ON THE THEORY OF ENVELOPE FUNCTIONS IN LATTICE-MATCHED HETEROSTRUCTURES

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 one-dimensional envelope function per band index n, in contradistinction to what is generally assumed. In the important case of [001] and [111] grown heterostructures it is shown that for parallel wave vectors contained in a region around the GAMMA-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.