BOUNDARY-CONDITIONS FOR ENVELOPE FUNCTIONS AT INTERFACES BETWEEN DISSIMILAR MATERIALS

In the effective-mass approximation, the validity of the boundary cond itions for an envelope function F across an interface between two diff erent materials is predicated on the similarity of the nearest band-ed ge Bloch functions. Such approximations break down when the two materi als are very dissimilar, e.g., a metal and a semiconductor. By studyin g 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'/m ) when similar band edges are aligned at abrupt interfaces in semicond uctor heterostructures. More importantly, we also derive appropriate b oundary conditions for the case when band edges with qualitatively dif ferent Bloch functions are aligned. In particular, we find unique boun dary conditions for interfaces between regions with and without period ic potentials. These boundary conditions are shown to apply to semicon ductor heterostructures described within the nested effective-mass app roximation and are arguably reasonable approximations for metal-semico nductor interfaces.