Electron dynamics in graphene with spin–orbit couplings and periodic potentials

Abstract

We use both continuum and lattice models to study the energy-momentum dispersion and the dynamics of a wave packet for an electron moving in graphene in the presence of spin–orbit couplings and either a single potential barrier or a periodic array of potential barriers. Both Kane–Mele and Rashba spin–orbit couplings are considered. A number of special things occur when the Kane–Mele and Rashba couplings are equal in magnitude. In the absence of a potential, the dispersion then consists of both massless Dirac and massive Dirac states. A periodic potential is known to generate additional Dirac points; we show that spin–orbit couplings generally open gaps at all those points, but if the two spin–orbit couplings are equal, some of the Dirac points remain gapless. We show that the massless and massive states respond differently to a potential barrier; the massless states transmit perfectly through the barrier at normal incidence while the massive states reflect from it. In the presence of a single potential barrier, we show that there are states localized along the barrier. Finally, we study the time evolution of a wave packet in the presence of a periodic potential. We discover special points in momentum space where there is almost no spreading of a wave packet; there are six such points in graphene when the spin–orbit couplings are absent.