Lattice dynamics of carbon-chain inside a carbon nanotube

Based on the density functional theory, we obtain the optimum geometry of carbon chain inside a carbon nanotube. The phonon spectrum and specific heat of such a chain and nanotube hybrid system are calculated in terms of lattice dynamics theory. Some new phonon branches that have been obtained come from the coupling vibrations of the nanotube and the chain. The bending and stretching modes of the chain appear at about 520 cm^-1 and 1935 cm^-1 at Γ point, respectively. It is found that the softening of G modes results mainly from the chain induced variations in the bond length on nanotube, independent of van der Waals interaction, while the stiffening of radial breathing mode is developed by the competition between the two factors. In the low-frequency region, the vibrational density of states are very different from that of the bare nanotube. Its specific heat implies the underlying quantized phonon structures and much large thermal conductivity in the hybrid system. In addition, the chain-length dependent vibration modes are calculated, from which it is expected that a finite chain of about 14 carbon atoms in the nanotube may produce the experimental Raman peak at about 1850 cm^-1.