Collective Properties of Excitons in Semiconductors

The problem of exciton interaction in semiconductors is considered in its multi-electron formulation. Expressions are obtained, in the approximation linear in the concentration, for the ground-state energy and for the law of dispersion of elementary excitations. Conditions for the Bose condensation of excitons are investigated and it is shown that low-density system of excitons behaves like a weakly nonideal Bose-gas. Furthermore, all quantities (the chemical potential, the rate of collective excitations) that depend on the two-particle scattering amplitude in the nonideal Bose-gas case are expressed in our analysis by the same formulas through the four-fermion interaction amplitude (two electrons and two holes) which includes, apart from the two-exciton scattering amplitude, the scattering amplitudes of two and three fermions as well as the terms connected with the Pauli statistics for the electrons and holes, and resulting from the fact that excitons are compound particles. These terms yield an essential positive contribution to the exciton scattering amplitude and may in principle ensure the stability of the ground Bose-condensed state even if there is a weak attraction between the excitons.