Envelope-function equation and variational method for superlattices

A description of the envelope-function equation of the superlattice (SL) is presented and then a review on the variational procedure for calculating e nergies and envelope functions of the SL is presented. The SL wave function and corresponding effective-mass equation are formulated in terms of a lin ear combination of Bloch states of the material with a smaller band gap. In this formulation, the difference in bulk Bloch states and band parameters between constituent materials is transformed to be a SL potential in the la rger band-gap material region, so that the effective-mass equation can be s olved as a whole. The present description provides easier implementation of numerical computations, because it does not require the interface matching as required in most of the previous approaches, and, in many cases, band p arameters are empirically known. On the other hand, Schlosser and Marcus's method has been used by Altarelli to include the interface-matching conditi on in the variational calculation for the SL structure in the multiband env elope-function approximation. This procedure is reexamined more thoroughly- and corresponding variational equations are presented in both general and r educed forms. The reduced form shows a difference from Altarelli's result. As an illustration of the application of the present approach, numerical co mputations are performed both for nonstrained GaAs/AlxGa1-xAs and strained In1-xGaxAs/In1-xGaxAsyP1-y SL's, and the results are compared with those re ported previously. [S0163-1829(98)00947-3].