Pinned states in Josephson arrays: A general stability theorem

TL;DR

This paper develops a general stability criterion for pinned states in 2D Josephson junction arrays, enabling prediction of depinning conditions based on the positive definiteness of a stiffness matrix.

Contribution

It introduces a universal algebraic stability criterion for superconducting pinned states, applicable to disordered arrays with arbitrary connectivity and flux patterns.

Findings

Pinned state stability is equivalent to the positive definiteness of the stiffness matrix.

The criterion predicts critical current and frustration levels for depinning.

The analysis neglects quantum, thermal, and inductive effects.

Abstract

Using the lumped circuit equations, we derive a stability criterion for superconducting pinned states in two-dimensional arrays of Josephson junctions. The analysis neglects quantum, thermal, and inductive effects, but allows disordered junctions, arbitrary network connectivity, and arbitrary spatial patterns of applied magnetic flux and DC current injection. We prove that a pinned state is linearly stable if and only if its corresponding stiffness matrix is positive definite. This algebraic condition can be used to predict the critical current and frustration at which depinning occurs.