Particle-hole–like symmetry and valley degeneracy splitting in graphene dot and antidot

Abstract

We report a combined effect of dot confinement, graphene-substrate coupling, spin-orbit interaction and applied magnetic field on particle-hole–like symmetry and valley degeneracy. General conditions to control and manipulate the bound-state solutions are described in terms of external and controllable parameters, such as repulsive and attractive voltages, the magnitude of the graphene-substrate coupling parameter, spin-orbit coupling (SOC) strength, and magnetic field. We find that the confinement potential alone does not lift the valley degeneracy. But it together with either the mass term or magnetic field splits the valley degeneracy. The SOC which couples the real spin of an electron and sublattice pseudospin interacts with magnetic field, leading to multiple splitting of Landau levels. Remarkably, changing the sign of voltage and the SOC, the particle-hole–like symmetry can be either broken or restored, which sheds light on the tunability of valley degeneracy and particle-hole symmetry by external controls.