Enhancement and spectral shift of optical gain in semiconductors from non-Markovian intraband relaxation

We will investigate some aspects of the phenomenological treatment of transition (line shape) and level broadening arising from phase and energy relaxation of Bloch states, respectively. Calculating the absorption/gain via the spontaneous emission formula and performing the broadening within the latter circumvents certain artifacts for both level and transition broadening. When using k-independent relaxation times, Gaussian (non-Markovian) broadening functions are superior to Lorentzian (Markovian) ones. In contrast to transition broadening, level broadening may even enhance the gain over its whole spectral width. In contrast to Lorentzian transition broadening, Gaussian transition broadening yields a blue shift of the gain maximum. The direction and magnitude of the spectral shift arising from Gaussian level broadening depends on the degree of degeneracy of the electron and hole bands involved. The level broadening can have a significant influence on the carrier statistics, which, consequently, has to be included into a consistent treatment. Thus, the phenomenological model functions depend distinctly on which kind of relaxation process is faster, energy or phase relaxation. For GaAs-like semiconductors, the application of transition broadening-even when using the spontaneous emission formula-to cases of dominant intraband relaxation yields significant numerical deviations from the correct treatment of level broadening. Broadening the energy levels requires an additional convolution integration. We will present an approximation, which yields excellent results for the gain in GaAs-like semiconductors. This enables one to include the significant effects of level broadening without increasing the numerical effort and leads to favorable formulae for experimental data fitting and device modeling.