The densities of states in the conduction and valence bands appropriate for the $p$ region of the junction have been calculated self-consistently in the screened potential and effective-mass approximations. Such a density of states for one particular band consists of a tail part taken from the theory of Halperin and Lax, an unperturbed parabolic density of states above the tail, and a smooth interpolation in between. The use of the unperturbed parabolic band is justified, since the perturbation technique of Bonch-Bruevich and a straightforward second-order perturbation calculation both show that the distortion of the band due to the presence of impurities at the concentration employed in a typical laser is less than 5%. Contrary to the generally accepted assumption and Stern's calculation using Kane's density of states of a long and reasonable large conduction band tail, our results show that the tail is negligibly small compared to the valence band tail. On the basis of this calculation, it is concluded that, for a typical laser, the electron quasi-Fermi level at lasing threshold for temperature above 77 °K should be in the parabolic portion of the band and not in the tail as is often assumed without justification. The approximations of using linear screening for the impurity potentials and the Gaussian statistics for the impurity distribution which are implied in the density-of-state functions of Kane and of Halperin and Lax are considered in detail.

• Received 27 March 1970

DOI:https://doi.org/10.1103/PhysRevB.2.4117