Quantum Theory of Electrical Transport Phenomena

TLDR

Conventional electrical conductivity theories, including the Boltzmann transport equation, have limitations beyond weak or dilute scattering regimes. The study aims to rigorously ground electrical transport theory using a simple model. A method is introduced that derives the full steady‑state density matrix for noninteracting electrons scattered by random impurities, expanding it in powers of the scattering potential. The approach yields higher‑order corrections to transport properties.

Abstract

The usual theories of electrical conductivity suffer from a number of weaknesses. This paper attempts, on the basis of a simple model, to put the theory on as rigorous a basis as possible. A technique is developed which gives the entire density matrix of the system of charge carriers in the steady state. Our model consists of noninteracting free (or Bloch) electrons being scattered by "random" rigid impurity centers. The density matrix is developed in ascending powers of the strength of the scattering potential. The familiar Boltzmann transport equation represents an approximation valid in the limiting cases of very weak or very dilute scatterers. Higher order corrections are given.