Collective spin excitations in trilayer nickelate La$_4$Ni$_3$O$_{10}$

This study utilizes Ni $L$-edge resonant inelastic X-ray scattering to reveal that while trilayer La$_4$Ni$_3$O$_{10}$ exhibits collective spin excitations with a bandwidth comparable to bilayer nickelates, its substantially suppressed spectral weight indicates weaker electronic correlations and a distinct three-dimensional, multi-orbital character that differentiates it from its bilayer counterparts.

The Gist

Imagine you are trying to build a super-fast, friction-free highway for electrons (the tiny particles that carry electricity). Scientists have long known that in certain materials called "cuprates," these electrons can team up and flow without resistance at surprisingly high temperatures. This is called superconductivity, and it's the holy grail of energy technology.

Recently, scientists discovered a new family of materials called nickelates that might do the same thing. Think of nickelates as a new, unexplored neighborhood in the city of superconductors.

This specific paper is like a detective story investigating the "traffic patterns" (magnetic excitations) in a specific type of nickelate building: the trilayer (a building with three floors of nickel atoms). The scientists wanted to see if this three-story building behaves like its famous two-story neighbor (the bilayer nickelate), which is already known to be a strong candidate for superconductivity.

Here is the breakdown of their findings using simple analogies:

1. The Two Buildings: Two-Story vs. Three-Story

• The Two-Story Building (Bilayer): Scientists already knew this building had strong "social connections" between its floors. The electrons on one floor talked loudly and clearly to the electrons on the other floor. This strong connection created a powerful "magnetic dance" (spin excitations) that helped the electrons pair up and superconduct.
• The Three-Story Building (Trilayer - La4Ni3O10): This is the new mystery. It looks similar on the outside, but it has an extra floor in the middle. The big question was: Does this extra floor make the magnetic dance stronger, weaker, or totally different?

2. The Investigation: Listening to the "Music"

To find out, the scientists used a high-tech tool called RIXS (Resonant Inelastic X-ray Scattering).

• The Analogy: Imagine you are in a dark room trying to understand how a group of people are interacting. You can't see them, so you shout a specific sound (X-rays) and listen to the echo.
• The Result: The echo tells you how the "people" (electrons) are moving and spinning.

3. What They Found: The "Quiet" Dance

The scientists discovered two very different types of "music" in the three-story building:

• The Localized "Solo" Acts (High Energy): They heard two distinct, sharp notes (at 100 and 200 meV).

  • What it means: These are like solo musicians playing on a single floor. They are stuck in one spot and don't travel. The scientists realized these come from the middle floor of the building, which acts a bit differently than the top and bottom floors. It's a "quiet" spot where the electrons aren't dancing with their neighbors.

• The Collective "Group" Dance (Low Energy): They also heard a low, rumbling sound (around 60 meV) that traveled across the whole building.

  • The Twist: In the two-story building, this group dance was loud and energetic (strong "spectral weight"). In the three-story building, the dance exists, but it is much quieter.
  • The Metaphor: Imagine a choir. In the two-story building, the whole choir sings at full volume. In the three-story building, the choir is there, but they are whispering.

4. The Big Conclusion: Why the Three-Story is Different

The fact that the "group dance" is quieter tells us something crucial about the material:

• Weaker Connections: The electrons in the three-story building are less "socially connected" to each other than in the two-story version. They are more independent.
• 3D vs. 2D: The three-story building is more "three-dimensional." The electrons can move up and down between floors more easily, which actually dilutes the strong magnetic force needed for superconductivity.
• The Temperature Limit: This explains why the three-story building has a lower "superconducting temperature" (it only works at 30 Kelvin, compared to 80 Kelvin for the two-story). The "whispering" magnetic dance isn't strong enough to hold the electron pairs together as tightly as the "singing" dance in the two-story building.

The Takeaway

This paper is a vital clue in the puzzle of high-temperature superconductivity. It tells us that adding more layers doesn't always make things better.

In fact, the "sweet spot" for superconductivity in these nickelates might be the two-story version. The three-story version shows us that while the materials are similar, the subtle differences in how the layers stack change the magnetic "personality" of the material, making it harder for superconductivity to happen.

In short: The scientists found that the three-story nickelate building is a bit more "disconnected" than its two-story cousin. The magnetic forces that help electrons team up are weaker, which explains why it's harder to make it superconduct. This helps scientists know exactly what ingredients they need to mix to build the perfect superconductor.

Technical Summary

1. Problem Statement

Ruddlesden-Popper (RP) nickelates ($RE_{n+1}Ni_nO_{3n+1}$) have emerged as a new family of high-temperature superconductors, with the bilayer compound La$_3$Ni$_2$O$_7$ ($n=2$) showing a superconducting transition temperature ($T_c$) of up to 80 K under pressure. In bilayer nickelates, strong antiferromagnetic spin fluctuations with large exchange couplings (bandwidth $\sim$70 meV) have been identified as potential mediators for superconductivity.

However, the magnetic properties of the trilayer compound La$_4$Ni$_3$O$_{10}$ ($n=3$) remain poorly understood. Despite structural similarities, trilayer nickelates exhibit:

• A significantly lower $T_c$ (30 K) compared to bilayers.
• A different magnetic ground state (incommensurate spin order vs. spin-stripe order).
• Enhanced three-dimensional (3D) electronic character due to the additional NiO$_6$ layer.

The central question is whether comparable strong spin correlations persist in trilayer nickelates and how the evolution of magnetic interactions from bilayer to trilayer systems influences the superconducting mechanism.

2. Methodology

The authors employed high-resolution Resonant Inelastic X-ray Scattering (RIXS) at the Ni $L$-edge to probe low-energy electronic and magnetic excitations in bulk La$_4$Ni$_3$O$_{10}$ single crystals.

• Sample: High-quality single crystals grown via the high-pressure optical floating zone method.
• Experimental Setup: Experiments were conducted at the ID32 beamline (ESRF) and 41A1 beamline (NSRRC) at 23–30 K. The energy resolution was set to 38 meV.
• Techniques:
  • X-ray Absorption Spectroscopy (XAS): Measured at the Oxygen $K$-edge and Ni $L$-edge to characterize the electronic structure and ligand hole character.
  • Polarimetric RIXS: Used to decompose scattering channels ($\sigma$-in/$\sigma$-out, $\sigma$-in/$\pi$-out, etc.) to distinguish between magnetic and orbital excitations.
  • Momentum Mapping: Scanned along high-symmetry directions in reciprocal space to determine the dispersion of excitations.
• Theoretical Modeling:
  • Exact Diagonalization (ED): Used to model localized single-ion excitations.
  • Linear Spin-Wave Theory (LSWT): Used to model the dispersive collective magnetic excitations based on a Heisenberg spin Hamiltonian with in-plane ($J_1, J_2$) and inter-layer ($J_\perp$) exchange couplings, assuming an incommensurate magnetic structure with a node at the inner layer.

3. Key Results

A. Electronic Structure and Localized Excitations

• Ligand Holes: Oxygen $K$-edge XAS confirmed the presence of ligand holes (characteristic of RP nickelates), similar to La$_3$Ni$_2$O$_7$.
• Localized Modes: Two sharp, non-dispersive spin-flip excitations were observed at ~100 meV and ~200 meV.
  • Polarimetric analysis confirmed their predominantly magnetic origin.
  • ED calculations attribute these to local dipolar (100 meV) and quadrupolar (200 meV) spin excitations of Ni$^{2+}$ ions in a high-spin state but without long-range ordered moments (consistent with the non-magnetic inner layer proposed in previous neutron studies).

B. Collective Spin Excitations (The Core Finding)

• Dispersive Mode: A third, weaker excitation mode was identified around 60 meV. Unlike the localized modes, this mode is dispersive.
• Bandwidth: The collective mode exhibits a bandwidth of approximately 60 meV, comparable to the $\sim$70 meV bandwidth observed in bilayer La$_3$Ni$_2$O$_7$.
• Spectral Weight Suppression: Crucially, the spectral weight (intensity) of these collective excitations in La$_4$Ni$_3$O$_{10}$ is substantially suppressed (nearly an order of magnitude weaker) compared to the bilayer counterpart.
• Dispersion Pattern: The intensity modulation increases away from the $\Gamma$ point toward the magnetic ordering wave vector $Q_{spin} = (0.31, 0.31)$, consistent with a spin-density-wave (SDW) state driven by Fermi surface nesting.

C. Exchange Couplings

LSWT modeling of the dispersive mode yielded the following effective exchange parameters:

• In-plane couplings: $SJ_1 \approx 12$ meV, $SJ_2 \approx 8$ meV.
• Inter-layer coupling: $SJ_\perp \approx 20$ meV.
• Significance: The inter-layer coupling is the strongest interaction, indicating enhanced three-dimensional magnetism in the trilayer system, contrasting with the more 2D nature of bilayers.

4. Key Contributions

1. First Comprehensive Mapping: Provided the first detailed momentum-resolved map of collective spin excitations in bulk La$_4$Ni$_3$O$_{10}$.
2. Identification of Weak Correlations: Demonstrated that while the energy scale (bandwidth) of magnetic excitations is similar to bilayers, the strength of electronic correlations is significantly weaker in trilayers, evidenced by the suppressed spectral weight.
3. 3D Magnetic Character: Quantified the exchange couplings, revealing that the inter-layer coupling ($J_\perp$) is dominant, confirming the 3D electronic and magnetic nature of the trilayer system.
4. Coexistence of Excitations: Established the coexistence of localized spin excitations (from non-magnetic inner layers) and itinerant collective modes (from outer layers), linked to orbital-selective correlations and SDW instability.

5. Significance and Implications

• Mechanism of Superconductivity: The findings suggest that the lower $T_c$ in trilayer La$_4$Ni$_3$O$_{10}$ (30 K) compared to bilayer La$_3$Ni$_2$O$_7$ (80 K) may be directly linked to the weaker electronic correlations and the suppressed spectral weight of spin fluctuations, despite the presence of strong 3D exchange couplings.
• Role of Dimensionality: The study highlights that increasing the layer number ($n$) in RP nickelates does not simply scale up the physics of the bilayer; it fundamentally alters the balance between 2D and 3D interactions and correlation strength.
• Theoretical Guidance: The results provide critical constraints for theoretical models, suggesting that superconductivity in nickelates is sensitive to the interplay between orbital selectivity, dimensionality, and the strength of spin-mediated pairing interactions. The data supports a scenario where strong correlations (Mott physics) are essential for high-$T_c$ superconductivity, which are less pronounced in the trilayer system.