Impurity Scattering in Carbon Nanotubes with Superconducting Pair Correlations

Effects of the superconducting pair potential on the impurity scattering processes in metallic carbon nanotubes are studied theoretically. The backward scattering of electrons vanishes in the normal state. In the presence of the superconducting pair correlations, the backward scatterings of electronand hole-like quasiparticles vanish, too. The impurity gives rise to backward scatterings of holes for incident electrons, and it also induces backward scatterings of electrons for incident holes. Negative and positive currents induced by such the scatterings between electrons and holes cancel each other. Therefore, the nonmagnetic impurity does not hinder the supercurrent in the regions where the superconducting proximity effects occur, and the carbon nanotube is a good conductor for Cooper pairs. Relations with experiments are discussed. INTRODUCTION Recent investigations (1,2) show that the superconducting proximity effect occurs when the carbon nanotubes contact with conventional superconducting metals and wires. The superconducting energy gap appears in the tunneling density of states below the critical temperature Tc. On the other hand, the recent theories discuss the nature of the exceptionally ballistic conduction (3) and the absence of backward scattering (4) in metallic carbon nanotubes with impurity potentials at the normal states. In this paper, we study the effects of the superconducting pair potential on the impurity scattering processes in metallic carbon nanotubes, using the continuum k · p model for the electronic states. We find the absence of backward scatterings of electronand hole-like quasiparticles in the presence of superconducting proximity effects, and the nonmagnetic impurity does not hinder the supercurrent in the regions where the superconducting proximity effects occur. Therefore, the carbon nanotube is a good conductor for Cooper pairs as well as in the normal state. This finding is interesting in view of the recent experimental progress of the superconducting proximity effects of carbon nanotubes (1,2). IMPURITY SCATTERING IN NORMAL NANOTUBES We will study the metallic carbon nanotubes with the superconducting pair potential. The model is as follows: H = Htube +Hpair, (1) Htube is the electronic states of the carbon nanotubes, and the model based on the k · p approximation (4,5) represents electronic systems on the continuum medium. The second term Hpair is the pair potential term owing to the proximity effect. The hamiltonian of a graphite plane by the k ·p approximation (4,5) in the secondly quantized representation has the following form: Htube = ∑ k,σ Ψ k,σ EkΨk,σ, (2) where Ek is an energy matrix: Ek = 