On the theory of envelope functions in lattice-matched heterostructures

Abstract

Starting from the envelope function formalism of Luttinger and Kohn (LK) for a three-dimensional crystal, exact one-dimensional envelope function equations are derived valid for lattice-matched heterostructures. It is shown that the transition to one-dimensional envelope functions generally leads to more than one-dimensional envelope function per band index n, in contradistinction to what is generally assumed. In the important case of [0 0 1] and [1 1 1] grown heterostructures it is shown that for parallel wave vectors contained in a reion around the Γ-point the number of one-dimensional envelope functions reduces to one, in accordance with the usual applied theory in the literature. Exact equations are derived governing the one-dimensional envelope functions. It is indicated to which form the one-dimensional envelope functions reduce for a perfect bulk crystal. This is of interest in a flat band approach in which such functions are used to construct wave functions which have appropriate continuity properties everywhere.