Role of phonon dimensionality on electron-phonon scattering rates

Electron-heating measurements were carried out to measure the electron-phonon scattering rates in CuCr thin films. We examined free-standing and supported films whose thickness and width span lengths comparable to the mean thermal-phonon wavelength. The measured scattering rate was proportional to ${\mathit{T}}^2$ from 0.5 to 10 K, independent of sample configuration. We observe that the quantization of the phonon spectrum required by sample dimensions has no effect on the temperature dependence of the electron-phonon scattering rate.