Densities of states and properties of spontaneous and stimulated emission in semiconductor lasers

Abstract

The spontaneous and stimulated absorption spectral functions are calculated using a band model consisting of a tail part given by Halperin and Lax, a parabolic part above the tail, and an optical model with an energy-dependent matrix element and no selection rule for the radiative transitions. The use of a parabolic band above the tail is justified since a second-order perturbation calculation shows negligible distortion of the band in this region. The parameters associated with the densities of states are determined self-consistently, involving no adjustable constants. Contrary to the generally accepted assumption and the results of Stern's calculation using Kane's density of states of a long and reasonably large-state density in the conduction-band tail, it is shown that the tail is negligibly small compared to the valence-band tail. Therefore, it is concluded that, for a typical laser, the electron quasi-Fermi level at threshold above 77°K should be in the parabolic portion of the conduction band instead of in the tail as is usually assumed without justification. The energy dependence of the matrix element is that for the average parabolic conduction band to an acceptor-level transition and should be a suitable one due to small conduction-band tail and the population of most of the holes in the vicinity of the acceptor ionization energy. General properties such as the gain current relationship and the temperature dependence of current and carrier quasi-Fermi levels are compared with those calculated without band tails and with band tails given by Kane's model. Considerable difference in each case is found and discussed. The calculations of the temperature dependence of the threshold current and the current dependence of the super radiance spectra are then applied to GaAs diffused diodes with substrate doping of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3 \times 10^{18}</tex> cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> , taking into account the temperature dependence of the cavity loss and the nonuniform accepter distribution in the p layer. Detailed comparison with experimental data is made and good quantitative agreement is obtained in either case, showing strong support to the conclusion on the band-tail structure. The approximations of using the linear screening and Gaussian statistics that were employed by Halperin and Lax, and Kane to obtain analytical expressions for the densities of states used in Stern's calculation are discussed.