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

Abstract

In this work, we have presented the spinful low energy six-band Hamiltonian representation of black phosphorene about the Γ point based on the k.p model via Löwdin 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., Γ4− and Γ2+ related to states B3u and B1g 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 D2h 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.

k·p theory for phosphorene: Effective g -factors, Landau levels, and excitons

“…There have already been applications of the k•p method to describe phosphorene. [19][20][21][22][23] However, to also study spin-orbit effects the effective models should fully exploit the symmetry of phosphorene by capturing the anisotropy of the interband dipole coupling. Effective g-factors illustrate this best.…”