Josephson diode effects in twisted nodal superconductors

Recent Josephson tunneling experiments on twisted flakes of high-Tc cuprate superconductor Bi2Sr2CaCu2O8+x revealed a nonreciprocal behavior of the critical interlayer Josephson current, i.e., a Josephson diode effect. Motivated by these findings we study theoretically the emergence of the Josephson diode effect in twisted interfaces between nodal superconductors, and highlight a strong dependence on the twist angle 0 and damping of the junction. In all cases, the theory predicts diode efficiency that vanishes exactly at 0 = 45 degrees and has a strong peak at a twist angle close to 0 = 45 degrees , consistent with experimental observations. Near 45 degrees , the junction breaks time -reversal symmetry T spontaneously. We find that for underdamped junctions showing hysteretic behavior, this results in a dynamical Josephson diode effect in a part of the T -broken phase. The direction of the diode is trainable in this case by sweeping the external current bias. This effect provides a sensitive probe of spontaneous T breaking. We then show that explicit T -breaking perturbations with the symmetry of a magnetic field perpendicular to the junction plane lead to a thermodynamic diode effect that survives even in the overdamped limit. We discuss an experimental protocol to probe the double -well structure in the Josephson free energy that underlies the tendency towards spontaneous T breaking even if T is broken explicitly. Finally, we show that in -plane magnetic fields can control the diode effect in the short junction limit, and predict the signatures of explicit T breaking in Shapiro steps.