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Multiorbital Anisotropy and Magnetic Excitations in Trilayer Nickelate Pr $_4$ Ni $_3$ O $_{10}$

By combining XAS and RIXS to study Pr $_4$ Ni $_3$ O $_{10}$ , researchers reveal that the material's multilayer architecture, rather than rare-earth substitution, is the primary determinant of its magnetic interactions and multiorbital electronic structure.

Original authors: Ziqiang Guan, Sophia F. R. TenHuisen, Xuefei Guo, Wei He, Aulden K. Jones, Marli dos Reis Cantarino, Xinglong Chen, Hong Zheng, J. F. Mitchell, Nicholas B. Brookes, Mark P. M. Dean, Matteo Mitrano

Published 2026-10-08

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CC BY 4.0

Original authors: Ziqiang Guan, Sophia F. R. TenHuisen, Xuefei Guo, Wei He, Aulden K. Jones, Marli dos Reis Cantarino, Xinglong Chen, Hong Zheng, J. F. Mitchell, Nicholas B. Brookes, Mark P. M. Dean, Matteo Mitrano

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to understand how materials conduct electricity or become superconductors, scientists often look to the arrangement of atoms as the blueprint for behavior. Imagine a crystal not as a solid block, but as a layered cake, where the specific number of layers and the way they stack determine how electrons move and interact. For decades, researchers have studied a family of materials called nickelates, which share a similar layered structure with the famous high-temperature superconductors known as cuprates. However, nickelates possess a unique twist: their internal architecture allows for different types of electron orbits to be active simultaneously, creating a complex electronic landscape that is still being mapped. A central mystery in this field is how the magnetic forces between atoms change when the material is altered, such as by swapping one type of atom for another or by changing the number of layers in the stack. Understanding these magnetic interactions is crucial because they are believed to be the glue that holds together the electron pairs responsible for superconductivity, a state where electricity flows with zero resistance.

A team of researchers recently turned their attention to a specific three-layer nickelate compound, where the rare-earth element praseodymium replaces the more common lanthanum. While previous studies had shown that this substitution changes the material's structure and raises the temperature at which it becomes superconducting, it remained unclear how this chemical change affected the underlying magnetic forces. To answer this, the scientists used powerful X-ray techniques to peer directly into the electronic and magnetic heart of the material. By firing beams of X-rays at the crystal and measuring how the energy of the scattered light changed, they were able to map out the specific ways electrons move between atoms and how the magnetic spins of nickel atoms interact with one another across the different layers.

The investigation revealed that the electronic structure of this material is highly directional. The researchers found that the electrons do not move equally in all directions; instead, they form distinct pathways. Some electrons prefer to hop sideways within the flat planes of the crystal, while others travel up and down, connecting the layers through oxygen atoms that sit between the nickel layers. This distinction is vital because it shows that the material's magnetic life is not confined to a single plane but is a three-dimensional network. The study confirmed that the praseodymium substitution, while it distorts the crystal lattice, does not fundamentally break this network. The magnetic forces that bind the layers together remain robust, suggesting that the material's ability to superconduct at higher temperatures is not simply a result of the chemical swap, but rather a consequence of its inherent three-layer design.

Perhaps the most significant finding concerns the strength of the magnetic connection between the layers. In a two-layer version of this material, the magnetic force linking the layers is extremely strong, dominating the behavior of the entire crystal. However, in the three-layer version studied here, the researchers measured the magnetic interaction between the outer layers and found it to be surprisingly comparable in strength to the forces acting within the layers themselves. This balance creates a unique magnetic environment where no single direction of interaction overwhelms the others. The data showed that the magnetic waves, or excitations, travel through the material with a specific pattern that softens as they approach a particular momentum, a signature of the material's ordered magnetic state. By analyzing these patterns, the team calculated that the magnetic coupling between the outer layers is roughly 16 units of energy, a value nearly identical to the strongest forces acting within the planes.

This result challenges the idea that rare-earth substitution alone is the primary driver of magnetic changes in these materials. Instead, the study points to the number of layers in the stack as the critical control knob. Whether the material has two layers or three, the magnetic hierarchy shifts dramatically, suggesting that the architecture of the crystal itself dictates how electrons and spins behave. The researchers also noted that the magnetic forces in this praseodymium-based compound are very similar to those found in the lanthanum-based version, despite the differences in their chemical composition and structural distortion. This consistency implies that the fundamental magnetic rules governing these materials are resilient to chemical changes, provided the layering remains the same.

The work provides a clear reference point for understanding how these materials behave under normal conditions, which is essential for interpreting what happens when they are squeezed under high pressure to induce superconductivity. By establishing that the magnetic interactions are strong and balanced across the layers, the study offers a concrete foundation for theories that link these magnetic forces to the emergence of superconductivity. The findings suggest that the path to higher-temperature superconductivity in these materials may lie in manipulating the layer architecture rather than just tweaking the chemical ingredients. Ultimately, the research paints a picture of a material where the three-dimensional arrangement of atoms creates a robust and balanced magnetic network, one that persists even as the chemical environment shifts around it.

Technical Summary: Multiorbital Anisotropy and Magnetic Excitations in Trilayer Nickelate Pr $_4$ Ni $_3$ O $_{10}$

Problem Statement
The emergence of high-temperature superconductivity in bilayer and trilayer Ruddlesden–Popper (RP) nickelates under pressure or epitaxial strain has raised critical questions regarding how the multilayer Ni–O architecture controls electronic and magnetic interactions. While isovalent rare-earth substitution (e.g., replacing La with Pr) is known to modify crystal structure, metallicity, and superconducting transition temperatures through chemical pressure, its specific impact on the underlying magnetic interactions remains unclear. In trilayer systems like La $_4$ Ni $_3$ O $_{10}$ , the presence of inner and outer Ni sites creates inequivalent orbital occupations and magnetic responses. However, in La-based compounds, the interpretation of Ni magnetic responses is complicated by the overlap of the La $M_4$ edge with the Ni $L_3$ edge. The authors investigate whether the multilayer architecture or rare-earth substitution is the primary control knob for magnetic interactions, using Pr $_4$ Ni $_3$ O $_{10}$ as a cleaner system to resolve these interactions without rare-earth spectral interference.

Methodology
The study employs a combination of X-ray Absorption Spectroscopy (XAS) and Resonant Inelastic X-ray Scattering (RIXS) at the Ni $L_3$ and O $K$ edges. Experiments were conducted on high-quality single crystals of Pr $_4$ Ni $_3$ O $_{10}$ grown via high-oxygen-pressure optical floating-zone growth, measured at 30 K (above the Pr–Ni magnetic coupling transition but below the density-wave transition).

Polarization-Dependent XAS: Measurements at the Ni $L_3$ and O $K$ edges utilized $\sigma$ - and $\pi$ -polarized incident x-rays to distinguish between in-plane (planar) and out-of-plane Ni–O hybridization channels.
Momentum-Resolved RIXS: Low-energy magnetic excitations were mapped using $\pi$ -polarized incident x-rays tuned to the Ni $L_3$ resonance (~855.8 eV). Scans were performed along in-plane directions ( $H, H$ ) and ( $H, 0$ ), as well as out-of-plane ( $L$ ) directions at fixed in-plane momenta.
Theoretical Modeling: The experimental dispersion was analyzed using:
- Linear Spin-Wave Theory (LSWT) based on an effective Heisenberg model for the outer NiO $_2$ layers, accounting for structural twins and the incommensurate spin-density-wave (SDW) wave vector.
- Entangled-dimer calculations to test the robustness of the interlayer coupling against local quantum correlations.
- Stripe-dimer models for comparative analysis.

Key Contributions and Results

Multiorbital Electronic Structure: Polarization-dependent XAS and RIXS revealed a clear anisotropy between planar and out-of-plane Ni–O hybridized states.
- $\pi$ -polarized spectra showed a distinct energy shift (~0.2 eV) and enhanced spectral weight compared to $\sigma$ -polarized spectra at both the Ni $L_3$ and O $K$ edges.
- This confirms the presence of active Ni $3d_{3z^2-r^2}$ –O $2p_z$ out-of-plane channels alongside the planar Ni $3d_{x^2-y^2}$ –O $2p_{x,y}$ channels.
- Unlike La $_4$ Ni $_3$ O $_{10}$ , Pr $_4$ Ni $_3$ O $_{10}$ allows direct access to the Ni $L_3$ resonance without the overlapping La $M_4$ contribution, providing a cleaner view of the trilayer electronic structure. The Pr compound exhibits broader orbital excitations, likely due to stronger structural distortions and inequivalent Ni sites.
Magnetic Excitations and SDW Softening: Momentum-resolved RIXS identified dispersive low-energy magnetic excitations (20–85 meV).
- The excitations soften significantly as the momentum approaches the incommensurate SDW wave vector ( $H \approx -0.31$ r.l.u.), where spectral weight accumulates near zero energy loss.
- The magnetic response is dominated by the two outer NiO $_2$ layers, with the inner layer acting as a magnetic node, consistent with diffraction data.
Quantification of Exchange Interactions: A simultaneous fit of the dispersion using LSWT yielded the following effective exchange parameters:
- In-plane nearest-neighbor coupling ( $J_1$ ): $15.7(3)$ meV
- Second-neighbor coupling ( $J_2$ ): $1.7(3)$ meV
- Third-neighbor coupling ( $J_3$ ): $9.8(1)$ meV
- Effective interlayer coupling ( $J_z$ ): $15.8(6)$ meV
Crucially, the effective coupling between the two outer layers ( $J_z$ ) is comparable to, and slightly larger than, the leading in-plane nearest-neighbor interaction ( $J_1$ ). This sizable out-of-plane interaction persists despite the presence of a non-magnetic inner layer.
Robustness Against Substitution: The extracted interaction scales in Pr $_4$ Ni $_3$ O $_{10}$ are comparable to those found in La $_4$ Ni $_3$ O $_{10}$ , despite the significant structural distortion and reduced metallicity in the Pr compound. This contrasts sharply with bilayer La $_3$ Ni $_2$ O $_7$ , where interlayer coupling is much stronger (~50–70 meV) and dominates the in-plane interactions.

Significance and Claims
The paper claims that the multilayer architecture, rather than rare-earth substitution alone, is the primary control parameter for the hierarchy of magnetic interactions in RP nickelates. The results demonstrate that the substantial participation of the out-of-plane Ni $3d_{3z^2-r^2}$ –O $2p_z$ orbital channel supports a robust magnetic coupling across the trilayer, even when the inner layer is magnetically suppressed.

The authors conclude that the dominant magnetic interaction scales in trilayer RP nickelates remain robust against rare-earth substitution, suggesting that the multiorbital nature of the system and the specific layer stacking sequence are more critical determinants of magnetic behavior than the specific rare-earth ion or the associated chemical pressure. These findings provide an ambient-pressure reference for understanding how magnetic hierarchies evolve as pressure suppresses the density-wave state and drives the system toward superconductivity.

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