Josephson current in carbon nanotubes with spin-orbit interaction.

We demonstrate that curvature-induced spin-orbit coupling induces a 0-π transition in the Josephson current through a carbon nanotube quantum dot coupled to superconducting leads. In the noninteracting regime, the transition can be tuned by applying a parallel magnetic field near the critical field where orbital states become degenerate. Moreover, the interplay between charging and spin-orbit effects in the Coulomb blockade and cotunneling regimes leads to a rich phase diagram with well-defined (analytical) boundaries in parameter space. Finally, the 0 phase always prevails in the Kondo regime. Our calculations are relevant in view of recent experimental advances in transport through ultraclean carbon nanotubes.