Interplay of the single particle and Josephson Cooper pair tunneling on supercurrent across the superconducting quantum dot junction

In the present paper the interplay of the single particle and Josephson Cooper pair tunneling on the Josephson supercurrent across the Superconducting Quantum dot junction has been investigated. For this purpose, we have used the renormalized Anderson model that includes the single particle coupling between the superconducting leads and quantum dot states, attractive BCS effective interaction in the superconducting leads responsible for pairing and Josephson Cooper pair tunneling responsible for the Cooper pair tunneling through such Superconducting Quantum dot junction. We have employed Green's function and the Ambegaokar-Baratoff formalism to analyse the Josephson supercurrent across such junction. It has been pointed out that the Josephson supercurrent across the superconducting Quantum dot junction depends on the competitive role of the single particle tunneling, the energy of the level on the dot with respect to binding energy of the Cooper pairs and also on the Josephson Cooper pair tunneling.