Direct derivation of effective-mass equations for microstructures with atomically abrupt boundaries

Abstract

A direct general method for deriving effective-mass equations for microstructures with atomically abrupt boundaries is presented. The principal assumption is that the envelope functions are slowly varying on the scale of the lattice period. The band-edge Bloch functions are not assumed to be the same on both sides of an interface and it is shown how the differences can be taken into account perturbatively. The particle in a box method is known to work well in many situations. To demonstrate why, a derivation of the effective-mass equation is carried out explicitly for the case of conduction-band states of a type-I microstructure composed of zinc-blende crystals without spin-orbit interaction. The derivation provides much insight into why the effective-mass method works so well. The method is illustrated by applying it to a one-dimensional superlattice problem. For this model problem, the effective-mass approximation to the wave function is seen to be good even for a quantum well one lattice period wide.