Absence of Casimir regime in two-dimensional nanoribbon phonon conduction

In stark contrast with three-dimensional (3D) nanostructures, we show that boundary scattering in two-dimensional (2D) nanoribbons alone does not lead to a finite phonon mean free path. If combined with an intrinsic scattering mechanism, 2D boundary scattering does reduce the overall mean free path, however the latter does not scale proportionally to the ribbon width, unlike the well known Casimir regime occurring in 3D nanowires. We show that boundary scattering can be accounted for by a simple Mathiessen type approach for many different 3D nanowire cross sectional shapes, however this is not possible in the 2D nanoribbon case, where a complete solution of the Boltzmann transport equation is required. These facts have strong implications for the thermal conductivity of suspended nanostructures.