Thermal expansion in carbon nanotubes and graphene: nonequilibrium Green's function approach

The nonequilibrium Green's function method is applied to investigate the coefficient of thermal expansion (CTE) in single-walled carbon nanotubes (SWCNT) and graphene. It is found that atoms deviate about 1% from equilibrium positions at T=0 K, resulting from the interplay between quantum zero-point motion and nonlinear interaction. The CTE in SWCNT of different sizes is studied and analyzed in terms of the competition between various vibration modes. As a result of this competition, the axial CTE is positive in the whole temperature range, while the radial CTE is negative at low temperatures. In graphene, the CTE is very sensitive to the substrate. Without substrate, CTE has large negative region at low temperature and very small value at high temperature limit, and the value of CTE at T=300 K is $-6\times 10^{-6}$ K$^{-1}$ which is very close to recent experimental result, $-7\times 10^{-6}$ K$^{-1}$ (Nat. Nanotechnol. \textbf{10}, 1038 (2009)). A very weak substrate interaction (about 0.06% of the in-plane interaction) can largely reduce the negative CTE region and greatly enhance the value of CTE. If the substrate interaction is strong enough, the CTE will be positive in whole temperature range and the saturate value at high temperature reaches $2.0\times 10^{-5}$ K$^{-1}$.