Limiting mechanism for critical current in topologically frustrated Josephson junctions

Eutectic Sr$_2$RuO$_4$-Ru samples with $\mu$m-sized Ru-metal inclusions support inhomogeneous superconductivity above the bulk transition of Sr$_2$RuO$_4$ in the so-called 3-Kelvin phase. In Pb/Ru/Sr$_2$RuO$_4$ Josephson junctions as realized by Maeno et al., a Pb film is indirectly coupled to the superconductor Sr$_2$RuO$_4$ mediated by the proximity-induced superconducting Ru-inclusions, yielding an extended Josephson contact through the interface between Ru and Sr$_2$RuO$_4$. Motivated by this experimental setup, we formulate a sine-Gordon model for the Josephson phase of the interface, assuming a simple cylindrical shape for the Ru-inclusion hosting the proximity-induced $s$-wave superconducting phase. Considering the Sr$_2$RuO$_4$ as a chiral $p$-wave superconductor, we discuss two types of Josephson junctions, a frustrated one due to the nature of the order parameter in Sr$_2$RuO$_4$, and an unfrustrated one for the topologically trivial 3-Kelvin phase. While the latter situation displays standard junction behavior, the former yields an unusual limiting mechanism for the critical current, based on a pinning-depinning transition of a spontaneously induced magnetic flux driven by an externally applied current. We analyze different coupling limits and show that different critical currents can arise for the two topologies. This concept fits well to recent experimental data obtained for the above setup showing an anomalous temperature dependence of the critical current at the transition temperature $T_c$ of bulk Sr$_2$RuO$_4$.