Theory of surface phonons in superlattices

We report the existence of surface-localized phonons for a superlattice consisting of alternating slabs (parallel to the surface) of two different crystals. The superlattice has a larger periodicity in the direction perpendicular to the slabs and therefore many phonon branches in the folded Brillouin zone. In the gaps existing between these phonon branches appear the surface localized modes. The theory is developed on a simple-cubic model. The simplicity of this model allows one to obtain in closed form the bulk and (001) surface Green's functions for this superlattice. The analytic knowledge of these functions enables us to study easily all the bulk and surface vibrational properties of a superlattice, which otherwise would require huge numerical calculations. We give here the analytic expression we obtained for the folded bulk phonon dispersion curves and also the expression that gives the surface-localized modes, which may appear within the extra gaps which exist between the folded bulk bands. A few figures for specific cases illustrate these results.