Josephson effects in twisted nodal superconductors

Motivated by the recent proposals for unconventional emergent physics in twisted bilayers of nodal superconductors, we study the peculiarities of the Josephson effect at the twisted interface between d-wave superconductors. We demonstrate that for clean interfaces with a twist angle θ0 in the range 0°C <θ0<45°C, the critical current can exhibit nonmonotonic temperature dependence with a maximum at a nonzero temperature as well as a complex dependence on the twist angle at low temperatures. These effects are shown to reflect the destructive interference between the d-wave order parameters near the nodes at nonzero twist angle. Close to θ0=45°C we find that the critical current does not vanish due to Cooper pair cotunneling, which can lead to the transition to a time-reversal breaking superconducting d+id phase, which can be suppressed by the interface roughness. We provide a comprehensive theoretical analysis of experiments that can reveal this cotunneling for twisted superconductors close to θ0=45°C. In particular, we demonstrate that both the emergence of the Fraunhofer interference pattern near θ0=45°C and fractional Shapiro steps yield unambiguous evidence of Cooper pair cotunneling, necessary for topological superconductivity.