Electronic states in graded-composition heterostructures

The purpose of this work is to develop and demonstrate the practical use of very efficient algorithms which can be readily used to study systems with significant inhomogeneities. We have developed two algorithms which can be used to study inhomogeneous discrete systems. The first one is an extension of known algorithms for homogeneous media and rests on the notion of transfer matrices, which are then used to evaluate the desired elements of the Green-function matrices to be employed in surface Green-function matching calculations. The second one is totally different and yields the Green-function matrices directly. Both work quite efficiently. When tested in practice for a graded-composition quantum well they give the same results for the local density of state at the interfaces. We apply the algorithms to the study of quantum wells consisting of AlAs in the barriers and ${\mathit{N}}_{\mathit{w}}$ layers ${\mathrm{Al}}_{\mathit{x}\left(\mathit{n}\right)}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}\left(\mathit{n}\right)}$As in the well region, and x varying linearly from x=0.3 to x=0. The ${\mathit{sp}}^3$ ${\mathit{s}}^{\mathrm{*}}$ empirical tight-binding model and the virtual-crystal approximation are used. We studied three wells of different thicknesses (${\mathit{N}}_{\mathit{w}}$=21,35,51). The ground-state and some excited-state energies of the conduction and valence bands are studied in detail: Spatial dependence and orbital composition of the corresponding spectral strengths show all the expected features.