Role of hydrodynamic viscosity on phonon transport in suspended graphene

Abstract

When phonon transport is in the hydrodynamic regime, the thermal conductivity exhibits peculiar dependences on temperatures ($T$) and sample widths ($W$). These features were used in the past to experimentally confirm the hydrodynamic phonon transport in three-dimensional bulk materials. Suspended graphene was recently predicted to exhibit strong hydrodynamic features in thermal transport at much higher temperature than the three-dimensional bulk materials, but its experimental confirmation requires quantitative guidance by theory and simulation. Here we quantitatively predict those peculiar dependences using the Monte Carlo solution of the Peierls-Boltzmann equation with an ab initio full three-phonon scattering matrix. Thermal conductivity is found to increase as ${T}^{\ensuremath{\alpha}}$ where $\ensuremath{\alpha}$ ranges from 1.89 to 2.49 depending on a sample width at low temperatures, much larger than 1.68 of the ballistic case. The thermal conductivity has a width dependence of ${W}^{1.17}$ at 100 K, clearly distinguished from the sublinear dependence of the ballistic-diffusive regime. These peculiar features are explained with a phonon viscous damping effect of the hydrodynamic regime. We derive an expression for the phonon hydrodynamic viscosity from the Peierls-Boltzmann equation, and discuss the fact that the phonon viscous damping explains well those peculiar dependences of thermal conductivity at 100 K. The phonon viscous damping still causes significant thermal resistance when a temperature is 300 K and a sample width is around 1 \textmu{}m, even though the hydrodynamic regime is not dominant over other regimes at this condition.