We consider the Josephson current through the carbon nanotube quantum dot with twofold orbital degeneracy connected by two superconductor leads. We show that the removal of orbital and spin degeneracies due to strong spin-orbit coupling and external magnetic field has a significant influence on the subgap Andreev bound states and the $0\text{−}\pi$ transition of the Josephson current. The $0\text{−}\pi$ transition point is determined by the level crossing of the lowest branch of the bound states and the Fermi level, and is given by $[T_K-{\Delta }_{\text{SO}}/2-(\mu +1)B]/\Delta \sim 1$, with $T_K$ as the Kondo temperature, ${\Delta }_{\text{SO}}$ the spin-orbit coupling, $\Delta$ the superconducting gap, $\mu$ the ratio of orbital moment and Bohr magneton, and $B$ the Zeeman splitting loaded by the magnetic field. The interplay of the Kondo effect and superconductivity in such a quantum dot device provides a route to manipulate the $0\text{−}\pi$ transition of the Josephson current.

• Received 15 November 2013
• Revised 16 June 2014

DOI:https://doi.org/10.1103/PhysRevB.89.245135