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

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 $4{\eta }^2{k}^2$, where $\eta$ is the rms roughness and $k$ is the acoustic wave vector, 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 ${\eta }^2{k}^4{L}^2$, 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.

Figure 1

Geometry of the problem.

Figure 2

Dependence of dimensionless parameters $I_{\mathrm{bulk}}$ and $I_R$ on the transverse-to-longitudinal velocities ratio $s$.

Figure 3

Normalized diffuse scattering probability vs the product of the acoustic wave vector and the correlation length for different values of the velocities ratio $s$. Contributions of scattering into bulk and Rayleigh surface waves are shown as indicated in the upper left panel.

Figure 4

Normalized diffuse scattering probability for a solid with $s=0.5$ vs a liquid as a function of $k_{l}L$.

Figure 5

Dependence of dimensionless parameters $J_{\mathrm{bulk}}$ and $J_R$ from Eq. (30) on the transverse-to-longitudinal velocities ratio $s$.

Figure 6

Normalized diffuse scattering probability for the normally incident transverse wave vs the product of the acoustic wave vector and the correlation length for different values of the velocities ratio $s$. Contributions of scattering into bulk and Rayleigh surface waves are shown as indicated in the upper left panel.