The dynamics of 1D Bloch electrons in constant electric fields

TL;DR

This paper rigorously analyzes the behavior of one-dimensional Bloch electrons in constant electric fields, demonstrating the absence of momentum localization for high-energy states and linking results to driven quantum rings.

Contribution

It provides a rigorous proof confirming the non-localization in momentum space for a broad class of initial conditions, based on physically motivated assumptions.

Findings

No momentum localization for high initial momentum

Energy increases in driven quantum rings with weak singularities

Validation of physical assumptions in electron dynamics

Abstract

We study the dynamics of a 1D Bloch electron subjected to a constant electric field. The periodic potential is supposed to be less singular than the $\delta $-like potential (Dirac comb). We give a rigorous proof of Ao's result \cite{Ao} that for a large class of initial conditions (high momentum regime) there is no localization in momentum space. The proof is based on the mathematical substantiation of the two simplifying assumptions made in physical literature: the transitions between far away bands can be neglected and the transitions at the quasi-crossing can be described by Landau-Zener like formulae. Using the connection between the above model and the driven quantum ring (DQR) shown by Avron and Nemirovski \cite{AvN}, our results imply the increase of energy for weakly singular such DQR and appropiate initial conditions.