3D bulk-resolved $g$-wave magnetic order parameter symmetry in the metallic altermagnet CrSb

Technical Summary: 3D Bulk-Resolved $g$-Wave Magnetic Order Parameter Symmetry in the Metallic Altermagnet CrSb

Problem and Context
The identification of order parameter symmetry is fundamental to distinguishing electronic phases of matter. While unconventional superconductors are characterized by nodal gap functions (e.g., $d$-wave), determining the symmetry of unconventional magnetic order parameters has remained experimentally challenging. Recently, the class of "altermagnets" was proposed: collinear, magnetically compensated materials that break time-reversal and lattice-translation symmetries individually but preserve a combined symmetry involving point-group operations (such as screw rotations). This symmetry breaking lifts Kramers spin degeneracy in momentum space, leading to momentum-dependent spin splitting ($\Delta(\mathbf{k})$) despite zero net magnetization. While numerous altermagnetic candidates exist in theory, robust experimental confirmation of bulk altermagnetic ordering and the determination of their specific order parameter symmetries (e.g., $d$-, $g$-, or $i$-wave) in metallic systems has been elusive. Previous candidates like RuO$_2$ faced controversy regarding the bulk nature of their magnetic order, highlighting the need for bulk-sensitive diagnostic tools.

Methodology
The authors employ magnetic quantum oscillation (QO) measurements, specifically the de Haas-van Alphen (dHvA) effect, to probe the bulk electronic structure of the metallic altermagnet candidate CrSb.

• Sample Preparation: High-quality single crystals of CrSb were grown via chemical vapor transport, exhibiting low residual resistivities ($\sim 2 \, \mu\Omega$cm) and high residual resistivity ratios (RRR $\approx 28$).
• Experimental Setup: Torque magnetometry was performed using cantilever beam magnetometry at the National High Magnetic Field Laboratory (NHMFL) in fields up to 41.5 T and temperatures down to 0.4 K. Additionally, contactless resistivity measurements using a proximity detector oscillator (PDO) were conducted in pulsed fields up to 65 T at the Dresden High Magnetic Field Laboratory (HLD).
• Angular Dependence: The magnetic field orientation ($\mathbf{H}$) was systematically rotated through high-symmetry nodal planes (e.g., $c-a$) and antinodal planes (e.g., $c-ab$), as well as a tilted low-symmetry plane. This allowed for the mapping of the spin-splitting function $\Delta(\theta, \phi)$ across the Brillouin zone.
• Analysis: The oscillatory component of the magnetic torque ($\Delta\tau$) was isolated via background subtraction (LOESS and polynomial fitting). Fast Fourier Transform (FFT) spectra were used to extract oscillation frequencies, which are proportional to the extremal cross-sectional areas of the Fermi surface.
• Theory: Density Functional Theory (DFT) calculations were performed using the WIEN2k code to simulate the Fermi surface and spin-splitting profiles, which were then compared against experimental data.

Key Results

1. Observation of Spin-Splitting: In nodal planes (e.g., $\mathbf{H} \parallel a$), the QO spectra exhibit a single frequency peak, indicating spin-degenerate Fermi surface sheets enforced by symmetry. In contrast, in antinodal planes (e.g., $\mathbf{H} \parallel ab$), the single peak splits into two distinct frequencies as the field is rotated slightly away from the nodal orientation. This splitting (up to $\sim 0.6$ kT for small rotations) corresponds to non-degenerate spin-up and spin-down Fermi sheets.
2. Mapping the Order Parameter: By tracking the frequency splitting across different rotation planes, the authors mapped the angular dependence of the spin splitting. The data reveals four nodal planes where splitting vanishes: three defined by azimuthal angles $\phi = 0^\circ, 60^\circ, 120^\circ$ and one by the polar angle $\theta = 90^\circ$.
3. Symmetry Identification: The observed angular profile of the spin splitting, $\Delta(\theta, \phi)$, matches the symmetry of the real spherical harmonic $Y^{-3}_4 \propto zy(3x^2 - y^2)$. This corresponds to a $g$-wave symmetry ($l=4$), analogous to the $g$-orbital in atomic physics. The splitting function transforms as the $B_{1g}$ irreducible representation of the $D_{6h}$ point group.
4. Effective Mass and Energy Splitting: Lifshitz-Kosevich analysis of the temperature dependence of the QO amplitudes yielded effective cyclotron masses of $m^*_1 = 1.97(5)m_e$ and $m^*_2 = 2.09(5)m_e$. The ratio of these masses corresponds to the square root of the frequency ratio, confirming that the two frequencies arise from spin-split daughters of the same parent Fermi sheet. The maximum energy splitting at the Fermi level is estimated to be $\sim 25$ meV.
5. Fermi Surface Topology: The primary Fermi surface sheet is identified as a closed, hole-like "dogbone" pocket. While DFT initially predicted open cylindrical sheets, the experimental observation of oscillations along the $c$-axis necessitates a closed topology, which was reconciled by shifting band edges in the simulation.

Significance and Claims
The paper claims to provide the first direct, bulk-sensitive, 3D mapping of the order parameter symmetry in an unconventional magnet. By demonstrating that the spin-splitting profile in CrSb follows a $g$-wave symmetry, the work conclusively establishes CrSb as a prototypical metallic altermagnet. The study validates magnetic quantum oscillations as a powerful diagnostic tool for determining the symmetry of unconventional magnetic order parameters, a task that has proven difficult in other systems like unconventional superconductors. The authors emphasize that the pristine, low-resistivity nature of their CrSb crystals makes the material a viable candidate for future spintronic applications, leveraging its ability to generate spin currents without stray magnetic fields. The work does not propose new applications but rather empirically grounds the potential of altermagnets by confirming their fundamental symmetry properties in a high-quality metallic system.