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

Limiting mechanism for critical current in topologically frustrated Josephson junctions