Impurity states associated with subsidiary energy-band minima

Abstract

Abstract Two results have been established concerning the existence of localized electronic states associated with a point impurity in a substance having a spherical energy band, E(k) = ϵ0 + ϵ1 × cosk + ϵ2cos2k, where k = ¦k¦, with a subsidiary minimum at k = 0. The 1st result is a generalization of Slater and Koster's result, that in 1 dimension an energy band with a subsidiary minimum has no localized states associated with this minimum if the impurity potential is of the δ-function type; i.e., it has only 1 matrix element between Wannier functions, a diagonal one referring to 1 site. We show that this result also holds for a spherical band in 3 dimensions. Our 2nd result is that, for the Coulomb-impurity potential screened by the static dielectric constant, and the above spherical band, there are hydrogen-like localized states built out of states near the subsidiary minimum, even when all powers of k in E(k) are taken into account in the equation for the envelope function. The deviation of the impurity potential from slow variation causes a long lifetime for decay of the localized state into conduction states of the same energy. For a typical shallow impurity state, the lifetime is ≅ 10−8−10−9 sec.