Gapless superconductivity from extremely dilute magnetic disorder in 2H-NbSe2-xSx

This study reveals that extremely dilute magnetic disorder induces gapless superconductivity in 2H-NbSe2-xSx, a phenomenon driven by Se-S substitution-induced band structure modifications that alter in-gap scattering mechanisms and challenge the conventional understanding of disorder robustness in superconductors.

The Gist

Imagine a superconductor as a perfectly synchronized dance floor. In a normal superconductor, every electron pairs up with a partner and moves in perfect unison. Because they are so coordinated, they can glide across the floor without any friction or resistance. This "perfect dance" creates a gap in the music: there are no solo dancers (electrons) allowed to move at the slowest, most energy-efficient speeds. This is called a superconducting gap.

Usually, if you throw a few random obstacles (impurities) onto this dance floor, the dancers just step around them. If the obstacles are non-magnetic (like a harmless rock), the dance continues perfectly. This is a famous rule in physics called Anderson's Theorem.

However, if the obstacles are magnetic (like tiny, spinning magnets), they act like rude dancers who try to break up the pairs. Usually, you need a lot of these magnetic troublemakers to ruin the dance floor completely and create a "gapless" state where solo dancers can exist at zero energy.

Here is the surprise in this paper:

The researchers took a material called 2H-NbSe₂ (a layered crystal that is naturally a superconductor) and did two things:

1. They swapped a tiny bit of Selenium (Se) atoms with Sulfur (S) atoms.
2. They added an incredibly tiny amount of magnetic Iron (Fe) atoms—so few that there is only one magnetic impurity for every 3,000 unit cells of the crystal.

The Result:
Even with such a tiny number of magnetic "troublemakers," the superconducting gap vanished completely. The dance floor became "gapless," allowing solo dancers to exist everywhere, even far away from the magnetic impurities.

The Analogy: The "Echo Chamber" Effect

To understand why this happened, imagine the crystal as a large, echoey concert hall.

1. The Pure Hall (Pure NbSe₂): The walls are smooth and reflective. If you shout (a magnetic impurity), the sound bounces around in a very specific, predictable pattern. The "echo" (the disturbance) stays localized near the shout. The rest of the hall remains quiet (the gap stays open).
2. The Renovated Hall (NbSe₂ with Sulfur): The researchers swapped some wall panels (Se for S). This changed the shape of the room and the acoustics. The walls became "flatter" in a specific way, and the sound waves started traveling differently.
3. The Tiny Shout (The Magnetic Impurity): When they added the single magnetic impurity, the sound didn't just bounce locally. Because the room's acoustics had changed, the "echo" of that single shout spread out much further and louder than before.

The "Cooperative" Disaster:
The paper shows that the Sulfur substitution didn't just change the room; it made the room extremely sensitive to the magnetic impurities.

• The Sulfur changed the "band structure" (the shape of the dance floor and the rules of movement).
• This new shape made the magnetic impurities' "ruining effect" spread out over a much larger area.
• Instead of the magnetic impurity only affecting its immediate neighbors, its influence stretched out, effectively "smearing" the gap across the whole material.

Key Takeaways in Plain English

• The "Gapless" Surprise: Usually, you need a heavy dose of magnetic junk to kill the superconducting gap. Here, a microscopic amount did it because the material's internal structure was tweaked.
• The Sulfur Switch: Swapping Selenium for Sulfur was the key. It didn't just add disorder; it fundamentally changed how electrons move, making them more "two-dimensional" (like moving on a flat sheet rather than a 3D block). This change made the electrons much more susceptible to being disrupted by the magnetic impurities.
• The CDW Connection: The original material has a "Charge Density Wave" (a static pattern of electrons, like a frozen ripple). The Sulfur substitution weakened this pattern. Once that pattern faded, the electrons became free to interact with the magnetic impurities in a new, more destructive way.
• The Lesson: You can't just look at the "bad guys" (magnetic impurities) to understand why a superconductor fails. You have to look at the "dance floor" (the material's band structure) too. A tiny change in the floor's design can make the material collapse under a tiny amount of pressure.

In summary: The researchers discovered that by slightly remodeling the "dance floor" of a superconductor with Sulfur, they made it so fragile that even a handful of magnetic "troublemakers" could destroy the superconducting gap entirely, turning a perfect superconductor into a "gapless" one.

Technical Summary

1. Problem Statement

Conventional superconductivity theory (Anderson's theorem) posits that nonmagnetic disorder does not significantly affect the superconducting gap, while magnetic impurities typically suppress superconductivity by breaking Cooper pairs. In conventional $s$-wave superconductors, a finite density of states (DOS) at the Fermi level (gapless superconductivity) is usually only observed at high concentrations of magnetic impurities (several atomic percent).

However, experimental observations in various systems have revealed a finite DOS at the Fermi level even when magnetic impurity concentrations are negligible. The origin of this phenomenon in single crystals with substitutional disorder remains unclear. Specifically, it is unknown how extremely low concentrations of magnetic impurities (fractions of a per mille) can induce a global gapless state, and how the interplay between magnetic disorder and nonmagnetic substitutional disorder (specifically in the presence of Charge Density Waves, CDW) drives this behavior.

2. Methodology

The study employs a multi-faceted approach combining experimental characterization, theoretical modeling, and first-principles calculations:

• Sample Synthesis: Single crystals of 2H-NbSe$_{2-x}$S$_x$ ($0 \le x \le 0.8$) were grown via solid-state reaction and vapor transport. All samples were doped with extremely low concentrations of randomly distributed Fe magnetic impurities ($n \approx 0.02$ at. %, or ~1 moment per 3000 unit cells).
• Scanning Tunneling Microscopy (STM):
  • Tunneling Conductance Maps: Measured at temperatures far below $T_c$ ($T/T_c < 0.05$) to visualize the local density of states (LDOS) with atomic resolution.
  • Quasiparticle Interference (QPI): Fourier transforms of LDOS maps were analyzed to identify scattering wave vectors and nesting features.
  • Vortex Lattice Imaging: Used to probe the bulk superconducting state and core states.
• Theoretical Modeling:
  • Self-Consistent Bogoliubov-de Gennes (BdG) Calculations: A microscopic model on a triangular lattice incorporating tight-binding parameters to reproduce the band structure. It included randomly placed point-like magnetic moments (Fe) and nonmagnetic repulsive potentials (S substitution).
  • Density Functional Theory (DFT): Performed using Quantum ESPRESSO to calculate the electronic band structure and Fermi surfaces of pure 2H-NbSe$_2$ and 2H-NbSe$_{1.8}$S$_{0.2}$.

3. Key Contributions and Results

A. Emergence of Gapless Superconductivity at Ultra-Low Impurity Levels

• Experimental Finding: In 2H-NbSe$_{1.8}$S$_{0.2}$ doped with only 0.014–0.032 at.% Fe, the tunneling conductance far from impurities exhibits a finite value at zero bias. This indicates a gapless superconducting state despite the magnetic impurities being well-separated (distances $\gg$ coherence length $\xi \approx 7$ nm).
• Cooperative Effect: The gapless state arises from the cooperative interplay between the magnetic impurities (creating Yu-Shiba-Rusinov or YSR states) and the nonmagnetic S substitutional disorder. Theoretical simulations confirm that while magnetic impurities alone at these low concentrations might not fully close the gap in a clean system, the addition of S disorder amplifies the effect, leading to a finite LDOS across the sample.

B. Modification of the Electronic Band Structure by S Substitution

• DFT Results: S substitution significantly alters the normal state band structure compared to pure 2H-NbSe$_2$:
  • Reduction of $k_z$ Warping: In pure NbSe$_2$, Nb-derived bands show significant warping along the $k_z$ direction. S substitution decouples the layers, resulting in highly two-dimensional sub-bands with nearly identical dispersion at the Brillouin zone center ($\Gamma-K-M$) and upper basal planes ($A-H-L$).
  • Fermi Surface Changes: The central Se-derived pocket shrinks slightly, while the Nb-derived pockets become well-separated.
• Impact on Scattering: This band structure modification shifts the dominant scattering mechanism. In pure NbSe$_2$, scattering is dominated by CDW-related wave vectors. In S-doped samples, new wave vectors ($q_1, q_2, q_3, q_4$) emerge in the QPI patterns, corresponding to nesting features of the modified Fermi surface (e.g., connecting flat portions of the hexagonal Nb pockets).

C. Mechanism of Gapless State Formation

• YSR State Extension: The YSR bound states induced by Fe impurities possess long tails. In the presence of S substitution, the normal state DOS is reduced, and the band structure changes facilitate the spatial extension of these YSR states.
• Disorder Cooperation: The nonmagnetic S disorder smears out the long tails of the YSR states, distributing the finite LDOS over the entire sample rather than keeping it localized near impurities. This leads to a global gapless state even when impurities are sparse.
• Quantitative Discrepancy: Theoretical models require slightly higher magnetic impurity concentrations than observed experimentally to reproduce the gapless state. The authors suggest this implies the model underestimates the cooperative influence of S disorder or that additional factors (e.g., enhanced RKKY coupling, CDW feedback, or electron-phonon variations) play a role.

D. Suppression of CDW and Evolution of $T_c$

• CDW Suppression: As the S concentration ($x$) increases, the CDW order is exponentially suppressed. The long-range coherence of the CDW in pure NbSe$_2$ is lost, reducing to small nanometer-sized patches.
• Superconducting Response: The critical temperature ($T_c$) decreases with increasing $x$. The study proposes that disorder shifts the dominant interaction from the specific CDW wave vector (phonon-mediated) to electron-electron interactions involving the new nesting features ($q_3$) that do not participate in CDW formation.

4. Significance

• Redefining Disorder Sensitivity: The work demonstrates that gapless superconductivity can emerge in conventional $s$-wave systems at extremely low magnetic impurity concentrations (orders of magnitude lower than previously thought necessary) when specific nonmagnetic substitutional disorder is present.
• Role of Band Structure: It highlights that understanding a superconductor's response to disorder requires incorporating material-specific band structure modifications. The change from 3D to 2D electronic character and the alteration of nesting vectors are crucial for the observed physics.
• Interplay of Orders: The study elucidates the complex relationship between superconductivity, CDW order, and magnetic impurities, showing how nonmagnetic disorder can decouple layers and fundamentally alter the scattering landscape, leading to unconventional macroscopic states (gapless superconductivity) from microscopic perturbations.

In conclusion, the paper reveals a novel mechanism where nonmagnetic substitutional disorder (S) cooperates with minute magnetic disorder (Fe) to induce a gapless superconducting state in 2H-NbSe$_2$ by fundamentally modifying the electronic band structure and enhancing the spatial reach of magnetic impurity states.