Zitterbewegung of nearly-free and tightly bound electrons in solids

TL;DR

This paper theoretically demonstrates that electrons in solids, both nearly-free and tightly bound, should exhibit Zitterbewegung, a trembling motion caused by periodic potentials, with implications for understanding electron behavior in crystals.

Contribution

It introduces a theoretical framework showing Zitterbewegung occurs in nonrelativistic electrons in solids, extending the concept beyond relativistic particles and including tightly bound electrons.

Findings

Zitterbewegung frequency is approximately E_g/ħ.

Amplitude of ZB can be on the nanometer scale.

Electrons behave as particles of finite size after band decoupling.

Abstract

We show theoretically that nonrelativistic nearly-free electrons in solids should experience a trembling motion (Zitterbewegung, ZB) in absence of external fields, similarly to relativistic electrons in vacuum. The Zitterbewegung is directly related to the influence of periodic potential on the free electron motion. The frequency of ZB is $\omega\approx E_g/\hbar$, where $E_g$ is the energy gap. The amplitude of ZB is determined by the strength of periodic potential and the lattice period and it can be of the order of nanometers. We show that the amplitude of ZB does not depend much on the width of the wave packet representing an electron in real space. An analogue of the Foldy-Wouthuysen transformation, known from relativistic quantum mechanics, is introduced in order to decouple electron states in various bands. We demonstrate that, after the bands are decoupled, electrons should be treated as particles of a finite size. In contrast to nearly-free electrons we consider a two-band model of tightly bound electrons. We show that also in this case the electrons should experience the trembling motion. It is concluded that the phenomenon of Zitterbewegung of electrons in crystalline solids is a rule rather than an exception.