Coulomb impurity under magnetic field in graphene: a semiclassical approach

TL;DR

This paper develops a semiclassical WKB approach to analyze Coulomb impurities in graphene under magnetic fields, accurately reproducing Landau levels and providing insights into impurity effects below the supercritical threshold.

Contribution

It introduces a WKB-based method to solve the Coulomb impurity problem in graphene with magnetic fields, extending understanding of impurity effects below the supercritical limit.

Findings

WKB energies match Landau levels without impurity.

Energy dependence on angular momentum aligns with exact results.

Series expansion in the effective fine structure constant converges.

Abstract

We address the problem of a Coulomb impurity in graphene in the presence of a perpendicular uniform magnetic field. We show that the problem can be solved below the supercritical impurity magnitude within the WKB approximation. Without impurity the semiclassical energies correctly reproduce the Landau level spectrum. For a given Landau level the WKB energy depends on the absolute value of angular momentum in a way which is consistent with the exact diagonalization result. Below the supercritical impurity magnitude, the WKB solution can be expanded as a convergent series in powers of the effective fine structure constant. Relevance of our results to validity of the widely used Landau level projection approximation is discussed.