Interplay between Kondo and Andreev-Josephson effects in a quantum dot coupled to one normal and two superconducting leads

Abstract

We study low-energy transport through a quantum dot coupled to one normal and two superconducting (SC) leads in a Y-shaped junction. In this geometry a crossover between Kondo-dominated and Cooper-pairing-dominated states occurs by tuning parameters such as the quantized energy level ${\ensuremath{\epsilon}}_{d}$ of the dot and the Josephson phase $\ensuremath{\phi}$, which induces a supercurrent flowing between the two SC leads through the dot. Furthermore, Andreev scattering takes place at the interface between the dot and normal lead. The low-lying energy states of this system can be described by a local Fermi-liquid theory for interacting Bogoliubov particles. In a description based on an Anderson impurity model, we calculate transport coefficients, renormalized parameters, and spectral function, using Wilson's numerical renormalization group approach, in the limit of a large SC gap. Our results demonstrate how the Andreev resonance level approaches the Fermi level in the crossover region between the Cooper-pairing singlet state and the strong-coupling situation as ${\ensuremath{\epsilon}}_{d}$ or $\ensuremath{\phi}$ are varied. The strong-coupling situation shows a Kondo effect with a significantly renormalized resonance width. The crossover is smeared when the coupling between the dot and the normal lead is large. Furthermore, asymmetry in the Josephson junction suppresses the cancellations of the SC proximity for finite $\ensuremath{\phi}$, and it favors the SC singlet state rather than the Kondo singlet.