Mechanism for Nodal Topological Superconductivity on PtBi$_2$ Surface

This paper proposes that anisotropic electron-phonon coupling combined with screened Coulomb repulsion drives nodal topological superconductivity on the PtBi$_2$ surface, while predicting that enhanced Coulomb screening could transform this state into a nodeless superconductor with a higher critical temperature.

The Gist

The Big Picture: A Superconductor with Holes

Imagine a material called PtBi2 (Platinum Bismuth). Scientists have discovered that the surface of this material acts like a superconductor—a substance that conducts electricity with zero resistance. But here's the twist: it's not a "perfect" superconductor. It has holes (called "nodes") in its energy gap where the superconductivity breaks down.

Think of a superconductor like a smooth, frictionless ice rink where electrons (the skaters) can glide effortlessly. In a normal superconductor, the whole rink is smooth ice. In this PtBi2 material, the rink is mostly smooth, but it has a few specific spots where the ice is broken or rough. The scientists wanted to know: Why do these holes exist, and how can we fix them?

The Cast of Characters

To understand the mechanism, we need to meet three main characters:

1. The Electrons (The Skaters): In this material, the electrons live mostly on the very surface, like skaters confined to a specific lane on the rink.
2. The Phonons (The Bouncers): These are vibrations in the crystal lattice (the atoms shaking). Usually, phonons help electrons pair up and skate together. Think of them as bouncers who gently push two skaters together so they hold hands and glide in sync.
3. The Coulomb Repulsion (The Grumpy Neighbors): Electrons are negatively charged, so they naturally hate being close to each other. They push each other away. This is the "Grumpy Neighbor" effect.

The Problem: The "Too Close" Problem

In most superconductors, the "Grumpy Neighbors" (Coulomb repulsion) are weak compared to the "Bouncers" (phonons). The bouncers win, and the electrons pair up easily.

However, in PtBi2, something special happens. The surface electrons are packed very tightly in the center of their path (the Fermi arc). Because they are so crowded, the "Grumpy Neighbors" are very strong right in the middle.

If the electrons tried to pair up in the center (where the bouncers are strongest), the Grumpy Neighbors would push them apart. The electrons are stuck in a dilemma:

• Option A: Pair up in the center (strong help from bouncers) but get pushed apart by neighbors.
• Option B: Stay away from the center to avoid neighbors, but then the bouncers aren't helping as much.

The Solution: The "Dance Around" Strategy

The paper proposes a clever solution. The electrons decide to dance around the problem.

Instead of pairing up head-on (which would put them right in the path of the Grumpy Neighbors), they pair up with a specific, complex twist. They choose a pairing style called "i-wave" (a fancy mathematical name for a specific spinning pattern).

The Analogy:
Imagine two people trying to hold hands while walking through a crowded room.

• If they walk straight toward each other, they might bump into people (Coulomb repulsion).
• Instead, they decide to walk in a spiral or a figure-eight pattern. They still hold hands (superconductivity), but their path is twisted so they avoid the most crowded spots.

This "twisted" path creates the holes (nodes) in the superconducting gap. The superconductivity is perfect everywhere except at the very center of the arc, where the Grumpy Neighbors are too strong to overcome, even with the twist.

Why is this special?

Usually, scientists think of electron-phonon interactions (the bouncers) as simple and short-range. But in this material, because the electrons are stuck on a thin surface layer, the "bouncers" act like they have long arms. They can reach out and help electrons pair up even when those electrons are trying to stay far apart to avoid the Grumpy Neighbors.

This unique combination—strong long-range help from phonons + strong short-range push from neighbors—forces the electrons into this twisted, nodal state.

The "Magic Fix": Coulomb Engineering

The paper ends with a prediction. What if we could make the Grumpy Neighbors less grumpy?

The scientists suggest "Coulomb Engineering." Imagine putting a special shield (a dielectric material) right next to the surface of the PtBi2. This shield would act like a soundproof wall, blocking the Grumpy Neighbors from shouting at the skaters.

The Result:

• If you shield the neighbors, the electrons no longer need to dance in a twisted spiral to avoid them.
• They can pair up in a simple, perfect circle.
• The holes disappear! The superconductor becomes "nodeless" (perfect ice rink).
• Bonus: The temperature at which this happens (Critical Temperature, $T_c$) goes up. It becomes a better superconductor.

Summary

1. The Mystery: PtBi2 has a superconducting surface with weird holes in it.
2. The Cause: The electrons are so crowded that they repel each other strongly in the center. To survive, they twist their pairing pattern (i-wave) to avoid the center, creating the holes.
3. The Mechanism: Surface vibrations (phonons) are strong enough to support this complex twisting dance.
4. The Future: If we can "shield" the electrons from their own repulsion (Coulomb engineering), we can remove the holes and make the material a stronger, more efficient superconductor, potentially useful for future quantum computers.

In short: The electrons are doing a complex dance to avoid their own bad temper, but with a little help from a shield, they could learn to dance perfectly.

Technical Summary

1. Problem Statement

The trigonal Weyl semimetal PtBi$_2$ exhibits unconventional superconductivity primarily localized on its topological surface states (Fermi arcs). Experimental evidence from Angle-Resolved Photoemission Spectroscopy (ARPES) and Scanning Tunneling Microscopy (STM) suggests:

• The material hosts intrinsic topological superconductivity.
• The superconducting gap function possesses nodes (specifically, 12 nodes consistent with $i$-wave pairing) located in the center of the Fermi arcs.
• Unlike high-$T_c$ cuprates, PtBi$_2$ does not show signs of strong electronic correlations, suggesting a weak-coupling mechanism is responsible.

The central theoretical challenge is to identify a microscopic mechanism that explains how nodal pairing arises in a weak-coupling regime where Coulomb repulsion is present, and why the gap nodes appear specifically in the center of the Fermi arcs where surface states are most localized.

2. Methodology

The authors employ a weak-coupling theoretical framework combining:

• Electronic Structure Model: An effective tight-binding model for PtBi$_2$ (based on Ref. [6]) describing a hexagonal slab geometry with $L$ layers. The model includes:
  • Two orbitals per site ($A$ and $B$).
  • Spin-orbit coupling (SOC) of the Rashba type with chiral $p$-wave momentum dependence.
  • Inversion symmetry breaking.
  • Calculation of surface state eigenstates and their layer-dependent weights ($W_{kn}$) to distinguish surface from bulk states.
• Interactions:
  • Electron-Phonon Coupling (EPC): Derived via a Taylor expansion of the hopping term around ionic displacements. Crucially, the authors highlight that out-of-plane EPC in a slab geometry does not vanish at zero momentum transfer (unlike in the jellium model), leading to a long-ranged attractive interaction component.
  • Coulomb Repulsion: Modeled using a two-orbital Hund's rule model for onsite repulsion ($U$) and nearest-neighbor (NN) density-density repulsion ($V$). The interaction is statically screened.
• Gap Equation: The authors solve the linearized gap equation near the critical temperature ($T_c$) within the Bardeen-Cooper-Schrieffer (BCS) formalism.
  • They consider the full first Brillouin zone (1BZ) and employ an adaptive quadrature integration scheme to handle the sharp peaks of the susceptibility function $\chi_k(T_c)$ near the Fermi surface (FS).
  • They analyze the eigenvalue problem to determine the dominant gap symmetry ($\Delta_k$) and the dimensionless coupling constant ($\lambda$).

3. Key Contributions and Physical Mechanism

The paper proposes a novel mechanism for nodal pairing driven by the interplay between anisotropic EPC and Coulomb repulsion under a specific energy scale condition:

• The "Comparable Bandwidth" Condition: The mechanism relies on the surface state bandwidth ($W$) being comparable to the maximum phonon energy ($\omega_D$). In PtBi$_2$, ARPES indicates $W \approx 10\text{--}20$ meV, and point-contact spectroscopy shows $\omega_D \approx 20$ meV.
• Failure of Radial Sign Change: In standard scenarios where $W \gg \omega_D$, the gap can avoid Coulomb repulsion by changing sign radially (positive near the FS, negative far away), effectively screening the repulsion (Morel-Anderson pseudopotential). However, when $W \approx \omega_D$, this radial sign-change strategy becomes ineffective because the phonon-mediated attraction is active across the entire surface state bandwidth.
• Angular Momentum Selection: To avoid the strong, short-range Coulomb repulsion (which is strongest where surface states are most compressed in real space, i.e., the center of the Fermi arcs), electrons are forced to pair with higher angular momentum.
• Role of SOC: The spin-orbit coupling imparts a chiral $p$-wave ($p_x + ip_y$) momentum dependence to the gap. The total gap is $\Delta_k = \tilde{\Delta}_k e^{i\phi_k}$.
  • Without Coulomb repulsion, the system favors fully gapped $p_x + ip_y$ pairing (anisotropic $s$-wave).
  • With Coulomb repulsion and $W \approx \omega_D$, the system transitions to nodal $i \times (p_x + ip_y)$-wave pairing (shortened as $i$-wave).
  • This corresponds to an even-parity $i$-wave function ($\tilde{\Delta}_k$) multiplied by the SOC-induced chiral factor.

4. Key Results

• Dominant Symmetry: Solving the gap equation across a parameter space of onsite ($U$) and NN ($V$) Coulomb repulsion reveals that nodal $i$-wave pairing is the dominant solution when $V/t \geq 0.24$ and $U/t \geq 0.1$.
• Gap Profile: The calculated gap magnitude $|\Delta_k|$ exhibits 12 nodes (consistent with experimental observations) and is maximal at the ends of the Fermi arcs, with nodes in the center. This matches the ARPES data for PtBi$_2$.
• Coupling Independence: Remarkably, the dimensionless coupling $\lambda_i$ for the nodal $i$-wave state is independent of both $U$ and $V$. This is because the pairing involves electrons separated by distances larger than the nearest-neighbor range, making it insensitive to the short-range Coulomb terms that suppress other symmetries ($p, f, h$-waves).
• Critical Temperature ($T_c$):
  • For realistic parameters, the nodal $i$-wave state yields a $T_c \approx 9.6$ K.
  • While lower than the fully gapped $p$-wave state ($T_c \approx 36.5$ K) in the absence of strong Coulomb repulsion, the nodal state remains robust.
  • The authors predict that Coulomb engineering (enhancing screening via dielectric environments or doping) can suppress the nodal state, leading to a nodeless gap and a significantly higher $T_c$.
• Topological Nature: The nodal $i$-wave state is identified as a weak topological superconductor. It hosts Majorana zero-energy flat bands on the hinges of the material, distinct from the chiral edge states of the fully gapped phases.

5. Significance

• Microscopic Explanation: The paper provides the first microscopic derivation of the nodal $i$-wave superconductivity observed in PtBi$_2$, resolving the puzzle of how nodal gaps arise in a weak-coupling, non-magnetic Weyl semimetal.
• New Mechanism: It establishes a new paradigm where the ratio of surface bandwidth to phonon energy dictates the pairing symmetry. This contrasts with traditional views where Coulomb repulsion only quantitatively reduces $T_c$; here, it qualitatively changes the gap symmetry to a nodal state.
• Experimental Predictions:
  • The model predicts that enhancing Coulomb screening (e.g., via proximity to a dielectric) should eliminate the nodes and increase $T_c$, offering a clear experimental pathway to optimize the material.
  • It confirms the topological nature of the surface states, suggesting PtBi$_2$ is a prime candidate for topological quantum computation via Majorana modes.
• Generalizability: The mechanism is not limited to PtBi$_2$ but applies to any Weyl semimetal where surface state bandwidths are comparable to phonon energies, broadening the search for intrinsic topological superconductors.