Degeneracy of the crystal states and departure from the rigid band model for a strongly localized impurity

Abstract

A standard perturbation calculation is performed for a set of q degenerate states whose perturbation is a strongly localized impurity. The free-electron approximation and the LCAO approximation for the electron states are considered. The calculation which seems not to have been done before shows that there is precisely no change in the energy for q−1 states but a strong shift of energy for one state. This effect is fully independent of the size of the perturbation potential providing the perturbation calculation is applicable. At the same time the wave function corresponding to the shifted state is not a damped state but a simple superposition of the unperturbed band states. For a crystal having a spherical shape it can be shown that the charge distribution of the shifted state has an oscillatory decrease with the distance from the position of the impurity Calcul de perturbation standard pour un ensemble de q etats degeneres perturbes par une impurete fortement localisee, a l'approximation des electrons libres et a l'approximation LCAO: un des etats est fortement perturbe en energie alors que les (q−1) autres ne le sont pas, independamment de la taille du potentiel perturbateur, et l'etat deplace n'est pas localise mais est une combinaison d'etats de bande non perturbes