Optical gain coefficients of silicon: a theoretical study

A theoretical model is presented and an explicit formula is derived for calculating the optical gain coefficients of indirect band-gap semiconductors. This model is based on the second-order time-dependent perturbation theory of quantum mechanics by incorporating all the eight processes of photon/phonon emission and absorption between the band edges of the conduction and valence bands. Numerical calculation results are given for Si. The calculated absorption coefficients agree well with the existing fitting formula of experiment data with two modes of phonons: optical phonons with energy of 57.73 meV and acoustic phonons with energy of 18.27 meV near (but not exactly at) the zone edge of the X-point in the dispersion relation of phonons. These closely match with existing data of 57.5 meV transverse optical (TO) phonons at the X₄-point and 18.6 meV transverse acoustic (TA) phonons at the X₃-point of the zone edge. The calculated results show that the material optical gain of Si will overcome free-carrier absorption if the energy separation of quasi-Fermi levels between electrons and holes exceeds 1.15 eV. (paper)