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We present a few recent or new theoretical results about surface acoustic localized and resonant modes associated with planar or deterministic rough surfaces. The following composite systems are considered: one or two adlayers deposited on a semi-infinite substrate and wires or grooves integrated near a planar surface. The surface modes can be obtained as well-defined features of the density of states resulting from a calculation of the Green functions in these heterostructures. In this work, the materials are described in the framework of the elasticity theory.
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Resonant Phonons in Adsorbates

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After a brief review on surface phonons, we focus on resonant phonons in adsorbed thin layers. Such resonances may in general be understood within the substrate bulk band as thin layer modes shiften and broaden by the interaction with the substrate phonons. These resonant phonons appear as well-defined peaks in the phonon density of states. Their experimental study started recently by Helium atom scattering spectroscopy. This review will end by a prospective revue done mostly after the presentation of two unpublished results. First the slab resonant phonons may almost be confined in the surface slab even for frequencies falling within the bulk phonon band, if the surface slab is separated from the substrate by a buffer layer. The other prospective unpublished result deals with a model system in which the propagating phonons reach a space point from which they can continue to propagate through two different finite in space channels before reaching again a homogeneous uniform path. Such a system exhibits some frequencies for which the transmission factor vanishes. This phenomenon is related to the resonances associated with the finite additional path offered to the propagation of the phonons.
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