We present a new mechanism of carbon nanotube superconductivity that originates from edge states which are specific to graphene. Using on-site and boundary deformation potentials which do not cause bulk superconductivity, we obtain an appreciable transition temperature for the edge state. As a consequence, a metallic zigzag carbon nanotube having open boundaries can be regarded as a natural superconductor/normal metal/superconductor junction system, in which superconducting states are developed locally at both ends of the nanotube and a normal metal exists in the middle. In this case, a signal of the edge state superconductivity appears as the Josephson current which is sensitive to the length of a nanotube and the position of the Fermi energy. Such a dependence distinguishs edge state superconductivity from bulk superconductivity.
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