Anion correlation induced nonrelativistic spin splitting in rutile antiferromagnets

Many studies of non-relativistic spin-splitting (NRSS), or altermagnetism, have focused on idealized, perfectly ordered crystals, relying on symmetry-based approaches to identify candidate materials. Here, we theoretically investigate how local short-range ordering (SRO) influences NRSS of energy bands in partially ordered collinear antiferromagnetic iron oxyfluoride (FeOF). Using the cluster expansion method, we identify four nearly degenerate structures (energy difference $\leq 8$ meV per formula unit) that represent distinct snapshots of local plane-to-plane O/F correlations. Our density functional theory (DFT) results show robust NRSS along the $Γ$-M direction in all four structures, despite the absence of long-range order. The magnitude and character of the splitting depend sensitively on the specific direction of anion correlations, effects that are not fully captured in high-symmetry average structures. Notably, two configurations ($Pmc2_1$ and $Pm$) exhibit $Γ$-point spin splitting absent in ordered FeF$_2$ and a virtual crystal approximation model of FeOF. We further predict distinct magneto-optical Kerr effect (MOKE) signatures, enabling experimental detection of SRO-driven electronic structure changes. These results highlight heteroanionic compounds as a promising design space for NRSS antiferromagnets, with experimentally synthesized FeOF already exhibiting a substantially higher Néel temperature (315,K) than FeF$_2$ (79,K).

💡 Research Summary

This paper investigates how short‑range ordering (SRO) of O and F anions influences non‑relativistic spin splitting (NRSS), also known as altermagnetism, in the rutile antiferromagnet FeOF. While most previous theoretical works on NRSS have assumed perfectly ordered crystals and relied on symmetry analysis to identify candidate materials, real compounds often contain substitutional disorder that can break local symmetries and modify electronic structure. FeOF is an ideal test case because it adopts the rutile framework of FeF₂ but hosts a mixed O/F sublattice that experimentally exhibits strong O/F correlations confined to {110} planes, without long‑range order.

The authors combine a cluster expansion model (CEM) with density‑functional theory (DFT) to capture the energetics of O/F configurations. They generate 238 symmetrically distinct Fe–(O/F) supercells (up to 3 × 1 × 1) and fit a LASSO‑regularized CEM containing 11 non‑zero effective cluster interactions, achieving a cross‑validation error of 0.013 eV per site. Canonical Monte‑Carlo simulated‑annealing runs on supercells ranging from 2 × 2 × 2 to 20 × 20 × 20 reveal four nearly degenerate low‑energy structures (energy spread ≤ 8 meV per formula unit). These structures belong to space groups Pmn2₁, P4₂/m, Pmc2₁ and Pm and differ in the relative stacking of O‑ and F‑rich layers along the c‑axis.

For each SRO configuration the authors perform spin‑polarized DFT calculations (VASP, PAW, PBESol, 650 eV cutoff, dense Γ‑centered k‑meshes) and unfold the resulting band structures onto the primitive rutile Brillouin zone. All four SRO structures display robust momentum‑dependent spin splitting along the Γ–M direction (the (110) plane), with magnitudes ranging from ~150 meV up to ~432 meV near the Fermi level—significantly larger than the ~159 meV splitting found in ordered FeF₂. Importantly, the Pmc2₁ and Pm configurations also exhibit a spin splitting at the Γ point, a feature absent in both perfectly ordered FeF₂ and the virtual crystal approximation (VCA) model of FeOF, where the mixed O/F site is replaced by an averaged pseudopotential preserving the P4₂/mnm symmetry.

The authors attribute these differences to the symmetry breaking introduced by specific O/F stacking patterns. In the ideal rutile lattice, combined translation and inversion operations together with time‑reversal (Θ) protect spin degeneracy on certain high‑symmetry lines and points. The SRO patterns remove key symmetry elements (e.g., Θ·C₄, Θ·Mₓ) that connect the two spin‑structure‑motif‑pairs (SSMPs), thereby allowing the spin bands to separate even at Γ. Moreover, the presence of O, which is larger and more electronegative than F, enhances Fe–X hybridization, strengthens super‑exchange pathways, and raises the calculated magnetic moment on Fe³⁺ to ~3.5 µB, consistent with an increased Néel temperature (experimentally 315 K for FeOF versus 79 K for FeF₂).

To provide an experimentally accessible probe of the SRO‑induced electronic changes, the authors compute magneto‑optical Kerr effect (MOKE) spectra using the independent‑particle Green‑Kubo formalism. By evaluating the antisymmetric part of the dielectric tensor (ε_xy) for opposite magnetization directions, they extract Kerr rotation angles θ_K(ω). The simulated spectra show distinct peak positions and amplitudes for each SRO configuration, demonstrating that MOKE can serve as a fingerprint of the underlying anion ordering.

Overall, the study establishes three key messages: (i) NRSS survives and can even be enhanced in the absence of long‑range order; (ii) specific short‑range anion correlations provide a powerful knob to tune the magnitude, momentum dependence, and even the presence of Γ‑point spin splitting; (iii) optical probes such as MOKE can directly detect these SRO‑driven modifications. By highlighting FeOF—a material already synthesized with a high Néel temperature—the work opens a broader design space of heteroanionic rutile compounds for altermagnetic spintronics, suggesting future directions that include experimental MOKE verification, strain or doping engineering to further amplify NRSS, and exploration of other mixed‑anion systems (e.g., Cl⁻, Br⁻) to tailor spin‑splitting characteristics.