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 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.

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

We propose an experiment allowing an observation of Zitterbewegung (ZB, trembling motion) of electrons in graphene in the presence of a magnetic field. In contrast to the existing theoretical work we make no assumptions concerning shape of the electron wave packet. A femtosecond Gaussian laser pulse excites electrons from the valence n = −1 Landau level into three other levels, creating an oscillating electron wave packet with interband and intraband frequencies. Oscillations of an average position of the packet are directly related to the induced dipole moment and oscillations of the average packet's acceleration determine emitted electric field. Both quantities can be measured experimentally. A broadening of Landau levels is included to make the description of ZB as realistic as possible. Criteria of realization of a ZB experiment are discussed.

In turgid multicellular organs, it is convenient to differentiate between the two kinds of tensile forces acting in cell walls as a result of turgor pressure. The primary forces occur both in situ and in cells isolated from the organ, whereas the secondary forces occur only in situ. The latter are an unavoidable physical consequence of the variation in mechanical parameters of tissues forming layers or strands. The most rigid tissue is under maximal tensile force, whereas the least rigid is under maximal compressive force. These forces cause tissue stresses (that is, certain tissues are under tensile stress, whereas others are under compressive stress in the organ). The primary and secondary forces result in primary and secondary stress in cell walls, respectively. The anisotropy of the primary stress is a function of cell shape. For instance, in cylindric cells the anisotropy expressed as the ratio of longitudinal to transverse stresses is 0.5. The anisotropy of the secondary stress is a function of the compound structure of the organ. For example, in the epidermis of sunflower hypocotyl, the longitudinal secondary stress is much higher than the transverse stress. The primary and secondary stresses are superimposed, and, as a consequence, the stress anisotropy in the outer thick walls of epidermal cells is greater than 1. These outer epidermal walls transmit most of the tissue stress. When the epidermis is peeled but remains turgid, only primary stress remains, but loading of the peel can reestablish the original stress anisotropy. We studied the effect of stress anisotropy changes on the orientation of cortical microtubules (CMTs) in the sunflower hypocotyl epidermis. We showed that changes in stress anisotropy cause the CMT orientation to change in the direction of maximal wall stress. In situ, the relatively high tensile tissue stress in the epidermis causes maximal stress in the longitudinal direction and relatively steep CMT orientation. When the tissue stress is removed from the epidermis by peeling, the CMTs tend to reorient toward the transverse direction, which is the direction of maximal stress in the primary component. On application of external longitudinal stress, to substitute for tissue stress, CMTs tend to reorient in the longitudinal direction. However, a relatively high rate of plastic strain is caused by the stress applied to the peel in an acid medium. This produces a less steep orientation of CMTs. It appears that the change in stress anisotropy orients the CMT in the direction in which the stress is maximal after the change, but there is also some effect of the growth rate on the orientation.