A Effective-Mass Theory of the Electronic Properties of Semiconductor Heterostructures.

A theory of the electronic properties of semiconductor heterostructures is developed which is simple enough to provide both intuitive and analytic analyses of these structures, yet rich enough to include all of their most important physical features. The usual effective-mass theorem in solids is only valid for infinite crystals. A multi-band effective -mass theorem is proven which is applicable in finite crystalline systems, regardless of any conditions imposed at their boundaries. Each layer of a heterostructure is modeled as a piece of homogeneous bulk crystal, so this theorem can describe the effects of slowly varying potentials within each layer. At the abrupt interfaces between the layers, theorem does not apply, and boundary conditions on the effective-mass envelope functions must be derived from a first-principles analysis of the interface. For narrow -gap semiconductors, this analysis requires a multi-band representation of the envelope functions. Using a fully three-dimensional(' )k(.)p method, one-band and two-band models of the interface are derived. Boundary conditions applicable to energies near the band gaps are found. Boundary conditions are also found for a six-band(' )k(.)p model, explicitly including spin, the strong spin -orbit interactions in narrow-gap semiconductors, and the coupling of the valence bands when the wavevector is not perpendicular to the interfaces. Analytic eigenvalue equations are obtained for sandwich and superlattice heterostructures, and are compared with experimental data for both Type I and Type II structures. The agreement with the data is excellent. A tight-binding theory of the interface gives boundary conditions for one-orbital and two-orbital models. A formalism of transfer and propagator matrices is developed which facilitates a comparison between the two theories. The differences between the two theories are slight, suggesting that the underlying physics of these systems is well-represented in this approach. Other heterostructure theories are analysed, and compared to the theories developed in this thesis. Several applications are suggested, including calculations involving spin-splitting and self-consistent potentials. Extensions are suggested which will allow the theory to encompass heterostructures composed of a very large class of materials.