🔬 mesoscale physics

High-field Josephson effect enabled by a moiré Hofstadter spectrum

This paper demonstrates that moiré-engineered graphene Josephson junctions can sustain phase-coherent superconductivity up to 6 Tesla by utilizing dispersive magnetic Bloch bands within the Hofstadter spectrum to overcome the magnetic field limitations that typically suppress conventional Josephson transport.

Original authors: A. Díez-Carlón, M. Cárdenes Wuttig, N. Wei, D. Ivanov, P. Altpeter, P. Hakonen, K. Watanabe, T. Taniguchi, L. I. Glazman, D. K. Efetov

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: A. Díez-Carlón, M. Cárdenes Wuttig, N. Wei, D. Ivanov, P. Altpeter, P. Hakonen, K. Watanabe, T. Taniguchi, L. I. Glazman, D. K. Efetov

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

Imagine a superhighway for electricity where the cars are electrons, and the traffic lights are magnetic fields. Usually, when you crank up the magnetic field, it's like a giant magnet sweeping through the highway, forcing every car into tight, spinning circles. They can't move forward; they just spin in place. In the world of superconductors, this is a disaster. It stops the "super-current" from flowing, effectively shutting down the device. For decades, scientists thought this was the end of the road: if you wanted to use superconductors to measure things or build quantum computers, you had to keep the magnetic field low.

But a team of researchers has just found a secret tunnel that lets the traffic flow even when the magnetic storm is raging at a massive 6 T (tesla).

Here is how they did it, and why it changes the rules of the game.

The Problem: The Magnetic Spin Cycle

In a normal superconductor, electrons pair up and zip through a material with zero resistance. But if you apply a strong magnetic field, those electrons get confused. They start orbiting in tiny circles (called cyclotron orbits) instead of moving straight. If the circle is bigger than the device itself, the electrons hit the walls and stop. They can't cross the gap. This usually kills the super-current at around 2.5 T.

The Solution: The Moiré Magic Carpet

The researchers built a special bridge using a sandwich of graphene (a single layer of carbon atoms) and hexagonal boron nitride (hBN). When they stacked these two materials at a tiny, specific angle (about 0.21°), they created a giant, repeating pattern called a "moiré superlattice."

Think of it like holding two window screens over each other at a slight angle. You see a new, giant pattern of hexagons appear. This pattern acts like a custom-built track for the electrons.

The Discovery: The Fractal Butterfly

When they turned up the magnetic field, they expected the electrons to get stuck in their tiny circles, just like in normal graphene. Instead, something magical happened. The moiré pattern interacted with the magnetic field to create a "fractal" energy landscape, known as the Hofstadter butterfly.

In this new landscape, the electrons didn't get stuck in flat, spinning circles. Instead, the moiré pattern turned those flat, dead-end orbits into "dispersive" bands—think of them as winding, sloping slides instead of flat parking spots. This gave the electrons a new kind of speed (a finite group velocity) that allowed them to zoom across the junction, even in the middle of a 6 T magnetic field.

The team measured this by looking at the "critical current" (the maximum super-current the device can hold). In their special device, the current didn't just survive; it oscillated in a complex, chaotic pattern all the way up to 6 T. This is far beyond the 2.5 T limit seen in standard graphene devices.

What It Is NOT

It is important to know what this isn't. The researchers explicitly ruled out a few other possibilities:

It's not the "Edge" trick: In some high-field experiments, super-currents sneak along the very edge of a material (like cars driving on the shoulder). The authors show that in their device, the current is flowing right through the middle (the bulk), not just the edges.
It's not just "flat" orbits: If the electrons were just stuck in the flat, non-moving orbits of a standard magnetic field, the current would have died. The fact that it survived proves the electrons were moving on these new, sloping "Hofstadter" tracks.

How They Knew

The team didn't just guess; they measured it. They built several devices and watched the current flow as they changed the magnetic field and the number of electrons (carrier density). They saw the super-current persist in specific "pockets" of the map, right where their computer simulations predicted the "Hofstadter minibands" would be.

They also checked the "normal" state (when the super-current is off) and confirmed that the electrons were behaving ballistically—bouncing back and forth like light in a mirror tunnel (Fabry-Pérot oscillations)—which proved the material was clean and the electrons were moving freely.

The Bottom Line

This paper suggests that by engineering the "track" electrons run on (using moiré patterns), we can protect superconductivity from magnetic fields that would normally destroy it. It's like building a rollercoaster that keeps running even when the wind is howling.

The authors measured super-currents surviving up to 6 T, a regime previously thought to be the exclusive domain of the "Quantum Hall" effect where superconductivity usually fails. They propose that this happens because the moiré potential transforms the electrons' motion from stuck orbits into flowing bands. While they have measured this effect and simulated the theory to match, they note that this opens a new door for studying quantum matter in high magnetic fields, potentially leading to new types of superconducting sensors or quantum devices that can operate in much stronger environments than ever before.

Technical Summary: High-field Josephson effect enabled by a moiré Hofstadter spectrum

Problem Statement
Conventional Josephson junctions (JJs) rely on phase-coherent supercurrents, which are typically suppressed by magnetic fields due to destructive interference and the bending of electron trajectories into cyclotron orbits. This limitation restricts superconducting interferometry to relatively low magnetic fields, preventing the direct phase-sensitive study of exotic quantum phases (such as quantum Hall states and topological superconductors) that emerge only in high-field regimes. In ballistic graphene JJs, the proximity effect is generally limited to fields below $\sim 2.5$ T, determined by mesoscopic device dimensions where the cyclotron diameter exceeds the junction length ( $2r_c > L$ ). Beyond this limit, bulk transport is suppressed, and supercurrents are typically restricted to fragile chiral edge states with negligible critical currents.

Methodology
The authors engineered ballistic Josephson junctions using monolayer graphene encapsulated in hexagonal boron nitride (hBN), where one hBN layer is aligned with the graphene to induce a moiré superlattice potential. The weak link of the junction consists of this graphene/hBN moiré superlattice, contacted by superconducting Molybdenum-Rhenium (MoRe) leads with a high critical field ( $\sim 8$ T).

The study employed the following experimental and theoretical approaches:

Device Fabrication: Van der Waals heterostructures were fabricated using dry transfer techniques, with one-dimensional edge contacts to the MoRe leads. Devices were characterized with varying twist angles ( $\theta \approx 0.21^\circ$ ) and junction lengths ( $L \approx 200$ nm).
Transport Measurements: Differential resistance ($dV/dI$) and critical current ( $I_c$ ) were measured as a function of carrier density ( $n$ ) and perpendicular magnetic field ( $B$ ) at base temperatures ($100$ mK).
Interferometry Analysis: The team analyzed Fabry-Pérot oscillations in the normal state and critical current to confirm ballistic transport and phase coherence. They mapped the Josephson interference patterns (Fraunhofer and high-field irregular oscillations) across the primary Dirac cone and the reconstructed moiré minibands.
Theoretical Modeling: The authors performed numerical calculations of the Hofstadter spectrum for the graphene/hBN system. They computed the average squared group velocity ( $\langle v^2 \rangle$ ) of the magnetic Bloch bands and compared these results with experimental inverse resistance ( $1/R$ ) maps to correlate superconducting survival with band dispersion.

Key Contributions and Results

Ballistic Transport in Moiré Minibands: The study established phase-coherent Andreev transport across both the primary Dirac cone and the reconstructed moiré minibands. Fabry-Pérot oscillations were observed in the critical current, persisting into the minibands ( $n/n_s < -1$ ), with a reduced Fermi velocity ( $v_F \sim 0.5 \times 10^6$ m/s) compared to the main Dirac cone.
High-Field Superconductivity: The most significant finding is the observation of robust, phase-coherent Josephson interference up to magnetic fields of $\sim 6$ T. This occurs specifically when the Fermi level enters the moiré minibands ( $n/n_s > 1$ ), a regime where conventional semiclassical models predict the suppression of bulk supercurrents.
Mechanism of Stabilization: The authors demonstrate that the survival of superconductivity in this high-field regime is not due to chiral edge states (which would yield much smaller critical currents, $I_c \lesssim 1$ nA). Instead, the critical currents reach values comparable to the quantum ballistic limit of a single Andreev bound state mode ( $I_c \sim 48$ nA).
Role of the Hofstadter Spectrum: Theoretical calculations reveal that in the moiré minibands, the moiré potential transforms the flat Landau levels (which have quenched group velocity) into dispersive magnetic Bloch bands. This "magnetic breakdown" restores a finite quasiparticle group velocity ( $v_H$ ). The authors show that the regions of high-field superconductivity coincide precisely with regions where the average squared group velocity $\langle v^2 \rangle$ remains significant, allowing for extended electron-hole Andreev trajectories across the junction.
Contrast with Conventional Graphene: In non-moiré ballistic graphene junctions, superconductivity is suppressed once $2r_c < L$ . In the moiré-engineered junctions, the superconducting phase extends well beyond this geometric limit, up to $\sim 6$ T, because the dispersive Hofstadter bands prevent the complete localization of charge carriers into flat Landau levels.

Significance
The paper claims to establish a new platform for high-field superconducting interferometry. By utilizing moiré-engineered magnetic Bloch bands, the authors demonstrate a route to stabilize phase-coherent supercurrents deep into the parameter domain conventionally associated with the quantum Hall regime. This work identifies a previously unexplored mechanism where the interplay between moiré potentials and magnetic fields creates dispersive subbands that support mesoscopic Andreev trajectories, thereby overcoming the field limitations of conventional ballistic Josephson junctions. The results provide a direct probe of macroscopic quantum coherence in regimes where Landau quantization is suppressed by the moiré potential, offering a pathway to study Andreev processes in fractal spectra.