Shear-strain-induced switchable spin splitting and piezomagnetic properties in altermagnetic materials

Abstract

Altermagnetic materials have recently attracted significant attention due to their unique intrinsic crystal symmetry. Breaking the lattice symmetry can yield a non-negligible piezomagnetic effect, which refers to a linear relationship between strain and magnetization. We proposed a method based on {\epsilon }_{xz} and {\epsilon }_{xy} shear strains that induced additional spin splitting along band paths that were originally spin-degenerate and generated weak ferromagnetic moments in altermagnetic CoF₂ and RuO₂. Additionally, the signs of spin splitting and weak ferromagnetic moments were switched by reversing the shear strain direction. The weak ferromagnetism had a substantial contribution from the orbital magnetic moments, with its y (z) component associated with the charge quadrupole moment. Our study highlights that shear strain provides a novel approach for exploring the physical properties of spin splitting, piezomagnetism, and multipolar order, while also broadening avenues for tuning the properties of altermagnetic materials.

Graphical abstract

Keywords

altermagnet spin splitting / piezomagnetic effect / charge quadrupole moment / shear strain

1 Introduction

Compared with ferromagnets, antiferromagnetic (AFM) materials exhibit high information storage density and unique terahertz (THz) spin dynamics [1, 2]; however, their weak response to external magnetic fields makes them challenging to manipulate effectively. In recent years, altermagnetism (AM), an emerging branch of magnetism, has attracted significant attention due to its distinctive combination of features — the spin-split band structure of ferromagnets and the zero net magnetization of antiferromagnets [3−9]. These unique properties enable various spintronic effects, including spin-polarized current, anomalous Nernst effects, giant magnetoresistance, and spin splitting (SS) torque [10−15]. AM arises from the breaking of combined time-reversal and spatial-inversion (TI) symmetry [16−19], which is precisely the symmetry required for piezomagnetism (PZM). The piezomagnetic effect describes a linear coupling between strain and external magnetic fields, meaning that applied strain can alter magnetization, conversely, magnetization can modify crystal shape. This effect has been first observed experimentally in CoF₂ and MnF₂ during the 1960s [20]. Recently, the THz piezomagnetic effect in these materials has been further detected through the excitation of Raman phonon modes [21]. Additionally, significant piezomagnetic effects have been observed in altermagnetic materials, including RuO₂ [22, 23], CrSb [24], and MnTe [25]. The piezomagnetic coefficient of MnTe, Q = 1.38 × 10⁻⁸ μ_B/Mn/MPa [25], provides a quantitative foundation for understanding the PZM in altermagnetic materials.

A significant advantage of the piezomagnetic effect is its capacity to enable the manipulation of the magnetic state of altermagnetic materials through strain engineering. In two-dimensional (2D) altermagnetic materials, theory predicts that the V₂Se₂O and V₂SeTeO monolayers have been exhibiting pronounced PZM under uniaxial strain with hole doping [26−28]. In three-dimensional (3D) altermagnetic materials, weak magnetization is typically observed under strain. For example, a large PZM has been observed in Mn₃SnN under biaxial strains [29], and Mn₃Sn under uniaxial strains [30]. In altermagnetic materials, ferroic-ordered magnetic multipoles have been identified as the order parameters and strongly correlate with PZM. Spaldin et al. demonstrated via first-principles calculations that the PZM in MnF₂ has been directly associated with the ferroic ordering of magnetic octupoles [19]. McClarty and Rau [31], based on Landau theory, have proposed that the PZM in rutile and hexagonal altermagnetic materials can be explained by multipolar secondary order parameters, thereby establishing a systematic theoretical framework for understanding the microscopic origin of PZM. Therefore, exploring the physical mechanisms underlying PZM and multipolar order in altermagnetic materials is of paramount importance.

In this work, we applied shear strain to altermagnetic CoF₂ and RuO₂ and calculated their SS, piezomagnetic effect and charge quadrupoles using the first-principles method. The shear strain reduced the crystal symmetry, and induced SS along additional paths in reciprocal space. The induced magnetization, which varied linearly with strain, comprised both spin and orbital components. We also identified the components of the charge quadrupole moment responsible for the PZM. The unique physical properties and tunable characteristics of altermagnetic materials promise significant advancements in future technological fields, including spin−orbit electronics, quantum computing, and information storage.

2 Computational method

We performed calculations based on density functional theory (DFT) [32, 33] using the VASP package [34, 35] and the projected augmented wave (PAW) [36] method. The exchange-correlation potential was described using the Perdew−Burke−Ernzerh (PBE) [37] formalism within the generalized gradient approximation (GGA) framework. A kinetic energy cutoff of 600 eV was set for the plane-wave basis, and a Monkhorst−Pack k-point mesh of 14 × 14 × 18 for CoF₂ and 16 × 16 × 24 for RuO₂ was used. To account for strong electron correlations in 3d and 4d orbitals, the GGA+U method was employed with effective U = 5.5 eV (Co) and 1.5 eV (Ru) [38−42]. The crystal structures were fully relaxed using convergence criteria of 0.01 eV/Å for the Hellmann−Feynman force on each atom and 10⁻⁷ eV for the total energy. The spin−orbit coupling (SOC) effect is incorporated into the calculations of both the piezomagnetic effect and the charge quadrupole moment.

3 Result and discussion

Both CoF₂ and RuO₂ have rutile structures, with Co (Ru) and F (O) ions occupying the 2a and 4f Wyckoff positions, respectively. Both materials exhibit a collinear AFM Néel ground state, as shown in Fig.1(a). The magnetic moments on the two sublattices align antiparallel. The Néel temperature are T_N = 39 K for CoF₂ [43, 44] and T_N > 300 K for RuO₂ [10, 45]. Their crystal symmetry can be described using the space group P4₂/mnm (No. 136) and the magnetic space group P4₂'/mnm' (136.499) [16, 43, 44].

We first calculated the band structure of CoF₂. The Brillouin zone and the high symmetry paths are shown in Fig.1(d). From the band structure of CoF₂ without SOC as shown in Fig.1(b), we can find that CoF₂ is a semiconductor with a band gap of 4.2 eV in the GGA+U calculations. However, the HSE06 hybrid functional calculations predict a much larger band gap of 4.9 eV as shown in Fig. S1 of the Supplementary Material (SM), consistent with previous reports in Refs. [38, 39]. From Fig.1(b), the bands along the Γ–X–M and Γ–Z–R–A paths are spin-degenerate, while significant SS emerges along the M–Γ and A–Z directions. The non-relativistic SS clearly shows the features of AM. Notably, such splitting is much larger compared with the typical relativistic Rashba-type SS and does not require any breaking of the inversion symmetry of the structure [46]. Fig.1(c) illustrates the band structure with SOC included. It is found that the SS observed without SOC still exists when SOC is included, with additional band splitting occurring along the Z–R–A path. Fig.1(e) and (f) present the band structure of metallic RuO₂. The band splitting in RuO₂ is similar to that in CoF₂ since they share the same crystal and magnetic structures.

SS unrelated to SOC in 3D systems requires the breaking of symmetries associated with the combination of time-reversal, spatial-inversion, and spatial translation (TIt), as well as the combination of spinor reversal and spatial translation (Ut) [16, 47]. To investigate the effect of symmetry breaking on SS, we applied shear strain. Specifically, shear strain {\epsilon }_{xz}=\tan \theta is applied by rotating the z-axis towards the x-axis by a small angle θ, while shear strain {\epsilon }_{xy}=\tan \phi is applied by rotating the y-axis towards the x-axis by a small angle φ. Under {\epsilon }_{xz} shear strain, the crystal symmetry is reduced to the P2₁/c space group, whereas under {\epsilon }_{xy} shear strain, it lowers to the P2/m space group (see Table 1 of the SM).

The band structures of CoF₂ without SOC under {\epsilon }_{xz} shear strains of −0.04 and 0.04 are shown in Fig.2(a) and (b), respectively. Compared with the band structure without strain, as shown in Fig.1(b), SS appears at additional nonhigh-symmetry lines, including X–Z, Y–A, and Γ–R. In contrast, the {\epsilon }_{xy} shear strain generates SS along the Γ–X–M and Γ–Z–R–A paths, which is different from {\epsilon }_{xz} shear strains [see Fig.2(d) and (e)]. In addition, the bands of the up and down spins are interchanged when the shear strain is reversed. Fig.2(c) and (f) display the SS as a function of the shear strains {\epsilon }_{xz} and {\epsilon }_{xy} strengths at the points B1/B2/B3 and C1/C2/C3/C4, which are labelled in Fig.2(a) and (d), respectively. Under {\epsilon }_{xz} shear strain, all high-symmetry points remain degenerate. Points B1−B3 are located at the midpoints between the Y–A and Γ–R nonhigh-symmetry paths. For {\epsilon }_{xy} shear strain, SS emerges throughout the entire Brillouin zone. Points C1−C4 comprise both high-symmetry points and midpoints along the Γ–Z high-symmetry path. We observe that the SS is linearly proportional to the strength of the shear strains, and its sign depends on the direction of the shear strains. The splitting at B2 is larger than that at B1, and their signs are opposite due to the different orbital components and spin-up/down order of the two states. The splitting values at C2 and C3 are nearly identical. The maximum SS of 156 meV, observed under {\epsilon }_{xy} shear strain of 0.04, occurs at point M (denoted as C2) along the valence band. We then analyse the SS near the Fermi energy based on symmetry, specifically for shear strains {\epsilon }_{xz} and {\epsilon }_{xy}, which primarily arises from the d_{xz}/d_{z^2} and d_{xz}/d_{x^2-y^2} orbitals of Co ions (see Fig. S2 of the SM).

The band structures of RuO₂ under different shear strains {\epsilon }_{xz} and {\epsilon }_{xy} are shown in Fig. S3 of the SM. The SS in RuO₂ shows the same characteristics as CoF₂, except that RuO₂ is a metal, and the band splitting is more pronounced. The maximum SS of 224 meV, observed under {\epsilon }_{xy} shear strain of 0.04, occurs at point X (denoted as C1) near the Fermi energy. The altermagnetic splittings are much larger in magnitude in the case of RuO₂ as compared to CoF₂. This SS is larger than many intrinsic altermagnets, such as Mn₅Si₃ (150 meV) [18] and MnF₂ (~70 meV) [19]. The significant SS energies are supposed to be directly observed in experiments through spin-and-angle-resolved photoemission spectroscopy.

The piezomagnetic effect is the linear relationship between weak ferromagnetism and strain [21, 31, 48], the ratio between them is referred to as the piezomagnetic tensor Λ. For CoF₂ and RuO₂, one has [20, 49]

(1)M_x={\mathrm{\Lambda }}_{xyz} {\sigma }_{yz},M_y={\mathrm{\Lambda }}_{yxz} {\sigma }_{xz},M_z={\mathrm{\Lambda }}_{zxy} {\sigma }_{xy},

where \mathit{M} is the magnetization induced by shear strain {\epsilon }_{ij}. The crystal symmetry guarantees that {\mathrm{\Lambda }}_{xyz}={\mathrm{\Lambda }}_{yxz}\ne {\mathrm{\Lambda }}_{zxy}. σ is expressed as shear stress, such that {\sigma }_{ij}=c_{ij}{\epsilon }_{ij}. The piezomagnetic effect results in a net moment along the y (z) direction when shear strain {\epsilon }_{xz} ({\epsilon }_{xy}) is applied.

We calculated the net magnetization of CoF₂ as a function of shear strains {\epsilon }_{xz} and {\epsilon }_{xy}, which are illustrated in Fig.3(a) and (b). The accuracy of these calculations is ensured by testing the convergence with respect to the number of k-points and the smearing width, which are provided in Fig. S4 of the SM. Applying shear strain {\epsilon }_{xz} to CoF₂ induces non-collinear net magnetic moments (NMM) in the y-direction (see Fig. S5 of the SM), which vary linearly with the strain, as shown in Fig.3(a). At a shear strain of 0.04, the total magnetic moment (TMM) reaches 0.029 μ_B per unit cell, significantly larger than MnF₂ (~0.001 μ_B per unit cell) [19]. The TMM of CoF₂ under {\epsilon }_{xz} shear strain can be decomposed into the spin magnetic moment (SMM) and orbital magnetic moment (OMM) [50]. The OMM competes with the SMM, leading to the SMM being more than twice that of the OMM. Importantly, reversing the direction of the shear strain can switch the sign of the magnetic moment. In contrast, the shear strain {\epsilon }_{xy} induces a magnetization that is ten times smaller than that induced by the {\epsilon }_{xz} shear strain, as shown in Fig.3(b). In this case, the PZM is solely due to the non-collinear OMM along the z-direction (see Fig. S5 of the SM), consistent with Ref. [51]. To demonstrate the PZM, we calculated the piezomagnetic coefficient of CoF₂ using Eq. (1), where the shear moduli of CoF₂ [52] are C₄₄ = 3.79 × 10¹¹ dyn/cm² and C₆₆ = 8.46 × 10¹¹ dyn/cm², and the corresponding {\mathrm{\Lambda }}_{yxz} and {\mathrm{\Lambda }}_{zxy} values are 0.033 and 0.002 Gauss/MPa, respectively. Experimental data of {\mathrm{\Lambda }}_{yxz} and {\mathrm{\Lambda }}_{zxy} for CoF₂ are 0.021 and 0.008 Gauss/MPa, respectively [20, 30]. The calculated piezomagnetic coefficients are of a similar magnitude to the experimental data.

Applying shear strain {\epsilon }_{xz} to RuO₂ also produces an non-collinear NMM in the y-direction that varies linearly with strain, as shown in Fig.3(c). In this case, PZM is solely due to the non-collinear SMM along the y-direction, which is quite different from the behaviour observed in CoF₂. We attribute this difference to the fact that RuO₂ is metallic, while CoF₂ is insulating (To validate this observation, the PZM of the semiconductor FeF₂, which has the same crystal and magnetic structure as CoF₂, was investigated under shear strain, as shown in Fig. S6 of the SM, and was found to be consistent with those observed for CoF₂). At a shear strain of 0.04, the SMM reaches 0.0036 μ_B per unit cell, much smaller than that of CoF₂ (~0.0187 μ_B per unit cell). In contrast, the shear strain {\epsilon }_{xy} induces a much larger magnetization, as shown in Fig.3(d), which is ten times larger than that induced by {\epsilon }_{xz} shear strain. Under {\epsilon }_{xy} shear strain, the TMM of RuO₂ is greater than that of CoF₂, owing to the stronger spin−orbit coupling effect in Ru atoms compared to Co atoms. The PZM of RuO₂ under {\epsilon }_{xy} is like that of CoF₂ under {\epsilon }_{xz} shear strain. However, due to the cooperative interaction between the spin and orbital components, both exhibit comparable magnitudes of SMM and OMM.

In CoF₂ and RuO₂, the inverse piezomagnetic effect involves the generation of shear stress due to the tilting of magnetization. In our calculations, the local spin moments are tilted along the y-axis and their directions are fixed during self-consistent calculations using constrained DFT. As shown in Fig. S7 of the SM, the calculated shear stress {\sigma }_{xz}, exhibits a linear dependence on the magnetization M_y.

The development of high-order multipole descriptions in AFM systems reveals that the ferroic ordering of local magnetoelectric multipoles breaks both T and I symmetries [49, 53, 54]. For example, in MnF₂, the ferroically ordered O_{{32}^{-}} magnetic octupoles magnitude can be directly modulated and quantified via the charge quadrupole moment Q_{{22}^{-}}. In MnF₂, the symmetries of the O_{{32}^{-}} and the toroidal moment ({\tau }_{x^2-y^2}) directly correspond to non-zero components of the piezomagnetic tensor, thereby establishing a one-to-one relationship between the piezomagnetic response and the magnetic octupoles [19]. These results demonstrate that strain modulates the charge quadrupole moment, which in turn influences variations in the magnetic octupoles or toroidal moments, ultimately governing the piezomagnetic response. For a general charge density {\mu }_{\text{e}}, the charge moments Q_{lm} are defined as, Q_{lm}=\sqrt{4\pi /(2n+1)}\int \mathrm{d}r[r^n{Y}_{lm}^{\ast }(\theta ,\phi )] {\mu }_e, where {\mu }_{\text{e}} is projected onto the spherical harmonics Y_{lm}^{\ast } [55]. Accordingly, we calculated the charge quadrupoles by projecting the self-consistent density matrices of the d-orbital manifold onto spherical harmonics. To gain a deeper understanding of the unconventional SS and PZM in CoF₂ and RuO₂, we analyzed the representation of the ferrotype charge quadrupole component.

The charge quadrupole components Q_{2,m} (where m = −2, −1, 0, 1, 2) represent the projection of atomic charge density onto real spherical harmonics Y_{2,m}. Our calculations show that the components Q_{21}=d_{xz}, Q_{{21}^{-}}=d_{yz}, and Q_{22}=d_{x^2-y^2} are non-zero in both CoF₂ and RuO₂ after summing over all transition metals in one unit cell. The other two components are either zero (indicating antiferroic ordering) or not linearly dependent on the strain, and can be considered negligible.

As the shear strain varies from −0.04 to 0.04, the calculated values of the charge quadrupoles Q_{21}, Q_{22}, and Q_{{21}^{-}} are shown in Fig.4(a)−(d). For CoF₂, both Q_{21} and Q_{{21}^{-}} vanish at zero strain. The shear strains {\epsilon }_{xz} and {\epsilon }_{xy} induce Q_{21} and Q_{{21}^{-}}, which vary linearly with strain. However, Q_{22} of CoF₂ is not zero even at zero strain and it remains almost unchanged under applied shear strains {\epsilon }_{xz} and {\epsilon }_{xy}. The atomic orbital of Q_{21}=d_{xz} is consistent with the result that shear strain {\epsilon }_{xz} produces PZM (including SMM and OMM) in the y-direction, while Q_{22} has little effect on the piezomagnetic direction. The atomic orbitals of Q_{{21}^{-}}=d_{yz}, and Q_{22}=d_{x^2-y^2}, in combination, cause shear strain {\epsilon }_{xy} to generate PZM (including OMM) in the z-direction. In contrast, the charge multipoles of RuO₂ exhibit different behavior. Q_{21} increases, while Q_{22} remains almost unchanged with shear strain {\epsilon }_{xz}. Q_{{21}^{-}} decreases, and Q_{22} increases with shear strain {\epsilon }_{xy}. Therefore, the resulting PZM is exactly the opposite of that in CoF₂, which is consistent with the previous analysis. Previous studies have reported that the quadrupoles in the isospace group compound Ba₂MgReO₆ have a strong spin magnetic dipole moments dependence [56], similar to our results. The magnitude of the charge quadrupole also depends on the shear strain ε, indicating that lattice distortion can modulate the charge distribution (see later).

To verify the PZM and charge quadrupoles in CoF₂ and RuO₂ discussed above, we also calculated the effects of other type of strain, including uniaxial, biaxial, and triaxial strains. None of these strains produced an NMM, confirming that PZM arises only from shear strain. However, all these strains produce charge quadrupoles Q_{20}=d_{z^2} and Q_{22}=d_{x^2-y^2} (see Fig. S8 of the SM), which contrasts with the behavior observed under the shear strain. Finally, we modified the Wyckoff site symmetry of F from 4f to 4g, while keeping the space group symmetry unchanged. The modified structure is equivalent to the original crystal structure of CoF₂, resulting in the CoF₆ octahedra rotating alternately by 90° [see Figs. S9(a, b) of the SM]. The calculation results for the shear strain {\epsilon }_{xz} in CoF₂ reveal that the magnitude of the piezomagnetic is opposite to that of the original structure [see Figs. S9(c, d) of the SM]. The NMMs for the original and modified structures are 0.0287 and −0.0291 μ_B per unit cell under {\epsilon }_{xz} = 0.04, respectively. This confirms that the piezomagnetic properties exhibit antisymmetry. The modified structure represents merely a rotation of the octahedron, while the fundamental properties of the material remain unchanged. The modified structure also produces a charge quadrupole Q_{21}, which is identical to that of the original structure. This indicates that the modified F environment does not affect the distortion of the CoF₆ octahedra along the xz direction, thus preserving the sign of the ferrotype Q_{21} quadrupole. Taking the {\epsilon }_{xz} of CoF₂ and RuO₂ as an example, we find that in the absence of SOC, the charge quadrupole is also non-zero, yielding Q_{21}, which is consistent with the conclusion regarding SOC (see Fig. S10 of the SM). This demonstrates that, compared to the higher-order charge multipoles in heavy fermion systems with SOC, the charge quadrupole in CoF₂ and RuO₂ has a non-relativistic origin. Therefore, Q_{21} is an important order parameter that induces the piezomagnetic effect in altermagnetic materials.

4 Conclusions

We have investigated the effect of shear strains on the SS and magnetization of prototypical altermagnetic materials, CoF₂ and RuO₂. First-principles calculations reveal SS along additional high symmetry paths in reciprocal space, resulting from the reduction of crystal symmetry under shear strains. Importantly, reversing the direction of the shear strain can switch the sign of the SS. The weak magnetization induced by shear strains arises from the contributions of both spin and orbital components. The Q_{21}, Q_{{21}^{-}}, and Q_{22} components of the charge quadrupole are identified as the origin of the linear piezomagnetic effect. Our work proposes an effective approach to inducing PZM in altermagnetic materials by breaking symmetry via shear strain. This approach and the corresponding analytical results can be generalized to other AB₂-type altermagnetic materials (A = Cr, Mn, Co, Fe, Ru, Rh; B = O, S, Se, F, Cl, Br), which share the same structure and AFM arrangement as CoF₂ and RuO₂.

References

[1] K. Olejník, T. Seifert, Z. Kašpar, V. Novák, P. Wadley, R. P. Campion, M. Baumgartner, P. Gambardella, P. Němec, J. Wunderlich, J. Sinova, P. Kužel, M. Müller, T. Kampfrath, and T. Jungwirth, Terahertz electrical writing speed in an antiferromagnetic memory, Sci. Adv. 4(3), eaar3566 (2018)

[2] W. B. Niu, G. L. Ding, Z. Q. Jia, X. Q. Ma, J. Y. Zhao, K. Zhou, S. T. Han, C. C. Kuo, and Y. Zhou, Recent advances in memristors based on two-dimensional ferroelectric materials, Front. Phys. (Beijing) 19(1), 13402 (2024)

[3] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Antiferromagnetic spintronics, Nat. Nanotechnol. 11(3), 231 (2016)

[4] O. Gomonay, V. Baltz, A. Brataas, and Y. Tserkovnyak, Antiferromagnetic spin textures and dynamics, Nat. Phys. 14(3), 213 (2018)

[5] P. Němec, M. Fiebig, T. Kampfrath, and A. V. Kimel, Antiferromagnetic opto-spintronics, Nat. Phys. 14(3), 229 (2018)

[6] J. Železný, P. Wadley, K. Olejník, A. Hoffmann, and H. Ohno, Spin transport and spin torque in antiferromagnetic devices, Nat. Phys. 14(3), 220 (2018)

[7] 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(6), 064435 (2022)

[8] S. D. Guo, Y. Liu, J. Yu, and C. C. Liu, Valley polarization in twisted altermagnetism, Phys. Rev. B 110(22), L220402 (2024)

[9] S. D. Guo, Valley polarization in two-dimensional zero-net-magnetization magnets, Appl. Phys. Lett. 126(8), 080502 (2025)

[10] H. Bai, L. Han, X. Y. Feng, Y. J. Zhou, R. X. Su, Q. Wang, L. Y. Liao, W. X. Zhu, X. Z. Chen, F. Pan, X. L. Fan, and C. Song, Observation of spin splitting torque in a collinear antiferromagnet RuO₂, Phys. Rev. Lett. 128(19), 197202 (2022)

[11] A. Bose, N. J. Schreiber, R. Jain, D. F. Shao, H. P. Nair, J. Sun, X. S. Zhang, D. A. Muller, E. Y. Tsymbal, D. G. Schlom, and D. C. Ralph, Tilted spin current generated by the collinear antiferromagnet ruthenium dioxide, Nat. Electron. 5(5), 267 (2022)

[12] S. Karube, T. Tanaka, D. Sugawara, N. Kadoguchi, M. Kohda, and J. Nitta, Observation of spin-splitter torque in collinear antiferromagnetic RuO₂, Phys. Rev. Lett. 129(13), 137201 (2022)

[13] L. Šmejkal, A. B. Hellenes, R. González-Hernández, J. Sinova, and T. Jungwirth, Giant and tunneling magnetoresistance in unconventional collinear antiferromagnets with nonrelativistic spin−momentum coupling, Phys. Rev. X 12(1), 011028 (2022)

[14] Q. R. Cui, B. W. Zeng, P. Cui, T. Yu, and H. X. Yang, Efficient spin seebeck and spin nernst effects of magnons in altermagnets, Phys. Rev. B 108(18), L180401 (2023)

[15] S. D. Guo, X. S. Guo, K. Cheng, K. Wang, and Y. S. Ang, Piezoelectric altermagnetism and spin-valley polarization in Janus monolayer Cr₂SO, Appl. Phys. Lett. 123(8), 082401 (2023)

[16] L. D. Yuan, Z. Wang, J. W. Luo, E. I. Rashba, and A. Zunger, Giant momentum-dependent spin splitting in centrosymmetric low-Z antiferromagnets, Phys. Rev. B 102(1), 014422 (2020)

[17] L. Šmejkal, J. Sinova, and T. Jungwirth, Beyond conventional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X 12(3), 031042 (2022)

[18] L. Šmejkal, J. Sinova, and T. Jungwirth, Emerging research landscape of altermagnetism, Phys. Rev. X 12(4), 040501 (2022)

[19] S. Bhowal and N. A. Spaldin, Ferroically ordered magnetic octupoles in d-Wave altermagnets, Phys. Rev. X 14(1), 011019 (2024)

[20] A. S. Borovik-romanov, Piezomagnetism in the antiferromagnetic fluorides of cobalt and manganese, Sov. Phys. JETP 11, 786 (1960)

[21] F. Formisano, R. M. Dubrovin, R. V. Pisarev, A. K. Zvezdin, A. M. Kalashnikova, and A. V. Kimel, Laser-induced THz piezomagnetism and lattice dynamics of antiferromagnets MnF₂ and CoF₂, Ann. Phys. 447, 169041 (2022)

[22] S. W. Lovesey, D. D. Khalyavin, and G. van der Laan, Magnetic structure of RuO₂ in view of altermagnetism, Phys. Rev. B 108(12), L121103 (2023)

[23] O. Fedchenko, J. Minár, A. Akashdeep, S. W. D’Souza, D. Vasilyev, O. Tkach, L. Odenbreit, Q. Nguyen, D. Kutnyakhov, N. Wind, L. Wenthaus, M. Scholz, K. Rossnagel, M. Hoesch, M. Aeschlimann, B. Stadtmüller, M. Kläui, G. Schönhense, T. Jungwirth, A. B. Hellenes, G. Jakob, L. Šmejkal, J. Sinova, and H. J. Elmers, Observation of time-reversal symmetry breaking in the band structure of altermagnetic RuO₂, Sci. Adv. 10(5), eadj4883 (2024)

[24] S. Reimers, L. Odenbreit, L. Šmejkal, V. N. Strocov, P. Constantinou, A. B. Hellenes, R. Jaeschke Ubiergo, W. H. Campos, V. K. Bharadwaj, A. Chakraborty, T. Denneulin, W. Shi, R. E. Dunin-Borkowski, S. Das, M. Kläui, J. Sinova, and M. Jourdan, Direct observation of altermagnetic band splitting in CrSb thin films, Nat. Commun. 15(1), 2116 (2024)

[25] T. Aoyama and K. Ohgushi, Piezomagnetic properties in altermagnetic MnTe, Phys. Rev. Mater. 8(4), L041402 (2024)

[26] H. Y. Ma, M. L. Hu, N. N. Li, J. P. Liu, W. Yao, J. F. Jia, and J. W. Liu, Multifunctional antiferromagnetic materials with giant piezomagnetism and noncollinear spin current, Nat. Commun. 12(1), 2846 (2021)

[27] Y. Zhu, T. K. Chen, Y. C. Li, L. Qiao, X. N. Ma, C. Liu, T. Hu, H. Gao, and W. Ren, Multipiezo effect in altermagnetic V₂SeTeO monolayer, Nano Lett. 24(1), 472 (2024)

[28] A. ullah, D. Bezzerga, and J. Hong, Giant spin seebeck effect with highly polarized spin current generation and piezoelectricity in flexible V₂SeTeO altermagnet at room temperature, Mater. Today Phys. 47, 101539 (2024)

[29] J. Zemen, Z. Gercsi, and K. G. Sandeman, Piezomagnetism as a counterpart of the magnetovolume effect in magnetically frustrated Mn-based antiperovskite nitrides, Phys. Rev. B 96(2), 024451 (2017)

[30] M. Ikhlas, S. Dasgupta, F. Theuss, T. Higo, S. Kittaka, B. J. Ramshaw, O. Tchernyshyov, C. W. Hicks, and S. Nakatsuji, Piezomagnetic switching of the anomalous Hall effect in an antiferromagnet at room temperature, Nat. Phys. 18(9), 1086 (2022)

[31] P. A. McClarty and J. G. Rau, Landau theory of altermagnetism, Phys. Rev. Lett. 132(17), 176702 (2024)

[32] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136(3B), B864 (1964)

[33] W. Kohn and L. J. Sham, Self−Consistent equations including exchange and correlation effects, Phys. Rev. 140(4A), A1133 (1965)

[34] G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6(1), 15 (1996)

[35] G. Kresse and J. Furthmuller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54(16), 11169 (1996)

[36] P. E. Blöchl, Projected augmented-wave method, Phys. Rev. B 50(24), 17953 (1994)

[37] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77(18), 3865 (1996)

[38] J. A. Barreda-Argüeso, S. López-Moreno, M. N. Sanz-Ortiz, F. Aguado, R. Valiente, J. González, F. Rodríguez, A. H. Romero, A. Muñoz, L. Nataf, and F. Baudelet, Pressure-induced phase-transition sequence in CoF₂: An experimental and first-principles study on the crystal, vibrational, and electronic properties, Phys. Rev. B 88(21), 214108 (2013)

[39] C. A. Corrêa and K. Výborný, Electronic structure and magnetic anisotropies of antiferromagnetic transition-metal difluorides, Phys. Rev. B 97(23), 235111 (2018)

[40] T. Berlijn, P. C. Snijders, O. Delaire, H. D. Zhou, T. A. Maier, H. B. Cao, S. X. Chi, M. Matsuda, Y. Wang, M. R. Koehler, P. R. C. Kent, and H. H. Weitering, Itinerant antiferromagnetism in RuO₂, Phys. Rev. Lett. 118(7), 077201 (2017)

[41] S. Liu, Q. Wang, M. Q. Long, and Y. P. Wang, Theoretical study of the nonlinear magnon–phonon coupling in CoF₂, J. Phys.: Condens. Matter 36(35), 355801 (2024)

[42] A. Smolyanyuk, I. I. Mazin, L. Garcia-Gassull, and R. Valentí, Fragility of the magnetic order in the prototypical altermagnet RuO₂, Phys. Rev. B 109(13), 134424 (2024)

[43] R. A. Erickson, Neutron diffraction studies of antiferromagnetism in manganous fluoride and some isomorphous compounds, Phys. Rev. 90(5), 779 (1953)

[44] J. W. Stout and L. M. Matarrese, Magnetic anisotropy of the iron-group fluorides, Rev. Mod. Phys. 25(1), 338 (1953)

[45] J. J. Lin, S. M. Huang, Y. H. Lin, T. C. Lee, H. Liu, X. X. Zhang, R. S. Chen, and Y. S. Huang, Low temperature electrical transport properties of RuO₂ and IrO₂ single crystals, J. Phys.: Condens. Matter 16(45), 8035 (2004)

[46] K. Yamauchi, P. Barone, and S. Picozzi, Bulk Rashba effect in multiferroics: A theoretical prediction for BiCoO₃, Phys. Rev. B 100(24), 245115 (2019)

[47] L. D. Yuan, Z. Wang, J. W. Luo, and A. Zunger, Prediction of low-Z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin−orbit coupling, Phys. Rev. Mater. 5(1), 014409 (2021)

[48] L. Šmejkal, R. González-Hernández, T. Jungwirth, and J. Sinova, Crystal time-reversal symmetry breaking and spontaneous hall effect in collinear antiferromagnets, Sci. Adv. 6(23), eaaz8809 (2020)

[49] C. Ederer and N. A. Spaldin, Towards a microscopic theory of toroidal moments in bulk periodic crystals, Phys. Rev. B 76(21), 214404 (2007)

[50] A. Scaramucci, E. Bousquet, M. Fechner, M. Mostovoy, and N. A. Spaldin, Linear magnetoelectric effect by orbital magnetism, Phys. Rev. Lett. 109(19), 197203 (2012)

[51] A. S. Disa, M. Fechner, T. F. Nova, B. Liu, M. Först, D. Prabhakaran, P. G. Radaelli, and A. Cavalleri, Polarizing an antiferromagnet by optical engineering of the crystal field, Nat. Phys. 16(9), 937 (2020)

[52] T. N. Gaydamak, G. A. Zvyagina, K. R. Zhekov, I. V. Bilich, V. A. Desnenko, N. F. Kharchenko, and V. D. Fil, Acoustopiezomagnetism and the elastic moduli of CoF₂, Low Temp. Phys. 40(6), 524 (2014)

[53] B. B. Van Aken, J. P. Rivera, H. Schmid, and M. Fiebig, Observation of ferrotoroidic domains, Nature 449(7163), 702 (2007)

[54] N. A. Spaldin, M. Fechner, E. Bousquet, A. Balatsky, and L. Nordström, Monopole-based formalism for the diagonal magnetoelectric response, Phys. Rev. B 88(9), 094429 (2013)

[55] M. T. Suzuki, H. Ikeda, and P. M. Oppeneer, First-principles theory of magnetic multipoles in condensed matter systems, J. Phys. Soc. Jpn. 87(4), 041008 (2018)

[56] A. M. Tehrani and N. A. Spaldin, Untangling the structural, magnetic dipole, and charge multipolar orders in Ba₂MgReO₆, Phys. Rev. Mater. 5(10), 104410 (2021)

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