Superconducting properties of transition metal dichalcogenides in proximity to a conventional superconductor

This study investigates the superconducting properties of transition metal dichalcogenide monolayers proximitized by a conventional $s$-wave superconductor, revealing that their multiorbital nature and strong Ising spin-orbit coupling induce complex hybridization gaps and robust mixed spin-triplet pair correlations comparable in magnitude to spin-singlet pairs, while Rashba coupling further introduces competing equal-spin triplet pairs.

The Gist

Imagine a world where electricity flows without any resistance at all. This is superconductivity, a magical state usually found in very cold, special materials. Scientists are always looking for new ways to create this state, especially in materials that are only one atom thick (like a single sheet of paper).

This paper explores what happens when you stack two specific types of "atomic sheets" on top of each other:

1. A Transition Metal Dichalcogenide (TMD): Think of this as a very special, thin sheet of material (like a single layer of MoS₂) that has a unique internal "magnetic compass" built into its atoms.
2. A Conventional Superconductor: Think of this as a standard, well-behaved sheet that already knows how to conduct electricity perfectly.

When you press these two sheets together, the "superpower" of the bottom sheet tries to leak into the top sheet. This is called the proximity effect. The authors wanted to see exactly what kind of superpower the top sheet would get.

Here is what they found, explained with simple analogies:

1. The "Internal Compass" (Ising Spin-Orbit Coupling)

The TMD sheet has a special feature called Ising Spin-Orbit Coupling. Imagine that every electron in this sheet is a tiny spinning top. Usually, these tops spin in random directions. But in this TMD sheet, the material acts like a giant, invisible magnetic field that forces all the tops to spin either "up" or "down" in a very specific way, depending on which side of the sheet they are on.

The paper found that this internal compass is so strong that it doesn't just organize the electrons; it actually forces the superconducting "leakage" from the bottom sheet to change its nature.

2. The "Hybrid Gaps" (The Traffic Jams)

When the two sheets touch, their energy levels mix. The authors discovered that this mixing creates "gaps" (areas where electrons can't exist) at two different places:

• The Main Gap: A large gap near zero energy, which is expected.
• The "Hybridization" Gaps: These are like unexpected traffic jams that appear at higher energies.

The Catch: In simpler models, you would expect to see these traffic jams clearly. But because the TMD sheet is complex (it has multiple "lanes" or orbitals for electrons) and the connections between them are uneven (anisotropic), these gaps get smeared out. It's like trying to spot a specific pothole in a road that is covered in thick fog and uneven gravel. You know the potholes are there because of the physics, but if you just look at the overall "density" of the road, they are hard to see.

3. The "Magic Trick": Creating New Partners

The most exciting discovery is about the partners the electrons form.

• Normal Superconductors: Electrons usually pair up as "Spin-Singlets." Imagine two dancers holding hands, spinning in opposite directions (one up, one down). They cancel each other out perfectly.
• The TMD Effect: Because of that strong internal compass (Ising SOC) mentioned earlier, the electrons in the TMD sheet are forced to pair up differently. They form Spin-Triplets. Imagine two dancers spinning in the same direction, or a mix of directions that don't cancel out.

The Analogy: Usually, you need a magnet to force electrons to dance in the same direction. But here, the TMD sheet's own internal structure acts as the magnet. The paper shows that this internal force is so strong that it creates these "same-direction" dancing pairs (Spin-Triplets) that are just as common as the normal "opposite-direction" pairs.

4. The "Double Trouble" (Rashba vs. Ising)

The authors also considered what happens at the very edge where the two sheets touch. This edge breaks the symmetry and creates a second type of force called Rashba Spin-Orbit Coupling.

• Ising Force: Creates "Mixed" Spin-Triplets (a specific type of same-direction dancing).
• Rashba Force: Creates "Equal" Spin-Triplets (a slightly different type of same-direction dancing).

The paper found that these two forces are in a tug-of-war. If you have both, they compete. However, even with this competition, the TMD sheet is still able to generate a massive amount of these special Spin-Triplet pairs.

Summary of the Findings

• Complexity Matters: You can't use simple models to understand these materials. You need to look at all the different "lanes" (orbitals) the electrons use, because they create complex, hard-to-see energy gaps.
• Strong Internal Magnetism: The TMD's internal "compass" is powerful enough to turn a standard superconductor into a source of exotic "Spin-Triplet" superconductivity.
• A New Platform: This suggests that stacking these specific atomic sheets is a promising way to create Spin-Triplet superconductivity without needing to use magnets or ferromagnets, which are usually required for this effect.

In short, the paper proves that by stacking a specific type of atomic sheet on a superconductor, you can naturally generate a rare and useful type of superconductivity, driven by the sheet's own internal magnetic rules.

Technical Summary

Technical Summary: Superconducting properties of transition metal dichalcogenides in proximity to a conventional superconductor

Problem Statement
Transition metal dichalcogenides (TMDs) are promising candidates for realizing exotic superconducting phases, particularly spin-triplet states, due to their inherent Ising spin-orbit coupling (SOC) and multiorbital nature. While previous studies have explored TMDs in the superconducting realm, they have largely relied on oversimplified low-energy $k \cdot p$ models that approximate band structures only around the $K$ and $K'$ valleys. These simplified models may fail to capture the full complexity of TMD monolayers when proximitized by a conventional superconductor. Specifically, the implications of a more elaborate description involving anisotropic couplings and the full Brillouin zone on the emergent superconducting properties remain underexplored.

Methodology
The authors investigate a heterostructure consisting of a TMD monolayer (specifically modeling MoS$_2$) placed in proximity to a conventional spin-singlet $s$-wave superconductor (SC). The study employs a realistic three-orbital tight-binding model that includes the $d_{z^2}$, $d_{xy}$, and $d_{x^2-y^2}$ orbitals of the transition metal atoms. This model reproduces first-principles band structures and incorporates:

1. Anisotropic couplings: Up to third-nearest neighbor hoppings.
2. Ising SOC: Intrinsic to the TMD monolayer due to broken inversion symmetry.
3. Rashba SOC: Induced by the breaking of out-of-plane mirror symmetry at the TMD-SC interface.
4. Tunneling: Coupling between the SC and the TMD's $d_{z^2}$ orbital.

The system is solved using the Bogoliubov-de Gennes (BdG) formalism. The authors calculate the spectral function, density of states (DOS), and superconducting pair amplitudes using anomalous Green's functions. They perform a full symmetry classification of the induced correlations based on the SPOT (Spin, Parity, Orbital, Time) framework to distinguish between spin-singlet and spin-triplet pairings.

Key Contributions and Results

• Hybridization Gaps and Spectral Signatures:
  The multiorbital nature of the TMD leads to the formation of hybridization gaps at higher energies where electron (hole) superconducting bands cross hole (electron) TMD bands. While these gaps are clearly visible in the spectral function, the authors find that anisotropic couplings cause them to be smeared out in the density of states (DOS). This challenges the identification of hybridization gaps as a clean experimental signature of odd-frequency superconductivity, contrasting with predictions from simpler multiband models.

• Induced Spin Polarization and Mixed Spin-Triplet Correlations:
  The inherent Ising SOC induces a finite spin splitting and spin polarization along the $z$-direction. The authors establish a direct link between this $z$-polarization and the emergence of mixed spin-triplet superconducting pair correlations (specifically the $F^{(z)}$ component) in the TMD.

  • Using realistic parameters for MoS$_2$, the induced mixed spin-triplet correlations are found to be of the same magnitude as the induced spin-singlet correlations.
  • In the energy regions between the spin-split hybridization gaps, the mixed spin-triplet correlations can even exceed the spin-singlet counterparts.
  • A characteristic signature is identified: the absence of a zero-crossing in the singlet correlation between spin-split peaks indicates the presence of dominant mixed spin-triplet correlations.

• Role of Rashba SOC and Competing Triplet Mechanisms:
  The inclusion of Rashba SOC, which arises naturally at the interface, breaks the $U(1)$ symmetry associated with $z$-spin conservation. This induces equal spin-triplet pairs ($F^{(x)}$ and $F^{(y)}$).

  • The study reveals a competing interplay between the two types of triplet pairings: Ising SOC drives mixed spin-triplet pairs, while Rashba SOC drives equal spin-triplet pairs.
  • Increasing the strength of one type of SOC tends to suppress the other.
  • Despite this competition, using realistic values for both Ising and Rashba SOC, both types of triplet correlations remain comparable in strength to the spin-singlet correlations within the TMD.

Significance and Claims
The paper claims that proximitized monolayer TMDs serve as a promising platform for realizing spin-triplet superconductivity, offering an alternative to the more commonly used superconductor-ferromagnet interfaces. The work demonstrates that the combination of multiorbital physics and strong intrinsic Ising SOC in TMDs is sufficient to induce robust spin-triplet correlations without the need for magnetic materials. The findings generalize to other TMDs and provide a deeper understanding of the interplay between strong spin-orbit coupling and superconductivity. The authors emphasize that their results help clarify the superconducting properties of TMDs in proximity to conventional superconductors, highlighting the complexity of spectral measurements and the significant magnitude of induced triplet pairing.