Boundary scattering of phonons: specularity of a randomly rough surface in the small perturbation limit

TL;DR

This paper analyzes how randomly rough surfaces affect phonon boundary scattering, deriving formulas for different correlation lengths and showing implications for thermal transport modeling accuracy.

Contribution

It provides a unified analysis of phonon boundary scattering across different surface correlation lengths, connecting known limits and revealing a maximum diffuse scattering at intermediate lengths.

Findings

Specularity reduction follows $4\,\eta^2k^2$ for large correlation lengths.

Specularity reduction follows $\eta^2k^4L^2$ for small correlation lengths.

Maximum diffuse scattering occurs at an intermediate correlation length.

Abstract

Scattering of normally incident longitudinal and transverse acoustic waves by a randomly rough surface of an elastically isotropic solid is analyzed within the small perturbation approach. In the limiting case of a large correlation length $L$ compared with the acoustic wavelength, the specularity reduction is given by $4\eta^2k^2$, where $\eta$ is the RMS roughness and $k$ is the acoustic wavevector, which is in agreement with the well-known Kirchhoff approximation result often referred to as Ziman's equation [J. M. Ziman, Electrons and Phonons (Clarendon Press, Oxford, 1960)]. In the opposite limiting case of a small correlation length, the specularity reduction is found to be proportional to $\eta^2k^4L^2$, with the fourth power dependence on frequency as in Rayleigh scattering. Numerical calculations for a Gaussian autocorrelation function of surface roughness connect these limiting cases and reveal a maximum of diffuse scattering at an intermediate value of $L$. This maximum becomes increasingly pronounced for the incident longitudinal wave as the Poisson's ratio of the medium approaches 1/2 as a result of increased scattering into transverse and Rayleigh surface waves. The results indicate that thermal transport models using Ziman's formula are likely to overestimate the heat flux dissipation due to boundary scattering, whereas modeling interface roughness as atomic disorder is likely to underestimate scattering.