Quantum transport for a Bloch electron quasiparticle in an inhomogeneous electric field scattering from a random distribution of impurities: A Wigner approach

Abstract

The quantum transport equation is derived in terms of the Wigner distribution function for a Bloch electron quasiparticle, that is, a Bloch electron in a single band, interacting with a random, inhomogeneous distribution of impurities, and subject to general homogeneous and inhomogeneous electric fields. The time dependent homogeneous electric field is described through the vector potential gauge. The derivation of the transport equation makes use of a unitary transformation of the Liouville equation based on the interaction picture to a form in which the scattering interaction appears quadratically, and utilizes accelerated Bloch states as basis states; the resulting generalized drift and diffusion terms are obtained exactly for an arbitrary band structure. In taking the limit of slowly varying inhomogeneous electric field and slowly varying scatterer density distribution, a quantum generalization of the Boltzmann-like Wigner transport equation is obtained which includes impurity-related intracollisional field effects in the collision term and a drift term comprising the total force due to both the homogeneous and inhomogeneous fields.