Edge dependence of the Josephson current in the quantum Hall regime

This study demonstrates that the Josephson current in the quantum Hall regime is mediated by counter-propagating edge states confined to physical edges, resolving previous ambiguities regarding its mechanism through a systematic comparison of various edge configurations in graphene devices.

The Gist

The Big Picture: A Traffic Jam in a Superhighway

Imagine you have a superhighway for electrons (the particles that carry electricity). Usually, in a special state called the Quantum Hall Regime, this highway is like a one-way street. The electrons are forced to run along the very edge of the road in a single direction, like cars on a racetrack. They can't turn around, and they can't cross to the other side.

Now, imagine you want to introduce Superconductivity (a state where electricity flows with zero resistance) into this system. You want to create a "Josephson Junction," which is basically a bridge where electrons can jump back and forth between two superconducting islands.

The Mystery:
Scientists have been trying to build this bridge, but they were confused.

1. The Puzzle: They saw electricity flowing across the bridge, but the "traffic rules" didn't make sense. If the electrons were strictly following the one-way edge rules, the bridge shouldn't work.
2. The Clue: The flow of electricity seemed to depend heavily on the edges of the road, not the middle. But nobody knew exactly why or how the electrons were getting across.

The Experiment: Testing Different Road Conditions

The researchers in this paper decided to play detective. They built several versions of this "electron highway" using graphene (a super-thin, strong material) and changed the condition of the road's edges to see what happened.

Think of it like testing different types of curbs on a racetrack:

1. The "Native Edge" (The Natural Curb): This is the edge left naturally when you cut the material. It's a bit rough and uneven, like a jagged cliff.
   • Result: The bridge worked! Electricity flowed easily.
2. The "Edge-Free" Design (The Smooth Floor): They made a device where there was no physical edge at all; the superconductors just sat on a smooth, flat sheet of graphene.
   • Result: Nothing happened. No electricity crossed. This proved that the "middle" of the road (the bulk) is an insulator and cannot carry the current. The action must happen at the edge.
3. The "Etched Edge" (The Smoothed Curb): They took a natural edge and used a laser/plasma to smooth it out perfectly.
   • Result: The bridge still worked, but it was weaker. The current was smaller. This was surprising! You'd think a smoother road is better, but in this quantum world, a slightly rough edge actually helps.
4. The "Gate-Defined Edge" (The Virtual Curb): Instead of cutting the material, they used an electric "fence" (a gate) to create a virtual edge.
   • Result: The bridge worked, but only if the "fence" created a specific kind of traffic jam where cars could go both ways.

The Solution: The "Two-Lane" Trick

So, how did the electrons cross the bridge? The paper solves the mystery with a clever mechanism called Counter-Propagating Edge States (CPES).

The Analogy: The Two-Lane River
Imagine the edge of the graphene isn't a single lane, but a tiny river with two lanes running right next to each other:

• Lane A: Electrons flowing forward (Downstream).
• Lane B: Electrons flowing backward (Upstream).

Normally, in a perfect Quantum Hall state, Lane B doesn't exist. But the researchers found that at the physical edges (especially the rough, natural ones), the electric field gets messy. This messiness creates a "bump" that forces some electrons to turn around and flow backward, creating that second lane.

How the Bridge Works:

1. An electron comes down Lane A.
2. It hits the superconducting bridge.
3. Instead of stopping, it turns into a "hole" (a missing electron) and jumps into Lane B, flowing backward.
4. This back-and-forth dance creates a "standing wave" of energy (called an Andreev Bound State).
5. This standing wave acts as the invisible bridge, allowing the supercurrent to flow across the gap.

Why Did the "Smooth" Edge Fail?

When the researchers smoothed out the edge (the Etched Edge), they removed the "bumps" that caused the traffic jam. Without the bumps, the electrons couldn't switch lanes to flow backward. They were stuck in a single lane, and the bridge collapsed.

The "Traffic Light" Experiment

To prove this, they built a device with a "traffic light" (a gate) that could control the lanes.

• Scenario 1: They set the gate so that the forward lane and backward lane cancelled each other out. Result: No bridge.
• Scenario 2: They set the gate so that both lanes existed. Result: The bridge appeared.

This confirmed that you need two lanes (one forward, one backward) running side-by-side at the edge to make the supercurrent work.

The Takeaway

This paper solves a long-standing mystery in physics. It tells us that:

1. Superconductivity in the Quantum Hall regime lives on the edges, not in the middle.
2. Roughness is good: A slightly imperfect, natural edge is actually better for this technology than a perfectly smooth, etched one because it creates the necessary "two-way traffic" for the electrons.
3. Future Tech: By understanding how to control these edge lanes, we can build better quantum computers and devices that use "Majorana particles" (a special type of particle needed for fault-tolerant quantum computing).

In short: The researchers found that to make electricity flow across a quantum bridge, you need a messy edge that forces electrons to run in circles, creating a secret two-way lane that acts as a superhighway for the current.

Technical Summary

1. Problem Statement

The coexistence of superconductivity and the quantum Hall (QH) effect in graphene Josephson junctions (GJJs) is a significant phenomenon for topological quantum computing, particularly for realizing Majorana zero modes. However, the microscopic mechanism driving the Josephson current (JC) in this regime has remained unresolved due to two primary contradictions in previous observations:

• Magnetic Interference Period: Experiments observed an interference period of $h/2e$, which contradicts the expected $h/e$ periodicity for chiral Andreev edge states (AES) traveling along a single edge.
• Conduction Channels: Deviations from precisely quantized Hall plateaus in the normal state suggested the existence of additional conduction channels, raising the question of whether the JC is mediated by the bulk of the QH graphene (which is imperfectly insulating) or by specific edge states.

Three potential mechanisms were proposed but unproven:

1. Chiral AES: Single-edge chiral states (expected $h/e$ period).
2. Counter-Propagating Edge States (CPES): Upstream and downstream modes on a single edge forming Andreev Bound States (ABS).
3. Imperfect Bulk Insulation: Current flowing through the bulk of the graphene during plateau transitions.

2. Methodology

The authors fabricated high-quality GJJs using graphene encapsulated in hexagonal boron nitride (hBN) on a silicon dioxide/silicon substrate. To systematically isolate the role of edge configurations, they created four distinct device types:

• Native Edge (NE): Devices with edges formed naturally during the exfoliation/transfer process.
• Etched Edge (EE): Devices where the graphene was physically etched (using plasma) to create a defined edge, introducing potential impurities.
• Edge-Free (EF): Devices where the graphene channel was suspended or defined without physical edges in the junction region, relying solely on bulk transport.
• Graphite Gate-Defined Edge (GGDE): Devices utilizing a narrow graphite strip as a local back-gate to define the edge potential and tune the local filling factor ($\nu_l$) independently of the global filling factor ($\nu_g$).

Experimental Conditions:

• Measurements were performed in a dilution refrigerator at a base temperature of ~20 mK.
• Magnetic fields were applied up to ~3.1 T.
• Differential resistance ($dV/dI$) and critical current ($I_c$) were mapped as functions of magnetic field ($B$) and filling factor ($\nu$).

3. Key Results

A. Exclusion of Bulk-Mediated Current

• Edge-Free (EF) Devices: In devices without physical edges, the resistance diverged in the QH regime (longitudinal conductance vanished), and no Josephson current was observed.
• Conclusion: This definitively rules out the "imperfect bulk insulator" hypothesis. The JC cannot be mediated by the bulk of the graphene, even in plateau transition regions.

B. Necessity of Physical Edges and Impact of Impurities

• Native vs. Etched Edges: When an Edge-Free device was etched to create an Etched Edge (EE) device, JC appeared. However, comparing Native Edge (NE) and Etched Edge (EE) devices on the same sample revealed that the JC in EE devices was significantly weaker.
• Mechanism: The plasma etching process introduces impurities near the edge. These impurities facilitate scattering between upstream and downstream modes, disrupting the formation of Andreev Bound States (ABS). The stronger JC in NE devices suggests that cleaner, atomically sharp edges are crucial for robust CPES formation.

C. Confirmation of Counter-Propagating Edge States (CPES)

• Magnetic Interference: The observed $h/2e$ interference period and the specific behavior of critical current oscillations (SQUID-like patterns at certain $\nu$, absence at others) are consistent with the CPES model.
• GGDE Device Control: In Graphite Gate-Defined Edge devices, the authors tuned the local ($\nu_l$) and global ($\nu_g$) filling factors.
  • Configuration 1 (Same polarity, $|\nu_g| > |\nu_l|$): Downstream modes from the global region canceled those of the local region, leaving only single-chirality states. Result: No JC.
  • Configurations 2 & 3 (Opposite polarity or $|\nu_l| > |\nu_g|$): Upstream and downstream modes coexisted, forming counter-propagating pairs. Result: JC observed.
• Localization: The JC pockets showed a strong vertical dependence on $\nu_l$ but weak dependence on $\nu_g$, proving that the ABS are confined to the local edge region defined by the sharp potential gradient of the graphite gate.

4. Key Contributions

1. Definitive Mechanism Identification: The study provides direct evidence that the Josephson current in the QH regime is mediated by counter-propagating edge states (CPES) on a single physical edge, not by chiral states or bulk conduction.
2. Edge Sensitivity: It demonstrates that the formation of ABS is highly sensitive to edge quality. Impurities introduced by etching suppress the JC, highlighting the need for pristine edges in topological devices.
3. Resolution of the $h/2e$ Puzzle: The findings explain the anomalous $h/2e$ interference period as a result of interference between Josephson currents mediated by CPES on opposite edges of the junction, or within a single edge where upstream/downstream modes interfere.
4. Engineering Strategy: The work establishes that CPES can be engineered and controlled via electrostatic gating (e.g., graphite side gates), offering a new pathway to manipulate topological superconductivity.

5. Significance

This research resolves a long-standing ambiguity in the physics of hybrid superconductor-quantum Hall systems. By confirming that CPES are the vehicle for Josephson coupling, the paper:

• Validates the theoretical framework for Andreev bound states in the QH regime.
• Provides a design rule for future devices: To realize robust Majorana zero modes or topological qubits, device fabrication must prioritize atomically sharp edges and minimize edge scattering.
• Opens new avenues for engineering topological phases by precisely controlling edge potentials, potentially enabling the realization of spin-polarized CPES for fault-tolerant quantum computing applications.