On the phase transitions for the electronic excitations in semiconductors

Abstract

A model Hamiltonian for a system of interacting electrons, holes and Wannier excitons is derived. This system of electronic excitations is assumed to be in a quasi-equilibrium state. With the aid of Bogolubov's variational principal the thermodynamic potential is calculated. Using the most general mean-field Hamiltonian as a trial Hamiltonian, a set of coupled integral equations is obtained for the self-energies. These equations are solved numerically for equal effective masses of the electrons and holes. Below a critical temperature ofk _B T _c≈0.65E ^b_ex whereE ^b_ex is the exciton binding energy, we find a first order phase transition from an exciton rich phase into a degenerate electron-hole phase. The mechanical and thermal stability of both phases is proven. Below a critical temperaturek _B T _c≈0.11E ^b_ex the exciton system becomes degenerate (Bose-Einstein condensation). A complete phase diagram of these three phases is given.