Theory of surface and interface transverse elastic waves in N-layer superlattices

N-layer superlattices are formed out of a periodic repetition of N different slabs. Such materials for N=3 and 4 are now easily grown by molecular-beam epitaxy. We present closed form expressions for localized and resonant transverse elastic waves associated with the surface of a semi-infinite N-layer superlattice, with or without a cap layer, and in contact either with vacuum or with a homogeneous substrate. We also calculate the corresponding Green's- function and densities of states. These general results are illustrated by a few applications to four-layer superlattices made of Nb-Fe-Nb-Cu and Nb-Cu-Nb-Cu. We show that increasing the number of the layers in each unit cell of the superlattice increases, in general, the number of the minigaps and surface modes. We generalize some of the results obtained previously in the case of two-layer superlattices, namely, (i) the creation from an infinite superlattice of a free surface gives rise to delta peaks of weight (-1/4) in the density of states, at the edges of the superlattice bulk bands; (ii) by considering together the two complementary semiinfinite superlattices obtained by cleavage of an infinite superlattice along a plane parallel to the interfaces, one always has as many localized surface modes as minigaps, for any value of the wave vector k(parallel to) (parallel to the interfaces). Finally, we investigate the localized and resonant modes associated with the presence of a cap layer on top of the superlattice.