🔬 mesoscale physics

Subgap Linear Thermoelectricity in Superconducting Quantum Hall Systems

This paper demonstrates that an integer quantum Hall system proximized by superconductors can exhibit a significant linear-response Seebeck effect mediated by Andreev processes, provided that both triplet superconducting correlations and spin polarization are present.

Original authors: Leonardo Pierattelli, Fabio Taddei, Alessandro Romito, Alessandro Braggio

Published 2026-06-23

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CC BY 4.0

Original authors: Leonardo Pierattelli, Fabio Taddei, Alessandro Romito, Alessandro Braggio

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 electrons, but instead of cars, we have tiny particles zooming along the very edge of a special material called a "Quantum Hall" system. Usually, these electrons are like disciplined soldiers marching in perfect lines. If you try to push them with heat (like warming one end of the road), they just keep marching straight. Because their speed is perfectly linked to their energy (a "linear" path), they don't naturally create an electrical voltage from heat. It's like trying to generate electricity by blowing on a perfectly smooth, frictionless slide; nothing happens.

Now, imagine we place a "Superconducting" blanket over a section of this highway. Superconductors are materials where electricity flows with zero resistance. Usually, when electrons hit this blanket, they bounce back as "holes" (a fancy way of saying they turn into the opposite of an electron). This process is called an "Andreev reflection."

The Big Discovery
The authors of this paper found a clever trick to make these disciplined electrons generate electricity from heat, even on that smooth slide. They discovered that if the superconducting blanket has a specific, hidden "twist" in its structure—called a triplet correlation—it breaks the perfect symmetry of the highway.

Think of it this way:

The Normal Highway: Electrons move left or right. If you heat one side, they just keep moving. No voltage is created.
The Twisted Blanket: The superconductor acts like a magical mirror that doesn't just reflect the electrons; it flips their "spin" (a property like a tiny internal compass) and changes their direction based on how fast they are moving.
The Result: Because the blanket treats fast electrons differently than slow ones, and treats "spin-up" electrons differently than "spin-down" ones, the heat creates a traffic jam that forces a voltage to build up.

The Key Ingredients
The paper explains that two things are absolutely necessary for this to work:

The "Twist" (Triplet Correlation): Without this specific type of superconducting connection, the electrons would just bounce back symmetrically, and the heat effect would cancel itself out. The "twist" breaks the symmetry, allowing the heat to push electrons in one direction more than the other.
The "Compass" (Spin Polarization): The electrons must already be sorted by their internal compass direction (spin). The setup uses a magnetic field to separate them, ensuring that only one type of electron hits the twisted blanket.

What They Found
The researchers built a mathematical model (a "minimal setup") to predict what would happen. Their main findings are:

It Works: They proved that a voltage can be generated from heat in this setup, even though the electrons are moving in a straight line and the energy is very low.
The Size of the Effect: The voltage generated (called the Seebeck coefficient) can be quite strong, reaching values that physicists consider significant for quantum devices.
The Rhythm: The strength of this voltage doesn't stay constant. It oscillates (goes up and down) depending on how long the superconducting blanket is. It's like a musical instrument where changing the length of the string changes the note.
The Angle Matters: The direction of the "twist" in the superconductor matters. If you rotate the twist, the voltage can flip from positive to negative, or even disappear.

Why It Matters (According to the Paper)
The paper suggests this is a new way to probe how superconductors and quantum materials interact. It shows that you don't need complex, high-energy setups to get thermoelectric effects; you just need the right combination of a quantum highway, a superconducting blanket, and a specific type of "twist."

In short, the paper claims that by adding a specific "twisted" superconducting layer to a quantum edge, we can turn heat into electricity in a regime where it was previously thought impossible, creating a new tool for studying these exotic quantum systems.

Problem Statement
The paper addresses a gap in the theoretical understanding of thermoelectric (TE) effects in hybrid quantum Hall (QH) and superconducting (SC) systems, specifically within the linear-response regime at the center of a QH plateau. While TE effects in QH systems and QH-SC interfaces have been studied previously, a full theoretical treatment for the subgap regime in the middle of the plateau ( $\nu=2$ ) is missing. Conventional wisdom suggests that linear TE effects should vanish in this regime because the chiral edge states possess a linear band dispersion, which typically precludes thermoelectricity. Furthermore, in the absence of superconductivity, the system lacks the necessary symmetry breaking to generate a Seebeck effect. The authors investigate whether a finite subgap linear TE response can emerge in such a setup and identify the specific mechanisms required to break the necessary symmetries.

Methodology
The authors propose an effective model for a QH bar at filling factor $\nu=2$ , proximized by a grounded superconductor of length $L$ . The system features spin-polarized edge states due to a strong Zeeman splitting. A metallic gate depletes the electron gas in the superconducting region to $\nu=1$ , ensuring that only one spin species (parallel to the magnetic field) enters the proximized region, while the antiparallel spin is reflected. This configuration allows edge states from two thermally biased, electrically floating contacts to interact with the superconductor.

Transport is modeled using the Landauer-Büttiker approach. The authors assume a non-zero singlet gap $\Delta$ and temperatures $k_B T \ll \Delta$ , ensuring no quasiparticle states exist in the superconductor. The system is described by a one-dimensional low-energy Bogoliubov–de Gennes Hamiltonian that includes:

Spin-resolved edge states with Fermi velocity $v$ and Zeeman energy $B$ .
A singlet superconducting order parameter $\Delta e^{i\psi}$ .
A triplet superconducting coupling vector $\vec{v}_\Delta$ .

The authors simplify the Hamiltonian by gauging out spin-orbit coupling into an effective triplet component and assuming unitary triplet pairing. They derive the transfer matrix $M$ for the proximized region to calculate the Andreev transmission probabilities $P^A_{2i}(E)$ . The Seebeck coefficient $S$ is then computed as the ratio of the open-circuit voltage to the thermal bias, derived from the TE coefficients $\alpha_{ij}$ and electrical conductances $G_{ij}$ .

Key Contributions and Results
The primary result is the demonstration that finite subgap linear TE coefficients arise in this hybrid system only when triplet superconducting correlations are present. The key findings include:

Necessity of Triplet Correlations and Spin Polarization: The authors analytically show that both triplet correlations and spin polarization are essential. In the absence of triplet pairing, the transfer matrix commutes with the energy term, resulting in an energy-independent scattering probability that preserves energy-inversion symmetry (EIS). Consequently, the TE response vanishes. The triplet component introduces momentum-dependent particle-hole mixing, breaking EIS in the Andreev scattering probabilities ( $P^A_{21}(E) \neq P^A_{21}(-E)$ ), which is a prerequisite for a non-zero Seebeck effect.
Magnitude of the Effect: The predicted Seebeck coefficient $S$ can reach values on the order of $k_B/e$ in the middle of the quantum Hall plateau.
Analytical Expression: For the specific case where the triplet vector is aligned with the spin-quantization axis ( $\theta=0$ ) and the superconducting order parameters are real ( $\psi=0$ ), the authors derive an explicit analytical expression for $S$ . This expression highlights the role of the triplet component magnitude ( $\tilde{v}_\Delta$ ) and the system length ( $\tilde{L}$ ).
Oscillatory Behavior: The thermopower $S$ $S$ exhibits oscillatory behavior as a function of:
- The length of the superconducting region ( $\tilde{L}$ ), arising from interference between crossed Andreev reflection and normal transmission.
- The magnitude of the triplet coupling ( $\tilde{v}_\Delta$ ).
- The orientation of the triplet vector relative to the magnetic field (zenith angle $\theta$ ).
- The relative phase ( $\psi$ ) between singlet and triplet components.
Temperature Dependence: Increasing temperature leads to thermal averaging over the energy window, which suppresses the amplitude of the oscillations in $S$ . The damping length decreases with increasing temperature and increasing triplet coupling strength.
Symmetry and Phase Sensitivity: The sign of the Seebeck coefficient reverses when the triplet vector is flipped (parallel vs. antiparallel to the magnetic field), effectively exchanging the roles of the spin channels. Furthermore, $S$ shows periodic oscillations with respect to the phase difference $\psi$ , suggesting it can serve as a probe for the relative phase between singlet and triplet components.

Significance and Claims
The paper claims that this work provides the first effective model describing thermoelectricity in a hybrid QH-SC device at the $\nu=2$ plateau. The significance lies in demonstrating that a linear thermoelectric effect can emerge in a regime (subgap, linear dispersion) where it is normally forbidden. The effect is uniquely enabled by the interplay between quantum Hall edge states and the triplet component of superconductivity.

The authors position their results as a theoretical foundation for interpreting recent experimental observations of oscillatory TE responses in QH-SC systems, which have previously been attributed to fluxon dynamics. They suggest their model offers an alternative interpretation based on triplet correlations. The paper concludes modestly, stating that while inter-edge Coulomb interactions might produce quantitative modifications, they are not expected to alter the qualitative oscillatory behavior predicted. The authors hope these results will stimulate further experimental investigations into linear TE effects in QH-SC hybrid systems.

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