Magnetic edge states in graphene in nonuniform magnetic fields

Abstract

We theoretically study the electronic properties of a graphene sheet on the $xy$ plane in a spatially nonuniform magnetic field, $B={B}_{0}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ in one domain and $B={B}_{1}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ in the other domain, in the quantum Hall regime, and in the low-energy limit. We find that the magnetic edge states of the Dirac fermions, formed along the boundary between the two domains, have features strongly dependent on whether ${B}_{0}$ is parallel or antiparallel to ${B}_{1}$. In the parallel case, when the Zeeman spin splitting can be ignored, the magnetic edge states originating from the $n=0$ Landau levels of the two domains have dispersionless energy levels contrary to those from the $n\ensuremath{\ne}0$ levels. Here, $n$ is the graphene Landau-level index. They become dispersive as the Zeeman splitting becomes finite or as an electrostatic step potential is additionally applied. In the antiparallel case, the $n=0$ magnetic edge states split into electronlike and holelike current-carrying states. The energy gap between the electronlike and holelike states can be created by the Zeeman splitting or by the step potential. These features are attributed to the fact that the pseudospin of the magnetic edge states couples to the direction of the magnetic field. We propose an Aharonov--Bohm interferometry setup in a graphene ribbon for an experimental study of the magnetic edge states.