Effects of Rashba spin-orbit coupling, Zeeman splitting and gyrotropy in two-dimensional cavity polaritons under the influence of the Landau quantization

We consider the energy spectrum of the two-dimensional cavity polaritons under the influence of a strong magnetic and electric fields perpendicular to the surface of the GaAs-type quantum wells (QWs) with p-type valence band embedded into the resonators. As the first step in this direction the Landau quantization (LQ) of the electrons and heavy-holes (hh) was investigated taking into account the Rashba spin-orbit coupling (RSOC) with third-order chirality terms for hh and with nonparabolicity terms in their dispersion low including as well the Zeeman splitting (ZS) effects. The nonparabolicity term is proportional to the strength of the electric field and was introduced to avoid the collapse of the semiconductor energy gap under the influence of the third order chirality terms. The exact solutions for the eigenfunctions and eigenenergies were obtained using the Rashba method [E.I. Rashba, Fiz. Tverd. Tela 2, 1224 (1960) [Sov. Phys. Solid State 2, 1109 (1960)]]. On the second step we derive in the second quantization representation the Hamiltonians describing the Coulomb electron-electron and the electron-radiation interactions. This allow us to determine the magnetoexciton energy branches and to deduce the Hamiltonian of the magnetoexciton-photon interaction. On the third step the fifth order dispersion equation describing the energy spectrum of the cavity magnetoexciton-polariton is investigated. It takes into account the interaction of the cavity photons with two dipole-active and with two quadrupole-active 2D magnetoexciton energy branches. The cavity photons have the circular polarizations σ k ± oriented along their wave vectors k, which has the quantized longitudinal component k z =± π/L c , where L c is the resonator length and another small transverse component k ∥ oriented in the plane of the QW. The 2D magnetoexcitons are characterized by the in-plane wave vectors k ∥ and by circular polarizations σ M arising in the p-type valence band with magnetic momentum projection M=± 1 on the direction of the magnetic field. The selection rules of the exciton-photon interaction have two origins. The first one, of geometrical-type, is expressed through the scalar products of the two-types circular polarizations. They depend on the in-plane wave vectors k ∥ even in the case of dipole-active transitions, because the cavity photons have an oblique incidence to the surface of the QW. Another origin is related with the numbers n e and n h of the LQ levels of electrons and heavy-holes taking part in the magnetoexciton formation. So, the dipole-active transitions take place for the condition n e =n h , whereas in the quadrupole-active transitions the relation is n e =n h ± 1. It was shown that the Rabi frequency Ω R of the polariton branches and the magnetoexciton oscillator strength f osc increase in dependence on the magnetic field strength B as Ω R ~ √B, and f osc ~ B. The optical gyrotropy effects may be revealed if changing the sign of the photon circular polarization at a given sign of the wave vector longitudinal projection k z or equivalently changing the sign of the longitudinal projection k z at the same selected light circular polarization. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015