The bound states of two interacting two-dimensional magnetoexcitons with electrons and holes on the lowest Landau levels (LLLs) moving in-plane of the layer with equal but opposite oriented wave vectors and forming a molecular-type structures with the resultant wave vector →k = 0 were investigated. Four possible spin structures of two electrons and of two holes forming the bound states were considered. Two of them lead to the formation of the para and ortho magnetoexcitons in the presence of the electron-hole (e-h) Coulomb exchange interaction. In this case we have studied the interaction of two para magnetoexcitons and of two ortho magnetoexcitons with the resultant spin equal to zero. Another two variant, are actual when the Coulomb exchange e-h interaction is negligible small and the spin of two electrons separately and the effective spin of two holes are interconnected and forms the singlet or the triplet states with zero spin projections on the magnetic field direction. The spin states of the four particles were constructed combining the singlet two electron state with the singlet two hole state as well as the triplet two electron state with the triplet two hole state. Only the bound states of two electrons and of two holes with singlet-singlet and with triplet-triplet spin structures were studied. It was shown that the spin structure of the type singlet-triplet and triplet-singlet do not exist due to the hidden symmetry of the magnetoexcitons. The orbital structure of the 2D magnetoexciton with wave vector →k ≠ 0 is similar with an in-plane electric dipole with the dipole moment perpendicularly oriented to the wave vector. The bimagnetoexciton with resultant wave vector →k = 0 is composed from two antiparallel oriented electric dipoles moving with antiparallel wave vectors →k ≠ 0. Their relative motion in the frame of the bound states is characterized by the variational wave functions φn(→k) depending on the modulus →k. It was shown that the stable bound state in the lowest Landau levels approximation do not exist in four investigated spin combinations. Instead of them a deep metastable bound state with an activation barrier comparable with the ionization potential of the magnetoexciton with →k = 0 was revealed in the triplet-triplet spin configuration. Its orbital structure in the momentum space representation is characterized by the maximal exciton density on the in-plane ring and with zero density in the center.
The energy spectrum of the two-dimensional cavity magnetoexciton-polaritons has been investigated previously, using exact solutions for the Landau quantization (LQ) of conduction electrons and heavy holes (hhs) provided by the Rashba method. Two lowest LQ levels for electrons and three lowest Landau levels for hhs lead to the construction of the six lowest magnetoexciton sates. They consist of two dipole-active, two quadrupole-active, and the two forbidden quantum transitions from the ground state of the crystal to the magnetoexciton states. The interaction of the four optical-active magnetoexciton states with the cavity-mode photons with a given circular polarization and with well-defined incidence direction leads to the creation of five magnetoexciton-polariton branches. The fifth-order dispersion equation is examined by using numerical calculations and the second-order dispersion equation is solved analytically, taking into account only one dipole-active magnetoexciton state in the point of the in-plane wave vector k→∥=0. The effective polariton mass on the lower polariton branch, the Rabi frequency, and the corresponding Hopfield coefficients are determined in dependence on the magnetic-field strength, the Rashba spin–orbit coupling parameters, and the electron and hole g-factors.
The energy spectrum of the collective elementary excitations of a 2D electrom-hole (e-h) system situated in a strong
perpendicular magnetic field in a state of Bose-Einstein condensation (BEC) with wave vector k=0 was investigated
in the frame of Bogoliubov theory of quasiaveraes. The starting Hamiltonian describing the e-h system contains not
only the Coulomb interaction between the particles lying on the lowest Landau levels(LLLs), but also the
supplementary interaction due to their virtual quantum transitions from the LLLs to the excited Landau levels and
return back. This supplementary interaction generates after the averaging on the ground BCS-type state wave
function the direct Hartree-type terms with attractive character, the exchange Fock-type terms giving rise to
repulsion as well as the similar terms arising after the Bogoliubov u - v transformation. The interplay of these three
parameters gives rise to the resulting different from zero interaction between the magnetoexcitons with wave vector
k=0 and to stability of their BEC as regards the collapse. It influences also on the single particle energy spectrum as
well as on the collective elementary excitations. It consists from six branches. Four of them are excitonic-type
branches, two of them being of exciton origin whereas the second two are the quasienergy branches representing the
mirror reflection of previous two branches. Another two branches are the optical and acoustical plasmon branches.
The indirect aftractive interaction between the electron and holes lying on the lowest Landau
levels on the surface of a two-dimensional structure in the presence of a strong perpendicular
magnetic field appears due to their virtual quantum transitions to excited Landau levels as a result
of the Coulomb scattering. The influence of this indirect interaction on the ground state energy and
on the chemical potential of the Bose-Einstein condensed magnetoexcitons is determined. The
corrections to the energy spectrum and to the wave function of the lowest magnetoexciton band due
to the influence of the first three excited exciton bands were investigated.
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