Electrical and thermal magnetotransport and the Wiedemann-Franz law in semimetals with electron-electron scattering

Abstract

We study the electrical and thermal transport properties and the violation of the Wiedemann-Franz (WF) law of two-carrier semimetals using exact treatments of the Boltzmann equation with the impurity and electron-electron scatterings in a magnetic field. For comparison, we also study those in the case of Baber scattering: a single-carrier system with an impurity scattering and phenomenological momentum-dissipative electron-electron scattering. In both systems, the longitudinal and transverse WF laws, $L=L_{\text{H}}=L_0={\pi }^2{k}_B^2/3e^2$, hold at zero temperature, where the Lorenz ratio $L$ and the Hall Lorenz ratio $L_{\text{H}}$ are ratios of thermal conductivity ${\kappa }_{\mu \nu }$ to electrical conductivity ${\sigma }_{\mu \nu }$ divided by temperature. However, the electron-electron scattering makes Lorenz ratios deviate from $L_0$ with increasing temperature. To describe the WF law in a magnetic field, we introduce another set of Lorenz ratios, $\tilde{L}$ and ${\tilde{L}}_{\text{H}}$, defined as the ratios of the resistivity and the Hall coefficient to their thermal counterparts. The WF laws for them, $\tilde{L}={\tilde{L}}_{\text{H}}=L_0$, and their violation are helpful for the discussion of $L$ and $L_{\text{H}}$. For Baber scattering, our exact result shows $L_{\text{H}}/L_0\sim (L/L_0{)}^2$ in a weak magnetic field. In semimetals, the violations of the WF laws are significant, reflecting the different temperature dependence between the electrical and thermal resistivities in a magnetic field. This is because the momentum conservation of the electron-electron scattering has a completely different effect on electrical and thermal magnetotransport. We sort out these behaviors using $\tilde{L}$ and ${\tilde{L}}_{\text{H}}$. We also provide a relaxation time approximation, which is useful for comparing theory and experiment.

32 pages, 15 figures