Introduction

Magnetoelectric (ME) materials are experiencing great interest because of their inherent cross-coupling between electric and magnetic degrees of freedom1,2,3,4. The systematic control of the magnitude and crystallographic direction of magnetic order parameters by an electric field is attractive for spintronics, next-generation memory devices, and other original devices. The first typical coupling between ferroelectric polarization (P) and magnetic moment (M) is via the existence of an energy term ΔE ~ P2M2 in some type-I multiferroics5,6,7,8. The second typical ME coupling occurs in the type-II multiferroics, such as TbMnO39,10,11, where the microscopic mechanism of polarization is connected with the spin-orbit interaction, via P ~ rij × [Si × Sj] (or other forms of energetic terms), where rij is the vector connecting neighboring spins Si and Sj12,13,14. The third typical ME coupling is an indirect coupling that occurs in the so-called hybrid improper multiferroics where two energy terms play a role, with the first involving the ferroelectric polarization and another structural parameter, and the second term coupling this structural parameter with the magnetic order15,16,17,18,19. However, these ME mechanisms still lack the achievement of practical potential applications based on them. The first typical ME coupling is weak and cannot reverse one ferroic parameter by switching the other ferroic degree of freedom. The second typical ME coupling occurs in type-II multiferroics which possess only very small polarization and usually possess multiferroicity only at low temperatures. The third typical ME mechanism exists in the hybrid improper ferroelectric materials, which are rare. Furthermore, most studies to describe the ME effect focus on the linear ME parameters which are usually very small20,21,22,23,24,25,26 and can not describe the complex ME switching process.

Here, we propose a ME mechanism of rotational switching magnetic vector by rotationally reversing the ferroelectric polarization from the first-principles calculations. The switching of both ferromagnetism27,28,29,30 and antiferromagnetism31,32,33,34 is investigated here, as the potential functional materials for spintronic applications. This ME coupling occurs in one class of orthorhombic two-dimensional (2D) proper multiferroics (CrX2Se3 and MnX2Te3, X = Sn, Ge) and orthorhombic perovskite multiferroics (EuTiO3 and BiFeO3/LaFeO3 superlattice). The in-plane ferroelectric polarization rotates from the + Pa state to the + Pb state, and then to the − Pa state in the switching pathway, where the Pa state and Pb state are energetically degenerate, possess a polarization being perpendicular to each other and interchange their in-plane lattice constants. Because of structural and magnetic anisotropies, the magnetic vector rotates from a to b, and then to -a direction, following the rotation of the ferroelectric polarization. The rotational switching of ferroelectric polarization and magnetic vector can be described by a cosine function.

Results and discussion

Atomic structure of 2D CrX 2Se3 and stability

First-principles method is used to study the structure, ferroelectric and magnetic properties, as well as the magnetoelectric coupling (see the details about Methods and Supplementary Information). It is important to recall that a variety of two-dimensional structures with various thicknesses from one atomic layer (graphene) to seven atomic layers (MA2Z4 family35,36) and even up to nine atomic layers MXenes37 has been recently discovered. In particular, two atomic layers 2D SnSe, GeSe, and SnTe adopt ferroelectric properties38,39,40, while CrSe and MnTe possess (anti)ferromagnetic properties41,42. It is thus legitimate to wonder if combining structures made by 2D ferroelectrics and (anti)ferromagnets could result in the formation of stable multiferroics43,44,45. We indeed found one class of stable three-atom-layer 2D structures that are multiferroics, namely CrX2Se3 (X = Ge, Sn) and MnX2Te3. CrX2Se3 possesses proper ferroelectric polarization and ferromagnetism, while MnX2Te3 exhibit proper polarization and antiferromagnetism. Various configurations of the three atomic layer structure are considered (see the detail in Supplementary Information). The most stable structure is such as the top and bottom layers are made of XSe (XTe) and the middle layer consists of CrSe (MnTe), as shown in Fig. 1, as well as Supplementary Table 4. Their resulting space group is Pmm2 with a point group of C2v.

Fig. 1: Structures of 2D CrX2Se3.
figure 1

a Side view of the structure with polarization along +a direction. b–d Top view of the structures with polarization along −a, +b and +a direction, respectively. e The blue and red arrows represent the corresponding directions of P and M, respectively.

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To investigate the stability of the 2D CrX2Se3 and MnX2Te3 (X=Ge, Sn), we first calculated the formation energy. The formation energy Ef is also calculated by \({E}_{{{{\rm{f}}}}}=\frac{1}{6}({E}_{{{{\rm{Cr}}}}{X}_{2}{{{{\rm{Se}}}}}_{3}}-{E}_{{{{\rm{CrSe}}}}}-2{E}_{X{{{\rm{Se}}}}})\), where \({E}_{{{{\rm{Cr}}}}{X}_{2}{{{{\rm{Se}}}}}_{3}}\), ECrSe, EXSe are the energies in the formula unit of 2D CrX2Se3, CrSe bulk46 and XSe bulk47,48, respectively, with X = Ge, Sn. Both orthorhombic (o-CrX2Se3) and hexagonal (h-CrX2Se3) structures are considered. The structures of CrX2Se3, CrSe bulk, and XSe bulk can be found in Supplementary Fig. 3. As shown in Table 1, the formation energies of orthorhombic 2D structures of CrX2Se3 and MnX2Te3 (X = Ge, Sn) are in the range of 30–64 meV atom−1, which is very similar or lower than those of experimentally stable VSe2 and 2H-WS2. The small formation energies in our study thus indicate the stability of 2D structures of CrX2Se3 and MnX2Te3 (X = Ge, Sn). The structural stability is further evaluated by phonon spectrum and ab-initio molecular dynamics calculations. The phonon spectrum of CrX2Se3 is shown in Supplementary Fig. 1. No significant imaginary frequency is found in the phonon dispersion, therefore confirming the dynamic stability of these compounds. We further performed ab-initio molecular dynamics (AIMD) calculations to study its stability under a high temperature of 600 K. As shown in Supplementary Fig. 2, both CrGe2Se3 and CrSn2Se3 are stable at high temperatures.

Table 1 Formation energy
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Spontaneous polarization and magnetism of 2D CrX 2Se3

We first consider CrX2Se3. The ferroelectric polarization of CrGe2Se3 and CrSn2Se3 are calculated to be 55 μC cm−2 and 26 μC cm−2 along a direction, respectively (Here we use the effective thckness 3d/2 to compute the polarization, because the distribution of electron cloud are considered), which are larger than or similar to that of prototypical ferroelectric BaTiO3 (26 μC cm−2)49. The polarization comes from a polar mode (see Supplementary Fig. 5) and the 2D structures are proper ferroelectrics. The polarization mainly originates from the top and bottom XSe layers, the middle layer CrSe provides the ferromagnetism with 4 μB on each Cr ion50. Different magnetic configurations are considered, the ferromagnetic configuration is found to be the lowest energy state (see Supplementary Information for details) and thus the ground state. Magnetic anisotropy calculations shows that 2D CrX2Se3 possesses an easy axis along a direction, which is parallel to the ferroelectric polarization.

In the 2D CrX2Se3 structures, each Cr ion in the middle layer has six nearest neighboring Se ions and is at the center of Se octahedra. There are three pairs of Cr-Se bonds along x, y, and z direction, respectively, where x is close to the direction of a-b, y is near a + b, and z is parallel to c. The bonds for CrI (see Fig. 2) along the three directions are 2.7 Å, 3.2 Å, and 2.6 Å, respectively. This distortion expanding bonds along y direction and shrinking bonds along the x and z direction are the Jahn-Teller one, which is induced by the eg orbital splitting where \({d}_{3{y}^{2}-{r}^{2}}\) (\({d}_{3{x}^{2}-{r}^{2}}\)) shifts to lower energy and is occupied, while \({d}_{{z}^{2}-{x}^{2}}\) (\({d}_{{y}^{2}-{z}^{2}}\)) shifts to higher energy and is empty. Therefore three electrons of 3d4 occupy the three t2g orbits, and the fourth occupies \({d}_{3{y}^{2}-{r}^{2}}\) (\({d}_{3{x}^{2}-{r}^{2}}\)). As shown in Fig. 2, the projected density of states (PDOS) displays occupations of \({d}_{3{y}^{2}-{r}^{2}}\) in CrI and \({d}_{3{x}^{2}-{r}^{2}}\) in CrII, indicating the orbital ordering of eg between CrI and CrII. In the middle layer of the 2D structures, the interaction between Cr ions with 180° Cr-Se-Cr bond angle is the double exchange interaction, and that between Cr ions with about 90° Cr-Se-Cr bond angle is the well-known Goodenough-Kanamori-Anderson (GKA) superexchange interaction51,52,53,54. The Jahn-Teller effect and corresponding orbit occupation result in a ferromagnetic exhagne interaction in CrX2Se3. In MnX2Te3, there is no Jahn-Teller effect and an antiferromagnetic exchange interaction is energetically favorable.

Fig. 2: Jahn-Teller distortion in 2D CrX2Se3.
figure 2

a Jahn-Teller distortion of the middle atomic layer compared to the high symmetry structure. b The sketch of \({d}_{3{x}^{2}-{r}^{2}}\) and \({d}_{3{y}^{2}-{r}^{2}}\) orbitals of Cr ions. c PDOS of CrI and CrII.

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The magnetic anisotropy properties

We now investigated the magnetocrystalline anisotropy energy (MAE). The MAE is a direction-related quantity which remains invariable for all symmetrical operations on the crystal (more details can be found in Supplementary Table 6). Based on the symmetry of C2v and completeness of trigonometric polynomials, MAE can be expanded into trigonometric series as \({{{\rm{MAE}}}}(\theta ,\varphi )={\sum }_{n,m}{K}_{nm}{\sin }^{2n}\theta {\cos }^{2m}\varphi\), where θ and φ are the azimuthal angle for a direction and polar angle for c direction, respectively, in the spherical coordinate system. Knm are magnetocrystalline anisotropy parameters, and n and m are nonnegative integers. In most cases, the second order is enough to describe the phenomena of MAE and the higher orders can be neglected (Fig. 3, Supplementary Figs. 7 and 8). When neglecting terms of six and higher orders, MAE can be written as

$${{{\rm{MAE}}}}={E}_{0}+{K}_{10}{\sin }^{2}\theta +{K}_{01}{\cos }^{2}\varphi +{K}_{20}{\sin }^{4}\theta +{K}_{02}{\cos }^{4}\varphi +{K}_{11}{\sin }^{2}\theta {\cos }^{2}\varphi .$$
(1)

Based on this Eq. (1), MAE in the ab plane is \({E}_{0}+{K}_{10}{\sin }^{2}\theta +{K}_{20}{\sin }^{4}\theta\). Figure 3a show the MAE of CrGe2Se3 for spins in the ab plane, which confirms that the phenomenological mode of Eq. (1) agrees very well with the first-principles calculations. The easy axis is along a direction and the energy difference between spins along b and a directions is about 20 μeV Cr−1. The calculations from both first principles and the model of Eq. (1) show that the MAE for a spin along the out-of-plane direction is larger than that along a direction by about 120 μeV Cr−1 (see the MAE in 3D space in Fig. 3b). CrSn2Se3 has the easy axis along a direction (see Supplementary Fig. 7).

Fig. 3: MAE (in μeV Cr−1) of CrGe2Se3 as a function of the ferromagnetic direction.
figure 3

a MAE for ferromagnetic direction in the ab plane. b MAE for ferromagnetic direction in the three-dimensional space. The energy of ferromagnetic direction along the (a) direction is set to zero.

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Similar to CrX2Se3, MnX2Te3 are ferroelectric with polarization along a direction, and its magnitude is 49.2 μC cm−2 for X = Ge and 50.5 μC cm−2 for X = Sn. MnX2Te3 possess the same point group of C2v with CrX2Se3 and thus exhibit MAE that can be modeled by Eq. (1). The easy axis of MnX2Te3 is along b direction, therefore perpendicular to its polarization, which is thus different from the case of CrX2Se3 where both the easy axis and polarization are along a direction (see details in Supplementary Figs. 8 and 9).

Rotational magnetoelectric switching in 2D CrX 2Se3

We then consider the switching of the ferroelectric polarization and magnetoelectric coupling. The energy barrier of the ferroelectric switching pathway is calculated from linearly changing the magnitude of polarization and also from the climbing-image solid-state nudged elastic band (ssNEB) method55 which is particularly well-suited for ferroelectric switching due to its ability to accurately capture the energy landscape and transition pathways. As shown in Fig. 4a, the energy barrier from the ssNEB method is much smaller than that of linear changing polarization. In the switching pathway from ssNEB calculations, there are two maximal energy barriers of 83 meV f.u.−1, and one minimal energy which is identical between the initial state +P and the final state -P. Figure 4c shows the polarization components of Pa along a and Pb along b direction. One can thus see that the polarization firstly is along +a direction at the reaction coordinate of 0, then rotates by 45° at the switching coordinate of 0.25, and rotates by 90° which is along +b direction at the switching coordinate of 0.50, then rotates by 135° at the coordinate of 0.75, and finally rotates by 180° which is along -a direction at the coordinate of 1.0. In this rotational switching of ferroelectric polarization, the polarization direction rotates by 180° from +a to −a and the magnitude of polarization is almost unchanged, rather different from the switching pathway where the polarization linearly changes and passes a zero polarization at the half of the pathway.

Fig. 4: The evolution of physical quantities in the switching pathway.
figure 4

a The energy barrier of the ferroelectric switching pathway of CrGe2Se3 from state + P to state − P by ssNEB method and linearly changing the polarization. The blue line is the fitting by Eq. (2). b The energy difference between EP,M and EP. The blue line is the fitting by Eq. (3). c Electrical polarization components along (a, b) directions. d The angle θM. e MAE (in μeV Cr−1) in the ab plane. The switching coordinate of 0.0 and 1.0 represent + Pa and − Pa, respectively. The insets of the panel c and d are the sketches of the directions of P and M at the corresponding intermediate structures of the switching coordinate.

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To understand the rotational switching ferroelectric polarization, we write the Landau free-energy potential as \({E}_{{{{\bf{P}}}}}={\sum }_{n,m}{A}_{nm}{P}_{{{{\bf{a}}}}}^{2n}{P}_{{{{\bf{b}}}}}^{2m}\), where Amn is parameter, n and m are nonnegative integers. The magnitude of polarization slightly changes in the switching pathway, the free energy can be written as

$${E}_{{{{\bf{P}}}}} \sim {A}_{11}{P}_{{{{\bf{a}}}}}^{2}{P}_{{{{\bf{b}}}}}^{2}={A}_{11}\frac{{P}^{4}}{8}[1-\cos (4{\theta }_{{{{\bf{P}}}}})],$$
(2)

where θP is the angle between P and a direction, and 0° ≤ θP ≤ 180°. Therefore, the energy barrier in the rotational switching pathway is dependent on the direction of the polarizations in the switching process. Figure 4a confirms that the model of Eq. (2) agrees very well with the first-principles calculations.

The ferromagnetic vector (easy axis) of CrGe2Se3 also changes in the ferroelectric switching process. As the ferromagnetism along the out-of-plane direction has much higher energy than that in-plane direction by more than 120 μeV f.u.−1 (see Supplementary Fig. 3), the easy axis remains within the ab plane when switching the ferroelectric polarization. Figure 4e shows the in-plane MAE surface of a series of intermediate structures during the ferroelectric switching. One can see that the easy axis continuously rotates clockwise in the ab plane by 180° from a to -a (another possible path is from -a to a for the equivalent parallel/antiparallel magnetic vectors), during the switching of polarization. Figure 4d displays the evolution of angle (θM) between the easy axis and a direction in the switching pathway. One can clearly see θM changes from 0 (at coordinate 0) to − π/2 (at coordinate 0.5), and then to − π (at coordinate 1.0). The insets in Fig. 4c and d (also see Fig. 1e) also display the directions of ferromagnetism and polarization. This rotational switching of ferromagnetism originates from the magnetic anisotropy in the switching pathway (see Fig. 4e) where both lattice parameters and polarization are changed. For example, the initial state of + Pa has the lattice parameter a = 8.45 Å and b = 7.87 Å, while the intermediate state of Pb (at the coordinate of 0.5) has the lattice parameter a = 7.87 Å and b = 8.45 Å.

In the switching pathway, the easy axis Me rotates clockwise by 180° and polarization P rotates anticlockwise by 180° (see Fig. 1e and the insets of Fig. 4c, d). To understand the rotational magnetoelectric switching, we determine the coupling between magnetism and polarization as

$${E}_{{{{\rm{ME}}}}}\propto -\alpha | {{{\bf{P}}}}\cdot {{{{\bf{M}}}}}_{{{{\bf{e}}}}}{| }^{2}-\beta | {{{\bf{P}}}}\times {{{{\bf{M}}}}}_{{{{\bf{e}}}}}{| }^{2}=-{\alpha }^{{\prime} }-{\beta }^{{\prime} }\cos (2{\theta }_{{{{\bf{PM}}}}}),$$
(3)

where α and β are positive parameters, \({\alpha }^{{\prime} }\)=(α + β)P2M2/2, \({\beta }^{{\prime} }\)=(α − β)P2M2/2, P and M are the magnitudes polarization of and magnetic moment, respectively. P slightly changes and M remains the same during the switching pathway. θPM is the angel between P and Me, and 0° ≤ θPM ≤ 360°. Note that the energy of magnetoelastic coupling is also included in Eq. (3). Figure 4b shows the energy difference between the calculations with spin-orbit coupling (SOC) (EP,M with magnetic vector) and without SOC (EP without magnetic vector). The fitting of Eq. (3) agrees very well with the DFT calculations, indicating that P and Me can be (anti)parallel or perpendicular to each other in the switching process. The lowest EME in Fig. 4b at coordinates 0.25 and 0.75 imply the largest magnetoelectric coupling during the switching, consistent with the fact that the very large response at the critical point of phase transition (the electrocaloric effect at Curie temperature56,57,58,59,60,61, the piezoelectric effect at morphotropic phase boundaries62,63 and so on).

The phenomenon of ferromagnetism in CrGe2Se3 being switched by 180° by switching polarization under an electric field is also found in CrSn2Se3 (see Supplementary Fig. 12). The ferroelectric antiferromagnets of MnX2Te3 also exhibit rotational magnetoelectric switching phenomena (see Supplementary Figs. 13 and 14) which is confirmed by the calculation of polarization switching and antiferromagnetism switching under an external electric field (see Supplementary Figs. 15 and 16).

Rotational magnetoelectric switching in perovskites

The rotational magnetoelectric switching mechanism can be also found in other orthorhombic multiferroics. For instance, the prototypical orthorhombic multiferroic EuTiO328,64 and BiFeO3/LaFeO3 superlattice15,65. We show the case of magnetoelectric switching of EuTiO3 in Fig. 5. At tensile strain, EuTiO3 exhibits ferromagnetic and ferroelectric properties. The ferroelectric polarization is along a direction and the easy axis is along b direction and the MAE is as large as 288 μeV Eu−1. As shown in Fig. 5a, the energy barrier for rotational ferroelectric switching is much lower than for linear ferroelectric switching. The ferroelectric switching energy fits Eq. (2) very well. As shown in Fig. 5d, the easy axis rotates gradually from \(\frac{\pi }{2}\) to \(-\frac{\pi }{2}\) when the ferroelectric polarization rotates from +P state to −P state. The energy surface of MAE shown in Fig. 5e also clearly exhibits the magnetization rotation in the ferroelectric switching pathway.

Fig. 5: The evolution of physical quantities in the ferroelectric switching pathway of EuTiO3 under the epitaxial strain 2%.
figure 5

a The energy barrier of the ferroelectric switching pathway from state +P to state -P by ssNEB method and linearly changing the polarization. The blue line is fitting by Eq. (2). b The energy difference between EP,M and EP. The blue line is fitting by Eq. (3). c Electrical polarization components along a and b directions. d The angle θM (red circles), and MAE (blue squares). e The energy surface of MAE in ab plane during the switching pathway. The bule and red arrows are the directions of P and M, respectively. The switching coordinate of 0.0 and 1.0 in the pathway represent +Pa and −Pa, respectively. The inset of (a) shows the structure of EuTiO3 under the epitaxial strain 2%.

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Note that the proposed ferroelectric and magnetoelectric switching mechanism is general in perovskite structures. To show the generality of the rotational switching, the ferroelectric switching of BaTiO3 and magnetoelectric switching of BiFeO3/LaFeO3 are shown in Supplementary Figs. 17 and 18, respectively. The rotational ferroelectric switching in prototypical ferroelectric BaTiO3, rather than the linear changing and reversing the polarization, further confirms the generality of the rotational switching mechanism.

In summary, we propose a rotational magnetoelectric switching mechanism in two-dimensional multiferroics and perovskite multiferroics by first-principles methods. The orthorhombic structures possess strong structural and magnetic anisotropies. Thesse strong anisotropies lead to the switching of proper polarization by rotating its direction by 180°, and result in the switching of (anti)ferromagnetism from rotating the magnetic vector by 180°. This magnetoelectric coupling mechanism opens a route to control spins by an electric field and broadens the potential application of spintronics. Note that this phenomenon may also occur in other structures with different symmetries66.

Methods

Ab initio calculations

Density functional theory (DFT) calculations are performed with the Vienna Ab-initio Simulation Package (VASP)67,68,69, using the projected augmented wave (PAW) method70 with the generalized gradient approximation (GGA) parametrized by Perdew, Burke, and Ernzerhof (PBE) for the exchange-correlation functional71. The energy cutoff of 350 eV is adopted for the plane-wave basis sets. The Brillouin zone integration is sampled by using a Monkhorst-Pack k-point sampling with a mesh of 6 × 6 × 1. A vacuum space larger than 15 Å was added along the out-of-plane direction to eliminate the interactions between replica artificial layers. The tolerance of 0.001 eV Å−1 is used for the Hellman-Feynman forces during the atomic optimization and the convergence criterion for the energy is 10−7 eV. To account for the strongly correlated effect of the d orbit, an effective Hubbard value Ueff of 4.1 eV for Cr and 3.0 eV for Mn is used, as provided by the linear response approach72,73,74 and as also chosen when comparing with results from the HSE06 hybrid functional75. The polarization was calculated using the Berry phase method76. The phonon dispersion spectrum is calculated with the finite displacement method (Frozen-phonon approach)77 on a 2 × 2 × 1 supercell with 96 atoms by using the PHONOPY package78. The climbing-image solid-state nudged elastic band (ssNEB) method was used for determining the ferroelectric switching paths and energy barrier55. For the magnetocrystalline anisotropy energy calculations, a dense k-point mesh of 8 × 8 × 1 and a more precise energy convergence criterion of 10−9 eV are used. The spin-orbit coupling (SOC) is included in the noncollinear magnetic calculations. The non-collinear calculations with SOC are all performed in a self-consistent manner. The electric field is applied by minimizing the approximate electric enthalpy8,19,79,80 (see the details in the Supplementary Information), which provides accurate results for the calculations in ferroelectric and multiferroic compounds. The supercell lattice and atomic position are relaxed to compute the energies of different magnetic configurations.