Josephson diode effects in twisted nodal superconductors

Abstract

Recent Josephson tunneling experiments on twisted flakes of high-$T_c$ cuprate superconductor Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+x}$ revealed a non-reciprocal 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 $\theta$ and damping of the junction. In all cases, the theory predicts diode efficiency that vanishes exactly at $\theta =45^{\circ }$ and has a strong peak at a twist angle close to $\theta =45^{\circ }$, consistent with experimental observations. Near $45^{\circ }$, the junction breaks time-reversal symmetry $\mathcal{T}$ spontaneously. We find that for underdamped junctions showing hysteretic behavior, this results in a \emph{dynamical} Josephson diode effect in a part of the $\mathcal{T}$-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 $\mathcal{T}$-breaking. We then show that explicit $\mathcal{T}$-breaking perturbations with the symmetry of a magnetic field perpendicular to the junction plane lead to a {\em 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 $\mathcal{T}$-breaking even if $\mathcal{T}$ 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 $\mathcal{T}$-breaking in Shapiro steps.

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