Quantum Hall effect and semiconductor-to-semimetal transition in biased black phosphorus

Abstract

We study the quantum Hall effect of two-dimensional electron gas in black phosphorus in the presence of perpendicular electric and magnetic fields. In the absence of a bias voltage, the external magnetic field leads to a quantization of the energy spectrum into equidistant Landau levels, with different cyclotron frequencies for the electron and hole bands. The applied voltage reduces the band gap, and eventually a semiconductor-to-semimetal transition takes place. This nontrivial phase is characterized by the emergence of a pair of Dirac points in the spectrum. As a consequence, the Landau levels are not equidistant anymore but follow the ${\ensuremath{\varepsilon}}_{n}\ensuremath{\propto}\sqrt{nB}$ characteristic of Dirac crystals as graphene. By using the Kubo-Bastin formula in the context of the kernel polynomial method, we compute the Hall conductivity of the system. We obtain a ${\ensuremath{\sigma}}_{xy}\ensuremath{\propto}2n$ quantization of the Hall conductivity in the gapped phase (standard quantum Hall effect regime) and a ${\ensuremath{\sigma}}_{xy}\ensuremath{\propto}4(n+1/2)$ quantization in the semimetallic phase, characteristic of Dirac systems with nontrivial topology.