Electron-phonon interaction in ultrasmall-radius carbon nanotubes

We perform analysis of the band structure, phonon dispersion, and electron-phonon interactions in three types of small-radius carbon nanotubes. We find that the (5,5) nanotube can be described well by the zone-folding method and the electron-phonon interaction is too small to support either a charge-density wave or superconductivity at realistic temperatures. For ultrasmall (5,0) and (6,0) nanotubes we find that the large curvature makes these tubes metallic with a large density of states at the Fermi energy and leads to unusual electron-phonon interactions, with the dominant coupling coming from the out-of-plane phonon modes. By combining the frozen-phonon approximation with the random phase approximation analysis of the giant Kohn anomaly in one dimension we find parameters of the effective Frohlich Hamiltonian for the conduction electrons. Neglecting Coulomb interactions, we find that the (5,5) carbon nanotube (CNT) remains stable to instabilities of the Fermi surface down to very low temperatures while for the (5,0) and (6,0) CNTs a charge density wave instability will occur. When we include a realistic model of Coulomb interaction we find that the charge-density wave remains dominant in the (6,0) CNT with T-CDW around 5 K while the charge-density wave instability is suppressed to very low temperatures in the (5,0) CNT, making superconductivity dominant with transition temperature around 1 K.