WKB Energy levels in gapped graphene under crossed electromagnetic fields

We consider a single layer of graphene subjected to a magnetic field H applied perpendicular to the layer and an in-plane constant radial electric field E. The Dirac equation for this configuration does not admit analytical solutions in terms of known special functions. Using the Wentzel–Kramers–Brillouin (WKB) approximation, we demonstrate that for gapped graphene the Bohr–Sommerfeld quantization condition for eigenenergies includes an additional valley-dependent geometrical phase. When this term is accounted for, the WKB approximation exhibits good agreement with results from the exact diagonalization method except for the lowest Landau level.