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

TL;DR

This work provides a comprehensive, exact Boltzmann-equation analysis of electrical and thermal magnetotransport in two-band semimetals, including impurity and electron–electron scatterings, in magnetic fields. By contrasting a Baber-scattering single-carrier limit with the full two-band model, it uncovers how momentum conservation in ee scattering differentially affects electrical versus thermal transport and the WF law, introducing tilde Lorenz ratios to analyze magnetotransport. The study reveals regime-dependent violations of the WF law: in compensated systems, L and L_H deviate significantly with T and B, while tilde ratios capture the dominant elastic vs inelastic contributions; in uncompensated systems, resistivity can saturate while the thermal response continues to evolve, leading to nuanced L and L_H behavior. These results, alongside a relaxation-time-approximation reference, offer a detailed framework for interpreting magnetotransport experiments in semimetals and related materials where electron–electron scattering plays a crucial role.

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, $L = L_{\text{H}} = L_{0}= π^2k_B^2/3e^2$, hold at zero temperature, where the Lorenz ratio $L$ and the Hall Lorenz ratio $L_{\text{H}}$ are ratios of thermal conductivity $κ_{μν}$ to electrical conductivity $σ_{μν}$ 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, $\widetilde{L}$ and $\widetilde{L}_{\text{H}}$, defined as the ratios of the resistivity and the Hall coefficient to their thermal counterparts. The WF laws for them, $\widetilde{L} = \widetilde{L}_{\text{H}} = L_{0}$, 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 $\widetilde{L}$ and $\widetilde{L}_{\text{H}}$. We also provide a relaxation time approximation, which is useful for comparing theory and experiment.

Figures (15)

• Figure 1: The schematic of two cases in this study: the two-band semimetal and Baber scattering. The black arrows represent how momentum is transferred. $I_{\text{e-e}}^{(ll')}$ indicates the electron-electron scattering between bands $l$ and $l'$ ($l = 1$ means electrons and $l = 2$ means holes) and $I_{\text{imp}}^{(l)}$ indicates the impurity scattering of the band $l$. They are introduced in Sec. \ref{['Sec:model_Boltzmann']}.
• Figure 2: An overview of the paper.
• Figure 3: An image of the two-band model Takahashi2023.
• Figure 4: A schematic of the angles $\theta$ and $\varphi$ with a possible combination of momenta fixed on the Fermi surfaces satisfying the momentum conservation $\bm{k} + \bm{k}_2 = \bm{k}_3 + \bm{k}_4$.
• Figure 5: (a) A comparison of the magnetic field dependence of $\sigma_{xx}$, $\sigma_{xy}$, $\kappa_{xx}$, and $\kappa_{xy}$ for Baber scattering divided by the RTA results without the impurity scattering ($\zeta^2 = 1-i(2/\pi^2)\omega_{\text{c}}\tau_{\text{e-e}}$) for $\lambda_{\sigma} = \lambda_{\kappa} = 1/3$. (b) The magnetic field dependence of the electrical resistivity $\rho$ and thermal resistivity $WT$ for the same parameters as in (a).