Phase sensitive information from a planar Josephson junction

Josephson tunneling across a planar junction generally depends on the relative twist angle, $\theta$, 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 $\pi/2$ rotation) then the bulk Josephson current vanishes for all $\theta$. Even in this case, we show that for a finite junction, the Josephson current, $J$, has a nonzero edge contribution that depends on $\theta$ 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 $J$.