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Influence of pairing and correlations on transport properties in three-terminal superconducting nanostructure

This paper investigates the interplay between induced pairing and correlation effects in a three-terminal superconducting nanostructure, revealing how the quantum dot's ground state (singlet vs. doublet) dictates distinct transport signatures, such as negative nonlocal conductance and Seebeck coefficient reversal for singlets versus Kondo-enhanced conductance for doublets, thereby enabling the identification of the doublet-to-singlet crossover.

Original authors: G. Górski, V. Pokorný, T. Domański

Published 2026-10-30

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CC BY 4.0

Original authors: G. Górski, V. Pokorný, T. Domański

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 microscopic world of modern electronics, scientists often build tiny islands of material, called quantum dots, to trap individual electrons. These islands are usually connected to wires made of metal, allowing electrons to flow in and out like cars on a highway. However, when one of these connections is replaced by a superconductor—a material that conducts electricity with zero resistance and has a unique ability to pair electrons together—the rules of the road change completely. In this superconducting state, electrons do not travel alone; they form pairs that move in perfect unison. When a quantum dot sits between normal metal wires and a superconductor, it becomes a stage where two competing forces play out: the natural tendency of electrons to repel each other due to their negative charge, and the superconductor's urge to force them into pairs. Understanding how these opposing forces shape the flow of electricity and heat through such a tiny system is crucial for developing future quantum technologies, from ultra-fast computers to sensitive sensors.

A team of researchers has now mapped out exactly how this tug-of-war influences the movement of charge and heat in a specific three-terminal setup. They studied a system where a single quantum dot is sandwiched between two normal metal leads and one superconducting lead. By using advanced theoretical calculations, they simulated how electrons behave in this arrangement, focusing on how the strength of the electron repulsion and the temperature of the system alter the electrical conductance and the thermoelectric power, which is the ability to generate voltage from a temperature difference. Their work reveals that the quantum dot can exist in two distinct ground states, depending on the balance of forces. In one state, the electrons on the dot are paired up, resembling the behavior of the superconductor itself. In the other, the dot holds a single, unpaired electron, creating a magnetic-like moment that interacts strongly with the passing electrons.

The researchers found that the way electricity flows between the two metal leads depends entirely on which of these two states the dot is in. When the dot is in the paired state, a surprising phenomenon occurs: the electrical conductance measured between the two metal leads becomes negative. This counterintuitive result happens because the dominant process involves a mechanism called crossed Andreev reflection. In this process, an electron entering from one metal lead pairs up with another electron on the dot to form a superconducting pair that enters the superconductor, while a "hole" (a missing electron) is reflected into the other metal lead. This creates a current flow that opposes the applied voltage, resulting in a negative conductance reading. The researchers also observed that the thermoelectric power, or Seebeck coefficient, reverses its sign in this same paired state, flipping from positive to negative as the system transitions between different configurations.

In contrast, when the quantum dot holds a single unpaired electron, the behavior changes dramatically. Here, the system enters a regime known as the Kondo effect, where the unpaired electron on the dot interacts with the sea of electrons in the metal leads, forming a complex, correlated state. This interaction acts like a bridge, significantly enhancing the ability of single electrons to tunnel through the dot, especially at low temperatures. The researchers saw a clear spike in the local electrical conductance in this region, a signature that the Kondo effect is active. Unlike the paired state, the nonlocal conductance between the metal leads becomes positive in this unpaired configuration, driven by the direct tunneling of single electrons rather than the paired reflection process. The transition between these two behaviors is not a sharp switch but a smooth crossover, and the researchers identified specific signatures in the conductance and thermopower that allow scientists to pinpoint exactly where the system sits on this spectrum.

The study also explored what happens when an external magnetic field is applied to the system. The magnetic field breaks the symmetry between electrons with different spins, effectively lifting the degeneracy of the energy states. In the paired state, this causes the energy levels of the electron pairs to split and shift, which eventually destroys the delicate pairing and suppresses the negative conductance. In the unpaired state, the magnetic field gradually weakens and eventually eliminates the Kondo enhancement, causing the high conductance to drop back down. The researchers noted that the magnetic field can shift the point where the system transitions from one state to the other, providing a way to tune the quantum dot's behavior with external controls.

These findings offer a detailed roadmap for identifying the quantum state of a nanoscale system by simply measuring how it conducts electricity and heat. The ability to distinguish between a paired, superconducting-like state and an unpaired, magnetic-like state through transport measurements is a powerful tool. The researchers suggest that these results could be verified in future experiments using similar three-terminal nanostructures. By observing the sign of the nonlocal conductance or the temperature dependence of the thermopower, experimentalists could determine the ground state of the quantum dot without needing to directly probe the internal electron configuration. This work extends previous studies that were limited to idealized conditions, such as infinite superconducting gaps or zero temperature, by providing a more realistic picture that includes finite energy gaps and varying temperatures. The results highlight the rich physics that emerges when superconductivity, electron repulsion, and magnetic fields compete in a confined space, offering new insights into the fundamental nature of quantum transport.

Technical Summary: Influence of Pairing and Correlations on Transport Properties in Three-Terminal Superconducting Nanostructure

Problem and Motivation
This study investigates the local and nonlocal charge transport properties of a three-terminal hybrid nanostructure comprising a correlated quantum dot (QD) embedded between two metallic leads (normal metals) and a superconducting electrode. The research focuses on the subgap energy regime, where transport is governed by the interplay between proximity-induced electron pairing (Andreev processes) and local Coulomb repulsion. A central objective is to understand how these competing interactions influence the ground state configuration of the QD—specifically the crossover between even-parity (BCS-type singlet) and odd-parity (doublet) states—and how this transition manifests in measurable transport coefficients such as conductance and thermopower. The work extends previous theoretical investigations, which often assumed an infinite superconducting gap or zero temperature, to a regime with finite pairing gaps and finite temperatures, while also incorporating thermoelectric properties.

Methodology
The authors employ the Anderson impurity model to describe the correlated QD coupled to normal and superconducting leads. To treat electron correlations, they utilize a self-consistent second-order perturbation theory (SOPT) to calculate the matrix self-energy. This approach allows for the determination of the retarded and advanced Green's functions within the Nambu spinor representation.

Transport properties are analyzed using linear response theory. The authors derive expressions for charge currents resulting from ballistic electron transfer (ET), direct Andreev reflection (DAR), crossed Andreev reflection (CAR), and quasiparticle tunneling. From these, they calculate:

Local and Nonlocal Conductance: $G_{LL}$ (local) and $G_{RL}$ (nonlocal).
Thermopower (Seebeck Coefficient): $S_{LL}$ (local) and $S_{RL}$ (nonlocal).

The study systematically varies model parameters including the Coulomb interaction strength ( $U$ ), the QD energy level ( $\epsilon$ ), the superconducting gap ( $\Delta$ ), and temperature ( $T$ ). The validity of the perturbative approach is cross-checked against unbiased Quantum Monte Carlo (QMC) calculations in the half-filled case, showing satisfactory agreement in the weak-to-intermediate correlation regime.

Key Results

Singlet-Doublet Crossover Signatures:
- BCS-Singlet Regime ( $U \lesssim \Gamma_S$ ): In the weak interaction limit, the QD ground state is a singlet. Here, crossed Andreev reflection (CAR) dominates the nonlocal transport. The authors predict negative values for the nonlocal conductance ( $G_{RL}$ ) and a reversal of the nonlocal Seebeck coefficient ( $S_{RL}$ ). These phenomena are direct signatures of the dominance of the CAR channel over single-electron tunneling.
- Doublet Regime ( $U \gtrsim \Gamma_S$ ): In the strong interaction limit, the QD ground state is a singly occupied doublet. The dominant transport mechanism shifts to single-electron tunneling (ET), causing the nonlocal conductance to become positive.
Subgap Kondo Effect:
- In the crossover region near half-filling ( $\epsilon \approx -U/2$ ) and for strong interactions, the subgap Kondo effect emerges at low temperatures.
- This effect is characterized by a significant enhancement of the local conductance ( $G_{LL}$ ), which can exceed the unitary limit ( $2e^2/h$ ) due to the combined contribution of ET and DAR channels.
- The Kondo effect also induces a reversal in the sign of the local Seebeck coefficient ( $S_{LL}$ ) at temperatures below the Kondo temperature ( $T_K$ ), altering the characteristic saw-tooth dependence on the QD energy level.
- Conversely, the nonlocal transport properties ( $G_{RL}$ and $S_{RL}$ ) are found to be less sensitive to the subgap Kondo effect, as the CAR mechanism is suppressed in the doublet region.
Magnetic Field Effects:
- The application of an external magnetic field lifts the spin degeneracy via Zeeman splitting.
- In the weak-interaction regime, the field shifts the internal quasiparticle states, leading to a crossing that enhances the zero-energy spectral function before suppressing it at higher fields.
- In the strong-interaction regime, the magnetic field suppresses the Kondo state, leading to a reconstruction of the spectral function from a central peak to a four-peak structure.
- The magnetic field induces a sign reversal in the nonlocal conductance from negative to positive as it drives the system away from the singlet-dominated CAR regime. It also generates spin-polarized thermopower components ( $S^S_{LL}$ and $S^S_{RL}$ ) which undergo sign reversals near critical Zeeman energies.

Significance and Claims
The paper claims that the calculated temperature-dependent local and nonlocal transport properties provide unambiguous signatures to identify the specific ground-state configuration of a correlated quantum dot. Specifically:

Negative nonlocal conductance serves as a robust indicator of the BCS-singlet ground state and dominant CAR processes.
Temperature-dependent enhancement of local conductance and sign reversal of local thermopower serve as hallmarks of the subgap Kondo effect in the doublet/crossover region.
The study generalizes previous results (often limited to $\Delta \to \infty$ ) to finite superconducting gaps and includes thermoelectric responses, offering a more comprehensive theoretical framework for interpreting experimental data in multiterminal hybrid structures.
The authors suggest that these findings could guide experimental investigations into the ground-state configuration of quantum dots and potentially aid in exploring topologically nontrivial states in more complex superconducting nanostructures.

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