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.