Conductivity relaxation time due to electron-hole collisions in optically excited semiconductors

In an electron-hole plasma, electron-hole collisions conserve the total momentum but relax the electric current. The corresponding relaxation time ${\tau }_{e\mathrm{-}h}$ is obtained from two coupled Boltzmann equations. We take into account multivalley structure and screening. An exact solution is given when the electron-hole plasma is in the quantum limit, while in the classical limit, we use a variational approach. We find that when the plasma density is increased at fixed temperature, ${\tau }_{e\mathrm{-}h}$ goes through a minimum reached roughly when the plasma becomes degenerate. The size of ħ/${\tau }_{e\mathrm{-}h}$ at the maximum is mainly controlled by the exciton binding energy.