Transient Zitterbewegung of charge carriers in graphene and carbon nanotubes

Observable effects due to trembling motion (Zitterbewegung, ZB) of charge carriers in bilayer graphene, monolayer graphene and carbon nanotubes are calculated. 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. Analytical results for bilayer graphene allow us to investigate phenomena which accompany the trembling motion. In particular, it is shown that the transient character of ZB in graphene is due to the fact that wave subpackets related to positive and negative electron energies move in opposite directions, so their overlap diminishes with time. This behavior is analogous to that of the wave packets representing relativistic electrons in a vacuum.Comment: 7 pages, 3 figures, augmented versio

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

One-dimensional semirelativity for electrons in carbon nanotubes

It is shown that the band structure of single-wall semiconducting carbon nanotubes (CNT) is analogous to relativistic description of electrons in vacuum, with the maximum velocity $u$= $10^8$cm/s replacing the light velocity. One-dimensional semirelativistic kinematics and dynamics of electrons in CNT is formulated. Two-band k.p Hamiltonian is employed to demonstrate that electrons in CNT experience a Zitterbewegung (trembling motion) in absence of external fields. This Zitterbewegung should be observable much more easily in CNT than its analogue for free relativistic electrons in vacuum.Comment: 4 pages no figure

Self-consistent equilibrium of a two-dimensional electron system with a reservoir in a quantizing magnetic field: Analytical approach

An analytical approach has been developed to describe grand canonical equilibrium between a three dimensional (3D) electron system and a two dimensional (2D) one, an energy of which is determined self-consistently with an electron concentration. Main attention is paid to a Landau level (LL) pinning effect. Pinning means a fixation of the LL on a common Fermi level of the 2D and the 3D systems in a finite range of the magnetic field due to an electron transfer from the 2D to the 3D system. A condition and a start of LL pinning has been found for homogeneously broadened LLs. The electronic transfer from the 3D to the 2D system controls an extremely sharp magnetic dependency of an energy of the upper filled LL at integer filling of the LLs. This can cause a significant increase of inhomogeneous broadening of the upper LL that was observed in recent local probe experiments.Comment: 12 pages, 2 figures, revtex

Conduction electrons localized by charged magneto-acceptors A2−^{2-} in GaAs/GaAlAs quantum wells

A variational theory is presented of A$^{1-}$ and A$^{2-}$ centers, i.e. of a negative acceptor ion localizing one and two conduction electrons, respectively, in a GaAs/GaAlAs quantum well in the presence of a magnetic field parallel to the growth direction. A combined effect of the well and magnetic field confines conduction electrons to the proximity of the ion, resulting in discrete repulsive energies above the corresponding Landau levels. The theory is motivated by our experimental magneto-transport results which indicate that, in a heterostructure doped in the GaAs well with Be acceptors, one observes a boil-off effect in which the conduction electrons in the crossed-field configuration are pushed by the Hall electric field from the delocalized Landau states to the localized acceptor states and cease to conduct. A detailed analysis of the transport data shows that, at high magnetic fields, there are almost no conducting electrons left in the sample. It is concluded that one negative acceptor ion localizes up to four conduction electrons.Comment: 8 pages, 5 figure

Temperature dependence of the electron spin g factor in GaAs

The temperature dependence of the electron spin $g$ factor in GaAs is investigated experimentally and theoretically. Experimentally, the $g$ factor was measured using time-resolved Faraday rotation due to Larmor precession of electron spins in the temperature range between 4.5 K and 190 K. The experiment shows an almost linear increase of the $g$ value with the temperature. This result is in good agreement with other measurements based on photoluminescence quantum beats and time-resolved Kerr rotation up to room temperature. The experimental data are described theoretically taking into account a diminishing fundamental energy gap in GaAs due to lattice thermal dilatation and nonparabolicity of the conduction band calculated using a five-level kp model. At higher temperatures electrons populate higher Landau levels and the average $g$ factor is obtained from a summation over many levels. A very good description of the experimental data is obtained indicating that the observed increase of the spin $g$ factor with the temperature is predominantly due to band's nonparabolicity.Comment: 6 pages 4 figure

Zitterbewegung of electrons in graphene in a magnetic field

Electric current and spacial displacement due to trembling motion [Zitterbewegung (ZB)] of electrons in graphene in the presence of an external magnetic field are described. Contributions of both inequivalent $K$ points in the Brillouin zone of graphene are considered. It is shown that, when the electrons are prepared in the form of wave packets, the presence of a quantizing magnetic field $B$ has very important effects on ZB. (1) For $B\neq 0$ the ZB oscillations are permanent, for B=0 they are transient. (2) For $B\neq 0$ many ZB frequencies appear, for B=0 only one frequency is at work. (3) For $B\neq 0$ both interband and intraband (cyclotron) frequencies contribute to ZB, for B=0 there are no intraband frequencies. (4) Magnetic field intensity changes not only the ZB frequencies but the entire character of ZB spectrum. An emission of electromagnetic dipole radiation by the trembling electrons is proposed and described. It is argued that graphene in a magnetic field is a promising system for an experimental observation of Zitterbewegung.Comment: 9 pages, 8 figure