WKB energy levels in gapped graphene under crossed electromagnetic fields

TL;DR

This paper investigates the energy levels of gapped graphene under crossed magnetic and electric fields using the WKB approximation, revealing a valley-dependent geometrical phase that improves energy level predictions.

Contribution

It introduces a valley-dependent geometrical phase into the WKB quantization condition for gapped graphene in crossed fields, enhancing the approximation's accuracy.

Findings

WKB approximation aligns well with exact diagonalization results for higher Landau levels.

An additional valley-dependent geometrical phase is identified in the quantization condition.

The approximation shows discrepancies only at the lowest Landau level.

Abstract

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 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 to the lowest Landau level.