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Contrasting Crystal-Field and Dipolar Anisotropies in the Triangular Antiferromagnet NaBaEr(BO $_3$ ) $_2$

By combining structural, magnetic, and neutron scattering measurements, this study reveals that the magnetic ground state of the triangular antiferromagnet NaBaEr(BO $_3$ ) $_2$ is determined by a competition between strong crystal-field-induced easy-axis anisotropy and dipolar interactions that favor in-plane ordering, ultimately resulting in an unexpected A-type antiferromagnetic structure with moments lying within the $ab$ plane.

Original authors: Cheng Su, Yonglin Wang, Zihang Gao, Xinyang Liu, Jianglong Zhang, Chin-Wei Wang, Michael Marek Koza, Juping Xu, Wen Yin, Hao Deng, Wei Li, Junsen Xiang, Wentao Jin

Published 2026-10-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cheng Su, Yonglin Wang, Zihang Gao, Xinyang Liu, Jianglong Zhang, Chin-Wei Wang, Michael Marek Koza, Juping Xu, Wen Yin, Hao Deng, Wei Li, Junsen Xiang, Wentao Jin

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

Magnetism is often thought of as a simple tug-of-war between invisible forces, but in the world of rare-earth minerals, the story is far more complex. Imagine a flat, triangular grid of tiny magnets, where each magnet wants to point in the opposite direction of its neighbors. Because of the triangular shape, it is impossible for all of them to satisfy this rule at once; this geometric deadlock is known as frustration. In these frustrated systems, the final arrangement of the magnets depends on a delicate balance between different types of energy. One force comes from the crystal itself, which acts like a rigid mold, forcing the atoms to align in a specific direction. Another force comes from the magnets' own long-range pull on one another, a subtle interaction that stretches across the entire material. Scientists are deeply interested in these materials because they can host exotic states of matter that defy our usual understanding of physics, potentially leading to new technologies for computing or energy storage. The question is: when these competing forces clash, which one wins?

A team of researchers has now answered this question for a specific mineral called NaBaEr(BO3)2, a compound containing the rare-earth element erbium. By combining measurements of how the material responds to cold temperatures and magnetic fields with detailed imaging using neutron beams, they discovered a surprising contradiction. The internal structure of the crystal strongly suggests that the magnetic moments should point straight up and down, like a flagpole. Yet, when the material finally settles into its ordered state at temperatures near absolute zero, the magnets actually lie flat, pointing sideways within the plane of the crystal. The researchers found that the long-range pull between the magnets, rather than the rigid crystal mold, dictates this final orientation, overturning what the local environment seemed to demand.

To understand this discovery, one must first look at the material's architecture. The researchers grew high-quality crystals of NaBaEr(BO3)2 and used neutron diffraction to map out the positions of every atom. They found that the oxygen atoms surrounding the erbium ions are slightly twisted, a detail that had been missed in previous studies which assumed a more symmetrical arrangement. This twist lowers the symmetry of the crystal structure, creating two distinct types of sites for the erbium atoms. This structural clarity was essential because the way the crystal holds the atoms determines how the electrons inside them behave. The team then used inelastic neutron scattering to probe the energy levels of these electrons, finding that the lowest energy state is well-separated from the next level by about 4 milli-electron volts. This large gap means that at the very low temperatures where the material becomes magnetic, the electrons behave effectively as simple spin-1/2 objects, making the system easier to analyze.

The behavior of these spins revealed a strong preference for direction. When the researchers measured how the material reacted to a magnetic field, they found that it was much easier to magnetize the sample along the vertical axis than along the horizontal plane. This "easy-axis" anisotropy suggested that the crystal field, the local environment of the atoms, was forcing the magnetic moments to stand upright. If this local force were the only thing at play, the material should have ordered with its magnets pointing up and down. However, when the team cooled the sample to just 0.1 Kelvin and used single-crystal neutron diffraction to watch the magnetic order form, they saw something different. The material developed an antiferromagnetic pattern where the layers of magnets stacked in an alternating fashion, but the magnets themselves were lying flat within the horizontal plane.

This result posed a puzzle: why would the magnets ignore the strong vertical preference of their local environment and choose to lie flat? The answer lay in the long-range dipolar interactions. Because the erbium ions have a large magnetic moment, they exert a significant magnetic pull on one another across the entire crystal. The researchers calculated the energy cost of different orientations for this specific stacking pattern and found that the long-range pull strongly favored the flat, in-plane arrangement. They determined that the energy scale of these dipolar interactions is comparable to the temperature at which the material orders, meaning this force is powerful enough to compete with the local crystal field. In fact, the calculation showed that the stronger the vertical preference from the crystal, the more the long-range dipolar forces pushed the magnets to lie flat to minimize their total energy.

The study confirms that in this rare-earth triangular magnet, the global interaction between the magnets wins over the local constraints of the crystal structure. The researchers ruled out the idea that the material simply followed the easy-axis rule, and they showed that the previously proposed, higher-symmetry crystal structure was incorrect, as it could not explain the observed magnetic behavior. By clarifying the crystal structure and measuring the energy scales involved, the team demonstrated that the magnetic ground state is a result of a competition where the long-range dipolar forces tip the balance. This finding provides a clear example of how different sources of magnetic anisotropy can clash, offering a new perspective on how to predict and control the magnetic properties of complex materials.

Technical Summary: Contrasting Crystal-Field and Dipolar Anisotropies in the Triangular Antiferromagnet NaBaEr(BO3)2

Problem Statement
The magnetic ground states of rare-earth frustrated magnets are governed by a complex interplay between crystal-field (CEF) anisotropy, exchange interactions, and long-range dipolar interactions. In triangular-lattice antiferromagnets (TLAFs), geometrical frustration often leads to exotic quantum phases. However, the specific competition between strong local CEF anisotropy (which typically dictates the easy axis of the magnetic moment) and long-range dipolar interactions (which introduce a competing energy scale) remains a critical area of investigation. The compound NaBaEr(BO3)2 (NBEB) presents a unique case where the Er $^{3+}$ ion possesses a large magnetic moment and strong spin-orbit coupling, yet its low-temperature magnetic properties and ground state had remained largely unexplored, with previous studies reporting no thermodynamic transitions down to 0.35 K.

Methodology
The authors employed a comprehensive multi-technique approach to investigate the structural, thermodynamic, and magnetic properties of NBEB:

Synthesis and Characterization: Polycrystalline and single-crystal samples were synthesized via solid-state reaction and flux methods, respectively. To minimize neutron absorption, $^{11}$ B-enriched precursors were used for scattering experiments.
Structural Analysis: Neutron powder diffraction (NPD) was performed at room temperature to resolve the crystal structure, specifically distinguishing between the $R\bar{3}$ and higher-symmetry $R\bar{3}m$ space groups.
Thermodynamic Measurements: DC magnetic susceptibility, magnetization, and specific heat measurements were conducted on single crystals using MPMS and PPMS systems with dilution refrigerator inserts, covering temperatures down to 0.05 K.
Neutron Scattering:
- Inelastic Neutron Scattering (INS): Performed at the Institut Laue-Langevin (ILL) using various incident neutron energies (7.5–110 meV) to map CEF excitations and determine the ground-state doublet properties.
- Single-Crystal Neutron Diffraction: Conducted at the Australian Nuclear Science and Technology Organisation (ANSTO) to determine the magnetic ordering vector and moment orientation below the transition temperature.
Theoretical Modeling: The CEF Hamiltonian was modeled using Stevens operators for the $S_6$ local symmetry. Classical dipolar energy calculations were performed for the experimentally determined magnetic structure to evaluate the competition between CEF and dipolar anisotropies.

Key Contributions and Results

Structural Refinement ( $R\bar{3}$ vs. $R\bar{3}m$ ):
NPD data revealed that the previously proposed $R\bar{3}m$ model was insufficient. The data are significantly better described by the lower-symmetry $R\bar{3}$ space group. This model accounts for rotations of the ErO $_6$ octahedra relative to the mirror-constrained configuration. Specifically, the ErO $_6$ octahedra adjacent to Na-containing layers exhibit a substantial rotation angle ( $\phi \approx 8.06^\circ$ ), while those near Ba-containing layers show minimal rotation ( $\phi \approx 0.43^\circ$ ). This structural distortion reduces the local point group symmetry from $D_{3d}$ to $S_6$ , which is crucial for the subsequent CEF analysis.
Effective Spin-1/2 Description and CEF Anisotropy:
INS measurements identified a well-isolated Kramers ground-state doublet separated from the first excited CEF level by approximately 4 meV (~46 K). This large energy gap, combined with the recovery of magnetic entropy close to $R \ln 2$ at low temperatures, supports an effective spin-1/2 description for the low-temperature magnetism. Magnetization data confirm a pronounced easy-axis anisotropy, with the $c$ -axis as the easy direction and a saturation moment of 4.26 $\mu_B$ /Er $^{3+}$ along this axis.
Magnetic Ground State and Ordering:
Below a Néel temperature ( $T_N$ ) of 0.10 K, the system undergoes a magnetic phase transition. Single-crystal neutron diffraction reveals an A-type antiferromagnetic order with a propagation vector $\mathbf{k} = (0, 0, 3/2)$ . Crucially, despite the strong paramagnetic easy-axis anisotropy, the ordered magnetic moments are found to lie predominantly within the $ab$ plane. The refined in-plane ordered moment is 0.9(1) $\mu_B$ /Er $^{3+}$ at 57 mK. The $c$ -axis component is constrained to be less than 0.35 $\mu_B$ /Er $^{3+}$ . The magnetic order is fragile, being suppressed by a small in-plane magnetic field of ~20 mT.
Competition between CEF and Dipolar Anisotropies:
The paper identifies a striking contradiction: the CEF ground state favors alignment along the $c$ -axis, yet the ordered moments align in the $ab$-plane. The authors calculate the energy scale of long-range dipolar interactions to be approximately 74 mK, which is comparable to $T_N$ and the in-plane Curie-Weiss temperature. Classical dipolar energy calculations for the observed A-type AFM stacking pattern demonstrate that dipolar interactions energetically favor an in-plane moment orientation ( $\theta = 90^\circ$ ). Furthermore, the calculations show that the strong easy-axis CEF anisotropy actually amplifies the dipolar preference for the in-plane orientation in this specific stacking geometry.

Significance
The paper establishes NaBaEr(BO3)2 as a rare-earth TLAF where distinct sources of magnetic anisotropy—local CEF effects and long-range dipolar interactions—compete on an exceptionally small energy scale. The primary significance lies in demonstrating that long-range dipolar interactions can overcome and reverse the moment orientation dictated by strong local CEF anisotropy. Specifically, the study reveals that for the A-type AFM structure observed, dipolar interactions favor the in-plane orientation, providing a plausible explanation for the counterintuitive ground state. This work highlights the necessity of considering dipolar interactions alongside CEF effects when determining the magnetic ground states of rare-earth frustrated magnets with large magnetic moments.

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