Field-induced diverse quantizations in monolayer and bilayer black phosphorus

Abstract

This report provides a comprehensive understanding of the magnetic quantization effects in phosphorene with the use of the generalized tight-binding model. Especially for bilayer systems, a composite magnetic and electric field can induce the feature-rich LL spectrum. We demonstrate the existence of two subgroups of Landau levels (LLs) near the Fermi level according to their distinguishable localization centers. The strong competition between the two subgroups induces unusual quantization behaviors, such as multiple anticrossings for the ${B}_{z}$- and ${E}_{z}$-dependent energy spectra. These results are clearly explained by the spatial distributions of subenvelope functions from which two types of LLs are characterized by being either the usual or the perturbed distribution modes. The detailed analysis of the diverse magnetic quantizations is quite important in understanding other physical properties, such as the dispersion relations of magnetoplasmons, magneto-optical selection rules, as well as electron transport properties. The unusual energy spectra are directly revealed by the special features of the density of states, which could be further validated by measurements employing scanning tunneling spectroscopy.