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Role of the band-gap parameter in the characterization of Landau levels in a gapped-phase semi-Dirac system: Monolayer phosphorene
Two-dimensional gapped semi-Dirac (GSD) materials are systems with a finite band gap that whose carriers behave relativistically in one direction and Schrödinger-like in the other. In the present paper, we show that besides the two well-known energy band features (curvature and chirality), the band-gap parameter also plays a crucial role in the index and magnetic-field dependence of the Landau levels (LLs) in a GSD system. We take monolayer phosphorene as a GSD representative example to explicitly provide physical insights into the role of this parameter in determining the index and magnetic-field dependence of LLs. We derive an effective one-dimensional Schrödinger equation for charge carriers in the presence of a perpendicular magnetic field and argue that the form of its effective potential is sensitive to a dimensionless band gap that is tunable by structural parameters. The theoretical magnitude of this effective gap and its interplay with oval shape k-space cyclotron orbits resolve the seemingly contradiction in determining the type of the quantum Hall effect in pristine monolayer phosphorene. Our results confirm that the dependence of LLs on the magnetic field in this GSD material is as for conventional two-dimensional semiconductor electron gases up to the very high field. Using the strain-induced band gap reduction, we show that the field dependence of the LLs continuously evolves into B2/3 behavior, which holds for a gapless semi-Dirac system. The highlighted role of the band-gap parameter may affect the consequences of the band anisotropy in the physical properties of a GSD material, including magnetotransport, optical conductivity, dielectric function, and thermoelectric performance.