Theoretical study on ambient pressure superconductivity in La$_3$Ni$_2$O$_7$ thin films : structural analysis, model construction, and robustness of $s\pm$-wave pairing

This theoretical study demonstrates that while the electronic structure of ambient-pressure La$_3$Ni$_2$O$_7$ thin films varies with crystal structure and computational details, the robustness of $s\pm$-wave pairing mediated by finite-energy spin fluctuations persists, though the observed reduction in critical temperature compared to pressurized bulk is best explained by models with small interlayer hopping derived from experimental lattice structures.

The Gist

The Big Picture: A Superconductor at Room Pressure

Imagine a material called La₃Ni₂O₇ (a type of nickel-based crystal) that can conduct electricity with zero resistance (superconductivity). Scientists recently discovered that if you squeeze this material with massive pressure, it becomes a superconductor at about 80 Kelvin (very cold, but warm for superconductors).

Recently, researchers found a way to make this happen without squeezing it. They grew the material as a very thin film on a specific type of crystal "floor" (a substrate). The floor was slightly smaller than the film, which squeezed the film from the sides, mimicking the effect of high pressure. This film became superconducting at around 40 Kelvin.

The Question: Why does the thin film work at a lower temperature (40 K) than the squeezed bulk material (80 K)? And what is the exact "recipe" inside the material that makes the electricity flow without resistance?

The Scientists' Approach: Building a Digital Model

The authors of this paper didn't just guess; they built a detailed computer simulation. Think of it like a video game engine where they tried to recreate the physics of this material from scratch.

1. The Blueprint (Structure): They looked at the "blueprint" of the atoms. They tried two different blueprints:
   • The Theoretical Blueprint: What their computer calculations said the atoms should look like.
   • The Experimental Blueprint: What scientists actually measured in the lab recently.
2. The Engine (FLEX): They used a complex mathematical engine called FLEX (Fluctuation Exchange Approximation). Imagine this engine as a super-accurate weather simulator. Instead of predicting rain, it predicts how electrons (the tiny particles carrying electricity) dance and interact with each other. It accounts for every possible move the electrons can make, not just the obvious ones.

Key Findings: The "Dance" of Electrons

1. The "γ-Pocket" Mystery

In the world of these materials, there is a specific shape of the electron crowd called the γ-pocket. Some scientists thought this pocket was essential for superconductivity; others thought it didn't matter.

• The Paper's Verdict: The authors found that whether this "γ-pocket" exists or disappears depends entirely on which blueprint you use (theoretical vs. experimental) and how you tweak the math.
• The Analogy: It's like looking at a crowd through different colored glasses. Through one lens, you see a distinct group of people (the pocket); through another, they blend in.
• The Result: Surprisingly, it didn't matter. Whether the pocket was there or not, the superconductivity remained strong. The "dance" of the electrons was robust enough to handle these structural changes.

2. The "Glue" Holding It Together

How do the electrons pair up to conduct electricity? Usually, they need a "glue."

• The Paper's Verdict: The glue here is spin fluctuations. Imagine the electrons are dancers spinning around. Sometimes, they wobble or fluctuate in their spin. These wobbles act like a rhythmic beat that helps the dancers pair up.
• The Twist: The paper argues that this "beat" comes from high-energy wobbles, not just the slow, obvious movements near the surface of the electron crowd. Because the glue is based on these high-energy wobbles, the superconductivity is very stable and doesn't break easily if the electron crowd's shape changes slightly.

3. Why is the Film Cooler (40 K) than the Bulk (80 K)?

This was the biggest puzzle. The thin film superconducts at half the temperature of the pressurized bulk material.

• The Paper's Verdict: The difference comes down to one specific number: $|t_\perp|$.
• The Analogy: Think of the material as a two-story building where electrons can jump between the floors.
  • In the pressurized bulk, the floors are perfectly aligned, and the jump between them is strong and easy (High $|t_\perp|$). This makes for a very efficient dance floor (80 K).
  • In the thin film, the experimental measurements show the floors are slightly misaligned or the jump is weaker (Low $|t_\perp|$).
• The Conclusion: When the authors used the "Experimental Blueprint" (which showed this weaker jump) in their simulation, the superconducting temperature dropped exactly to the observed 40 K. When they used the "Theoretical Blueprint" (which predicted a stronger jump), the temperature stayed high at 80 K.
• The Takeaway: The reason the film is "weaker" is likely because the actual physical structure of the film has a weaker connection between its layers than the theory predicted.

Summary in a Nutshell

The scientists built a high-tech simulation to understand why a new superconducting film works at room pressure. They found that:

1. The pairing mechanism is tough: The electrons pair up using high-energy "wobbles" (spin fluctuations), making the superconductivity very robust against small changes in the material's shape.
2. The "γ-pocket" doesn't matter: Whether a specific electron shape exists or not doesn't change the outcome.
3. The temperature drop is structural: The film only reaches 40 K (instead of 80 K) because the actual physical distance between atomic layers in the film is slightly different than what theory predicted, making the "jump" between layers weaker.

The paper essentially says: "We know the recipe for superconductivity in this material. The reason the film is slightly less efficient than the pressurized block is simply because the film's layers aren't quite as perfectly connected as we thought they would be."

Technical Summary

1. Problem Statement

The discovery of high-temperature superconductivity ($T_c \approx 80$ K) in pressurized bilayer nickelate $La_3Ni_2O_7$ sparked intense interest. Recently, ambient-pressure superconductivity ($T_c \approx 40$ K) was observed in thin films of $La_3Ni_2O_7$ grown on substrates like $SrLaAlO_4$ (SLAO). However, a significant theoretical challenge remains:

• The $T_c$ Discrepancy: The $T_c$ in thin films is roughly half that of the pressurized bulk system.
• Structural Ambiguity: There is a large deviation between crystal structures determined by Density Functional Theory (DFT) optimization and those determined experimentally. Specifically, the experimental structure shows a significantly smaller interlayer hopping integral ($|t_\perp| \approx 0.4$ eV) compared to DFT predictions ($|t_\perp| \approx 0.6$ eV).
• Fermi Surface Controversy: Experimental and theoretical studies disagree on the existence of the "$\gamma$-pocket" (a Fermi surface originating from the top of the $d_{3z^2-r^2}$ bonding band). Its presence or absence is debated regarding its impact on pairing symmetry.
• Goal: The authors aim to theoretically reproduce the ambient-pressure superconductivity in thin films, determine the origin of the reduced $T_c$, and investigate the robustness of the pairing symmetry against structural and electronic variations.

2. Methodology

The study employs a multi-step approach combining first-principles calculations with many-body perturbation theory:

• Structural Optimization:

  • First-principles calculations were performed using VASP with the PBEsol functional.
  • The in-plane lattice constants ($a$) were fixed to match three experimental substrates: LSAT, LAO, and SLAO.
  • The out-of-plane lattice constant ($c$) and atomic positions were optimized to minimize total energy.
  • Two symmetry assumptions were tested: the experimentally suggested $I4/mmm$ (tetragonal) and the theoretically optimized $Amam$ (orthorhombic).
  • An additional model was constructed using the experimentally determined structure from Ref. [24] (Yue et al.), which features a distinct Ni-Ni distance.

• Model Construction:

  • Wannier Functions: Maximally localized Wannier functions were extracted using WANNIER90, focusing on the Ni-$3d_{x^2-y^2}$ and $3d_{3z^2-r^2}$ orbitals.
  • Hamiltonian: A two-orbital bilayer tight-binding model was constructed.
  • Interaction Parameters: On-site Coulomb repulsion ($U$), interorbital repulsion ($U'$), Hund's coupling ($J$), and pair hopping ($J'$) were adopted from previous constrained RPA calculations ($U \approx 3.6-3.8$ eV).
  • Band Structure: Calculations were performed with and without the +U correction (Dudarev's formulation, $U=3$ eV) to assess sensitivity.

• Superconductivity Analysis (FLEX):

  • The Fluctuation Exchange (FLEX) approximation was applied to the two-orbital Hubbard model.
  • This method self-consistently calculates the self-energy and pairing interaction mediated by spin fluctuations, taking into account full momentum and frequency dependencies of the Green's function.
  • The linearized Eliashberg equation was solved to obtain the eigenvalue $\lambda$ (a proxy for $T_c$) and the superconducting gap function $\Delta(k)$.

3. Key Contributions and Results

A. Structural and Electronic Trends

• Bond Angle: As the in-plane lattice constant $a$ decreases (increasing compressive strain), the interlayer Ni-O-Ni bond angle ($\alpha$) approaches $180^\circ$, consistent with the transition toward $I4/mmm$ symmetry. However, even for the SLAO substrate, DFT optimization does not reach exactly $180^\circ$ ($Amam$ symmetry persists slightly), though the energy difference between $Amam$ and $I4/mmm$ is negligible.
• Orbital Occupancy: Reducing $a$ increases the apical oxygen height and the level offset ($\Delta E$) between $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals. This shifts electrons into the $d_{3z^2-r^2}$ orbital, bringing it closer to half-filling (occupancy $\approx 0.95$ for SLAO).
• Hopping Integrals:
  • The interlayer hopping $|t_\perp|$ increases as $a$ decreases in DFT-optimized structures (reaching $\sim 0.6$ eV).
  • Crucially, the experimentally determined structure yields a much smaller $|t_\perp| \approx 0.45$ eV due to a larger Ni-Ni distance.

B. Robustness of $s_{\pm}$-Wave Pairing

• Symmetry Stability: The study finds that the $s_{\pm}$-wave pairing symmetry is robust. It remains the dominant solution regardless of:
  • The specific crystal symmetry ($Amam$ vs. $I4/mmm$).
  • The presence or absence of the $\gamma$-pocket (which depends on $U$ correction and band filling).
  • The specific details of the Fermi surface topology.
• Gap Function Characteristics:
  • In the orbital representation, the gap function is dominated by the interlayer pairing of $d_{3z^2-r^2}$ orbitals.
  • This interlayer gap is nearly momentum-independent (intra-unit-cell pairing).
  • The gap function in the band representation reflects the orbital weight: it is large where $d_{3z^2-r^2}$ character is present and changes sign between bonding and antibonding bands.

C. Origin of Reduced $T_c$ in Thin Films

• Role of $t_\perp$: The primary reason for the halved $T_c$ in thin films (compared to pressurized bulk) is identified as the reduction of the interlayer hopping integral $|t_\perp|$.
  • Models based on DFT-optimized structures (large $|t_\perp| \approx 0.6$ eV) predict a high $\lambda$ similar to the pressurized bulk.
  • Models based on the experimentally determined structure (small $|t_\perp| \approx 0.45$ eV) yield a significantly suppressed $\lambda$, consistent with the observed $T_c \approx 40$ K.
• Mechanism: The suppression of $T_c$ with reduced $|t_\perp|$ is attributed to the nature of the pairing glue. The pairing is mediated by finite-energy spin fluctuations (rather than low-energy Fermi surface nesting).
  • High-energy spin fluctuations (which drive the pairing) are suppressed when $|t_\perp|$ is reduced.
  • Conversely, low-energy fluctuations (associated with Fermi surface nesting) are enhanced by reduced $|t_\perp|$, but these are less effective for $s_{\pm}$ pairing in this system.

4. Significance and Conclusion

• Resolution of the $T_c$ Puzzle: The paper provides a coherent theoretical explanation for the reduced $T_c$ in ambient-pressure thin films: the experimental crystal structure inherently possesses a smaller interlayer hopping ($t_\perp$) than predicted by standard DFT optimization, which directly suppresses the pairing strength mediated by finite-energy spin fluctuations.
• Validation of the Pairing Mechanism: The study reinforces the view that $La_3Ni_2O_7$ superconductivity is driven by finite-energy spin fluctuations mediating an interlayer $d_{3z^2-r^2}$ pairing. This mechanism is robust against variations in Fermi surface topology (such as the presence of the $\gamma$-pocket) and structural symmetry details.
• Theoretical Consistency: The results align with non-perturbative approaches (cDMFT, DCA, VMC), confirming that the system can be viewed as a bilayer Hubbard model (formed by $d_{3z^2-r^2}$ orbitals) embedded in a multiorbital environment.
• Future Directions: The large discrepancy between DFT-predicted and experimentally observed crystal structures (specifically the Ni-Ni distance) remains an open question. The authors suggest that competing orders (e.g., density waves) not captured in the current FLEX scheme might also play a role in the actual material's stability and properties.

In summary, the paper establishes that while the pairing symmetry ($s_{\pm}$) is robust, the magnitude of $T_c$ is highly sensitive to the interlayer hopping $t_\perp$, which is dictated by the precise atomic structure of the thin film.