The $s\pm$ pairing symmetry in the pressured La$_3$Ni$_2$O$_7$ from electron-phonon coupling

This study investigates the electron-phonon mediated pairing symmetry in pressured La$_3$Ni$_2$O$_7$ using two distinct coupling models, revealing that interlayer coupling favors an $s\pm$-wave state while intralayer coupling promotes an $s++$-wave symmetry.

The Gist

Imagine you've just discovered a new kind of "magic metal" called La₃Ni₂O₇. When you squeeze it really hard (like a vice grip with 14 times the pressure of the Earth's atmosphere), it suddenly starts conducting electricity with zero resistance. It becomes a superconductor at a temperature of about 80 Kelvin (-193°C). That's incredibly hot for a superconductor!

Scientists are scrambling to figure out how it does this. Is it like the copper-based superconductors we've known for decades? Or is it something entirely new?

This paper asks a specific question: What if the "glue" holding the electrons together is actually sound waves (vibrations) inside the metal? The authors, a team of physicists, decided to test this idea using a computer simulation.

Here is the story of their findings, explained simply:

The Setup: A Two-Layer Sandwich

Think of this material not as a solid block, but as a two-layer sandwich.

• The Layers: The electrons live in two main types of "rooms" (orbitals) within the atoms: one shaped like a flat doughnut (called $d_{x^2-y^2}$) and one shaped like a dumbbell standing up (called $d_{3z^2-r^2}$).
• The Goal: For superconductivity to happen, electrons need to pair up and dance together. The question is: How do they hold hands?

The Two Scenarios (The Models)

The researchers built two different "rulebooks" to see how the electrons might dance, assuming the vibrations (phonons) are the music.

1. The "Full-Coupling" Rulebook (The Egalitarian Party)

• The Idea: In this scenario, the electrons in the flat doughnut rooms and the vertical dumbbell rooms are treated exactly the same. They can talk to each other both within their own layer and across to the other layer.
• The Result:
  • If the electrons only talk across the layers (between the top and bottom slice of the sandwich), they form a special, slightly rebellious dance called $s_{\pm}$.
  • The Analogy: Imagine two groups of dancers. The group on the top floor is clapping their hands, while the group on the bottom floor is stomping their feet. They are moving in sync, but with opposite rhythms. This "opposite sign" dance is the $s_{\pm}$ state.
  • If they only talk within their own layer, they do a standard, happy dance where everyone claps at the same time. This is the $s_{++}$ state.

2. The "Half-Coupling" Rulebook (The Specialized Party)

• The Idea: This is more realistic based on how the atoms actually move. Here, the flat doughnut electrons only talk to their neighbors in the same layer, while the vertical dumbbell electrons only talk to their neighbors across the layers.
• The Result: It turns out the outcome is almost the same!
  • The across-the-layer connection still pushes the electrons toward that rebellious $s_{\pm}$ dance (opposite rhythms).
  • The within-layer connection still pushes them toward the standard $s_{++}$ dance (same rhythm).

The Big Showdown: Who Wins?

The material is a battleground. The "across-layer" force wants the electrons to dance with opposite rhythms ($s_{\pm}$), while the "within-layer" force wants them to dance in unison ($s_{++}$).

• The Winner: The paper suggests that the across-layer connection is the stronger force. Just like a strong wind can blow out a candle, the inter-layer vibration is strong enough to force the electrons into the $s_{\pm}$ state.
• The Twist: They also looked at a "pair-hopping" term (where an electron pair jumps from one type of room to another). They found that if this hopping is "repulsive" (like two magnets pushing apart), it actually helps the rebellious $s_{\pm}$ dance win even more.

Why Does This Matter?

Think of the electrons as a crowd at a concert.

• If they all wave their hands up and down together, that's the $s_{++}$ state.
• If half the crowd waves up while the other half waves down, that's the $s_{\pm}$ state.

This paper argues that in this high-pressure nickel metal, the "sound waves" of the crystal lattice are strong enough to force the crowd into that split, opposite-wave pattern ($s_{\pm}$).

The Takeaway:
If the superconductivity in this new material is indeed caused by sound waves (electron-phonon coupling), then the electrons are likely dancing in that complex, opposite-rhythm $s_{\pm}$ style. This helps scientists distinguish between different theories about how these high-temperature superconductors work and brings us one step closer to understanding how to make superconductors that work at room temperature (which would revolutionize our power grids and technology).

Technical Summary

1. Problem Statement

The discovery of high-temperature superconductivity ($T_c \approx 80$ K) in the bilayer Ruddlesden-Popper nickelate La$_3$Ni$_2$O$_7$ (LNO) under high pressure ($>14$ GPa) has sparked intense debate regarding its pairing mechanism. While many theoretical studies focus on electronic correlations (e.g., spin fluctuations), recent experimental and theoretical evidence suggests that electron-phonon coupling (EPC) may play a pivotal role.

The central question addressed by this work is: If the superconductivity in pressurized LNO is mediated by electron-phonon coupling, what is the resulting pairing symmetry? Specifically, the authors investigate how the interplay between intralayer and interlayer couplings, involving the dominant Ni-$d_{x^2-y^2}$ and Ni-$d_{3z^2-r^2}$ orbitals, dictates the sign structure of the superconducting gap.

2. Methodology

The authors employ a BCS (Bardeen-Cooper-Schrieffer) theoretical framework to solve linearized gap equations in momentum space. Their approach involves the following steps:

• Electronic Structure Model: They utilize a minimal bilayer two-orbital tight-binding model adapted from Density Functional Theory (DFT) results. The basis includes Ni-$d_{x^2-y^2}$ and Ni-$d_{3z^2-r^2}$ orbitals in both top and bottom layers. The model captures orbital hybridization and interlayer hopping.
• Interaction Models: Two distinct pairing interaction models are introduced to simulate EPC:
  1. Full-Coupling Case: Both $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals participate equally in both intralayer and interlayer interactions.
  2. Half-Coupling Case: Intralayer coupling is restricted to the $d_{x^2-y^2}$ orbital (associated with in-plane $B_{1g}$ phonon modes), while interlayer coupling is restricted to the $d_{3z^2-r^2}$ orbital (associated with out-of-plane $A_{1g}$ phonon modes).
• Pair Hopping: The study also incorporates a "pair-hopping" term ($g_P$) representing the scattering of Cooper pairs between the two orbital types.
• Numerical Solution: The linearized gap equations are solved numerically for various chemical potentials (doping levels), including hole-doped ($\mu = -0.328$ eV), undoped ($\mu = 0$ eV), and electron-doped ($\mu = 0.1$ eV) regimes. The transition temperature ($T_c$) is determined by finding the eigenvalue of the interaction kernel equal to 1.

3. Key Contributions

• Systematic Comparison of Coupling Regimes: The paper provides a rigorous comparison between "full" and "half" coupling scenarios to isolate the effects of specific orbital-phonon interactions on pairing symmetry.
• Mechanism for s± Symmetry: It identifies interlayer coupling as the primary driver for the $s_{\pm}$-wave symmetry (sign-changing gap between Fermi pockets), contrasting it with the $s_{++}$-wave (sign-preserving) symmetry favored by intralayer coupling.
• Role of Pair Hopping: The study elucidates how pair-hopping interactions between orbitals can tune the system between $s_{\pm}$ and $s_{++}$ states, depending on the sign of the hopping strength.
• Feasibility of Phonon-Mediated SC: The authors provide a quantitative estimate showing that the interlayer EPC strength is sufficient to overcome Coulomb repulsion and generate the observed high $T_c$.

4. Key Results

A. Full-Coupling Case

• Interlayer Dominance: When only interlayer coupling is active, the system exhibits an $s_{\pm}$-wave state. The pairing gap has the same sign within the $\alpha$ and $\gamma$ Fermi pockets but opposite signs between the $\alpha$ and $\beta$ pockets. This is driven by negative Josephson coupling between these pockets.
• Intralayer Dominance: When only intralayer coupling is active, the system stabilizes a conventional $s_{++}$-wave state with a uniform gap sign across all Fermi pockets.
• Competition: Increasing the intralayer coupling strength ($g_1$) relative to the interlayer strength ($g_2$) drives a phase transition from $s_{\pm}$ to $s_{++}$.

B. Half-Coupling Case

• Consistency: The results mirror the full-coupling case. Interlayer coupling (acting on $d_{3z^2-r^2}$) favors $s_{\pm}$, while intralayer coupling (acting on $d_{x^2-y^2}$) favors $s_{++}$.
• Robustness: The $s_{\pm}$ phase occupies a vast region of the parameter space in the half-coupling scenario, suggesting it is a robust outcome if the interlayer phonon modes dominate.

C. Pair Hopping Effects

• Negative $g_P$: A repulsive pair-hopping interaction ($g_P < 0$) favors the $s_{\pm}$ state. This is physically intuitive as the $\gamma$ pocket is dominated by $d_{3z^2-r^2}$ and the $\alpha/\beta$ pockets by $d_{x^2-y^2}$; repulsion between these orbitals induces a sign change.
• Positive $g_P$: An attractive pair-hopping interaction favors the $s_{++}$ state.

D. Quantitative Estimates

• Using DFT-derived parameters, the authors estimate the interlayer coupling strength $g_2/A \approx 0.6$ eV. This value exceeds the typical interlayer antiferromagnetic exchange ($J_\perp \sim 0.32$ eV) and is strong enough to overcome the onsite Hubbard $U$ suppression, making a phonon-mediated mechanism physically plausible.
• Calculated $T_c$ values align with the experimental $\approx 80$ K under realistic coupling constants.

5. Significance

This work offers a compelling alternative to the purely electronic (spin-fluctuation) theories for LNO superconductivity. Its significance lies in:

1. Validating the Phonon Mechanism: It demonstrates that electron-phonon coupling, specifically via interlayer modes, is strong enough to mediate high-$T_c$ superconductivity in this material.
2. Predicting Symmetry: It predicts an $s_{\pm}$-wave symmetry as the likely ground state, which has distinct experimental signatures (e.g., in neutron scattering or tunneling spectroscopy) compared to the $s_{++}$ state often predicted by spin-fluctuation models.
3. Orbital Selectivity: It highlights the critical role of orbital-selective electron-phonon coupling, where specific phonon modes couple to specific Ni orbitals to determine the global pairing symmetry.
4. Guiding Future Experiments: The findings suggest that manipulating interlayer distances or phonon modes (via pressure or strain) could tune the pairing symmetry, providing a roadmap for further experimental verification.

In conclusion, the paper establishes that under the assumption of electron-phonon mediation, the interplay between intralayer and interlayer couplings in La$_3$Ni$_2$O$_7$ naturally leads to an $s_{\pm}$-wave superconducting state, offering a unified explanation for the material's high transition temperature and complex electronic structure.