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.

We review the spin splitting of subband energies caused by bulk and structure inversion asymmetries in semiconductor III-V and II-VI heterostructures. We present both theoretical and experimental aspects of the problem, and we discuss the spin splitting in the absence of external fields as well as its dependence on magnetic and electric fields. The theoretical description of conduction and valence subbands is based on a multiband k • p formalism. Experimental results are summarized, as obtained by beatings of the Shubnikov-de Haas oscillations, magnetoconductance in antilocalization regime, Raman scattering, spin resonance and cyclotron resonance. This review article is motivated by recent interest in spin properties of heterostructures in view of spintronic applications.

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An analogy between the band structure of narrow gap semiconductors and the Dirac equation for relativistic electrons in vacuum is used to demonstrate that semiconductor electrons experience a Zitterbewegung (trembling motion). Its frequency is ωZ ≈ Eg/ and its amplitude is λZ, where λZ = /m * 0 u corresponds to the Compton wavelength in vacuum (Eg is the energy gap, m * 0 is the effective mass and u ≈ 1.3 × 10 8 cm/sec). Once the electrons are described by a two-component spinor for a specific energy band there is no Zitterbewegung but the electrons should be treated as extended objects of size λZ. Possible consequences of the above predictions are indicated. It was noted in the past that the E(k) relation between the energy E and the wavenumber k for electrons in narrow-gap semiconductors (NGS) is analogous to that for relativistic electrons in vacuum [1][2][3][4]. The analogy was also shown to hold for the presence of external fields which was experimentally confirmed on InSb [5]. This "semirelativity in semiconductors" is valid for time dependent phenomena as well, so that the cyclotron resonance of conduction electrons in InSb could be interpreted in terms of the time dilatation between a moving electron and an observer [5]. The semirelativistic phenomena appear at electron velocities of 10 7 − 10 8 cm/sec, much lower than the light velocity c. The reason for this is that the maximum velocity u in semiconductors, which plays the role of c in vacuum, is about 10 8 cm/sec. Until present the semirelativistic considerations for semiconductors were concerned with phenomena related mostly to classical mechanics. The purpose of this contribution is to investigate the quantum domain described by the Hamiltonian for energy bands in NGS, which bears close similarity to the Hamiltonian for relativistic electrons in vacuum. The effects we predict should be much more readily observable in NGS than in vacuum so this investigation is of interest not only for the solid state physics but also for the high energy physics.We begin with the k · p approach to InSb-type III-V semiconductor compounds, first written down by Kane [6]. Taking the limit of large spin-orbit energy ∆, the resulting dispersion relation for the conduction and the light-hole bands is E = ±E p , whereHere E g is the energy gap and m * 0 is the effective mass at the band edge. This expression is identical with the relativistic relation for electrons in vacuum, with the correspondence: E g → 2m 0 c 2 and m * 0 → m 0 . The electron velocity v in the conduction band described by Eq.(1) reaches the saturation value as p increases. This can be seen directly by calculating v i = ∂E p /∂p i and taking the limit of large p i , or by using the analogy:Taking the experimental parameters E g and m * 0 we calculate very similar value of u ≈ 1.3 × 10 8 cm/sec for different III-V compounds.Now we define an important quantitywhich we call the length of Zitterbewegung for reasons given below. Here we note that it corresponds to the Compton wavelength λ c = /m 0 c for elect...

We review recent research on Zitterbewegung (ZB, trembling motion) of electrons in semiconductors. A brief history of the subject is presented, the trembling motion in semirelativistic and spin systems is considered and its main features are emphasized. Zitterbewegung of charge carriers in monolayer and bilayer graphene as well as in carbon nanotubes is elaborated in some detail. We describe effects of an external magnetic field on ZB using monolayer graphene as an example. Nature of electron ZB in crystalline solids is explained. We also review various simulations of the trembling motion in a vacuum and in semiconductors, and mention ZB-like wave phenomena in sonic and photonic periodic structures. An attempt is made to quote all the relevant literature on the subject.