Semiclassical approach to the physics of smooth superlattice potentials in graphene

Abstract

Due to the chiral nature of the Dirac equation, governing the dynamics of electrons in graphene, overlying of an electrical superlattice (SL) can open new Dirac points on the Fermi surface of the energy spectrum. These lead to novel low-excitation physical phenomena. A typical example for such a system is neutral graphene with a symmetrical unidirectional SL. We show here that in smooth SLs, a semiclassical approximation provides a good mathematical description for particles. Due to the one-dimensional nature of the unidirectional potential, a wave-function description leads to a generalized Bohr-Sommerfeld quantization condition for the energy eigenvalues. In order to pave the way for the application of semiclassical methods to two-dimensional SLs in general, we compare these energy eigenvalues with those obtained from numerical calculations and with the results from a semiclassical Gutzwiller trace formula via the beam-splitting technique. Finally, we calculate ballistic conductivities in general point-symmetric unidirectional SLs with one electron and one hole region in the fundamental cell showing only Klein scattering of the semiclassical wave functions.