Engineering superconductivity on the surface of Weyl semimetals

This paper proposes a method to engineer high-temperature surface superconductivity in Weyl semimetals by depositing an additional layer to induce surface van Hove singularities, which, when combined with the material's topological Fermi arcs, significantly enhances the critical temperature.

The Gist

Imagine a Weyl semimetal as a special kind of crystal that acts like a highway for electrons. On the inside of this crystal, electrons move normally, but on the very surface, they are forced to travel along unique, one-way "roads" called Fermi arcs. These roads are special because they are protected by the crystal's internal geometry; you can't easily erase them or block them with small bumps or dirt.

The paper asks a simple question: Can we make these surface roads superconducting (carrying electricity with zero resistance) at much higher temperatures than the rest of the crystal?

Here is the story of how the authors figured out how to do it, explained through everyday analogies:

1. The Problem: The Road is Too Straight

In a normal Weyl semimetal, the surface Fermi arcs are like a perfectly straight, empty highway. While electrons can travel on it, the "traffic density" (how many electrons are packed into a specific energy level) isn't high enough to trigger a superconducting party. The authors wanted to create a traffic jam of a specific kind—a Van Hove Singularity (VHS).

Think of a Van Hove Singularity like a traffic bottleneck or a sharp curve in the road. When electrons hit this curve, they slow down and pile up. This pile-up creates a massive spike in the number of electrons available to pair up and become superconductors. The more electrons you can pack into this "bottleneck," the easier it is to get the whole system to superconduct.

2. The Solution: Building a Detour

The authors realized that to create this "traffic bottleneck" (the VHS) on the surface, they needed to change the shape of the road. They couldn't just dig up the whole crystal (which is hard and would ruin the internal structure). Instead, they proposed a clever trick: lay a new layer of material on top.

Imagine the surface of the crystal is a row of houses (atoms) connected by short fences (short-range connections). Electrons usually just hop from one house to the next.

• The Trick: The authors suggest placing a new layer of "helper" material on top of these houses.
• The Effect: This new layer acts like a bridge or a detour. It allows an electron to jump from House A, go up to the bridge, and land on House C (skipping House B).
• The Result: This "long jump" changes the shape of the road. Instead of being a straight line, the road now curves sharply, creating the perfect traffic bottleneck (Van Hove Singularity) right where the electrons are.

3. The Payoff: A Superconducting Party

Once this "bottleneck" is created, the authors ran the numbers (simulations) to see what happens.

• The Spike: When the energy of the electrons matches the location of this new bottleneck, the ability to superconduct skyrockets.
• The Temperature: In the specific material they studied (PtBi2), the inside of the crystal becomes superconducting at a very cold 0.6 Kelvin. However, with their engineered surface "bottleneck," the surface layer could theoretically superconduct at around 13 Kelvin.
• Why the difference? It's like having a regular street vs. a super-highway. The surface "highway" with the bottleneck is so efficient at pairing electrons that it stays superconducting at temperatures more than 20 times higher than the rest of the material.

4. Why This Matters (According to the Paper)

The paper explains that this mechanism solves a mystery. Scientists have been seeing superconductivity on the surface of these materials, but it's been inconsistent—sometimes it's there, sometimes it's not, and the temperature varies wildly.

The authors argue that this is because the "traffic bottleneck" (the Van Hove Singularity) is extremely sensitive. If you add even a tiny bit of impurity (like a speck of dust) to the surface, it shifts the "traffic" slightly. If the traffic shifts into the bottleneck, superconductivity explodes. If it shifts away, it disappears. This explains why different samples behave so differently.

Summary

The paper proposes a recipe for engineering high-temperature superconductivity on the surface of special crystals:

1. Start with a Weyl semimetal (a crystal with protected surface roads).
2. Add a thin layer of a different material on top.
3. Let this layer act as a bridge, forcing electrons to take "long jumps" between atoms.
4. Result: This creates a sharp curve in the electron path (a Van Hove Singularity), causing electrons to pile up and superconduct at much higher temperatures than the bulk material.

The authors emphasize that this is a theoretical blueprint. They show that by choosing the right "bridge" material, we can tune these surface states to create a robust, high-temperature superconducting layer, essentially "engineering" a new state of matter right on the surface of an existing crystal.

Technical Summary

1. Problem Statement

The paper addresses the challenge of realizing and understanding surface-only superconductivity in Weyl semimetals (WSMs), specifically motivated by experimental observations in PtBi₂ and ZrAs₂.

• The Phenomenon: Experiments have revealed superconducting states confined to the surface of these materials with critical temperatures ($T_c$) significantly higher than their bulk counterparts (e.g., bulk $T_c \approx 0.6$ K vs. surface $T_c \approx 13$ K in PtBi₂).
• The Mystery: While topological Fermi arcs are known to mediate this surface superconductivity, existing theoretical models struggle to explain:
  1. The magnitude of the $T_c$ enhancement.
  2. The large sample-to-sample variations in superconducting gap sizes observed experimentally, despite samples having similar nominal purity.
• The Gap: Standard Bardeen-Cooper-Schrieffer (BCS) arguments suggest Fermi arcs increase the local density of states (DOS), but they do not fully account for the extreme sensitivity to chemical potential shifts or the specific mechanism required to engineer these states.

2. Methodology

The authors employ a combination of tight-binding modeling, effective Hamiltonian theory, and variational free-energy minimization to investigate the interplay between surface topology and superconductivity.

• Model System: They utilize a four-band tight-binding model for the time-reversal invariant Weyl semimetal PtBi₂ (based on Ref. [17]), defined on a trigonal lattice.
• Surface Engineering Strategy:
  • Instead of modifying the bulk, they propose depositing a suitable atomic overlayer on the WSM surface.
  • They model this by introducing long-range (third-nearest-neighbor) hoppings ($t'$) specifically on the top surface layer.
  • Mechanism: They demonstrate that an overlayer with strong chemical affinity to specific surface atoms can create an effective path for electrons to hop between next-nearest neighbors, effectively generating the required $t'$ without altering the bulk topology.
• Superconductivity Calculation:
  • Since deriving an exact 2D Hamiltonian for Fermi arcs is difficult, they use a variational approach based on free-energy minimization (Refs. [22, 23]).
  • They solve the Bogoliubov–de Gennes equations for a semi-infinite slab using a mean-field Hamiltonian with a BCS spin-singlet pairing interaction ($U$).
  • The superconducting order parameter is modeled as $\Delta_z = \Delta_0 + \Delta_1 e^{-z/\lambda}$, separating bulk ($\Delta_0$) and surface ($\Delta_1$) contributions.

3. Key Contributions

A. Engineering Van Hove Singularities (VHS)

The authors show that introducing long-range hoppings ($t'$) on the surface layer breaks the monotonic dispersion of the Fermi arcs along the $k_x$ direction.

• This modification creates a saddle point in the energy-momentum dispersion, resulting in a surface Van Hove Singularity (VHS).
• Crucially, this VHS appears only on the surface, leaving the bulk topology intact. The Fermi arc distorts into a "horseshoe" shape, consistent with Density Functional Theory (DFT) calculations for PtBi₂.

B. Theoretical Explanation of Sample Variability

The paper posits that the large experimental fluctuations in $T_c$ and gap sizes arise because the surface chemical potential ($\mu$) is extremely sensitive to minor impurities or doping.

• When $\mu$ lies in the vicinity of the engineered VHS, the surface DOS diverges (logarithmically), drastically boosting the superconducting pairing strength.
• Small shifts in $\mu$ away from the VHS cause a sharp drop in $T_c$, explaining why nominally identical samples show vastly different superconducting properties.

C. Interface Engineering Protocol

The authors provide a concrete theoretical framework for interface engineering:

• By depositing a film (Material C) on a WSM substrate (Materials A and B), where C has strong affinity for A but weak coupling to B, an effective long-range hopping ($t' \approx -t_{AC}^2 / \epsilon_C$) is induced.
• They propose a screening criterion based on the ratio $t'/t$ derived from DFT calculations to identify candidate materials for high-temperature surface superconductivity.

4. Key Results

• Enhanced Critical Temperature: Self-consistent calculations reveal a massive spike in the superconducting gap at the surface when the chemical potential aligns with the VHS.
  • The surface gap is >30 times larger than the bulk gap.
  • Extrapolating to experimental parameters for PtBi₂ ($t \approx 0.75$ eV), the model predicts a surface $T_c \approx \mathbf{13 \text{ K}}$, matching experimental STM and ARPES data, compared to a bulk $T_c \approx 0.6$ K.
• Asymmetric Response: The onset of surface superconductivity is sharply asymmetric with respect to the chemical potential relative to the VHS energy. This asymmetry is driven by the specific geometry of the Fermi arc (conical shape below the VHS vs. opened dispersion above).
• Penetration Depth: The superconducting state is exponentially localized at the surface (penetration depth $\lambda$), confirming the "surface-only" nature of the phase.
• Robustness: The topological protection of the Fermi arcs ensures that the surface states persist even with the overlayer, while the overlayer provides the necessary spectral weight redistribution to induce the VHS.

5. Significance

• Resolution of Experimental Discrepancies: The work provides a unified explanation for the high $T_c$ and the sample-to-sample variability in PtBi₂, attributing both to the proximity of the chemical potential to a surface VHS.
• New Design Paradigm: It establishes a roadmap for engineering high-temperature 2D superconductivity not by changing the bulk material, but by manipulating surface electronic states via interface design.
• Experimental Accessibility: The proposed mechanism (depositing a specific atomic layer) is experimentally feasible. The authors offer a practical criterion ($t'/t$ ratio) to guide high-throughput searches for new materials exhibiting this phenomenon.
• Broader Impact: The findings extend beyond Weyl semimetals, suggesting that topological protection can be leveraged to manipulate surface states for exotic phases, potentially opening routes to room-temperature superconductivity in 2D systems.

In summary, Vocaturo and Trama demonstrate that topological protection combined with interface engineering can create surface Van Hove singularities, which act as a powerful lever to tune and significantly enhance surface superconductivity, offering a solution to long-standing puzzles in topological superconductivity.