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Ferroelectric switchable altermagnetism
Mingqiang Gu, 1 Yuntian Liu, 1 Haiyuan Zhu, 1 Kunihiro Yananose, 2 Xiaobing Chen, 1 Yongkang Hu, 1 Alessandro Stroppa, 3 , ∗ and Qihang Liu 1 , 4, †
arXiv:2411.14216v2 [cond-mat.mtrl-sci] 22 Jan 2025
1 Department of Physics and Guangdong Basic Research Center of Excellence for Quantum Science,
Southern University of Science and Technology, Shenzhen 518055, China
2 Korea Institute for Advanced Study, Seoul 02455, Republic of Korea 3 CNR-SPIN, c/o Dip.to di Scienze Fisiche e Chimiche - Università degli Studi dell’Aquila - Via Vetoio - 67100 - Coppito (AQ), Italy 4 Guangdong Provincial Key Laboratory of Computational Science and Material Design,
Southern University of Science and Technology, Shenzhen 518055, China
Southern University of Science and Technology, Shenzhen 518055, China (Dated: January 23, 2025) We propose a novel ferroelectric switchable altermagnetism effect, the reversal of ferroelectric polarization is coupled to the switching of altermagnetic spin splitting. We demonstrate the design principles for the ferroelectric altermagnets and the additional symmetry constraints necessary for switching the altermagnetic spin splitting through flipping the electric polarization based on the state-of-the-art spin-group symmetry techniques. 22 ferroelectric altermagnets are found by screen- ing through the 2001 experimental reported magnetic structures in the MAGNDATA database and 2 of them are identified as ferroelectric switchable altermagnets. Using the hybrid improper ferro- electric material [C(NH 2 ) 3 ] Cr(HCOO) 3 as an example, we show how the altermagnetic spin splitting is tightly coupled to the ferroelectric polarization, providing an ideal platform for designing electric- field-controllable multiferroic devices. Finally, we find that such manipulation of altermagnetism can be detected by monitoring the physical quantities that are related to the non-vanishing Berry curvature dipole, such as the linearly polarized photogalvanic spin current. Introduction - Multiferroic materials exhibit more than one type of ferroic order simultaneously, such as ferroelec- tricity (spontaneous electric polarization) and magnetism (ferromagnetism or antiferromagnetism) [1]. In partic- ular, magnetoelectrically coupled multiferroic materials have attracted significant interest due to their poten- tial for controlling magnetic properties through electric fields, and vice versa, opening pathways for innovative applications in memory storage, sensors, and spintron- ics [2–5]. Multiferroics are broadly categorized into two types [6]. In type-I multiferroics, ferroelectric and mag- netic orders originate from distinct mechanisms [7, 8], which often results in weak coupling between these or- der parameters. In contrast, type-II multiferroics demon- strate stronger coupling because ferroelectricity arises di- rectly from magnetic ordering [9]. Historically, the de- sign and manipulation of multiferroicity primarily rely on the interaction between net magnetic moments and electric polarization [10]. For example, in multiferroic antiferromagnet Ca 3 M n 2 O 7 , the multiferroic coupling stems from the ferroelectric polarization and the net mag- netic moment induced by spin-orbit coupling (SOC), i.e., Dzyaloshinskii–Moriya interaction [11]. However, the small weak FM magnetization (0.045 µ B /Mn in theory and 0.0025 µ B /Mn in experiment) [12] limits the strength of magnetoelectric coupling and thus the multiferroic ap- plications. Recently, altermagnetism has garnered considerable attention [13–16] as a promising avenue for achieving novel spintronic properties. Altermagnets are a class of collinear antiferromagnetic (AFM) materials charac- terized by alternating spin polarization across recipro- cal space due to breaking of certain spatial symmetries, even though they have no net macroscopic magnetiza- tion. The order parameter S featuring altermagnetism can be represented by the energy splitting between the two spin channels in certain paths of the Brillouin zone ∆ E s k = E ↑ k − E ↓ k . The spin-split bands of Bloch electrons in altermagnets create the possibility of designing new types of spintronic devices, e.g., spin-filtering magnetic tunnel junctions [17, 18]. Due to the combination of the advantages of FM and AFM in terms of spin splitting, al- termagnetism is also considered a third type of collinear magnetism [14, 19]. In this Letter , we introduce the ferroelectric switch- able altermagnets , a new class of materials which shows not only the coexistence of ferroelectricity and altermag- netism, but also the synergistic coupling of the two or- der parameters, leading to a reversal of spin-character in the splitted energy bands by switching the ferroelec- tric polarization. As illustrated in Fig. 1(a), in specific circumstances, the altermagnetic spin degree of freedom is tightly correlated to the electric polarization, indicat- ing a distinct form of magnetoelectric coupling. Such a mechanism enables the control of a nonvolatile spin- split electronic dispersion through the external electric field, providing an efficient route to achieve robust all- in-one multiferroic memory devices. For example, Fig. 1(b) illustrates an electric-field-controllable spin-filtering altermagnetic tunnel junction device without the need of switching the N´eel vector . By considering whether the spin-polarized Fermi surface between the free layer and the fixed layer match or not [14], the conductance σ can be switched between high and low by simply switching
2
(a) (b) +
P
High -
P
Low Free Fixed Free Fixed FIG. 1. (a) Schematic illustration of ferroelectric switch- able altermagnetism, where the altermagnetic spin splitting S strongly couples to the ferroelectric polarization P . (b) A non-volatile spin filtering tunnel junction device design with ferroelectric switchable altermagnets. The Fermi surfaces of the fixed layer and the free layer are also illustrated. The electric polarization P can be used to control the spin polar- ization of the free layer, leading to a transition between high- and low-conductance ( σ ) states. the ferroelectric polarization. By applying the state- of-the-art spin-group symmetry analyses, we searched through all the MAGNDATA database [20] and screened out 22 ferroelectric altermagnets , in which only two can- didates support the synergistic switching of altermag- netic spin splitting through ferroelectric polarization. In order to discuss a representative case of the fer- roelectric switchable altermagnets, we perform first- principles calculations on the metal-organic framework (MOF) system [C(NH 2 ) 3 ] Cr(HCOO) 3 , which we will de- note as Cr-MOF hereafter. This compound is isostruc- tural to Cu-MOF found in MAGNDATA and is partic- ularly interesting since it expands the search for poten- tially interesting altermagnets to the extremely tunable and versatile class of hybrid organic-inorganic perovskites for functional materials design. Cr-MOF has been exper- imentally synthesized, and it has been proposed as hy- brid improper ferroelectric [21, 22]. Here, we predict that the altermagnetic spin splitting in this material is tightly coupled to the ferroelectric polarization, in both the mag- nitude and the sign of ∆ E s k , providing an ideal platform for designing electric-field-controllable spintronic devices. Finally, we find that the manipulation of altermagnetism can be monitored by the nonlinear optical generation of spin current, which originates from the non-zero Berry curvature dipole. Material design principles - Altermagnetic spin split- ting is a non-relativistic property that manifests with- out the need for spin-orbit coupling (SOC), where the symmetry-theoretical framework should adopt spin space group (SSG) [15, 23–27] instead of the commonly used magnetic space group (MSG). We first consider ferroelec- tric altermagnets where the two order parameters P and S are not necessarily coupled. Two rules under the frame- work of SSG emerge: i) For a collinear antiferromagnet manifesting spin splitting in the momentum space, the symmetries of joint parity and time-reversal ( PT ) and T τ with τ being the fractional lattice translation, should be broken; ii) For ferroelectric material, the spatial part of the parent space group (SG) of the SSG should belong to a polar group. We diagnosed the SSGs of the 2001 experimentally re- ported magnetic structures in the MAGNDATA database by using our homemade online program FINDSPIN- GROUP [28], and found 22 systems that meet the above criteria. One immediately finds that the filtered mate- rials can be divided into two classes by examining the MSG of the materials: the MSG of the first class allows weak spin canting, leading to a non-zero residual ferro- magnetic (FM) moment, while the second class forbids net magnetic moment in the system. For the first class with P-M coupling, the electronic polarization can inter- play with the weak FM moment, leading to a classical magnetoelectric coupling [11]. Note that SOC plays an essential role (e.g., Dzyaloshinskii-Moriya interaction) in slightly tilting the spins from the collinear AFM con- figuration. For the second class without P-M coupling, since the net magnetic moment is forbidden by MSGs, the flipping of electric polarization is not supposed to in- duce any macroscopic effect on the magnetization. How- ever, both two classes are potential candidates for ferro- electric switchable altermagnets if the real-space electric polarization and the momentum-space spin splitting are coupled, even without SOC. In general, the coexistence of electric polarization P and altermagnetic spin splitting ∆ E S k (denoted by S ) does not guarantee their coupling. Therefore, we next focus on the search of symmetry operations that syner- gistically switch P and S , and if these operations cor- respond to reasonable flipping paths under an external electric field, which does not change the N´eel order. For example, considering an initial state ψ i k ( P, S ), one can always flip both electric and spin polarization by ap- plying a joint operation PT to the system to get a re- versed state PT ψ i k ( P, S ) = ψ f k ( −P, −S ). However, in the transition path connecting ψ f k ( −P, −S ) and ψ i k ( P, S ) phases, the N´eel vector may also be reversed, indicat- ing a transition path that cannot be achieved by elec- tric field only. By going through the list in TABLE SI in Supplementary Materials, we found that the sign of ∆ E S k for most candidates remains unchanged during the polarization reversal, as exemplified by PbNiO 3 in Sup-
3
Gua (HCOO) - Cr 2 + Γ X S R Y 𝑘 𝑧 𝑘 𝑦 𝑘 𝑥 + + - - (a) (b) (c) Cr-MOF ( Pna2 1 ) Sgn(S) P
z
(e) up down up down 𝑘 𝑥 𝜋 𝑎 𝑘
𝑦
𝜋
𝑏
0 0.5 -0.5 -0.5 0 0.5 𝑘 𝑥 𝜋 𝑎 𝑘
𝑦
𝜋
𝑏
0 0.5 -0.5 -0.5 0 0.5 (d) (𝑃, 𝑆) (−𝑃, −𝑆) FIG. 2. (a) Crystal structure of Cr-MOF. (b) Spin-split band structure of Cr-MOF. (c) Spin-resolved isoenergy con- tour ( E = − 0 . 05 eV) for the ( P, S ) state at the k z = 0 . 25 plane, blue and red denote the spectral weight for spin up and spin down, respectively. (d) Same as (c), but for the ( −P, −S ) state. (e) The sign of ∆ E S k for the VBM in recip- rocal space. plementary Materials Sec. III. Interestingly, there are two hybrid improper ferroelectric materials, Ca 3 M n 2 O 7 and [C(NH 2 ) 3 ] Cr(HCOO) 3 , as prototypical candidates of ferroelectric switchable altermagnets. Altermagnetism in hybrid improper ferroelectric Cr- MOF - Hybrid improper ferroelectricity arises from a combination of two or more lattice modes that individually preserve inversion symmetry but collec- tively breaks it, resulting in a spontaneous polariza- tion. Several candidate materials from Class 1 in TA- BLE SI belong to this category, including Ca 3 M n 2 O 7 (SG Cmc 2 1 ) [29] and the hybrid organic-inorganic per- ovskite [C(NH 2 ) 3 ] M(HCOO) 3 (SG P na 2 1 ) with M being transition-metal elements Cu or Cr. Below we use Cr- MOF (see Fig. 2(a)) to illustrate the design principles of ferroelectric switchable altermagnets for two reasons: (a) it exhibits the largest spin splitting near the Fermi level among this material family, and (b) various AFM config- urations can be realized in Cr-MOF through compressive strain, offering an additional degree of freedom for phase manipulation [30]. Moreover, the compositional tunabil- ity in this family offers more flexibility in designing ma- terials for spin-splitting magnitude optimization. While A-, C- and G-type AFM Cr-MOF are all altermagnets, here we focus on the C-type configuration, as it manifests the largest spin splitting near the Fermi level. Further discussions on other MOFs and AFM configurations can be found in the Supplementary Materials Sec. IV. The spin-polarized band structure obtained from den- sity functional theory (DFT, see Supplementary Materi- als Sec. I for details) calculations is shown in Fig. 2(b). The band gap is about 3 eV, ensuring a good insulat- ing background that is favorable for ferroelectrics. No- tably, a large spin splitting is observed at the valence band maximum (VBM), with the largest energy differ- ence being approximately 20 meV. Such spin splitting occurs at the relatively low-symmetry points in recipro- cal space, such as along the Γ(0 , 0 , 0) → R (0 . 5 , 0 . 5 , 0 . 5) path. The spin-polarized isoenergy contour at E = − 0 . 05 eV is shown in Fig. 2(c), showing a deviated feature compared with the so-called “d-wave” altermagnetism because of the absence of four-fold symmetry. Further- more, the sign of ∆ E S k , a well as the spin polarization of a given spin-polarized band, are odd under mirror opera- tions { 1 ||m 100 } and { 1 ||m 010 } , and even under { 1 ||m 001 } , as illustrated in Fig. 2(e). Synergistically switch of electric and spin polarizations - Here, a state with the coexistence of polarization P ( P//z ) and altermagnetism S is denoted as the ( P, S ) state. The SSG operations that map P → −P and S → −S are analyzed and listed in Fig. S3. While the SSG op- erations connecting P to −P state can be one of the fol- lowing: { 1 || -1 } , { 1 || 2 1 00 } , { 1 || 2 0 10 } , { -1 || -1 } , { -1 || 2 1 00 } , and { -1 || 2 0 10 } , only three of them map the ( P, S )state to the ( −P, −S ) state, i.e., { 1 || 2 1 00 } , { 1 || 2 0 10 } , and { -1 || - 1 } . In addition, whether the ( −P, −S ) state can be ex- perimentally realized upon the application of an electric field, however, depends on the energy barrier between the two phases, which is ultimately governed by the internal atomic displacements during the phase transition. The search for the reversed polarization state can also be achieved by considering the mode decomposition of hybrid improper ferroelectricity. The electric polariza- tion of Cr-MOF along the z direction is activated by two non-polar lattice distortion modes through a trilinear coupling F = γQ X − 1 Q X + 4 Q P , where Q X − 1 , Q X + 4 and Q P represent the amplitudes of the two zone-boundary non- polar distortion modes and the polar distortion mode, respectively, relative to the high symmetry Imma phase [21]. Reversing either of the zone-boundary modes flips the polarization P → −P , while, reversing both preserves P . As shown in Fig. 3(a), the X − 1 mode consists of two major distortions: a “scissor”-like distortion for the equatorial Cr-O bonds and a rotation of the guanidinium molecule by about 42°. The X + 4 mode corresponds pri- marily to a Q 2 Jahn-Teller (JT) distortion of the Cr oc- tahedra, with minimal impact on the organic molecules. Starting from the Imma structure, imposing only the X − 1 ( X + 4 ) mode results in a centrosymmetric Pnna ( Pnma ) structure. The combination of both X − 1 and X + 4 modes leads to the Pna2 1 structure, a polar SG that induces the softening of the polar distortion. Since the polar distor- tion is a secondary effect arising from the coupling of the zone-boundary modes, the interplay between these two modes fully determines the final structure and its related properties in this system. For simplicity, and without loss of generality, we will use these two unstable modes ( X − 1 and X + 4 ) as variables in the discussion below.
4 {1||2 100 }
X 1 − X 4 + -1 Imma Pnma Pnna Pna2 1 (P, S) (-P, -S) X 1 − X 4 + (a) (b) Ε 1 1
(-P, S)
(c) (d)
Δ
Ε
~
0
.
1
e
V
-1 (P, -S)
{1||m 001 }
FIG. 3. (a) The hybrid improper coupled zone-boundary distortion modes in Cr-MOF. (b) Energy landscape for the coupling of the X − 1 and X + 4 modes, where different transition paths are marked by yellow and purple dashed lines. (c) Energy profiles along the two paths denoted in (b). (d) Spin splitting as a function of the interpolation structural parameter ξ . The insets in (d) denote the Q 2 JT pattern at the endpoints, with the pink and blue diamonds denote the octahedra centered with the spin-up and spin-down Cr, respectively. The elongated octahedra form an antiferro-distortive JT distortion pattern associated to the orbital-ordering which “pseudo-rotate” from the two endpoints ( P, S ) and ( −P, −S ). Fig. 3(b) shows the energy landscape of the coupled zone-boundary modes relative to the Imma phase. Start- ing from the energetically favored ( P, S ) ground state, if the amplitude of X − 1 mode is continuously reduced and eventually reversed its sign, the structure transforms to a ( −P, S ) state. This transition path, indicated by the purple line in Fig. 3(b), passes through a centrosym- metric Pnma phase. The ( −P, S ) state and the original ( P, S ) state are related by { 1 ||m 001 } SSG operation. The energy barrier is about 4 eV per unit cell, as shown in the left panel of Fig. 3(c). Such a high energy barrier is at- tributed to the large atomic distortion amplitude ( ∼ 6.3 ˚ A) along this path, primarily due to the rotation of the guanidinium molecule. In sharp contrast, when the X + 4 mode is continuously tuned to the reversed amplitude, the transition occurs via a Pnna reference phase (orange line in Fig. 3(b)). This path leads to the ( −P, −S ) state, connected by { 1 || 2 1 00 } SSG operation, with a signifi- cantly lower energy barrier of approximately 0.1 eV (the right panel of Fig. 3(c)), nearly 40-times smaller than that for the Pnma path. Thus, the ( P, S ) → ( −P, −S ) structural transition through the Pnna path is energeti- cally favored and represents a possible realistic transition pathway when polar direction is flipped by an external electric field. The spin reversal of the isoenergy contour for the ( −P, −S ) state shown in Fig. 2(d) strongly supports the possibility to control altermagnetic spin splitting by flipping the ferroelectric polarization in this system. To quantify this transition, we define a normalized inter- polating structural parameter ξ , where ξ = 1 repre- sents the ( P, S ) structure and ξ = − 1 corresponds to either the ( −P, S ) or ( −P, −S ) structure. The evolu- tion of ∆ E s k during the structural transition is plotted in Fig. 3(d). We found that ∆ E s k changes monoton- ically during the ( P, S ) → ( −P, −S ) transition, while
5 0.4 Spin LPGE σ zyy (μA/V 2 )
in the ( P, S ) → ( −P, S ) transition it reduces to zero at the Pnma phase and then regresses to its original value. Such behavior arises because ∆ E s k is odd under { 1 || 2 1 00 } but even under { 1 ||m 001 } , consistent with the symme- try presented in Fig. 2(e). Furthermore, the microscopic mechanism of the reversal of the altermagnetic spin split- ting by reversing the X + 4 mode can be understood by the change of the Q 2 JT distortion pattern, as illustrated in the insets of Fig. 3(d). Since the JT distortion is known to couple with the orbital ordering in oxide perovskites [31] and MOF systems [32], our results provide a con- crete example of how altermagnetism can be entangled with orbital ordering [33], offering new insights into the control of spin splitting through orbital as well as lattice degrees of freedom. To illustrate the electric-field-controllable altermag- netic tunnel junction device, we consider a prototypical homojunction system proposed in Fig. 1(b), where both the fixed and free layers are composed of Cr-MOF. We evaluate the giant magnetoresistance (GMR) by the ratio between the DFT-calculated longitudinal conductivity of the spin-up and spin-down channels al ong th e [110] di- rection [34]. Based on a 184-atom √ 2 × √ 2 supercell, our results suggest that the maximum GMR rate for this system could reach approximately 53% (see Fig. S5 and Supplementary Materials Sec. VI for details), validating the potential of ferroelectric switchable altermagnetism for non-volatile spintronic devices.
Detection of the switch of altermagnetic spin split-
Detection of the switch of altermagnetic spin split- ting - The direct experimental detection of the AFM- induced spin splitting typically requires advanced tech- niques such as spin-resolved angle-resolved photoemis- sion spectroscopy [35, 36]. Since the N´eel vector re- mains unchanged through ( P, S ) → ( −P, −S ) transi- tion, we propose a way to detect the switch of alter- magnetic spin splitting through the linear-polarized pho- togalvanic spin current effect (spin LPGE), which di- rectly couples to the spin polarization through nonlinear optical response. By using the SSG symmetry analy- sis implemented in FINDSPINGROUP, we find that six non-zero Berry curvature dipole (BCD) tensor elements ( D x xz , D x zx , D z xx , D y yz , D y zy , D z yy ) are allowed with- out SOC in Cr-MOF. All the other tensor elements, such as the quantum metric dipole and inverse mass dipole [37], are forbidden. The non-zero BCD can induce a nonlinear Hall current in transport measurement, as well as the spin LPGE from an optical perspective. Due to the broken spatial inversion symmetry, a shift current is expected under linear-polarized light excitation [38]. In Cr-MOF system, both charge and spin currents can be generated: the charge current reflects the direction of the electric polarization P , while the spin current serves as a signature of the difference between the two spin channels, i.e. the spin splitting. The spin LPGE conductivity component σ z yy ↑−↓ for the ( P, S ) and ( −P, −S ) states is shown in Fig. 4. A non-
0.3 (-P,-S) 0.2 0.1
0 -0.1 (P,S) -0.2 -0.3 -0.4 3.0 3.2 3.4 3.6
Photon Energy (eV)
3.8 4.0
FIG. 4. The spin LPGE conductivity component σ z yy ↑ −↓ for the ( P, S ) and ( −P, −S ) states. zero dc current begins to emerge when the photon energy exceeds the band gap, reaching the first peak at about 3.2 eV with a value of about 0.23 µA/V 2 . The second peak, near 4 eV, is attributed to the spin splitting at the higher conduction bands. During the ( P, S ) → ( −P, −S ) structure transition, the sign of the spin LPGE current reverses. Therefore, the spin LPGE current can serve as a signature of the reversal of the altermagnetic spin splitting. Conclusion and discussion - Recently, the gate-field control of different spin channels in layered altermag- nets through spin-valley-layer locking effect has been pro- posed [39, 40], highlighting the importance for the ma- nipulation of spin degree of freedom in altermagnets. However, such a non-versatile strategy requires the con- stant application of an external electric field, which is not favorable for storage devices unless further integra- tion with other materials. Sliding ferroelectricity [41] or phase transition between ferroelectric and antiferroelec- tric states [42] can also be used to tune the PT symme- try in the two dimensional van der Waals systems, which eventually control the emergence of the altermagnetic state. Considering that the hybrid improper ferroelec- tric materials can be artificially designed by chemistry or strain engineering [43], our study provides a wide config- uration space for designing versatile ferroelectric switch- able altermagnets. Our approach provides a novel method for control- ling spin properties through ferroelectric polarization, which differs from traditional magnetoelectric multifer- roics where ferroelectricity is used to control the net mag- netic moment. For the Class 1 candidate Ca 3 M n 2 O 7 , the energy splitting ( ∼ 100 meV) is even larger than that of Cr-MOF. Spin canting is allowed in this system, leading to a weak net ferromagnetic moment M . Our symmetry
6 analysis shows that the order parameter for spin splitting S follows the same transformation rules as that for M [11], which can be detected by the anomalous Hall effect. Therefore, an additional order parameter M is added to the coupled degrees of freedom, i.e., ( P, M, S ), offering multiple tuning knobs for material manipulation and pro- viding enhanced versatility in device design (see Supple- mentary Materials Sec. V for details). In contrast, for the Class 2 candidates with no net magnetic moment, the most traditional magnetic detection method such as magneto-optical Kerr effect [16, 44] could be effective, our approach still offers a viable pathway to achieve magne- toelectric coupling by correlating the electric polariza- tion with spin splitting in reciprocal space. To conclude, ferroelectric switchable altermagnetism is a new type of magnetoelectric coupling beyond what is achievable in traditional multiferroics based solely on the spin-orbit interaction, paving a new functional materials platform for designing all-electric-controlled spintronic devices. Note added - Recently, we became aware of a related study on Ca 3 M n 2 O 7 for the “altermagnetoelectric effect” [45], a concept similar to the ferroelectric switchable al- termagnetism proposed in our work. A.S. thanks useful discussions with P. Radaelli (Ox- ford University), P. Barone (CNR-SPIN), Sang-Wook Cheong (Rutgers University) and Xinfeng Chen (Xi’an Jiaotong University). This work was supported by the National Key R&D Program of China under Grant No. 2019YFA0704900, the National Natural Science Foundation of China under Grant No. 12274194 and 12474232, Guangdong Provincial Key Laboratory for Computational Science and Material Design under Grant No. 2019B030301001, Shenzhen Science and Technol- ogy Program (Grants No. RCJC20221008092722009 and No. 20231117091158001), the Innovative Team of General Higher Educational Institutes in Guang- dong Province (Grant No. 2020KCXTD001), Guang- dong Provincial Quantum Science Strategic Initiative un- der Grant No. GDZX2401002, the Science Technology and Innovation Commission of Shenzhen Municipality (JCYJ20210324104812034), Natural Science Foundation of Guangdong Province (2021A1515110389) and Center for Computational Science and Engineering of South- ern University of Science and Technology. This work has been funded by the European Union - NextGenera- tionEU, Mission 4, Component 1, under the Italian Min- istry of University and Research (MUR) National Inno- vation Ecosystem grant ECS00000041-VITALITY-CUP B43C22000470005. We also acknowledge the support by bilateral agreement for scientific collaboration CNR and NSFC (China), “Ferroelectric and chiral hybrid organic inorganic perovskites”, for the years 2024-2025 (CUP B53C2300716000). K.Y. was supported by an individual grant (No. CG092501) at Korea Institute for Advanced Study. ∗ [email protected] † [email protected] [1] H. Schmid, Multi-ferroic magnetoelectrics, Ferroelectrics 162 , 317 (1994). [2] N. A. Spaldin, Multiferroics: Past, present, and future, MRS Bulletin 42 , 385 (2017). [3] R. Ramesh and N. A. Spaldin, Multiferroics: progress and prospects in thin films, Nat Mater 6 , 21 (2007). [4] M. Fiebig, T. Lottermoser, D. Meier, and M. 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