Exciton Condensation Under High Magnetic Field

The new results in the theory of Bose-Einstein condensation (BEC) of the two-dimensional (2D) magnetoexcitons formed by the high-density electron–hole (e–h) pairs created on the semiconductor mono-layer in a strong perpendicular magnetic field are reviewed. One of them is the metastable dielectric liquid phase (MDLP) formed by the 2D magnetoexcitons BEC-ed on the single-particle state with sufficiently large values of the wave vector k, so that its product kl with the magnetic length l equals about kl ≈ 3–4. This state was revealed in the conditions when the electrons and holes are situated on the lowest Landau levels (LLLs) and the polarizability of the Bose gas was calculated on the base of the Anderson-type coherent excited states. They give rise to correlation energy and to chemical potential displaying a nonmonotonous dependence on the filling factor v ² with a relative minimum and with positive compressibility in its vicinity. The influence of the excited Landau levels (ELLs) on the quantum states of the e–h system is due to the virtual quantum transitions of particles from the LLLs to ELLs during the Coulomb scattering processes and to their subsequent return back. These quantum transitions were taken into account in the frame of the second order perturbation theory giving rise to an effective Hamiltonian describing the supplementary indirect interactions between the particles lying on the LLLs. This interaction is characterized by a small parameter equal to the ratio r of the magnetoexciton ionization potential I _ex(0) to the Landau quantization energy ħω _c . The parameter r = I _ex(0)/ħω _c , decreases as H ^−1/2 with the increasing the magnetic field strength H. The supplementary interaction is attractive, making the magnetoexcitons in the Hartree approximation more robust. Nevertheless its exchange, Fock terms as well as the Bogoliubov u–v transformation terms give rise to positive, repulsion-type contributions to the chemical potential. The Bose gas of magnetoexcitons with k = 0 becomes weakly nonideal when the ELLs are taken into account. The collective elementary excitations of two ground states corresponding to BEC-ed magnetoexcitons forming either a nonideal Bose gas with k = 0 or the MDLP with kl ≈ 3–4 were studied in the frame of the perturbation theory with the infinitesimal parameter v ² (1 – v ²), chosen as a product of the filling factor v ² and of the phase space filling factor (1 – v ²). The collective elementary excitations in both cases consist from the exciton and plasmon branches. Due to the presence of the condensate there are energy and quasi-energy branches. The self-energy parts containing the unknown frequency in denominators increase the degree of the dispersion equations and give rise to mixed exciton-plasmon and exciton–exciton elementary excitation branches.