Topological and disorder corrections to the transverse Wiedemann-Franz law and Mott relation in kagome magnets and Dirac materials

The Wiedemann-Franz law and Mott relation are textbook paradigms on the ratios of the thermal and thermoelectric conductivities to electrical conductivity, respectively. Deviations from them usually reveal insights for intriguing phases of matter. The recent topological kagome magnets $\mathrm{Tb}{\mathrm{Mn}}_6{\mathrm{Sn}}_6$ and ${\mathrm{Mn}}_3\mathrm{Ge}$ show confusingly opposite derivations in the Hall measurement. We calculate the topological and disorder corrections to the Wiedemann-Franz law and Mott relation for the Hall responses in topological kagome magnets and Dirac materials. The calculation indicates the dominance of the topological correction in the experiments. More importantly, we derive analytic correction formulas, which can universally capture the two opposite experiments with the chemical potential as the only parameter and will be a powerful guidance for future explorations on the magnetic topological matter and Dirac materials.

Figure 1

(a) The electric Hall conductivity ${\sigma }_{xy}\equiv J_y/E_x$, thermoelectric Hall coefficient ${\alpha }_{xy}\equiv J_y/(-{\partial }_{x}T)$, and thermal Hall conductivity ${\kappa }_{xy}\equiv J_y^Q/(-{\partial }_{x}T)$, as the response of the transverse electric current $J_y$ or thermal current $J_y^Q$ to the longitudinal electric field $E_x$ or temperature gradient $-{\partial }_{x}T$. Our analytical corrections (solid lines) to the classical behaviors (dashed lines) of (b) the ratios ${\kappa }_{xy}^{\mathrm{in}}/{\sigma }_{xy}^{\mathrm{in}}$ and (c) ${\alpha }_{xy}^{\mathrm{in}}/{\sigma }_{xy}^{\mathrm{in}}$ can capture two types of experiments (scatters), with the chemical potential $\mu$ as the only tuning parameter. We choose $\mu$ = 0.11 and 0.05 eV for the $\mathrm{Tb}{\mathrm{Mn}}_6{\mathrm{Sn}}_6$ [31] (Exp. 1) and ${\mathrm{Mn}}_3\mathrm{Ge}$ [32] (Exp. 2) experiments, respectively. In the $\mathrm{Tb}{\mathrm{Mn}}_6{\mathrm{Sn}}_6$ experiment, $\mu =0.13\mathrm{eV}$.

Figure 2

Two typical topological kagome magnets. (a) $\mathrm{Tb}{\mathrm{Mn}}_6{\mathrm{Sn}}_6$ with the out-of-plane magnetization [31] can be described by the 2D massive Dirac model (b), as verified by the fan diagram of the Landau levels under magnetic fields [30, 34]. (c) The electronic behaviors of the layered (red for one sublayer, blue for the other) in-plane noncollinear antiferromagnet ${\mathrm{Mn}}_3\mathrm{Ge}$ [32] is believed to be described by the Weyl semimetal [35], which can be viewed as layers of coupled 2D massive Dirac models (d).

Figure 3

(a) The total (topological $+$ disorder) contribution to the deviation of the Wiedemann-Franz law ${\kappa }_{xy}^{\mathrm{tot}}/{\sigma }_{xy}^{\mathrm{tot}}-L_{0}T$ in units of $k_B^2/e^2$, and (b) ${\alpha }_{xy}^{\mathrm{tot}}/{\sigma }_{xy}^{\mathrm{tot}}$ in units of $k_B/e$ for the 2D massive Dirac model. The dashed lines represent $m=0.017\mathrm{eV}$ for the $\mathrm{Tb}{\mathrm{Mn}}_6{\mathrm{Sn}}_6$ [31]. (c),(d) The chemical potential $\mu$ dependence of two ratios for the topological (red) and total (black) contributions with $m=0.017\mathrm{eV}$. The parameters are $v=1\mathrm{eV}\mathrm{nm}, n_i{V}_0^4/V_1^3=1\mathrm{eV}$, and $T=100\mathrm{K}$.

Figure 4

(a),(b) For the 3D Dirac model [Eq. (9)], ${\kappa }_{xy}^{\mathrm{in}}/{\sigma }_{xy}^{\mathrm{in}}$ (a) and ${\alpha }_{xy}^{\mathrm{in}}/{\sigma }_{xy}^{\mathrm{in}}$ (b) as functions of the chemical potential $\mu$ at 100 K. The dashed lines represent the classical Wiedemann-Franz law. The purple and blue circles indicate $\mu =0.06\mathrm{eV}$ and $\mu =0.13\mathrm{eV}$, corresponding to the two curves in (c) and (d). (c),(d) ${\kappa }_{xy}^{\mathrm{in}}/{\sigma }_{xy}^{\mathrm{in}}$ (c) and ${\alpha }_{xy}/{\sigma }_{xy}$ (d) as functions of temperature $T$, for $\mu =0.06\mathrm{eV}$ and $\mu =0.13\mathrm{eV}$ (the solid curves). Inset of (d) depicts the dispersion and Berry curvature $\mathrm{\Omega }$ of the energy bands for the 3D Dirac model with the Zeeman energy, which lifts the degeneracy of the bands. The parameters are $v=0.1\mathrm{eV}\mathrm{nm}, m=-0.04\mathrm{eV}, b=-0.18\mathrm{eV}{\mathrm{nm}}^2$, and $M=5.788\times {10}^{-3}\mathrm{eV}$ [66].