Classification and design of two-dimensional altermagnets

TL;DR

This article surveys two-dimensional altermagnets, a class of collinear antiferromagnets with zero net moment but nonrelativistic spin splitting governed by spin-group symmetry. It establishes a symmetry-based classification, catalogs monolayer candidates, and details strategies to engineer altermagnetism in 2D via stacking, multicomponent design, adsorption, electric fields, structural distortion, strain, and organic frameworks. The work highlights potential spintronic, valleytronic, and topological applications, including altermagnetically induced tunneling magnetoresistance and quantum anomalous Hall effects, while candidly addressing experimental and theoretical challenges such as ground-state determination and SOC effects. By providing design rules and a roadmap for synthesis and characterization, the paper aims to accelerate experimental realization and device integration of 2D altermagnets with high spin splitting and tunable functionalities.

Abstract

Altermagnets -- newly identified collinear antiferromagnets -- carry zero net moment with non-relativistic, spin-polarized bands, distilling the best of ferromagnets and antiferromagnets into a single spintronic platform. Shrunking to the two-dimensional limit, they inherit the tunability of two-dimensional crystals while adding symmetry-protected spin splitting, a combination now driving intense experimental interest. Here, we review the symmetry classification of two-dimensional altermagnets based on spin-group theory and survey the growing list of candidate materials, emphasizing those with large spin splitting for experimental realization. We then examine strategies for engineering two-dimensional altermagnetism. This Review aims to consolidate theoretically proposed candidate materials and realization strategies for two-dimensional altermagnets, providing insights for future experimental efforts in this emerging field.

Figures (9)

• Figure 1: Schematic illustration of the crystal structures and nonrelativistic band structures of three types of two-dimensional (2D) collinear magnets, and the advantages of two-dimensional altermagnets. (a) Ferromagnets possess a nonzero net magnetization and exhibit spin-split band structures. (b) and (c) Antiferromagnets consist of two magnetic sublattices with opposite spin orientations, resulting in a zero net magnetization. Depending on whether spin splitting appears in the nonrelativistic band structure, they can be classified into spin-degenerate antiferromagnets (i.e., conventional antiferromagnets) and spin-splitting antiferromagnets (i.e., altermagnets). (b) In spin-degenerate antiferromagnets, sublattices are connected by symmetries such as inversion ($\mathcal{P}$), translation ($\tau$), twofold rotation perpendicular to the material plane ($C_{2z}$), or mirror reflection parallel to the material plane ($m_z$), which enforce spin-degenerate bands. (c) In altermagnets the sublattices are related by other proper or improper rotations $A$, leading to spin-split band structures. (d) Schematic diagram illustrating the advantages of 2D altermagnets. The left side highlights the key merits of altermagnets, while the right side showcases the unique advantages of 2D materials. 2D altermagnets retain the key merits of bulk altermagnets, but in contrast to three-dimensional altermagnets, they also leverage the unique advantages of 2D materials.
• Figure 2: Schematic illustration of the design routes for two-dimensional (2D) altermagnets. We will introduce six strategies proposed for engineering 2D altermagnetism, including stacking (Section \ref{['sec:level4']}), multicomponent design (Section \ref{['sec:level5']}), surface adsorption (Section \ref{['sec:level6']}), electric-field control (Section \ref{['sec:level7']}), structural distortion (Section \ref{['sec:level8']}), and strain (Section \ref{['sec:level9']}).
• Figure 3: Schematics of how different spin and spatial symmetries act on the $\varepsilon(s,\mathbf{k})$ spectrumPhysRevB.110.054406. In reciprocal space, time reversal ($T$), inversion ($\overline{E}$), and twofold rotation about $z$ ($C_{2z}$) reverse $\mathbf{k}$, whereas identity ($E$), translation ($\tau$), and mirror symmetry about the $xy$ plane ($m_z$) leave $\mathbf{k}$ unchanged. In spin space, inversion ($\overline{E}$) and twofold rotations about axes perpendicular to the spins flip the spin ($C_2$), while identity ($E$) and the combined operation of a perpendicular twofold rotation with spin-space inversion ($\overline{C_2}$) keep it invariant.
• Figure 5: The band structure of altermagnets exhibiting large spin splitting in the absence of spin-orbit coupling: (a) CrOguo2023quantumchen2023giant, (b) Ca(CoN)$_2$PhysRevLett.133.056401, (c) V$_2$Se$_2$Oma2021multifunctional, (d) Fe$_2$WTe$_4$tan2024bipolarized, (e) Fe$_2$WSe$_4$PhysRevB.111.094411, (f) Nb$_2$SeTeOxie2025piezovalley, (g) Cr$_2$BAlsattigeri2025dirac, (h) Mn$_2$Te$_3$O$_8$wang2024electric, (i) Nb$_2$Se$_2$Oxie2025piezovalley. All of them exhibit spin splitting greater than 500 meV.
• Figure 6: The crystal structure and band structure without spin-orbit coupling (SOC) of bilayer altermagnets. (a)-(c) and (g) The crystal structure of bilayer NiZrCl$_6$PhysRevB.110.174410, twisted NiCl$_2$ (with a twisting angle of $21.79^\circ$)PhysRevB.110.174410, twisted MnPSe$_3$ (with a twisting angle of $21.79^\circ$)PhysRevMaterials.8.L051401 and twisted CrSBr (with a twisting angle of $90^\circ$)PhysRevLett.130.046401, respectively. (d)-(f) and (h) The band structure without SOC of bilayer NiZrCl$_6$, twisted NiCl$_2$ (with a twisting angle of $21.79^\circ$), twisted MnPSe$_3$ (with a twisting angle of $21.79^\circ$) and twisted CrSBr (with a twisting angle of $90^\circ$) respectively. (i) The orientations of magnetic moments in the orthogonal bilayer CrSBrboix2024multistep. Red arrows denote the spin orientations along the easy magnetic axis, assuming that interlayer magnetic interactions are negligible.