Optical gain and co-stimulated emissions of photons and phonons in indirect bandgap semiconductors

A model calculation on optical gain and co-stimulated emission of photons and phonons in indirect bandgap semiconductors such as silicon is presented. An analytical expression for optical gain via phonon-assisted optical transitions in indirect bandgap semiconductors is presented. Population inversion can occur when the difference between the quasi-Fermi levels for electrons and holes is greater than the photon energy. The rate equations and their steady state solutions for electron, photon, and phonon involved in the phonon-assisted optical transitions are presented. It is shown that co-stimulated emissions of photons and phonons will occur when the threshold condition for laser oscillation is satisfied. The magnitude of optical gain in bulk crystalline silicon is calculated and shown to be smaller than the free carrier absorption at room temperature. However, it is shown, for the first time, that the optical gain is greater than the free carrier absorption in bulk crystalline silicon at the temperature below 23 K. Thus, the calculation predicts that the co-stimulated emissions of photons and phonons could take place in bulk crystalline silicon at the low temperature.