Anomalous magneto-optical response of black phosphorus thin films

Abstract

We theoretically investigate the Landau levels (LLs) and magneto-optical conductivity (MOC) of black phosphorus thin films under a perpendicular magnetic field based on an effective $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ Hamiltonian and linear-response theory. We obtain the analytical expression for LLs, which agrees well with the numerical calculations, and find that the LLs sublinearly depend on the magnetic field and LL index. By using the Kubo formula, we evaluate the longitudinal and Hall optical conductivities as functions of the photon energy and the magnetic field. The analytical optical transition matrix elements reveal unusual selection rules for the interband (intraband) optical transitions between the LLs $\mathrm{\ensuremath{\Delta}}n=0,\ifmmode\pm\else\textpm\fi{}2$ $(\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}3)$. The MOC shows strongly anisotropic behaviors of the band structure. For the interband transition, the MOC for linearly polarized light along the armchair direction is three orders of magnitude larger than that along the zigzag direction. Interestingly, we find a beating pattern in the interband MOC due to the interference among the three kinds of optical transitions. For the intraband transition, the MOC can be used to determine the band parameters such as the effective masses and the interband coupling at zero magnetic field. Our results about the MOC can also be applied to the monolayer black phosphorus.