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<Subscript>c</Subscript>(H<Subscript>a</Subscript>) is maximum at non-zero applied magnetic field H<Subscript>a</Subscript>, and depends strongly on the ratio of Josephson penetration depth λ<Subscript>J</Subscript> to facet length L<Subscript>f</Subscript>. If λ<Subscript>J</Subscript>/L<Subscript>f</Subscript> ≫1 and the number of facets is large, there is a broad range of H<Subscript>a</Subscript> where J<Subscript>c</Subscript>(H<Subscript>a</Subscript>) 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<Subscript>J</Subscript>/L<Subscript>f</Subscript> →∞, our model reduces to a previously-obtained analytical superposition result for J<Subscript>c</Subscript>(H<Subscript>a</Subscript>). In the same limit, we also obtain an analytical expression for the effective field-dependent quality factor Q<Subscript>J</Subscript>(H<Subscript>a</Subscript>), finding that <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$Q_J(H_a) \propto \sqrt{J_c(H_a)}$</EquationSource> </InlineEquation>. We suggest that measuring the field-dependence of Q<Subscript>J</Subscript>(H<Subscript>a</Subscript>) 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
