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

We show theoretically that non-relativistic nearly-free electrons in solids should experience a trembling motion (Zitterbewegung, ZB) in the absence of external fields, similarly to relativistic electrons in a vacuum. The ZB is directly related to the influence of the periodic potential on the free electron motion. The frequency of the ZB is , where E g is the energy gap. The amplitude of the ZB is determined by the strength of periodic potential and the lattice period, and it can be of the order of nanometres. We show that the amplitude of the ZB does not depend much on the width of the wavepacket 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 in this case also the electrons should experience the trembling motion. It is concluded that the phenomenon of ZB of electrons in crystalline solids is the rule rather than the exception.