Anomalous transverse response of ${\mathrm{Co}}_2\mathrm{MnGa}$ and universality of the room-temperature ${\alpha }_{ij}^A/{\sigma }_{ij}^A$ ratio across topological magnets

The off-diagonal (electric, thermal, and thermoelectric) transport coefficients of a solid can acquire an anomalous component due to the nontrivial topology of the Bloch waves. We present a study of the anomalous Hall effect (AHE), anomalous Nernst effect (ANE), and thermal Hall effect in the Heusler-Weyl ferromagnet ${\mathrm{Co}}_2\mathrm{MnGa}$. The anomalous Wiedemann-Franz law, linking electric and thermal responses, was found to be valid over the whole temperature window. This indicates that the AHE has an intrinsic origin and the Berry spectrum is smooth in the immediate vicinity of the Fermi level. We extract ${\alpha }_{ij}^A$ from our ANE data and find that the ${\alpha }_{ij}^A/{\sigma }_{ij}^A$ ratio approaches $k_B/e$ at room temperature. Scrutinizing all topological magnets previously explored, we observe that this ratio is a sizable fraction of $k_B/e$ at room temperature. We provide a rough explanation for this feature by arguing that the two anomalous transverse coefficients depend on universal constants, the Berry curvature averaged over a window set by either the Fermi wavelength (for Hall) or the de Broglie thermal length (for Nernst). The universal scaling indicates that the widths of the two windows approaches each other at room temperature.

Figure 1

Basic transport and anomalous transverse responses at $\left(T=300\mathrm{K}\right)$ in ${\mathrm{Co}}_2\mathrm{MnGa}$: Temperature dependence of (a) electric and (b) thermal conductivity. (c) Magnetization $M$ at room temperature. (d) Anomalous Hall effect, (e) anomalous Nernst effect, and (f) anomalous thermal Hall effect at room temperature. In all cases in this Rapid Communication, $B\parallel \left[110\right],I\parallel \left[001\right]$. The insets in (d)–(f) show the experimental configurations for Hall, Nernst, and thermal Hall measurements.

Figure 2

Temperature evolution of Hall conductivity, thermal Hall conductivity, and anomalous transverse Lorenz number in ${\mathrm{Co}}_2\mathrm{MnGa}$: (a) and (b) Field-dependent Hall conductivity ${\sigma }_{xy}$ and transverse thermal conductivity divided by $T{\kappa }_{xy}/T$ at selected temperatures. (c) Temperature-dependent anomalous Hall conductivity and transverse thermal conductivity. (d) The anomalous transverse Lorenz ratio $(L_{xy}^A={\kappa }_{xy}^A/{\sigma }_{xy}^{A}T)$.

Figure 3

Anomalous transverse coefficients: (a) The anomalous Lorenz number $L_{xy}^A={\kappa }_{xy}^A/T{\sigma }_{xy}^A$ remains close to the Sommerfeld value of $L_0$. The anomalous Lorenz number of ${\mathrm{Mn}}_3\mathrm{Sn}$ [6] and ${\mathrm{Mn}}_3\mathrm{Ge}$ [10] are also shown. The dashed horizontal line represents $L_0$. (b) The anomalous Hall conductivity ${\sigma }_{xy}^A$ as a function of temperature. (c) The anomalous transverse thermoelectric conductivity ${\alpha }_{xy}^A$ as a function of temperature. For comparison with data reported previously [8, 9], see the Supplemental Material [22].

Figure 4

The universal relation found in the ${\alpha }_{ij}^A/{\sigma }_{ij}^A$ ratio and the pondering functions: (a) The ${\alpha }_{ij}^A/{\sigma }_{ij}^A$ ratio as a function of temperature in different magnets including ${\mathrm{Mn}}_3\mathrm{Sn}$ [6], Fe [6], ${\mathrm{Mn}}_3\mathrm{Ge}$ [10], ${\mathrm{La}}_{0.7}{\mathrm{Sr}}_{0.3}{\mathrm{CoO}}_3$ [27], and ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ [13]). $k_B/e$ is represented by a red solid line. (b) Room-temperature ${\alpha }_{ij}^A/{\sigma }_{ij}^A$ in different magnets $({\mathrm{Fe}}_{0.7}{\mathrm{Ge}}_{0.3}$ [28]). All the points are shown at 300 K $({\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ is taken at 170 K because of the magnetic order). They lie all between $k_B/5e$ (blue line) and $k_B/e$ (red line). Note the range of ${\sigma }^A$ between 8 and $1000{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$. (c)–(f) Pondering functions in the Fermi sea (c) and (e) and the Fermi surface (d) and (f) expressions of the anomalous Hall coefficients. (c) Fermi-Dirac distribution $f\left(k\right)$ used for Fermi-sea ${\sigma }_{ij}^A$. (d) Its energy derivative, used for Fermi-surface ${\sigma }_{ij}^A$. (e) The entropy density of electrons $s\left(k\right)$, used for Fermi-sea ${\alpha }_{ij}^A$. (f) Its energy derivative, used for Fermi-surface ${\alpha }_{ij}^A$.