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Classification and design of two-dimensional altermagnets

Sike Zeng1 , Dong Liu1 , Hongjie Peng1 , Chang-Chun He1 ,2 , Xiao-Bao Yang1 , and Yu-Jun Zhao1 ∗ 1 Department of Physics, South China University of Technology, Guangzhou 510640, China 2

arXiv:2601.10183v2 [cond-mat.mtrl-sci] 12 Apr 2026

Guangdong Provincial Key Laboratory of Functional and Intelligent Hybrid Materials and Devices,

South China University of Technology, Guangzhou 510640, China

South China University of Technology, Guangzhou 510640, China
(Dated: April 14, 2026)
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 driv-
ing 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, em-
phasizing those with large spin splitting for experimental realization. We then examine strategies
for engineering two-dimensional altermagnetism. This Review aims to consolidate theoretically pro-
posed candidate materials and realization strategies for two-dimensional altermagnets, providing
insights for future experimental efforts in this emerging field.
Keywords: altermagnet; two-dimensional material; spin group; spintronics; unconventional magnetism

I. INTRODUCTION Magnetism is a time-honored yet continually thriv- ing field that constitutes one of the core research areas in condensed matter physics and underpins a broad range of modern technologies. Most tradi- tional studies of magnetism have focused on ferromag- nets and antiferromagnets. Recently, a new type of collinear antiferromagnet known as altermagnet has been proposed[1], sparking growing interest in both theoretical and experimental research[2–6]. Altermag- nets are characterized by vanishing net magnetization, as in antiferromagnets, and the non-relativistic spin- polarized band structure, as in ferromagnets. They are therefore also referred to as spin-splitting anti- ferromagnets. This phenomenon was initially pre- dicted by several research groups[7–12]. Due to its non-relativistic origin, altermagnetism is governed by spin-group symmetry[1, 13]. From a symmetry per- spective, the opposite-spin sublattices in altermagnets are connected by proper or improper rotations, rather than by spatial inversion or pure translations, as is the case in conventional antiferromagnets. As a result, conventional antiferromagnets exhibit spin-degenerate band structures due to the protection of PT or T τ symmetry, where P denotes spatial inversion, T de- notes time-reversal, and τ is a fractional translation. In contrast, the absence of these symmetries in alter- magnets allows for spin splitting in the band structure, similar to ferromagnets. From an application perspective, altermagnets hold great promise as efficient platforms for next- generation spintronic devices. Conventional spintron- ics has been primarily based on ferromagnets, ow- ing to their broken time-reversal symmetry and the associated physical responses. However, the pres- ence of stray fields in ferromagnets poses signifi- cant constraints on both storage density and scal- ∗ [email protected] ability. In addition, the presence of a finite net magnetization renders them sensitive to perturba- tions from external magnetic fields. In contrast, antiferromagnetic materials offer distinctive advan- tages, including robustness against external mag- netic field perturbations, the absence of stray fields, ultrafast spin dynamics. Consequently, antiferro- magnetic spintronics, which replaces ferromagnets with antiferromagnets as the spin-dependent com- ponents, has attracted considerable attention and emerged as a highly promising field[14, 15]. Never- theless, the absence of strong spin currents and time- reversal symmetry-breaking responses has limited the practical applications of conventional antiferromag- nets. As time-reversal symmetry-breaking antifer- romagnets, altermagnets integrate the key merits of both ferromagnets and antiferromagnets, offering new opportunities for advancing spintronics. A wide range of materials have been recognized as altermagnets[3], providing a solid foundation for fur- ther theoretical and experimental studies. Through angle-resolved photoemission spectroscopy (ARPES), several materials have been experimentally confirmed as altermagnets, including MnTe[16, 17], CrSb[18– 21], KV2 S e2 O [22], and Rb1 −δ V2 T e2 O [23]. Among the experimentally established altermagnets, MnTe stands out as a particularly promising system. Recent nanoscale x-ray magnetic circular dichroism (XMCD) imaging on single-crystal MnTe has provided direct evidence of bulk altermagnetic order, confirming its intrinsic nature beyond thin-film or surface effects[24]. Spin-dependent transport phenomena, such as the gi- ant magnetoresistance (GMR) effect and the tunnel magnetoresistance (TMR) effect, have been investi- gated in altermagnets[25–30], paving the way for their potential applications in spintronics. In addition, altermagnets have attracted growing research inter- est in diverse areas, including topological physics[31, 32], ultrafast dynamics[33], valleytronics[34–36], superconductivity[37–42], thermoelectricity[27, 43– 48] and multiferroic[49–52].

2 FIG. 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 mag- netic 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 antiferro- magnets, sublattices are connected by symmetries such as inversion (P), translation (τ ), twofold rotation perpendicular to the material plane (C2 z ), or mirror reflection parallel to the material plane (mz ), which enforce spin-degenerate bands. (c) In altermagnets the sublattices are related by other proper or improper rotations A, leading to spin-split band struc- tures. (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. Following the initial experimental demonstration of two-dimensional (2D) van der Waals (vdW) mag- nets in 2017[53, 54], 2D magnets have attracted widespread attention in both fundamental mag- netism and spintronic applications[55]. Compared to three-dimensional (3D) magnets, they exhibit unique advantages[56], as summarized in the right side of Fig. 1(d). Owing to the weak interlayer vdW interaction, 2D magnets can be combined with any other 2D ma- terials to form heterostructures without suffering from lattice mismatch issues, thereby enabling a wide range of tunable physical properties[57, 58]. Even bilayers or multilayers composed of the same material can exhibit emergent phenomena[59, 60]. Moreover, 2D magnets exhibit enhanced tunability under external stimuli, such as gating or strain. Furthermore, 2D magnetic materials offer significant promise for enhancing the efficiency of electrical control in magnetic devices[56]. Building on the concept of two-dimensional mag- netic materials, the idea of two-dimensional altermag- nets has emerged naturally. 2D collinear magnets can be categorized into three distinct types, with their differences in crystal structures and band structures illustrated in Figs. 1(a)-(c). 2D altermagnets are an- ticipated to retain the key merits of bulk altermag- nets while offering additional advantages that emerge uniquely in reduced dimensions, making them a com- pelling platform for spintronic applications and for fundamental studies of magnetism, as illustrated in Fig. 1(d). Recently, research on two-dimensional al- termagnets has been rapidly growing. The first re- ported two-dimensional altermagnet is V2 S e2 O [12], which exhibits a giant noncollinear spin current. As a class of two-dimensional materials, 2D altermag- nets have emerged as an important platform for val- leytronics research[61–63], including studies on val- ley polarization[35, 36, 64, 65] and the valley Hall effect[34]. Anomalous transport phenomena have also been theoretically predicted in 2D altermagnets[66– 68]. Moreover, circularly polarized light has been pro- posed as an effective means to selectively tune the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in 2D d -wave altermagnets[69]. By combining ferro- electricity, two-dimensional multiferroic altermagnets have been explored[70, 71], where sliding ferroelectric- ity serves as an efficient method to control the spin splitting in altermagnets[50, 72–74].The interplay be- tween altermagnetism and superconductivity in a two- dimensional system has been explored[40, 75]. It has been shown that when a superconductor is proximi- tized to an altermagnet, the altermagnetic anisotropic spin splitting can induce a gapless superconducting state[76, 77]. Yet, the field is still dominated by theory, with experimental work lagging behind. Al-

3 though the 2D limit of altermagnet CdAlSi has been investigated[78], this film is not an altermagnet in principle. Motivated by this gap, we survey all theo- retically predicted 2D candidates and the routes that may lead to their laboratory realization, offering ex- perimenters a concise roadmap. Broader discussions of altermagnetism can be found in the dedicated reviews cited herein[2–6], which sur- vey the conceptual foundations of altermagnetism, summarize recent experimental and theoretical ad- vances, and highlight the key physical properties, ma- terial realizations, and experimental probes associated with this emerging magnetism. Nonetheless, their dis- cussions are centered predominantly on bulk systems. In the 2D limit, enhanced sensitivity to external per- turbations, the presence of stacking degrees of free- dom, and extensive chemical tunability create design rules and material realizations fundamentally distinct from those in bulk systems, yet these aspects have not been comprehensively consolidated. In this review, we address this gap by introducing a unified symmetry classification for 2D altermagnets based on spin-group theory, compiling all reported monolayer candidates, and summarizing the chemical and structural design strategies that are used in the 2D regime. The Review is organized as follows. We begin by outlining the symmetry framework based on spin groups, which underpins the classification of 2D alter- magnets. We then provide an overview of reported 2D altermagnets, summarized in Table II, and high- light representative materials with large spin split- ting as promising candidates for experimental realiza- tion. In addition, we discuss various strategies pro- posed for engineering 2D altermagnetism, including stacking, multicomponent design, surface adsorption, electric-field control, structural distortion and strain. A schematic diagram is provided for quick reference, as shown in Fig. 2. Finally, we provide an outlook on the future opportunities and challenges in this emerg- ing field. II. SYMMETRY CLASSIFICATION OF TWO-DIMENSIONAL ALTERMAGNETS Altermagnetism is a nonrelativistic phenomenon and is therefore governed primarily by spin group sym- metries. Spin group theory, originally proposed in the last century[79–81], has recently been further devel- oped as a powerful framework for describing the sym- metries of magnetic structures[82–84]. Compared to magnetic groups, spin groups allow for a partial de- coupling of real-space and spin-space symmetry op- erations. In this section, we introduce the symmetry description of two-dimensional altermagnets based on spin group theory. A spin group is conventionally expressed as the di- rect product rs ⊗ Rs , where rs denotes the spin-only group consisting of symmetry operations acting solely in spin space, and Rs represents the nontrivial spin group composed of pairs of operations [Ri ∥ Rj ], in FIG. 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 IV), multicomponent design (Section V), surface adsorption (Section VI), electric-field control (Section VII), structural distortion (Section VIII), and strain (Section IX). which the operation to the left of the double bar acts exclusively on spin space, while the one on the right acts only on real space. For collinear spin arrange- ments, the spin-only group is the same and primarily includes two types of symmetry operations: (i) con- tinuous rotations around the common spin axis (C∞ ) ; and (ii) a twofold rotation around an axis perpendic- ular to the spin direction, combined with the inver- sion symmetry in spin space, which is always realized through time-reversal symmetry, denoted as [C2 ∥ T ]. The second symmetry operation strictly constrains collinear magnetic systems to exhibit only even-parity wave magnetism. Different types of collinear magnetic orders are characterized by distinct nontrivial spin groups. As three-dimensional groups with two-dimensional trans- lations, layer groups offer a complete symmetry framework for describing the crystal symmetries of quasi-two-dimensional materials. Therefore, for two- dimensional magnetic materials, the real-space part of the nontrivial spin group should be a layer group[85]. In ferromagnetic systems, only a single spin sublat- tice is present, as shown in Fig. 1(a). As a re- sult, the corresponding nontrivial spin group takes the form [E ∥ G], where G denotes layer group. Al- though the ferrimagnets contain multiple spin sub- lattices, there are no symmetry operations connect- ing them. As a result, their spin groups are identical to those of ferromagnets. Recently, two-dimensional fully compensated ferrimagnets have also attracted increasing attention[86]. These ferrimagnets exhibit zero net magnetic moment without symmetry protec-

4 FIG. 3. Schematics of how different spin and spatial symmetries act on the ε(s, k) spectrum[85]. In recipro- cal space, time reversal (T ), inversion (E), and twofold rotation about z (C2 z ) reverse k, whereas identity (E), translation (τ ), and mirror symmetry about the xy plane (mz ) leave k unchanged. In spin space, inversion (E) and twofold rotations about axes perpendicular to the spins flip the spin (C2 ) , while identity (E) and the combined operation of a perpendicular twofold rotation with spin- space inversion (C2 ) keep it invariant. tion, along with nonrelativistic spin-split band struc- tures. Compared to altermagnets, half-metallicity can emerge in the fully compensated ferrimagnets[87]. For collinear antiferromagnets, two spin sublattices are present and are connected by a spin-group symme- try operation, which guarantees that the net magne- tization vanishes. Therefore, the corresponding non- trivial spin group takes the form [E ∥ H] + [C2 ∥ G − H] = [E ∥ H] + [C2 ∥ AH], where H is a halving sub- group of the layer group G, and A is a coset rep- resentative. In this case, the operations in the sub- group H connect sites with the same spin orientation, while those in the coset AH connect sites with opposite spin orientations. In two-dimensional systems, spin- degenerate band structures can be protected by four spin-group operations, including [C2 ∥ P], [C2 ∥ τ ], [C2 ∥ mz ] and [C2 ∥ C2 z ], which is more than in three- dimensional systems[85]. Here, P denotes spatial in- version, τ represents a translation, mz is the mirror re- flection with respect to the plane of the material, C2 z denotes a two-fold rotation around the out-of-plane axis. The origin of the spin degeneracy is that each of these operations pairs the spin-up and spin-down states at the same k point[85]. Fig. 3 provides a schematic illustration to aid the understanding of how various spin-group symmetries affect the ε(s, k) spec- trum. First, [C2 ∥ τ ] maps the two magnetic sublat- tices onto each other through a translation τ , and since spin degeneracy does not depend on real-space translations, [C2 ∥ τ ] is equivalent to [C2 ∥ E]. This directly enforces [C2 ∥ τ ] ε(s, k) = ε(−s, k) = ε(s, k). (1) The second equality holds because this operation is a symmetry of the system. Moreover, [C2 ∥ P] combines with the spin-only operation [C 2 ∥ T ] to produce [C2 ∥ P] [C 2 ∥ T ] = [E ∥ PT ], (2) which corresponds to PT symmetry that guarantees Kramers degeneracy. In two dimensions, two addi- tional operations—[C2 ∥ mz ] and [C2 ∥ C2 z ]—protect the spin degeneracy. This is because [C2 ∥ mz ] leads to [C2 | |mz ]ε(s, k) = ε(−s, k) = ε(s, k), (3) and [C2 ∥ C2 z ] combining with [C 2 ∥ T ] leads to [E||T C2 z ]ε(s, k) = ε(−s, k) = ε(s, k). (4) Taken together, these symmetry relations reveal that reduced dimensionality naturally expands the set of operations able to protect spin degeneracy in 2D ma- terials. Therefore, when A belongs to {P, τ, mz , C2 z }, the nonrelativistic band structure of the antiferromagnet is spin-degenerate, as shown in Fig. 1(b). In other words, two dimensional collinear antiferromagnets in which opposite-spin sites are connected by one of the operations in {P, τ, mz , C2 z } exhibit spin-degenerate band structures in the nonrelativistic limit. Recently, antiferromagnets in which opposite-spin sites are con- nected by mz or C2 z have been classified as type-IV two-dimensional collinear magnets[88]. These systems are characterized by spin-degenerate band structures in the nonrelativistic limit, and simultaneously ex- hibiting time-reversal symmetry-breaking responses. Since the symmetry operations of the form [C2 ∥ A], with A ∈ {P, τ, mz , C2 z }, enforce spin degeneracy, they are incompatible with the nonrelativistic spin splitting that characterizes altermagnets, as shown in Fig. 1(c). Therefore, the spin groups that describe 2D altermagnets must exclude such symmetry operations. Since our classification of 2D altermagnets focuses on spin-momentum locking, translational operations can be neglected. For all collinear magnets, regardless of whether they possess real-space inversion symmetry, the nonrelativistic band structure remains invariant under real-space inversion operations[85]. Therefore, Laue groups driven from layer groups are used to be the real-space part of nontrivial spin group for describ- ing 2D altermagnets. All nontrivial spin Laue groups describing 2D alter- magnets are listed in Table I. They are classified into three types, including d -, g-, i -wave, according to the basic characteristic of spin-momentum locking. When traversing a closed loop in momentum space around the Γ-point within the xy-plane, the spin undergoes a full 360◦ rotation through two discrete reversals and the total number of such rotations defines a charac- teristic even integer. When the characteristic integer is 2, 4, or 6, the corresponding types of spin-orbit coupling are respectively classified as d -wave, g-wave, and i -wave[1], as listed in Table I. From a symmetry perspective, the characteristic integer in 2D altermag- nets is always determined by how many mirror planes perpendicular to the 2D plane or in-plane twofold ro- tations are accompanied by a spin rotation. For ex- ample, a d -wave altermagnet typically hosts two such symmetries, yielding a characteristic integer of 2. The

5 TABLE I. All nontrivial spin Laue groups for two di- mensional altermagnets, the basic characteristic of spin- momentum locking of the materials described by these groups and their type[85]. Here we adopt Litvin’s notation of the spin groups. Type Spin-momentum locking (kx , ky ) spin Laue group d -wave 2 2/2 mx 2 m2 m1 m 2 4/1 m 2 4/1 m2 m1 m g-wave 1 4/1 m2 m2 m i-wave 1 32 m 1 6/1 m2 m2 m dicted to occur only in low-symmetry d -wave alter- magnets, while it is prohibited in higher-symmetry g-wave and i -wave altermagnets[25]. In 2D altermagnets, a spin-degenerate nodal line is protected by a [C2 ∥ G − H] symmetry when it leaves the wave vector on the line invariant. More generally, if this symmetry maps a wave vector to itself or to another vector differing by a reciprocal lattice vector, the corresponding band structure at this k-vector will exhibit spin degeneracy. In contrast, opposite spins emerge at distinct k-vectors related by the G-H sym- metry. Symmetry plays a crucial role not only in the un- derstanding and classification of two-dimensional al- termagnets, but also in their discovery and rational design. Here, we summarize the symmetry condi- tions required for the emergence of two-dimensional altermagnetism. To realize two-dimensional altermag- netism, the system must host a collinear-compensated magnetic order, meaning that it contains two sublat- tices with antiparallel magnetic moments. The atoms belonging to these opposite sublattices must not be connected by translation symmetry, inversion sym- metry, a mirror plane parallel to the xy plane, or a twofold rotation about the z axis, assuming the sys- tem lies in the xy-plane. Additionally, there must ex- ist at least one symmetry operation, either a proper or improper rotation, that maps one sublattice onto the other. Based on these criteria, numerous two- dimensional altermagnets have been theoretically pre- dicted, encompassing d -, g-, and i -wave types. Mean- while, diverse approaches have emerged for realizing altermagnetism from both magnetic and nonmagnetic materials. III. SURVEY OF TWO-DIMENSIONAL

same symmetry argument applies to other types of al- termagnets. Such symmetry operations correspond to spin-degenerate nodal lines passing through the Γ- point. Consequently, the type of an given altermag- net determines the appropriate k-path for both the- oretical band-structure calculations and spin-resolved angle-resolved photoemission spectroscopy (SARPES) measurements, when verifying its spin splitting. Different types of 2D altermagnets give rise to dis- tinct experimentally accessible responses. The pos- sibility of an in-plane anomalous Hall effect (AHE), where the N´eel vector, Hall current, and electric field all lie within the 2D plane, has recently been ex- plored in 2D altermagnets. Such an in-plane AHE is predicted to be allowed only in d -wave altermag- nets with nontrivial spin Laue group 2 2/2 mx and in i -wave altermagnets with spin Laue group 1 32 m, while it is strictly forbidden in g-wave altermagnets[66]. In addition, giant magnetoresistance (GMR), derived from the anisotropic spin-momentum coupling and the anisotropic spin-dependent conductivities, is pre-

ALTERMAGNETS

Theoretical efforts have identified a wide range of two-dimensional altermagnets, which span metallic, semimetallic, and semiconducting electronic proper- ties and are predominantly characterized by d -wave spin-momentum locking. These predicted candidates are summarized in Table II. In terms of electronic transport properties, the majority of these materials are semiconductors. With respect to the character- istics of spin-locking, most of them exhibit d -wave altermagnetism. Notably, only 2H-FeBr3 has been identified to exhibit i -wave altermagnetism. Under well-defined symmetry conditions for the emergence of altermagnetism, screening materials databases pro- vides an efficient approach to discover 2D altermag- nets. A systematic screening has been conducted across four databases[89–92]: the Computational 2D Materials Database (C2DB), 2DMatPedia, Materi- als Cloud two-dimensional crystals database (MC2D), and Database for Pentagon-Based Sheets. Meanwhile, several 2D functional materials exhibit- ing altermagnetism have also been proposed. Some materials have been identified as ferroelectric alter-

6 magnets, such as VOI2 [ 70, 93] and MgFe2 N 2 [ 94], where ferroelectric polarization enables the control of spin splitting in altermagnets. This mechanism has recently been proposed as a novel type of magneto- electric coupling[50–52, 70, 93]. CuMoP2 S (Se)6 and CuWP2 S (Se)6 have been identified as antiferroelec- tric altermagnets, in which a weak electric field can drive a phase transition between the antiferroelectric altermagnet and the ferroelectric antiferromagnet[49]. Fe2 W Te4 and Fe2 M oZ4 (Z = S, Te) have been iden- tified as Weyl semimetals, which exhibit the crys- tal valley Hall effect[95]. V2 S e2 O is not only an al- termagnet but also a novel valleytronic material, in which the valleys are connected by crystallographic symmetries rather than by time-reversal symmetry, and can therefore be modulated by external fields or geometric engineering[12, 65, 96]. Building on this paradigm, a symmetry-guided chemical design frame- work combined with extensive high-throughput calcu- lations has been developed, through which 670 mono- layer altermagnets have been identified[97]. Organic materials with altermagnetism have also been inves- tigated, such as Cr(diz)2 (diz = 1,3-diazete) and t- Cr2 [ Pyc-O8 ] (Pyc=pyracylene). Altermagnets with large spin splitting are more accessible to experimen- tal observation and hold greater potential for applica- tions. Therefore, we summarize nine altermagnets ex- hibiting spin splitting larger than 500 meV, and their band structures in the absence of spin-orbit coupling (SOC) are presented in Fig. 4. TABLE II: Two-dimensional altermagnets with their types and electronic properties.

No. Material Type Electronic Property Ref.

1 AgF2 d -wave semiconductor [89] 2 RuF4 d -wave semiconductor [89, 92, 98] 3 VF4 d -wave semiconductor [89, 92] 4 OsF4 d -wave semiconductor [89] 5 V2 C lBrI2 O 2 d -wave semiconductor [89] 6 OsNNaSCl5 d -wave semiconductor [89] 7 MnTeMoO6 d -wave semiconductor [85] 8 Ca2 R uO4 d -wave semiconductor [99] 9 V2 S e2 O d -wave semiconductor [12] 10 CoS2 d -wave semiconductor [91] 11 CoPSe d -wave semiconductor [91] 12 VOX2 (X=Cl,Br,I) d -wave semiconductor [70, 93] 13 VX4 (X=Cl,Br,I) d -wave semiconductor [68] 14 Fe2 W Te4 d -wave semimetal [95] 15 Fe2 M oZ4 (Z=S,Se,Te) d -wave semimetal [95] 16 V2 T e2 O d -wave metal [67, 100] 17 LiMnF5 d -wave semiconductor [90] 18 LiVSiO4 d -wave semiconductor [90] 19 Co2 T e3 O 8 d -wave semiconductor [90] 20 Mn2 T e3 O 8 d -wave semiconductor [90] 21 LiFeSiO4 d -wave semiconductor [90] 22 LiMnPO4 d -wave semiconductor [90] 23 TcF4 d -wave semiconductor [90] 24 TcCl4 d -wave semiconductor [90] 25 TcBr4 d -wave semiconductor [90] 26 TcI4 d -wave metal [90] 27 Cr2 S iO4 d -wave semiconductor [90] 28 FeMgSn d -wave metal [90] 29 MnMgSn d -wave metal [90] 30 M(pyz)2 (M = Ca and Sr, pyz = pyrazine) d -wave semiconductor [101] 31 Ca(CoN)2 d -wave semiconductor [102] 32 Ca(FeN)2 d -wave semiconductor [103] Continued on next page

7 TABLE II – continued from previous page

No. Material Type Electronic Property Ref.

33 CrO d -wave semimetal [104, 105] 34 Cr2 T e2 O d -wave semiconductor [27] 35 Cr2 S e2 O d -wave semiconductor [27] 36 Fe2 S e2 O d -wave semiconductor [62] 37 Cr2 B Al d -wave metal [106] 38 Cr(diz)2 (diz = 1,3-diazete) d -wave semiconductor [107] 39 Cr(c-pyr)2 (c-pyr = pyrrolo[3,4-c]pyrrole) d -wave semiconductor [107] 40 Cr(f-pid)2 (f-pid = pyrrolo[3,4-f]isoindole) d -wave semiconductor [107] 41 Cr2 S 2 d -wave semiconductor [108] 42 CuMoP2 S (Se)6 d -wave semiconductor [49] 43 CuWP2 S (Se)6 d -wave semiconductor [49] 44 FeS (110) d -wave semiconductor [109] 45 FeSe (110) d -wave semiconductor [109] 46 FeSe (001) d -wave metal [110] 47 Cr2 S iCS2 d -wave semiconductor [111] 48 Cr2 S iCSe2 d -wave semiconductor [111] 49 Nb2 S e2 O d -wave semiconductor [112] 50 Nb2 S eTeO d -wave semiconductor [112] 51 MgFe2 N 2 d -wave semiconductor [94] 52 t-Cr2 [ Pyc-O8 ] (Pyc=pyracylene) d -wave semiconductor [113] 53 SrRuO3 (010) d -wave semiconductor [114] 54 VSX2 (X=Cl,Br,I) d -wave semiconductor [70] 55 Fe2 W S4 d -wave semiconductor [34] 56 Fe2 W Se4 d -wave semiconductor [34] 57 Co2 M oSe4 d -wave semiconductor [115] 58 MoO d -wave metal [116] 59 MnC2 g-wave semiconductor [91] 60 MnS2 g-wave semiconductor [91] 61 Li2 M nP2 O 7 g-wave semiconductor [90] 62 VP2 H 8 ( NO4 ) 2 g-wave semiconductor [85] 63 2H-FeBr3 i-wave semiconductor [89] Yet, with only a handful of two-dimensional alter- magnets identified beside the 200-plus bulk crystals already catalogued [3], the field is clearly in its in- fancy. This scarcity leaves ample room—from materi- als discovery and property engineering to application prototyping—for the next wave of exploration. For most of the materials listed in Table II, the magnetic ground state has been determined merely by compar- ing the energies of a few collinear magnetic config- urations. This approach, however, is insufficient for identifying the true ground state due to the complex magnetic energy landscape and the vast configuration space. Therefore, employing systematic and effec- tive approaches to determine magnetic ground states, such as spin spiral calculation[89, 117] and Machine Learning[118], is essential for the future research on the prediction of 2D altermagnets. IV. STACKING ALTERMAGNET Interlayer stacking—whether between identical lay- ers or heterogeneous combinations—serves as a versa- tile symmetry-engineering strategy that can generate, modulate, and even switch altermagnetism in 2D sys- tems. Compared to three-dimensional counterparts, two-dimensional materials possess an extra ’stacking’ degrees of freedom, offering a distinctive and tunable knob for symmetry engineering. Owing to the weak interlayer van der Waals (vdW) interactions, two- dimensional vdW materials can be freely stacked into either homostructures or heterostructures, thereby en- abling a wide variety of physical properties[55]. From a symmetry perspective, different stacking configura- tions can result in structures with distinct crystallo- graphic symmetries, regardless of whether they in-

8 FIG. 4. The band structure of altermagnets exhibiting large spin splitting in the absence of spin-orbit coupling: (a) CrO[104, 105], (b) Ca(CoN)2 [ 102], (c) V2 S e2 O [12], (d) Fe2 W Te4 [ 95], (e) Fe2 W Se4 [ 34], (f) Nb2 S eTeO[112], (g) Cr2 B Al[106], (h) Mn2 T e3 O 8 [ 90], (i) Nb2 S e2 O [112]. All of them exhibit spin splitting greater than 500 meV. volves identical or different materials. The bilayer stacking theories provide guidance on how van der Waals monolayers with known symmetries can be stacked to yield bilayers with desired crystallographic symmetries[119]. This establishes the theoretical ba- sis for rationally designing 2D systems with targeted symmetries, while ongoing advances in precise stack- ing techniques now make their laboratory assembly an experimental reality[120–123]. Because altermag- netism is symmetry-governed[1, 85], layer stacking be- comes a readily accessible experimental lever for cre- ating it in two dimensions. Here we survey practi- cal stacking protocols—using identical or distinct 2D magnets—and highlight the device opportunities they unlock. Although experimental demonstrations of altermag- netism realized through stacking in 2D materials are still absent, existing fabrication and assembly techniques[122, 124–126] clearly show that this ap- proach is experimentally feasible. Traditional ap- proaches to constructing stacked 2D bilayer or mul- tilayer can be grouped broadly into two categories: ’top-down’ transfer and assembly, and ’bottom-up’ direct growth. ’Top-down’ methods first produce monolayers by mechanical or chemical exfoliation and then stack them using an appropriate trans- fer technique[127, 128]. Common transfer strategies include dry-transfer[123, 129] and wet-transfer[130]. This method allow flexible and precise stacking of dis- similar layers, enabling controlled layer sequence, rela- tive twist angle, and atomically clean heterointerfaces. On the other hand, ’bottom-up’ direct growth tech- niques, such as chemical vapor deposition (CVD) and molecular beam epitaxy (MBE), avoids contamination or damage associated with transfer steps and is bet- ter suited to large-area, scalable fabrication[131, 132]. Notably, wafer-scale, uniform bilayer and multilayer 2D materials (e.g., graphene and various transition- metal dichalcogenides) have already been success- fully synthesized[131–133]. Taken together, although stacked 2D altermagnets have not yet been realized ex- perimentally, the existing stacking technologies for 2D materials, ranging from finely controlled ’top-down’ assembly to scalable ’bottom-up’ growth, are already well established, providing a solid technological foun- dation for their future experimental implementation.

9

A. Homostructure Stacking The realization of a bilayer altermagnet in a ho- mostructure can be achieved either by stacking two antiferromagnetic layers or by stacking two ferro- magnetic layers with interlayer antiferromagnetic cou- pling, thereby ensuring a zero net magnetic mo- ment. In previous studies, twisted bilayer altermag- nets have been proposed[135, 136], which can be re- garded as stacking altermagnets, since twisting con- stitutes a particular stacking operation. In addi- tion, several special stacking configurations have been proposed, including the flipped[138] and reversed[71] types. Moreover, an approach was proposed to sys- tematically derive the stacking configurations of bi- layer A-type altermagnets[134, 139]. Recent studies in multiferroics have proposed novel magnetoelectric coupling mechanisms, where sliding ferroelectricity is employed to modulate spin splitting[50, 73, 140]. In this subsection, we outline the derivation of bilayer A- type altermagnets, introduce representative examples of 2D altermagnets formed by homostructure stack- ing, and discuss their potential applications. We first introduce the bilayer A-type altermagnet, characterized by ferromagnetic coupling within each layer and antiferromagnetic coupling between layers, i.e., an altermagnet realized through stacking ferro- magnetic monolayers[134, 139]. In A-type altermag- nets, sublattices with opposite spins cannot be con- nected by translational symmetry, a consequence of the intrinsic absence of out-of-plane translational sym- metry in 2D materials. Moreover, since the intralayer magnetic moments are aligned in parallel, they also cannot be connected by a twofold rotational symmetry around the z -axis. According to Section II, the spin- degenerate band structure is protected by the sym- metry operations [C2 ∥ A], where A ∈ {P, τ, mz , C2 z }. Therefore, the emergence of A-type altermagnetism merely requires the breaking of [C2 ∥ P] and [C2 ∥ mz ]. Therefore, A-type altermagnets are more likely to emerge. Owing to the nonrelativistic nature of altermag- netism, real space and spin space are decoupled. Since the magnetic order in A-type altermagnets is predeter- mined, the symmetry analysis is confined to real-space operations alone[134]. Therefore, the stacking model, originally proposed in the field of ferroelectrics[119], can also be applied in the present context. The stack- ing model can be formulated as follows: S = ˆRs S , ∀ ˆ R s ∈ Gs B = ˆRB B, ∀ ˆ R B ∈ GB B = S + S′ S ′ = ˆτz ˆOS (5) Here, S denotes a single ferromagnetic layer with layer group Gs = { ˆ R s } , where ˆRs represents the space- group symmetry operation. The second ferromagnetic layer S′ is generated by applying the stacking opera- tion ˆτz ˆO on S, where ˆτz is an arbitrary trivial trans- lation operation along z direction and cannot affect the symmetry of the bilayer B whose layer group is GB = { ˆ R B }. ˆO is the non-trivial part of stacking operation. The bilayer B is composed of two ferro- magnetic layers, S and S′ . The emergence of altermagnetism is primarily de- termined by the symmetry operations between sublat- tices, as discussed in Section II, specifically the inter- layer symmetry operations in A-type stacking. There- fore, it is sufficient to consider only the symmetries of the stacked bilayer that interchange the two layers. The complete set of 2D point-group symmetry oper- ations capable of reversing the z -coordinate can be expressed as {E, mz , C2 α, Sn z }. For a ferromagnetic layer with a known layer group, the stacking model can be used to determine the stacking operations re- quired to generate or preserve each of the symmetry operations {E, mz , C2 α, Sn z } in the stacked bilayer. By further excluding the stacking operations that yield {E, mz } from those giving rise to {C2 α, Sn z }, one ensures that the sublattices in the stacked bilayer are not connected by E or mz , while still being re- lated by at least one symmetry operation. Through the above procedure, all stacking operations capable of generating 2D altermagnetism can be identified for a given monolayer. A detailed derivation is available in Ref.[134]. In addition, an alternative approach has also been proposed[139]. For a bilayer composed of monolayers S and S′ , the symmetry operations can be divided into two sets. The intersection set Zi = Gs ∩ Gs ′ (6) contains the symmetry elements common to both monolayers, where Gs denotes the layer group of monolayer S, and Gs ′ is obtained by applying the stacking operator ˆO to Gs , i.e., Gs ′ = ˆOGs ˆ O − 1 . The exchange set Ze = ˆOGs ∩ Gs ˆ O − 1 (7) consists of operations that interchange S and S′ . The symmetry group of the bilayer is then given by GB = Zi ∪ Ze , (8) which can be determined directly from the mono- layer layer group Gs and the stacking operator ˆO. By considering all symmetry operations of the mono- layer and the stacking operator ˆO, the symmetry of the bilayer under different stacking configurations can be obtained, and subsequently examined to deter- mine whether the conditions for 2D altermagnetism are satisfied. A more detailed discussion can be found Ref.[139]. Based on the above theories, the main conclusions regarding A-type altermagnets are drawn and sum- marized as follows. Intrinsic A-type altermagnetism, which arises from the direct stacking of two identi- cal monolayers, can only be realized in 17 specific layer groups (8-10, 19, 22, 50, 53-54, 57-60, 67-68,

10 FIG. 5. The crystal structure and band structure without spin-orbit coupling (SOC) of bilayer altermagnets. (a)-(c) and (g) The crystal structure of bilayer NiZrCl6 [ 134], twisted NiCl2 (with a twisting angle of 21.79◦ )[134], twisted MnPSe3 (with a twisting angle of 21.79◦ )[135] and twisted CrSBr (with a twisting angle of 90◦ )[136], respectively. (d)-(f) and (h) The band structure without SOC of bilayer NiZrCl6 , twisted NiCl2 (with a twisting angle of 21.79◦ ), twisted MnPSe3 (with a twisting angle of 21.79◦ ) and twisted CrSBr (with a twisting angle of 90◦ ) respectively. (i) The orientations of magnetic moments in the orthogonal bilayer CrSBr[137]. Red arrows denote the spin orientations along the easy magnetic axis, assuming that interlayer magnetic interactions are negligible. and 76). Accordingly, A-type antiferromagnetic bilay- ers belonging to these groups inherently exhibit alter- magnetism. The sublattices of A-type altermagnets can be connected only through S4 z or C2 α operations. Moreover, a C2 α rotation serves as a general stack- ing operation capable of generating an A-type alter- magnet from an arbitrary monolayer. In addition, if the initial monolayer possesses an in-plane C2 rota- tion symmetry, then its twisted bilayer with a twist angle θ must possess an in-plane C2 axis. In this case, the angle between the C2 axes of the monolayer and the twisted bilayer is θ/2, which generally preserves altermagnetism in the bilayer. Several bilayer altermagnets are illustrated in Fig. 5. Panels (a) and (b) show the crystal structures of bilayer NiZrCl6 and NiCl2 , respectively, the latter pos- sessing a twisting angle of 21.79◦ . Both systems ex- hibit A-type altermagnetism[134]. In these systems, the spin sublattices are connected by an in-plane C2 symmetry. The corresponding nonrelativistic band structures of bilayer NiZrCl6 and NiCl2 are presented in Figs. 5(d) and (e), indicating that bilayer NiCl2 exhibits a more pronounced spin splitting compared with bilayer NiZrCl6 . We next provide some qualita- tive insights to understand this difference. Owing to the weak vdW interaction between layers, the band structure of a bilayer can be approximated as sim- ply composed of the electronic structures of its con- stituent monolayers. For each layer of an A-type al- termagnet, when considered individually, the spin-up (spin-down) band structure of the top layer is identi- cal to the spin-down (spin-up) band structure of the bottom layer. Because bilayer NiZrCl6 is formed by direct stacking, it exhibits only a very small spin split- ting. This splitting originates from perturbations in- duced by interlayer vdW interactions, whose influence on the band structure is not significant. This point is further supported by the fact that the spin splitting in

11 bilayer NiZrI6 increases with the strength of the van der Waals interactions. In contrast, bilayer NiCl2 is obtained by rotating the top layer by 21.79◦ before stacking, which leads to a misalignment between the band structures of the two layers and consequently results in a large spin splitting. Furthermore, if the bandwidth of the monolayer is narrow, the spin split- ting in the bilayer will be small regardless of the stack- ing configuration. Therefore, for A-type altermagnets, a larger bandwidth of band structure in the monolayer generally favors a more pronounced spin splitting in the bilayer. Fig. 5(h) reveals that 90◦ -twisted bi- layer CrSBr—whose crystal structure is depicted in Fig. 5(g)—displays a pronounced spin splitting[136]. This arises not only from the large bandwidth but also from the strong anisotropy of the band structure of the monolayer CrSBr. Therefore, the magnitude of spin splitting in bilayer A-type altermagnets is governed by two key factors: the bandwidth and the anisotropy of the band structure in the monolayers. Although altermagnetism is a nonrelativistic effect and studies on altermagnets usually neglect SOC, this approximation appears to be problematic for bilayer CrSBr. As is well known, CrSBr exhibits strong in-plane magnetocrystalline anisotropy origi- nating from the SOC effect[141]. When construct- ing a twisted bilayer of CrSBr with a 90◦ rotation, the a axis of one layer is aligned parallel to the b axis of the other, rendering the easy magnetic axes of the two layers mutually perpendicular and resulting in a coplanar magnetic configuration, as illustrated in Fig. 5(i)[137]. Such an insight appears to be in conflict with the previous report claiming that bilayer CrSBr is an altermagnet[136]. Therefore, the impact of SOC on this system should be further investigated, and whether it qualifies as an altermagnet remains an open question. Furthermore, this issue generally arises when stacking 2D materials with an in-plane spin orientation. In addition to A-type altermagnets, we present a bilayer of the PT antiferromagnet MnPSe3 , stacked with a twisting angle of 21.79◦ , which exhibits 2D al- termagnetism, as shown in Fig. 5(c)[135]. Because each layer in this bilayer is a spin-degenerate antifer- romagnet, the spin splitting originates mainly from the perturbations induced by interlayer vdW inter- actions and is therefore very small, as illustrated in Fig. 5(f). From a qualitative perspective, 2D alter- magnets constructed from spin-degenerate antiferro- magnets are not expected to exhibit pronounced spin splitting. In the field of ferroelectrics, an interesting phe- nomenon has attracted considerable attention: when two nonpolar vdW layers are stacked in a noncen- trosymmetric configuration, an out-of-plane electric polarization emerges, which can be reversed via in- terlayer sliding[142, 143]. This phenomenon is re- ferred to as sliding ferroelectricity. Compared with conventional ferroelectrics, sliding ferroelectrics of- fer notable advantages, such as enhanced fatigue resistance and low energy consumption, rendering them highly promising for next-generation spintronic applications[144]. Stacking altermagnets provide a promising platform for exploring the interplay be- tween sliding ferroelectricity and altermagnetism, paving the way toward the realization of uncon- ventional magnetoelectric couplings. Recent stud- ies have primarily focused on the coupling between the nonrelativistic spin splitting and sliding ferro- electricity in stacking altermagnets, as shown in Fig. 6(a)[72, 73, 140]. To achieve the coexistence of altermagnetism and ferroelectricity, the stacked configuration must fulfill the symmetry criteria of both. Specifically, the real- space part of the spin group of a stacking altermag- net should correspond to a polar group. In such sys- tems, two distinct cases arise: (i) the nonrelativis- tic spin splitting remains unchanged upon ferroelec- tric polarization reversal, and (ii) the spin splitting reverses concomitantly with the ferroelectric polar- ization. From a symmetry perspective, the two op- posite ferroelectric states connected by sliding can be regarded as being related through a specific spin- group operation. The influence of this operation on the spin texture determines which case the stacking altermagnet belongs to. For example, stacking con- figurations {m0 01| (0.5, −0.16)} and {m0 01| (0.5, 0.16)} correspond to two opposite ferroelectric states of bi- layer AgF2 , which are related by the spin-group oper- ation [C2 | |mz ], as shown in Fig. 6(b)[73]. This opera- tion reverses the spin splitting, i.e., [C2 | |mz ]ε1 ( s, k) = ε2 ( −s, k), where ε1 and ε2 denote the energies of the two distinct stacking configurations. Consequently, the nonrelativistic spin splitting reverses in tandem with the reversal of ferroelectric polarization, as il- lustrated in Fig. 6(c). For bilayer MnPSe3 , the AB ({m0 01| ( 13 , 0)}) and BA ({m0 01| (− 13 , 0)}) stacking configurations correspond to two opposite ferroelectric states, which are related by the [E||mz ] symmetry op- eration, as shown in Fig. 6(d)[140]. This operation unchange the spin splitting, i.e., [E||mz ]ε AB (s, k) = ε BA( s, k). Therefore, sliding between the two fer- roelectric states leads to a reversal of polarization, whereas the spin splitting remains invariant. Although significant progress has been made in exploring bilayer altermagnets, existing studies have largely focused on how stacking, twisting, or sliding can generate altermagnetism, while the underlying in- terlayer exchange-coupling mechanisms remain poorly understood. Drawing insights from previous work on conventional magnetic bilayers, such as the Bethe- Slater-curve-like behavior reported in transition-metal dichalcogenide bilayers[145], it is reasonable to ex- pect that similar interlayer-coupling phenomena may also emerge in 2D altermagnets. In those systems, the competition between Pauli (Coulomb) repulsion and kinetic-energy gain across the van der Waals gap determines whether the interlayer interaction favors ferromagnetic or antiferromagnetic alignment. Such mechanisms, if present in altermagnets, would enable fine control of the overall magnetic state through ver- tical pressure or stacking engineering, offering an addi-

12 FIG. 6. (a) Schematic diagram of sliding-ferroelectricity-controlled spin-splitting switching, including configurations (upper) and bands (lower)[73]. (b) Energy barrier and evolution of out-of-plane polarization along the sliding path between AgF2 bilayers B1 and B2, with the inset showing the corresponding B1 and B2 configurations. B1 and B2 are related by the [C2 ∥ Mz ] symmetry[73]. (c) The band structures of B1 (left) and B2 (right) without spin-orbit coupling[73]. (d) Side view of the MnPSe3 bilayer in the ground states AB (left) and BA (right), which are related by the [E ∥ Mz ] symmetry[140]. (e) The crystal structure of heterostructure MnPSe3 / MoTe2 [ 110]. (f) Device configuration of the altermagnetic tunneling junction (AMTJ) CSe2 / MnS2 / CSe2 [ 91]. (g) Lowest decay rates of MnS2 evanescent states along the [001] direction, resolved with respect to spin and in-plane momentum (k∥ ) [91]. (h) Stacking configuration of the trilayer heterostructure SnS2 / MnPSe3 / SnS2 and two ferroelectric transition paths in the heterostructure[72]. (i) Mechanism and working states of the two-dimensional AMTJ, with red and blue representing opposite spins[91]. tional degree of freedom for designing multifunctional spintronic devices. Looking ahead, elucidating the in- terlayer exchange pathways, the dominant interaction mechanisms, and their dependence on structural de- grees of freedom in bilayer altermagnets will open a new avenue for achieving highly tunable altermagnetic states. B. Heterostructure Stacking In addition to homostructure stacking, heterostruc- ture stacking can also break the symmetry, thereby transforming an antiferromagnet into an altermagnet. Evidently, bilayer heterostructure stacking breaks both spatial inversion and mirror symmetry with re- spect to the xy plane, thereby lifting the spin degen- eracy. For example, MnPSe3 , a PT antiferromagnet, can acquire altermagnetism when stacked with MoTe2 to form a heterostructure that breaks PT symme- try, as shown in Fig. 6(e)[110]. In contrast, trilayer stacking is more intricate, as it does not inherently break any specific symmetry. For instance, the tri- layer heterostructure SnS2 / MnPSe3 / SnS2 retains PT symmetry under both AB and BA stacking[72]. In the AA and BB stacking configurations, the breaking of PT symmetry induces both ferroelectric polarization and altermagnetism in the heterostructure, with the polarization primarily arising from interlayer charge transfer between S and P atom pairs, as shown in Fig. 6(h). These two stacking configurations pos- sess opposite ferroelectric polarizations. They are re- lated by the PT operation, which in turn results in the two ferroelectric states exhibiting opposite spin splittings. Therefore, when sliding from the AA to the BB stacking configuration, both the ferroelec- tric polarization and the nonrelativistic spin splitting are simultaneously reversed. Because both the upper and lower SnS2 monolayers can slide relative to the MnPSe3 layer, the heterostructure exhibits two tran- sition pathways for switching the polarization from AA to BB stacking, as shown in Fig. 6(h). From the perspective of the energy barrier, Path 1 is more favorable for polarization switching. From an application perspective, heterostructures can also be employed to realize giant magnetoresis- tance or tunneling magnetoresistance[91]. An alter- magnetic tunneling junction (AMTJ) is illustrated in Fig. 6(f), where the altermagnet MnS2 serves as the insulating layer and the nonmagnetic metal CSe2 forms the electrodes on both sides. In MnS2 , k∥ - dependent evanescent states have spin-dependent de-

13 cay rates, as illustrated in Fig. 6(g). Therefore, in the parallel (P) state, where all layers share the same N´eel orientation, spin-up and spin-down carriers tun- nel through relatively low barriers, yielding a large current, while reversing the N´eel order in half of the altermagnetic insulating layers (AP state) drastically suppresses the current. A schematic illustration of the AMTJ mechanism is shown in Fig. 6(i). A giant TMR of 1.5 × 105 % at the Fermi level is predicted in the AMTJ CSe2 / MnS2 / CSe2 . In addition, uncon- ventional proximity effects have been explored in het- erostructures composed of altermagnets and nonmag- netic materials, where the characteristic momentum- alternating spin splitting of the altermagnet is directly transferred to the adjacent nonmagnetic layers[146]. V. MULTICOMPONENT ALTERMAGNET Chemical alloying and compositional asymmetry emerge as a potent strategy for unlocking altermag- netism in two-dimensional multicomponent systems, where they enable precise control over symmetry breaking. Multicomponent altermagnets are a type of altermagnet constructed by combining two materials with the same symmetry and similar chemical com- position, following the alloying concept[147]. As dis- cussed in Section II, the spin-degenerate band struc- ture in spin-degenerate antiferromagnets is protected by the four symmetry operations [C2 ∥ A], where A ∈ {P, τ, mz , C2 z }.By constructing multicomponent structures from spin-degenerate antiferromagnets and excluding these symmetry operations, one can gener- ate candidate structures from which a wide range of altermagnets can be identified. For materials with similar element and crystal structure, a series of rich multicomponent structures can be obtained by mixing two or more components in different concentrations. For example, CrSiS and CrSiSe share the same crystal structure and have sim- ilar chemical compositions, as illustrated in Fig. 7(a). Based on this crystal structure, multicomponent com- pounds can be constructed by varying the concentra- tions of S and Se, as shown in Figs. 7(b)-(d). Specifi- cally, Cr2 S i2 S 5 S e corresponds to an S:Se ratio of 5:1, Cr2 S i2 S 4 S e2 corresponds to 2:1, and Cr2 S i2 S 3 S e3 cor- responds to 1:1. These multicomponent compounds may have different symmetries. Through symmetry analysis, candidate structures that satisfy the sym- metry requirement of altermagnetism can be screened out. Furthermore, by calculating the band structure to verify their spin splitting, a series of multicompo- nent altermagnets can be obtained. The band struc- ture of Cr2 S i2 S 3 S e3 , calculated without SOC, is pre- sented in Fig. 7(e). Janus engineering has emerged as a popular strat- egy for obtaining altermagnets[110, 148–152]. In fact, the altermagnets realized through this approach con- stitute a subset of multicomponent altermagnets. In Janus structures, both spatial inversion and mirror symmetry in the xy-plane are broken. As a result, the spin degeneracy protected by the combination of these operations with spin reversal is lifted, thereby offering a route to realize altermagnetism through Janus engineering. For example, MnPSe3 is a two- dimensional antiferromagnetic semiconductor whose spin degeneracy is protected by PT symmetry[153]. Replacing one Se layer with S yields the Janus struc- ture Mn2 P 2 S 3 S e3 , as shown in Fig. 7(f), in which the breaking of PT symmetry results in a nonrela- tivistic spin-split band structure[110]. It can also be regarded as a multicomponent altermagnet formed by combining MnPS3 and MnPSe3 with an S:Se ratio of 1:1. Janus engineering can also be applied to extend 2D altermagnets based on the reported altermagnets. For example, V2 S eTeO, a Janus derivative of V2 S e2 O and V2 T e2 O , exhibits 2D altermagnetism[150]. To enable experimental synthesis, it is crucial to evaluate the stability of the proposed crystals. A por- tion of the multicomponent structures generated by the aforementioned methods may lack stability and, as a result, may be challenging to realize experimentally. To assess dynamic stability, it is necessary to analyze the phonon spectra of multicomponent altermagnets. A structure is deemed dynamically stable if its phonon spectrum exhibits no imaginary frequencies. For ther- modynamic stability, the formation energy should be calculated. Assuming that in the multicomponent material, A is a fixed component, B and C are ad- justable components, and the multi-component mate- rial is ABx C ( 1−x), the formation enthalpy is calculated by ∆HF = E(ABx C ( 1−x)) − xE(AB) − (1 − x)E(AC) in which E(ABx C ( 1−x)) denotes the total energy of ABx C ( 1−x) obtained from density functional theory (DFT) calculations and E(AB) and E(AC) the cor- responding total energies of the binary compounds. If the formation enthalpy of a structure is less than 0 and, at the same time, is the smallest among all multi- component structures at the same concentration, then the multicomponent structure ABx C ( 1−x) is thermo- dynamically stable, and the phase separation fails to appear. Previous studies have predominantly focused on dynamic stability, whereas the evaluation of ther- modynamic stability is equally essential. VI. ACHIEVING ALTERMAGNETISM VIA ADSORPTION Surface adsorption serves as a versatile strat- egy to engineer altermagnetism in 2D systems. Two-dimensional antiferromagnetic materials can host altermagnetism through surface adsorption[154]. Based on symmetry analysis, altermagnetism can be achieved in a 2D antiferromagnets by adsorbing atoms or molecules at specific sites to break the symmetry that protects spin degeneracy in the band structure, while preserving the rotational or mirror symmetry connecting opposite-spin sublattices. As shown in

14 FIG. 7. (a) The crystal structure of CrSiS3 or CrSiSe3 [ 147]. (b) The crystal structure of Cr2 S i2 S 5 S e, which is a multicomponent derivative formed from CrSiS3 and CrSiSe3 with a sulfur-to-selenium ratio of 5:1[147]. (c) and (d) The crystal structure of Cr2 S i2 S 4 S e2 and Cr2 S i2 S 3 S e3 [ 147]. (e) The band structure of Cr2 S i2 S 3 S e3 without spin-orbit coupling[147]. (f) The crystal structure of Mn2 P 2 S 3 S e3 [ 110]. Figs. 8(a) and (b), taking N´eel-type antiferromag- nets VPS3 and MnPSe3 with spin point group 2 32 m as examples, O atoms and NH3 molecules were tested at three possible adsorption sites. Calculations of adsorption energy and chemical potential phase dia- grams indicate that the highest-coverage adsorption is the most stable. The lowest-energy adsorption site lies exactly on a mirror plane, which breaks the PT sym- metry protecting the spin degeneracy. Since the mir- ror symmetry connecting the opposite-spin sublattices remains unbroken, a momentum-dependent alternat- ing spin polarization emerges in the absence of net magnetization, and the spin texture exhibits i -wave symmetry. 2D hydrognated graphyne (HGY) consists of alter- nating hexagonal and octagonal carbon rings. Its unique nonlinear sp bonds and π-conjugated frame- work allow a smooth transition from sp to sp 2 hy- bridization upon adsorption of nonmagnetic sp im- purity atoms (e.g., B, N, P, S). When adsorbed at specific sites in the HGY unit cell—namely, at the center bridging two adjacent octagonal rings and di- rectly above a hexagonal ring—these impurity atoms can induce local magnetic moments[155], as illustrated in Fig. 8(c). By bridging the octagonal rings and triggering electronic redistribution, they effectively in- duce collinear compensated spin order with zero net magnetization. Theoretical calculations reveal that B, N, and P atoms convert graphyne from a semiconduc- tor to a metal, while the S-adsorbed system remains semiconducting due to the even number of electrons contributed by S. Moreover, the spin splitting is ob- served after the adsorption of B and S atoms[155]. This momentum-dependent spin splitting has mag- nitudes of 177.4 meV for B and 82.3 meV for S, as shown in Figs. 8(d) and (e). In contrast, N and P atoms fail to induce altermagnetism due to next- nearest-neighbor ferromagnetic coupling. Moreover, recent work has demonstrated that the adsorption of hydrogen atoms on graphene can in- duce magnetic moments[156]. Controlling the num- ber of adsorbed H atoms and symmetry tuning enable the realization of multiple magnetic phases, including ferromagnetic, antiferromagnetic, ferrimagnetic, and altermagnetic states. Specifically, the covalent bond- ing between the hydrogen s orbital and the carbon pz orbital (out-of-plane) effectively removes an electron from the corresponding graphene sublattice, leaving behind an unpaired electron that generates a mag- netic moment localized on the complementary sublat- tice. To achieve the altermagnetic phase, it is neces- sary to adsorb at least four H atoms. This ensures the

15 FIG. 8. (a) Structural configuration and corresponding band structure of a monolayer VPS3 with an adsorbed oxygen atom[154]. (b) Structural configuration and corresponding band structure of a monolayer MnPSe3 with an adsorbed NH3 molecule[154]. (c) Schematic illustration of two-dimensional hydrogenated graphyne (HGY) along with its ferromagnetic and antiferromagnetic magnetic configurations[155]. (d) and (e) Band structures of HGY after adsorption of B and S atoms[155]. (f)-(g) Atomic magnetization and energy bands corresponding to a distinct spin configuration with spin groups 2 m and 1 32 m[156]. removal of PT symmetry. Two spin groups are com- patible with hydrogenated graphene: one is 2 m, which has a mirror plane perpendicular to the lattice paired with spin inversion; the other is 1 32 m, which includes a threefold rotation not paired with spin inversion, and three vertical mirror symmetries that are paired with spin inversion. As illustrated in Figs. 8(f) and (g), the first spin group requires a minimum of four H atoms to realize the altermagnetic phase, while the second requires at least six. Scanning tunneling mi- croscopy (STM) manipulation of individual H atoms has already demonstrated this concept, but the tech- nique cannot yet deliver the periodic arrays required for device-scale films. VII. ALTERMAGNETISM INDUCED BY ELECTRIC FIELD An out-of-plane electric field, a widely used and ef- ficient means to modulate physical properties in 2D materials[158], can break PT symmetry in 2D an- tiferromagnets, inducing reversible and tunable spin splitting characteristic. When an out-of-plane elec- tric field is applied, the top and bottom surfaces of the two-dimensional magnet acquire different elec- trostatic potentials, thereby breaking PT symme- try. Consequently, in PT -symmetric antiferromag- nets whose bands are spin-degenerate, a perpendic- ular electric field can lift the degeneracy; if the lat- tice also retains an in-plane rotation or mirror axis, the same field converts the material into an altermag- net. For example, CaMnSi is a PT antiferromagnet that possesses the [C2 ∥ Mx y ] symmetry, as illustrated in Fig. 9(a). Applying an out-of-plane electric field induces momentum-dependent spin splitting in the band structure, a defining feature of altermagnetism, as illustrated in Fig. 9(b)[90]. This occurs because the field breaks PT symmetry while preserving the [C2 ∥ Mx y ] symmetry. Moreover, as illustrated in Fig. 9(b), reversing the field direction simultaneously reverses the nonrela-

16 FIG. 9. (a) The crystal structure of CaMnSi[90]. (b) The spin-resolved band structure under opposite out-of-plane electric fields[90]. (c) the magnitude of spin splitting varies linearly with the strength of electric field[90]. (d) and (e) A schematic illustration of structure-distortion-induced altermagnetism[70]. (d) Ferroelectric antiferromagnets (FEAFMs) exhibit spin-degenerate band structures protected by translational symmetry. (e) Upon structural distortion, this translational symmetry is broken and replaced by mirror symmetry that links opposite spin sublattices, leading to the emergence of ferroelectric altermagnet (FEu A M). When an external electric field is applied, the spin splitting reverses concurrently with the polarization reversal. (f) The crystal structure of VOI2 before and after distortion[70]. (g) Energies of monolayer VOI2 in both undistorted and distorted configurations as a function of lattice constant, showing that strain can drive a phase transition between the FEAFM phase and the ferroelectric altermagnet (FEAM) phase[70]. (h) Total energies of the four magnetic configurations in the Mn2 S e2 O as a function of biaxial strain, evaluated relative to the energy of the FM phase. (i) Exchange interaction parameters of the Mn2 S e2 O as a function of biaxial strain[157]. tivistic spin splitting, as the two states are connected by the PT operation, i.e., PT ε+ E (s, k) = ε− E (−s, k). Furthermore, the magnitude of spin splitting varies linearly with the strength of electric field, as shown in Fig. 9(c). By screening two-dimensional mate- rial databases, 24 antiferromagnets are predicted to exhibit altermagnetism under an out-of-plane electric field[90]. In addition, MnPSe3 [ 110] and VPSe3 [ 152] are also proposed to realize altermagnetism via this mechanism. VIII. ALTERMAGNETISM INDUCED BY STRUCTURE DISTORTION Structural distortions in 2D materials can break translational symmetry while preserving ferroelec- tric order, providing a route to realize ferroelectric altermagnetism[70, 74]. It always can arise from Jahn- Teller effects, Peierls-type dimerization, or structural modifications externally tuned by temperature, pres- sure, or chemical doping[159]. In ferroelectric systems, inversion symmetry is intrinsically broken. However, the magnetic sublattices are typically connected by translational symmetry, giving rise to conventional an- tiferromagnets, as shown in Fig. 9(d)[70]. To real- ize ferroelectric altermagnets, translational symmetry must be broken while preserving ferroelectric polar-

17 ization, accompanied by the emergence of a symme- try operation that connects the magnetic sublattices. This can be accomplished through lattice distortions, as illustrated in Fig. 9(e). When these distortions occur perpendicular to the polarization, they break the translational symmetry while preserving the FE order, and emerge mirror symmetry that connects the magnetic sublattices, thereby giving rise to ferroelec- tric altermagnetism. In this case, the spin splitting of the altermagnet is coupled to the direction of the fer- roelectric polarization. Under external fields, the spin splitting reverses concurrently with the ferroelectric polarization reversal. Fig. 9(f) illustrates an exam- ple in VOI2 , where V-V dimerization breaks transla- tional symmetry while preserving mirror symmetry, thereby driving a transition from the antiferromag- netism to the altermagnetism[70]. This approach pro- vides a platform for exploring the interplay between ferroelectricity and altermagnetism. IX. ALTERMAGNETISM INDUCED BY STRAIN Strain can induce structure distortions and modu- late magnetic exchange interactions, enabling the con- version of ferromagnets or spin-degenerate antiferro- magnets into altermagnets. Because 2D materials ex- hibit exceptional flexibility and strong strain sensi- tivity, strain engineering emerges as an effective and promising strategy for tuning their electrical, mag- netic, and optical properties[160]. Meanwhile, the ex- perimental feasibility of applying controllable strain to 2D materials is well established[161], for example through substrate-induced lattice mismatch. There- fore, it is feasible to generate and modulate altermag- netism in 2D materials via strain. Recently, there has been growing interest in strain- tuning the properties of 2D altermagnets, including studies on the multipiezo effect[108, 115, 162, 163]— which encompasses piezovalley responses, piezomag- netism, and piezoelectricity—as well as on strain- induced modulation of their electronic structures[164]. Furthermore, studies have shown that strain can in- duce topological phase transitions and give rise to a rich variety of topological states[162, 163, 165]. In addition, a strain-induced transition from g-wave to d -wave altermagnetism has also been proposed[166]. Strain not only can tune the properties of existing 2D altermagnets, but also serves as an effective mech- anism for inducing altermagnetism. From a mecha- nistic perspective, strain can induce altermagnetism through lattice distortions[70] and the modulation of exchange interactions[157]. As discussed in the previ- ous section, lattice distortions can break specific sym- metries, thereby giving rise to altermagnetism. Such lattice distortions can be generated by applying strain. We next take VOI2 as an illustrative example. As il- lustrated in Fig. 9(f), VOI2 exhibits two structural phases, whose stability is strongly strain dependent. As shown in Fig. 9(g), tensile strain expands the V- V distances and drives the system into the antiferro- magnetism, whereas compressive strain stabilizes the altermagnetism. In this way, strain effectively con- trols the structural and magnetic landscape of VOI2 , enabling reversible switching between the antiferro- magnetism and altermagnetism. Lattice distortions can break the relevant symme- tries and convert a spin-degenerate antiferromagnet into an altermagnet, while the modulation of exchange interactions can drive a transition from a ferromag- netic state to an altermagnetic one. Notably, strain- driven transitions between ferromagnetic and antifer- romagnetic phases have been widely investigated in earlier works[167–169]. Recent studies have shown that the ferromagnet Mn2 S e2 O undergoes a transition to an altermagnetic state when applying a compressive strain of about 5%[157]. As shown in Fig. 9(h), com- pressive strain progressively reduces the energy dif- ference between the ferromagnetic and altermagnetic states, and once the strain exceeds 5%, the altermag- netic state becomes energetically favored. From the perspective of exchange interactions, Fig. 9(i) reveals that the nearest-neighbor coupling J1 changes from positive to negative under compression, which pro- motes the stabilization of the altermagnetic state. X. TWO-DIMENSIONAL ORGANIC ALTERMAGNET Recently, there has been increasing interest in 2D organic altermagnets[101, 107, 113, 170, 171]. Based on H¨uckel theory and nonbonding molecu- lar orbitals (NBMOs), a reverse design strategy has been developed, in which organic ligands containing NBMOs are selected and assembled with transition- metal to construct a series of 2D MOF monolay- ers with altermagnetism[107]. In addition, alter- magnetism has also been explored in π-conjugated nanographenes. Nonalternant nanographene units, such as dibenzo[ef,kl]heptalene (DBH), can serve as magnetic building blocks. When assembled into 2D lattices, a 90◦ rotation of the DBH units induces the characteristic momentum-dependent spin splitting of altermagnets[170]. XI. PERSPECTIVE In this section, we provide a forward-looking per- spective on two-dimensional altermagnets, highlight- ing potential future milestones, promising applica- tions across diverse fields, and possible directions for future research. A. Experimental: Synthesis and Characterization There is no doubt that the field of two-dimensional altermagnets is still in its infancy. Although theoreti-

18
cal advances have been rapid, experimental confirma- netic and antiferromagnetic systems, are anticipated
tion remains scarce. The most significant milestone in to be highly promising for the future spintronic ap-
this field is the first experimental realization of a 2D plications. A prototypical spintronic device is the
altermagnet together with definitive evidence of its al- spin valve, comprising two magnetic conducting lay-
termagnetism. Moreover, as outlined in this review, ers separated by a nonmagnetic spacer. Depending on
numerous strategies for realizing 2D altermagnetism whether the spacer is metallic or insulating, the device
have been proposed at the theoretical level. Yet com- exhibits giant magnetoresistance (GMR) or tunneling
pelling experimental evidence supporting their fea- magnetoresistance (TMR), respectively. Recent stud-
sibility and validity is still lacking. Demonstrating ies indicate that, in contrast to conventional magnetic
these strategies experimentally would constitute an- tunnel junctions (MTJs), altermagnetic tunnel junc-
other major milestone, as it would indicate that 2D al- tions (AMTJs) exhibit an anomalous scaling behav-
termagnetism can be engineered directly from conven- ior, where TMR diminishes as the barrier thickness
tional two-dimensional magnets. Such progress would increases[177]. Consequently, the TMR is expected
substantially expand the materials space of 2D alter- to achieve its largest value in the monolayer limit of
magnets, providing not only a broad platform for ex- the barrier. This is especially beneficial in 2D ma-
ploring their interplay with other physical properties, terials, which allow precise layer control and nearly
but also an enriched reservoir of candidate materials perfect interfaces in van der Waals heterostructures.
for designing multifunctional devices. From an appli- Taken together, these features suggest that AMTJs
cation perspective, it is also crucial to find air-stable based on 2D altermagnets can achieve superior perfor-
2D altermagnets with N´eel temperatures at or above mance compared to those based on 3D altermagnets.
room temperature. Equally important is the ability to Further experimental confirmation of this anomalous
produce high-quality atomic layers over large areas. scaling behavior would be highly valuable, as it could
All these developments, if realized, will help propel contribute to guiding the design of AMTJs based on
this emerging field toward its next stage of growth. 2D altermagnets. In addition, large TMR has been
Owing to its sensitivity to electron momentum, theoretically predicted in various AMTJs based on 2D
angle-resolved photoemission spectroscopy (ARPES) altermagnets[91, 177], which call for experimental val-
enables direct measurements of the electronic struc- idation. A natural next step for research is to realize
ture of crystalline materials. Previous studies have these AMTJs as building blocks in magnetic random
established ARPES as the most direct and power- access memories (MRAM) devices, unlocking their po-
ful technique for identifying altermagnetism[20–23, tential for high-performance, robust, and high-density
172, 173]. One of the key characteristics of alter- spintronic memory applications. Without a doubt, re-
magnets is the nonrelativistic spin splitting of elec- alizing this would constitute a significant milestone in
tronic bands. Consequently, spin-resolved ARPES, the application of 2D altermagnets.
which can directly distinguish between spin-up and Recent pioneering work has identified spin-split
spin-down states in momentum space, has become floating edge states in two-dimensional altermagnetic
the most widely used experimental tool for unambigu- second-order topological insulators[178]. Exploiting
ously demonstrating altermagnetism[21–23, 172, 173]. these states has led to the proposal of an unconven-
ARPES has also been extensively employed in the tional edge tunnel magnetoresistance effect, which of-
study of 2D systems[174, 175]. However, whether fers an electrically switchable platform with an ul-
ARPES techniques, including soft X-ray ARPES and trahigh on/off ratio for next-generation spintronic
spin-resolved ARPES, can be effectively exploited to devices[178]. In addition, spin-selective quantum
provide unambiguous experimental evidence for al- transport in heterostructures composed of a normal
termagnetism in 2D materials remains an open and metal and a two-dimensional d -wave altermagnet has
largely unexplored question. In 2D altermagnets, been theoretically investigated, revealing a Fermi-
both the magneto-optical Kerr effect[176] and anoma- surface-geometry-driven mechanism for achieving per-
lous transport properties[67] have been theoretically fect spin polarization[179]. Building on this mecha-
predicted. Although these responses do not constitute nism, all-electrically controlled spin filters and spin
sufficient proof of altermagnetism, they can serve as valves have been proposed, establishing a promising
indirect or initial experimental evidence. platform for scalable, magnetic-field-free spintronic
devices[179]. These results further highlight the ad-
vantages of 2D altermagnets for spintronic applica-
B. Spintronics tions and provide valuable insights for the design of
next-generation spintronic devices.
Altermagnets hold significant promise for applica- The successful realization of 2D spintronic de-
tions in spintronics. Conventional ferromagnet-based vices and circuits requires the ability to achieve
spintronic devices face inherent limitations in storage spin-charge conversion, spin transport, and spin
density, operation speed, and robustness. Meanwhile, manipulation[180]. However, current studies on 2D al-
conventional antiferromagnets, with spin-degenerate termagnets have predominantly focused on spin trans-
electronic structures, are limited in their ability to port, while comparatively little attention has been
generate large spin currents. As a result, altermag- paid to the realization of spin-charge conversion and
nets, integrating key advantages of both ferromag- spin manipulation, as well as to the corresponding de-

19 vice concepts. Therefore, corresponding studies de- serve increased attention in future research. Another promising direction that deserves attention is mag- netic skyrmions. Because of the vanishing topolog- ical charge, the skyrmion Hall effect has long been considered absent in antiferromagnets. However, re- cent studies predict that this effect can emerge in 2D altermagnets[181]. This intriguing prediction awaits experimental validation. As the most prominent ap- plication of skyrmions is racetrack memory[182], de- veloping racetrack devices based on 2D altermag- netic skyrmions should become a key focus of future research. Whether other two-dimensional magnetic topological quasiparticles can be realized in 2D alter- magnets remains an open question. It is also worth exploring whether phenomena previously realized in 2D magnetic systems can be extended to 2D alter- magnets. For example, four distinct resistance states have been achieved in the tunnel junctions composed of Fen G eTe2 layers separated by a 2D ferroelectric In2 S e3 barrier[183], raising the question of whether analogous functionalities can be realized in altermag- netic counterparts. C. Topology, Cornertronics and Superconductivity Substantial progress has been made in understand- ing the interplay between magnetism and nontriv- ial band topology[184–186], providing an established context for discussing topological responses in 2D altermagnets. To date, a growing number of two- dimensional altermagnetic topological materials have been proposed, including, but not limited to, CrO[104, 105], V2 T e2 O [67, 100], MnS2 [ 91], and Cr2 B Al[106]. The quantum anomalous Hall effect (QAHE) is a topological quantum transport phenomenon in the ab- sence of an external magnetic field, and it holds signifi- cant promise for low-dissipation electronics and topo- logical quantum devices. Recently, the QAHE has been theoretically predicted in 2D altermagnets[104, 116], representing the first realization of this effect in collinear antiferromagnetic systems. These predic- tions call for experimental efforts to realize the QAHE in 2D altermagnets, which would not only confirm the theoretical proposals but also open new avenues for antiferromagnetic topological electronics. In ad- dition, the symmetry requirements for topologically protected spin-polarized Dirac points in 2D altermag- nets have been systematically analyzed, and the cor- responding altermagnetic spin-wallpaper groups have been identified[187]. In the future, high-throughput first-principles calculations could be employed to sys- tematically search for 2D altermagnetic materials with specific topological properties. It remains an open question whether stacking configurations and exter- nal electric fields can modulate the topological phases of 2D altermagnets. Cornertronics is an emerging research field that ex- ploits symmetry-protected electronic states localized at the corners of higher-order topological insulators, known as corner states, as controllable degrees of free- dom for emerging electronic functionalities. In this context, 2D altermagnets have recently demonstrated unique potential in this direction. On the one hand, a spin-corner locking mechanism has been identified in second-order topological insulators (SOTIs) realized in 2D altermagnetic systems[91, 188, 189]. As a result, 2D altermagnets constitute an ideal platform for re- alizing spin-mediated higher-order topological corner states. Meanwhile, intrinsic spin-corner coupling is re- garded as a characteristic of 2D altermagnetic SOTIs. In bilayer NiZrI6 nanodisks, a spin-corner-layer lock- ing mechanism is realized by introducing the layer de- gree of freedom[190]. On the other hand, corner states can be flexibly manipulated by external fields, such as electric fields and strain. For example, in bilayer NiZrI6 nanodisks, a vertical electric field enables si- multaneous reversal of the spin and layer polarizations of corner states, thereby modulating the absorption, emission, and polarization of terahertz waves[190]. Furthermore, in 2D altermagnets CrO and Cr2 S e2 O , uniaxial strain breaks the Mx y symmetry, leading to spin-resolved corner modes and a corner-polarized second-order topological insulator[189]. These results suggest that 2D altermagnets provide a fertile play- ground for cornertronics. Looking ahead, the rich in- terplay between topology and altermagnetism is ex- pected to further expand the scope of cornertronics in two-dimensional systems. The interplay between altermagnetism and super- conductivity has attracted growing research interest in recent years, owing to the intrinsic spin-momentum locking characteristic of altermagnets[191]. In this context, a variety of intriguing phenomena have been theoretically predicted, including the supercon- ducting diode effect[38, 41], Josephson effects[192], Andreev reflection[39, 193], and finite-momentum superconductivity[194–197]. Meanwhile, altermag- nets provide a fertile platform for exploring uncon- ventional and topological superconductivity. A strong exchange field can drive the emergence of Bogoli- ubov Fermi surfaces in altermagnets, which sup- press conventional spin-singlet s-wave pairing and ultimately induce chiral p-wave superconductivity or finite-momentum superconductivity[195]. In 2D metals with d -wave altermagnetism and Rashba spin-orbit coupling, the system favors mixed spin- singlet s-wave and spin-triplet p-wave pairings, and when p-wave pairing dominates, it can host var- ious topological superconducting phases[75]. Re- cent studies on altermagnet-superconductor interfaces and heterostructures have revealed a variety of non- trivial superconducting interface effects, including orientation-dependent Andreev reflection[193], finite- momentum Cooper pairing[196], gapless supercon- ducting states[76, 77], as well as higher-order topo- logical superconductivity supporting Majorana corner modes[198]. In the two-dimensional limit, their atomi- cally sharp interfaces and flexible van der Waals stack- ing markedly enhance proximity coupling, positioning

20
2D altermagnets as a promising platform for super- doping offers an alternative route to achieving electri-
conducting spintronic devices. cal control of magnetism[55]. However, this approach

D.

doping offers an alternative route to achieving electri-
cal control of magnetism[55]. However, this approach
remains largely unexplored in two-dimensional alter-
magnets.

Valleytronics and multiferroics

Valleytronics is a field of research that exploits the valley degree of freedom in momentum space as an information carrier, with the goal of enabling low- power and high-efficiency information encoding, pro- cessing, and storage. 2D materials are regarded as an ideal platform for valleytronics[199], which provides a solid basis for exploring valley-dependent phenomena in 2D altermagnets and their potential valleytronic applications. In 2D altermagnets, spin-momentum locking protected by spin-group symmetries naturally gives rise to a theoretical basis for crystal-symmetry- protected spin-valley locking, also referred to as C- paired spin-valley locking[12]. Since the spin-valley locking is protected by crystal symmetries, various ex- ternal approaches, such as strain and electric fields, can be employed to manipulate the valley degree of freedom, thereby giving rise to a variety of intriguing physical phenomena. For example, valley polariza- tion can be realized in 2D altermagnets by applying strain[36, 164] or external electric fields[35]. Mean- while, the crystal valley Hall effect has been proposed and is predicted to be realized in 2D altermagnets Fe2 W Se4 and Fe2 W S4 [ 34]. These phenomena call for further experimental investigation. Conventional val- leytronic materials have long faced challenges includ- ing limited valley lifetimes, insufficient material qual- ity and scalability, and low operating temperatures, which collectively hinder the practical implementa- tion of valleytronic applications[199]. Whether 2D valleytronic altermagnets suffer from similar limita- tions, or instead offer viable routes to overcome these challenges, remains an open question that warrants further investigation. Electrical control of magnetism provides an energy- efficient route for tuning magnetism-related physical properties, because the absence of heating by cur- rents. In multiferroic materials, this control can be re- alized through magnetoelectric coupling mechanisms. At present, theoretically proposed 2D multiferroic al- termagnets include ferroelectric altermagnets[70, 74], in which ferroelectricity coexists with altermagnetism, and antiferroelectric altermagnets[49], where antifer- roelectric order is combined with altermagnetism. Among these systems, the ferroelectric altermagnets, which conmbine altermagnetism and sliding ferroelec- tricity, is particularly promising for applications, ow- ing to its enhanced fatigue resistance and low energy consumption. Moreover, an unconventional magne- toelectric coupling mechanism has been proposed, in which the reversal of nonrelativistic spin splitting is accompanied by ferroelectric switching[72, 73, 140]. A detailed discussion of these can be found in Section IV. Experimental validation of these results would be a crucial step toward enabling practical applications of multiferroic altermagnets. In addition, electrostatic

E. Miscellaneous

Future research should not only aim to expand the library of candidate materials, but also to deepen our understanding of their underlying physics. De- veloping symmetry-guided effective theories—such as k · p models for d -wave altermagnets[200]—and benchmarking them against first-principles calcula- tions will be essential for uncovering the micro- scopic origins of intrinsic transport phenomena and for guiding experimental probes. Moreover, the de- sign of novel spintronic devices based on 2D alter- magnets remains to be explored. Furthermore, in- tegrating 2D altermagnets with other research areas could yield a wider variety of emergent physical phe- nomena and represents a promising direction for fu- ture studies. In addition, it remains an open ques- tion whether recently proposed symmetry-driven un- conventional magnetism[201], such as 2D fully com- pensated ferrimagnets[86], Type IV 2D Collinear Magnets[88] and p-wave magnets[202], can be realized using the design strategies summarized in this review. Continued exploration of 2D altermagnetism promises to unveil new physics and to spawn spintronic devices whose efficiency and functionality exceed current lim- its, positioning 2D altermagnets at the forefront of future magnetism research and quantum technologies.

XII. CONCLUSION AND CHALLENGES

This review centers on the symmetry principles and
material realizations of two-dimensional altermagnets.
It highlights their symmetry descriptions and clas-
sifications, compiles the theoretically predicted can-
didates reported so far, and discusses diverse design
strategies for achieving altermagnetism in 2D materi-
als. Finally, we offer a perspective for the field and
highlight several promising directions where future
breakthroughs may emerge. Again, the field is still in
its infancy: most predictions rest on calculations, and
the key experiments that would confirm—or refute—
two-dimensional altermagnetism in the proposed can-
didates have yet to be performed. Therefore, ad-
vancing experimental efforts will be crucial for future
progress.
Research on 2D altermagnets has already made no-
table progress, and the rapidly growing body of theo-
retical and computational studies reflects the surging
interest and momentum in this emerging area. Nev-
ertheless, many challenges still remain in achieving a
deeper understanding of 2D altermagnetism and in
predicting new 2D altermagnets. In magnetic sys-
tems, particularly those containing transition-metal
or rare-earth elements, incorporating a Hubbard U

21 correction is often essential in first-principles calcu- lations. However, the choice of U value can strongly influence the calculated electronic and magnetic prop- erties. Key magnetic quantities, such as exchange splitting, local magnetic moments, interlayer exchange interactions, and even the relative energetic stability of different magnetic configurations, can exhibit pro- nounced U-dependence. For 2D altermagnets, where the prediction fundamentally relies on an unambigu- ous identification of the magnetic ground state, this U sensitivity introduces non-negligible uncertainty. Moreover, the choice of the DFT+U scheme, whether the Dudarev’s or the Liechtenstein’s approach, can also affect the correct prediction of the magnetic ground state, as demonstrated in the case of mono- layer CrOCl[203]. At the same time, a broadly applicable strategy for predicting the true magnetic ground state is still lacking. As summarized in Section III, most pre- dicted altermagnets have had their ground states de- termined by comparing the energies of only a few collinear spin configurations. However, the mag- netic phase space is inherently large, encompassing collinear, coplanar, and noncoplanar arrangements. If the sampled configurations fail to include the ac- tual ground state, the prediction is inevitably incor- rect. Although spin-spiral calculations have shown promise in identifying magnetic ground states in two- dimensional magnets[89, 117], this approach is limited by its requirement of a primitive cell containing only one magnetic atom, which limits its applicability to more complex materials. In most theoretical studies of two-dimensional al- termagnets, spin-orbit coupling (SOC) is typically ne- glected. However, this approximation can lead to sub- stantial inaccuracies in certain cases. For example, it has been reported that materials predicted to be fer- romagnetic in the absence of SOC actually develop a chiral spin-spiral ground state once SOC is taken into account[117]. As discussed in Section IV, a sim- ilar concern arises in the twisted 90◦ bilayer CrSBr, where the easy axes of the two layers are mutually orthogonal. Consequently, whether a collinear spin configuration can remain stable after SOC is included remains an unresolved issue that warrants careful ex- amination. Further studies on altermagnets should take into account whether the effects induced by SOC can be neglected. ACKNOWLEDGMENTS This work is financially supported by the Na- tional Natural Science Foundation of China (Grant No. 12474229 and 12504268), the Fundamental Research Funds for the Central Universities(No. 2025ZYGXZR039) and the Guangdong Provincial Key Laboratory of Functional and Intelligent Hy- brid Materials and Devices (Grant No. 2023- GDKLFIHMD-04). [1] L. ˇSmejkal, J. 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