High throughput discovery of 2D ferromagnetic and multiferroic transition metal oxyhalides and nitrogen halides

Abstract

Two-dimensional (2D) transition metal oxyhalides and nitrogen-halides (T_MBXs, where T_M = transition metal, B = O-group and N-group elements, X = halogen) have emerged as promising candidates for exploring multiferroic orders and spintronic applications. In this study, we conduct a systematic first-principles high-throughput screening combined with machine learning to identify novel 2D ferromagnetic and multiferroic materials within T_MBX family. From a comprehensive dataset comprising 672 T_MBX monolayers, we identify 78 ferromagnetic systems, of which 38 exhibit high Curie temperatures (T_C ≥ 200 K), significantly expanding the known library of 2D magnetic materials. A machine learning model is developed to elucidate the key factors governing ferromagnetism, revealing that the second-nearest neighbor exchange interaction (J₂) plays a dominant role in determining T_C. Furthermore, we discovered seven ferromagnetic-ferroelectric multiferroic systems, revealing unique polarization switching pathways. Notably, spin transport simulations using the nonequilibrium Green’s function formalism demonstrate exceptional spin filtering capabilities (~100%) and giant bias-dependent tunneling magnetoresistance (>10⁵%). These findings deepen the fundamental understanding of 2D multiferroics and establish a desired platform for future experimental exploration and the development of next-generation spintronic devices.

Introduction

Two-dimensional (2D) magnetic materials have garnered significant attention as promising candidates for next-generation high-density spintronics applications^1,2. Various 2D magnetic materials have been discovered and synthesized, including transition metal halides^3,4, chalcogenides⁵, carbonitrides⁶, covalent organic framework⁷, and their heterostructures^8,9. However, a persistent challenge undermining their practical utility is their intrinsically low Curie temperature (T_C), typically below 200 K, as exemplified by materials such as CrI₃³, Cr₂Ge₂Te₆¹⁰, and Fe₃GeTe₂¹¹, which greatly limits their potential for room-temperature device applications.

The Heisenberg model offers foundational insights into strategies for enhancing the T_C, emphasizing the importance of strengthening the exchange interaction J and magnetic anisotropy energy (MAE) or increasing the coordination number of neighboring spins. In honeycomb lattice systems, such as CrI₃ and Cr₂Ge₂Te₆, the limited coordination number of three nearest neighbors constrains the cumulative magnetic interaction strength. Conversely, triangular lattice systems, with six nearest neighbors, benefit from a potentially higher interaction density; however, their high symmetry often diminishes MAE, reducing the magnetic stability. Moreover, the exchange interactions in most triangular lattice materials remain relatively weak, further limiting their capacity to achieve high T_C. In comparison, rectangular lattice magnetic materials remain significantly underexplored relative to their honeycomb and triangular counterparts. These systems may offer a promising avenue for achieving an optimal trade-off among the number of nearest neighbors, exchange interaction strength, and MAE, potentially facilitating the design of high-T_C 2D magnets.

Among the prospective candidates, ternary transition metal oxyhalides and nitrogen-halides T_MBXs (where T_M represents transition metals; B denotes elements form the oxygen or nitrogen groups; and X denotes halogen elements), emerge as a compelling system for investigating rectangular magnetic lattices. Bulk T_MBX materials exhibit a layered orthorhombic structure characterized by a variety of magnetic ground states. Their rectangular monolayer derivatives display pronounced uniaxial in-plane magnetic anisotropy, attributable to substantial in-plane lattice anisotropy¹². Despite this promise, the T_C of experimentally synthesized T_MBX monolayers has yet to surpass 200 K, a constraint primarily arising from the limited diversity of systems explored thus far. A notable example is the CrSBr monolayer^13,14, a ferromagnetic semiconductor with a T_C of ~150 K¹⁵. This material has revealed lots of intriguing properties, including magnetoelastic coupling¹⁶, a magnetic high-order topological insulator phase¹⁷, multiferroic behavior^18,19, magnetic bimerons²⁰, giant magnetoresistance²¹, and spin-spiral magnetic ordering^22,23. Meanwhile, theoretical studies have intermittently predicted T_C values exceeding room temperature for certain T_MBX monolayers, such as MnNI (~310 K) and VNI (~500 K)^24,25, suggesting significant untapped potential within this material family.

Given the challenges associated with fast experimental synthesis, first-principles high-throughput calculation could greatly accelerate the study of the fundamental electronic and magnetic properties of T_MBX system. Several primary questions of this ternary system could be answered through the calculations, including the upper bound of the ferromagnetic T_C, the magnetic phase diagram, the key physical parameters governing magnetism, and possible couplings with ferroelectric polarization. By systematically mapping these properties, this study aims to unlock the full potential of rectangular lattice T_MBX materials, paving the way for the rational design of 2D magnetic materials with enhanced T_C and functionalities for spintronics applications.

High-throughput density functional theory (DFT) calculations were performed for 672 T_MBX monolayers, in which we screened out 78 ferromagnetic monolayers, including 38 candidates with T_C ≥ 200 K. Contemporary data mining techniques were employed to analyze the principal physical and chemical factors influencing T_C. Additionally, a predictive model was developed using the sure independence screening and sparsifying operator (SISSO) method²⁶ to predict T_C for this ternary system. Among these high-T_C candidates, six systems were identified as multiferroic, concurrently displaying ferromagnetism and in-plane ferroelectricity. Their ferroelectric switching pathways and the interplay between their ferroelectric and magnetic properties were thoroughly assessed. Furthermore, a two-probe spin transistor based on T_MBXs was designed, and its spin-polarized transport properties were investigated using the non-equilibrium Green’s function (NEGF) formalism. The results demonstrate remarkable spin-filtering efficiency and substantial tunneling magnetoresistance, highlighting the considerable promise of these materials for advanced nanomagnetic device applications.

Results

High-throughput and data screening

T_MBXs monolayer manifests a rectangle lattice as shown in Fig. 1a. Its monolayer comprises four atomic layers, with transition metal atoms adopting a distorted octahedron coordination. The structure follows the prototype structure of CrSBr system (with point group symmetry of D_2h), where the oxygen-group and nitrogen-group atoms are located in the middle two atomic layers and forming ionic bonds with the transition metal, while the halogen atoms are located in the outermost atomic layer. The unequal bond lengths of T_M-B and T_M-X lead to a large distortion of the octahedron and strong in-plane anisotropy. As a result, the doubly degenerate e_g orbitals split into two distinct orbitals, while the triply degenerate t_2g orbitals split into three separate orbitals. For example, in VSeF system, e_g orbitals split into two a₁ orbitals, and t_2g orbitals split into b₁, b₂, and a₂ orbitals¹⁹. The detailed d-state splitting configurations of various T_M element can be found in Table S1 (Supporting Information).

Fig. 1: High-throughput and screening workflows.

a Schematic diagram of T_MBX monolayer structure, where the blue, red and yellow balls are transition metal, oxygen or nitrogen groups and halogen atoms, respectively. J₁, J₂, and J₃ are the spin exchange interactions. b Workflow diagram of high-throughput calculation and the screening of magnetic/multiferroic materials. c The phase diagram of ferromagnetic (FM), antiferromagnetic (AFM) and non-magnetic (NM) monolayers. Red, gold, and dark cyan squares represent FM, AFM, and NM, respectively. The white square represents unstable systems. The inserted diamond marker represent that they are semiconductors. d The number of paraelectric (PE), ferroelectric (FE) and antiferroelectric (AFE) structures. The arrow corresponds to the in-plane polar displacement.

Figure 1b gives the workflow of the high-throughput calculation and data screening for the T_MBX family. Due to this study focus on the magnetic properties, we only select the element set of {V, Cr, Mn, Fe, Co, Ni, Cu, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Ta, W, Re, Os, Ir, Pt, Au} with 21 elements for the T_M site. It should be pointed out that Tc is a radioactive element. Here, for the continuity of the element’s properties, we did not exclude Tc for data analysis. While 4 oxygen-group and 4 nitrogen-group elements were selected for the B site, and 4 halogen elements were used for the X site, as shown in the inset element table of Fig. 1b. In this way, 672 T_MBXs can be constructed as the initial compound set through element substitution based on the prototype rectangular structure of CrSBr. We discarded those nonmagnetic (NM) systems and applied formation energy stability filtering to the entire set (see Fig. S1), leaving 310 stable T_MBXs, among which 78 FM compounds and 232 AFM compounds were identified. Figure 1c shows the relative stability and magnetic property distributions of the entire set. After the electronic structure calculation, 78 FM compounds can be further divided into 41 FM half metal/metal and 37 FM semiconductors (Fig. S2).

The second branch of the high-throughput workflow is to identify the possible ferroelectric (FE)/multiferroic materials. We introduced artificial slight distortion into the nonpolar high symmetry structure (with point group of D_2h), as a result, the structure symmetry was reduced to a polar symmetry of C_2v. By comparing the relative energy, we screened out 7 ferroelectric and 4 antiferroelectric (AFE) T_MBXs from the 78 FM compounds. The 2D ferroelectric T_MBXs possess in-plane polarization along the b-axis as shown in the inset of Fig. 1d. Then, we calculated their spin exchange interaction parameters (listed in Table S2). The Monte Carlo (MC) simulations of anisotropic Heisenberg model were performed for all 78 FM systems. Three nearest-neighbor interactions were considered, which have been roughly labeled in Fig. 1a, respectively. The details of the MC simulation and spin exchange interaction were given in section 2.1 of the Supporting Information. In the end, we screened out 38 high-T_C FM candidates with T_C ≥ 200 K (listed in Table S3). Among them, 7 compounds possess polar structures, whose dynamical and thermodynamic stabilities were confirmed (Figs. S3–S5).

Ferromagnetism and High-T _C

In Fig. 2a, 672 T_MBX compounds were divided into FM, AFM, and NM subsets according to their T_M-site elements. In line with the cognition, common NM transition metal systems, such as those containing noble metals and Cu, Nb, etc., do not appear magnetism, while Mo and Ta are the exceptions. To surprise, T_MBXs with Fe, Co, and Os are all AFM. As for NiBXs, the number of AFM cases is also more than twice that of FM cases. Moreover, the number of AFM materials accounts for 34.2% of the total T_MBX, while FM materials only account for 11.6%. As shown in Fig. 2b, AFM is almost three times the number of FM. In contrary, from existing crystal databases, such as the Materials Project²⁷, AFM materials account for less than 10% of magnetic materials. This implies that the rectangle T_MBX lattice system must have super exchange characteristics that are more suitable for AFM. In addition, these AFM T_MBXs all favor the high-symmetry D_2h lattice, while FM T_MBXs allow for polar distortions.

Fig. 2: Ferromagnetic properties of T_MBX monolayers.

a The number of FM, AFM, and NM T_MBXs with the same T_M. b The proportion of FM, AFM, and NM T_MBXs. The occurrence of 3d-4d T_M in FM systems. c High-T_C (T_C ≥ 200 K) FM T_MBX monolayers, simulated using the anisotropic Heisenberg model.

Among those FM compounds, the proportions of 3d-, and 4d-T_M-based systems are 71%, and 29%, respectively. Figure 2c shows the specific T_C of 38 screened-out high-T_C (T_C ≥ 200 K) FM candidates, which are 15 semiconductors and 23 half-metals, respectively. It’s worth mentioning that 26 of them are room-temperature ferromagnets with T_C ≥ 300 K. Due to the strongest localization of 3d electrons, 89% of high-T_C compounds are 3d-T_MBX, i.e., T_M = {V, Cr, Mn, Ni}-based compounds. The left 11% are MoNXs, where each Mo has the magnetic moment of 2 μ_B. The MoNXs were also predicted to be high T_C monolayers in previous works²⁸. Another interesting trend is that, for most T_MBXs that have the same T_M and B, the larger the electronegativity of halogen X, the lower the T_C, indicating the important role of charge transfer in determining the T_C (Table S4). In addition, as the electronegativity of X increases, the system tends to transform from a metal to a semiconductor (Fig. S2). For the example shown in Fig. S6, the band gaps of MnOXs increase from 0 to 1.5 eV as the electronegativity of X increases. Though the analysis of the electronic structure (Figs. S7–S9), when B is oxygen-group element, the band structures near the Fermi level are primarily contributed by T_M-d orbitals, B, and X-p orbitals. Conversely, when B is nitrogen-group element, the dominant contributions are only from the T_M and B.

Machine learning models in predicting T _C and J

To gain a more comprehensive understanding of the T_C of 78 FM T_MBXs, we used the gradient boosting method²⁹ to evaluate the relative importance of the parameters in the effective Hamiltonian and the fundamental properties of the elements. As shown in Fig. 3a, the second-nearest exchange parameter J₂ plays a dominating role in determining the T_C, rather than the J₁. For most monolayer T_MBXs, J₂ manifests the diagonal super-exchange interaction along the T_M-X-T_M path. Thereby, we see the significant effect of halogen elements on modulating the T_C as we discussed in the last section. However, even J₂ itself contains too much information, including from the electronic structure, occupancy, and lattice. In Fig. 3b, we see that the intensity of J₂ relates to the magnet moment (M), nearest-neighbor T_M-T_M distance length (L₁), and band gap at the same time, so it’s hard to quantify J₂ with a few simple parameters.

Fig. 3: Machine learning of the magnetic ground states.

Featuring relative importance in predicting (a) T_C and (b) J₂, respectively. The descriptions of the features were given in Table S5. Inset is the corresponding Pearson correlation (pc) coefficients between the feature and T_C (J₂). The comparation of SISSO predicted and DFT calculated (c) T_C and (d) J₂, respectively.

Therefore, we derived two SISSO machine learning models to predict the T_C and J₂, respectively. This model allows us to maximize the data utilization of a small dataset by constructing high-dimensional descriptors from simple fundamental features²⁶. 15 simple features, including the electronic, lattice- and element-related features, were used for SISSO, which were summarized in Table S5. The produced high-dimensional descriptors were given in Supplementary Materials. Then, linear model using these SISSO descriptors can be constructed. For predicting T_C and J₂, our linear models reach well accuracy with the root mean square error (RMSE) of 33.59 K and 5.36 meV, respectively.

Multiferroic properties

The T_MBX family also provides an ideal setting for studying 2D multiferroic and exploring the coupling between electric, magnetic, and structural order parameters³⁰. To make the discussion more concise, we will refer to these polar metallic structures as “FEs” just for discussions despite that their polarization cannot be switched by an electric field. The AFE-FM systems are also called multiferroics for discussions, whose polarization cannot be switched as well.

Table 1 has listed 11 selected high-T_C T_MBXs that possess rich order parameters. They are VNXs (X = F and Cl), CrNF, MnOXs, and MoNXs (X = F, Cl, Br, and I), respectively. Among them, MnOXs and MoNCl are conventional FE-FM multiferroics with the band gaps ranging from 0.16 to 1.58 eV. VNXs (X = F and Cl) and MoNXs (X = Br and I) are AFE-FM multiferroics. Here, MoNF is more special than other systems. Its ground state coexists FE-FM and AFE-FM multiferroics, which will be addressed in detail later. The in-plane polarizations of these FE T_MBX systems range from 2.4 to 9.0 μC/cm², and they need overcome moderate energy barriers from 16 to 252 meV/f.u. to switch the polarization. Switching the polarization of metallic T_MBXs needs overcome a much higher energy barrier. To our surprise, the switching energy barrier of AFE systems is even lower than most FE systems. The magnetic easy axes, MAE and T_C are also given in Table 1. Interestingly, these 11 high-T_C multiferroic systems happen to be the only ones with low lattice symmetry (C_2h and C_2v) among the 78 FM systems.

Table 1 The HSE06 band gap (eV), ferroelectric polarization (μC/cm²) and polarization direction, the type of polarization switching path, the energy barrier ΔE (meV/f.u.) of polarization switching. The magnetic easy axis, MAE (μeV/T_M), and T_C (K) of selected systems

Figure 4a shows three types of the polarization switching paths existing in T_MBX systems, which are FE-PE-FE, FE-AFE-FE, and AFE-PE-AFE. MnOX (X = F, Cl, and Br) systems favor the FE-AFE-FE path, while MoOX (X = F, Br, and I) systems favor the AFE-PE-AFE path (Fig. S10). The envelope of the polarization switching energy of MnOF and MoNF selected as examples, was displayed in Fig. 4b. For MnOF, the energy barrier ΔE of the FE-AFE-FE path is much lower than the FE-PE-FE path. Moreover, the middle point of its switching path (corresponds to the AFE phase) possesses similar energy with respect to its ground-state FE structure, indicating the polymorphism nature of this system. The comparation of two switching paths for MnOCl and MnOBr was displayed in Fig. S11, which possesses similar characteristics to that of MnOF.

Fig. 4: Multiferroic properties.

a Three types of switching paths of the displacement mode (indicated with the arrows). b Free energy envelope of the displacement switching. c The normalized magnetic moment as a function of temperature of AFE and FE MoNF monolayer. d Selected snapshots of AFE MoNF during the Monte-Carlo (MC) simulations as the temperature decreases from 500 K to 0.1 K. The right panels are enlarged local magnetic vortex and antivortex. Here, red and blue represent spin up and spin down, respectively, while the arrows are the in-plane spin projections.

The MoNF system exhibits unique characteristics that distinguish it from other systems. The relative energy difference between the FE and AFE phases is remarkably small, within 10 meV per formula unit (f.u.). The ground-state structure of MoNF is identified as a ferroelectric phase, as illustrated in Fig. S10. The ferroelectric polarization is oriented along the b-axis, determined by analyzing the imaginary mode of the paraelectric (PE) structure. Both the FE and AFE phases are dynamically stable. The FE phase possesses a 9.0 μC/cm² in-plane polarization, and favors the FE-PE-FE switching path with an energy barrier of 114 meV/f.u.. While the AFE phase favors the AFE-PE-AFE switching path, which has a much lower energy barrier of 55 meV/f.u.. It is also worth mentioning that there are numerous AFM-FE systems with large net polarizations, such as PtOCl, OsOCl, PdOF, and MoOF monolayers, which also deserve further in-depth investigations.

From the perspective of magnetic properties, the magnetic easy axes of T_MBX systems also have diverse orientations, enabling them to display distinct magnetoelectric coupling behaviors. For MnOX systems, their magnetic easy axes are along the c-axis but with a small canting in the bc-plane. The easy axes of the VNF and CrNF are along the b-axis (polar axis). The magnetic easy axis of AFE-MoNX systems is along the a-axis, while that of FE-MoNF is along the b-axis (polar axis). This means that if the ferroelectric polarization of FE MoNF is switched, the magnetic easy axis will also be switched. The simulated T_C of FE and AFE MoNF are shown in Fig. 4c, and their MAE and exchange coupling parameters were listed in Table 2. The FE MoNF shows a T_C of 480 K, while the AFE MoNF shows a lower T_C of 360 K. This is due to that FE shows a much larger J₂ and MAE, which play the dominant role in determining T_C. More interesting, AFE MoNF could form stable Bloch and Néel type merons near the domain wall at approximately 110 K as shown in Figs. 4d and S12.

Table 2 Magnetic parameters of FE and AFE MoNF monolayer, including the exchange coupling parameter (J_i, in meV), perpendicular magnetic anisotropy constant K (meV/Mo), and saturation magnetic moment M (μ_B/Mo)

Spin-polarized transport

Next, we investigate the spin-polarized transport properties of T_MBXs. A two-probe spin transistor was constructed, comprising a central region made of a FM semiconductor CrSBr monolayer sandwiched by left and right electrodes made of a FM half metal CrNF, as shown in Fig. 5a. The electrodes extend to y = ±∞ where the current is collected. The transport direction of Fig. 5 is along the b-axis, while the transport simulation along the a-axis was given in Fig. S13. Two types of magnetic configuration of the CrNF electrodes were considered, which are spin parallel configuration (PC) and spin anti-parallel configuration (APC), respectively. The current-voltage dependences of two spin configurations were shown in Fig. 5c. We have considered a bias voltage from 0.0 to 0.6 V, where the linear current-voltage relationship was maintained. On the other hand, compared to the PC device, APC device exhibits significantly higher resistance due to the existing tunnel magnetoresistance effect. The parallel spin carriers can easily tunnel through the semiconductor layer, indicating the great potential of T_MBX systems in spin filtering applications. To quantify the spin transport performance, we defined the spin injection efficiency (SIE) and tunneling magnetoresistance (TMR) as following: ${\rm{SIE}}=\frac{{I}_{\uparrow }\,\left(V\right)-{I}_{\downarrow }\,(V)}{{I}_{\uparrow }\left(V\right)+{I}_{\downarrow }\,(V)}$; ${\rm{TMR}}=\frac{{I}_{{PC}}\,(V)-{I}_{{APC}}(V)}{{I}_{{APC}}(V)}$, where I_↑, I_↓, I_PC, and I_APC are the spin-up current, the spin-down current, the current of PC device, and the current of APC device, respectively. Since I_↓ is almost equal to zero in our systems, a nearly fully spin-polarized current is generated (SIE ~100%, shown in the inset of Fig. 5d). Then, TMR shows a large dependence on the bias voltage (Fig. 5d). There is a maximum TMR of 5.2 × 10⁵% along a-axis and 2.5 × 10⁶% along b-axis when under a bias of 0.1 V, which is much larger than those measured in the ZrTe₂/CrOCl/CrOCl/ZrTe₂ (2.5 × 10⁴%)³¹, graphene/CrI₃/graphene (1.9 × 10⁴%)³² and (Fe_0.8Co_0.2)₃GaTe₂/WSe₂/Fe₃GaTe₂ (1.8 × 10²%). The transmission spectra under a bias of 0.1 V were presented in Fig. 5e, f. Under a bias of 0.1 V, the transmission coefficient T_PC approaches 0.5, while T_APC approximately equals to 0. Moreover, the transmission spectrum implies that almost completely spin-polarized transmission can be achieved in a broad energy window from −0.8 to 3 eV.

Fig. 5: Spin transport properties.

Schematic diagram of the two-probe spin transistor of CrNF/CrSBr/CrNF with (a) spin parallel configuration (PC) and (b) spin antiparallel configuration (APC). c I-V curves of PC (dark green circles) and APC (dark orange squares) devices. d Tunneling magnetoresistance (TMR) as a function of the bias voltage. Inset is the corresponding spin injection efficiency (SIE). The spin-polarized transmission spectra of (e) PC and (f) APC devices under 0.1 V bias.

Discussion

In this work, we have conducted a comprehensive high-throughput investigation of the T_MBXs, uncovering their rich ferromagnetic and multiferroic properties. By screening 672 T_MBX systems, 78 ferromagnetic monolayers were identified, 38 of which had high Curie temperatures (T_C ≥ 200 K), greatly expanding the known 2D magnets. Machine learning models were developed to reveal the critical role of the second-nearest neighbor exchange interaction in determining T_C, offering a new insight for designing 2D magnets. Additionally, we discovered 7 FM-FE multiferroics, exhibiting unique polarization switching mechanisms. Spin transport simulations demonstrate that T_MBX monolayers can exhibit exceptional spin filtering and tunneling magnetoresistance (>10⁵%), making them highly promising for spintronic applications. These findings provide a solid foundation for future experimental studies and the development of next-generation 2D multiferroics.

Methods

Density functional theory

First-principles calculations were performed within generalized gradient approximation (GGA) in the form proposed by Perdew, Burke, and Ernzerhof (PBE)³³ using Vienna ab initio simulation package (VASP)³⁴. The lattice relaxations adopted a vdW functional optB88-vdW³⁵. Atomic positions were relaxed until the maximum force on each atom was less than 10^-3eV/Å. A 600 eV energy cutoff and a 10 × 10 × 1 Γ-centered k grid of were used. And a vacuum layer more than 25 Å was applied to avoid the interaction between periodic images. To reduce the self-interaction error of T_M-d orbitals, we have considered the Hubbard correction³⁶. The effective Hubbard U parameter (U_eff) for V, Cr, Mn, Fe, Co, Ni, and Mo are 3.3 eV, 3.7 eV, 4.0 eV, 5.3 eV, 4.0 eV, 6.2 eV, and 4.0 eV, respectively. The Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional was used to correct the problem of severe underestimation of the band gap by PBE³⁷ method. Spin-orbit coupling (SOC) was taken into account in determining the magnetization easy axis. Phonon dispersion was calculated using density-functional perturbation theory (DFPT) with VASP-PHONOPY interface^38,39. The Born effective tensor is defined as: Z^∗_i,j = $\frac{\varOmega }{e}\frac{\partial {p}_{i}}{\partial {u}_{j}}$, in which e is the charge of electron, p is polarization, and u is the atomic displacement from its high-symmetry nonpolar position. The calculated polarization of 2D materials strongly depends on the thickness of slab we chose. Here, we added two extra 1 Å thcikness along the c-axis on both sides of T_MBX monolayers when calcauling the volume Ω. The polarization was calculated using Born effective charge method.

Quantum transport

Transport properties of the device were calculated using Nanodcal⁴⁰ within the NEGF-DFT framwork⁴¹. The DZP atomic orbital basis was used to expand all the physical quantities; the exchange and correlation were treated at the level of PBE functional; and atomic cores were defined by the standard norm conserving nonlocal pseudopotentials, and 16 × 16 × 1 k-points were used. Under certain bias voltage V_b, the spin current I was calcualted as follows: ${I}_{\sigma }\left(V\right)=e/h\int T(\epsilon ,{V}_{b})\left[{f}_{L}\left(\epsilon -{\mu }_{L}\right)-{f}_{L}\left(\epsilon -{\mu }_{R}\right)\right]{dE}$. σ ≡ ↑, ↓ is the spin index. The integration is over the bias window between μ_L and μ_R which are the electrochemical potentials of the top/bottom leads. The total current is given by I_total = I_↑+ I_↓.