Origins of electromagnetic anisotropy in monolayer black phosphorus

TL;DR

This paper investigates the origins of electromagnetic anisotropy in monolayer black phosphorus, revealing that higher-order effects in the $k ext{ extperiodcentered} ext{ extperiodcentered}p$ theory explain experimental observations of zigzag-polarized optical transitions.

Contribution

The authors derive a higher-order $k ext{ extperiodcentered} ext{ extperiodcentered}p$ Hamiltonian including symmetry considerations, explaining anisotropic optical responses in phosphorene.

Findings

Third-order $k ext{ extperiodcentered} ext{ extperiodcentered}p$ perturbation dominates zigzag optical transitions

Spin-orbit interaction effects are negligible in this material

Higher-order corrections explain experimental anisotropy

Abstract

Contrary to empirical observations, lowest-order $k\cdot\hat{p}$ theory predicts that monolayer black phosphorus ("phosphorene") is completely immune to zigzag-polarized optical excitation at the bandgap energy. Using symmetry arguments, we derive a $2\times2$ Hamiltonian under the $k\cdot\hat{p}$ formalism including higher-order corrections, which is used to show that the experimentally-measured band-gap transition with zigzag polarization is dominated by the third order $k\cdot\hat{p}$ perturbation in the interband optical matrix element, whereas the effects of spin-orbit interaction are negligible in this material, consistent with a trivial orbital diamagnetic contribution to the $\textit{g}$-factor.