Josephson current through a nanoscale quantum dot contacted by conventional superconductors

We have performed a numerical simulation for the Josephson current through a quantum dot connected to two s-wave BCS superconductors. In order to estimate the Josephson current at finite temperatures, we have used a quantum Monte-Carlo method by the Hirsh-Fye and Bogoliubov-de Gennes (HFBdG) algorithm, where we take into account the effect of the superconducting pair potentials on the dot. The characteristics of the Josephson current for noninteracting case without on-site Coulomb repulsion U on the dot is in a good agreement with those obtained by the Ambegaokar-Baratoff current formula. On the other hand, for small U with the low transparency at the superconductor-dot interface, the temperature dependence of the current has a sign change, where the junction becomes the π-junction caused by the Coulomb blockade. For the large magnitude of U with high transparency, the amplitude of the Josephson current has a maximum at low temperatures around π-0 crossover points.