Phase sensitive information from a planar Josephson junction

Josephson tunneling across a planar junction generally depends on the relative twist angle, θ, between the two layers. However, if under a discrete rotation, the order parameter in one layer is odd and the other is even (as, e.g., for a s-wave to $${d}_{{x}^{2}-{y}^{2}}$$ -wave junction under a π/2 rotation) then the bulk Josephson current vanishes for all θ. Even in this case, we show that for a finite junction, the Josephson current, J, has a nonzero edge contribution that depends on θ and the orientation of the junction edges in ways that can serve as an unambiguous probe of the order parameter symmetry of any time-reversal preserving system (including multiband systems and those in which spin-orbit coupling is significant). We also analyze the microscopic considerations that determine the magnitude of J.