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

Abstract

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