Boundary conditions and spurious solutions in envelope-function theory

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Abstract
Spurious, non-normalizable wave functions can occur in the solution of the effective-mass equations for carriers confined in quantum wells. We show that the problem arises when too many boundary conditions are imposed on a wave function constructed from an overcomplete set of basis functions, as used in some applications of the k⋅p method. Correct boundary conditions that prevent this problem can be obtained directly from Schrödinger’s equation. The results obtained depend on the form of the potential at the interface, and so differ, in general, from the results of the usual effective-mass method; in particular, the envelope function need not be continuous at an interface. For a simple one-dimensional model we show that the boundary conditions reduce to those of conventional effective-mass theory in the limit of small band gaps. With the use of symmetry we deduce the correct number of boundary conditions on the envelope functions at a (100) interface between materials with the zinc-blende structure. © 1996 The American Physical Society.