Boundary conditions for envelope functions at interfaces between dissimilar materials

In the effective-mass approximation, the validity of the boundary conditions for an envelope function F acoss an interface between two different materials is predicated on the similarity of the nearest band-edge Bloch functions. Such approximations break down when the two materials are very dissimilar, e.g., a metal and a semiconductor. By studying one-dimensional model potentials we derive more accurate functional relations between the total wave function and its envelope. From these the usual boundary conditions are restored (continuity of F and F’/${\mathit{m}}^{\mathrm{*}}$) when similiar band edges are aligned at abrupt interfaces in semiconductor heterostructures. More importantly, we also derive appropriate boundary conditions for the case when band edges with qualitatively different Bloch functions are aligned. In particular, we find unique boundary conditions for interfaces between regions with and without periodic potentials. These boundary conditions are shown to apply to semiconductor heterostructures described within the nested effective-mass approximation and are arguably reasonable approximations for metal-semiconductor interfaces.