Josephson Coupling through a Quantum Dot

We derive, via fourth order perturbation theory, an expression for the Josephson current through a gated interacting quantum dot. We analyze our expression for two different models of the superconductor-dot-superconductor (SDS) system. When the matrix elements connecting dot and leads are featureless constants, we compute the Josephson coupling Jc as a function of the gate voltage and Coulomb interaction. In the limit of a diffusive dot, we compute the probability distribution P (Jc) of Josephson couplings. In both cases, π junction behavior (Jc < 0) is possible, and is not simply dependent on the parity of the dot occupancy. Magnetic impurities in Josephson junctions tend to degrade the critical current [1–7]. This result has been known since the work of Kulik [1], who analyzed a simple model in which spin-preserving as well as spin-flip hopping processes (with amplitudes t and t, respectively) are present. The critical current in such a junction is given by the expression Ic = I AB c · (|t| − |t|)/(|t| + |t|), where I c is the Ambegaokar-Baratoff critical current. Since spinflip tunneling results in a sign change of the Cooper pair singlet (↑↓ to ↓↑), the spin-flip hopping contribution is negative, and reduces Ic [5]. When Ic < 0, one has a π junction, for which the ground state energy is minimized when the superconducting phase difference is δ = π. A ring containing a single π junction will enclose trapped flux [3]. A related effect occurs in Josephson tunneling through a ferromagnetic layer [8–11], and recent experiments on Nb-CuxNi1−x-Nb junctions suggest that Ic < 0 states have been observed [12]. In this paper, we investigate Josephson coupling mediated by a quantum dot [13], generalizing the case of a single impurity to a system with many quantized energy levels. We derive first a general expression, within fourth-order perturbation theory, for the Josephson coupling Jc (Ic = 2eJc/h̄). We then consider two models for the tunneling amplitudes tαj from the leads to the dot. As we shall see, in contrast to the single impurity case, the parity of the number of electrons on the dot, N0, does not uniquely determine the sign of the Josephson coupling. It is well-known in the theory of elastic co-tunneling that if the Coulomb repulsion U is large then the conductance through a dot is independent of U . Correspondingly, we find the critical current is insensitive to U in this regime, and furthermore when the system is close to a charge degeneracy point, the probability distribution P (Jc) for a disordered dot has universal properties. Our Hamiltonian is a sum of three terms: 1