Tunneling-states model and the propagation characteristics of surface phonons in amorphous solids at low temperatures

The propagation characteristics of the surface phonon in an amorphous film are theoretically investigated at low temperatures based on the tunneling-states model. It is shown that the attenuation rate of the surface phonon is proportional to ω tanh ?ω/2k_BT and the relative velocity variation to ln (T/T₀) for the thickness of the amorphous film comparable to or larger than the surface phonon wavelength. Below 3K, our result can well account for recent measurements of the relative velocity variation by Hartemann et al.