Hot electrons and curves of constant gain in long wavelength quantum well lasers

In long wavelength quantum well lasers the effective electron temperature (T_e) is often a strong function of the pump current and hence the T_e correlates with the carrier concentration n in the active region. On the other hand, the material gain g in the active layer depends on both variables, g=g(n, T_e), We discuss a convenient way of analyzing this situation, based on considering the contours of constant gain g on the surface g(n,T_e). This is qualitatively illustrated with two model examples involving quantum well lasers, the long-wavelength quantum well laser with current dominated by the Auger recombination and the unipolar quantum cascade laser.