Multilayers black phosphorus: from tight-binding to continuum description

TL;DR

This paper develops an analytical framework for understanding the electronic properties of multilayer black phosphorus, providing explicit formulas for energy gaps, effective masses, and Landau levels as functions of the number of layers.

Contribution

It introduces a simple, accurate analytical method based on a tight-binding Hamiltonian to study multilayer black phosphorus, including effective models and physical property expressions.

Findings

Derived explicit expressions for energy gaps and effective masses.

Showed multilayer problem reduces to effective monolayer problems in long wavelength limit.

Calculated Landau levels as a function of magnetic field.

Abstract

We investigate the electronic properties of $N$-layer black phosphorus by means of an analytical method based on a recently proposed tight-binding Hamiltonian involving $14$ hopping parameters. The method provides simple and accurate general expressions for the Hamiltonian of $N$-layer phosphorene, which are suitable for the study of electronic transport and optical properties of such systems, and the results show the features that emerge as the number of layers increases. In addition, we show that the $N$-layer problem can be translated into $N$ effective monolayer problems in the long wavelength approximation and, within this analytical picture, we obtain expressions for the energy gap and the effective masses for electrons and holes along the $N$-layer black phosphorus plane directions as function of the number of layers, as well as for the Landau levels as function of perpendicular magnetic field.