Theory of generalized Wannier functions for nearly periodic potentials

There is a great interest in the properties of electrons in nanostructures with superposed potentials varying slowly in space, or with slowly graded composition, or both. The quantum mechanics of these electrons, moving in nearly periodic potentials, is conveniently described in a basis of localized generalized Wannier functions, similar to the conventional Wannier functions for strictly periodic potentials. This has led us to consider generalized Wannier functions for (i) weakly perturbed and (ii) compositionally graded crystals. We first construct generalized Wannier functions for uniform crystals with an everywhere weak perturbation. The corrections to an unperturbed Wannier function of a given band may be chosen to involve only the unperturbed Wannier functions from other bands. We then consider crystals with slowly varying composition, and construct generalized Wannier functions from the Wannier functions of uniform crystals.