Theory of planar quasi-ballistic Josephson junctions

TL;DR

This work develops a quasiclassical framework based on the Eilenberger equations to describe planar quasi-ballistic Josephson junctions with multiple superconducting leads on a large crystal. It introduces three transport models to capture impurity-driven, specular, and diffusive surface scattering across thickness regimes, and employs a trajectory-based Green's-function approach to compute interlead currents. The study reveals strong coupling between junctions in multi-terminal setups and explains nonmonotonic $I_c$ behavior with interlayer length, along with diverse magnetic-field responses from Fraunhofer-like to monotonic decay, in agreement with companion experiments. Together, these results provide a foundational theory for designing and understanding complex multi-terminal Josephson networks in the quasi-ballistic planar regime.

Abstract

We develop a theoretical framework for planar quasi-ballistic Josephson junctions, where multiple superconducting leads are coupled through a large, nearly ballistic normal metal crystal. Our approach, based on quasiclassical Eilenberger equations, accounts for the dominant role of electron reflections from the crystal surfaces or single impurities, a mechanism distinct from both purely ballistic and diffusive limits. We calculate the critical current between superconducting leads for various geometries, examining its dependence on temperature and magnetic field. Crucially, we demonstrate that in multi-terminal setups, the junctions are not independent but form a strongly coupled system. The theory successfully explains key experimental observations from a companion work, including a non-monotonic dependence of the critical current on the interlayer length, providing a foundation for designing and understanding complex multi-terminal Josephson systems.

Figures (7)

• Figure 1: (a) Collored SEM image of the studied complex system of Josephson junctions (JJs) fabricated by placing the Al superconducting electrodes on top of the Au monocrystalline sample. (b)-(d) Different models used in the work for description of the quasi-ballistic planar JJs. The main contribution to the electric current is given via (b) single scattering event at an impurity in the bulk of the semi-infinite N crystal; (c) multiple specular reflections at the N layer surfaces; (d) single diffusive reflection at the bottom N layer surface.
• Figure 2: Sketch of the trajectories between two small superconducting elements S and R contributing to the current flow between S and R via a single scattering event at the scattering point P.
• Figure 3: Sketch of the trajectories between two small superconducting elements S and R contribution to the current flow between S and R via a multiple mirror reflections.
• Figure 4: Dependence of the critical Josephson current on temperature calculated for a JJ between two superconducting leads in the absence of other leads on top of the N crystal. Different curves correspond to different JJs between the leads marked by the same letters in Fig. \ref{['fig:setup']}. (a) The model of a single scattering at a bulk impurity. $D=0.004$. (b) The model of multiple specular reflections from the surfaces of the N crystal. $D=0.013$. (c) The model of a single diffusive reflection from the bottom boundary of the N layer. $D=0.03$. The transparency for each of the models is chosen to fit the experimental value of the critical current Polevoy2025_joint.
• Figure 5: Dependence of the critical Josephson current on the applied perpendicular magnetic field. Different curves correspond to the same JJs as in Fig. \ref{['fig:one_temperature']}. The results are calculated in the framework of the model of a single diffusive reflection from the bottom N layer surface. $T=0.7K$.