Unconventional superconductivity in monolayer transition metal dichalcogenides

This paper proposes a theoretical pairing model mediated by spin and charge fluctuations, combined with Ising spin-orbit coupling and even-odd parity mixing, to explain the unconventional superconductivity, nodal gap, large upper critical field, and gap anisotropy observed in monolayer transition metal dichalcogenides like TaS$_2$.

The Gist

The Big Picture: A New Kind of Superconductor

Imagine a material that conducts electricity with zero resistance. That's a superconductor. Usually, these materials are like a well-organized dance floor where everyone moves in perfect, predictable steps (this is called "conventional" superconductivity).

However, scientists have found that when you take a specific type of material called a Transition Metal Dichalcogenide (TMD) and shave it down to a single atom-thick layer (a "monolayer"), the dance floor changes. The electrons start behaving in a weird, "unconventional" way. This paper focuses on one specific material, TaS2 (Tantalum Disulfide), and tries to figure out why it dances so differently.

The Setting: The "Ising" Lock

In normal 3D materials, electrons can spin in any direction. But in these ultra-thin 2D sheets, there is a special force called Ising Spin-Orbit Coupling.

• The Analogy: Imagine the electrons are dancers wearing magnetic boots. In a normal room, they can spin left or right. But in this 2D material, the "floor" is so magnetic that it forces all the dancers to lock their boots pointing straight up or straight down. They can't tilt sideways.
• The Result: This locking mechanism protects the superconducting state, allowing it to survive in much stronger magnetic fields than usual.

The Mystery: What's the Glue?

For superconductivity to happen, electrons need to pair up (like dance partners). In normal materials, the "glue" that holds them together is vibrations in the crystal lattice (like the floor shaking slightly).

But in TaS2, experiments suggest the glue might be something else: fluctuations in spin and charge.

• The Analogy: Instead of the floor shaking, imagine the dancers are constantly reacting to each other's moods. If one dancer gets excited (a spin fluctuation), it triggers a reaction in the neighbor, pulling them together. The authors propose that these "mood swings" (fluctuations) are the primary force pairing the electrons, rather than just the floor vibrations.

The Discovery: A "Nodal" Dance

The authors built a computer model to simulate this dance. Here is what they found:

1. The "Nodal" Gap: In a perfect superconductor, there is a uniform "gap" (a safe zone) where electrons can't break apart. But in TaS2, the authors found that this gap has "holes" or "nodes" in it.

   • The Analogy: Imagine a safety net for trapeze artists. A normal net is solid everywhere. A "nodal" net has specific weak spots where the net is missing. The authors' model shows that the superconducting state in TaS2 has these weak spots, which matches what scientists see when they look at the material with a super-microscope (STM).

2. Mixing Parities (The Odd Couple): Because the material lacks a center of symmetry, the electron pairs are a mix of "even" and "odd" behaviors.

   • The Analogy: Think of a dance couple where one partner is wearing a tuxedo (even) and the other is wearing a t-shirt (odd). They are a mismatched pair, but they dance together perfectly. The paper shows that this "mismatched" pairing is actually the strongest and most stable state for TaS2.

3. The Magnetic Field Test: When you apply a magnetic field to a normal superconductor, it usually breaks the pairs apart quickly.

   • The Analogy: It's like a strong wind blowing the dancers off the floor.
   • The Result: Because of the "magnetic boots" (Ising coupling) and the "mismatched pairs" (even-odd mixing), the TaS2 dancers are incredibly tough. They can withstand a magnetic wind that is much stronger than what would blow away a normal superconductor. The paper explains why this happens: the specific way the spins are locked and mixed creates a shield against the magnetic wind.

The Conclusion: Solving the Puzzle

The paper argues that if you combine the "mood swing" glue (spin fluctuations) with the "magnetic boots" (Ising coupling), you get a perfect explanation for all the weird things scientists have observed in TaS2:

• Why it survives strong magnetic fields.
• Why the "safety net" has holes (nodal gaps).
• Why the resistance changes in a specific two-fold pattern when a magnetic field is applied.

The authors also checked a similar material, NbSe2, and found that while the rules are similar, TaS2 is even more extreme in its behavior. Their theory successfully ties together all the different experimental clues into one consistent story: TaS2 is an unconventional superconductor held together by electron mood swings, protected by magnetic locks, and dancing in a unique, mixed-up style.

Technical Summary

Technical Summary: Unconventional Superconductivity in Monolayer Transition Metal Dichalcogenides

Problem Statement
Monolayer transition metal dichalcogenides (TMDs), specifically 1H-TaS$_2$, exhibit superconductivity that diverges from the conventional electron-phonon mediated mechanism observed in their bulk counterparts. Experimental observations in these Ising superconductors suggest an unconventional pairing mechanism, characterized by:

• An in-plane upper critical field significantly exceeding the Pauli paramagnetic limit.
• The observation of Leggett modes and nodal features in the superconducting gap via Scanning Tunneling Microscopy (STM).
• A twofold symmetric gap anisotropy in magnetoresistance measurements.
  While previous studies have proposed various mechanisms (pure electron-phonon, pure repulsive interactions, or mixed models), a comprehensive theoretical framework that reconciles these diverse experimental findings in monolayer TaS$_2$—specifically accounting for the strong Ising spin-orbit coupling (SOC) and multiorbital nature of the electronic structure—remains necessary.

Methodology
The authors employ a comprehensive theoretical framework based on a multiorbital tight-binding Hamiltonian derived from relativistic Density Functional Theory (DFT) calculations. The model incorporates:

1. Electronic Structure: A basis of three Ta $d$-orbitals ($d_{z^2}$, $d_{x^2-y^2}$, $d_{xy}$) to describe the low-energy bands. The Hamiltonian explicitly includes Ising SOC, which locks electron spins out of the 2D plane, splitting the Fermi surface into spin-momentum locked bands.
2. Pairing Mechanism: The study assumes superconducting pairing is driven by spin and charge fluctuations within the Random Phase Approximation (RPA). The interaction is parametrized by on-site Hubbard repulsion ($U$) and Hund's coupling ($J$).
3. Gap Equation: The authors solve the linearized BCS gap equation derived from Gor'kov Green's functions. This equation accounts for the mixing of spin and orbital indices induced by the lack of inversion symmetry and the presence of Ising SOC.
4. Hybrid Interactions: The model investigates the interplay between the repulsive spin-fluctuation-mediated pairing and an attractive, conventional electron-phonon coupling (parameterized by $\Lambda$ and a mixing parameter $\alpha$).
5. Magnetic Field Effects: The effect of an in-plane magnetic field is analyzed by solving the gap equation with Zeeman coupling, calculating the upper critical field ($H_{c2}$) relative to the Pauli limit.

Key Results

• Dominant Instability: In the presence of Ising SOC, the dominant superconducting instability in monolayer TaS$_2$ belongs to the two-dimensional $E'$ irreducible representation of the $D_{3h}$ point group. This state represents a mixture of even ($E_{2g}$) and odd ($E_{1u}$) parity symmetry components from the $D_{6h}$ group (effectively an $s+f$ or $d+f$ wave mixture).
• Gap Structure and Nodal Behavior: The calculated superconducting gap exhibits significant anisotropy. While the gap magnitude does not possess explicit nodes, the anisotropy increases with decreasing Coulomb interaction strength ($U$). This results in a nodal-like density of states (DOS) that aligns with STM observations of a nodal local DOS. The addition of spin-fluctuation pairing transforms a conventional full-gap structure (characteristic of pure electron-phonon coupling) into this unconventional nodal-like form.
• Upper Critical Field: The theory successfully reproduces the large in-plane upper critical field observed experimentally. This enhancement arises from the combination of Ising SOC (which pins spins out-of-plane, suppressing the Zeeman effect) and the even-odd parity mixing in the superconducting state.
• Leggett Modes and Parity Mixing: The model identifies a significant admixture of even and odd parity states in the ground state, quantified by the mixing parameter $\eta$. For $U \geq 0.6$ eV, the odd-parity component dominates. This parity mixing provides a theoretical basis for the observation of Leggett modes in STM experiments, which are attributed to the breaking of inversion symmetry.
• Magnetoresistance Anisotropy: The application of an in-plane magnetic field splits the degeneracy of the superconducting ground state. The resulting twofold symmetric superconducting order parameter offers a theoretical explanation for the twofold gap anisotropy observed in magnetoresistance experiments.
• Comparison with NbSe$_2$: The study highlights that monolayer TaS$_2$ possesses a stronger Ising SOC ($\sim 250$ meV) compared to NbSe$_2$ ($\sim 130$ meV) and a shifted susceptibility peak. Despite these differences, the dominant instability remains the $E'$ irrep, suggesting a unified mechanism for Ising superconductivity in these materials.

Significance and Claims
The authors claim that their proposed theoretical pairing model, driven by spin and charge fluctuations and incorporating Ising SOC, successfully reconciles diverse experimental observations in monolayer TaS$_2$. Specifically, the theory:

• Stabilizes a superconducting ground state with a nodal-like density of states consistent with STM data.
• Explains the large in-plane upper critical field through the mechanism of even-odd parity mixing and Ising SOC.
• Accounts for the twofold symmetric gap anisotropy observed in magnetoresistance via the splitting of the ground state degeneracy by in-plane magnetic fields.
• Remains consistent with observations in other dichalcogenide superconductors like monolayer NbSe$_2$.

The paper concludes that the interplay between spin-fluctuation pairing and conventional electron-phonon interactions is crucial for governing the specific gap structure, transforming a conventional full gap into the unconventional, nodal-like state observed in experiments.