Contrasting Spin Excitations in Octahedral and Square-Planar n=8 Ruddlesden-Popper Nickelates

Using Ni $L_3$-edge resonant inelastic x-ray scattering, this study reveals fundamental differences in spin excitations between non-superconducting octahedral Nd$_9$Ni$_8$O$_{25}$, which exhibits a spin-density-wave ground state with dispersive paramagnons, and its superconducting square-planar counterpart Nd$_9$Ni$_8$O$_{18}$, which displays dispersionless magnetic excitations and a distinct elastic peak.

The Gist

Imagine you are trying to build a super-fast elevator (superconductivity) that can carry electricity without any friction or energy loss. For decades, scientists have been obsessed with a specific type of building material called cuprates (copper-based) because they are the best at this job.

Recently, scientists discovered a new family of materials called nickelates (nickel-based) that look a lot like cuprates and might do the same thing. But there's a catch: these nickelates come in two very different "architectural styles," and until now, we didn't know how their internal "traffic systems" (electron and spin movements) compared.

This paper is like a detective story where researchers use a high-tech X-ray camera to peek inside two specific nickelate buildings to see how their internal traffic behaves.

The Two Buildings: Octahedral vs. Square-Planar

Think of the two materials as two different skyscrapers made of the same bricks (Nickel and Oxygen) but built with different blueprints:

1. The "Parent" Building (Octahedral): This is the original structure. Imagine a tall tower where every room (Nickel atom) is surrounded by oxygen atoms on all six sides, like a cube. This building is non-superconducting (it doesn't carry electricity perfectly).
2. The "Reduced" Building (Square-Planar): This is the renovated version. Scientists took the "Parent" building and surgically removed the top and bottom oxygen atoms from every room. Now, the rooms are flat squares. This building does show signs of superconductivity (it can carry electricity perfectly at very low temperatures).

The big question was: What happens to the internal "traffic" (the spins of the electrons) when you switch from the cube-shaped rooms to the flat square rooms?

The Detective Tool: RIXS

To answer this, the team used a technique called RIXS (Resonant Inelastic X-ray Scattering).

• The Analogy: Imagine throwing a ping-pong ball (an X-ray) at a crowd of people (the electrons).
• If the crowd is just standing still, the ball bounces back with the same energy.
• If the crowd is moving or dancing (excited), the ball bounces back with less energy.
• By measuring exactly how much energy the ball loses, the scientists can map out how the electrons are dancing and interacting.

The Findings: Two Different Dance Floors

Here is what the scientists discovered when they looked at the "dance floor" of these two buildings:

1. The Parent Building (The Cube):

• The Vibe: It's a rigid, organized dance. The electrons are marching in a very specific, repeating pattern called a Spin Density Wave.
• The Dance Move: They found a specific "beat" (a magnetic wave) that travels through the building. It's like a wave of people doing "The Wave" in a stadium. This wave is slow and moves in a predictable, wavy pattern.
• The Result: This rigid, ordered state is why this building cannot be a superconductor. It's too stiff.

2. The Reduced Building (The Flat Square):

• The Vibe: The rigid order is gone! The "Wave" of the stadium has collapsed.
• The Dance Move: Instead of a slow, traveling wave, the electrons are doing something else. They are vibrating in place with high energy, but they aren't moving across the floor. It's like a crowd of people jumping up and down in place (dispersionless) rather than running in a line.
• The Result: This "jumping in place" behavior is much more chaotic and energetic. The scientists believe this specific type of magnetic vibration is actually helpful for superconductivity. It's the "sweet spot" that allows electricity to flow without friction.

The "Secret Ingredient"

The paper reveals that the key difference isn't just the shape of the room (cube vs. square), but the oxygen atoms that were removed.

• In the Parent building, the oxygen atoms act like heavy anchors, locking the electrons into a rigid, marching pattern.
• When you remove the top and bottom oxygens (to make the Reduced building), you release those anchors. The electrons are free to jump around in that high-energy, "dispersionless" way.

Why This Matters

Think of superconductivity like a highway.

• In the Parent building, the cars (electrons) are stuck in a traffic jam, marching in a single file line.
• In the Reduced building, the traffic jam clears up. The cars start moving in a way that allows them to zip past each other without crashing.

This paper is crucial because it proves that changing the shape of the atomic "rooms" changes the fundamental rules of how electrons dance. It suggests that to build better superconductors, we need to engineer materials that encourage this specific "jumping in place" magnetic behavior, rather than the rigid marching of the old style.

In short: By taking a "cube" nickelate and flattening it into a "square," scientists unlocked a new, energetic dance move for electrons that is likely the secret sauce for superconductivity.

Technical Summary

1. Problem Statement

The discovery of superconductivity in reduced square-planar nickelates ($RNiO_2$) has established them as structural and electronic analogs to high-$T_c$ cuprates. However, the recent observation of superconductivity in parent octahedral Ruddlesden–Popper (RP) nickelates (specifically under pressure) challenges existing theories. These parent phases possess a distinct electron count ($d^{7+1/n}$) compared to the cuprate-like reduced phases ($d^{9-1/n}$).

A critical gap in understanding exists regarding the relationship between crystal structure (octahedral vs. square-planar), electron filling, and the nature of magnetic excitations (spin fluctuations) in these systems. While previous studies on $n=2, 3, 5$ layers showed conflicting trends (dispersive vs. non-dispersive excitations), a comprehensive comparison of the ground states and spin dynamics in a high-order member ($n=8$) of both the parent (p-RP) and reduced (r-RP) families was lacking. This study aims to resolve whether the magnetic excitation spectra correlate with the emergence of superconducting correlations and how structural changes (removal of apical oxygens) alter the electronic and magnetic ground states.

2. Methodology

The authors employed a multi-faceted approach combining experimental spectroscopy and theoretical calculations:

• Sample Synthesis: High-quality thin films of the parent octahedral $n=8$ RP ($Nd_9Ni_8O_{25}$, p-RP) and its reduced square-planar counterpart ($Nd_9Ni_8O_{18}$, r-RP) were grown using ozone-assisted molecular beam epitaxy (MBE) on $NdGaO_3$ substrates.
• Resonant Inelastic X-ray Scattering (RIXS):
  • Experiments were conducted at the Ni $L_3$-edge (853 eV) at the ESRF (ID32 beamline).
  • Polarimetric RIXS (pol-RIXS): Measurements were performed with both $\sigma$ and $\pi$ incident polarizations to distinguish between magnetic (cross-polarized, $\pi-\sigma'$) and phononic (non-cross-polarized, $\pi-\pi'$) excitations.
  • Energy Resolution: Combined energy resolution of $\sim$40 meV.
  • Temperature: Measurements taken at $T=25$ K.
• Theoretical Calculations:
  • DFT: Kohn-Sham density-functional theory calculations were performed using the WIEN2k code (APW+lo basis) to determine electronic band structures, density of states (DOS), and Fermi surfaces.
  • Susceptibility: Calculations of bare magnetic susceptibility were used to predict ordering vectors.

3. Key Contributions

• Systematic Comparison: This is the first comprehensive RIXS study comparing the $n=8$ parent and reduced RP nickelates, bridging the gap between bilayer ($n=2$) and infinite-layer ($n=\infty$) systems.
• Decoupling Magnetic and Lattice Modes: The use of polarimetric RIXS allowed the authors to unambiguously separate magnon excitations from phonon modes, a crucial step often difficult in nickelate studies due to overlapping energy scales.
• Mapping the $n=8$ Phase Diagram: The study establishes the electronic and magnetic ground states for the $n=8$ limit, providing a benchmark for understanding the evolution of superconductivity across the RP series.

4. Key Results

A. Electronic Structure (DFT)

• Parent (p-RP, $Nd_9Ni_8O_{25}$): Exhibits a charge-transfer energy $\Delta_{CT} \approx 3.3$ eV with significant Ni(3d)-O(2p) overlap, indicating a $3d^8\bar{L}$ (ligand hole) character. The Fermi surface consists of hole-like sheets with mixed $e_g$ character.
• Reduced (r-RP, $Nd_9Ni_8O_{18}$): Removal of apical oxygens increases $\Delta_{CT}$ to $\sim$4.5 eV. The system features self-doping bands formed by Nd(5d)-Ni(3d) hybridization, effectively reducing the Ni filling to $d^{8.875}$. The Fermi surface is dominated by $Ni-d_{x^2-y^2}$ hole pockets, placing the system in the underdoped regime of the cuprate phase diagram ($x=0.125$).

B. Magnetic Order and Elastic Scattering

• p-RP (Octahedral): Develops a long-range Spin Density Wave (SDW) ground state with an ordering wave vector $q_{SDW} = (1/4, 1/4)$. This is analogous to the bilayer $n=2$ parent phase ($La_3Ni_2O_7$).
• r-RP (Square-Planar): Long-range magnetic order is suppressed. A weak elastic peak appears at $q^* = (1/3, 0)$, which the authors attribute to oxygen-rich impurity phases formed during reoxidation, rather than intrinsic magnetic order.

C. Inelastic Spin Excitations (RIXS)

• p-RP: The low-energy spectrum is dominated by weakly dispersive paramagnons (optical magnons) with an energy of $\sim$55 meV along both $(H, 0)$ and $(H, H)$ directions.
• r-RP: The spectrum is characterized by dispersionless (flat) magnetic excitations at a higher energy of $\sim$60–65 meV.
• Damping: Despite the structural and electronic differences, the damping (linewidth) of the excitations remains nearly constant across momentum for both phases.

D. Interaction Strengths

• The transition from p-RP to r-RP involves the removal of apical oxygens, which weakens the interlayer exchange interaction ($J_\perp$).
• The r-RP shows a slight enhancement in the in-plane exchange interaction ($J_\parallel$), evidenced by the hardening (higher energy) of the paramagnon mode. This dominance of $J_\parallel$ favors theories where the $d_{x^2-y^2}$ orbital plays a central role in superconductivity.

5. Significance

• Structural Control of Magnetism: The study demonstrates that the presence or absence of apical oxygens (octahedral vs. square-planar coordination) is the critical tuning parameter for the magnetic ground state, switching the system from a long-range SDW ordered state to a state with short-range, dispersionless spin fluctuations.
• Distinction from Cuprates: While the r-RP ($n=8$) shares the hole-doping level ($x=0.125$) with underdoped cuprates, its magnetic excitations are dispersionless and narrow, contrasting sharply with the overdamped, dispersive paramagnons observed in cuprates. This suggests that the mechanism driving superconductivity in nickelates may differ fundamentally from that in cuprates, or that the specific electronic structure of the $n=8$ layer introduces unique constraints.
• Implications for Superconductivity: The suppression of long-range order and the hardening of spin excitations in the reduced phase correlate with the emergence of superconducting correlations ($T_c \approx 5$ K). The results highlight that the $p_z$ orbital of the apical oxygen is essential for tuning the collective ground states, suggesting that optimizing the removal of these oxygens is key to enhancing $T_c$ in RP nickelates.

In conclusion, the paper provides a definitive map of spin excitations in $n=8$ nickelates, revealing that while both families share magnetic origins, their ground states and excitation dynamics are fundamentally distinct, driven by structural dimensionality and orbital hybridization.