Frustrated Frustration of Arrays with Four-Terminal Nb-Pt-Nb Josephson Junctions

TL;DR

This work demonstrates a 30×30 array of four-terminal Nb-Pt-Nb Josephson junctions in which an alternating checkerboard flux texture $f=\Phi/\Phi_0$ and $f'=\Phi'/\Phi_0$ stabilizes a superconducting phase via the BKT mechanism even at irrational flux values. A classical overdamped RCSJ network model with two interleaved flux patterns captures the observed dc-resistance beating and the resulting frustrated-frustration pattern, including a beating period linked to the area ratio $A/A'$. The authors extract the central weak-link area as $A'\approx(165\ \text{nm})^2$ using the measured beat frequency, and they show reproducibility across two arrays on the same chip as well as a reference 2TJJ. The study reveals a new avenue for exploring vortex configurations and quasiperiodic flux textures in multi-terminal Josephson junction systems, with implications for understanding flux control and weak-link engineering in superconducting devices. Key findings include the identification of a robust frustrated-frustration pattern, the estimation of the central weak-link area, and the demonstration of BKT-stabilized superconductivity at incommensurate flux textures.

Abstract

We study the frustration pattern of a square lattice with in-situ fabricated Nb-Pt-Nb four-terminal Josephson junctions. The four-terminal geometry gives rise to a checker board pattern of alternating fluxes f, f' piercing the plaquettes, which stabilizes the Berezinskii-Kosterlitz-Thouless transition even at irrational flux quanta per plaquette, due to an unequal repartition of integer flux sum f+f' into alternating plaquettes. This type of frustrated frustration manifests as a beating pattern of the dc resistance, with state configurations at the resistance dips gradually changing between the conventional zero-flux and half-flux states. Hence, the four-terminal Josephson junction array offers a promising platform to study previously unexplored flux and vortex configurations, and provides an estimate on the spatial expansion of the four-terminal Josephson junction central weak link area.

Figures (14)

• Figure 1: a) Schematics of the Josephson junction array where each 4TJJ is described by four interconnected two-terminal Josephson junctions (see dashed circle) draelos_supercurrent_2019agraziano_transport_2020arnault_multiterminal_2021aarnault_dynamical_2022. Their weak link material is platinum (see inset). The array is formed by connecting the superconducting arms of the 4TJJs. Upon application of a magnetic field each large plaquette area ($A$) of the array is penetrated by a magnetic flux $\Phi$ and the 4TJJ weak link area ($A^\prime$) by a flux $\Phi^\prime$. b) Theoretical representation of the array shown in a) as a 2TJJ array with square plaquettes having alternating frustrations $f = \Phi/ \Phi_0$ (grey) and $f^\prime = \Phi^\prime/ \Phi_0$ (blue). c) False-color scanning electron micrograph of a $5\times5$ four-terminal Josephson junction array with partially removed stencil mask. The platinum (blue) is deposited under rotation and, thus, covers a larger area. The niobium (red) forms the superconducting contacts. The dashed square represents unit cell area $A_\text{uc}$. d) Scanning electron micrograph of the $30\times30$ 4TJJ array presented in this study. Due to the shadow evaporation process, the 4TJJ array is slightly deformed at the left and right ends of the array.
• Figure 2: Measurement data of the $30\times30$ 4TJJ array at 80 mK. a) Differential resistance as a function of bias current and magnetic field. A periodic oscillation of the critical current is clearly visible. b) Resistance as a function of magnetic field with an applied dc bias of 30 $\mu$A. The resistance oscillations correspond to a periodicity of 6.25 mT. At around $\pm5\;f_\mathrm{uc}$, the resistance oscillations are damped. c) Resistance oscillations under magnetic field ranging to $\pm140$ mT with an applied dc bias of 30 $\mu$A. In total, 30 flux quantum oscillations are present.
• Figure 3: Results of the theoretical analysis. In all curves (a,c-g), the $x$-axis is $f$, and the $y$-axis is $R_\text{dc}/R$. (a) $R_\text{dc}$ as a function of $f$ for the checker board model with $f/f^\prime=10.9$ and $I/I_c=5.0$. (b) Checker board ($f\neq f^\prime$) versus regular ($f=f^\prime$) lattice models. (c) Zoomed out version of (a) showing the beating pattern. (d) $R_\text{dc}$ for the regular lattice model $f=f^\prime$, with all other parameters the same as in (a,c). (e,f) $R_\text{dc}$ including Fraunhofer effect (red curves, see main text), where for the checker board model $f_0=30$ (e) and for the regular lattice $f_0=10.9$ (f). (g) $R_\text{dc}$ for $f/f^\prime=10.9$ (no Fraunhofer) for decreasing bias current. Data shifted for clarification. (h) Equilibrium loop current configuration ($I=0$) for different values of $f,f^\prime$. The dots indicate counter- (red) or clockwise (blue) going currents and their size represents the magnitude of the loop current (relative linear scale).
• Figure S1: Data of the second 30$\times$30 four-terminal Josephson junction array. a) Differential resistance-bias current-magnetic field diagram. The oscillations of the critical current are clearly visible. b) Scanning electron microscopy image of the device. c) I-V measurement at 80 mK. d) Differential resistance calculated from the data in c).
• Figure S2: Scanning electron microscope images of the reference two-terminal Josephson junction.