Theoretical proposal of superconductivity in hole-doped reduced bilayer nickelate La3Ni2O6: a manifestation of orbital-space bilayer model with incipient bands

This paper theoretically proposes that hole-doped reduced bilayer nickelate La$_3$Ni$_2$O$_6$ can exhibit $s\pm$-wave superconductivity driven by interorbital interactions within an orbital-space bilayer model, where the absence of apical oxygens creates a large orbital energy offset analogous to interlayer hopping in bilayer systems.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Idea: Finding a New Way to Make Superconductors

Imagine you are trying to build a highway where cars (electrons) can travel without any friction or traffic jams. This is what a superconductor does: it conducts electricity with zero resistance. Scientists have been hunting for materials that can do this at higher temperatures (so we don't need expensive liquid nitrogen to cool them down).

Recently, scientists found a new family of materials called nickelates (made of nickel and oxygen) that can superconduct. There are two main types:

1. The "Single-Layer" Nickelates: Like a single-lane road. They work, but they have a speed limit.
2. The "Bilayer" Nickelates (La₃Ni₂O₇): Like a double-decker highway. Because the electrons can hop between the two layers, they can move faster and superconduct at much higher temperatures.

The Problem: The famous double-decker highway (La₃Ni₂O₇) only works under massive pressure (like being crushed by a giant hydraulic press). Scientists want to find a material that works like the double-decker highway but doesn't need that crushing pressure.

The New Proposal: La₃Ni₂O₆

This paper proposes a new candidate: La₃Ni₂O₆.

Think of the famous double-decker highway (La₃Ni₂O₇) as a building with two floors connected by a very strong elevator. The electrons love using that elevator to zip back and forth, which helps them superconduct.

The new material, La₃Ni₂O₆, is like a building where they removed the elevator shaft in the middle. At first glance, you might think, "Oh no, no elevator! The electrons can't hop between floors anymore!"

But here is the twist: The authors argue that even without the physical elevator, the building has a different kind of "magic."

The "Orbital-Space" Trick (The OSBM)

In the world of atoms, electrons live in different "rooms" called orbitals.

• In most materials, these rooms are all on the same floor (energy level).
• In this new material, the authors found that one specific room (the $d_{x^2-y^2}$ orbital) is on a very high floor, while the other four rooms are on a lower floor.

This huge gap between the floors is called $\Delta E$ (Delta E).

The Analogy:
Imagine a dance floor (the lower orbitals) that is almost full of people, and a VIP balcony (the upper orbital) that is almost empty.

• In the old "Double-Decker" model, the magic happened because people could jump between two physical floors.
• In this new "Orbital-Space" model, the magic happens because the "VIP balcony" is so high up and the "dance floor" is so full that the electrons on the floor are desperate to jump up, and the ones on the balcony are desperate to jump down.

The authors call this the Orbital-Space Bilayer Model (OSBM). It's like having a two-story building where the "layers" aren't physical floors, but different types of electron rooms stacked on top of each other in energy.

The Secret Sauce: "Incipient Bands"

The paper argues that superconductivity is strongest when the system is in a state called "incipient band."

• Imagine a bathtub: If the water (electrons) is exactly at the rim, it spills over. If it's too low, it's dry.
• The Sweet Spot: The best superconductivity happens when the water is just below the rim. The water is about to spill, but hasn't yet. The "lower floor" is full, and the "upper floor" is just barely touching the water line.

The authors propose that if you dope the material (add a few extra "holes" or remove a few electrons), you push the system into this perfect "almost-spilling" state. This triggers a special kind of superconductivity where the electrons pair up in a specific way (called $s\pm$-wave), allowing them to flow without resistance.

Why This Material is Special

1. It's "Reduced": La₃Ni₂O₆ is a "reduced" version of the famous La₃Ni₂O₇. It's missing some oxygen atoms. This missing oxygen is actually a good thing here because it creates that huge energy gap ($\Delta E$) between the electron rooms, which is the key to the OSBM trick.
2. It's Different: Even though it looks chemically similar to the high-pressure superconductor, the mechanism is totally different. One relies on physical layers; the other relies on energy gaps between electron rooms.
3. Stability: The authors checked if this material would fall apart or change shape. They found that by tweaking the ingredients (swapping some atoms for slightly bigger or smaller ones) or applying a little pressure, they can stabilize the structure needed for this superconductivity.

The Conclusion

The paper is a theoretical proposal. They haven't made the superconductor yet, but they have done the math and the simulations to say:

"Hey, if you take this specific nickel material (La₃Ni₂O₆), remove some oxygen, add a few holes, and maybe squeeze it a little bit, you might create a new type of superconductor that works at high temperatures without needing the crushing pressure of the current record-holders."

It's like finding a new blueprint for a super-fast car that uses a different engine than the ones we've been building for decades. If this works, it could be a huge step toward making room-temperature superconductors a reality.

Technical Summary

Here is a detailed technical summary of the paper "Theoretical proposal of superconductivity in hole-doped reduced bilayer nickelate La$_3$Ni$_2$O$_6$: a manifestation of orbital-space bilayer model with incipient bands."

1. Problem Statement

The discovery of high-temperature superconductivity in bilayer nickelates (specifically La$_3$Ni$_2$O$_7$ under pressure) has sparked intense interest in the mechanisms driving these states. While La$_3$Ni$_2$O$_7$ is understood to rely on strong interlayer hopping ($t_\perp$) between $d_{3z^2-r^2}$ orbitals (a Real-Space Bilayer Model, or RSBM), the reduced bilayer nickelate La$_3$Ni$_2$O$_6$ presents a different scenario.

• Structural Difference: La$_3$Ni$_2$O$_6$ lacks the inner apical oxygen found in La$_3$Ni$_2$O$_7$, resulting in a square-planar coordination for Ni ions and a significantly reduced interlayer hopping for $d_{3z^2-r^2}$ orbitals.
• Theoretical Gap: Previous studies suggested that without strong interlayer coupling, La$_3$Ni$_2$O$_6$ would not support the RSBM-based superconductivity seen in La$_3$Ni$_2$O$_7$. Furthermore, La$_3$Ni$_2$O$_6$ is an insulator at ambient pressure, and its superconducting potential under hole doping remains unexplored.
• Core Question: Can La$_3$Ni$_2$O$_6$ host superconductivity via a different mechanism, specifically the Orbital-Space Bilayer Model (OSBM), where orbital level offsets ($\Delta E$) play the role of interlayer hopping?

2. Methodology

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

• First-Principles Calculations (DFT):
  • Used VASP with the PBEsol functional for structural optimization of both T-type and T'-type crystal structures.
  • Employed GGA, GGA+U ($U=3$ eV for Ni-3d), and Quasi-particle Self-consistent GW (QSGW) methods to calculate band structures and assess correlation effects.
  • Validated the choice of $U$ by comparing GGA+U results with the parameter-free QSGW method.
• Model Construction:
  • Constructed five-orbital tight-binding models (including all Ni 3d orbitals: $d_{x^2-y^2}$, $d_{3z^2-r^2}$, $d_{xz}$, $d_{yz}$, $d_{xy}$) using Wannier functions derived from the DFT bands.
  • Calculated interaction parameters (Coulomb repulsion $U, U'$, Hund's coupling $J$, pair hopping $J'$) using the constrained Random Phase Approximation (cRPA).
• Superconductivity Analysis:
  • Applied the Fluctuation Exchange (FLEX) approximation to solve the linearized Eliashberg equation.
  • Analyzed the eigenvalue $\lambda$ (where $\lambda \to 1$ at $T_c$) as a function of band filling (hole doping) to determine the pairing strength and symmetry.
• Stability Analysis:
  • Calculated enthalpy differences ($\Delta H$) between T and T' structures under various pressures and A-site substitutions (Sr, Ba, Nd, Sm) to predict structural transitions.
  • Performed phonon dispersion calculations (using the frozen-phonon method) to verify dynamical stability.

3. Key Contributions

1. Proposal of OSBM in La$_3$Ni$_2$O$_6$: The paper establishes that La$_3$Ni$_2$O$_6$ is a prime candidate for the Orbital-Space Bilayer Model (OSBM). In this model, the large energy offset ($\Delta E$) between the $d_{x^2-y^2}$ orbital and the other four $d$-orbitals acts analogously to the interlayer hopping ($2t_\perp$) in real-space bilayer systems.
2. Mechanism Identification: It demonstrates that superconductivity in this system is driven by interorbital interactions in an incipient-band regime, rather than interlayer $d_{3z^2-r^2}$ coupling.
3. Structural Stability Prediction: The study predicts that hole doping (via A-site substitution) and external pressure can induce a structural transition from the ambient-pressure T' phase to the T phase, which is energetically favorable under these conditions.

4. Key Results

A. Electronic Structure and OSBM Conditions

• Large Orbital Offset ($\Delta E$): Due to the absence of inner apical oxygens, a large energy separation ($\Delta E \approx 2.2 - 2.5$ eV) exists between the $d_{x^2-y^2}$ orbital and the lower-lying $d_{3z^2-r^2}$, $d_{xz/yz}$, and $d_{xy}$ orbitals.
• Incipient Band Regime: Upon hole doping (reducing electron count $n$ from 8.5 to ~8.1), the Fermi level ($E_F$) moves down. The lower bands (composed of $d_{3z^2-r^2}$ and $t_{2g}$ orbitals) become fully filled and lie just below $E_F$, while the $d_{x^2-y^2}$ band remains nearly empty. This creates the incipient-band situation crucial for OSBM-enhanced superconductivity.

B. Superconductivity Properties

• Pairing Symmetry: The FLEX calculations reveal $s_{\pm}$-wave superconductivity. The gap function changes sign between the $d_{x^2-y^2}$ band and the lower bands.
• Doping Dependence: The superconducting eigenvalue $\lambda$ peaks significantly in the hole-doped regime ($n \approx 8.1 - 8.2$).
  • For the T' structure (GGA+U), $\lambda$ reaches 0.91 at $n=8.1$.
  • For the T structure, $\lambda$ reaches 0.70 at $n=8.1$.
• Role of Interorbital Interactions:
  • When interorbital interactions are removed, superconductivity drops significantly, confirming that the pairing is mediated by interactions between the $d_{x^2-y^2}$ orbital and the other four orbitals.
  • The "RSBM-like" contribution (from $d_{x^2-y^2}$ bilayer splitting) is found to be weak compared to the OSBM contribution.

C. Structural and Dynamical Stability

• Energetic Stability:
  • At ambient pressure, the T' structure is the ground state.
  • Hole doping (Sr/Ba substitution) and hydrostatic pressure shift the enthalpy balance, making the T structure energetically more stable.
  • Larger ionic radii (e.g., Ba) favor the T structure more strongly than smaller ones (Sr). Conversely, lanthanide substitution (Nd, Sm) stabilizes the T' structure due to reduced internal pressure.
• Dynamical Stability: Phonon calculations show no imaginary modes for the T' structure at 0 GPa (consistent with experiment) and for both T and T' structures at 15 GPa, confirming that the predicted structural phases are dynamically stable.

5. Significance

• New Pathway for High-$T_c$ Superconductivity: This work proposes a distinct mechanism for high-$T_c$ superconductivity in nickelates that does not rely on the specific interlayer coupling found in La$_3$Ni$_2$O$_7$. Instead, it leverages the orbital-space bilayer physics driven by large orbital offsets.
• Material Design Strategy: It suggests that hole-doped reduced bilayer nickelates (like La$_3$Ni$_2$O$_6$) are viable candidates for realizing OSBM superconductivity. The study provides a roadmap for experimentalists: apply hole doping (via A-site substitution) and pressure to metallize the insulator and potentially induce superconductivity.
• Theoretical Unification: It reinforces the theoretical correspondence between real-space bilayer models and multi-orbital models, showing that the "incipient band" mechanism is a robust route to enhancing superconductivity across different material classes.
• Resolution of Contradictions: The paper addresses previous conflicting theoretical views on La$_3$Ni$_2$O$_6$ by showing that while RSBM-based pairing is weak, the OSBM mechanism offers a strong alternative pathway for superconductivity in the hole-doped regime.

In conclusion, the authors theoretically propose that hole-doped La$_3$Ni$_2$O$_6$ can host robust $s_{\pm}$-wave superconductivity driven by the Orbital-Space Bilayer Model, offering a promising new direction for the search for high-temperature superconductors beyond the infinite-layer and standard bilayer nickelates.