WKB analysis of edge states in graphene in a strong magnetic field

TL;DR

This paper uses WKB semiclassical analysis to analytically study the edge state energy spectra in graphene ribbons under strong magnetic fields, providing new insights into their fine structure and classical orbit quantization.

Contribution

It develops a WKB-based semiclassical formalism for edge states in graphene, extending analytical understanding beyond previous numerical studies.

Findings

Analytical energy spectra match numerical results closely.

Effective Hamiltonian describes edge states with double well potential.

Classical orbit quantization offers qualitative spectral insights.

Abstract

We investigate the fine structure of the edge states energy spectrum for zigzag and armchair ribbons of graphene in a strong magnetic field. At low energy, the spectra can be described by an effective Schrodinger Hamiltonian with a double well potential, symmetric in the zigzag case and asymmetric in the armchair case. We develop a semiclassical formalism based on the WKB approximation to calculate analytically the energy spectrum for the two types of edges, including regions which were not studied earlier. Our results are in very good quantitative agreement with numerical calculations. This approach leads to a qualitative description of the spectra in terms of the quantization of unusual classical orbits in the real space.