Nonlinear optical Hall effect of few-layered NbSe2

Abstract

${\mathrm{NbSe}}_{2}$ is one of metallic two-dimensional (2D) transition-metal dichalcogenide (TMDC) materials. Because of broken crystal inversion symmetry, large spin splitting is induced by Ising-type spin-orbit coupling in odd-number-layered ${\mathrm{NbSe}}_{2}$, but absent for even-number-layered ${\mathrm{NbSe}}_{2}$ with the inversion symmetry. In this paper we numerically calculate nonlinear optical charge and spin Hall conductivities of few-layered ${\mathrm{NbSe}}_{2}$ based on an effective tight-binding model which includes ${d}_{{z}^{2}}$, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$, and ${d}_{xy}$ orbitals of Nb atoms. We show that the nonlinear optical Hall conductivity for the second harmonic generation (SHG) process has a nonvanishing value in odd-number-layered ${\mathrm{NbSe}}_{2}$. Also, we provide a nonlinear optical selection rule in few-layered ${\mathrm{NbSe}}_{2}$ and their polarization dependencies. Furthermore, for the even-number-layered case, the nonlinear optical Hall currents can be generated by applying electric fields which break inversion symmetry. We also discuss that the nonlinear optical Hall effect is expected to occur in TMDC materials in general. Thus, our results will serve to design potential opt-spintronics devices based on 2D materials to generate the spin Hall current by SHG.