Quantum transport in crystals: effective-mass theorem and k.p Hamiltonians

TL;DR

This paper rigorously analyzes the effective mass approximation and k.p models for electron quantum dynamics in crystals, demonstrating their accuracy and convergence properties under certain conditions.

Contribution

It provides a rigorous mathematical justification for the effective mass and k.p models, showing their closeness to exact electron dynamics in crystals.

Findings

Effective mass and k.p models are close to exact dynamics for smooth external potentials.

Position density converges weakly to the effective mass approximation.

Homogenization techniques validate the models in the asymptotic limit.

Abstract

In this paper the effective mass approximation and k.p multi-band models, describing quantum evolution of electrons in a crystal lattice, are discussed. Electrons are assumed to move in both a periodic potential and a macroscopic one. The typical period of the periodic potential is assumed to be very small, while the macroscopic potential acts on a much bigger length scale. Such homogenization asymptotic is investigated by using the envelope-function decomposition of the electron wave function. If the external potential is smooth enough, the k.p and effective mass models, well known in solid-state physics, are proved to be close (in strong sense) to the exact dynamics. Moreover, the position density of the electrons is proved to converge weakly to its effective mass approximation.