Theory of Electrons in Nearly Periodic Potentials

This dissertation presents work on the theory of electrons in nearly periodic potentials. The subject is motivated by recent interest in the electronic properties of nanometer-scale semiconductor structures grown by modern molecular-beam epitaxy methods. The nearly periodic potentials arise either from uniform crystals in the presence of scalar potentials varying slowly in space, or from compositionally graded crystals with or without applied potentials. The quantum mechanics of electrons moving in these nearly periodic potentials is conveniently described in a basis of localized generalized Wannier functions, similar to the conventional Wannier functions for strictly periodic potentials. I begin by constructing generalized Wannier functions for both weakly perturbed crystals and compositionally graded crystals. The basis of generalized Wannier functions is then used to construct an effective Hamiltonian describing the motion of electrons in compositionally graded crystals. This Hamiltonian is valid throughout a given energy band. Near the edges of a simple or composite band, this effective Hamiltonian reduces to an effective mass Hamiltonian with position dependent effective mass. I then examine more general states--not restricted to the vicinity of a band edge--in crystals with composition and applied potential variation in one direction. I obtain a WKB-type solution for the envelope functions, as well as the appropriate turning point connection rules and bound state quantization condition. Finally, I turn to the problem of the dynamics of electrons in graded crystals when driven by a time-dependent electric field.