Effective mass theorems with Bloch modes crossings

TL;DR

This paper analyzes the behavior of electrons in crystals using Bloch modes, showing that conical crossings do not trap energy and allowing for a better understanding of wave dispersion in quantum systems.

Contribution

It provides a rigorous analysis of energy dispersion in Schrödinger equations with Bloch mode crossings, including the effects of conical degeneracies.

Findings

Conical crossings do not trap energy or prevent dispersion.

The limit of time-averaged energy densities is characterized.

Interactions at degenerate crossings are investigated.

Abstract

We study a Schr{\"o}dinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wavelength comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.