Envelope-function formalism for electrons in abrupt heterostructures with material-dependent basis functions

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Abstract
An envelope-function model is derived for electrons in abrupt semiconductor heterostructures. It uses material-dependent basis functions that diagonalize the bulk zone-center Hamiltonian in each unit cell of the crystal. The initial formalism is exactly equivalent to the one-electron Schrödinger equation; approximations suitable for abrupt junctions are then developed. The abrupt change in microscopic potential at an ideal interface is shown to introduce no new interband coupling; all such coupling arises from the kinetic energy, specifically from a momentumlike matrix element containing the gradient of the basis functions with respect to changes in material composition. This generates interface effects not included in conventional envelope-function theories, such as zone-center coupling between heavy and light holes. An effective-mass equation is derived for the conduction band of a layered zinc-blende structure; it exhibits both envelope discontinuities and δ-function potentials, in agreement with the transfer matrices derived from other microscopic theories. © 1996 The American Physical Society.