🔬 mesoscale physics

Aharonov-Bohm interference probing of chiral Andreev edge states via scanning gate microscopy

This paper theoretically demonstrates that scanning gate microscopy can probe chiral Andreev edge states at quantum Hall-superconductor interfaces by locally decoupling electron and hole paths to induce robust, disorder-resistant Aharonov-Bohm conductance oscillations, with distinct behaviors observed across different filling factors.

Original authors: S. Maji, M. P. Nowak

Published 2026-09-24

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CC BY 4.0

Original authors: S. Maji, M. P. Nowak

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

In the strange world of quantum physics, electrons do not always behave like tiny billiard balls bouncing off obstacles. Under the right conditions, specifically when a material is cooled to near absolute zero and subjected to a powerful magnetic field, these electrons organize themselves into a rigid, one-way traffic system known as the quantum Hall effect. They are forced to travel only along the very edges of the material, moving in a single direction without ever turning back. This behavior is so robust that it is protected by the fundamental laws of topology, making it nearly impossible to disrupt. However, when scientists bring a superconductor—a material that conducts electricity with zero resistance—into contact with this edge, a fascinating transformation occurs. The superconductor has a unique ability to turn an incoming electron into a "hole," which is essentially the absence of an electron, and vice versa. When this conversion happens repeatedly at the boundary between the two materials, it creates a hybrid state called a chiral Andreev edge state. These states are a mixture of electron and hole, traveling together along the interface. Understanding how these hybrid states move and interact is crucial for developing future technologies in quantum computing, but observing them directly has proven difficult because their signals are often hidden by the noisy background of the material itself.

Researchers at the AGH University of Krakow have developed a new theoretical method to see these elusive states clearly, using a technique called scanning gate microscopy. Imagine a very fine, charged needle, like the tip of an atomic force microscope, hovering just above the surface of the material. As this tip moves across the device, it acts like a localized electric gate, creating a small hill of electrical potential that the electrons must navigate around. The team simulated this process on a computer to see how the chiral Andreev edge states would react to this moving obstacle. They found that when the tip approaches the interface where the superconductor meets the normal material, it does something unexpected: it forces the mixed electron and hole parts of the state to separate. Instead of staying together as a single hybrid entity, the electron and the hole are pushed apart and forced to take different paths around the tip's electric field.

This separation creates a remarkable effect. Because the electron and the hole travel along slightly different routes around the tip, they accumulate a difference in their quantum phase, which is a property related to their wave-like nature and the magnetic field they are moving through. When they eventually recombine after passing the tip, this difference in phase causes them to interfere with each other, much like ripples in a pond meeting and creating a pattern of peaks and valleys. The researchers observed that this interference shows up as a rapid, rhythmic oscillation in the electrical conductance of the device. By measuring these oscillations as the tip moves, they can map out exactly where the chiral Andreev edge states are located and how they behave. The team confirmed that this pattern is not just a random fluctuation but a direct signature of the Aharonov-Bohm effect, a phenomenon where charged particles are influenced by a magnetic field even when they do not pass directly through it.

The study went further to test how this method holds up in more complex and messy situations. In the real world, materials are rarely perfect; they contain impurities and defects that usually scramble delicate quantum signals. The researchers simulated a device filled with disorder to see if their method would still work. They found that while the disorder destroyed the clean, repeating patterns seen in perfect materials, the scanning gate technique still managed to reveal the presence of the edge states. Even in a dirty system, the tip could still induce the separation of the electron and hole paths, creating a localized signal that stood out against the noise. This suggests that the technique is robust enough to be used in actual experiments, even if the materials are not flawless.

The researchers also explored what happens when the material is set to a higher "filling factor," a condition where more distinct paths for the electrons exist. In this scenario, the tip interacts with multiple edge channels at once. As the tip moves closer to the interface, it first disturbs the outermost path, creating a localized interference loop, and then progressively affects the inner paths. This dynamic interaction changes how the different channels mix and transport electricity, creating a complex but detectable pattern of oscillations. The team demonstrated that by analyzing the amplitude and behavior of these signals, one can distinguish between different numbers of edge channels. This ability to differentiate between simple and complex states is vital for understanding the underlying physics of these hybrid systems.

Ultimately, this work provides a clear roadmap for how to detect and study chiral Andreev edge states in the laboratory. By using a moving electric tip to gently nudge the electrons and holes apart, scientists can force these hidden quantum states to reveal themselves through a distinct pattern of conductance oscillations. The findings suggest that this method is a powerful tool for probing the interface between superconductors and quantum Hall materials, offering a way to visualize and manipulate these states even in imperfect, real-world conditions. This clarity could be a significant step forward in the quest to build stable quantum computers and explore new forms of topological matter.

Technical Summary: Aharonov-Bohm Interference Probing of Chiral Andreev Edge States via Scanning Gate Microscopy

Problem Statement
The coupling between chiral edge states in a quantum Hall (QH) device and a superconductor (SC) creates chiral Andreev edge states (CAESs) at the interface. These hybrid states are of significant interest for exploring topological superconductivity and chiral Majorana modes. However, characterizing CAESs presents a challenge: while transport measurements reveal conductance oscillations indicative of CAES interference, these signatures are often masked by disorder-induced noise. Furthermore, standard transport measurements cannot directly visualize the spatial presence of CAESs or distinguish them from other phenomena. In normal QH systems, topological protection prevents backscattering unless counter-propagating edges are connected, making local probing difficult. The authors aim to theoretically investigate whether Scanning Gate Microscopy (SGM) can serve as a direct tool to detect and map the spatial response of CAESs, specifically looking for unique signatures that distinguish them from standard QH edge states.

Methodology
The study employs a theoretical framework based on the Bogoliubov-de Gennes (BdG) formalism within a tight-binding approximation.

System Model: The device consists of a normal region (inducing a QH phase) coupled to a superconducting region. The system is described by a Hamiltonian incorporating a perpendicular magnetic field (via the Landau gauge), a superconducting pairing potential ( $\Delta$ ), and a chemical potential ( $\mu$ ). Spin degeneracy is assumed, and Zeeman interaction is neglected.
SGM Simulation: The SGM tip is modeled as a localized electrostatic perturbation using a Lorentzian potential distribution. The tip acts as a local gate, moving across the system to alter charge carrier trajectories.
Transport Calculation: The Landauer-Büttiker approach is used to calculate non-local conductance ( $G_{bottom,top}$ ) at zero temperature. The scattering matrix is computed using the Kwant package, focusing on electron-to-electron and electron-to-hole transmission coefficients.
Analytical Modeling: For the $\nu=2$ regime, an analytical model is developed to explain the observed conductance oscillations. This model treats the SGM tip as a hard-wall circular potential that decouples the CAESs, forcing electron and hole components to detour around the tip, thereby accumulating a relative Aharonov-Bohm (AB) phase.

Key Contributions and Results

Decoupling Mechanism and AB Interference ( $\nu=2$ ):
The authors demonstrate that in the lowest filling factor regime ( $\nu=2$ ), the SGM tip locally decouples the chiral Andreev edge states from the superconductor. Unlike normal QH edge states which are topologically protected against backscattering, the CAESs are split into independent electron and hole paths when the tip approaches the interface. These quasiparticles are forced to detour around the tip-induced potential in the normal region. During this detour, they accumulate a relative magnetic phase difference ( $\Delta\phi_{AB}$ ) proportional to the enclosed magnetic flux. Upon recombining at the superconductor, this phase difference results in rapid Aharonov-Bohm-type conductance oscillations.
- Validation: The numerical results for conductance oscillations as a function of tip position show nearly perfect agreement with a geometric analytical model (Eq. 11) that relates the conductance to the enclosed area and the AB phase.
Higher Filling Factors ( $\nu=4$ ):
In the $\nu=4$ regime, where two pairs of CAESs exist, the SGM tip interacts with spatially distinct edge channels. The tip progressively affects the modes based on their spatial span. As the tip approaches the interface, it first induces localized AB loops in the outermost edge mode before fully decoupling both channels. This dynamic interaction alters the pre-existing Andreev-induced mode mixing, redistributing transmission probabilities between channels and amplifying transport resonances beyond simple two-mode interference.
Robustness in Disordered Systems:
The study investigates the system in the presence of disorder (modeled by a finite mean free path). While disorder destroys the clean, periodic conductance oscillations seen in pristine systems and masks non-local transport signatures, the SGM technique remains effective.
- Local Signatures: Even in disordered systems, the SGM reveals highly localized interferometric signatures near the interface where CAESs propagate.
- Differentiation: The technique can distinguish between different filling factors ( $\nu=2$ vs. $\nu=4$ ) by monitoring the amplitude of the conductance oscillations, which corresponds to multiples of the conductance quantum ( $2e^2/h$ ). The oscillation amplitude for $\nu=4$ exceeds that of $\nu=2$ .

Significance
The paper claims that Scanning Gate Microscopy provides a unique and robust method for detecting chiral Andreev edge states, particularly in non-ideal experimental environments where conventional transport measurements fail. The primary significance lies in the identification of a specific spatial response: the ability of the SGM tip to locally decouple CAESs into electron and hole paths, generating AB-type interference. This signature is strictly absent in standard unidirectionally propagating QH edge modes, which do not exhibit conductance changes until a strong perturbation connects counter-propagating edges. The authors conclude that these localized conductance variations serve as direct signatures capable of revealing both the presence of hybrid electron-hole states and the system's underlying filling factor, even in the presence of significant disorder.

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