Localized modes at extended defects in crystals

Abstract

Abstract Perturbations in the homogeneity of a crystal can give rise to localized modes of vibration. We have discussed the simplest cases of extended defects, namely planes of impurity atoms with special directions (001, 011, etc.) in a simple cubic crystal with nearest neighbour interaction. Extended defects, such as planes of impurity atoms, will have localized modes with exponentially decreasing amplitudes in the direction perpendicular to the planes and wave-like character in directions parallel to them. The different modes of localized vibrations have been analyzed group-theoretically. It comes out that there will be in general an acoustical and an optical branch of localized modes for a plane defect, the occurrence of which and the frequencies relative to the band of the ideal lattice frequencies depend on the defect-parameters (changes in mass and force constants). In the limit of vanishing coupling between defect plane and “host” lattice we get a free surface, which has been considered by Wallis et al. This limiting case has only an acoustical branch, which is identical with the Rayleigh-surface modes for long waves (provided the force-constants fulfill the conditions allowing localized states at all). Also lines of defects can have two branches of modes. The details depend as in other cases on the defect-parameters. If the homogeneity of a crystal lattice is disturbed by a defect, some of the eigenvibrations can be localized modes, i.e. modes the vibration amplitudes of which decrease exponentially with increasing distance from the defect. The occurrence of such localized modes at point defects has been investigated in a large number of cases.(1–5) A free surface can be considered as a perturbation of the infinite lattice and localized modes can occur; this has been discussed in some cases too.(6–8) But in lattices there might be other sorts of defects (e.g. stacking faults, dislocations, etc.) of one and two dimensions. As an attempt to study the localized modes at such defects we have investigated the simplest cases, namely a plane and a line of impurity atoms in a simple cubic lattice; we will explain here only the results concerning the plane defects, the calculation of which is very simple. Line defects are somewhat more involved, but show in general the same features.(9)