Theory of optical gain in strained-layer quantum wells within the 6×6 Luttinger-Kohn model

We study the effects of the spin-orbit (SO) split-off band coupling with the heavy- and the light-hole bands on the band structure, density of states, dipole moment, and the linear and nonlinear optical gains of strained-layer quantum wells by comparing the 6×6 and the 4×4 Luttinger-Kohn models. First, a unitary transformation is found that block diagonalizes the 6×6 Hamiltonian into two 3×3 blocks that are real symmetric in the finite difference formulation. We find that the peak gains are overestimated, especially in the case of a tensile quantum well, in the 4×4 model, for the InGaAs-InP quantum-well systems which show weak SO coupling. The change of the density of states is suggested to be the dominant factor for the discrepancy of the results between the 6×6 and the 4×4 models in the linear gain. For nonlinear gain the discrepancy between the two models is smaller than that of the linear gain because the change of the fourth-order dipole moment in the nonlinear gain compensates that of the density of states. We also considered InGaP-In(AlGa)P quantum wells which have very narrow SO split-off energy. In this case, because of the strong coupling between the SO bands and the hole bands, the InGaP-In(AlGa)P quantum-well systems show more complicated behavior. The peak gain of strained quantum well is overestimated in the low carrier injection region and is underestimated in the high injection, in the 4×4 model. On the other hand, the peak gain of an unstrained quantum well is overestimated in the 4×4 model over the wide range of carrier densities.