$d$-Wave Surface Altermagnetism in Centrosymmetric Collinear Antiferromagnets

Abstract

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.

Figures (3)

• Figure 1: Bulk and surface magnetic structures and symmetry conditions. (a) Bulk magnetic configurations of the centrosymmetric AFMs V$_3$Al and BaMn$_2$Sb$_2$ (G-type) and MnPt (C-type). All bulk phases preserve combined$P T$ symmetry, enforcing spin-degenerate electronic bands in the absence of spin--orbit coupling. Arrows indicate magnetic moment orientations. (b) Top view of the surface and subsurface layers at a representative surface termination. The surface is taken along the (001) direction of the bulk unit cell, for which each atomic layer contains two antiferromagnetically coupled magnetic atoms. For V$_3$Al and BaMn$_2$Sb$_2$, the surface layers form a checkerboard arrangement that breaks the equivalence of the two antiferromagnetically coupled surface sublattices ($A\neq B$), thereby breaking both $PT$ and $tT$ symmetry. Although a fourfold rotation combined with time reversal ($C_4T$) remains, this symmetry does not enforce spin degeneracy, allowing a symmetry-permitted nonrelativistic d-wave altermagnetic spin splitting, schematically indicated by the surface Fermi surfaces. In contrast, the MnPt surface preserves $tT$ symmetry despite broken inversion and therefore remains spin-degenerate. The indicated symmetry operations summarize the distinct symmetry constraints in each case.
• Figure 2: Surface altermagnetic electronic structure of centrosymmetric AFMs. (a) Bulk and surface band structures of the G-type AFM V$_3$Al calculated along the in-plane X--$\Gamma$--Y high-symmetry path. The surface band structure is obtained from a 20-layer slab, while the bulk reference is calculated using a corresponding 20-layer bulk unit cell to enable direct comparison. The bulk bands are strictly spin-degenerate due to combined $PT$ symmetry, whereas the surface bands exhibit pronounced nonrelativistic spin splitting in surface-localized states. (b) Same as (a), but for the G-type AFM BaMn$_2$Sb$_2$. (c), (d) Spin-resolved two-dimensional Fermi surfaces of the full slabs for V$_3$Al and BaMn$_2$Sb$_2$, respectively. The characteristic four-lobe angular dependence reflects the d-wave symmetry of the altermagnetic surface states.
• Figure 3: Layer-resolved surface altermagnetic states in V$_3$Al. Fat-band representations of the asymmetric 20-layer V$_3$Al slab projected onto the first three magnetic V layers, shown along the in-plane X--$\Gamma$--Y path. The surface magnetic layer (Layer 1) hosts pronounced nonrelativistic d-wave altermagnetic spin splitting near the Fermi level with dominant spectral weight. The same split surface-derived states persist in the subsurface layers (Layers 2 and 3) with rapidly decreasing weight.