Zitterbewegung of nearly-free and tightly-bound electrons in semiconductors

Abstract

We show theoretically that non-relativistic nearly-free electrons in solids should experiencea trembling motion (Zitterbewegung, ZB) in the absence of external fields, similarly torelativistic electrons in a vacuum. The ZB is directly related to the influence of the periodicpotential on the free electron motion. The frequency of the ZB is , where Eg is the energy gap. The amplitude of the ZB is determined by the strength of periodicpotential and the lattice period, and it can be of the order of nanometres. We show thatthe amplitude of the ZB does not depend much on the width of the wavepacketrepresenting an electron in real space. An analogue of the Foldy–Wouthuysentransformation, known from relativistic quantum mechanics, is introduced in order todecouple electron states in various bands. We demonstrate that after the bands aredecoupled electrons should be treated as particles of a finite size. In contrast to nearly-freeelectrons we consider a two-band model of tightly-bound electrons. We show that in thiscase also the electrons should experience the trembling motion. It is concluded that thephenomenon of ZB of electrons in crystalline solids is the rule rather than the exception.