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Ferroelasticity tunable altermagnets

Ning Ding, Haoshen Ye, Shan-Shan Wang,∗ and Shuai Dong†

arXiv:2510.14193v2 [cond-mat.mtrl-sci] 24 Dec 2025

Key Laboratory of Quantum Materials and Devices of Ministry of Education,

School of Physics, Southeast University, Nanjing 21189, China

School of Physics, Southeast University, Nanjing 21189, China (Dated: December 25, 2025) Altermagnets have garnered great interest due to their non-relativistic spin splitting and novel physical properties. However, the control of altermagnetic states remains underexplored. Here, we propose a unique multiferroic state, i.e. ferroelastic altermagnetic state, in which ferroelastic strain couples directly to the spin-splitting. Through symmetry analysis and first-principles calculations, we identify the ferroelastic d-wave altermagnetism of puckered pentagonal CoSe2 monolayer. Inter- estingly, uniaxial stress can induce a ferroelastic phase transition, accompanied by a 90◦ rotation of the spin-splitting bands. Cooperative rotation of the lattice and N´eel vectors preserves the sign of Kerr angle, whereas noncooperative rotation reverses it. Our work provides a general strat- egy for manipulating altermagnetism in multiferroic systems and opens other avenues for exploring emergent magnetoelastic phenomena. Altermagnetism represents an emerging research fron- tier of condensed matter research, which hosts unconven- tional magnetic phenomena and harnesses their unique physical properties for technological innovation [1–4]. With antiferromagnetic spin configurations, altermag- nets exhibit non-relativistic spin splitting in Brillouin zone, and thus combine the advantageous features of both ferromagnets and antiferromagnets. Specifically, they can display a magneto-optical response similar to ferro- magnets while maintaining the negligible stray fields and field robustness of antiferromagnets [5, 6]. Recent experimental researches have discovered var- ious altermagnetic materials, including MnTe [7–9], CrSb [10–12], α-MnTe2 [13], KV2 S e2 O [14], Mn5 S i3 [15], and CoNb4 S e8 [16]. Moreover, many theoretical works have predicted the existence of altermagnetism in many van der Waals systems [17–24]. Stacking and twisting engineering has been proposed to be a promising route to produce two-dimensional (2D) altermagnets [25–29]. Multiferroic systems combining more than one fer- roic orders, usually magnetic and ferroelectric orders, hold promise for controlling magnetism through nonmag- netic fields [30]. Inspired by this prospect, magneto- electric coupling has been proposed in various ferroelec- tric/antiferroelectric altermagnets [31–42], where ferro- electricity can be used to control altermagnetism, offering opportunities for electrically tunable spintronics. Elec- tric field control is indeed experimentally straightforward to implement. However, this approach becomes ineffec- tive when the system exhibits metallic behavior. Nev- ertheless, the metallic nature of altermagnetic systems enables promising functionalities such as spin-polarized current generation, prompting the need to develop al- ternative manipulation strategies for manipulating alter- magnetism. In fact, multiferroics are not limited to fer- roelectric magnets. Ferroelasticity is also a kind of fer- roicity, broader than ferroelectricity [43–45], which can ∗ [email protected] † [email protected] be coupled with altermagnets. This coupling could en- able the control of spin-related properties such as the magneto-optical Kerr effect (MOKE) through ferroelas- tic strain engineering. However, research on the coupling between altermagnetism and ferroelasticity remains rare and urgently needed. In this Letter, we theoretically study the ferroelastic altermagnets, which enable ferroelastic control of spin splitting in electronic bands, as depicted in Fig. 1. The material candidate is CoSe2 monolayer, a 2D pentagonal lattice with unequal lattice parameters [46, 47]. Con- nected by rotation symmetry, the opposite spin sublat- tices violate the parity-time reversal symmetry, and the time-reversal after fractional translation. As a result, the spin splitting occurs, which can be rotated by 90◦ through ferroelastic switching. Methods. First-principles calculations were performed with the projector augmented-wave (PAW) potentials as implemented in Vienna ab initio Simulation Pack- age (VASP) [48]. The Perdew-Burke-Ernzerhof (PBE) parameterization of the generalized gradient approxima- tion (GGA) was used for the exchange correlation func- tional [49]. The Hubbard U (Ue ff = 3 eV) was applied using the Dudarev parametrization to better describe the correlation effects in transition-metal magnetic ions [50] according to the previous works [47, 51]. In addition, the cases of different Hubbard Ue ff were shown in Fig. S1 in the Supplemental Materials (SM) [52]. A vacuum layer thicker than 15 ˚A was used to avoid artificial interactions from periodic boundary conditions. The plane-wave cut- off energy was 500 eV, and Co’s 4s3d3p electrons were treated as valence states. The k-point grid of 6 × 6 × 1 was employed for structural relaxation and static cal- culations. The energy convergence criterion was set to 10− 6 eV and the Hellman-Feynman forces criterion dur- ing structural relaxation was 0.005 eV/˚A. The calculation of the phonon spectrum was used by the finite-difference method [53, 54]. Ab initio molecular dynamics (AIMD) simulations were performed at 300 K by using an N V T ensemble that lasts 10 ps with a time step of 4 fs [55]. The tight-binding Hamiltonian, which incorporates

2 FIG. 1. Schematic illustration of ferroelastic altermagnetism and its switching. We postulate the lattice constants (a0 and b0 ) of the ground state of the ferroelastic system, with a0 < b0 . The horizontal coordinate is a′ = (2a − a0 − b0 ) /(b0 − a0 ) , where a is lattice constant of a-direction during the ferroe- lastic switching. To enhance the visual clarity, the unit cells of ferroelastic states are represented by transparent colored boxes: light red denotes ferroelastic state I with lattice con- stant a = a0 (a′ = −1), purple represents the paraelastic state with a = (a0 + b0 ) /2 (a′ = 0), and light blue indicates ferroelastic state II with a = b0 (a′ = +1). The following diagrams are spin-momentum locking diagrams of two ferroe- lasctic states. Co-d and Se-p orbitals was constructed using maximally localized Wannier functions (MLWF) generated by the WANNIER90 code [56]. The spin-resolved conductivity σsi j was calculated by using the BoltzWann module with a k-point mesh of 500 × 500 × 1 [57]. The optical conduc- tivity was calculated within the MLWF basis employing a high-density k-point grid of 200 × 200 × 1 [58–60]. CoSe2 monolayer has the weak anisotropy, which can be confirmed by the nearly equal optical conductivity tensors (ξx x and ξy y) as shown in Fig. S2 [52]. Thereby, the MOKE can be described by the Kerr angle θK and ellipticity ηK , which can be quantitatively characterized by the following equation [61]: ΦK = θK + iηK = i 2ωD c ξx y ξs ub x x , (1) where c represents the speed of light in vacuum, ω is frequency of incident light, and D is the thickness of cal- culated material (D = 3 ˚A in our system). ξs ub x x is the diagonal element of the optical conductivity tensor for SiO2 substrate, which equals i(1 − n2 )ω/4π with refrac- tive index n = 1.546. To determine the magnetic ground state, the Monte Carlo (MC) method with the Metropolis algorithm was employed. A 20 × 20 lattice with periodic boundary con- ditions was used in the MC simulation, and larger lat- tices were also tested to verify the physical results. The initial 5 × 105 MC steps were discarded for thermal equi- FIG. 2. (a) Top and side views of pentagonal CoSe2 mono- layer. d: the thickness. (b) Phonon spectrum of monolayer CoSe2 . (c) Orbital-projected density of state (PDOS) of one spin-up Co ion. Inset: schematic of electron occupation of Co2 + ’s 3d orbitals under a quasi-tetragonal crystal field. The unpaired dz 2 orbital can be visualized by the spin polarized electron cloud colored in green. librium, followed by 5 × 105 MC steps used for statistical averaging at each temperature. Temperature scanning was performed by using a quenching process. Structure and magnetism. The space group of pen- tagonal CoSe2 monolayer is P 21 / c (No. 14), identical to the previously predicted structural analogue CoS2 mono- layer [47]. As shown in Fig. 2(a), CoSe2 monolayer ex- hibits a buckled pentagonal structure. The buckling leads to a thickness d = 1.39 ˚A, with each Co ion sandwiched between two top and two bottom Se ions. Another con- sequence of buckling is that the in-plane four-fold rota- tional symmetry is broken, which results in the inequal- ity of the in-plane lattice constants i.e. anisotropy (a = 5.577 ˚A, b = 5.770 ˚A) and two kinds of Co-Se bonds with unequal lengths (2.343 ˚A and 2.357 ˚A), according to our calculations. The phonon spectrum (Fig. 2(b)) shows no obvious imaginary modes, confirming its dy- namical stability. The elastic constants Ci j have been calculated using energy-strain method by VASPKIT [62] as shown in Table S1 [52]. The pentagonal CoSe2 mono- layer meets the Born criteria: C1 1 > 0, C6 6 > 0, and C1 1 × C2 2 > C1 2 × C1 2, indicating the mechanical stabil- ity [63]. Furthermore, angle-dependent Young’s modulus

3 and Poisson’s ratio derived from elastic constants Ci j can be found in Fig. S3 [52]. As depicted in Fig. S4 [52], AIMD simulation at 300 K has been performed to check the thermal stability of CoSe2 monolayer. There is no obvious distortion after a 10 ps AIMD simulation at 300 K, demonstrating the thermodynamic stability of CoSe2 monolayer. Unpaired 3d orbitals may exist in the pentagonal CoSe2 monolayer, potentially inducing magnetism. Our DFT calculation indeed finds 1 µB / Co for the ferromag- netic state, implying the low-spin state of 3d5 or 3d7 . The orbital-projected density of state (PDOS) of one Co ion is shown Fig. 2(c). It seems that the t2 g orbitals (dx y /dy z /dx z ) are fully occupied, while the dx 2 −y2 orbital is fully empty. The dz 2 orbital is only occuplied by the spin-up electron. The shape of spin-polarized electron cloud also confirms the 3d7 scenario, i.e. Co2 + & Se1 − . This unusual valence can be attributed to the relatively small electronegativity difference between Co (Pauling χ = 1.88) and Se (Pauling χ = 2.55). Indeed, Se can ex- hibit the oxidation state of −1 in certain systems, such as sodium diselenide (Na2 S e2 ) [64]. The orbital split- ting and filling of Co2 + ion is summarized as the inset of Fig. 2(c). Four magnetic configurations in a 2 × 2 supercell (Fig. S5 in SM [52]), including ferromagnetic (FM), N´eel antiferromagnetic (N-AFM), stripe antiferromagnetic (S- AFM), and double-stripe antiferromagnetic (D-AFM) have been calculated to determine the magnetic ground state. Our calculations indicate the N-AFM state is the energetically most favorable, while the energies of FM, S-AFM, and D-AFM states are 36.9, 21.8, and 17.5 meV per unit cell (u.c.) higher than the ground state. The magnetic moments originate primarily from Co2 + ions, while the contribution of Se1 − ions are very small. Al- though the electron number of each Se1 − ion is odd, the formation of Se-dimers [Fig. 2(a)] quenches their local moments via covalent bonding. According to the energy profile obtained via calcu- lation, the exchange interactions are extracted, includ- ing the nearest-neighbor coupling (J1 ) , second nearest- neighbor coupling (J2 ) , third nearest-neighbor coupling (J3 ) , which can be summarized in Table S2 in SM [52]. The magnetocrystalline anisotropy of pentagonal CoS2 monolayer has been calculated by including spin-orbit coupling (SOC) effect. The angle-dependent magnetic anisotropy energies (MAE) are depicted in Fig. 3(a). The energy variation for in-plane (ab-plane) spin rota- tion is one order of magnitude smaller than that for out- of-plane (ac-plane) spin rotation. For the N-AFM state, the easy magnetization axis is along the a-axis, with an energy lower than that along the b-axis and c-axis for 18 µeV/u.c. and 242 µeV/u.c. respectively. Then a spin model Hamiltonian can be written in the FIG. 3. (a) The MAE of CoSe2 monolayer as a function of spin orientation in the ab and ac planes. The value of ab-plane has been amplified tenfold for better comparison. The radial axis represents the angle difference of the spin orientation relative to the x-direction. And θ=0◦ indicates the x directon in the ab and ac planes. (b) The MC simulation of antiferromagnetic- paramagnetic phase transition, indicated by the peak of heat capacity, which represents the total specific heat rather than per lattice site. Inset: a MC snapshot at 3 K. FIG. 4. (a) Spin charge density and symmetry operations connecting two opposite spins of pentagonal CoSe2 monolayer with N-AFM order. The red and green sites are spin-up and spin-down. (b) The electronic band structure in the first Bril- louin zone. Left panel: without splitting along high symmetry k-paths. Inset: 2D Brillouin zone. Right panel: with split- ting along other k-paths. Inset: magnified view around the Γ point. following form: H = − J1 ⟨i,j⟩ Si · Sj − J2 ⟨⟨m,n⟩⟩ Sm · Sn − J3 ⟨⟨⟨k,l⟩⟩⟩ Sk · Sl − A i (Szi )2 , (2) where S represents the normalized spin (i.e. |S| = 1) and A represents the single-axis magnetic anisotropy term. Note that here the much weaker in-plane MAE is ne- glected for simplify. With this model Hamiltonian, MC simulation has been performed. The peak of heat ca- pacity appears at TN ∼ 41 K [Fig. 3(b)], which indicates the magnetic phase transition. The low-temperature MC snapshot confirms the N-AFM as the magnetic ground state. Altermagnetism & magnetoelastic coupling. To clarify

4 the possible altermagnetism, the spin group is adopted to describe the symmetry operations. In 2D materials, four symmetry operations [Ri | |Rj ] protect spin degen- eracy in non-relativistic band structure throughout the Brillouin zone, where Ri and Rj represent the spin and spatial symmetry operations, respectively. These opera- tions are [C2 | |τ ], [C2 | | ¯ E ], [C2 | |mz ], and [C2 | |C2 z ]. CoSe2 monolayer possesses a nontrivial spin layer group 2 2/2 mx that lacks these four symmetry operations, and thus the spin splitting is expectable. The relevant symmetry op- erations for this system can be described as: [E||P] + [C2 | |Mx , C2 x] (3) where E represents the identity operation for spins, which links the sites with same spin. C2 denotes the an- tisymmetric spin flip operation that links the sites with opposite spin (Fig. 4(a)). Operators P, Mx , and C2 x in- dicate the space inversion, mirror symmetry, and twofold rotation symmetry, respectively. For high-symmetry lines Γ-Y (kx = 0) and Γ-X (ky = 0 ), the symmetry operation [C2 | |Mx ] and [C2 | |C2 x] ensures the Kramers spin degeneracy, respectively. Thereby, one have spin degenerate bands: [C2 | |Mx ] E(s, 0, ky ) = E(−s, 0, ky ), [C2 | |C2 x] E(s, kx , 0) = E(−s, kx , 0). (4) Consequently, one can obtain: E(s, 0, ky ) = E(−s, 0, ky ), E(s, kx , 0) = E(−s, kx , 0). (5) This explains the spin degeneration along the high- symmetry lines Γ-X and Γ-Y. While for other general k-paths, e.g. Γ-S, the band structure exhibits alternat- ing momentum-dependent spin splitting, as depicted in Fig. 4(b). Notably, the spin-splitting energy at the top of valence band can reach up to 213 meV, comparable to other 2D altermagnetic materials like VBr4 (150 meV) and RuF4 (∼ 240 meV) [17, 18, 43]. Besides the aforementioned intriguing magnetic prop- erties, pentagonal CoSe2 monolayer also exhibits ferroe- lastic and magnetoelastic behavior due to the buckling ef- fect. Ferroelastic materials own two or multiple energeti- cally degenerate states that undergo reversible switching between variants under external mechanical strain. Here the two degenerate states can be defined as the structures with lattice constants a < b (FES1) and a > b (FES2), as shown in Fig. 5(a). These two ferroelastic states can be connected via one of two symmetry operations, i.e. the 90◦ clockwise or counterclockwise rotations (C + 4 z vs C − 4 z ), which can be described mathematically as: C + 4 z = 0 −1 0 1 0 0 0 0 1 , C − 4 z = 0 1 0 − 1 0 0 0 0 1 . (6) The C + 4 z and C − 4 z operations lead to two different crys- talline structures, which can be interconnected via Mz FIG. 5. (a) The energy barrier of ferroelastic switching esti- mated by the CI-NEB method. The insets illustrate the two states before (a¡b) and after (a¿b) ferroelastic switching, along with a hypothetic intermediate state (a=b). The red (green) atoms represent spin-up (spin-down) magnetic moments. (b) Spin-resolved band structures of the top of valence band for the FES1 and FES2 states without spin-orbit coupling. Inset: the first Brillouin zone. symmetry operation. The key distinction lies in the posi- tions of the Se atoms, whereas the magnetic order on the two Co ions remains identical (right insets of Fig. 5(a)). Thus, the spin-polarized band structures of two states of FES2 (C + 4 z and C − 4 z ) are also identical. Being energeti- cally degenerate, they show identical switching barriers, as illustrated in Fig. 5(a). The hypothetic paraelastic state (PES) occurs at a = b = 5.673 ˚A. The climbing image-budged elastic band (CI-NEB) method has been performed to simulate the ferroelastic switching process. The calculated energy barrier (Eb ) is ∼ 72 meV/atom, which is comparable to that of AgF2 (51 meV/atom) [65], but much lower than those of Nb2 S iTe4 (∼ 237 meV/atom) [66] and phosphorene (∼ 200 meV/atom) [67]. The ferroelastic strain in CoSe2 monolayer can be char- acterized by the anisotropy, determined as a/b − 1, which is 3.3% for the FES1 state. This value is comparable to that of typical 2D ferroelastic material SnSe (2.1%) [67]. The ferroelastic switching involves a 90◦ rotation of the crystal, which correspondingly rotates both the band structure and associated spin splitting features, as de- picted by Fig. S6 in SM [52]. As specifically illustrated in Fig. 5(b), along non-high-symmetry paths, the ferroe- lastic switching can also switch the altermagnetic spin- splitting, which would induce changes in other physical properties.

5 FIG. 6. (a) Energy-dependent magneto-optical Kerr spectra of different states inferred from Wannier90 calculations. (b) FES1 state and four possible states emerging after ferroelas- tic switching when taking SOC into consideration. C4 z and N4 z indicate crystal and spin rotate 90◦ respectively, then the positive (+) and negative (-) signs control clockwise and coun- terclockwise rotation respectively. (c) Schematic diagram of symmetry operations connecting different states, where red (bule) lines indicate pairs of states with opposite (identical) magneto-optical Kerr angles. In altermagnets, spin-polarized current can be gen- erated by electric field due to spin-splitting, which can be characterized as spin-resolved conductivity tensor σsi j (s is ↑, ↓ for spin index) [3, 68, 69]. As depicted in Fig.S7 [52], the diagonal components of the conductiv- ity are spin-degenerate (σ↑i i= σ↓i i, i = x, y), whereas the off-diagonal component is spin-polarized (σ↑x y ̸= σ↓x y) . In addition, ferroelasticity can reverse the sign of off- diagonal spin-polarized conductivity σx y. Namely, σ↑x y and σ↓x y change in opposite directions. The anisotropic spin polarization can be defined as SP = σ↑i j −σ↓i j σ↑i j +σ↓i j = 2σ↑x ys in(2φ) σx xc os2 φ+σy ys in2 φ , where φ denotes the angle between the direction of the electric field and the x-axis. And we further observe that ferroelasticity also reverse its sign. These results indicate the ferroelasticity can effectively manipulate spin-polarized currents, enabling the design of strain-controlled spintronic devices for information en- coding and sensing. Switching of magnetic-optical Kerr effect. Taking spin- orbit coupling into consideration, MOKE may be occured in altermagnets. For CoSe2 monolayer with the N-AFM order, the corresponding magnetic space group (MSG), magnetic point groups (MPG), and optical conductivity tensor ξi j can be found in Table S3 in SM [52]. The MOKE response in this system exhibits strong N´eel vec- tor orientation dependence. When the magnetization di- rection aligns along the x-axis, there are two nonzero off-diagonal components of optical conductivity tensor, namely ξx y and ξy z. However, due to the 2D nature, only ξx y survives. When the magnetization direction turns to y-axis or z-axis, MOKE becomes forbidden. Since the easy magnetization direction of CoSe2 monolayer is along the x-axis as established earlier, the MOKE can be ac- tivated spontaneously. As shown in Fig. 6(a), the Kerr angle can reach up to 0.2◦ , which is readily detectable experimentally. Considering the direction and phase of N´eel vector N , defined as the difference of the magnetic moments M be- tween Co1 and Co2 in the unit cell, i.e. N =MC o1 -MC o2 . And N + 4 z (N − 4 z ) can describe the 90◦ clockwise (counter- clockwise) rotation of the N´eel vector. Thus four possible energy-degenerate FES2 states that can be derived from the FES1 state, labeled as C + 4 z N + 4 z , C + 4 z N − 4 z , C − 4 z N + 4 z , and C − 4 z N − 4 z , as shown in Fig. 6(b). Here C4 z and N4 z rep- resent the 90◦ rotations of crystal axes and N´eel vec- tors respectively, and the superscripts + and − indicate the clockwise and counterclockwise rotations. Interest- ingly, these four states exhibit equal occurrence probabil- ity during the ferroelastic switching, but they contribute distinct MOKE responses, as depicted in Fig. 6(a). Our analysis reveals that the MOKE response remains un- changed compared to the FES1 state when N´eel and crystal vectors rotate homodirectionally, but the sign of MOKE signal is reversed when their rotations oppose each other. In most cases, the inversion of MOKE signal can be achieved via the 180◦ spin rotation. Here this inversion may be done via the 90◦ spin rotation, with the help of ferroelasticity. This phenomenon can be fully explained by symmetry analysis as depicted in Fig. 6(c). R+ (R− ) represents a 90◦ cooperative clockwise (counterclockwise) rotation of both crystal axes and N´eel vectors, preserving the Kerr angle sign. In contrast, time reversal operation (T ) reverses the Kerr angle according to Onsager’s rela- tion [70]. Therefore, the homodirectional rotation of the lattice and N´eel vectors maintains the Kerr angle sign, while opposite-directional rotation reverses it, establish- ing a distinct magnetoelastic coupling mechanism. In summary, based on first-principles calculations, the electronic and magnetic properties of pentagonal CoSe2 monolayer have been revealed, which exhibits a N´eel-type antiferromagnetic ground state with TN = 41 K. Notably, prominent spin splitting (214 meV) in the absence of SOC confirms its altermagnetic nature. Remarkably, the observed ferroelastic phase transition may induce a con- comitant 90◦ rotation of the spin splitting bands. Accom- panying the ferroelastic switching, the homodirectional rotation of the lattice axes and N´eel vectors can preserve the Kerr angle sign, while the opposite-directional rota- tion inverts it, unveiling a novel magnetoelastic coupling

6 mechanism. Our work not only establishes a fundamental approach for manipulating altermagnetic states in more general multiferroic systems but also paves the way for in- vestigating unconventional magenetoelastic phenomena. Importantly, these findings are generalizable to analogous systems, including RuF4 , VX4 (X = F, Br, I) and MnF4 monolayers. Note added. Recently, we have noticed two works on ferroelastic altermagnets [71, 72]. The work by Guo et al. coincides with our findings, demonstrating that ferroelas- ticity can reverse the spin-splitting in altermagnets [71]. The study by Huang et al. also emphasizes the modulat- ing effect of ferroelasticity on the band structure of alter- magnetic systems, which can induce a transition from an altermagnetic to a compensated ferrimagnetic state [72]. ACKNOWLEDGMENTS We thank Professor Shengyuan A. Yang for his in- sightful and helpful discussions. This work was sup- ported by the National Natural Science Foundation of China (Grants No. 12274069, No. 12325401 and No. 12104089), Jiangsu Funding Program for Excellent Post- doctoral Talent under Grant Number 2024ZB001, China Postdoctoral Science Foundation under Grant Number 2024M760423, the Fundamental Research Funds 356 for the Central Universities (Grant No. 2242025K30023) and the Big Data Computing Center of Southeast University. [1] L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond conven- tional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X 12, 031042 (2022). [2] L. ˇSmejkal, J. Sinova, and T. Jungwirth, Emerging re- search landscape of altermagnetism, Phys. Rev. X 12, 040501 (2022). [3] H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and J. Liu, Multifunctional antiferromagnetic materials with giant piezomagnetism and noncollinear spin current, Nat. Commun. 12, 2846 (2021). [4] L. Bai, W. Feng, S. Liu, L. ˇSmejkal, Y. Mokrousov, and Y. Yao, Altermagnetism: Exploring new frontiers in mag- netism and spintronics, Adv. Funct. Mater. 34, 2409327 (2024). [5] C. Song, H. Bai, Z. Zhou, L. Han, H. Reichlova, J. H. Dil, J. Liu, X. Chen, and F. Pan, Altermagnets as a new class of functional materials, Nat. Rev. Mater. 10, 473 (2025). [6] R. D. Gonzalez Betancourt, J. Zub´aˇc, R. Gonzalez- Hernandez, K. Geishendorf, Z. ˇSob´aˇn, G. Springholz, K. Olejn´ık, L. ˇSmejkal, J. Sinova, T. Jungwirth, S. T. B. Goennenwein, A. Thomas, H. Reichlov´a, J. ˇZelezn´y, and D. Kriegner, Spontaneous anomalous Hall effect arising from an unconventional compensated magnetic phase in a semiconductor, Phys. Rev. Lett. 130, 036702 (2023). [7] J. Krempask´y, L. ˇSmejkal, S. W. D’Souza, M. Ha- jlaoui, G. Springholz, K. Uhl´ıˇrov´a, F. Alarab, P. C. Constantinou, V. Strocov, D. Usanov, W. R. Pudelko, R. Gonz´alez-Hern´andez, A. Birk Hellenes, Z. Jansa, H. Reichlov´a, Z. ˇSob´aˇn, R. D. Gonzalez Betancourt, P. Wadley, J. Sinova, D. Kriegner, J. Min´ar, J. H. Dil, and T. Jungwirth, Altermagnetic lifting of Kramers spin degeneracy, Nature 626, 517 (2024). [8] S. Lee, S. Lee, S. Jung, J. Jung, D. Kim, Y. Lee, B. Seok, J. Kim, B. G. Park, L. ˇSmejkal, C.-J. Kang, and C. Kim, Broken Kramers degeneracy in altermag- netic MnTe, Phys. Rev. Lett. 132, 036702 (2024). [9] A. Hariki, A. Dal Din, O. J. Amin, T. Yamaguchi, A. Badura, D. Kriegner, K. W. Edmonds, R. P. Cam- pion, P. Wadley, D. Backes, L. S. I. Veiga, S. S. Dhesi, G. Springholz, L. ˇSmejkal, K. V´yborn´y, T. Jungwirth, and J. Kuneˇs, X-ray magnetic circular dichroism in alter- magnetic α-MnTe, Phys. Rev. Lett. 132, 176701 (2024). [10] J. Ding, Z. Jiang, X. Chen, Z. Tao, Z. Liu, T. Li, J. Liu, J. Sun, J. Cheng, J. Liu, Y. Yang, R. Zhang, L. Deng, W. Jing, Y. Huang, Y. Shi, M. Ye, S. Qiao, Y. Wang, Y. Guo, D. Feng, and D. Shen, Large band splitting in g-wave altermagnet CrSb, Phys. Rev. Lett. 133, 206401 (2024). [11] M. Zeng, M.-Y. Zhu, Y.-P. Zhu, X.-R. Liu, X.-M. Ma, Y.- J. Hao, P. Liu, G. Qu, Y. Yang, Z. Jiang, K. Yamagami, M. Arita, X. Zhang, T.-H. Shao, Y. Dai, K. Shimada, Z. Liu, M. Ye, Y. Huang, Q. Liu, and C. Liu, Observation of spin splitting in room-temperature metallic antiferro- magnet CrSb, Adv. Sci. 11, 2406529 (2024). [12] Z. Zhou, X. Cheng, M. Hu, R. Chu, H. Bai, L. Han, J. Liu, F. Pan, and C. Song, Manipulation of the al- termagnetic order in CrSb via crystal symmetry, Nature 638, 645 (2025). [13] Y.-P. Zhu, X. Chen, X.-R. Liu, Y. Liu, P. Liu, H. Zha, G. Qu, C. Hong, J. Li, Z. Jiang, X.-M. Ma, Y.-J. Hao, M.-Y. Zhu, W. Liu, M. Zeng, S. Jayaram, M. Lenger, J. Ding, S. Mo, K. Tanaka, M. Arita, Z. Liu, M. Ye, D. Shen, J. Wrachtrup, Y. Huang, R.-H. He, S. Qiao, Q. Liu, and C. Liu, Observation of plaid-like spin split- ting in a noncoplanar antiferromagnet, Nature 626, 523 (2024). [14] B. Jiang, M. Hu, J. Bai, Z. Song, C. Mu, G. Qu, W. Li, W. Zhu, H. Pi, Z. Wei, Y.-J. Sun, Y. Huang, X. Zheng, Y. Peng, L. He, S. Li, J. Luo, Z. Li, G. Chen, H. Li, H. Weng, and T. Qian, A metallic room-temperature d- wave altermagnet, Nat. Phys. 21, 754 (2025). [15] H. Reichlova, R. Lopes Seeger, R. Gonz´alez-Hern´andez, I. Kounta, R. Schlitz, D. Kriegner, P. Ritzinger, M. Lammel, M. Leivisk¨a, A. Birk Hellenes, K. Olejn´ık, V. Petˇriˇcek, P. Doleˇzal, L. Horak, E. Schmoranzerova, A. Badura, S. Bertaina, A. Thomas, V. Baltz, L. Michez, J. Sinova, S. T. B. Goennenwein, T. Jungwirth, and L. ˇSmejkal, Observation of a spontaneous anomalous Hall response in the Mn5 S i3 d-wave altermagnet candidate, Nat. Commun. 15, 4961 (2024). [16] R. B. Regmi, H. Bhandari, B. Thapa, Y. Hao, N. Sharma, J. McKenzie, X. Chen, A. Nayak, M. El Gazzah, B. G. M´arkus, L. Forr´o, X. Liu, H. Cao, J. F. Mitchell, I. I. Mazin, and N. J. Ghimire, Altermagnetism in the layered

7 intercalated transition metal dichalcogenide CoNb4 S e8 , Nat. Commun. 16, 4399 (2025). [17] J. Sødequist and T. Olsen, Two-dimensional altermag- nets from high throughput computational screening: Symmetry requirements, chiral magnons, and spin-orbit effects, Appl. Phys. Lett. 124, 182409 (2024). [18] H. Jin, Z. Tan, Z. Gong, and J. Wang, Anomalous Hall effect in two-dimensional vanadium tetrahalogen with al- termagnetic phase, Phys. Rev. B 110, 155125 (2024). [19] M. Milivojevi´c, M. Orozovi´c, S. Picozzi, M. Gmitra, and S. Stavri´c, Interplay of altermagnetism and weak ferro- magnetism in two-dimensional RuF4 , 2D Materials 11, 035025 (2024). [20] Y. Che, Y. Chen, X. Liu, H. Lv, X. Wu, and J. Yang, Inverse design of 2D altermagnetic metal–organic frame- work monolayers from H¨uckel theory of nonbonding molecular orbitals, JACS Au 5, 381 (2025). [21] Y.-Q. Li, Y.-K. Zhang, X.-L. Lu, Y.-P. Shao, Z.-Q. Bao, J.-D. Zheng, W.-Y. Tong, and C.-G. Duan, Ferrovalley physics in stacked bilayer altermagnetic systems, Nano Lett. 25, 6032 (2025). [22] Y. Zhu, T. Chen, Y. Li, L. Qiao, X. Ma, C. Liu, T. Hu, H. Gao, and W. Ren, Multipiezo effect in altermagnetic V2 S eTeO monolayer, Nano Lett. 24, 472 (2024). [23] D. Wang, H. Wang, L. Liu, J. Zhang, and H. Zhang, Electric-field-induced switchable two-dimensional alter- magnets, Nano Lett. 25, 498 (2025). [24] C.-Y. Tan, Z.-F. Gao, H.-C. Yang, Z.-X. Liu, K. Liu, P.- J. Guo, and Z.-Y. Lu, Crystal valley Hall effect, Phys. Rev. B 111, 094411 (2025). [25] R. He, D. Wang, N. Luo, J. Zeng, K.-Q. Chen, and L.- M. Tang, Nonrelativistic spin-momentum coupling in an- tiferromagnetic twisted bilayers, Phys. Rev. Lett. 130, 046401 (2023). [26] Y. Liu, J. Yu, and C.-C. Liu, Twisted magnetic van der Waals bilayers: An ideal platform for altermagnetism, Phys. Rev. Lett. 133, 206702 (2024). [27] B. Pan, P. Zhou, P. Lyu, H. Xiao, X. Yang, and L. Sun, General stacking theory for altermagnetism in bilayer systems, Phys. Rev. Lett. 133, 166701 (2024). [28] W. Sun, H. Ye, L. Liang, N. Ding, S. Dong, and S.- S. Wang, Stacking-dependent ferroicity of a reversed bi- layer: Altermagnetism or ferroelectricity, Phys. Rev. B 110, 224418 (2024). [29] S. Zeng and Y.-J. Zhao, Bilayer stacking A-type al- termagnet: A general approach to generating two- dimensional altermagnetism, Phys. Rev. B 110, 174410 (2024). [30] S. Dong, J.-M. Liu, S.-W. Cheong, and Z. R. and, Multi- ferroic materials and magnetoelectric physics: symmetry, entanglement, excitation, and topology, Adv. Phys. 64, 519 (2015). [31] X. Duan, J. Zhang, Z. Zhu, Y. Liu, Z. Zhang, I. ˇZuti´c, and T. Zhou, Antiferroelectric altermagnets: Antiferro- electricity alters magnets, Phys. Rev. Lett. 134, 106801 (2025). [32] M. Gu, Y. Liu, H. Zhu, K. Yananose, X. Chen, Y. Hu, A. Stroppa, and Q. Liu, Ferroelectric switchable alter- magnetism, Phys. Rev. Lett. 134, 106802 (2025). [33] Z. Zhu, X. Duan, J. Zhang, B. Hao, I. ˇZuti´c, and T. Zhou, Two-dimensional ferroelectric altermagnets: From model to material realization, Nano Lett. 25, 9456 (2025). [34] W. Sun, W. Wang, C. Yang, R. Hu, S. Yan, S. Huang, and Z. Cheng, Altermagnetism induced by sliding ferro- electricity via lattice symmetry-mediated magnetoelec- tric coupling, Nano Lett. 24, 11179 (2024). [35] W. Sun, C. Yang, W. Wang, Y. Liu, X. Wang, S. Huang, and Z. Cheng, Proposing altermagnetic- ferroelectric type-III multiferroics with robust magneto- electric coupling., Adv. Mater. 37, 2502575 (2021). [36] P.-J. Guo, Y. Gu, Z.-F. Gao, and Z.-Y. Lu, Altermagnetic ferroelectric LiFe2 F 6 and spin-triplet excitonic insulator phase (2023), arXiv:2312.13911. [37] I. Khan, D. Bezzerga, and J. Hong, Coexistence of al- termagnetism and robust ferroelectricity in a bulk MnO wurtzite structure, Mater. Horiz. 12, 2319 (2025). [38] S. Wang, W.-W. Wang, J. Fan, X. Zhou, X.-P. Li, and L. Wang, Two-dimensional dual-switchable ferroelectric altermagnets: Altering electrons and magnons, Nano Lett. 25, 14618 (2025). [39] L. ˇSmejkal, Altermagnetic multiferroics and altermagne- toelectric effect (2024), arXiv:2411.19928. [40] Y. Yang, Giant bulk spin photovoltaic effect in the two- dimensional altermagnetic multiferroics VOX2 , Phys. Rev. B 112, 104427 (2025). [41] Y. Zhu, M. Gu, Y. Liu, X. Chen, Y. Li, S. Du, and Q. Liu, Sliding ferroelectric control of unconventional magnetism in stacked bilayers, Phys. Rev. Lett. 135, 056801 (2025). [42] A. Urru, D. Seleznev, Y. Teng, S. Y. Park, S. E. Reyes- Lillo, and K. M. Rabe, G-type antiferromagnetic BiFeO3 is a multiferroic g-wave altermagnet, Phys. Rev. B 112, 104411 (2025). [43] N. Wang, J. Chen, N. Ding, H. Zhang, S. Dong, and S.-S. Wang, Magneto-optical Kerr effect and magnetoelasticity in a weakly ferromagnetic RuF4 monolayer, Phys. Rev. B 106, 064435 (2022). [44] S. Xu, F. Jia, X. Cheng, and W. Ren, Predicting intrinsic antiferromagnetic and ferroelastic MnF4 monolayer with controllable magnetization, J. Mater. Chem. C 9, 17152 (2021). [45] L. Zhang, C. Tang, and A. Du, Two-dimensional vana- dium tetrafluoride with antiferromagnetic ferroelastic- ity and bidirectional negative Poisson’s ratio, J. Mater. Chem. C 9, 95 (2021). [46] J. Wang, X. Yang, Z. Yang, J. Lu, P. Ho, W. Wang, Y. S. Ang, Z. Cheng, and S. Fang, Pentagonal 2D alter- magnets: Material screening and altermagnetic tunnel- ing junction device application., Adv. Funct. Mater. 35, 2505145 (2025). [47] L. Liu, I. Kankam, and H. L. Zhuang, Single-layer antifer- romagnetic semiconductor CoS2 with pentagonal struc- ture, Phys. Rev. B 98, 205425 (2018). [48] G. Kresse and J. Furthm¨uller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996). [49] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996). [50] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Electron-energy-loss spec- tra and the structural stability of nickel oxide: An LSDA+U study, Phys. Rev. B 57, 1505 (1998). [51] S. Wang, X. Lv, and F. Li, Ternary penta-CoXS (X = Se and Te): 2D monolayers with coexistence of antiferro- magnetic, auxetic, photocatalytic, and piezoelectric fea- tures, J. Phys. Chem. C 128, 3064 (2024). [52] See Supplementary Materials [url] for more detials about the mechanical properties, possible magnetic orders and

8 the spin splitting before and after ferroelastic switching. [53] Y. Le Page and P. Saxe, Symmetry-general least-squares extraction of elastic data for strained materials from ab initio calculations of stress, Phys. Rev. B 65, 104104 (2002). [54] X. Wu, D. Vanderbilt, and D. R. Hamann, Systematic treatment of displacements, strains, and electric fields in density-functional perturbation theory, Phys. Rev. B 72, 035105 (2005). [55] R. N. Barnett and U. Landman, Born-oppenheimer molecular-dynamics simulations of finite systems: Struc- ture and dynamics of (H2 O )2 , Phys. Rev. B 48, 2081 (1993). [56] G. Pizzi, V. Vitale, R. Arita, S. Bl¨ugel, F. Freimuth, G. G´eranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Iba˜nez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Ponc´e, T. Pon- weiser, J. Qiao, F. Th¨ole, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A. Mostofi, and J. R. Yates, Wannier90 as a community code: new fea- tures and applications, J. Phys. Condens. Matter. 32, 165902 (2020). [57] G. Pizzi, D. Volja, B. Kozinsky, M. Fornari, and N. Marzari, Boltzwann: A code for the evaluation of thermoelectric and electronic transport properties with a maximally-localized Wannier functions basis, Comput. Phys. Commun. 185, 422 (2014). [58] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010). [59] Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D.-s. Wang, E. Wang, and Q. Niu, First principles calculation of anomalous Hall conductivity in ferromagnetic bcc Fe, Phys. Rev. Lett. 92, 037204 (2004). [60] X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation, Phys. Rev. B 74, 195118 (2006). [61] X. Yang, X. Zhou, W. Feng, and Y. Yao, Tun- able magneto-optical effect, anomalous Hall effect, and anomalous Nernst effect in the two-dimensional room- temperature ferromagnet 1T -CrTe2 , Phys. Rev. B 103, 024436 (2021). [62] V. Wang, N. Xu, J.-C. Liu, G. Tang, and W.-T. Geng, Vaspkit: A user-friendly interface facilitating high- throughput computing and analysis using VASP code, Comput. Phys. Commun. 267, 108033 (2021). [63] C. Long, Y. Liang, H. Jin, B. Huang, and Y. Dai, PdSe2 : Flexible two-dimensional transition metal dichalco- genides monolayer for water splitting photocatalyst with extremely low recombination rate, ACS Appl. Energy Mater. 2, 513 (2019). [64] A. Krief and M. Derock, Synthesis of diselenides and se- lenides from elemental selenium, Tetrahedron Lett. 43, 3083 (2002). [65] X. Xu, Y. Ma, T. Zhang, C. Lei, B. Huang, and Y. Dai, Prediction of two-dimensional antiferromagnetic ferroe- lasticity in an AgF2 monolayer, Nanoscale Horiz. 5, 1386 (2020). [66] T. Zhang, Y. Ma, X. Xu, C. Lei, B. Huang, and Y. Dai, Two-dimensional ferroelastic semiconductors in Nb2 S iTe4 and Nb2 G eTe4 with promising electronic prop- erties, J. Phys. Chem. Lett. 11, 497 (2020). [67] M. Wu and X. C. Zeng, Intrinsic ferroelasticity and/or multiferroicity in two-dimensional phosphorene and phosphorene analogues, Nano Lett. 16, 3236 (2016). [68] R. Gonz´alez-Hern´andez, L. ˇSmejkal, K. V´yborn´y, Y. Ya- hagi, J. Sinova, T. c. v. Jungwirth, and J. ˇZelezn´y, Efficient electrical spin splitter based on nonrelativis- tic collinear antiferromagnetism, Phys. Rev. Lett. 126, 127701 (2021). [69] M. Dou, X. Wang, and L. L. Tao, Anisotropic spin- polarized conductivity in collinear altermagnets, Phys. Rev. B 111, 224423 (2025). [70] H. Rathgen, M. I. Katsnelson, O. Eriksson, and G. Zwicknagl, Polar magneto-optical Kerr effect for low- symmetric ferromagnets, Phys. Rev. B 72, 014451 (2005). [71] D. Guo, J. Dai, R. Wang, C. Wang, and W. Ji, Mechani- cally and electrically switchable triferroic altermagnet in a pentagonal FeO2 monolayer, Phys. Rev. B 112, 195410 (2025). [72] Y. Huang, C. Hua, R. Xu, J. Liu, Y. Zheng, and Y. Lu, Spin inversion enforced by crystal symmetry in ferroelas- tic altermagnets, Phys. Rev. Lett. 135, 266701 (2025).