Hybrid quantum dot-superconducting systems: Josephson current and Kondo effect in the narrow-band limit

The case of a quantum dot connected to two superconducting leads is studied by using the narrow-band limit to describe the superconducting degrees of freedom. The model provides a simple theoretical framework, almost analytical, to analyze the interplay between the Kondo effect, superconductivity, and finite temperature. In the quantum dot Kondo regime, the model is completely characterized by the ratio Delta/J, with Delta the superconducting gap and J an effective antiferromagnetic exchange coupling between the dot and the leads. The model allows us to calculate, at any temperature T, the equilibrium Josephson current through the dot in a very straightforward way as a function of Delta/J. The behavior of the current allows us to distinguish the four types of hybrid junctions: 0, 0', pi', and pi. The presence of the 0- and 0'-junction configurations are intrinsically linked to the Kondo effect in the quantum dot, while the pi-and pi'-junction configurations are driven by the superconductivity in the leads. The Josephson critical current has a non-monotonic behavior with temperature, that may be used for the experimental characterization of the fundamental 0 - pi transition. The model allows us to obtain easily a phase diagram Delta/J vs temperature, from where we can obtain an overall picture on the stability of the different types of junctions. From the explicit analytical expressions for the ground-state, low-energy excitations, free energy, and Josephson current, it is easy to understand the physical nature of the main features of the critical current and the phase diagram. The results, obtained with a minimum of numerical effort, are in a good qualitative agreement with more demanding calculational approaches aimed to solve the full model.