Spin-dependent k.p Hamiltonian of black phosphorene based on Lowdin partitioning method

In this work, we have presented the spinful low energy six-band Hamiltonian representation of black phosphorene about the Gamma point based on the k.p model via Lowdin partitioning method. To this end, we have first extracted the spinless low energy two-band Hamiltonian of black phosphorene from the spin-independent sixteen-band Hamiltonian using group theory. Regarding this purpose, we have focused on the band edges of the black phosphorene structure, i.e., Gamma(-)(4) and Gamma(+)(2) related to states B-3u and B-1g which stand for the lowest sub-band in the conduction band and the highest sub-band in the valence band, respectively. The results show that the energy dispersion is linear in one direction of a crystalline lattice and nonlinear in the other direction. This is due to the orthorhombic lattice and special symmetries of the black phosphorene structure presented in the D-2h point group. This is in agreement with our previous results, in which the band structure of black phosphorene was calculated via the infinitesimal basis transformations method. We then derived the spinful Hamiltonian of black phosphorene. Final results have shown that with considering spin-orbit interaction, the conduction and valence bands have been split into six new sub-bands with different degeneracies. Published by AIP Publishing.