Altermagnetic Monolayers: Symmetry & Spintronics

• Altermagnetic monolayers are 2D materials with collinear magnetic order and zero net magnetization, where crystalline symmetry induces momentum-dependent spin splitting.
• They are designed using rational chemical synthesis, high-throughput DFT screening, and symmetry analysis to enable tunable d-, i-, and g-wave spin-splitting phases.
• Their unique spin textures facilitate applications in spintronics, valleytronics, and quantum devices through strain engineering, electric gating, and proximity effects.

Altermagnetic monolayers are atomically thin materials exhibiting collinear magnetic order with zero net magnetization, yet their electronic bands display pronounced spin splitting as a direct consequence of crystalline symmetries, rather than spin–orbit coupling or externally applied magnetic fields. Distinguished from both conventional antiferromagnets and ferromagnets, altermagnets are defined by a class of symmetry operations—such as high-order rotations and noncentrosymmetric mirror operations—that connect opposite-spin sublattices and permit momentum-dependent, nonrelativistic spin splitting throughout the Brillouin zone. In recent years, a concerted effort combining theoretical modeling, first-principles calculations, symmetry analysis, and experimental synthesis has converged to uncover a wide range of 2D altermagnetic monolayers, elucidating their underlying design principles, unique physical responses, and applicability in next-generation quantum, spintronic, and valleytronic devices.

1. Fundamental Symmetry Principles and Classes

Altermagnetism in monolayers arises from the absence of joint parity–time (𝒫𝒯) or other symmetries that would otherwise enforce spin degeneracy and from the presence of operations such as even-fold rotations ($C_2$, $C_4$, $C_6$) or glide mirror symmetries that relate magnetic sublattices without restoring global inversion or time-reversal invariance. The general symmetry requirement is that the magnetic space group must lack the elements {I|0}θ, {C₂z|0}θ, {m_z|0}U, and {m_z|τ}U (Sødequist et al., 11 Jan 2024). Monolayers commonly realize nonrelativistic, momentum-alternating spin splitting via the connection of opposite-spin sites through these symmetries.

Altermagnetic monolayers exhibit various classes of spin-splitting symmetry in momentum space, notably:

• d-wave: $\Delta E(\mathbf{k}) \propto k_x^2 - k_y^2$, yielding two sign changes around the Brillouin zone center.
• i-wave: $\Delta E(\mathbf{k}) \propto \cos(6\theta_k)$ or configurations with 12 sign changes, associated with specific mirror and rotational operations.
• g-wave: Higher-order mixed symmetries, such as $$\Delta E_n(\mathbf{k}) \propto k_x k_y (k_x^2 - k_y^2)$$, as in Cairo pentagonal monolayers (Li et al., 22 Dec 2024).

The minimal model for spin splitting can be encapsulated as: $$\Delta E_{n}(\mathbf{k}) = S(\mathbf{k})\{\text{rotational, mirror, and sublattice symmetry dictated terms}\}$$ with $S(\mathbf{k})$ acquiring a momentum-dependent sign structure in accordance with the retained symmetries.

2. Chemical and Structural Design Strategies

Rational chemical design for altermagnetic monolayers is guided by sublattice symmetry, specifically the use of layered templates with magnetic sublattices connected via rotations (not inversion) and diagonal mirrors (Xu et al., 21 May 2025). The blueprint involves four framework types (M₂A₂B, M₂A₂, Janus M₂AA′B, Janus M₂AA′) and comprehensive valence-adaptive substitution recipes for magnetic (M), chalcogen/pnictide/halide (A, A′), and oxygen/chalcogen (B) sites. High-throughput DFT screening of 2600 candidates identified 670 altermagnetic monolayers, with a subset manifesting crystal-symmetry paired spin–momentum locking (CSML) and Dirac cone features.

Synergy between topology and magnetism can also be engineered via lattice design. For instance, design of pentagonal Cairo lattices with both magnetic and nonmagnetic sites yields models that sustain g-wave altermagnetism and enable strain-driven transitions to d-wave phases (Li et al., 22 Dec 2024). Similarly, monolayers constructed from self-assembled Q1D magnetic chains (XY_n) allow control over inter-chain coupling (by spacing or stacking sequence), enabling toggling between altermagnetic and nodal-line phases (Guo et al., 9 Feb 2025).

Janus engineering—chemical modification where top and bottom ligands are made inequivalent—breaks inversion symmetry and is a key route to functionalizing conventional 2D antiferromagnets into monolayer altermagnets, as with MnP(S,Se)₃ and FeSe (Mazin et al., 2023).

3. Methods of Induction: Symmetry Engineering and External Controls

Several distinct mechanisms for inducing or tuning altermagnetism have been demonstrated:

• External electric field: Out-of-plane gating shifts ligand layers in compounds such as MnP(S,Se)₃, breaking 𝒫𝒯 but preserving crucial mirror symmetry, leading to six (i-wave) or more nodal lines in spin splitting (Mazin et al., 2023, Guo, 29 Mar 2025).
• Strain engineering: Uniaxial or biaxial strain can break diagonal mirrors ($M_{xy}$), induce valley polarization (“piezovalley effect”), and produce significant effective magnetic fields and piezomagnetic responses (Chen et al., 23 Oct 2024, Xie et al., 8 Nov 2024). Strain can drive transitions between g and d-wave altermagnetic phases (Li et al., 22 Dec 2024).
• Symmetry-selective adsorption: Adsorbing atoms or molecules (e.g., O on VPS₃, NH₃ on MnPSe₃) breaks specific symmetry operations that otherwise protect spin degeneracy (such as [C₂∥E]), allowing altermagnetism to emerge without net magnetization (Liu et al., 13 Jul 2025).
• Proximity effect (AMPE): Placing a monolayer altermagnet (e.g., V₂Se₂O) in contact with a nonmagnetic material (e.g., PbO, PbS, or NbSe₂) can imprint the momentum-alternating spin texture onto the adjacent layer, yielding proximitized altermagnets with uniquely tunable properties (Zhu et al., 8 Sep 2025).
• Ferroelastic and multi-ferroic coupling: In systems such as RuF₄ and CuF₂, ferroelastic reorientation switches the altermagnetic spin-splitting nonvolatilely (“altermagnetoelastic effect”) (Peng et al., 27 May 2025). In pentagonal FeO₂, electric-field or mechanical strain interlinks the altermagnetic, ferroelectric, and ferroelastic orders, forming a “triferroic” platform (Guo et al., 23 Jul 2025). Multistate switching is possible by mechanically or electrically reconnecting polarization and strain axes, directly reversing spin splitting.

4. Band Structure, Valley Polarization, and Topological Phenomena

Monolayer altermagnets frequently display complex spin-split band structures:

• Momentum-alternating spin splitting: As detailed for RuF₄ (d-wave) and FeBr₃ (i-wave), bands split with symmetry-imposed patterns, with high-symmetry lines either preserving or reversing the sign of splitting (Sødequist et al., 11 Jan 2024).
• Valley polarization and piezovalley effect: Strain or magnetization direction can modulate the energy difference between inequivalent valleys (e.g., X/Y points in Nb₂Se₂O, Nb₂SeTeO, or Cr₂S₂), enabling robust and tunable valley polarization even with negligible SOC (Xie et al., 8 Nov 2024, Chen et al., 23 Oct 2024). Valley splitting under strain can result in effective magnetic fields exceeding 100 T.
• Topological responses: Altermagnetic order leads to zone-specific gap closings, nodal lines, and topological transitions. For example:
  • Monolayer FeSe can exhibit both unconventional AM time-reversal symmetry breaking and quantized spin Hall conductivity in the presence of SOC (Mazin et al., 2023).
  • Proximity-induced alternating spin splitting in NbSe₂ can drive topological superconductivity, supporting chiral or helical Majorana modes, with the relevant BdG Hamiltonian incorporating the momentum-alternating AMPE term $J_A(\cos k_x - \cos k_y)\sigma_z\tau_0$ (Zhu et al., 8 Sep 2025).
  • In C₄₀ Shastry–Sutherland fullerene networks, a d-wave AM phase can be continuously tuned into quantum spin liquid, plaquette, or dimer phases through moderate strain (Wu et al., 28 Aug 2025).
• Nonlinear optical effects: Ferroelectric altermagnets such as VOX₂ produce exceptionally large spin shift currents—up to 330 μA/V²—with switchable sign via electric-field-driven polarization reversal (Yang, 17 Mar 2025).

5. Dimensionality, Magnetic Competing States, and Phase Diagrams

The transition to the monolayer limit often plays a decisive role in the emergence or suppression of altermagnetism:

• Dimensional Crossover: In GdAlSi and GdAlGe, reduction to a monolayer breaks bulk symmetry, enabling spontaneous anomalous Hall effects linked to nonrelativistic spin splitting (Parfenov et al., 11 Jun 2024, Parfenov et al., 19 May 2025). In GdAlGe, the interplay between the AM and a weak FM state is thickness-tunable, producing intrinsic exchange bias in intermediate films.
• Suppression by Frustration: In MnTe, monolayer structures relax into geometries where frustration or disorder suppresses collinear AM order, favoring spin-glass behavior, while bilayers maintain robust layered AFM (Cuxart et al., 15 Apr 2025).
• Multistate and controllable regimes: In RuF₄ (2-state) and CuF₂ (3-state), ferroelastic states cycle the orientation and sign of the AM spin-splitting, allowing encoding of spin conductivities in a nonvolatile fashion (Peng et al., 27 May 2025). Odd-even effects in multilayer AMs, as in Cr₂SX, relate to the persistence or cancellation of spin-valley locking depending on stacking sequence and symmetry (Tian et al., 21 Oct 2024).
• Phase Tunability: In Q1D monolayer prototypes (e.g., CrBr₃ or MnBr₃), inter-chain spacing or electric fields manipulate the exchange sign, toggling between altermagnetic (FM-coupled) and nodal-line (AFM-coupled) states (Guo et al., 9 Feb 2025).

6. Applications in Spintronics, Valleytronics, and Quantum Devices

Monolayer altermagnets, due to their momentum-dependent spin splitting and net-zero magnetization, offer several advantages:

• Stray-field-free spintronics: AM monolayers avoid stray fields endemic to FMs, enabling ultra-compact, energy-efficient memory and logic (e.g., nonvolatile nanomechanical spin switches via ferroelastic control (Peng et al., 27 May 2025)).
• Valleytronic circuits: Robust, room-temperature valley polarization and piezovalley effect as realized in Nb₂Se₂O and Cr₂S₂ provide a new degree of freedom for encoding, reading, and manipulating information with strain and magnetic orientation (Xie et al., 8 Nov 2024, Chen et al., 23 Oct 2024).
• Opto-spintronics: Ferroelectric AMs such as VOX₂ combine large, switchable shift currents with magnetoelectric coupling, enabling photo-induced spin current generation and the design of photonic switches or detectors (Yang, 17 Mar 2025).
• Quantum platforms: Chiral magnonic states (C₄₀ monolayers), spin liquids, exchange bias, and topological superconductivity (NbSe₂ under AMPE) suggest new architectures for quantum gates and robust memory.

7. Current Challenges and Outlook

Despite diverse theoretical advances and an increasing roster of candidate monolayer altermagnets, several questions and technical challenges remain:

• Experimental Realization: While synthesis and exfoliation of monolayers such as GdAlSi (Parfenov et al., 11 Jun 2024), CoNb₄Se₈ (Regmi et al., 16 Aug 2024), and CrX₃ (Guo, 29 Mar 2025) are well established, the controlled induction of AM states (via substrates, gating, adsorption, or strain) still faces materials-specific constraints on symmetry breaking, environmental stability, and defect control.
• Symmetry Identification and Classification: A universal symmetry-engineering framework—using spin group theory and comprehensive classification of spin point groups—now allows for systematic prediction and tailoring of 2D AFM hosts ready for functionalization via adsorption, gating, or stacking (Liu et al., 13 Jul 2025, Pan et al., 11 Sep 2024).
• Interplay of Competing Phases and Disorder: Dimensional reduction or interface effects may favor competing magnetic ground states (e.g., spin-glass in thin-layer MnTe (Cuxart et al., 15 Apr 2025)), requiring subtle tuning of strain, substrate, and field conditions.
• Integration with Device Architectures: Harnessing the full range of AM functionalities for complex device stacks, hybrid quantum circuits, or opto-spintronic interfaces will require advances in nanomanufacturing, proximity control, and highly resolved spectroscopies to validate predicted properties and responses.

Overall, the rapid development of symmetry-based chemical design, high-throughput computational screening, and multifunctional engineering (mechanical, electrical, and chemical controls) is positioning altermagnetic monolayers as a versatile platform for both fundamental quantum science and the realization of low-power, high-speed spintronic and valleytronic devices.