Critical Current Calculations For Long $0$-$\pi$ Josephson Junctions

TL;DR

This paper models the critical current behavior of long 0-π Josephson junctions with zigzag boundaries, revealing how magnetic field and junction parameters influence current and quality factor, aligning with experimental observations.

Contribution

It introduces a discretized RCSJ model for long 0-π Josephson junctions, providing analytical and numerical insights into their field-dependent critical current and quality factor behaviors.

Findings

Critical current peaks at non-zero magnetic fields.

Critical current can be less than 2% of maximum over a range of fields.

Analytical relation between quality factor and critical current.

Abstract

A zigzag boundary between a $d_{x^2-y^2}$ and an $s$-wave superconductor is believed to behave like a long Josephson junction with alternating sections of $0$ and $\pi$ symmetry. We calculate the field-dependent critical current of such a junction, using a simple model. The calculation involves discretizing the partial differential equation for the phase difference across a long $0$-$\pi$ junction. In this form, the equations describe a hybrid ladder of inductively coupled small $0$ and $\pi$ resistively and capacitively shunted Josephson junctions (RCSJ's). The calculated critical critical current density $J_c(H_a)$ is maximum at non-zero applied magnetic field $H_a$, and depends strongly on the ratio of Josephson penetration depth $\lambda_J$ to facet length $L_f$. If $\lambda_J/L_f \gg 1$ and the number of facets is large, there is a broad range of $H_a$ where $J_c(H_a)$ is less than $2\%$ of the maximum critical current density of a long $0$ junction. All of these features are in qualitative agreement with recent experiments. In the limit $\lambda_J/L_f \to \infty$, our model reduces to a previously-obtained analytical superposition result for $J_c(H_a)$. In the same limit, we also obtain an analytical expression for the effective field-dependent quality factor $Q_J(H_a)$, finding that $Q_J(H_a) \propto \sqrt{J_c(H_a)}$. We suggest that measuring the field-dependence of $Q_J(H_a)$ would provide further evidence that this RCSJ model applies to a long $0$-$\pi$ junction between a d-wave and an s-wave superconductor.