Josephson effects in twisted cuprate bilayers

TL;DR

This paper investigates Josephson effects in twisted cuprate bilayers, revealing signatures of topological phases with broken time-reversal symmetry through unconventional Josephson relations and experimental signatures in Fraunhofer patterns and Shapiro steps.

Contribution

It provides theoretical predictions of Josephson phenomena indicating topological phases in twisted cuprate bilayers, including a transition to a $ ext{pi}$-periodic Josephson relation near 45° twist.

Findings

Non-monotonic temperature dependence of critical current at intermediate twist angles

Transition from 2π to π periodicity in Josephson relation near 45° twist

Altered Fraunhofer patterns and Shapiro steps as signatures of the T-breaking phase

Abstract

Twisted bilayers of high-$T_c$ cuprate superconductors have been argued to form topological phases with spontaneously broken time reversal symmetry ${\cal T}$ for certain twist angles. With the goal of helping to identify unambiguous signatures of these topological phases in transport experiments, we theoretically investigate a suite of Josephson phenomena between twisted layers. We find an unusual non-monotonic temperature dependence of the critical current at intermediate twist angles which we attribute to the unconventional sign structure of the $d$-wave order parameter. The onset of the ${\cal T}$-broken phase near $45^\circ$ twist is marked by a crossover from the conventional $2\pi$-periodic Josephson relation $J(\varphi)\simeq J_c\sin{\varphi}$ to a $\pi$-periodic function as the single-pair tunneling becomes dominated by a second order process that involves two Cooper pairs. Despite this fundamental change, the critical current remains a smooth function of the twist angle $\theta$ and temperature $T$ implying that a measurement of $J_c$ alone will not be a litmus test for the ${\cal T}$-broken phase. To obtain clear signatures of the ${\cal T}$-broken phase one must measure $J_c$ in the presence of an applied magnetic field or radio-frequency drive, where the resulting Fraunhofer patterns and Shapiro steps are altered in a characteristic manner. We discuss these results in light of recent experiments on twisted bilayers of the high-$T_c$ cuprate superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$.