Pairing mechanism and superconductivity in pressurized La$_5$Ni$_3$O$_{11}$

Using density functional theory and random phase approximation, this study reveals that superconductivity in pressurized La$_5$Ni$_3$O$_{11}$ primarily originates from an $s^\pm$-wave pairing mechanism within the bilayer subsystem, while the single-layer subsystem facilitates inter-bilayer coherence via weak Josephson coupling, collectively explaining the experimentally observed dome-shaped pressure dependence of the critical temperature.

The Gist

The Big Picture: A New Kind of Superconductor

Imagine you have a material that conducts electricity with zero resistance (superconductivity) when you squeeze it with a giant hydraulic press. Scientists recently found a new material, La5Ni3O11, that does exactly this. It's a "hybrid" material, meaning it's built like a sandwich: it has layers of double nickel-oxide sheets alternating with layers of single nickel-oxide sheets.

The mystery the scientists wanted to solve was: How does this sandwich become a superconductor, and why does its performance look like a hill (a "dome") when you increase the pressure?

The Cast of Characters

To understand the mechanism, think of the material as a city made of two types of neighborhoods:

1. The "Super" Neighborhood (The Bilayer): These are the double-layer sheets. They are the energetic, social hubs where the magic happens. Electrons here love to pair up and dance together without bumping into anything.
2. The "Grumpy" Neighborhood (The Single Layer): These are the single-layer sheets. They are a bit isolated and grumpy. At normal pressure, they are almost like an insulator (a wall that stops electricity). They don't really want to participate in the superconducting dance.

The Discovery: Who is Doing the Dancing?

The researchers used powerful computer simulations (like a digital microscope) to look at the electrons. They found that:

• The Super Neighborhood does the work: The actual superconductivity (the pairing of electrons) happens almost entirely inside the double-layer sheets.
• The Grumpy Neighborhood is just a bridge: The single layers are so disconnected from the double layers that they barely talk to them. However, they act as a bridge connecting the different double-layer neighborhoods.

The Analogy: Imagine a series of islands (the double layers) surrounded by a moat. The islands have people dancing (superconductivity). The moat is filled with a thick, sticky mud (the single layers). The people on the islands can't jump the mud easily. But, if they can find a way to send a signal across the mud to the next island, the whole archipelago can dance in sync.

The Mechanism: The "Josephson Bridge"

For the whole material to be a superconductor, all the islands must dance in perfect rhythm (phase coherence). This happens through a process called Interlayer Josephson Coupling (IJC).

Think of it like a whispering gallery.

• The "Super" islands are whispering a secret (the superconducting state).
• The "Grumpy" mud is very thick, so the whisper is very faint.
• Because the whisper is so faint, the islands are out of sync. They are dancing to different beats.

The Pressure Effect: Why the "Dome" Shape?

When you apply pressure (squeeze the material), two things happen, creating a "Goldilocks" zone that looks like a hill on a graph:

1. The Low Pressure Side (Climbing the Hill):
When you start squeezing, the "mud" (the single layer) gets compressed. The gap between the islands shrinks.

• Result: The whisper gets louder! The connection between the islands strengthens.
• Outcome: The islands start dancing in sync much better. The superconducting temperature ($T_c$) goes up.

2. The High Pressure Side (Sliding Down the Hill):
If you squeeze too hard, something else happens. The electronic structure of the "Super" islands changes. The number of available dancers (electrons) starts to drop.

• Result: Even though the connection between islands is perfect, there aren't enough people on the islands to dance.
• Outcome: The superconducting temperature ($T_c$) starts to go down.

The "Dome":

• Start: Weak connection, few dancers = Low performance.
• Middle: Strong connection, many dancers = Peak Performance (The Dome Top).
• End: Strong connection, but too few dancers = Performance drops.

The Pairing Style: The "s±" Dance

The paper also identified how the electrons pair up. They found it's an s± wave.

• Simple Analogy: Imagine the dancers on one island are wearing Red shirts, and the dancers on the neighboring island are wearing Blue shirts. They are holding hands, but they are doing opposite moves (one steps forward, the other steps back). This specific "opposite" coordination is what makes the superconductivity stable in this material, similar to what was found in a related material called La3Ni2O7.

Why This Matters

This paper solves a puzzle. Previously, scientists were confused because the single layers looked like they should be insulators, yet the whole material became a superconductor.

• The Solution: The single layers aren't the dancers; they are the bridges.
• The Lesson: In hybrid materials, you can't just look at the "active" part. You have to understand how the "inactive" parts connect the active parts.

In a nutshell: The material is a team of super-dancers (double layers) separated by a thick wall (single layers). Squeezing the wall makes it easier for the teams to communicate and dance together, but squeezing it too much removes the dancers themselves. The perfect amount of squeezing creates the peak superconductivity we see in experiments.

Technical Summary

1. Problem Statement

The discovery of high-temperature superconductivity (SC) in pressurized Ruddlesden-Popper (RP) phase nickelates, particularly the bilayer $La_3Ni_2O_7$ ($T_c \approx 80$ K), has sparked intense research. Recently, the hybrid RP phase nickelate $La_5Ni_3O_{11}$ (the "1212" phase), which features an alternating stacking of single-layer ($La_2NiO_4$) and bilayer ($La_3Ni_2O_7$) $NiO_2$ planes, was reported to exhibit SC under pressure with a dome-shaped $T_c$ dependence peaking at $\approx 64$ K.

However, the microscopic mechanism remains controversial:

• Does SC originate from the bilayer subsystem, the single-layer subsystem, or their hybridization?
• Previous Random Phase Approximation (RPA) studies suggested $d$-wave pairing from the single layer, while Density Functional Theory + Dynamical Mean-Field Theory (DFT+DMFT) studies suggested the single layer is a Mott insulator and cannot carry SC.
• The experimentally observed dome-shaped pressure dependence of $T_c$ in $La_5Ni_3O_{11}$ contrasts with the monotonic or right-triangle behavior seen in pure bilayer systems, requiring a unified theoretical explanation.

2. Methodology

The authors employed a multi-scale theoretical approach combining first-principles calculations and many-body perturbation theory:

• Density Functional Theory (DFT): Used to calculate the electronic band structure of $La_5Ni_3O_{11}$ under various pressures (12–42 GPa). The calculations utilized the PBE functional and included an effective Hubbard $U$ ($3.5$ eV) to account for Ni-3d correlations.
• Wannier Downfolding & Tight-Binding (TB) Model: Based on DFT results, a six-orbital ($d_{z^2}$ and $d_{x^2-y^2}$ for both single and bilayer subsystems) TB model was constructed. This allowed for the separation of the system into distinct bilayer (BL) and single-layer (SL) subsystems.
• Random Phase Approximation (RPA): Applied to the isolated bilayer subsystem to investigate pairing instabilities driven by electron-electron interactions (Hubbard $U$, Hund's coupling $J_H$, and inter-orbital repulsion $V$).
• Josephson Coupling Analysis: A semi-phenomenological model was developed to analyze the interlayer Josephson coupling (IJC) between superconducting bilayers mediated by the non-superconducting single layers.

3. Key Contributions and Results

A. Electronic Structure and Subsystem Decoupling

• Decoupled Subsystems: The DFT and TB analysis reveal that the coupling between the single-layer and bilayer subsystems is extremely weak (hopping integral $|t_{z,BS}| \approx 0.01$ eV).
• Separate Treatment: Due to this weak coupling, the two subsystems can be treated independently. The single-layer subsystem acts primarily to equilibrate chemical potentials but does not significantly hybridize with the bilayer bands.
• Fermi Surface (FS): The full system has five FS pockets ($\alpha, \alpha', \beta, \gamma, \gamma'$). The $\alpha, \beta, \gamma$ pockets originate from the bilayer, while $\alpha', \gamma'$ come from the single layer. The $\gamma$ pocket (dominated by $d_{z^2}$) is crucial for pairing.

B. Pairing Mechanism in the Bilayer Subsystem

• Origin of SC: RPA analysis on the isolated bilayer subsystem confirms that superconducting pairing occurs predominantly within the bilayer subsystem. The single-layer subsystem is close to a Mott-insulating state (consistent with DMFT results) and serves as a "bad metal" that does not support intrinsic SC.
• Pairing Symmetry: The leading pairing symmetry is $s_{\pm}$-wave.
  • The pairing is mediated by spin fluctuations.
  • The spin susceptibility peaks at nesting vectors $Q_1 = (\pi, 0)$ and $Q_2$, connecting the $\beta$ and $\gamma$ pockets.
  • The gap function exhibits opposite signs on the $\alpha$ and $\gamma$ pockets, similar to the mechanism in pressurized $La_3Ni_2O_7$.

C. Explanation of the Dome-Shaped $T_c$

The paper resolves the discrepancy between the decreasing pairing eigenvalue ($\lambda$) with pressure and the experimental dome-shaped $T_c$:

1. Intralayer Pairing ($\lambda$): As pressure increases, the density of states (DOS) on the critical $\gamma$-pocket decreases, causing the intrinsic pairing strength ($\lambda$) and the 2D pairing temperature ($T_c^{RPA}$) to decrease monotonically.
2. Interlayer Josephson Coupling (IJC): The bulk 3D superconductivity requires phase coherence between bilayers. In $La_5Ni_3O_{11}$, this coherence is established via the Interlayer Josephson Coupling (IJC) across the insulating/bad-metal single layers.
   • The IJC is extremely weak ($\eta \sim 10^{-8}$) because it involves a high-order tunneling process through the single layer.
   • Under pressure, the interlayer distance ($d_\perp$) decreases, causing the IJC to exponentially increase ($\eta \propto e^{-4d_\perp/\xi}$).
3. Competition:
   • Low Pressure: The rapid enhancement of IJC dominates, overcoming the slight reduction in pairing strength, leading to an increase in bulk $T_c$.
   • High Pressure: The IJC enhancement saturates, and the reduction in DOS (and thus $\lambda$) becomes the dominant factor, causing $T_c$ to decrease.
   • This competition naturally reproduces the experimental dome-shaped pressure dependence.

4. Significance

• Unified Framework: The study provides a unified mechanism explaining why hybrid RP nickelates ($La_5Ni_3O_{11}$) exhibit different pressure dependencies compared to pure bilayer systems ($La_3Ni_2O_7$). The key distinction is the role of the single layer as a barrier requiring pressure-enhanced Josephson coupling for 3D coherence.
• Clarification of Roles: It definitively identifies the bilayer subsystem as the source of pairing (with $s_{\pm}$ symmetry) and the single-layer subsystem as a passive bridge for phase coherence, resolving previous contradictions regarding the electronic state of the single layer.
• General Applicability: The proposed mechanism of "weak IJC enhancement vs. DOS reduction" offers a potential framework for understanding pressure-induced superconductivity in other hybrid or layered quantum materials.

Conclusion

The paper concludes that superconductivity in pressurized $La_5Ni_3O_{11}$ is driven by $s_{\pm}$-wave pairing in the bilayer subsystem, while the dome-shaped $T_c$ arises from the competition between the pressure-induced reduction of the pairing eigenvalue and the exponential enhancement of the interlayer Josephson coupling necessary for bulk phase coherence.