Effective-mass Hamiltonian and boundary conditions for the valence bands of semiconductor microstructures

Using the recently developed exact envelope-function theory, an explicit form for the effective-mass Hamiltonian is derived for the valence bands (including the spin-orbit split-off band) of a semiconductor quantum well or superlattice. It is shown that the correct form of the Hamiltonian gives physically reasonable results, while the commonly used ‘‘symmetrized’’ form can produce nonphysical solutions for the heavy-hole subbands in which the quantum-well effective mass is very sensitive to the difference in Luttinger parameters between the well and the barrier. This problem arises because the correct boundary conditions for the heavy-hole states are determined exclusively through interaction with other p states, while the symmetrized boundary conditions implicitly incorporate the much larger s-state interaction, hence they substantially overestimate the magnitude of the interband coupling.