Specularity of longitudinal acoustic phonons at rough surfaces

The specularity of phonons at crystal surfaces is of direct importance to thermal transport in nanostructures and to dissipation in nanomechanical resonators. Wave scattering theory provides a framework for estimating wavelength-dependent specularity, but experimental validation remains elusive. Widely available thermal conductivity data presents poor validation since the involvement of the infinitude of phonon wavelengths in thermal transport presents an underconstrained test for specularity theory. Here, we report phonon specularity by measuring the lifetimes of individual coherent longitudinal acoustic phonon modes excited in ultrathin (36--205 nm) suspended silicon membranes at room temperature over the frequency range $\ensuremath{\sim}20--118$ GHz. Phonon surface scattering dominates intrinsic Akhiezer damping at frequencies $\ensuremath{\gtrsim}60$ GHz, enabling measurements of phonon boundary scattering time over wavelengths $\ensuremath{\sim}72--140\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$. We obtain detailed statistics of the surface roughness at the top and bottom surfaces of membranes using HRTEM imaging. We find that the specularity of the excited modes are in good agreement with solutions of wave scattering only when the TEM statistics are corrected for projection errors. The often-cited Ziman formula for phonon specularity also appears in good agreement with the data, contradicting previous results. This work helps to advance the fundamental understanding of phonon scattering at the surfaces of nanostructures.