Quantum dot attached to superconducting leads: Relation between symmetric and asymmetric coupling

Alžběta Kadelcová, Martin Žonda, Tomáš Novotný

Published 2016-10-26Version 1

We study the Anderson single-level quantum dot attached to two BCS superconducting leads with the same gap size. We reveal that a system with asymmetric tunnel coupling to the leads ($\Gamma_{L}\neq\Gamma_{R}$) can be related to the symmetric system with the same net coupling strength $\Gamma=\Gamma_{L}+\Gamma_{R}$. Surprisingly, it is the symmetric case which is the most general, meaning that all physical quantities in case of asymmetric coupling are fully determined by the symmetric ones. We give ready-to-use conversion formulas for the $0-\pi$ phase transition boundary, on-dot quantities, and the Josephson current, and illustrate them on the NRG results of [Oguri, Tanaka and Bauer, Phys. Rev. B 87, 075432 (2013)] for the three-terminal setup. We apply our theory to the recent $0-\pi$ transition measurement [Delagrange et al., Phys. Rev. B 93, 196437 (2016)] and justify the universal arccosine phase-dependence of the boundary by a linearization procedure which, moreover, allows to determine the asymmetry of the experimental setup from the measured transition width. Finally, we establish that the widely assumed Kondo "universality" of physical quantities depending only on the ratio of the Kondo temperature and the superconducting gap $T_{K}/\Delta$ cannot hold for asymmetric junctions.