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

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.