Pairing mechanism and superconductivity in 1313 phase La$_3$Ni$_2$O$_7$

Using DFT+DMFT and RPA methods, this study reveals that superconductivity in pressurized 1313-phase La$_3$Ni$_2$O$_7$ originates from its metallic trilayer subsystem with $s^{\pm}$-wave pairing, while its critical temperature is significantly suppressed by hole doping and interlayer Josephson coupling through insulating single-layer bridges, suggesting that the high-$T_c$ phase in this family actually belongs to the 2222 structure rather than the 1313 phase.

The Gist

The Big Picture: A Mystery in the Nickel Family

Imagine a family of materials called Nickelates (made of Nickel, Oxygen, and Lanthanum). Recently, scientists discovered that one specific member of this family, La₃Ni₂O₇, can become a superconductor (a material that conducts electricity with zero resistance) when you squeeze it with immense pressure. This is exciting because it happens at relatively "warm" temperatures for superconductors, potentially near the temperature of liquid nitrogen.

However, there is a confusion. The material comes in different "architectural styles" (phases). One style is called 2222 (two layers of nickel), and another is 1313 (a mix of one layer and three layers).

• The Mystery: Early reports suggested the 1313 version was the superconducting superstar. But newer experiments showed that thin films of 1313 don't superconduct well, while the 2222 version does.
• The Goal: This paper acts like a detective story. The authors used powerful computer simulations to figure out: Why is the 1313 version so weak at superconducting compared to its 2222 cousin?

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The Investigation: Peeling Back the Layers

The authors looked at the 1313 structure under a microscope (via computer models). They found that this structure is like a sandwich made of two very different types of bread:

1. The "Trilayer" (TL) Subsystem: This is a stack of three nickel layers.
2. The "Single-Layer" (SL) Subsystem: This is a single nickel layer sitting between the trilayers.

1. The "Bad Metal" vs. The "Good Metal"

When they analyzed the electrons inside:

• The Single-Layer (SL) turned out to be a "Bad Metal." It's almost an insulator (it blocks electricity). The electrons here are stuck in a traffic jam (a phenomenon called Mott physics), unable to move freely. It's like a road where every car is parked; no traffic can flow.
• The Trilayer (TL) is a "Good Metal." The electrons here are moving freely. This is where the superconductivity actually lives.

The Analogy: Imagine a relay race. The Trilayer is the fast runner who can actually run the race. The Single-Layer is a runner who is tied to a chair. The race (superconductivity) happens on the track, but the tied runner is in the way.

2. The "Hole" in the Team

The authors found that the "Good Metal" (Trilayer) in the 1313 structure has a slightly different chemical makeup than the famous 2222 structure.

• The 1313 version has been "hole-doped." In physics, a "hole" is like a missing electron.
• The Analogy: Think of the electrons as players on a soccer team. The 2222 team has the perfect number of players to win. The 1313 team has a few players missing (holes). Because the team is unbalanced, they can't play as well, and their "winning temperature" (how hot they can get before losing superconductivity) drops significantly.

3. The "Broken Bridge" (The Josephson Junction)

This is the most critical finding. For a material to be a superconductor, the electrons need to move in perfect unison across the whole block of material.

• In the 1313 structure, the "Good Metal" (Trilayer) blocks are separated by the "Bad Metal" (Single-Layer).
• To get from one Trilayer to the next, the superconducting electrons have to tunnel through the "Bad Metal."
• The Analogy: Imagine a series of islands (the Trilayers) connected by bridges (the Single-Layers). In the 2222 world, the bridges are wide, sturdy highways. In the 1313 world, the bridges are tiny, shaky footbridges over a deep canyon.
• Because the "Bad Metal" bridge is so weak, the islands cannot coordinate their movements. The "phase coherence" (the team working together) breaks down. This is called an S-N-S Josephson Junction (Superconductor-Normal-Superconductor). The "Normal" (N) part acts as a bottleneck, killing the overall superconductivity.

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The Verdict: Who is the Real Superconductor?

The paper concludes with a clear answer to the mystery:

1. The 1313 Phase is Weak: It has two problems. First, its "team" is unbalanced (hole doping). Second, its "bridges" are too weak to let the whole structure work together. This explains why its superconducting temperature is only 3.6 K (very cold).
2. The 2222 Phase is Strong: The 2222 structure doesn't have the "Bad Metal" layer blocking the way. It has strong bridges and a balanced team. This is why it can superconduct at much higher temperatures (potentially up to 80 K or more in some experiments).

The Final Takeaway:
The authors suggest that when people talk about the "high-temperature superconductivity" in the La₃Ni₂O₇ family, they are likely looking at the 2222 phase, not the 1313 phase. The 1313 phase is essentially a "spoiler" that gets in the way of the real magic.

Summary in One Sentence

The paper explains that the 1313 version of this nickel superconductor is like a relay team with a tied-up runner and broken bridges, which is why it fails to superconduct at high temperatures, whereas the 2222 version is the true champion with a clear path and a balanced team.

Technical Summary

1. Problem Statement

The discovery of superconductivity (SC) in pressurized nickelates, particularly $La_3Ni_2O_7$, has sparked intense research. However, there is significant ambiguity regarding the intrinsic superconducting phase within the Ruddlesden-Popper (RP) family.

• The Conflict: Recent experiments report SC in the 1313 phase ($La_3Ni_2O_7$) with a low $T_c \approx 3.6$ K, while other studies suggest the 2222 phase (bilayer $La_3Ni_2O_7$) is the true high-$T_c$ host.
• The Question: What is the electronic structure of the hybrid 1313 phase? Does the single-layer (SL) or trilayer (TL) subsystem host superconductivity? Why is the $T_c$ in the 1313 phase so drastically lower than in bulk trilayer $La_4Ni_3O_{10}$ or the 2222 phase?
• Theoretical Discrepancy: Previous RPA studies suggested the SL subsystem might host pairing, while DFT+DMFT studies suggested the SL is a Mott insulator. A comprehensive understanding of the pairing mechanism and the suppression of $T_c$ in this hybrid structure was lacking.

2. Methodology

The authors employed a multi-scale theoretical approach combining first-principles calculations with many-body physics:

• DFT+DMFT (Density Functional Theory + Dynamical Mean-Field Theory): Used to calculate the correlated electronic structure of the 1313 phase under high pressure (20 GPa). This method captures strong electronic correlations and orbital-selective behavior, distinguishing between the SL and TL subsystems.
• Effective Model Construction: Based on DFT+DMFT results, a renormalized Trilayer (TL) two-orbital tight-binding model was constructed. This model incorporates the specific band renormalization and orbital occupations derived from the many-body calculations.
• RPA (Random Phase Approximation): Applied to the effective TL model to investigate the superconducting instability, pairing symmetry, and the dependence of $T_c$ on interaction strength and doping.
• Josephson Junction Analysis: Theoretical modeling of the interlayer coupling between superconducting TL layers separated by the non-superconducting SL layer to explain global phase coherence.

3. Key Contributions and Results

A. Electronic Structure and Layer-Dependent Correlations

• SL Subsystem (Single Layer): The DFT+DMFT calculations reveal that the SL subsystem (analogous to $La_2NiO_4$) behaves as a nearly insulating "bad metal."
  • The $d_{z^2}$ orbital exhibits Mott physics (insulating behavior).
  • The $d_{x^2-y^2}$ orbital lacks well-defined quasiparticle excitations.
  • Conclusion: The SL subsystem is not the host of superconductivity.
• TL Subsystem (Trilayer): The TL subsystem remains metallic and is the primary candidate for superconductivity.
  • It exhibits hole doping relative to bulk $La_4Ni_3O_{10}$. The Ni-$e_g$ orbital occupancy is reduced by $\approx 0.17$ electrons per Ni.
  • The outer layers show stronger correlations than the inner layers.

B. Pairing Mechanism and Symmetry

• Symmetry: RPA analysis on the renormalized TL model identifies an $s_{\pm}$-wave pairing symmetry.
• Mechanism: The pairing is mediated by spin fluctuations arising from the nesting between the electron-like $\epsilon$ pocket (near $\Gamma$) and the hole-like $\gamma$ pocket (near $M$).
• Doping Effect: The hole doping in the TL subsystem (relative to bulk $La_4Ni_3O_{10}$) leads to a weaker nesting condition and a reduced pairing eigenvalue ($\lambda$), inherently lowering the intrinsic $T_c$ of the TL layer compared to the bulk compound.

C. Suppression of Global $T_c$ (The "S-N-S" Mechanism)

The paper identifies two critical factors suppressing the global $T_c$ in 1313 $La_3Ni_2O_7$ compared to bulk $La_4Ni_3O_{10}$:

1. Intrinsic Weakening: Hole doping in the TL subsystem reduces the pairing strength.
2. Interlayer Decoherence (S-N-S Josephson Junction):
   • The structure consists of alternating Superconducting (S) TL layers and Normal (N) SL layers.
   • The SL layer acts as a weak link (insulating spacer) between adjacent TL superconductors.
   • Global phase coherence relies on Interlayer Josephson Coupling (IJC) via virtual tunneling across the SL.
   • The effective inter-trilayer hopping ($t_z^{TL}$) is extremely small ($\sim 10^{-4}$ eV), two orders of magnitude smaller than in bulk $La_4Ni_3O_{10}$.
   • This weak coupling leads to a severe suppression of the global $T_c$ according to the relation $T_c \propto 1/\ln(1/\eta)$, where $\eta$ is the anisotropy parameter proportional to the square of the interlayer hopping.

4. Significance and Conclusions

• Resolution of the Phase Debate: The study provides strong theoretical evidence that the 2222 phase (pure bilayer $La_3Ni_2O_7$) is the genuine high-$T_c$ phase, while the 1313 phase is intrinsically limited by its hybrid structure.
• Experimental Corroboration: The findings align with recent experimental observations where 2222 thin films show SC, but 1313 thin films do not, and where 1313 bulk samples show only low $T_c$ ($\sim 3.6$ K).
• Design Rules: The work establishes that in hybrid layered nickelates, the presence of insulating or Mott-localized spacer layers (like the SL in 1313) is detrimental to high-$T_c$ superconductivity because it disrupts interlayer phase coherence.
• Implication: To achieve higher $T_c$ in nickelate superconductors, future material design should focus on pure-phase structures (like 2222 or 3333) that maximize interlayer coherence and avoid "bad metal" or insulating spacer layers that act as Josephson weak links.

In summary, the paper elucidates that while the 1313 phase of $La_3Ni_2O_7$ does exhibit superconductivity, it is a "frustrated" state where the intrinsic pairing in the trilayer blocks is severely suppressed by both hole doping and the decoupling effect of the intervening single-layer Mott insulator.