New approach to the k·p theory of semiconductor superlattices

Along with the growing interest in semiconductor superlattices, various theoretical schemes have been proposed to study the nature of the electronic states within these structures. The work presented here highlights a new method to investigate the electronic and optical properties of semiconductor superlattices. The backbone of the theory rests on a realistic description of the complex-k band structure of the constituent semiconductors coupled with a suitable set of boundary conditions for the superlattice wave function. The bulk Bloch solutions, propagating and evanescent, in each semiconductor are described within a full-zone k · p Hamiltonian that provides an accurate description of the solutions up to the first Brillouin zone edge. An attractive feature of the present treatment is that the complex-k bulk Bloch solutions of each constituent semiconductor are expanded on the same set of zone-center basis functions. A new technique for constructing the k · p Hamiltonian of each constituent semiconductor is presented. The superlattice wave function is described by a linear combination of propagating and evanescent bulk Bloch solutions. The expansion amplitudes are determined by imposing a set of boundary conditions on the superlattice wave function across the superlattice interfaces. These boundary conditions are used to formulate an eigenvalue problem whose solution yields directly the corresponding superlattice states associated with real or complex superlattice wave vector q. This method provides an accurate technique to treat superlattices where one of the constituent semiconductors has an indirect energy band gap. An exposition of the formalism is presented, and the physical origin of the superlattice states is studied. The test case of the GaAs–AlAs (100) superlattice is presented. Pertinent applications are also discussed