Electron and Phonon Bound States and Scattering Resonances for Extended Defects in Crystals

Abstract

By making use of the translational symmetry associated with line and plane defects in crystals, we define certain subbands of the unperturbed electron and phonon bands. Certain Wannier functions and Green's functions associated with these subbands are defined and are used to study the existence of localized electron and phonon states and scattering resonances associated with extended defects. By firstly considering very simple examples of such perturbations, of arbitrary strength and secondly considering perturbations of quite general form but of small strength, we establish the general existence of electron and phonon bound states for line and plane defects, and the existence of electron and phonon scattering resonances for line defects. The effects of different characteristics of the unperturbed band structure are indicated. In contrast to the above results, bound states for point defects and scattering resonances for point and plane defects do not occur unless the perturbation exceeds a certain minimum strength. Our results underline the basic importance of including the band structure in scattering problems of this type and also the dangers of relying on perturbation approaches. Attempts which have been made to arrive at properties of crystal defects by the study of one dimensional models should also be reviewed in the light of our results. The above-mentioned bound levels form continuous bands which may lie partly between or within the allowed bands of the unperturbed crystal, and those electron levels that lie in the forbidden regions should have an important influence on the properties of semiconductors and insulators. Such effects have long been observed, and have been interpreted usually in terms of the "dangling bond" theory of Shockley and Read; our theory gives a much more general basis for their existence and, although the difficulties are considerable, seems to offer a means of quantitative investigation which previously did not exist. The electron and phonon scattering resonances seem to afford a natural explanation of the long-standing discrepancy between theory and experiment on the subject of dislocation contributions to electrical and thermal resistivities.