Fermi-surface studies of altermagnetic CrSb from Shubnikov-de Haas oscillations

1. Problem Statement

Altermagnets (AMs) represent a newly predicted third class of magnetic materials, distinct from ferromagnets and antiferromagnets. They feature a compensated collinear magnetic structure coexisting with large, momentum-dependent spin splitting in their electronic bands, driven by unconventional spin-space group symmetries rather than spin-orbit coupling (SOC).

• Specific Challenge: While CrSb is identified as a prototypical room-temperature altermagnet with high spin-splitting energy (1.2 eV) and a high Néel temperature (700 K), its detailed bulk electronic structure remains incompletely understood.
• Gap in Knowledge: Previous studies at lower magnetic fields (up to ~41.5 T) reported Shubnikov-de Haas (SdH) oscillations and nontrivial Berry phases, but the detailed evolution of these oscillations with field orientation and temperature, and their precise mapping to the theoretical Fermi surface (FS), required higher field resolution and comprehensive angular analysis to confirm the altermagnetic nature of the band structure.

2. Methodology

The study employed a combination of high-field experimental magnetotransport measurements and first-principles density-functional-theory (DFT) calculations.

Experimental Approach:

• Sample Preparation: High-quality single crystals of CrSb were synthesized via chemical vapor transport. Microstructures were fabricated using Gallium-focused ion beam (FIB) lithography to create specific geometries for probing current flow along different crystallographic directions ($[2\bar{1}\bar{1}0]$, $[\bar{1}2\bar{1}0]$, and $[0001]$, labeled $a$, $a'$, and $c$).
• Measurement Conditions: Magnetotransport measurements (longitudinal and Hall resistivity) were conducted in both steady fields (up to 14 T) and pulsed fields (up to 68 T).
• Angular Dependence: Systematic measurements were performed by rotating the magnetic field relative to the crystal axes (specifically within the $a'-c$ and $a-c$ planes) to map the anisotropy of the Fermi surface.
• Data Analysis: SdH oscillations were extracted by subtracting a polynomial background. Fast-Fourier-transform (FFT) analysis was used to identify oscillation frequencies. The temperature dependence of oscillation amplitudes was analyzed using the Lifshitz–Kosevich (LK) formula to extract effective cyclotron masses ($m^*$).

Theoretical Approach:

• DFT Calculations: Electronic band structures and Fermi surfaces were calculated using two software packages: VASP (using the Projector Augmented Wave method) and FPLO (full-potential local-orbitals).
• Models: Calculations were performed for non-magnetic, altermagnetic (without SOC), and altermagnetic (with SOC) states. The study utilized the Local Spin-Density Approximation (LSDA) and Generalized Gradient Approximation (GGA).
• Comparison: Calculated extremal Fermi surface cross-sections and SdH frequencies were directly compared with experimental data to validate the altermagnetic model.

3. Key Contributions

• High-Field Mapping: The study extends SdH measurements to 68 T, significantly surpassing previous limits, allowing for the resolution of complex, multi-band features that are obscured at lower fields.
• Microstructure Fabrication: The successful fabrication of FIB-patterned microstructures enabled precise control over current orientation and contact geometry, facilitating the separation of anisotropic transport properties from geometric artifacts.
• Validation of Altermagnetism: By comparing experimental SdH frequencies against DFT models, the authors definitively confirmed that the observed spin splitting is a result of the altermagnetic order (collinear antiferromagnetism with broken Kramers degeneracy) rather than a non-magnetic or standard ferromagnetic state.
• Comprehensive Angular Analysis: The study provides a complete angular map of the Fermi surface, distinguishing between quasi-2D and 3D characteristics of different electron and hole pockets.

4. Key Results

• Transport Properties: CrSb exhibits a non-saturating magnetoresistance (reaching ~550% at 4.2 K) and a strongly nonlinear Hall effect, indicative of multiband transport with multiple carrier pockets of distinct mobilities.
• SdH Oscillations:
  • A dominant oscillation frequency ($F_1$) of approximately 1640 T was observed, corresponding to a Fermi surface cross-sectional area of $\approx 0.156$ Å$^{-2}$.
  • The effective mass for this orbit was determined to be $m^* \approx 1 m_e$ (free electron mass).
  • The amplitude of oscillations was observable up to ~12 K.
• Angular Dependence:
  • The frequency $F_1$ remained nearly constant as the magnetic field was tilted from the $c$-axis towards the $a$-axis, suggesting a spherical, three-dimensional Fermi surface shape for this specific orbit.
  • Other frequencies (e.g., ~250 T, ~600 T, ~2000 T) appeared at specific angles, indicating complex, anisotropic pockets.
• DFT Comparison:
  • Non-magnetic calculations failed to reproduce the experimental frequencies, confirming the necessity of magnetic order.
  • Altermagnetic calculations without SOC showed discrepancies in angular dependence.
  • Altermagnetic calculations with SOC (LSDA+SOC) provided an excellent match. The dominant experimental frequency ($F_1 \approx 1640$ T) corresponds to the calculated $\alpha^+_2$ branch (hole-like spin-split band).
  • The calculated effective mass ($0.78 m_e$) is slightly lower than the experimental value ($1 m_e$), which is attributed to many-body interaction enhancements not included in standard DFT.
• Band Structure Confirmation: The results confirm the existence of spin-split $\alpha$ (hole-like) and $\beta$ (electron-like) bands crossing the Fermi level, consistent with the predicted g-wave altermagnetic symmetry.

5. Significance

• Fundamental Physics: This work provides robust experimental evidence for the existence of altermagnetism in CrSb, validating theoretical predictions that unconventional spin-space symmetries can lift Kramers degeneracy without strong SOC.
• Methodological Benchmark: It demonstrates that extreme magnetic fields are essential for accurately mapping the Fermi surfaces of unconventional emergent materials, particularly those with complex, multi-band topologies.
• Material Potential: By confirming the high-quality, room-temperature altermagnetic state with substantial spin splitting, CrSb is reinforced as a prime candidate for future spintronic applications and for probing the interplay between magnetic order, symmetry, and band topology.
• Topological Insights: The observation of specific SdH frequencies and their agreement with topological band structure calculations supports the classification of CrSb as an altermagnetic Weyl semimetal, opening avenues for studying topological surface states and Berry phase effects in compensated magnetic systems.