  The superconductivity of the 4-angstrom single-walled carbon nanotubes
(SWCNTs) was discovered more than a decade ago, and marked the breakthrough of
finding superconductivity in pure elemental undoped carbon compounds. The van
Hove singularities in the electronic density of states at the Fermi level in
combination with a large Debye temperature of the SWCNTs are expected to cause
an impressively large superconducting gap. We have developed an innovative
computational algorithm specially tailored for the investigation of
superconductivity in ultrathin SWCNTs. We predict the superconducting
transition temperature of various thin carbon nanotubes resulting from
electron-phonon coupling by an ab-initio method, taking into account the effect
of radial pressure, symmetry, chirality (N,M) and bond lengths. By optimizing
the geometry of the carbon nanotubes, a maximum Tc of 60K is found. We also use
our method to calculate the Tc of a linear carbon chain embedded in the center
of (5,0) SWCNTs. The strong curvature in the (5,0) carbon nanotubes in the
presence of the inner carbon chain provides an alternative path to increase the
Tc of this carbon composite by a factor of 2.2 with respect to the empty (5,0)
SWCNTs.
