Critical current calculations for long 0-π Josephson junctions

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 π 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-π junction. In this form, the equations describe a hybrid ladder of inductively coupled small 0 and π 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 λ J to facet length L f . If λ J /L f ≫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 λ J /L f →∞, 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-π junction between a d-wave and an s-wave superconductor. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007