Spontaneous and Stimulated Recombination Radiation in Semiconductors

Spectral line shapes of the radiation produced by band-to-band recombination of excess carriers in semi-conductors are calculated under the assumption that the momentum matrix element is the same for all initial and final states, i.e., that there is no momentum selection rule. The peak of the stimulated radiation falls at a lower photon energy than does the peak of the spontaneous radiation, except when $T=0$\ifmmode^\circ\else\textdegree\fi{}K. Some numerical results are given for simple parabolic bands, specifically for the case of electron injection into $p$-type GaAs, and are used to deduce the temperature dependence of the forward current which is necessary to maintain a fixed gain in the active region of a diode. The result is closely related to the temperature dependence of the threshold current in an injection laser, and gives reasonable agreement with experiment. The effect of a conduction band tail is briefly considered.