Theoretical model for the optical gain coefficient of indirect-band-gap semiconductors

A theoretical model for the calculation of the optical gain coefficient of indirect-band-gap semiconductors is presented. An explicit formula for the calculation is derived from the second-order time-dependent perturbation theory of quantum mechanics, which incorporates all the eight possible processes of photon/phonon emission and absorption. Conditions for achieving the population inversion and optical gain are investigated. The results show that, under the assumptions of constant interaction matrix elements and a simplified band structure with equal energy gaps at the conduction band minimum and the valence band maximum, a rather simple result can be obtained, in which the conditions for achieving population inversion and optical gain are in phonon-emission-dominant processes, the Fermi energy difference between electrons and holes is required to be larger than the photon energy plus the phonon energy and in phonon-absorption-dominant processes, the Fermi energy difference between electrons and holes is required to be larger than the photon energy minus the phonon energy. Since phonon absorption and emission processes coexist in a real situation, the realistic conditions for achieving optical gain are far more complicated and should be calculated numerically. We calculated the optical gain coefficient of bulk crystalline silicon. The results show that, without the suppression of other second-order optical-loss processes (e.g., free-carrier absorption), it is unlikely that the magnitude of optical gain will overcome the optical loss to achieve the lasing condition for bulk crystalline silicon.