Evaluation of various approximations used in the envelope-function method

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Abstract
We investigate a number of issues related to the application of the envelope-function method to calculate confined-state energies and subband structure in quantum-well structures. We first consider zone-center confined-state energies and show how the explicit elimination of spurious solutions from the envelope-function band structure leads to a slightly modified form of the standard result through which the conduction-band confined-state energies are calculated using a one-band model and an energy-dependent effective mass. We show that the effects of nonparabolicity can be predicted directly from the bulk band structure in an infinitely deep quantum well, and demonstrate how the bulk band structure can also be used to predict the errors in calculated confinement energies in wells of finite depth. The correct choice of boundary conditions still remains controversial for the calculation of valence-subband structure using the Luttinger-Kohn Hamiltonian. We compare the valence-band structure calculated with the lowest conduction band included either explicitly or treated as a remote band, using perturbation theory. We demonstrate that the boundary conditions recently derived by Burt and Foreman are correct. Finally, we compare the valence-band structure calculated using the 4×4 and 6×6 Luttinger-Kohn Hamiltonians. We show how the warping of the highest valence band is markedly different at both intermediate and large wave vectors when the spin-split-off band is included. The use of the axial model to calculate valence-band density of states is therefore questionable with the 6×6 Hamiltonian. The calculated warping is very sensitive to the values of the Luttinger γ parameters used, indicating the importance of investing more effort to determine these parameters accurately.