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

Abstract

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 [1]. 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 kz = ±π/Lc, where Lc 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 σM 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 ne 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 ne = n h , whereas in the quadrupole-active transitions the relation is ne = n h ± 1. It was shown that the Rabi frequency ΩR of the polariton branches and the magnetoexciton oscillator strength fosc increase in dependence on the magnetic field strength B as ΩR ∼ √ B, and fosc ∼ 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 kz or equivalently changing the sign of the longitudinal projection kz at the same selec...

Metastable bound states of the quasi–bimagnetoexcitons in the lowest Landau levels approximation

“…The interaction coefficients depend only on the difference ( pq  ) for the electron-electron ( ee  ) and the hole-hole ( hh  ) interactions, and on the sum ( pq  ) for the electron-hole ( eh  ) interactions. The magnetoexciton creation operator introduced in [12] but with spin labels [15,16] The wave function of the magnetoexciton is…”

Dispersion laws of the two-dimensional cavity magnetoexciton-polaritons

“…The Hamiltonians of the 2D magnetoexcitons, of the cavity photons and of their interaction have been obtained in Refs. [2,3], see e.g. Egs.…”

“…The magnetoexciton-cavity-photon interaction Hamiltonian taking into only the resonance terms and neglecting by the antiresonance ones is described by the formula (79) of Ref. [2] ( ) (…”

“…The corresponding energy levels calculated for the GaAs-type QW with their dependences on the magnetic and electric field strengths depicted in Figs.5-7 of Ref. [2] 3 represent: two dipole-active branches, two quadrupole-active branches, and two forbidden branches in both approximations. These results will be used in the present study.…”

“…The unit vector 3 a  is perpendicular to the QW plane. The energy spectrum of the cavity photons propagating with arbitrary 3D wave vector of a small in-plane component | | k  of the photon wave vector k  as compared with the quantized longitudinal component | | C k L π = the cavity photon dispersion law is given by the formula (64) of the Ref [2].…”