Landau levels of single-layer and bilayer phosphorene

Abstract

In this work we introduce a low-energy Hamiltonian for single-layer and bilayer black phosphorus that describes the electronic states at the vicinity of the $\mathrm{\ensuremath{\Gamma}}$ point. The model is based on a recently proposed tight-binding description for electron and hole bands close to the Fermi level. We calculate expressions for the Landau-level spectrum as function of magnetic field, and in the case of bilayer black phosphorus we investigate the effect of an external bias on the electronic band gap. The results showcase the highly anisotropic character of black phosphorus, and in particular for bilayer BP, the presence of bias allows for a field-induced semiconductor-metal transition.