Theory of phonon thermal transport in single-walled carbon nanotubes and graphene

A theory is presented for describing the lattice thermal conductivities of graphene and single-walled carbon nanotubes. A phonon Boltzmann transport equation approach is employed to describe anharmonic phonon-phonon, crystal boundary, and isotopic impurity scattering. Full quantum mechanical phonon scattering is employed and an exact solution for the linearized Boltzmann transport equation is determined for each system without use of common relaxation time and long-wavelength approximations. The failures of these approximations in describing the thermal transport properties of nanotubes is discussed. An efficient symmetry based dynamical scheme is developed for carbon nanotubes and selection rules for phonon-phonon scattering in both graphene and nanotubes are introduced. The selection rule for scattering in single-walled carbon nanotubes allows for calculations of the thermal conductivities of large-diameter and chiral nanotubes that could not be previously studied due to computational limitations. Also due to this selection rule, no acoustic-only umklapp scattering can occur, thus, acoustic-optic scattering must be included in order to have thermal resistance from three-phonon processes. The graphene selection rule severely restricts phonon-phonon scattering of out-of-plane modes. This restriction leads to large contributions to the total thermal conductivity of graphene from the acoustic, out-of-plane modes which have been previously neglected. Empirical potentials used to model interactions in carbon-based materials are optimized to better describe the lattice dynamics of graphene-derived systems. These potentials are then used to generate the interatomic force constants needed to make calculations of the thermal conductivities of graphene and carbon nanotubes. Calculations of the thermal conductivities of single-walled carbon nanotubes and graphene for different temperatures and lengths are presented. The thermal conductivities of SWCNTs saturate in the diffusive regime when the effects of higher-order scattering processes are estimated and correctly reproduce the ballistic limit for short-length nanotubes at low temperatures. The effects of isotopic impurity scattering on the thermal conductivities of graphene and SWCNTs are explored. Isotopic impurities have little effect in the low (high) temperature regime where boundary (umklapp) scattering dominates the behavior of the thermal conductivities. In the intermediate temperature regime, modest reductions in the thermal conductivities, 15-20%, occur due to impurities. The thermal conductivities of a wide-range of SWCNTs are explored. The thermal conductivities of successively larger-diameter, one-dimensional nanotubes approach the thermal conductivity of two-dimensional graphene.