Researchers at the University of Cambridge and Helmholtz-Zentrum Dresden-Rossendorf have mapped the internal magnetic structure of chromium antimonide (CrSb) with detail, revealing an altermagnetic order parameter. This unconventional magnetism, where spin orientations are non-degenerate around the Fermi surface, challenges traditional classifications of magnetic materials. The work demonstrates that magnetic quantum oscillation measurements provide a high-resolution, 3D understanding of the order parameter, establishing CrSb as a prototypical metallic altermagnet; the exchange splitting between up and down spins follows the profile of a real spherical harmonic, analogous to a g-orbital of the hydrogen atom.
3D Mapping Reveals Altermagnetic Order in CrSb
Magnetic quantum oscillation measurements have revealed the g-wave symmetry of the altermagnetic order parameter within chromium antimonide (CrSb), providing a three-dimensional understanding of its internal magnetic structure. The work, conducted by researchers at the Cavendish Laboratory, University of Cambridge, and the Helmholtz-Zentrum Dresden-Rossendorf, establishes a direct link between quantum oscillation frequency spectra and the underlying magnetic order, mapping its k-dependence as Δ(k). This mapping is justified by considerations detailed in the Supplementary Information, which also accounts for electronic correlations within the material.
The study demonstrates a novel application of quantum oscillation measurements as a diagnostic tool for ascertaining order parameter symmetry in unconventional metallic magnets. The team equated a quantum oscillation frequency to a reciprocal space area, utilizing the Onsager relation to establish that F(E) = ℏ/(2π e)A(E).
Further analysis, detailed in the published work, determined that the energy difference associated with frequency splitting of two bands can be calculated as Δ E~Δ FdE/dF = eℏ/m_cΔ F, allowing for precise determination of the energy scales involved in the magnetic ordering. By examining the rate of change of orbital area with energy, researchers calculated the cyclotron mass, , revealing information about the band structure and spin-splitting within CrSb. This approach offers a bulk-sensitive method for probing the magnetic order, surpassing limitations of surface-sensitive techniques.
This connection extends the concept of nodal symmetry, typically found in these superconducting systems, to a magnetic material, suggesting a previously unobserved relationship between the two areas of condensed matter physics. “Our study thereby demonstrates the sensitivity of QO measurements to spin-split band structures,” the authors write, emphasizing the power of their technique. The findings establish CrSb as a magnetic system where up- and down-spin species are non-degenerate around the Fermi surface.
Quantum Oscillations Map CrSb’s Order Parameter
This bulk-sensitive technique provides a high-resolution understanding of the material’s internal magnetic structure, extending concepts of nodal symmetry, typically observed in superconductors, to a magnetic system. The detailed analysis, as presented in the supplementary information, explicitly accounts for electronic correlations, justifying the simplified symmetry picture employed. The symmetry of the observed magnetic order parameter, Δ(θ,φ), closely matches that of a real spherical harmonic, specifically Y₄⁻³(θ,φ) ∝ P₄³(cosθ) sin 3φ, where Pℓᵐ(x) represents the associated Legendre polynomial.
This harmonic transforms as the B₁(g) irreducible representation of the D₆h point group of CrSb, and can be expressed in Cartesian coordinates as yz(3x²-y²) or, in spherical polars, as r⁴ sin³(θ) cos(θ) sin(3φ). The authors state, highlighting the power of this technique for characterizing complex magnetic materials. The ability to map the order parameter in three dimensions provides a new framework for understanding altermagnetic systems. This connection suggests a previously unobserved relationship between unconventional superconductivity and magnetism, potentially opening new avenues for materials design and discovery.
CrSb Exhibits g-Wave Altermagnetic Spin-Splitting
Quantum oscillation measurements confirm a specific symmetry within the magnetic order of chromium antimonide, revealing a g-wave altermagnetic spin texture previously suggested by surface-sensitive studies of other materials. While photoemission spectra of ruthenium oxide and KV₂Se₂O₄ initially hinted at altermagnetic spin-splitting, subsequent bulk measurements questioned whether those materials truly exhibited this unconventional magnetism; CrSb’s behavior, now mapped in three dimensions, offers a clearer case.
Researchers established how the spin-split quantum oscillation frequency profile of its primary Fermi pocket evolves under rotation to understand the unconventional magnetic order parameter symmetry of CrSb. The team mapped the evolution of quantum oscillation frequency profiles across different rotation planes to establish the location of nodal planes.
The g-wave symmetry profile, visually depicted as a real spherical harmonic, characterizes the altermagnetic spin-splitting in CrSb, and the azimuthal angle φ is defined as the inclination from a to ab in the hexagonal basal plane. The polar angle θ is measured from the c axis towards the basal ab plane, providing a complete spatial description of the magnetic order. This work firmly establishes CrSb as a material exhibiting k-dependence of the spin-splitting per Fermi pocket, Δ(k), and provides a detailed understanding of its unconventional magnetic order.
Momentum-Dependent Spin-Splitting Defines Altermagnetic Order
The energy difference between spin-up and spin-down bands in chromium antimonide (CrSb) can be estimated from quantum oscillation frequency splitting, revealing a value of approximately 0.41 kT. This calculation, detailed in recent work by Mengmeng Long of the University of Cambridge and colleagues, links observed frequencies directly to the material’s altermagnetic spin-splitting, a departure from traditional magnetism where spin orientations are typically degenerate.
The team utilized the Onsager relation to translate frequency differences into a difference in orbital areas, and explored how this relates to the spin polarization within CrSb. By relating quantum oscillation frequency to reciprocal space area, the researchers established a connection between the experimental data and the underlying altermagnetic order parameter Δ(k), which describes how spin-splitting varies with momentum.
Nodal Planes Bisect Spin-Split Fermi Surfaces in CrSb
CrSb exhibits nodal planes within its spin-split Fermi surface, a configuration identified through detailed magnetic quantum oscillation measurements. The presence of these nodal planes is analogous to the behavior seen in unconventional superconductors, where similar nodes appear in the energy gap, suggesting a previously unobserved connection between magnetism and superconductivity. Analysis of Δτ curves and corresponding Fast Fourier Transforms confirmed the observed spin-splitting. This detailed analysis provides a high-resolution, three-dimensional understanding of the magnetic structure within CrSb, demonstrating the sensitivity of quantum oscillation measurements to spin-split band structures, as the team previously noted.
Distinguishing Bulk Altermagnetism from Surface Effects
Surface-sensitive measurements alone can be misleading when characterizing unconventional magnetism; studies of ruthenium dioxide and potassium vanadium selenoxide revealed apparent altermagnetic signatures at surfaces that subsequent bulk-sensitive experiments refuted as arising from topological surface states or conventional antiferromagnetism within the material’s interior. These instances underscore the necessity for probing the intrinsic magnetic properties of candidate altermagnets using techniques that extend beyond surface analysis, a challenge the current work addresses with CrSb.
Analysis of data collected with magnetic fields oriented along different planes revealed frequencies. The authors state, highlighting the discrepancy between theoretical predictions and experimental results. A further quantum oscillation was also clearly detected, but the absence of predicted frequency-splitting at high fields, up to 64 tesla, suggests a robust altermagnetic order, maintained even at substantial magnetic field strengths and high temperatures. The material’s properties appear to prevent this transition.
Y_4^(-3) Spherical Harmonic Characterizes CrSb’s Symmetry
By rotating a magnetic field and observing resulting quantum oscillations, scientists mapped the spin-splitting of CrSb’s Fermi sheets, revealing a reduction in symmetry for each spin species away from nodal orientations in momentum space. This precise mapping of magnetic order parameter symmetry, achieved through quantum-oscillatory quasiparticle spectroscopy, establishes CrSb as a primary example of a g-wave metallic altermagnet.
Defining real spherical harmonics, which vary in magnitude as a function of φ, allows for a mapping of the harmonic to the g-wave symmetry profile of the altermagnetic order parameter in CrSb, a technique useful because complex spherical harmonics are defined up to a phase factor and do not change with φ. The work establishes a new benchmark for understanding momentum-dependent spin-splitting and provides a framework for characterizing unconventional magnetic materials using quantum oscillation measurements, a technique that reveals the intricate interplay between symmetry and magnetism at the quantum level.
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