Properties of Electron Gas in Constant Magnetic and Electric Fields.

摘要

A method is developed for defining localized states and effective Hamiltonians in perturbed crystals. The method is based on the localization ideas in the KQ representation for perfect lattices. An equation is derived defining localized states for perturbations caused by an impurity, the magnetic and electric fields. First, the impurity problem is considered in detail. A correction term is obtained to the one-band Koster-Slater effective Hamiltonian. It is shown to be significant for bound states and scattering cross sections of a localized impurity. Second, an orthonormal set of localized states for a crystal with a perturbation is developed. It includes the impurity problem, surface states, superlattices and other perturbations. These localized states are used for deriving one-band effective Hamiltonians up to second order of the perturbation expansion. Relatively simple results for localized states and one-band Hamiltonians are obtained in the cases of wide and narrow energy gap crystals. The orthonormal set of states that diagonalize the Hamiltonian is also used for deriving an expression for local charge densities in a perturbed crystal which can directly be compared with experiment.