🔬 condensed matter

$d$ -wave superconductivity in quintuple-layer bulk superconductor La $_6$ Ni $_5$ O $_{12}$

Using variational Monte Carlo calculations on an effective five-layer Hubbard model, this study predicts that the bulk quintuple-layer nickelate La $_6$ Ni $_5$ O $_{12}$ exhibits a spin-singlet $d_{x^2-y^2}$ -wave superconducting state with characteristic nodal signatures, suggesting a common $d$ -wave pairing tendency across the reduced Ruddlesden--Popper nickelate family.

Original authors: Zhan Wang, Yuxin Wang, Hui-Ke Jin, Kun Jiang, Jiangping Hu

Published 2026-10-16

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CC BY 4.0

Original authors: Zhan Wang, Yuxin Wang, Hui-Ke Jin, Kun Jiang, Jiangping Hu

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

Superconductivity is a state of matter where electricity flows without any resistance, a phenomenon that usually requires temperatures near absolute zero. For decades, scientists have been fascinated by a specific family of materials called cuprates, which are made of copper and oxygen, because they can superconduct at relatively higher temperatures. Recently, researchers discovered a new family of superconductors based on nickel instead of copper. These nickel-based materials, known as reduced Ruddlesden-Popper nickelates, share a similar layered structure with cuprates but offer a fresh playground to understand how these exotic materials work. The central mystery in this field is determining the "pairing symmetry" of the electrons. In a superconductor, electrons pair up to move freely; the shape of this pair's wave function—whether it looks like a sphere, a dumbbell, or something else—dictates the material's properties and how it might be used in the future. While thin films of these nickelates have shown promise, they are difficult to study with standard tools because they are too thin. The recent discovery of a bulk, or thick, version of a specific five-layer nickel compound has opened a new door, allowing scientists to probe the inner workings of these materials with the same precision used for traditional superconductors.

A team of researchers has now used advanced computer simulations to investigate this new five-layer material, known as La6Ni5O12. By building a detailed mathematical model of the material's atomic structure and running millions of calculations to find the most stable state, they discovered that the electrons in this compound pair up in a specific pattern called d-wave. This pattern is the same type of pairing found in the famous copper-based superconductors, suggesting a deep connection between the two families of materials. The researchers explicitly ruled out another common possibility, an extended s-wave state, finding it to be far too weak to be the true nature of the superconductivity in this system. Their simulations indicate that the material is a "nodal" superconductor, meaning the energy gap that protects the electron pairs is not uniform but has specific points where it drops to zero, allowing certain low-energy excitations to exist.

To make their findings useful for real-world experiments, the team translated their theoretical model into predictions for measurements that can be taken in a laboratory. They calculated what would happen if scientists measured the density of states, which is essentially a count of how many electron energy levels are available at different energies. Their results predict a distinct V-shaped dip at low energies, a signature that would be clearly visible in tunneling experiments. They also predicted how the material would respond to magnetic fields and temperature changes. Specifically, they found that the superfluid density, which measures how many electrons are participating in the superconducting flow, would decrease in a straight line as the temperature rises from absolute zero. Furthermore, they predicted that the nuclear spin-lattice relaxation rate, a measure of how quickly atomic nuclei lose their magnetic alignment, would follow a specific mathematical curve proportional to the square of the temperature. These combined signatures form a consistent fingerprint of a clean nodal superconductor, providing a clear checklist for experimentalists to verify the theory.

The researchers did not stop at just this one material; they applied the same analytical framework to other members of the nickelate family to see if this d-wave behavior was unique or common. When they looked at the infinite-layer version of the material, which consists of a single repeating layer, their simulations again pointed strongly toward a d-wave pairing state. However, when they examined a three-layer version of the compound, the results were more ambiguous. The three-layer material appeared to sit right on a boundary between being a superconductor and not being one at all, suggesting that its superconducting properties might be very sensitive to small changes in its composition. This comparison highlights that while the tendency to form d-wave pairs seems to be a common thread across the family, the strength of the superconductivity depends heavily on the number of layers and the concentration of charge carriers.

The significance of this work lies in the fact that the five-layer material is a bulk crystal, meaning it is thick enough to be studied with powerful tools like nuclear magnetic resonance and penetration depth measurements, which are impossible to perform on the thin films that have been the focus of previous research. By providing a robust theoretical prediction for what these bulk experiments should see, the study establishes this material as a crucial platform for finally settling the question of pairing symmetry in nickelate superconductors. The findings suggest that the physics governing these new nickel-based superconductors is likely very similar to that of the older copper-based ones, driven by the same underlying mechanisms. This insight not only helps clarify the nature of the current materials but also offers a roadmap for understanding how to potentially enhance superconductivity in related compounds, such as the three-layer version, by adjusting their electron counts.

Technical Summary: d-wave Superconductivity in Quintuple-Layer Bulk Superconductor La $_6$ Ni $_5$ O $_{12}$

Problem Statement
The recent discovery of bulk superconductivity in the quintuple-layer nickelate La $_6$ Ni $_5$ O $_{12}$ (6512) presents a critical opportunity to resolve the pairing symmetry of reduced Ruddlesden–Popper (RP) nickelates. While infinite-layer nickelates (112) have shown evidence of nodal superconductivity, definitive characterization has been hindered by the fact that superconductivity in these systems has primarily been realized in epitaxial thin films. This limitation restricts the application of conventional bulk-sensitive probes such as nuclear magnetic resonance (NMR), penetration depth measurements, and thermodynamic studies. Furthermore, while theoretical studies on infinite-layer and finite-layer nickelates have predominantly favored spin-singlet $d_{x^2-y^2}$ -wave pairing mediated by antiferromagnetic spin fluctuations, a comprehensive understanding of how pairing tendencies evolve across the reduced RP family (from infinite-layer to finite-layer compounds) remains incomplete.

Methodology
The authors employ a realistic theoretical framework combining density functional theory (DFT) with variational Monte Carlo (VMC) calculations to investigate the ground state of La $_6$ Ni $_5$ O $_{12}$ .

Model Construction: Based on DFT calculations, the authors construct a two-dimensional five-band tight-binding (TB) model focusing on the Ni $d_{x^2-y^2}$ orbitals. The model treats the five NiO $_2$ layers (labeled outer–inner–center–inner–outer) and includes a La-derived band acting as a self-doping reservoir. Interlayer hopping is parameterized to match the momentum dependence used in multilayer cuprates.
Interaction and Variational Approach: The system is modeled using a five-layer Hubbard Hamiltonian with an onsite Hubbard repulsion $U$ . The ground state is investigated using VMC with a Gutzwiller-projected wave function ( $|\Psi\rangle = P_G |\Phi_{mf}\rangle$ ). The variational parameters include interlayer hopping amplitudes, layer-dependent chemical potentials, and intralayer pairing amplitudes.
Comparative Analysis: To determine the favored pairing symmetry, the authors compare three mean-field ansatzes: a $d_{x^2-y^2}$ -wave state, an extended $s$ -wave state, and an unpaired reference state. The same framework is extended to trilayer La $_4$ Ni $_3$ O $_8$ (438) and infinite-layer LaNiO $_2$ (112) to examine trends across the family.
Observable Calculation: Using the optimized variational state, the authors calculate experimentally accessible observables, including the tunneling density of states (TDOS), superfluid density, Knight shift, and nuclear spin-lattice relaxation rate ( $1/T_1T$ ), incorporating Gutzwiller renormalization factors to account for correlation effects.

Key Results

Pairing Symmetry in 6512: The VMC calculations reveal that the spin-singlet $d_{x^2-y^2}$ -wave state is energetically favored over the non-superconducting reference. In contrast, the extended $s$ -wave pairing amplitude is strongly suppressed, being roughly two orders of magnitude weaker than the $d$ -wave amplitude. The layer-resolved electron density remains nearly uniform across the five layers.
Experimental Signatures: The calculated observables for the $d$ $d$ -wave state provide consistent signatures of a clean nodal superconductor:
- TDOS: Exhibits a characteristic V-shaped low-energy spectrum with linearly vanishing density of states, reflecting the nodal structure.
- Superfluid Density: Shows a linear low-temperature depletion ( $\rho_s(0) - \rho_s(T) \propto T$ ), indicative of thermally excited quasiparticles near gap nodes.
- Knight Shift: Decreases below $T_c$ with an approximately linear temperature dependence, consistent with spin-singlet pairing.
- Relaxation Rate: The spin-lattice relaxation rate follows $1/T_1T \propto T^2$ (equivalent to $1/T_1 \propto T^3$ ), a hallmark of clean nodal superconductors.
Comparison Across the Family:
- Infinite-layer (112): The framework confirms a robust $d_{x^2-y^2}$ -wave pairing tendency, consistent with previous studies.
- Trilayer (438): At its nominal hole filling ( $x=1/3$ ), the system lies close to the boundary between a $d$ -wave superconducting state and a non-superconducting state. The extended $s$ -wave amplitude is negligible. The authors suggest that electron doping (reducing hole concentration) might stabilize superconductivity in this compound.

Significance and Claims
The paper establishes bulk La $_6$ Ni $_5$ O $_{12}$ as a promising and experimentally accessible platform for determining the pairing symmetry of nickelate superconductors. By providing direct predictions for bulk-sensitive probes (such as penetration depth and NMR), the study offers a pathway to definitively characterize the order parameter, which has been difficult in thin-film systems.

The authors claim that their results suggest a common $d_{x^2-y^2}$ -wave pairing tendency across the reduced RP nickelate family, with the strength of superconductivity controlled by carrier concentration and layer structure. The study positions bulk 6512 as a bridge connecting the established superconductivity of infinite-layer 112 with the emerging superconducting regime of finite-layer compounds like 438. The work does not propose new experimental setups but rather interprets the potential of existing bulk probes to validate the theoretical predictions of $d$ -wave symmetry in this new class of materials.

Technical Summary: d-wave Superconductivity in Quintuple-Layer Bulk Superconductor La6_66​Ni5_55​O12_{12}12​

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Technical Summary: d-wave Superconductivity in Quintuple-Layer Bulk Superconductor La $_6$ Ni $_5$ O $_{12}$