$k \cdot p$ theory for phosphorene: Effective g-factors, Landau levels, and excitons

Phosphorene, a single layer of black phosphorus, is a direct-band gap two-dimensional semiconductor with promising charge and spin transport properties. The electronic band structure of phosphorene is strongly affected by the structural anisotropy of the underlying crystal lattice. We describe the relevant conduction and valence bands close to the $\Gamma$ point by four- and six-band (with spin) $k \cdot p$ models, including the previously overlooked interband spin-orbit coupling which is essential for studying anisotropic crystals. All the $k \cdot p$ parameters are obtained by a robust fit to {\it ab initio} data, by taking into account the nominal band structure and the $k$-dependence of the effective mass close to $\Gamma$-point. The inclusion of interband spin-orbit coupling allows us to determine dipole transitions along both armchair and zigzag directions. The interband coupling is also key to determine the effective g-factors and Zeeman splittings of the Landau levels. We predict the electron and hole g-factor correction of $\approx 0.03$ due to the intrinsic contributions in phosphorene, which lies within the existing range of experimental data. Furthermore, we investigate excitonic effects using the $k \cdot p$ models and find exciton binding energy (0.81 eV) and exciton diameters consistent with experiments and {\it ab initio} based calculations. The proposed $k \cdot p$ Hamiltonians should be useful for investigating magnetic, spin, transport, optical properties and many-body effects in phosphorene.