The one-electron states of imperfect crystals

Abstract

The effect of localized structural flaws on the electron states of ideal crystals is treated generally. The work is based on the one-electron approximation, and on the assumption that the states are all generated by the same Hamiltonian operator. The conclusions are independent of further approximation. The crystal symmetry properties are used both to classify problems, and to determine for any given problem, those Wannier functions most appropriate for use as base functions. Unlocalized states have energies lying in the bands of the ideal crystal, but within certain limits whose positions depend on symmetry factors and which in general do not coincide with the band edges. States with energies outside these limits are localized on the flaws to an extent decreasing as a limit is approached. As the limit is reached the state becomes unlocalized. The procedure for obtaining results specific to any given class of problems is illustrated by reference to flaws having plane symmetry (e.g. clean surfaces, stacking faults and twin boundaries), line symmetry (e.g. wires and certain dislocations), and point symmetry (e.g. impurity and interstitial atoms, lattice site vacancies and a surface with an adsorbed molecule).