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

Phosphorene, a single layer of black phosphorous, 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 (including 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 not only the nominal band structure but also the $k$-dependence of the effective mass. 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 g-factor $g_{c1} \approx 2.17$, rather close to the hole $g_{v1}\approx 2.20$ for which there exists a range of xperimental data (our value falls within the range). Furthermore, we investigate excitonic effects using the $k \cdot p$ models and find exciton binding energy (0.82 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, and optical properties of phosphorene.