🔬 mesoscale physics

Magnetic field-bias current interplay in HgTe-based three-terminal Josephson junctions

This study investigates HgTe-based three-terminal Josephson junctions, revealing how the interplay between bias current and perpendicular magnetic fields enables tunable interference patterns, symmetry control, and strongly enhanced Josephson diode efficiency, with experimental findings quantitatively validated by resistively shunted junction simulations.

Original authors: J. Thieme, W. Himmler, F. Dominguez, G. Platero, N. Hüttner, S. Hartl, E. Richter, D. A. Kozlov, N. N. Mikhailov, S. A. Dvoretsky, D. Weiss

Published 2026-07-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: J. Thieme, W. Himmler, F. Dominguez, G. Platero, N. Hüttner, S. Hartl, E. Richter, D. A. Kozlov, N. N. Mikhailov, S. A. Dvoretsky, D. Weiss

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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Superconducting Dance Floor

Imagine a world where electricity flows without any friction at all, like a skater gliding forever on perfectly smooth ice. This is the realm of superconductivity, a state of matter that happens when certain materials get incredibly cold. In this frozen world, electrons pair up and move in perfect unison, creating a current that never loses energy. But what happens when you try to push this perfect flow through a tiny gap or a junction? Usually, the current stops, but if the gap is small enough, the electrons can "tunnel" through, creating a Josephson junction. Think of this junction as a magical bridge where the current can flow without resistance, but only if the "dance steps" (called phases) of the electrons on both sides are in sync.

Now, imagine you have not just one bridge, but three bridges connected in a triangle or a cross. This is a multiterminal Josephson junction. It's like a complex dance floor where three partners (the electrical leads) are trying to coordinate their moves. The paper we are exploring today investigates what happens when you change the music (the magnetic field) and the tempo (the bias current) on this dance floor. Scientists care about this because these tiny, multi-bridge systems are the building blocks for future quantum computers and ultra-sensitive sensors. By understanding how to control the flow of supercurrents in these complex shapes, we might be able to build devices that can do things ordinary electronics never could, like solving problems too hard for today's supercomputers or detecting the faintest magnetic whispers in the universe.

The Magnetic Tug-of-War

In this study, the researchers built tiny superconducting devices shaped like the letters "T" and "X" using a special material called HgTe (Mercury Telluride) sandwiched between layers of other materials. They connected these shapes to three superconducting leads made of Niobium (Nb). Their goal was to map out the "Critical Current Contour" (CCC). If you imagine the CCC as a map, it shows the exact boundary where the device stays superconducting (zero resistance) versus where it turns normal and resists the flow.

The team discovered that this boundary isn't just a simple circle or square; it's a living, breathing shape that changes dramatically depending on two things: how much current they push into the device (the bias current) and how they tilt a magnetic field around it.

The Shape-Shifting Boundary
When the researchers applied a magnetic field parallel to the device (lying flat), the superconducting region shrank exponentially, looking like the first big lobe of a wave pattern known as a Fraunhofer pattern. However, when they tilted the magnetic field slightly so it poked through the device (an out-of-plane component), the pattern changed completely. It turned into an oscillating, SQUID-like pattern (SQUID stands for Superconducting QUantum Interference Device).

Here is the magic trick: by adjusting the bias current, they could smoothly switch between these two different patterns. It's as if they had a dial that could turn the device from a simple wave-maker into a complex interferometer. They found that the magnetic field doesn't just shrink the superconducting area; it actually deforms the shape of the CCC in the (I1, I2) plane. Depending on the magnetic flux, the shape could become symmetric or wildly asymmetric, stretching and squashing in the current space.

The Superconducting Diode Effect
One of the most exciting findings is the superconducting diode effect. Normally, a diode is a one-way street for electricity: it lets current flow easily in one direction but blocks it in the other. In standard superconductors, current flows both ways perfectly. However, in these multi-terminal junctions, the researchers found they could create a "one-way" supercurrent.

By applying a specific bias current and a small magnetic field, they made the device allow a large supercurrent to flow in one direction (say, positive) while blocking it almost entirely in the opposite direction (negative). They measured an efficiency (how good the diode is) of up to η ≈ 0.8 at low bias and low magnetic fields. This means the device is highly effective at rectifying supercurrents, a feat that usually requires breaking specific symmetries in the material. The paper shows that this effect is tunable; you can turn the "diode-ness" up or down just by changing the current or the magnetic field.

Simulating the Dance
To make sure they understood what was happening, the team used a computer model called the Resistively Shunted Junction (RSJ) model. This model treats the junctions like a circuit with resistors and ideal superconductors. The simulations matched the experimental data almost perfectly. This allowed them to "disentangle" the contributions of each individual junction within the complex T and X shapes. They confirmed that the strange deformations and the diode effect weren't random glitches but were the result of the magnetic flux changing the phase relationship between the three junctions.

What It's Not
It is important to note what this paper is not claiming. The authors explicitly state that despite using HgTe, a material often associated with "topological" physics (exotic states of matter), they do not see evidence of topological superconductivity or Majorana bound states in their results. The dimensions of their devices are too large for the specific quantum interference effects of topological surface states to dominate. Instead, the behavior they observed is fully explained by standard superconducting physics and the geometry of the junctions. They are not discovering a new particle; they are mastering the control of existing supercurrents in a new way.

The Takeaway
This paper demonstrates that multiterminal Josephson junctions are incredibly versatile platforms. By playing with the "knobs" of magnetic field and bias current, scientists can engineer the landscape of supercurrents, creating shapes and behaviors that were previously hard to predict. They showed that you can engineer a highly efficient superconducting diode and control the interference patterns of the current simply by tuning these external parameters. This level of control is a powerful step forward for designing future superconducting circuits, potentially leading to more stable quantum bits (qubits) and new types of precision sensors, all without needing to rely on exotic, hard-to-find topological effects.

Technical Summary: Magnetic Field – Bias Current Interplay in HgTe-based Three-Terminal Josephson Junctions

Problem and Motivation
Multiterminal Josephson junctions (MTJJs) offer a versatile platform for realizing topological states of matter and novel superconducting circuit applications, such as the superconducting diode effect. In these systems, superconducting phase differences between multiple terminals act analogously to crystal momentum in a Brillouin zone, enabling the definition of synthetic Chern numbers and the hosting of topological phases like Weyl singularities and Majorana bound states. While previous research has characterized the effects of magnetic fields, gate dependence, and diode effects in fully coupled junctions, the specific relationship between the collective system behavior and its individual junction components remains largely unexplored. This work addresses the gap in understanding how bias currents and magnetic fields jointly determine the collective Josephson behavior in three-terminal devices.

Methodology
The study investigates HgTe/Nb-based three-terminal Josephson junctions fabricated in T-shaped and X-shaped geometries. The devices utilize a topological insulator HgTe grown lattice-matched between Cd $_{0.7}$ Hg $_{0.3}$ Te and CdTe layers, with Nb contacts deposited to form high-transparency interfaces ( $D \approx 0.7$ ).

Experimental Setup: Measurements were conducted at a base temperature of $\sim 20$ mK in a dilution refrigerator equipped with a rotatable sample holder to apply magnetic fields at various angles. A quasi-four-point measurement scheme was employed, allowing for the independent measurement of voltage drops between all pairs of contacts. This enabled the calculation of differential resistance ( $dU_{ij}/dI_1$ ) for each junction individually as a function of bias currents ( $I_1, I_2$ ) and magnetic field ( $B$ ).
Theoretical Modeling: The experimental data were quantitatively reproduced using a Resistively Shunted Junction (RSJ) model adapted for multiterminal systems. This model incorporates current conservation, the Josephson equations, and the gauge-invariant phase relations modified by magnetic flux. The model accounts for both in-plane fields (causing exponential suppression via the Doppler effect or kinetic energy) and out-of-plane fields (inducing phase shifts and SQUID-like interference).

Key Results
The investigation focuses on the Critical Current Contours (CCCs) in the $(I_1, I_2)$ plane and their evolution under magnetic fields.

Decomposition of CCCs: The CCCs were decomposed into contributions from individual junctions. In the intermediate bias regime, the CCC exhibits a six-lobed pattern where each arm corresponds to a specific junction remaining in the superconducting state. The width and slope of these arms were analytically derived and found to depend on both the individual critical currents ( $I_{c,ij}$ ) and the relative resistances ( $R_{ij}$ ) of the junctions.
Magnetic Field Interplay:
- In-plane Fields: Induce an exponential suppression of the critical current, resembling the first lobe of a Fraunhofer pattern.
- Out-of-plane Fields: Introduce a tunable crossover between SQUID-like and Fraunhofer-like interference patterns. A perpendicular magnetic field creates a SQUID-like oscillatory pattern superimposed on the exponential suppression. The amplitude of this pattern scales with the minimum critical current of the three junctions.
- Bias Current Control: The applied bias current ( $I_2$ ) controls the crossover between these regimes. A finite $I_2$ shifts the superconducting region, distorting the SQUID pattern and creating asymmetry in the $(I_1, I_2)$ plane.
Symmetry and Deformation: Magnetic flux induces pronounced deformations of the CCC. Depending on the flux value ( $\Phi/\Phi_0$ ), the CCC transitions between symmetric and asymmetric shapes. For half-integer flux quanta, the functional form of the supercurrent changes (sine to cosine), leading to asymmetric lobes in the current plane.
Superconducting Diode Effect: The combined breaking of time-reversal symmetry (by bias current) and inversion symmetry (by geometry) results in a tunable superconducting diode effect. The efficiency ( $\eta$ ) of this effect reaches values up to $\eta \approx 0.8$ in the low-bias and low-magnetic-field regime. The efficiency is maximized when the CCC is asymmetrically deformed by the magnetic flux and bias current.

Significance and Claims
The authors claim that their work demonstrates that multiterminal Josephson junctions constitute a versatile platform where magnetic flux and bias currents act as interdependent control parameters to engineer the supercurrent landscape.

Predictive Capability: The quantitative agreement between the experimental CCCs and the RSJ model allows for the accurate prediction of device behavior based on the intrinsic gaps and resistances of individual junctions. This provides a foundation for designing multiterminal superconducting circuits with improved reproducibility and engineered Josephson couplings.
Diode Efficiency: The identification of a regime with high diode efficiency ( $\eta \approx 0.8$ ) without requiring external magnetic fields (or with minimal fields) highlights the potential of MTJJs for non-reciprocal superconducting devices.
Fundamental Physics: While the specific HgTe devices studied did not exhibit topological superconductivity features (such as $4\pi$ periodicity), the framework established bridges fundamental superconducting physics with practical device engineering. The ability to disentangle individual junction contributions from collective behavior offers a pathway toward controlled, tunable, and scalable superconducting quantum and sensing architectures.

The paper concludes that the interplay of geometry, current bias, and magnetic fields in MTJJs creates rich, tunable behaviors that extend beyond simple sums of individual critical currents, offering new mechanisms for symmetry control and diode effect enhancement.

The Superconducting Dance Floor

The Magnetic Tug-of-War

More like this