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

We study the quantum Hall effect of 2D 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 $\varepsilon_n\propto \sqrt{nB}$ 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 $σ_{xy}\propto 2n$ quantization of the Hall conductivity in the gapped phase (standard quantum Hall effect regime), and a $σ_{xy}\propto 4(n+1/2)$ quantization in the semimetalic phase, characteristic of Dirac systems with non-trivial topology.