Landau levels for relativistic particles: Einstein–Brillouin–Keller quantization approach

The effects of a transverse magnetic field on relativistic particles in two dimensions are treated by using the semiclassical quantization rules and the role played by the spin is emphasized. The Landau levelsʼ energies are analyzed by focusing on the square-root dependence on level index obtained for relativistic spinless particles. This result will be compared to the energies calculated for relativistic particles with spin that are governed by the Dirac equation in relativistic quantum mechanics. Then relativistic massless fermions are discussed. The approach provides a conceptual and intuitive introduction to the grounds of quantum Hall effect in carbon based nanostructures.

► The Letter provides insight into the origins of grapheneʼs properties.
► The role played by the spin on the semiclassical quantization rules is emphasized.
► The approach is capable to explain two quantum Hall properties of graphene:
► The n dependence of the energy in terms of the LL number n due to the relativity;
► The existence of a gapless lowest Landau state due to the spin.