ direction, and the Nernst signal is measured along the $i$ direction.

Figure 2

(a)–(c) Nernst signal of the ${\mathrm{Mn}}_3\mathrm{Ge}$ single crystals with respect to the applied magnetic field $B$ in different configurations. $S_{ij}$ is obtained by measuring the voltage along the $i$ direction while applying a thermal gradient $\mathbf{\nabla }T$ and a magnetic field $B$ along the $j$ and $k$ directions, respectively.

Figure 3

Temperature dependence of (a) $S_{ij}$ at $B=14$ T, (b) ${\alpha }_{xz}$ obtained by using the relation ${\alpha }_{xz}=\left({\rho }_{zz}{S}_{xz}-{\rho }_{xz}{S}_{zz}\right)/\left({\rho }_{xx}{\rho }_{zz}+{\rho }_{xz}^2\right)$ [30] (blue dots) and the fit provided by our theoretical model (red line); (c) the two terms contributing to ${\alpha }_{xz}$.

Figure 4

Schematic picture of the band dispersion in a Weyl semimetal. The Weyl points of opposite chirality are placed along the $k_y$ direction. Saddle point energy, energy of the Weyl points with respect to the Fermi level, and their distance from $k_y=0$ are labeled as $E_S, \mu$, and $\mathrm{\Delta }k$, respectively.

Figure 5

${\alpha }_{xz}$ in ${\mathrm{Mn}}_{3.06}{\mathrm{Sn}}_{0.94}$ (black squares) and ${\mathrm{Mn}}_{3.09}{\mathrm{Sn}}_{0.91}$ (blue triangles) from Ref. [17]. The corresponding fits of the data are represented by red lines. We extracted values of ${\mu }_0=45\mathrm{meV}, C_{\alpha }=0.004\mathrm{V}{\left(\mathrm{K}\mathrm{\Omega }\mathrm{m}\right)}^{-1}$ for ${\mathrm{Mn}}_{3.06}{\mathrm{Sn}}_{0.94}$ and ${\mu }_0=108\mathrm{meV}, C_{\alpha }=0.010\mathrm{V}{\left(\mathrm{K}\mathrm{\Omega }\mathrm{m}\right)}^{-1}$ for ${\mathrm{Mn}}_{3.09}{\mathrm{Sn}}_{0.91}$.

Figure 6

Temperature-dependent resistivity along the $x$ and $z$ directions.

Figure 7

Plot of the Hall conductivity ${\sigma }_{xz}$ vs temperature. Blue dots represent experimental data. The data were fitted (red line) by using our theoretical approach as well as the fixed parameters (${\mu }_0=8.3$ meV, $E_s=143$ meV, and $N_W=22.3$) extracted from the fit of ${\alpha }_{xz}$.

Figure 8

Temperature-dependent Seebeck coefficient measurement along the $y$ and $z$ directions.

Figure 9

(a) Zoom-in of $S_{xz}\left(B\right)$ with a clearly visible hysteresis of the Nernst signal at different selected temperatures. (b) Comparison of the hysteresis curves of $S_{xz}\left(B\right)$ and the magnetization $M$ in the $y$ direction at $T=100\mathrm{K}$. $B$ denotes the external magnetic field. (c) $S_{xz}$ vs $M$. There is no obvious scaling of the anomalous Nernst signal on the magnetization of the sample.

Figure 10

Magnetic moment of ${\mathrm{Mn}}_3\mathrm{Ge}$ in the (a) $x$ direction and (b) $y$ direction. The magnetization has been corrected for geometry effects. However, demagnetization corrections are small and have not been applied. $B$ denotes the external magnetic field.