$d$-Wave Surface Altermagnetism in Centrosymmetric Collinear Antiferromagnets

Broken inversion symmetry at the surfaces of centrosymmetric collinear antiferromagnets lifts the combined inversion and time-reversal symmetry ($PT$) and can generate nonrelativistic d-wave spin splitting, termed surface altermagnetism. Combining symmetry analysis with first-principles calculations, we show that surface inversion breaking, while necessary, is not sufficient for this effect. Surface altermagnetism emerges only when the surface termination simultaneously breaks both $PT$ and translation–time-reversal symmetry ($tT$), thereby inducing magnetic sublattice inequivalence between antiferromagnetically coupled surface moments. We demonstrate this mechanism explicitly for the G-type antiferromagnets V$_3$Al and BaMn$_2$Sb$_2$, and show that the same symmetry criterion applies broadly across distinct structural families of centrosymmetric antiferromagnets. These results establish a general, symmetry-based route to realizing robust, exchange-driven spin polarization at antiferromagnetic surfaces and interfaces.

💡 Research Summary

The manuscript investigates a previously unexplored route to generate non‑relativistic spin splitting at the surfaces of centrosymmetric collinear antiferromagnets (AFMs). In bulk, such materials preserve the combined inversion‑time‑reversal symmetry (PT) and therefore exhibit spin‑degenerate bands when spin–orbit coupling (SOC) is neglected. The authors demonstrate that merely breaking inversion at a surface is insufficient to lift this degeneracy. A second symmetry, translation‑time‑reversal (tT), must also be broken. tT is a symmetry operation that combines a half‑lattice translation with time reversal; when it is present, the two magnetic sublattices (spin‑up and spin‑down) remain equivalent in the surface plane, enforcing spin degeneracy even though PT is absent. Consequently, surface altermagnetism—characterized by a d‑wave momentum‑dependent spin splitting—appears only when both PT and tT are simultaneously absent, i.e., when the surface termination makes the two antiferromagnetically coupled sublattices inequivalent.

To substantiate this symmetry criterion, the authors perform density‑functional theory (DFT) calculations within the generalized‑gradient approximation, explicitly neglecting SOC to isolate the exchange‑driven effect. They study (001) slabs of three representative materials: the G‑type Heusler alloy V₃Al, the G‑type 122 pnictide BaMn₂Sb₂, and the C‑type metallic AFM MnPt. In V₃Al and BaMn₂Sb₂ the surface consists of chemically distinct sites (A and B), which destroys the half‑translation symmetry and thus breaks tT. In MnPt, by contrast, the surface sites are equivalent (both Pt atoms), preserving tT despite the loss of inversion symmetry.

The calculated band structures reveal pronounced non‑relativistic spin splittings localized at the outermost magnetic layer. In V₃Al, Dirac‑like surface bands shift toward the X and Y points and acquire a spin splitting up to ~0.32 eV, forming metallic surface states with a characteristic four‑lobed d‑wave angular dependence. BaMn₂Sb₂, a small‑gap semiconductor, shows two surface Dirac points ~20 meV above the Fermi level along Γ–X and Γ–Y; each direction carries opposite spin polarization, yielding a spin splitting of ~0.35 eV. Spin‑resolved two‑dimensional Fermi surfaces display alternating spin lobes, directly visualizing the d‑wave symmetry imposed by the surface magnetic point group. Layer‑resolved fat‑band analysis confirms that the split states are strongly confined to the top magnetic layer, with rapidly diminishing weight in subsurface layers, while deeper layers revert to bulk‑like spin degeneracy.

The authors compile a broader materials survey (Table I and supplemental material) showing that the simultaneous breaking of PT and tT is the universal prerequisite for surface altermagnetism across diverse structural families (Heuslers, pnictides, perovskites, etc.). Materials where tT remains intact—such as MnPt, LaFeO₃, and K₂NiF₄—exhibit no surface spin splitting despite inversion breaking.

By establishing that the spin splitting originates purely from exchange interactions rather than SOC, the work distinguishes surface altermagnetism from Rashba/Dresselhaus effects and positions it as a robust, symmetry‑controlled phenomenon. The findings open new avenues for spintronic device engineering: interfaces or heterostructures can be designed to deliberately break tT, enabling controllable, momentum‑dependent spin polarization without relying on heavy elements or strong SOC. Potential applications include spin filters, non‑volatile magnetic memory elements, and novel spin‑orbit‑free spin‑torque mechanisms.

In summary, the paper provides a clear symmetry‑based framework for realizing d‑wave surface altermagnetism in centrosymmetric collinear AFMs, validates the concept with first‑principles calculations on V₃Al and BaMn₂Sb₂, and delineates the essential role of translation‑time‑reversal symmetry breaking. This work expands the catalog of altermagnetic materials to include surface and interface phenomena, offering a versatile platform for exchange‑driven spin polarization in future spintronic technologies.