Effect of boundary roughness on the attenuation of specular phonon reflection in graphene

Abstract

The reduced phonon specularity $p$ from boundary roughness scattering plays a major role in the lower thermal conductivity in semiconducting and insulating nanowires and films. Although the well-known Ziman formula $p=\exp(-4\sigma^{2}q_{x}^{2})$, where $\sigma$ and $q_{x}$ denote the root-mean-square boundary roughness and the normal component of the incident phonon wave vector, respectively, and its variants are commonly used in the literature to estimate how roughness attenuates $p$, their validity and accuracy remain poorly understood, especially when the effects of mode conversion cannot be ignored. In this paper, we investigate the accuracy and validity of the more general Ogilvy formula, from which the Ziman formula is derived, by comparing its predictions to the $p$ values computed from Atomistic Green's Function (AGF) simulations for an ensemble of rough boundaries in single-layer graphene. The effects of phonon dispersion, incident angle, polarization, mode conversion, and correlation length are analyzed. Our results suggest that the Ogilvy formula is remarkably accurate for $0<q_{x}<\frac{\pi}{4\sigma}$ when the lateral correlation length $L$ is large or the phonon is at normal incidence. At large $q_{x}$ in the short-wavelength limit, the $q_{x}$-dependence of $p$ becomes significantly weaker. In the large-$L$ limit, the numerical results suggest the existence of a minimum $p$ for short-wavelength phonons, given by $p\sim p_{0}\exp(-\pi^{2}/4)$, where $p_{0}$ is the baseline specularity for the ideal boundary.