Surface-phonon-assisted tunneling in two-dimensional amorphous systems at low temperatures

We investigate theoretically the effects of surface-phonon on the tunneling states in amorphous thin films at low temperatures. The attenuation rate of surface-phonon is estamated as functions of frequency and temperature. It is shown that the attenuation is proportional to $\text{ω}{}^3\text{T}$ for surface-phonon frequency small compared to k_BT. This dependence is quite different from that of amorphous bulk materials.