🔬 mesoscale physics

Theory of planar quasiballistic Josephson junctions

This paper develops a quasiclassical theoretical framework for planar quasiballistic Josephson junctions that accounts for surface and impurity reflections, demonstrating strong coupling in multiterminal systems and successfully explaining non-monotonic critical current behaviors observed in recent experiments.

Original author: G. A. Bobkov, I. V. Bobkova, A. M. Bobkov, K. B. Polevoy, V. S. Stolyarov

Published 2026-11-03

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CC BY 4.0

Original author: G. A. Bobkov, I. V. Bobkova, A. M. Bobkov, K. B. Polevoy, V. S. Stolyarov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of quantum physics, there is a peculiar phenomenon where electricity can flow without any resistance at all, a state known as superconductivity. This happens when certain materials are cooled to temperatures near absolute zero, allowing electrons to pair up and move in perfect unison. When two such superconducting materials are separated by a thin barrier of normal metal, a special kind of current can still tunnel through the gap, linking the two sides together. This is called a Josephson junction. While simple versions of these junctions have been studied for decades, scientists are now building much more complex structures. Imagine placing several superconducting electrodes on top of a single, large sheet of metal. In this setup, the supercurrent does not just flow between two points; it weaves through a network of connections, creating a system where the behavior of one part depends heavily on the others. Understanding how this current moves through such a crowded, multi-terminal landscape is essential for developing new types of quantum devices, but the physics involved is notoriously difficult to pin down.

A team of researchers from Moscow and surrounding institutions has now developed a new theoretical framework to describe exactly how this current behaves in a specific type of these complex devices. They focused on "planar quasiballistic" junctions, which are essentially flat sandwiches where superconducting leads sit on top of a large, nearly perfect crystal of normal metal. In these systems, the electrons travel so fast and the metal is so pure that they rarely bump into impurities. Instead, their paths are dominated by how they bounce off the top and bottom surfaces of the metal sheet. The researchers found that the standard ways of describing these systems were insufficient. The old models assumed the electrons were either moving in a perfectly straight line without hitting anything, or that they were constantly scattering off impurities like billiard balls in a chaotic game. Neither of these extremes fit the reality of their experiment, where the electrons travel in a middle ground: they move ballistically for long distances but are occasionally reflected by the crystal's surfaces or a single impurity.

To solve this, the team created a detailed mathematical model based on the way electrons move through the metal, accounting for these specific surface reflections. They tested three different scenarios to see which one matched the physical reality of the experiment. The first scenario considered a thick crystal where electrons might hit a single impurity deep inside the metal. The second looked at a very thin crystal with perfectly smooth surfaces, where electrons would bounce back and forth like light in a mirror. The third scenario involved a thin crystal with rough surfaces, where electrons would scatter in random directions after hitting the bottom. By calculating the critical current—the maximum amount of supercurrent the junction can carry before it stops being superconducting—the researchers could predict how the system would behave under different temperatures and magnetic fields.

The results of their calculations were then compared against real experimental data from a companion study involving gold crystals and aluminum electrodes. The experiment showed something surprising: the critical current did not simply get smaller as the distance between the electrodes increased. Instead, it followed a non-monotonic pattern, meaning it changed in a complex, up-and-down way depending on the specific geometry. The new theory successfully explained this behavior. It demonstrated that the junctions in these multi-terminal setups are not independent islands; rather, they form a strongly coupled system where the current flowing between any two points is influenced by all the other connections in the network. The researchers found that the model assuming electrons scatter diffusively from the rough bottom surface of the thin gold crystal provided the best match to the experimental data, particularly in how the current changed with temperature.

One of the most significant findings was that the relationship between the critical current and the distance between the leads depends heavily on the shape and size of the superconducting electrodes. In some cases, as the temperature dropped, the current grew in a way that was faster than a simple straight line, a behavior driven by the planar geometry of the setup. This happens because, at lower temperatures, the "coherence length"—a measure of how far the superconducting state extends—gets longer, allowing more of the superconducting electrodes to participate in the current flow. The team also showed that applying a magnetic field affects these junctions differently depending on their width. Wide junctions showed a classic interference pattern, while narrow ones showed a steady, monotonic decrease in current, a behavior distinct from what is seen in purely ballistic systems.

This work provides a crucial foundation for designing future quantum devices. By proving that these complex, multi-terminal systems can be accurately described using a theory that accounts for surface reflections and quasiballistic transport, the researchers have offered a tool for engineers to predict how these circuits will behave before they are built. The study confirms that in these delicate quantum structures, the whole is indeed greater than the sum of its parts, with the collective behavior of the network dictating the flow of electricity in ways that simple two-terminal models could never predict.

Technical Summary: Theory of Planar Quasiballistic Josephson Junctions

Problem Statement
The paper addresses the theoretical description of complex planar Josephson junctions (JJs) where multiple superconducting (S) leads are coupled through a large, nearly ballistic normal metal (N) crystal. While mesoscopic multi-terminal Josephson systems have been experimentally realized in diffusive, hybrid semiconductor-superconductor, and topological materials, a specific experimental implementation using a quasi-ballistic gold (Au) single crystal with superconducting aluminum (Al) electrodes deposited on top (SN/N/SN geometry) lacked a corresponding theoretical framework.

Existing theoretical descriptions for planar SN/N/SN structures are typically limited to the diffusive limit ( $l \ll \xi_d$ ), which is inapplicable here because the mean free path ( $l \approx 680$ nm) and the ballistic coherence length ( $\xi \approx 1.7$ µm) are comparable to or larger than the crystal thickness ( $d \approx 80$ nm) and interlayer lengths. Conversely, purely ballistic theories are also insufficient because no direct ballistic trajectory connects one superconducting lead to another in this planar geometry; current transport relies on electron trajectories involving reflections from the crystal surfaces or impurities. The authors aim to develop a theory that bridges these limits to explain experimental observations, specifically the non-monotonic dependence of critical current on interlayer length and the strong coupling between junctions in multi-terminal setups.

Methodology
The authors develop a theoretical framework based on the quasiclassical Eilenberger equations for the Green's functions $\check{g}(\mathbf{n}, \mathbf{r}, \omega)$ . The approach assumes a low-transparency interface between the superconductor and normal metal ( $D \ll 1$ ), allowing for a perturbative expansion in the inverse scattering time.

Three distinct scattering models are formulated to describe the quasiballistic transport, depending on the crystal thickness and surface roughness:

Bulk Impurity Scattering (Thick Crystal): Applicable when $d \gtrsim (\xi, l)$ . Transport is dominated by single scattering events at impurities within the bulk of the semi-infinite normal crystal.
Multiple Specular Reflections (Thin Crystal, Smooth Surfaces): Applicable when $d \ll (\xi, l)$ and surfaces are smooth. Transport occurs via trajectories undergoing multiple specular reflections from the upper and lower surfaces of the normal layer. This model generalizes previously studied 1D translationally invariant systems to arbitrary lead geometries using the method of image sources.
Single Diffusive Reflection (Thin Crystal, Rough Surfaces): Applicable when $d \ll (\xi, l)$ and surfaces are rough. The model considers a single isotropic diffusive scattering event from the bottom surface, neglecting multiple diffusive reflections.

For all models, the Josephson current between two specific leads is calculated by summing contributions from all pairs of small superconducting elements (source and receiver) across the leads, utilizing the principle of linear superposition. The total critical current for a multi-terminal system is determined by numerically solving for the phase distribution that satisfies Kirchhoff's current law (zero current divergence) at all internal nodes, given fixed phases at the input and output leads.

Key Contributions and Results

Theoretical Framework: The paper provides the first quasiclassical description of planar quasi-ballistic JJs that accounts for surface and impurity reflections, distinct from both purely ballistic and diffusive limits.
Temperature Dependence ( $I_c(T)$ ): The theory predicts two types of temperature dependence for the critical current:
- Linear growth saturating at low temperatures (similar to the Kulik-Omelyanchuk result for short junctions).
- Super-linear growth at low temperatures before saturation. This behavior arises because the temperature coherence length $\xi_T$ increases as $T$ decreases, causing a larger effective area of the superconducting leads (of order $\xi_T$ ) to contribute to the current. The specific behavior depends on the scattering model and the geometry of the leads.
Magnetic Field Dependence ( $I_c(B)$ ): The critical current response to a perpendicular magnetic field varies from the standard Fraunhofer pattern (for wide junctions, $w \gg \xi$ ) to a monotonic decrease (for narrow junctions, $w < \xi$ ). The absence of magnetic interference patterns in narrow junctions is attributed to the smearing of closed-loop Andreev bound states in the planar geometry.
Multi-Terminal Coupling: A crucial finding is that in multi-terminal setups, junctions are not independent. The measured critical current between any two leads is a result of a strongly coupled system involving all possible paths between all leads. The theory successfully reproduces the experimental observation that the critical currents of inner junctions ( $A_2-B_2, A_3-B_3$ ) are slightly larger than those of outer junctions ( $A_1-B_1, A_4-B_4$ ) in a specific 4-terminal configuration.
Quantitative Agreement: When applied to the experimental data from a companion work (Polevoy et al., Phys. Rev. Lett. 137, 166001 (2026)), the model of single diffusive reflection from the bottom surface (and to a lesser extent, multiple specular reflections) provides good quantitative agreement. The theory reproduces the ratios of critical currents between different lead pairs without fitting parameters, with only the interface transparency $D$ adjusted to match the absolute amplitude of the critical current.

Significance
The paper claims to provide a foundational theoretical framework for understanding and designing complex multi-terminal Josephson systems fabricated on large quasi-ballistic crystals. By demonstrating that these junctions form a strongly coupled system rather than a collection of independent devices, the theory explains key experimental anomalies, such as the non-monotonic dependence of critical current on interlayer length. The work establishes that the specific scattering mechanism (surface vs. bulk) and the geometry of the leads are critical determinants of the system's transport properties, offering a necessary tool for interpreting experiments in the emerging field of topological matter and artificial band structures in Josephson devices.

Technical Summary: Theory of Planar Quasiballistic Josephson Junctions

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