The temporal picture for Bloch electron dynamics in homogeneous electric fields

TL;DR

This paper develops a detailed transient analysis of Bloch electron dynamics in time-dependent electric fields, revealing how electrons transition from free-mass to effective-mass behavior during acceleration.

Contribution

It introduces a comprehensive temporal framework for Bloch electron evolution under arbitrary electric fields using the Schrödinger equation and eigenstate basis, highlighting mass transition effects.

Findings

Momentum expectation value varies from real to effective mass dynamics.

Field-dependent velocity exhibits Zitterbewegung-like behavior.

Electron dressing leads to changing inertia during acceleration.

Abstract

The transient picture for a Bloch electron accelerating in an arbitrarily time-dependent homogeneous electric field is developed. The temporal sequence for the analysis includes the instant after electron injection, followed by the time required for a small change in electron wavenumber away from initial injection, leading to the final time evolution over many Bloch periods. The time-dependent behavior is studied using the properties of the Schr\"{o}dinger equation. The electric field is described through the vector potential gauge, and the instantaneous eigenstates of the Bloch, electric-field-dependent Hamiltonian are used as basis states in describing the Bloch dynamics in the electric field. For each temporal sequence considered, the solution to the Schr\"{o}dinger equation is established and comparatively discussed. The expectation value of the momentum is obtained for the special case of first order in a constant electric field; the resulting velocity derived is a field-dependent generalization of the natural Zitterbewegung-like behavior discussed in the recent literature. The early-time and long-time limits of the momentum expectation value and its time derivative demonstrate that the resistance to Bloch acceleration after initial band injection varies from real mass to effective mass dynamics as the electron accelerates through the band under the influence of electric field. This changing inertia from early injection of a free-mass electron is the result of the {\it real mass} electron {\it dressing-up} into the states of the crystal to become {\it an effective mass} electron. The ramifications of this temporal {\it dressing} behavior are discussed in considering the general dynamics of Bloch electrons subject to ultrastrong electric fields.