Lattice Dynamics and Electron-Phonon Interaction in (3,3) Carbon Nanotubes

We present a detailed study of the lattice dynamics and electron-phonon coupling for a (3,3) carbon nanotube which belongs to the class of small diameter based nanotubes which have recently been claimed to be superconducting. We treat the electronic and phononic degrees of freedom completely by modern ab initio methods without involving approximations beyond the local density approximation. Using density functional perturbation theory we find a mean-field Peierls transition temperature of $\approx 240\mathrm{K}$ which is an order of magnitude larger than the calculated superconducting transition temperature. Thus in (3,3) tubes the Peierls transition might compete with superconductivity. The Peierls instability is related to the special $2k_F$ nesting feature of the Fermi surface. Because of the special topology of the ($n,n$) tubes we also find a phonon softening at the $\Gamma$ point.