Time dependent behavior of a localized electron at a heterojunction boundary of graphene

We develop a finite-difference time-domain (FDTD) method for simulating the dynamics of graphene electrons, denoted GraFDTD. We then use GraFDTD to study the temporal behavior of a single localized electron wave packet, showing that it exhibits optical-like dynamics including the Goos-Hänchen effect [F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947)] at a heterojunction, but the behavior is quantitatively different than for electromagnetic waves. This suggests issues that must be addressed in designing graphene-based electronic devices analogous to optical devices. GraFDTD should be useful for studying such complex time-dependent behavior of a quasiparticle in graphene.