Zitterbewegung (trembling motion) of electrons in narrow gap semiconductors

Theory of trembling motion [Zitterbewegung (ZB)] of charge carriers in various narrow-gap materials is reviewed. Nearly free electrons in a periodic potential, InSb-type semiconductors, bilayer graphene, monolayer graphene and carbon nanotubes are considered. General features of ZB are emphasized. It is shown that, when the charge carriers are prepared in the form of Gaussian wave packets, the ZB has a transient character with the decay time of femtoseconds in graphene and picoseconds in nanotubes. Zitterbewegung of electrons in graphene in the presence of an external magnetic field is mentioned. A similarity of ZB in semiconductors to that of relativistic electrons in a vacuum is stressed. Possible ways of observing the trembling motion in solids are mentioned.Comment: 8 pages, 5 figure

Zitterbewegung of nearly-free and tightly bound electrons in solids

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.Comment: 22 pages, 6 figures Published version, minor changes mad

Cyclotron motion in graphene

We investigate cyclotron motion in graphene monolayers considering both the full quantum dynamics and its semiclassical limit reached at high carrier energies. Effects of zitterbewegung due to the two dispersion branches of the spectrum dominate the irregular quantum motion at low energies and are obtained as a systematic correction to the semiclassical case. Recent experiments are shown to operate in the semiclassical regime.Comment: 6 pages, 1 figure include