Researchers have, for the first time, fully characterized a phenomenon that obscures a fundamental point in the study of magnetism, the ferromagnetic quantum critical point, within the metallic compound NbFe2. Using facilities Laue Langevin in Grenoble, France and the Isis Neutron Source in Didcot, UK, the team demonstrated that this wave arises from the arrangement of ferrimagnetic building blocks with antiparallel ferromagnetic sheets. This discovery reveals an unconventional magnetic order emerging from underlying quantum criticality, a phenomenon important for understanding non-Fermi liquid states and unconventional superconductivity.

NbFe₂ Exhibits Masked Ferromagnetic Quantum Criticality

The first fully refined example of a longitudinal spin-density wave obscuring a ferromagnetic quantum critical point has been observed in the compound NbFe₂ by an international team of scientists. This masking effect, where a modulated magnetic order hides a fundamental transition, is rare and complicates the search for elusive quantum critical points in materials. Researchers determined the structure of this wave by meticulously varying the iron concentration within a narrow range, allowing them to probe the low-temperature threshold of ferromagnetism with high precision.

Detailed analysis using spherical neutron polarimetry and high-intensity single-crystal neutron diffraction revealed the spin-density wave is characterised by a large-wavelength, incommensurate modulation of a low average moment. The structure is built from ferrimagnetic building blocks, consisting of antiparallel ferromagnetic sheets stacked along the material’s easy c-axis.

These sheets create a unique arrangement where neighbouring iron layers align oppositely, differing in moment size, and the niobium sublattice also exhibits magnetic order aligned with the iron lattice. Models proposing moments exclusively within the ab-plane failed to adequately fit the experimental data, with refinement of the Γ₄ representation yielding significantly poorer results.

The unconventional nature of this spin-density wave stems from the cancellation of magnetisation occurring only over mesoscopic length scales of 50 to 90 Å. This limited cancellation demonstrates a local similarity between the spin-density wave and the ferromagnetic phase from which it emerges, suggesting a direct link to underlying ferromagnetic quantum criticality. Large single crystals of NbFe₂ with compositions adjusted across the iron-rich side of the homogeneity range were grown using an ultra-high vacuum optical floating zone furnace, starting from polycrystals prepared by induction melting.

This precise control over composition allowed researchers to identify the masked quantum critical point. The exploration of ferromagnetic quantum phase transitions has previously led to the discovery of non-Fermi liquid states and unconventional superconductivity, but the question of whether a clean band magnet can host a true ferromagnetic quantum critical point remains contentious.

This research on NbFe₂ provides a rare, fully refined example of a system where such a point is masked, offering new insights into the complex interplay between magnetism and quantum criticality in metallic materials. The team’s findings suggest that the modulated order in NbFe₂ is an intrinsic property of the material, emerging from the delicate balance between ferromagnetic tendencies and the formation of the spin-density wave.

Longitudinal Spin-Density Wave Characterized by Neutron Polarimetry

High-intensity single-crystal neutron diffraction refined a longitudinal spin-density wave within the metallic magnet NbFe₂, revealing a magnetic order that obscures an essential ferromagnetic quantum critical point. This detailed characterization, a rare achievement in the field, employed spherical neutron polarimetry to exclude previously proposed spiral or transverse spin arrangements, establishing the wave’s unique longitudinal nature. The spin-density wave is characterised by a moment of 0341 μB per atom, a value determined through precise neutron scattering analysis.

Researchers identified candidates for the spin-density wave’s irreducible representations using the SARAh Refine program, based on the material’s crystal structure and its SDW propagation vector. The paper details the structural findings. While the current data strongly suggests a ferromagnetic parent phase, a direct observation of that structure will likely require polarised neutron diffraction. The discovery builds on previous work showing non-Fermi liquid behavior in NbFe₂ at low temperatures, a phenomenon often associated with quantum criticality and, potentially, unconventional superconductivity.

Incommensurate Modulation Defines SDW Ordering Wavevector

This detailed refinement was possible despite the material exhibiting a relatively weak average magnetic moment of 0. The ordering wavevector, measured as (0 0 0. 156) in a nearly stoichiometric sample, and (0 0 0. 10) in a slightly iron-rich sample, demonstrates a dependence on both temperature and iron concentration. Researchers examined two samples, one with a Néel temperature (T_N) of 15. 3 K and another with $T_{N}$ at 32. 5 K, alongside a Curie temperature (T_C) of 19 K, to observe these changes in the SDW’s characteristics.

Refinement results yielded goodness-of-fit parameters with R values below 15%, indicating a robust and reliable model of the magnetic structure. The average magnitude of the net unit-cell magnetic moment was calculated to be 0. 0249 ± 0. 0062 μB/atom, a value that aligns with measurements obtained through muon-spin rotation (μSR) techniques.

This local finite net magnetisation within the crystal-lattice unit-cell building block is not fully compensated, but instead cancelled only over mesoscopic length scales of 50-90 Å, a phenomenon that mimics ferromagnetic behavior at microscopic scales. The researchers excluded topological frustration as a primary driver of this behavior, noting that its role tends to be reduced in itinerant magnets like NbFe₂.

The ability to fully refine the magnetic structure, despite the weak magnetic moments, represents an advance in characterizing these complex magnetic systems and understanding the interplay between modulated magnetic order and quantum phase transitions. The full magnetic structure of the spin-density-wave phase was revealed through the combined use of these advanced neutron scattering techniques, providing a detailed picture of the magnetic arrangement within a crystal-lattice unit cell.

Fe Concentration Impacts SDW & FM Phase Boundaries

Spherical neutron polarimetry revealed that the spin-density wave phase in NbFe₂ arises from magnetic moments aligned along the easy c-axis, a finding important to understanding the material’s complex magnetic order. Detailed analysis of scattering factors obtained from two independent numerical methods, one analytical, demonstrated agreement within error margins, bolstering the reliability of the structural refinement process.

These factors, derived from measurements at 25K within the spin-density wave state and confirmed by background measurements at 50K in the paramagnetic phase, were essential for characterizing the magnetic structure. The refinement process considered four irreducible representations for the Fe₂ₐ site, six for Fe₆ₕ and a further four for each of the two niobium sites, all within the P6₃/mmc space group.

Constraints applied included the observation from spherical neutron polarimetry that the spin-density wave is formed by moments along the c-axis, and the existence of a single critical temperature separating the SDW and paramagnetic phases, indicating a common irreducible representation for all magnetic sites. Corrected spin-density wave peak intensities, adjusted for detector saturation and peak width, were then used with the FullProf software to achieve excellent agreement with the data, confirming the Γ₂ representation as the most accurate model.

This level of refinement is notable given the weak magnetic moments present in the material. The presence of ferromagnetic sheets within the ab-plane mirrors observations in TiFe₂, another Laves phase compound sensitive to stoichiometry. In TiFe₂, these sheets couple ferromagnetically, concentrating magnetic moments parallel to the c-axis.

However, unlike TiFe₂, assuming zero magnetic moment on the Fe₂ₐ sites in NbFe₂ resulted in poor refinement of neutron diffraction data, despite the small magnitude of those moments. Previous muon-spin rotation (μSR) data could not distinguish between ferromagnetic and ferrimagnetic order due to the location of the stopping site, while Mößbauer results have been interpreted as indicating ferromagnetic order.

Magnetisation measurements in Fe-rich samples of the parent phase detect an average magnetisation of approximately 0.06 μB per atom, a value that aligns with the refined magnetic structure. These findings demonstrate that the spin-density wave modulates a pre-existing ferromagnetic order, masking the underlying quantum critical point and revealing an unconventional magnetic order emerging from ferromagnetic quantum criticality.

Ferrimagnetic Building Blocks Reveal Unconventional SDW Nature

This masking effect is rare, as most modulated orders simply coexist with, rather than obscure, such critical points. The spin-density wave exhibits a large-wavelength incommensurate modulation along the material’s easy c-axis, a characteristic revealed through spherical neutron polarimetry which excluded previously proposed spiral or transverse SDW orderings. Despite a low average magnetic moment of 0.0341 μB/atom, the team achieved full refinement of the structure, a feat enabled by the intensity of the neutron beams used.

The resulting model shows the wave constructed from stacked, antiparallel ferromagnetic sheets. The existence of ferromagnetic sheets and cancellation of the magnetisation only over mesoscopic length scales show local similarity between the spin-density wave and the ferromagnetic parent phase and indicate the spin-density wave’s unconventional nature as emerging from underlying ferromagnetic quantum criticality.

Density functional theory calculations support the possibility of a ferrimagnetic ground state for NbFe₂, though several candidate magnetic states exist with minimal energy differences. The work demonstrates that NbFe₂’s modulated structure is an incommensurate long-wavelength spin-density wave, formed from these ferrimagnetic blocks stacked along the easy c-axis.

This research on NbFe₂ offers a new perspective, suggesting that such points can exist, but may be hidden by complex magnetic ordering. The refined understanding of this material’s magnetic structure provides an essential step toward unraveling the conditions necessary for realizing and characterizing these elusive quantum critical points in metallic magnets.

👉 More information
🗞 Spin-density wave of ferrimagnetic building blocks masking the ferromagnetic quantum critical point in NbFe₂
✍️ T. Poulis et al.
🧠 DOI: http://link.aps.org/doi/10.1103/lcrw-ptwv

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