Semiclassical response of disordered conductors: Extrinsic carrier velocity and spin and field-corrected collision integral

Abstract

The semiclassical equations of motion are widely used to describe carrier transport in conducting materials. Nevertheless, the substantial challenge of incorporating disorder systematically into the semiclassical model persists, leading to quantitative inaccuracies and occasionally erroneous predictions for the expectation values of physical observables. To address this issue, in this paper we provide a general prescription for reformulating the semiclassical equations of motion for carriers in disordered conductors by taking the quantum mechanical density matrix as the starting point. We focus on the case when only external electric fields are present, without magnetic fields, and the disorder potential is spin independent. The density matrix approach allows averaging over impurity configurations, and the trace of the velocity operator with the disorder-averaged density matrix can be reinterpreted as the semiclassical velocity weighted by the Boltzmann distribution function. Through this rationale the well-known intrinsic group and anomalous velocities are trivially recovered, while we demonstrate the existence of an extrinsic velocity, namely, a disorder correction to the semiclassical velocity of Bloch electrons, mediated by the interband matrix elements of the Berry connection. A similar correction is present in the nonequilibrium expectation value of the spin operator, contributing to spin-orbit torques. To obtain agreement with diagrammatic approaches, the scattering term in the Boltzmann equation must be corrected to first order in the applied electric field, and the Boltzmann equation itself must be solved up to subleading order in the disorder potential. Our prescription ensures that all vertex corrections present in diagrammatic treatments are taken into account, and to illustrate this, we discuss model cases in topological insulators, including the anomalous Hall effect as well as spin-orbit torques.