Frustration in d-wave Superconducting Circuits: $π$-Ring Behaviour

Abstract

A closed superconducting circuit containing an odd number of $π$-junctions, a $π$-ring, has a finite current in the ground state. We explicitly construct such rings for d-wave superconductors and demonstrate the existence of spontaneous currents by direct self-consistent solutions to the Bogoliubov de Gennes equations. We show that the current has a topological origin due to the frustration of the d-wave order parameter, which is only partially explained by the Sigrist-Rice tunneling formula.