Theory of the luminescence spectra of high-density electron–hole systems: crossover from excitonic Bose–Einstein condensation to electron–hole BCS state

We present a unified theory of luminescence spectra for highly excited semiconductors, which is applicable both to the electron–hole BCS state and to the exciton Bose–Einstein condensate. The crossover behavior between electron–hole BCS state and exciton Bose–Einstein condensate clearly manifests itself in the calculated luminescence spectra. The analysis is based on the Bethe–Salpeter equation combined with the generalized random-phase-approximation, which enables us to consider the multiple Coulomb scattering and the quantum fluctuation associated with the center-of-mass motion of electron–hole pairs. In the crossover regime, the calculated spectra are essentially different from results obtained by the BCS-like mean-field theory and the interacting Boson model. In particular, it is found that the broad spectrum, arising from the recombination of electron–hole BCS state, splits into the P- and P2-luminescence bands with decreasing particle density. The dependence of these bands on the carrier density is in good agreement with experiments for highly excited semiconductors.