Thermal Relaxation in Metal Films Limited by Diffuson Lattice Excitations of Amorphous Substrates

We examine the role of a silicon-based amorphous insulating substrate in the thermal relaxation in thin $\mathrm{Nb}\mathrm{N}$, ${\mathrm{In}\mathrm{O}}_{x}$, and $\mathrm{Au}$/$\mathrm{Ni}$ films at temperatures above 5 K. The samples studied consist of metal bridges on an amorphous insulating layer lying on or suspended above a crystalline substrate. Noise thermometry is used to measure the electron temperature ${T}_{e}$ of the films as a function of Joule power per unit area ${P}_{2\mathrm{D}}$. In all samples, we observe a ${P}_{2\mathrm{D}}\ensuremath{\propto}{T}_{e}^{n}$ dependence, with exponent $n\ensuremath{\simeq}\phantom{\rule{0.2em}{0ex}}2$, which is inconsistent with both electron-phonon coupling and Kapitza thermal resistance. In suspended samples, the functional dependence of ${P}_{2\mathrm{D}}({T}_{e})$ on the length of the amorphous insulating layer is consistent with the linear temperature dependence of the thermal conductivity, which is related to lattice excitations (diffusons) for a phonon mean free path shorter than the dominant phonon wavelength. Our findings are important for understanding the operation of devices embedded in amorphous dielectrics.