(Anti-)Altermagnetism from Orbital Ordering in the Ruddlesden-Popper Chromates Sr$_{n+1}$Cr$_n$O$_{3n+1}$

Abstract

Altermagnets are collinear antiferromagnets with spin-split electronic states. We introduce Ruddlesden-Popper chromates Sr$_{n+1}$Cr$_n$O$_{3n+1}$ (including SrCrO$_3$) as candidate materials in which altermagnetism can emerge from spontaneous orbital ordering rather than crystal symmetry. First-principles calculations reveal a layer-dependent spin splitting: if the spin and orbital orders align in adjacent layers, the system exhibits non-relativistic spin splitting, and thus altermagnetism. In contrast, if either the spin or the orbital order is reversed in adjacent layers, we observe a layerwise uncompensated spin splitting, that is compensated in the adjacent layer, giving rise to the concept of anti-altermagnetism. In the RP series, odd $n$ members support coexistence of altermagnetism and anti-altermagnetism, whereas even $n$ and the perovskite limit are strictly anti-altermagnetic. In both cases, larger $n$ favors metallicity, and in odd $n$ compounds strain can further stabilize altermagnetism.

Figures (6)

• Figure 1: (a) Crystal field splitting of the Cr$^{4+}$ in an octahedral environment results in the splitting into $e_g$ and $t_{2g}$ manifolds. The symmetry breaking of the Ruddelsden-Popper geometry due to the spacers, lowers the energy of the $d_{xy}$ orbitals and splits the $t_{2g}$ manifold into $d_{xy}$ and $d_{xz/yz}$. The same situation can be promoted in the perovskite case via epitaxial strain. (b) The electronic interactions of the $d_{xz/yz}^1$ manifold further favor the formation of orbital ordering, where either $d_{xz}$ or $d_{yz}$ is occupied on neighboring Cr ions.
• Figure 2: (a) Spin and orbital order parameters of perovskite units (A) and (B) in layer $i$, (b) Calculated magnetization density of SrCrO$_3$ ($\rho_\uparrow-\rho_\downarrow$), with red (blue) for spin up (down), showing the $t\mathcal{R}\mathcal{T}$ symmetry connecting magnetic sublattices, (d) Calculated magnetization density for C-OO/C-AFM and G-OO/C-AFM order, (f) Calculated altermagnetic splitting (C-C) and (g) anti-altermagnetic (G-C) splitting and projected bands onto each perovskite layer. The isosurfaces are 0.02 $e^-\AA^{-3}$.
• Figure 3: Schematic of magnetic symmetry and band splitting for different magnetic systems: Ferromagnets show a net spin polarization. Antiferromagnets have spin-degenerate bands due to the translational symmetry between sublattices. Altermagnets non-relativistic spin splitting due to combined translation and rotation. Anti-altermagnets host multiple inequivalent sublattice pairs: each displays local altermagnetism, but an additional translation cancels the splitting globally.
• Figure 4: (a) Crystal structure of single-layer RP Sr$_2$CrO$_4$. (b–c) Isosurfaces (0.02 $e^-\AA^{-3})$ of the magnetic charge density (red = $\uparrow$, blue = $\downarrow$) reveals orbital ordering, yielding either an altermagnetic state (b) or an anti-altermagnetic state (c), where a translation connecting the two layers is recovered. This results in altermagnetic band splitting with $\mathcal{A}_0 \neq 0$ (d), or in anti-altermagnetic splitting with $\mathcal{A}_1 \neq 0$ (e), where the total splitting vanishes but projected bands on Cr atoms in different layers still show local spin polarization.
• Figure 5: Electronic band structures for different numbers of layers $n$ in Sr$_{n+1}$Cr$_n$O$_{3n+1}$: Compounds with odd numbers of layers $n$ exhibit altermagnetism (red and blue are spin-up and spin-down bands respectively), while compounds with even $n$ show degenerate spin-up and spin-down bands.