Interface connection rules for effective-mass wave functions at an abrupt heterojunction between two different semiconductors

The problem of the connection rules for effective-mass wave functions across an abrupt heterojunction is investigated by expressing the results of a one-dimensional tight-binding approximation in terms of effective-mass wave functions. The widely used conventional connection rules of continuous-wave function and first derivative are only an approximation, invalid in all but the simplest limiting case. The connection-rule problem is reformulated by first extrapolating the effective-mass wave functions on the two sides of the heterojunction across the interface, as if the semiconductor were homogeneous. In each of the two lattice planes adjacent to the interface, the extrapolated wave function must then be proportional to the true wave function, with two proportionality coefficients that depend on certain matrix elements. By suitably renormalizing the wave function, the connection rules for type-I heterojunctions become, to the first order, the same as if a $\delta$-function scatterer were superimposed on the band-edge discontinuity. The effects of the new connection rules on the ground state of a symmetric square well are discussed as an example.