Galerkin Method for Calculating Valence-Band Wavefunctions in Quantum-Well Structures Using Exact Envelope Theory

The ability to calculate accurately the valence-band structure in semiconductors is important in the design of quantum-well (QW) semiconductor devices. The Galerkin method for calculating accurate analytic approximations for multicomponent valence-band wavefunctions is computationally fast and efficient. The Galerkin method is, in fact, an improved version of an earlier Raleigh–Ritz-type variational method. In this paper, we remark that both the variational method and the more recently proposed Galerkin method are formulated such that they imply symmetrized boundary constraints at material interfaces, with the symmetrized nature of the constraints arising from neglecting the ordering of the operators. Burt's exact envelope-function theory for semiconductor microstructures has been used to demonstrate, however, that the commonly used symmetrized boundary constraints for material interfaces are unphysical. We therefore present a modified version of the Galerkin method that implicitly assumes physically reasonable, exact envelope-function boundary constraints. Simulations show that the modified Galerkin method successfully produces physical, semi-analytic results that are consistent with exact envelope theory.