BOUNDARY-CONDITIONS AND SPURIOUS SOLUTIONS IN ENVELOPE-FUNCTION THEORY

Spurious, non-normalizable wave functions con 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 bas is functions, as used in some applications of the k . p method. Correc t boundary conditions that prevent this problem can be obtained direct ly From Schrodinger's equation. The results obtained depend on the for m of the potential at the interface, and so differ, in general, from t he results of the usual effective-mass method; in particular, the enve lope function need not be continuous at an interface. For a simple one -dimensional model we show that the boundary conditions reduce to thos e of conventional effective-mass theory in the limit of small band gap s. With the use of symmetry we deduce the correct number of boundary c onditions on the envelope functions at a (100) interface between mater ials with the zinc-blende structure.