Introduction

Two-dimensional (2D) materials with multiferroic properties and possessing both ferromagnetism (FM) and ferroelectricity (FE) have attracted increasing amounts of attention among researchers for their fascinating physical properties and promising multifarious applications. FM and FE are crucial because of technological applications such as magnetic memory and ferroelectric memory1,2,3. In conventional magnetoelectric multiferroics, magnetism usually originates from the partially filled d/f orbital of transition metal atoms, while electric polarization is usually caused by the off-center displacements of empty d/f orbital atoms4. The mutually exclusive requirements for the fully electronic occupation of the outer valence shell make it particularly challenging to achieve FM and FE simultaneously in the bulk material5,6,7,8,9,10,11,12,13,14,15,16,17.

2D multiferroic semiconductors with atomic thickness have received considerable attention due to their ability to overcome these mutually exclusive requirement18. The out-of-plane FE in 2D magnetic materials is generally considered to decrease the depolarization field2,19,20. Furthermore, polarization reversal in 2D ferroelectric materials with out-of-plane electric polarization can be achieved by an electric field in the vertical direction. To date, many 2D ferromagnetic materials, such as CrI3, VI3, VSe2, Cr2Ge2Te6, and Fe3GeTe2, etc., have been studied on the basis of theoretical or experimental results21,22,23,24,25,26,27,28,29,30,31,32,33. Likewise, considerable progress has been achieved in ferroelectric materials34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53. 2D ferroelectric materials such as SnTe, SnSe, and SnS thin films (several unit cells) have been shown to possess in-plane FE with critical transition temperature higher than those of their bulk phase45,46,47,48. Other structural analogs of 2D SnTe MX (M = Ge, Sn; X = S, Se) or elemental ferroelectric bismuth were also predicted to exhibit in-plane FE51,52,53,54,55. Only a few 2D ferroelectric materials with intrinsic out-of-plane polarization have been confirmed theoretically and experimentally, including 1T-MoTe2, CuInP2S6 (CIPS), and α-In2Se335,36,56,57,58,59,60. In view of the large family of 2D magnetic materials, 2D multiferroic materials with out-of-plane electric polarization by made of 2D magnetic semiconductors are desirable for integrated nanodevices.

Monolayer CrX3 (X = Cl, Br, I) is a prototypical Ising-type ferromagnet with out-of-plane magnetization21,61. Many studies on the regulation of CrX3 FM through strain, electron doping, and atom substitution doping have been conducted in recent years62,63,64. Currently, most CrX3-based multiferroic devices are heterostructures composed of CrX3 and an out-of-plane ferroelectric monolayer with a large thickness and nondegenerate polarized states65,66,67,68,69. To our knowledge, little attention has been given to the effect of interstitial decoration on the ferroelectric and ferromagnetic properties of 2D CrX370,71,72.

Herein, the large family of 2D M(CrX3)2 (M = transition metal elements, X = Cl/Br/I) materials was systematically studied by first-principles calculation. We identified 12 stable FM + FE out-of-plane single-layer multiferroic materials by decorating chromium trihalides using 28 transition metal elements. As a concrete example, we reveal that the Cu-decorated monolayer CrX3 exhibits out-of-plane multiferroicity as a result of symmetry-controlled p–d coupling due to the crystal field effects. The decorated Cu atom fills one side of the hollow region, resulting in large permanent electric dipoles whose direction can be switched because of the lower transition barriers. In addition, the incorporation of Cu enhances the intrinsic FM of monolayer CrX3. Therefore, Cu-decorated monolayer CrX3 is an ideal atomically thick 2D material, that possesses both permanent out-of-plane FE and FM, and has broad application prospects in nonvolatile memory nanodevices like electrets.

Results

Emergence of out-of-plane ferroelectricity in Cu-decorated chromium trihalides

Pristine CrX3 monolayers belong to the \(P\bar{3}1m\) space group which is centrosymmetric with no spontaneous electric polarization. The Cr atoms are arranged in a honeycomb lattice and are hexa coordinated by X atoms. We decorated a Cu atom at the center of the Cr honeycomb lattice. Figure 1a shows the atomic structure of monolayer Cu(CrX3)2 (X = Cl/Br/I). The dynamic stability of the Cu(CrX3)2 monolayer was subsequently confirmed by its phonon spectrum (without imaginary phonon modes) (Fig. S3). Cu(CrX3)2 is in the \(P31m\) space group, and its spatial inversion symmetry is broken compared to that of CrX3. Due to the breaking of the inversion symmetry induced by Cu decoration, the spatial separation of positive and negative charge centers in the out-of-plane direction (Fig. 2b) results in an unequal electrostatic potential at the top and bottom surfaces, respectively, producing an out-of-plane electric polarization vector.

Fig. 1: The structure and ferroelectric properties of Cu(CrX3)2.
figure 1

a Top and side views of monolayer Cu(CrX3)2 atomic structure. b The ferroelectric switching pathways of Cu in CrX3 monolayers. The projected density of states (PDOS) for c Cu(CrBr3)2, and d Ag(CrBr3)2 monolayer calculated by DFT + U method. The upper and lower panels represent the PE and FE phases, respectively. The Br↑ and Br↓ represent the upper and lower surfaces of Br atoms, respectively. The Br↓ atoms in the FE phase are slightly below Cu atoms. The contributions of Cu d and Ag d orbitals at the Fermi level (Ef) are highlighted.

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Fig. 2: The ferroelectric transiton pathway, polarization origin and band structure of Cu(CrBr3)2.
figure 2

a Kinetics pathways of polarization reversal processes for Cu(CrBr3)2 structure. One FE state is taken as the initial state (left inset). b The reduced electron density difference along the z direction ρ(z) for FF and PE states. The band structures of CrBr3 (c) and Cu(CrBr3)2 (d). The inset in (d) is the spin–orbit coupling band structure of Cu(CrBr3)2.

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As illustrated in Fig. 1b and Fig. S5, we label two typical sites of the Cu atom in the lattice according to its out-of-plane displacement: (i) the bond center site surrounded by six nonmetal atoms, labeled PE, and (ii) the in-plane (out-of-plane) tetrahedral site with three nonmetal atoms in close proximity, labeled FE and FE′ (FE1 and FE1′). To elucidate the origin of the polarization of Cu(CrX3)2, two factors must be considered. First, the two sites have different local point symmetries. Second, the energy of the Cu d orbitals is similar to that of the X p orbitals (as shown in Fig. 1c). We name these p orbitals as the host p orbitals. Taking Cu(CrBr3)2 as an example, we classify all these sites into two types of symmetries:

(i) When the Cu atom occupies the bond center site (PE), the system has a Oh symmetry, which includes the irreducible representations A1g, A2g, Eg, T1g, T2g, A1u, A2u, Eu, T1u, and T2u. At the center of the Brillouin zone Γ, the Cu d orbitals are described by the Eg and T2g representations, while the host unoccupied p state is described by the A1g representation. Since there are no common representations between the Cu d orbitals and the host p orbital, the p–d coupling defined as \({V}_{p-d}={|\langle p|\varDelta V|{d}_{0}\rangle |}^{2}\,/({\epsilon }_{p}-{\epsilon }_{d})\) is symmetrically forbidden, i.e., Vp–d = 073. (ii) When the Cu atom is an out-of-plane (in-plane) tetrahedral site (FE or FE′ or FE1 or FE1′), the system reduces to C3v due to the crystal field effect, in which the Cu d orbitals are described by the A1g and Eg representations, and the host p state is described by the A1g representation. Consequently, the Cu d and host p orbitals could couple with each other (Vp–d ≠ 0) because they are now described by a common representation (A1g) (see Fig. S7).

A summary of the symmetry analysis is given in Table 1. When the Cu atom remains at the PE site there is no p–d coupling; when the Cu atom is at the in-plane (out-of-plane) tetrahedral site, p–d coupling is allowed. In addition, the strength of the p–d coupling increases as the ionicity of Cu(CrX3)2 increases since Vp–d is proportional to the difference between the anion and cation potentials. Note that, although s–d coupling may also be allowed in these systems, it does not have much effect on the ferroelectric transitioning since both the s and d orbitals are fully occupied.

Table 1 Irreducible representations of Cu d and host p orbitals in the Oh and C3v point groups, respectively
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To confirm the emergence of FE, we compute the potential energy along the ferroelectric transition path of Cu-decorated monolayer CrX3 compounds. As shown in Fig. 1b, the ferroelectric transition paths are exactly along the \(\left\langle 001\right\rangle\) direction of the 2D material, i.e., from the FE′ to the PE to the FE site as shown in Fig. 1b. Deviation of the Cu atom from the PE sites leads to p–d coupling decreasing the total energy, which is confirmed by the calculated electronic structure of Cu(CrBr3)2 as followed. Figure 1c shows the projected density of states (PDOS) of the Br p orbital and Cu d orbital in the paraelectric and ferroelectric phases, respectively. The Br↑ and Br↓ represent the upper and lower surface Br atoms, respectively. The results show that the Cu d orbital and the neighboring Br p orbital strongly interact, resulting in p–d coupling around the Fermi level. Strong p–d coupling reduces the energy of the Br p and Cu d orbitals at the neighboring Fermi level (the first peak to the left of the light green area in Fig. 1c), thereby reducing the energy of the system. For the FE phase, the Br↓ atoms are close to the Cu plane. Therefore, the p–d coupling between the lower Br atoms and Cu atoms is stronger in the ferroelectric phase.

From the Cu(CrI3)2 to Cu(CrCl3)2 monolayer compounds, the p–d coupling strength and magnitude of charge transfer from the cation to the anion further increase due to the increased ionicity. For Cu atoms with high-lying d orbitals, the enhanced p–d coupling significantly increases the energy at the PE site, leading to an increase in the ferroelectric transition energy barrier (see Fig. 1b). Thus, in Cu-decorated chromium trihalides, the ferroelectric polarization originates from strong p–d coupling at the Fermi level.

Atomic selectivity of out-of-plane ferroelectricity

In addition to Cu atoms, Ag and Au atoms belong to the same family and have the same number of valence electrons as Cu. So, Ag and Au atoms were also used to decorate CrBr3. Figure S5b shows the ferroelectric switching energy curves of single-layer CrBr3 decorated with Cu, Ag, and Au atoms. The double-well potential energy curve shows that Cu and Au atom decoration can form two switchable ferroelectric phases. However, Ag decoration results in the formation of a paraelectric phase with a single potential well. Why does Ag atom decoration not result in a ferroelectric phase? To reveal the mechanism of the paraelectric state by Ag decoration, the calculated PDOS of Ag(CrBr3)2 is shown in Fig. 1d and Fig. S6. It is shown that Ag has lower 4d orbital levels (the 4d orbital energy of Ag is lower than that of the Cu 3d orbital74). Consequently, the p–d coupling at the Fermi level is very weak. Figure S5a shows the atomic structure of the FE phase and PE phase of metal atoms decorated with monolayer CrBr3. In the FE phase, the decorated atoms form a tetrahedral structure with three Br atoms. In the PE phase, the decorated atoms form an octahedral structure with six Br atoms. In the Ag(CrBr3)2 system, the p–d coupling is weak, and the Coulomb interaction competitive relationship between tetrahedral and octahedral fields plays a dominant role. From the PDOS (Fig. 1c, d and Fig. S6), we can see that the energy splitting (the gapped region shaded by the light green) of the tetrahedral crystal field of Cu and Au is much greater than that of the octahedral crystal field. In the Ag(CrBr3)2 system, the energy splitting of the tetrahedral crystal field is almost the same as that of the octahedral crystal field. This is beneficial for the formation of a PE phase with higher symmetry.

The above results indicate that in the M(CrBr3)2 (M = Cu/Ag/Au) system, if the p–d coupling is strong, an FE phase will form. If the p–d coupling is weak, a PE phase will be formed due to the effect of the octahedral crystal field Coulomb interaction. The emergence and stability of out-of-plane FE with strong atomic selectivity are intrinsic in transition-metal-decorated 2D magnetic semiconductors CrX3, which has a great advantage over prior strontium oxide with soft phonon instability. The strong atomic selectivity discussed above is not limited to Cu and Au. After screening 28 transition metal atoms in the periodic table of elements, we identify 12 intrinsic single-layer multiferroic materials M(CrX3)2 possessing out-of-plane FE (see Fig. S1 and Table S1).

Single-layer out-of-plane ferroelectricity induced by decoration atoms

The intrinsic out-of-plane FE of nanometer-scale films is required to miniaturize electronic devices. We obtained the single-layer intrinsic out-of-plane ferroelectric material Cu(CrX3)2 by decorating Cu in monolayer CrX3. Compared with that of CrX3 (CrCl3 = 6.061 Å, CrBr3 = 6.448 Å, and CrI3 = 7.081 Å), the lattice constant of Cu(CrX3)2 increased (see Table S3, Cu(CrCl3)2, Cu(CrBr3)2, and Cu(CrI3)2 increased by 4.37%, 3.44%, and 1.20%, respectively). The translation pathway, i.e., the ferroelectric transition pathway (dE3), was computed by the climbing image nudged elastic band (CI-NEB) method75, as shown in Fig. 2a and Fig. S9. The barrier of the Cu(CrCl3)2 system is about 125 meV/f.u., which is close to that of 2D ferroelectric In2Se3 (66 meV/f.u.) revealing the feasibility of ferroelectric switching under ambient conditions36. The transition energy barrier of Cu(CrBr3)2 and Cu(CrI3)2 is slightly increased to 144 and 181 meV/f.u., respectively.

According to the charge center method, the calculated vertical polarization of Cu(CrCl3)2, Cu(CrBr3)2, and Cu(CrI3)2 are 14.98 × 10−12 C/m, 7.56 × 10−12 C/m, and 5.72 × 10−12 C/m, respectively (see Table S3), more than three times greater than that (2.08 × 10−12 C/m) of AB stacked BN reported previously, which is beneficial for the experimental measurements39,76. Figure 2c, d shows the energy band structures of CrBr3 and Cu(CrBr3)2, respectively. Cu(CrBr3)2 exhibits half-metallic properties compared with pristine magnetic semiconductor CrBr3. This peculiar characteristic is highly desirable for spin-based devices (to be presented in Fig. 5)77, such as magnetic tunneling junctions78,79, spin field-effect transistors80, and spin-light-emitting diodes81.

We also considered the antiferroelectric (AFE) phase of Cu(CrBr3)2, in which opposite electric dipole moments aligned in one lattice direction, as shown in Fig. S12. The AFE1 configuration is lower in energy than the FE and AFE2 phases, which is similar to the cases of FE Sc2CO2 and CuMP2X6 (M = Cr, V; X = S, Se)15,38. The AFE1 phase belongs to the P21/m space group with inversion symmetry, so the overall electric polarization is zero. The energy barrier (dE2 in Fig. 2a) for the transition from AFE1 to FE is lower than that of the previously reported Cu(CrI3)2, which demonstrates the AFE1 phases can be converted into FE phases using an external electric field72.

Enhanced magnetism and magnetic phase transition

Based on the above discussion, we have obtained single-layer intrinsic out-of-plane ferroelectric materials Cu(CrX3)2. Considering the intrinsic magnetism of CrX3, it is necessary to evaluate the magnetic properties of Cu(CrX3)2. To assess the magnetic ground state, we analyzed the total energies of the different magnetic configurations (see Fig. S13), and the results are summarized in Table S4. The FM configuration is energetically favorable for all the Cu(CrX3)2 considered in our work, which indicates that the 2D FE Cu(CrX3)2 inherits the ferromagnetic ground state of CrX3. The magnetic anisotropic energy (MAE) is evaluated as MAE = Ein − Eout, where Ein and Eout are the energies of CrX3 systems with in-plane and out-of-plane magnetism, respectively. Positive and negative MAE values indicate out-of-plane and in-plane magnetic anisotropy, respectively. As shown in Table S4, the MAEs of CrCl3, CrBr3, and CrI3 are 0.03, 0.21, and 0.77 meV/Cr atom, respectively, reflecting the out-of-plane magnetization of CrX3, which is in agreement with previous results63. However, the MAE of Cu(CrCl3)2 and Cu(CrI3)2 per Cr atom becomes negative, indicating that the spin magnetization axis is switched to the in-plane direction, which is distinct with that of Cu(CrBr3)2.

The Curie temperature (TC) is estimated via Monte Carlo (MC) simulations based on the 2D Heisenberg model. The spin Hamiltonian is given by

$$H=-J\sum _{\left\langle i,j\right\rangle }{S}_{i}\cdot {S}_{j}-\sum _{i}A\left({S}_{i}^{2}\right),$$
(1)

where J is the nearest-neighbor exchange parameter, A is the MAE parameter, and Si and Sj are the spin operators which are assumed to be half of the calculated magnetic moment in this work. A supercell of 70 × 70 × 1 is adopted and 2 × 105 steps are performed in the MC simulations. As shown in Fig. 3a, c and Table S4, the obtained curie temperature TC of pristine CrCl3, CrBr3, and CrI3 are 21, 34, and 43 K, respectively, which are in agreement with experimental TC data21,22,23. Then, we calculate the TC of single-layer Cu(CrX3)2. Compared with those of CrX3, the obtained TC of Cu(CrX3)2 is significantly enhanced due to the increased intralayer FM coupling.

Fig. 3: The magnetic properties of CrX3 and Cu(CrX3)2.
figure 3

a, c The magnetic moment and specific heat capacity for monolayer CrX3 and Cu(CrX3)2. b Schematic diagrams of the super-exchange interaction and FM coupling (d) in Cu(CrX3)2 monolayer.

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Due to the octahedral crystal field, the Cr d orbitals in pristine CrX3 systems split into threefold t2g and twofold eg manifolds. The lower t2g orbitals are partially occupied, while the higher eg orbitals are empty, making these systems magnetically semiconducting (Fig. 3d). Furthermore, in these monolayer CrX3 systems, the Cr–X–Cr bond angle is close to 90°, and the FM semiconducting ground state is established via super-exchange interactions within the Cr–X–Cr configuration according to the well-known Goodenough–Kanamori–Anderson rules (Fig. 3b, d)82,83,84. However, due to the competitive AFM interaction (i.e., Cr–Cr direct exchange), the overall FM coupling driven by super-exchange is quite weak. An in-depth study of the electronic structure of the CrX3 monolayer may reveal similarities between them, that is the large energy gap between the occupied t2g↑ and the empty eg↑ orbitals (Fig. S7a–c). Huang et al. used a double-orbital model to prove that a larger virtual exchange gap (Eex) is the key factor for weak FM coupling (Fig. 3d)85. When the Cu decoration is introduced, the eg↑ orbitals become partially occupied, enabling a decreased crystal field splitting energy (Ec); hence, Eex decreases, and the FM coupling of Cu(CrX3)2 increases (see Fig. 3d and Table S2).

Magnetoelectric coupling

Given the prediction of the coexistence of FE and FM in Cu(CrX3)2, it is necessary to discuss the magnetoelectric coupling effect of Cu(CrX3)2. To study the polarization switching of Cu(CrX3)2 under an external electric field, we investigate how applying a vertical electric field can reduce the transition barrier by breaking the degeneracy of the two polarization states. Notably, the energy barrier decreases significantly with increasing electric field (Fig. 4a and Fig. S14). We find that as the positive electric field increases, the energy of the P↓ state increases, while the energy of the P↑ state decreases, resulting in a sharp decrease in the energy barrier from P↓ to P↑. The results show that an electric field of 1.0 V/Å can produce an energy difference of 1.0 eV/f.u. in the Cu(CrCl3)2 system, which suggests that the critical electric field for reversing polarization is ~1.0 V/Å.

Fig. 4: Magnetoelectric coupling properties of Cu(CrX3)2.
figure 4

a Energy versus polarization under a vertical external electric field in Cu(CrCl3)2. b Theoretical ferroelectric hysteresis loops obtained from the first-order Landau–Ginzburg model for Cu(CrCl3)2, Cu(CrBr3)2, and Cu(CrI3)2. c The P–E hysteretic loop with the corresponding electric and magnetic ordering.

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The intrinsic FE coercivity field can be estimated by determining the external electric field required to exceed the depolarization field. Landau–Devonshire phenomenology is used to determine the intrinsic coercive force field for continuous FE‒PE first-order phase transitions. Taking the polarization P as the order parameter, the free energy F can be written as a power series expansion with respect to the polarization P86:

$$F=\sum _{i}\left[\frac{\alpha }{2}{P}_{i}^{2}+\frac{\beta }{4}{P}_{i}^{4}+\frac{\gamma }{6}{P}_{i}^{6}\right]-E\cdot {P}_{i},\,$$
(2)

where Pi is the polarization of the ith unit cell, α, β, and γ are expansion coefficients, in which α takes the Curie–Weiss form α = α0(T − T0), and T0 is the Curie temperature87. The electric field E is calculated from the minimum free energy density, which may be described by the Landau–Devonshire theory expanded in the electric polarization P: E = αP + βP3 + γP5, where the derivative relation gives the critical electric field exceeding the intrinsic coercive field EV at the turning points of the function P(E). The magnitude of the intrinsic coercive field EV is easily calculated from the extrema of Eq. (2).

The coercive field values of Cu(CrCl3)2, Cu(CrBr3)2, and Cu(CrI3)2, are calculated to be 1.0, 1.4, and 2.2 V/Å, respectively as shown in Fig. 4b. Notably, the magnitude of the critical electric field is similar to that of 2D In2Se3 (0.66 V/Å)36, suggesting that the electric polarization of Cu(CrX3)2 can also be reversed by an appropriate vertically applied electric field59. Interestingly, in the Cu(CrBr3)2 system, different electric ordering coupling occurs with different magnetic ordering, and the magnetism is determined by electric ordering which can be controlled by a gate field (Fig. 4c). Specifically, there are two phases in the Cu(CrBr3)2 system, namely, AFE-FMin (in-plane FM) phase and the FE-FMout (out-of-plane FM) phase as shown in Fig. 4c and Fig. S15. Unlike traditional multiferroics, where the magnetoelectric coupling allows one to adjust the amplitude, here the coupling behaves like a logic operation, allowing one to radically change the ordering and phase transitions.

Spin-polarized transport of Cu(CrBr3)2 devices

In view of the semiconducting and half-metallic nature of CrX3 and their ferroelectric characteristics, a two-probe device with distinct ferroelectric polarity in each electrode is constructed to study the spin-polarized transport. The PE-FE and FE-AFE configurations are denoted in subpanels A and B of Fig. 5a, respectively. As shown in Fig. 5b, the spin-down current of PE-FE or FE-AFE configuration is completely forbidden throughout the bias range for any device or transport direction, indicating that both devices can filter tunneling electrons based on their spin state and potentially serve as spin filters. Additionally, within the bias range from −0.1 to 0.1 V, the PE-FE device exhibits a larger non-equilibrium electric current at a certain bias than the FE-AFE Cu(CrBr3)2 device, which corresponds to the larger differential conductance (dI/dV) in the PE-FE device as shown in Fig. 5c. On the other hand, both devices can exhibit negative differential resistance (NDR) in a larger range of positive or negative bias, which is the basis for many device applications including high-frequency oscillators, analog-to-digital converters, and logic gates88,89,90. The PE-FE Cu(CrBr3)2 device exhibits significant NDR effects in the X and Y directions when the bias voltage exceeds 0.15 and 0.1 V, respectively. This observation can be further validated by the ratio of dI/dV with respect to the applied voltage. As shown in Fig. 5c, the PE-FE Cu(CrBr3)2 device exhibits negative dI/dV values for both the X and Y directions at 0.15 and 0.1 V, respectively. A small NDR effect is observed in the Y direction of the FE-AFE Cu(CrBr3)2 device at 0.15 and −0.1 V, and this effect in both the X and Y directions is almost completely quenched to zero at a negative bias of −0.25 V. Although NDR behaviors have been realized in many nanoscale systems, they usually occur at relatively high biases. Then the Cu(CrBr3)2 devices with low-bias NDR are desirable from the point view of power consumption.

Fig. 5: Spin-polarized transport of Cu(CrBr3)2 devices.
figure 5

a Schematic diagram of the devices of the PE-FE and FE-AFE Cu(CrBr3)2 with spin-resolved current–voltage (I–V) curves (b) and differential conductance (dI/dV) curves (c) in X and Y directions. The solid lines correspond to the nonzero spin-up current and the hollow lines correspond to the zero spin-down current. A and B represent PE-FE and FE-AFE Cu(CrBr3)2 devices, respectively.

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Discussion

In summary, based on the combination of first-principles calculations and group symmetry analysis, we discussed the important role of Coulomb interactions in octahedral crystal fields and crystal symmetry forced p–d coupling in determining the decorating behavior of Cu, Ag, and Au in chromium trihalide semiconductors. When CrX3 is decorated with Cu or Au, the p–d coupling effect is stronger than the Coulomb interaction of the octahedral crystal field, so Cu(CrX3)2 and Au(CrX3)2 possess intrinsic ferroelectric phases due to their structural stability. However, when Ag is decorated with CrX3, Ag(CrX3)2 takes a paraelectric phase since the p–d coupling is weaker than the octahedral Coulomb interaction. After screening 28 transition metal elements, we identify that 12 M(CrX3)2 possesses the out-of-plane FE exhibiting the strong atomic selectivity. Atomic decoration can not only result in large out-of-plane electric polarization with a low energy barrier and tunable reversal electric field, but also enhanced intralayer FM coupling, giving rise to a higher Curie temperature. There are two phases AFE-FMin and FE-FMout in the Cu(CrBr3)2 system, which inspires ideas for the design of logical magnetoelectric coupling devices. In addition, Cu(CrBr3)2 devices demonstrate outstanding spin-filtering transmission and exhibit a low-bias NDR effect for low power consumption. This study provides theoretical guidance for the application of 2D ferromagnetic materials such as 2D CrX3 in the design of multiferroic devices.

Methods

First-principles calculations of atomistic and electronic structure

Spin-polarized density-functional theory (DFT) first-principles calculations were carried out with the Vienna ab initio simulation package (VASP)91,92. The projector augmented wave method was employed to describe the interaction between the core and valence electrons93. The Perdew, Burke, and Emzerhof parameterized gradient approximation is chosen to describe the exchange-correction functional94. The CI-NEB method is used to calculate the energy barriers of transition states75. In addition, for all the calculations, the cut-off energy for the basis set is chosen to be 500 eV. All geometric structures are fully relaxed until the force on each atom is <0.001 eV/Å, and the convergence criterion is 10−6 eV for energy. A vacuum space of 20 Å along the z direction was adopted to avoid spurious interactions between adjacent layers. The k-points sampling was performed with Monkhorst-Pack grids of 12 × 12 × 1. The on-site effective Hubbard interaction U was taken to be 0.5 eV in considering the electron correlation of 3d electrons of Cr atoms. Phonon dispersion relations are then acquired employing the PHONOPY code95 with the finite displacement approach and the adapted 3 × 3 × 1 supercell is large enough to evaluate the dynamical stability. Then, we chose the Heisenberg model to simulate the Curie temperature of our systems based on the MC simulation. The software package developed by Zhang et al.96, and the 70 × 70 × 1 lattice is used in the MC simulation.

Calculation of the electron transport properties in Cu(CrBr3)2

The calculation of the electron transport properties is implemented by Quantum Atomistix ToolKit (ATK) package97, which is based on the DFT and the non-equilibrium Green’s function. The spin-polarized generalized gradient approximation (SGGA) with the parametrization of Perdew–Burke–Ernzerhof (PBE), the cut-off energy of 310 Ry and a basis set of double-zeta polarized are used. A vacuum slab of 15 Å is added between the nearest neighboring layers to eliminate their coupling. The Monkhorst-Pack k-mesh is chosen to be 1 × 5 × 150. Using the non-equilibrium Green’s function formalism, the transmission coefficient T(E) is calculated as98

$$T\left(E\right)={{\rm{T}}}_{{\rm{r}}}\left[{\Gamma }_{{\rm{L}}}{G}_{r}{\Gamma }_{{\rm{R}}}{G}^{a}\right],$$
(3)

where Gr(a) is the retarded (advanced) Green’s function, and ΓL(ΓR) is the coupling of the central region to the left (right) lead. And the current is calculated by the Landauer–Büttiker formula, which is written as99

$$I\left(V\right)=\frac{2e}{h}{\int }_{\!{\mbox{-}}\infty }^{\infty }T\left(E\right)[f\left(E-{\mu }_{\rm{L}}\right)-f(E-{\mu }_{\rm{R}})]{dE},$$
(4)

where f(E − μL(R)) is the Fermi–Dirac distribution function of the left (right) electrode, where μL(R) is the electrochemical potential of the left (right) electrode.