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 dx2−y2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d}_{{x}^{2}-{y}^{2}}$$\end{document}-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.