Secondary proximity effect in a side-coupled double quantum dot structure

Semiconductor quantum dots in close proximity to superconductors may provoke localized bound states within the superconducting energy gap known as the Yu-Shiba-Rusinov state, which is a promising candidate for constructing Majorana zero modes and topological qubits. Side-coupled double quantum dot systems are ideal platforms revealing the secondary proximity effect. Numerical renormalization group calculations show that if the central quantum dot can be treated as a noninteracting resonant level, it acts as a superconducting medium due to the ordinary proximity effect. The bound state in the side dot behaves as the case of a single impurity connected to two superconducting leads. The side dot undergoes a quantum phase transition between a spin-singlet state and a doublet state as the Coulomb repulsion, the interdot coupling strength, or the energy level sweeps. Phase diagrams indicate that the phase boundaries could be well illustrated by $\mathrm{\ensuremath{\Delta}}\ensuremath{\approx}c{T}_{K2}$ in all cases, where $\mathrm{\ensuremath{\Delta}}$ is the superconducting gap, ${T}_{K2}$ is the side Kondo temperature and $c$ is of the order 1.0. These findings offer valuable insights into the secondary proximity effect, which is a promising approach for realizing superconducting couplings between quantum dots and reducing the random-disorder potential via quantum interferences.