Josephson effect in a multiorbital model for Sr2RuO4

We study Josephson currents between $s$-wave/spin-triplet superconductor junctions by taking into account details of the band structures in ${\mathrm{Sr}}₂{\mathrm{RuO}}₄$, such as three conduction bands and spin-orbit interactions in the bulk and at the interface. We assume five superconducting order parameters in ${\mathrm{Sr}}₂{\mathrm{RuO}}₄$: a chiral $p$-wave symmetry and four helical $p$-wave symmetries. We calculate the current-phase relationship $I(\ensuremath{\varphi})$ in these junctions, where $\ensuremath{\varphi}$ is the macroscopic phase difference between the two superconductors. The results for a chiral $p$-wave pairing symmetry show that a $cos(\ensuremath{\varphi})$ term appears in the current-phase relation because of time-reversal symmetry (TRS) breaking. On the other hand, this $cos(\ensuremath{\varphi})$ term is absent in the helical pairing states that preserve TRS. We also study the dependence of the maximum Josephson current ${I}_c$ on an external magnetic flux $\mathrm{\ensuremath{\Phi}}$ in a corner junction. The calculated ${I}_c(\mathrm{\ensuremath{\Phi}})$ obeys ${I}_c(\mathrm{\ensuremath{\Phi}})\ensuremath{\ne}{I}_c(\ensuremath{-}\mathrm{\ensuremath{\Phi}})$ in a chiral state and ${I}_c(\mathrm{\ensuremath{\Phi}})={I}_c(\ensuremath{-}\mathrm{\ensuremath{\Phi}})$ in a helical state. We calculate ${I}_c(\mathrm{\ensuremath{\Phi}})$ in a corner superconducting quantum interference device (SQUID) and a symmetric SQUID geometry. In the latter geometry, ${I}_c(\mathrm{\ensuremath{\Phi}})={I}_c(\ensuremath{-}\mathrm{\ensuremath{\Phi}})$ is satisfied for all the pairing states and it is impossible to distinguish a chiral state from a helical one. On the other hand, a corner SQUID always gives ${I}_c(\mathrm{\ensuremath{\Phi}})\ensuremath{\ne}{I}_c(\ensuremath{-}\mathrm{\ensuremath{\Phi}})$ and ${I}_c(\mathrm{\ensuremath{\Phi}})={I}_c(\ensuremath{-}\mathrm{\ensuremath{\Phi}})$ for a chiral and a helical state, respectively. Experimental tests of these relations in corner junctions and SQUIDs may serve as a tool for unambiguously determining the pairing symmetry in ${\mathrm{Sr}}₂{\mathrm{RuO}}₄$.