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 thermal relaxation 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 metal bridges on an layer lying or suspended crystalline substrate. Noise thermometry is used to measure electron temperature ${T}_{e}$ as function Joule power per unit area ${P}_{2\mathrm{D}}$. In all samples, we observe ${P}_{2\mathrm{D}}\ensuremath{\propto}{T}_{e}^{n}$ dependence, with exponent $n\ensuremath{\simeq}\phantom{\rule{0.2em}{0ex}}2$, which inconsistent both electron-phonon coupling Kapitza resistance. functional dependence ${P}_{2\mathrm{D}}({T}_{e})$ length consistent linear conductivity, related lattice excitations (diffusons) for phonon mean free path shorter than dominant wavelength. Our findings are important understanding operation devices embedded dielectrics.