Zitterbewegung (Trembling Motion) of Electrons in Graphene

Abstract

This chapter describes a somewhat mysterious phenomenon known in the literature under the German name of “Zitterbewegung” (trembling motion). Both the phenomenon and its name were conceived by Erwin Schrodinger who, in 1930, published the paper Ueber die kraeftefreie Bewegung in der relativistischen Quantenmechanik in which he observed that in the Dirac equation, describing relativistic electrons in a vacuum, the 4× 4 operators corresponding to components of relativistic velocity do not commute with the free-electron Hamiltonian (Barut & Bracken, 1981; Schroedinger, 1930). In consequence, the electron velocity is not a constant of the motion also in absence of external fields. Such an effect in a vacuum must be of a quantum nature as it does not obey Newton’s first law of classical motion. Schrodinger calculated the resulting time dependence of the electron velocity and position concluding that, in addition to classical motion, they experience very fast periodic oscillations which he called Zitterbewegung (ZB). Schrodinger’s idea stimulated numerous theoretical investigations but no experiments since the predicted frequency hωZ ≃ 2m0c ≃ 1 MeV and the amplitude of about λc = h/mc ≃ 3.86× 10−3A are not accessible to current experimental techniques. It was recognized that the ZB is due to an interference of states corresponding to the positive and negative electron energies resulting from the Dirac equation (Bjorken & Drell, 1964; Greiner, 1994; Sakurai, 1997). Lock (1979) showed that, if an electron is represented by a wave packet, its ZB has a transient character, i.e. it disappears with time. It was conceived years later that the trembling electron motion should occur also in crystalline solids if their band structure could be represented by a two-band model reminiscent of the Dirac equation (Cannata et al., 1990; Lurie & Cremer, 1970; Vonsovskii et al., 1990; Zawadzki, 1997). An intense interest in ZB of electrons in semiconductors was launched only in 2005 (Schliemann et al., 2005; Zawadzki, 2005). There followed a wave of theoretical considerations for various semiconductor and other systems. It was recognized that the phenomenon of ZB occurs every time one deals with two or more interacting energy bands (Cserti & David, 2006; Rusin & Zawadzki, 2007a; Winkler et al., 2007). As mentioned above, in a vacuum the ZB characteristics are not favorable. In solids, the ZB characteristics are much better but it is difficult to observe the motion of a single electron. However, recently Gerritsma et al. (2010) simulated experimentally the Dirac equation and 15