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.