Operator ordering and boundary conditions for valence-band modeling: Application to [110] heterostructures

A general method to derive the operator ordering and boundary conditions within the framework of the recently developed exact envelope-function theory is presented. An ordered form of the familar Luttinger-Kohn Hamiltonian is derived which can be regarded as the natural starting point for calculations involving heterostructures, such that transforming with an appropriate angular momentum basis, the derived ordering provides an unambiguous prescription for the boundary conditions across the interface. As an example, specific expressions are derived for structures with confinement along the [110] direction where the resulting boundary conditions are found to differ significantly from those currently used in the literature. The effect on the calculated in-plane dispersions is examined for two material systems, where it is found the greatest differences occur in systems with a large difference in Luttinger parameters. Typically, the results show that employing the traditional boundary conditions tends to exaggerate any negative curvature of the valence bands near the zone center.