Josephson diode effects in twisted nodal superconductors

TL;DR

This paper theoretically investigates the Josephson diode effect in twisted nodal superconductors, revealing its dependence on twist angle, damping, and time-reversal symmetry breaking, and proposes experimental methods to probe these phenomena.

Contribution

It provides a comprehensive theoretical framework for understanding the Josephson diode effect in twisted superconductors, including the role of spontaneous and explicit time-reversal symmetry breaking.

Findings

Diode efficiency peaks near 45° twist angle, matching experiments.

Spontaneous ${ m T}$-breaking leads to a dynamical diode effect in underdamped junctions.

Explicit ${ m T}$-breaking perturbations cause a thermodynamic diode effect observable even in overdamped junctions.

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 ${\cal T}$ spontaneously. We find that for underdamped junctions showing hysteretic behavior, this results in a \emph{dynamical} Josephson diode effect in a part of the ${\cal T}$-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 ${\cal T}$-breaking. We then show that explicit ${\cal T}$-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 ${\cal T}$-breaking even if ${\cal T}$ 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 ${\cal T}$-breaking in Shapiro steps.