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Rui Peng 1 , Shibo Fang 1 , Pin Ho 2 , Tong Zhou 3 , Junwei Liu 4, * , Yee Sin Ang 1, †
1 Science, Mathematics and Technology (SMT) Cluster, Singapore University of Technology and
Design (SUTD), Singapore 487372, Singapore. 2 Institute of Materials Research and Engineering (IMRE), Agency for Science, Technology and
Research (A*STAR), Singapore 138634, Singapore. 3 Eastern Institute for Advanced Study, Eastern Institute of Technology, Ningbo, Zhejiang 315200,
China 4 Hong Kong University of Science and Technology, Hong Kong, China SAR.
Emails: *[email protected], † [email protected]
Combination of altermagnetism and ferroic orders, such as ferroelectric switchable
altermagnetism [Phys. Rev. Lett. 134, 106801 (2025) and Phys. Rev. Lett. 134, 106802
(2025)], offers a powerful route to achieve nonvolatile switching of altermagnetic spin
splitting. In this work, by synergizing altermagnetism and ferroelasticity, we propose the
concept of ferroelastic altermagnets in which the ferroelastic crystal reorientation can drive
multistate nonvolatile switching of the altermagnetic spin splitting via altermagnetoelastic
effect. Using monolayers RuF 4 and CuF 2 as material candidates, we demonstrate 2-state
and 3-state altermagnetic spin splitting switching as driven by ferroelastic strain states.
Transport calculation shows that multistate spin conductivities can be ferroelastically
encoded in an ferroelastic altermagnet, thus suggesting the potential of ferroelastic
altermagnetic as nonvolatile nanomechanical spin switches. The proposed concept of
ferroelastic altermagnetism enriches the emerging landscape of multiferroic altermagnetism
and shall pave a way towards straintronic-spintronic device applications.
1
Together with ferromagnet (FM) and conventional antiferromagnet (AFM), altermagnet
represents an emerging new family of collinear magnetic materials [1-5]. Altermagnets
exhibit zero net magnetization like conventional AFM, but host nonrelativistic band spin
splitting like FM. Such FM-AFM dichotomy endows altermagnets with the advantages of
AFM such as robustness against external fields and compatible with ultrahigh-speed device
operation while still exhibiting broken spin degeneracy of FM, which are critical for spin
manipulation and information processing [6-21]. Altermagnets exhibit a plethora of intriguing
physical phenomena such as spin current generation [1-2], spin Hall effect [6], spin Nernst
effect [7], anomalous Hall effect [5], and tunneling magnetoresistance effects [6], which can
be harnessed for various device applications. A large variety of altermagnets have been
theoretically proposed [3-4,17,22], and some of them have been experimentally confirmed
[23-26].
The physics of altermagnets can be further enriched by multiferroicity. Multiferroics are
singular materials that exhibit two or more ferroic orders among (anti)ferromagnetism,
(anti)ferroelectricity and (anti)ferroelasticity [27-35]. For instance, the coupling between
ferroelectricity and magnetism yields magnetoelectric effect where ferroelectric polarization
and magnetization are coupled and can be mutually switched. Recent studies have unveiled
the altermagnetic counterpart of magnetoelectric effect, namely the altermagnetoelectric
effect [Fig. 1(a)] in which the altermagnetic spin splitting can be tuned by various forms of
ferroelectric switching [36-43], thus unveiling a route towards electrical-based nonvolatile
switching of altermagnetism. The multiferroic integration of ferroelectricity and
altermagnetism immediately raises the following question: Can altermagnetism be
integrated with ferroelasticity [44-48] – magnetoelastic effect – to achieve nonvolatile
nanomechanical deformation-induced switching of the altermagnetic spin splitting?
Here we propose the concept of ferroelastic altermagnetism in which altermagnetism
and ferroelasticity are synergized. As ferroelastic switching is equivalent to a rotation
operation on the lattice, such lattice rotation changes the spin-momentum coupling in a
ferroelastic altermagnetism, thus enabling an altermagnetoelastic effect in which the
altermagnetic spin splitting can be mechanically switched [Figs. 1(b,c)]. Ferroelastic 2 altermagnetism represents a mechanical multiferroic counterpart of the recently proposed
ferroelectric altermagnetism [36-40]. Using monolayers RuF 4 [49] and CuF 2 as proof-of-
concept, 2-state and 3-state nonvolatile switching of the altermagnetic spin splitting are
demonstrated, respectively, via first-principles calculations. We further show that the
multistate ferroelastic switching lead to nonvolatile switchable spin transport. These findings
reveal a previously unexplored mechanism to achieve nonvolatile strain-based
altermagnetic switching that enables multistate nonvolatile information encoding. This
concept of ferroelastic altermagnet shall serve as a harbinger of straintronic-spintronic
devices uniquely enabled by the union of altermagnetism and ferroelasticity.
2.1 Ferroelastic-driven altermagnetic spin switching and material candidates Two-dimensional (2D) materials can be classified according to layer groups [50], and those
belonging to layer groups L 8 –L 48 may host ferroelasticity due to their rectangular 2D Bravais
lattices. Altermagnetic monolayers within these layer groups may thus exhibit coexisting
ferroelasticity and altermagnetism [18]. To illustrate the concept of ferroelastic
altermagnetism, we use monolayers RuF 4 and CuF 2 (layer group L 17 ) as a proof-of-concept
3 material platform for achieving 2-state and 3-state ferroelastic-driven altermagnetic spin
switching. Monolayers RuF 4 and CuF 2 can potentially be mechanically exfoliated from bulks
RuF 4 and CuF 2 due to their layered structures [51-52]. As shown in Fig. 2(a), the crystal
lattice of monolayer RuF 4 has a rectangular lattice with a space group of P2 1 /c. The Ru
atoms are encapsulated in a distorted RuF 6 octahedron, and each Ru atom is connected to
six F atoms. The optimized lattice constants of monolayer RuF 4 is a = 5.42 Å and b = 5.08
Å. The crystal structure of monolayer CuF 2 shares the same space group P2 1 /c with
monolayer RuF 4 [Fig. 2(b)]. Also, the Cu atoms are encapsulated in a distorted CuF 6
octahedron, and each Cu atom is connected to six F atoms. The difference is that the
adjacent RuF 6 octahedrons are connected via the corner-sharing manner. The optimized
lattice constants of monolayer CuF 2 is a = 5.32 Å and b = 3.56 Å. The stability of monolayers
RuF 4 and CuF 2 are confirmed by phonon and ab initio molecular dynamics simulations (Fig.
Under the octahedral crystal field, the d orbitals split into t 2g triplets and e g doublets. In
monolayer RuF 4 , Ru atoms lose four electrons and transform into Ru 4+ ions with four 4d
electrons. Two of them occupy one t 2g orbital, one of them occupy another t 2g orbital, and 4 one of them occupy the remaining t 2g orbital, giving rise to a local magnetic moment of 2 μ B .
The inequivalent distribution of the electrons on the three t 2g orbitals leads to the
compressed Jahn–Teller distortions of the crystal structure, which further splits the t 2g and
e g orbitals (Fig. S2 [53]). In monolayer CuF 2 , Cu atoms lose two electrons and transform into Cu 2+ ions with nine 3d electrons. Six of them occupy the three t 2g orbitals, two of them occupy
one e g orbital, and one of them occupy the other e g orbital, giving rise to a local magnetic
moment of 1 μ B . The inequivalent distribution of the electrons on the two e g orbitals leads to
the elongated Jahn–Teller distortions of the crystal structure, making t 2g and e g orbitals split
in a different way with those in monolayer RuF 4 (Fig. S2 [53]). The calculated magnetic
moments on Ru and Cu atoms are 1.67 and 0.64 μ B , respectively, which is consistent with
the above analysis.
The ground states of monolayers RuF 4 and CuF 2 are determined by comparing the
energy among three magnetic configurations: FM, Néel antiferromagnetic (AFM-Néel), and
stripe antiferromagnetic (AFM-stripe) (Fig. S3 [53]). It is found that the energies of FM and
AFM-stripe are 30.28 (7.47) and 11.69 (4.27) meV/unit cell higher than the AFM-Néel in
monolayer RuF 4 (CuF 2 ), respectively, thus indicating AFM-Néel is the energetically most
favorable state for both RuF 4 and CuF 2 . The spin charge densities of AFM-Néel RuF 4 and
CuF 2 are shown in Figs. 2(c-d). After considering magnetism, the two opposite-spin
sublattices in RuF 4 and CuF 2 can only be transformed to each other by screw or glide
symmetry, which is typical for altermagnets.
Monolayer RuF 4 has an indirect band gap of 0.82 eV, with CBM located at M/M’ point
and VBM located along Γ-M/M’ line [Fig. 2(e)]. The spin-up and spin-down states are
degenerate along M-X-Γ-Y-M, while an alternating spin splitting appears along M’-Γ-M,
forming a spin-momentum locking physics. We also plotted the two highest valence bands
and the amplitude of the spin splitting ΔE between them over the entire Brillouin zone. The
spin-momentum locking physics can also be observed at generic k points. The amplitude of
the spin splitting reaches up to 205 meV. For monolayer CuF 2 , its indirect band gap is 0.69
eV, with CBM located at X point and VBM located at M/M’ point [Fig. 2(f)]. Similarly, the spin-
up and spin-down states are degenerate along M-X-Γ-Y-M, while an alternating spin splitting
appears along M’-Γ-M. The amplitude of the spin splitting reaches up to 98 meV. The spin-
up and spin-down density of states of RuF 4 and CuF 2 are equal [Figs. 2(g-h)].
5 2.2 Symmetry analysis The underlying mechanism of the spin-momentum locking in monolayers RuF 4 and CuF 2
can be elucidated through symmetry analysis. When omitting the nonmagnetic F atoms, the
systems have PT and t 1/2 T symmetry, thus guaranteeing that the band structures are spin-
degenerate over the entire Brillouin zone. The F atoms in the noncentrosymmetric sites
break the PT and t 1/2 T symmetry, yielding spin splitting in the band structures. The space
group of monolayers RuF 4 and CuF 2 , P2 1 /c, contains four symmetry operations: {E, P, t 1/2 M y ,
t 1/2 C 2y }. The former two connects the same-spin sublattices, while the latter two connects
the opposite-spin sublattices. Considering the crystal and spin rotation symmetries remain
decoupled when considering AFM coupling in the absence of spin-orbital coupling, the
symmetry connecting two opposite-spin sublattices can be represented as [C 2 ||R], where
the transformations on the left (right) of the double vertical bar act only on the spin (real)
space. When the [C 2 ||t 1/2 M y ] and [C 2 ||t 1/2 C 2y ] is operated on monolayer CuF 2 , respectively,
we obtain:
(1)
Consequently, we have
(2)
This explains the emergence of alternating spin-split bands along M’-Γ-M and at generic k
points. We also have
(3)
6 After considering the Bloch theorem, the bands are spin-degenerate when k x = mπ and k y =
nπ, thus explaining the spin-degenerate bands along high-symmetry paths M-X-Γ-Y-M.
2.3 Multistate nonvolatile ferroelastic-altermagnetic spin splitting switching
Interestingly, RuF 4 and CuF 2 are also ferroelastic monolayers. RuF 4 has two energy-
equivalent states F1 (a > b) and F2 (a < b), and one paraelastic state P (a = b) [Fig. 3(a)].
The ferroelastic switching from F1 to F2 is equivalent to a π/2 rotation of the lattice. In
contrast, CuF 2 has three ferroelastic states. For convenience, we choose the deformed
hexagon to be the structure unit. The three energy-equivalent ferroelastic phases can be
described as F1 (a 1 > a 2 /a 3 ), F2 (a 2 > a 1 /a 3 ) and F3 (a 3 > a 1 /a 2 ), where a 1 , a 2 and a 3 are the
diagonal lines of the deformed hexagon [Fig. 3(b)]. The paraelastic phase P is the T-phase
CuF 2 with a 1 = a 2 = a 3 . The ferroelastic switching from F1 to F2 (F3) is equivalent to a 2π/3
(4π/3) lattice rotation. The ferroelastic reversible strain of RuF 4 and CuF 2 , which are defined
as (a/b - 1) × 100% and (a i /a j - 1) × 100%, are calculated to be 6.8% and 11.4%, respectively,
We simulate the ferroelastic switching process of RuF 4 and CuF 2 using the nudged
elastic band (NEB) method. The overall switching barrier of RuF 4 and CuF 2 are 4 and 54
meV/atom [Figs. 3(c-d)], respectively, which is lower than those of phosphorene (200
meV/atom) [44] and borophane (100 meV/atom) [45]. The low switching barrier suggests
that monolayers RuF 4 and CuF 2 are potentially compatible with fast and low-energy
ferroelastic switching operation. Upon ferroelastic switching, the reoriented lattice changes
spin-momentum locking fashion, thus realizing multistate nonvolatile switching of the
In altermagnets, the spin-polarized current can be generated by an external in-plane
electric field due to the anisotropic spin splitting [1,2,40]. In ferroelastic altermagnets, the
spin transport current can be mechanically switched via different ferroelastic states. We
calculate the transport current in RuF 4 and CuF 2 when driven by an external field [53]. The 8 longitudinal and transverse spin conductivities exhibit an angular dependence on φ with a
period of π, where φ is the direction of the in-plane electric field relative to x axis [Fig. 4(a)]
and φ can be experimentally controlled by changing the relative angle between the source-
drain electrode axis [see Fig. 4(b) for a device schematic] and the crystal structure of
ferroelastic altermagnet sample. Upon a ferroelastic switching, the lattice re-orientation
changes the relative orientation between the altermagnetic spin-split band structure and the
electric field, thus leading to spin current switching effect. In RuF 4 , the carrier transport under
F1 and F2 states exhibits equal but opposite transverse and longitudinal spin conductivities
[Figs. 4(c-d)]. The 2-state ferroelastic switching between F1 and F2 states thus lead to spin
reversal. For CuF 2 , the spin conductivities exhibit even richer switching effect via the three
ferroelastic states F1, F2 and F3. For instance, when the electric field is directed at φ=0, the
longitudinal spin conductivities under F1, F2 and F3 states can be switched between positive,
zero and negative magnitudes, respectively [Figs. 4(e-f)]. The ferroelastic altermagnetic
broaden the application scenario of altermagnets, enabling nonvolatile spin information
encoding and hybrid straintronic-spintronic device application.
In summary, we have proposed the concept of ferroelastic altermagnets that synergizes ferroelasticity with altermagnetism. The switchable ferroelastic states in ferroelastic altermagnets drive multistate nonvolatile switching of the altermagnetic spin splitting, leading to an altermagnetoelastic effect. Using monolayers RuF 4 and CuF 2 as proof-ofconcept, 2-state and 3-state nonvolatile switching of the altermagnetic spin splitting are demonstrated. We further show that the multistate ferroelastic phase transitions are accompanied by a switching of the spin current associated with the altermagnetic spin splitting. We note that the ferroelastic altermagnetism can also be achieved in other 2D ferroelastic altermagnets such as the two-state ferroelastic monolayers of VF 4 [11,13,54,55] and OsF 4 [13], and the three-state ferroelastic monolayer AgF 2 [13,56,57].
First-principles calculations are performed based on density functional theory (DFT) [58] as implemented in Vienna ab initio simulation package (VASP) [59]. Exchange-correlation interaction is described by the Perdew-Burke-Ernzerhof (PBE) parametrization of 9 generalized gradient approximation (GGA) [60]. Structures are relaxed until the force on each atom is less than 0.01 eV/Å. The cutoff energy and electronic iteration convergence criterion are set to 600 eV and 10 −6 eV, respectively. To sample the 2D Brillouin zone, Monkhorst–Pack (MP) k-grid messes [61] of 7 × 7 × 1 and 5 × 9 × 1 are used for monolayers RuF 4 and CuF 2 , respectively. To avoid the interaction between adjacent layers, a vacuum space of 20 Å is added. Phonon calculations are carried out using the PHONOPY code with a 4 × 5 × 1 and 4 × 4 × 1 supercell for monolayers RuF 4 and CuF 2 , respectively [62]. Ab initio molecular dynamics (AIMD) simulations are performed at 300 K for 5 ps with a time step of 1 fs with a 4 × 5 × 1 and 4 × 4 × 1 supercell for monolayers RuF 4 and CuF 2 , respectively. Ferroelastic switching barrier is obtained by nudged elastic band (NEB) method [63]. The spin-resolved transport properties are calculated using a housing-made code, in which the electron energy and electron group velocity are evaluated from the Wannier-based tight binding Hamiltonian [64].
This work is supported by the Singapore Ministry of Education Academic Research Fund
(AcRF) Tier 2 under the Project ID MOE-000834-00 (Proposal ID T2EP50224-0006). P.H.
acknowledges the supports by the RIE2025 Manufacturing, Trade and Connectivity (MTC)
Individual Research Grants (Grant No. M23M6c0101) and (Grant No. M24N7c0086),
administered by A*STAR. T.Z. acknowledges the supports by the National Natural Science
Foundation of China (12474155) and the Zhejiang Provincial Natural Science Foundation of
China (LR25A040001). The computational work for this article was performed on resources
of the National Supercomputing Centre, Singapore (https://www.nscc.sg).
10
The authors declare no conflict of interest.
The data that support the findings of this study are available in the supplementary material
of this article.
Altermagnetism, Ferroelasticity, Multiferroicity, First-principles calculation, Nonvolatile
switching 11
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