⇦Chinese Physics Letters, 2021, Vol. 38, No. 7, Article code 077501Express Letter Strong Coupled Magnetic and Electric Ordering in Monolayer of Metal Thio(seleno)phosphates Chenqiang Hua (华陈强)1, Hua Bai (白桦)1, Yi Zheng (郑毅)1, Zhu-An Xu (许祝安)1, Shengyuan A. Yang (杨声远)2, Yunhao Lu (陆赟豪)1*, and Su-Huai Wei (魏苏淮)3 Affiliations 1Zhejiang Province Key Laboratory of Quantum Technology and Device, State Key Laboratory of Silicon Materials, Department of Physics, Zhejiang University, Hangzhou 310027, China 2Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore 3Beijing Computational Science Research Center, Beijing 100193, China Received 6 May 2021; accepted 7 June 2021; published online 18 June 2021 Supported by the National Key R&D Program of China (Grant No. 2019YFE0112000), the Zhejiang Provincial Natural Science Foundation of China (Grant No. LR21A040001), and the National Natural Science Foundation of China (Grant No. 11974307, 12088101, 11991060, and U1930402).
*Corresponding author. Email: luyh@zju.edu.cn Citation Text: Hua C Q, Bai H, Zheng Y, Xu Z A, Yang S Y A et al. 2021 Chin. Phys. Lett. 38 077501    Abstract The coupling between electric ordering and magnetic ordering in two-dimensional (2D) materials is important for both fundamental research of 2D multiferroics and future development of magnetism-based information storage and operation. Here, we introduce a scheme for realizing a magnetic phase transition through the transition of electric ordering. We take CuMoP$_{2}$S$_{6}$ monolayer as an example, which is a member of the large 2D transition-metal chalcogen-phosphates family. Based on first-principles calculations, we find that it is a multiferroic with unprecedented characters, namely, it exhibits two different phases: an antiferroelectric-antiferromagnetic phase and a ferroelectric-ferromagnetic phase, in which the electric and magnetic orderings are strongly coupled. Importantly, the electric polarization is out-of-plane, so the magnetism can be readily switched by using the gate electric field. Our finding reveals a series of 2D multiferroics with special magnetoelectric coupling, which hold great promise for experimental realization and practical applications. DOI:10.1088/0256-307X/38/7/077501 © 2021 Chinese Physics Society Article Text Multiferroics, the materials with coexisting ferroelectricity and magnetism, have been a subject of extremely active research in the past twenty years.[1,2] Besides their intriguing fundamental physics, the interest on these materials is highly motivated by their potential for functional device applications, particularly, the prospects of manipulating magnetism purely via electric field, without involving electric currents (which would cause undesired dissipation and heating problems). The multiferroics discovered to date are conventionally classified into two groups.[3] In type-I multiferroics, the ferroelectricity and magnetism have different sources, e.g., originated from different ions, so the coupling between them is typically rather weak. As for type-II multiferroics, the ferroelectricity is generated by the magnetism that breaks the inversion symmetry. In this case, the coupling is strong, but the ferroelectricity is generally weak, with low critical temperatures.[4] The current challenge in multiferroics is to find candidates that can have strong coupling between ferroelectric and magnetic orderings as well as robust ordering. In addition, for device applications, the materials are usually made into layered forms, and it is highly preferred to have out-of-plane electric polarizations such that the orderings can be controlled by using gate electric field. The advance in two-dimensional (2D) materials opens new possibilities to address this problem. The robust ferroelectric and magnetic orderings have been confirmed in 2D materials in recent few years,[5–9] even some 2D material with multiferroic properties have been proposed.[10–16] The reduced dimension may help to enhance the coupling between different degrees of freedom.[17–19] Indeed, a very recent work predicted that in monolayer ReWCl$_{6}$, the orientation of the in-plane ferroelectric polarization can exhibit a strong coupling with different magnetic orderings.[20] However, it is technically difficult to control in-plane ferroelectricity, particularly, impossible with gate field. Thus, it remains a challenge to search for strongly coupled multiferroics with out-of-plane ferroelectricity. In this work, we address this challenge by proposing a scheme to realize 2D strongly coupled multiferroics with out-of-plane electric polarization, which exhibits simultaneous transition in magnetic and electric orderings. Here, different electric orderings couple with different magnetic orderings, and the magnetism is determined by the electric ordering which can be controlled by gate field [Fig. 1(a)]. Specifically, we will show that there are two phases in our systems, one is the antiferroelectric (AFE)-antiferromagnetic (AFM) phase and the other is the ferroelectric (FE)-ferromagnetic (FM) phase, as shown in Fig. 1(b). Unlike conventional multiferroics, in which the coupling only allows one to tune the magnitude, here, the coupling acts like a logic operation, allowing one to fundamentally change the orderings and to drive a phase transition [Another scenario shown in Fig. S1 in the Supporting Information (SI) with AFE-FM phase and FE-AFM phase may also be realized in certain systems].
cpl-38-7-077501-fig1.png
Fig. 1. (a) Schematic diagrams of the magnetic ordering versus electric gate field. Different electric orderings correspond to different magnetic exchange paths and thus result in the different magnetic orderings. (b) $P$–$E$ loop with the corresponding electric and magnetic ordering.
The idea is motivated by the experimental demonstration of out-of-plane ferroelectricity in 2D few-layered CuInP$_{2}$S$_{6}$.[21] This material belongs to a large family of compounds known as the metal thio(seleno)phosphates (MTPs). MTPs are van der Waals layered materials, with each layer formed by metal cations inserted into a hexagonal [P$_{2}$S(Se)$_{6}$]$^{4-}$ framework.[22–24] A variety of metal elements and their combinations can be accommodated in this material family, including the transition metals, lanthanide and even actinide elements. Out-of-plane ferroelectricity is common in MTPs containing Cu$^{1+}$ ions, as they tend to be driven by the symmetry-reduction-enhanced strong coupling between occupied Cu 3$d$ orbitals and unoccupied $s$ orbitals to deviate from mid-plane of the layer, e.g., as in CuInP$_{2}$S$_{6}$ and CuInP$_{2}$Se$_{6}$.[25,26] Clearly, Cu$^{1+}$ ion with fully filled $d$-shell is nonmagnetic. To simultaneously have magnetism, besides Cu$^{1+}$, we may replace the group-III metal by a transition metal ion $M^{3+}$, which is magnetic and has valence $+$3, to form a quaternary compound Cu$M$P$_{2}$S(Se)$_{6}$, which has been achieved in experiment.[27] From construction, the obtained material should be a type-I multiferroic: the ferroelectricity is from the Cu$^{1+}$ ions, whereas the magnetism is from the $M^{3+}$ ions. Nevertheless, unlike typical type-I multiferroics, we show that in the 2D limit, the polarization of Cu$^{1+}$ ions can strongly affect the magnetic super-exchange interaction between the $M^{3+}$ ions. Taking monolayer CuMoP$_{2}$S$_{6}$ as an example, from first-principles calculations, we find that the change in the electric polarization can even drive a magnetic phase transition: its ground state is (AFE + AFM) ordering; and under a gate electric field, it can be driven into a metastable state which is (FE + FM) ordering. As mentioned above, for conventional multiferroics, the electric control of magnetism is mostly referring to the relatively small change in the magnitude of magnetization or the period of magnetic spirals, etc.[4] This is incomparable with the case here, where the change is in a sense singular as it corresponds to a magnetic phase transition. Thus, we have achieved a truly strongly coupled multiferroic, a breakthrough of both fundamental and practical significances. The 2D character, the out-of-plane electric polarization, and the gate-field control of magnetism endow these new multiferroics with great potential for a wide range of applications.
cpl-38-7-077501-fig2.png
Fig. 2. (a) and (b) The structure of s-AFE CuMoP$_{2}$S$_{6}$. (d) and (e) The structure of FE CuMoP$_{2}$S$_{6}$. In FE, $C_{3}$ rotation symmetry (solid triangle) is kept. Mo atoms are represented by red balls and Cu (S) atoms on up/down sides are represented by light/deep blue (green) balls. For high symmetry phase, metals are locating at A/B sites, i.e., the center of $M$S$_{6}$ octahedra. (c) and (f) The perspective view of local octahedral structure at s-AFE state and FE states. The CuS$_{3}$ tetrahedral is labeled by dash triangle. P atoms are skipped for clearness.
Let us start by examining the lattice structure of MTP materials. Bulk MTPs have been known for over a hundred years.[22] They are layered materials with layers bonded by weak van der Waals interactions. Hence, 2D MTPs should be readily obtained from the bulk, e.g., by the mechanical exfoliation method, which was indeed demonstrated in recent experiments.[17] Figures 2(a) and 2(d) show the structure of a monolayer MTP in AFE and FE configurations. A common backbone shared by this material family is a hexagonal sublattice of the ethane-like [P$_{2}X_{6}$]$^{4- }$ ($X$ = S, Se) anionic unit. The metal cations fill the interstitial sites of this framework, which form another honeycomb sublattice. For a quaternary MTP compound, as the Cu$M$P$_{2}$S(Se)$_{6}$ studied here, Cu$^{1+}$ and $M^{3+}$ ions naturally occupy the A and B sites of this honeycomb lattice, as shown in Figs. 2(a) and 2(d). In this structure, each Cu and $M$ ion is located in an octahedron formed by six surrounding $X$ ions. In a high-symmetry reference state, they should sit at the center of this octahedron. However, as mentioned above, the monovalent Cu$^{1+}$ is known for its tendency towards lower coordination, which is driven by a symmetry-reduction-enhanced coupling between the fully filled 3$d^{10}$ orbits and the empty 4$s$ orbits.[26,28] As a result, the Cu ions exhibit a sizable vertical shift from the mid plane of the 2D layer. This shift of the Cu ion creates a local electric dipole moment in the out-of-plane direction. It should be mentioned that the $M$ ions generally would display a compensatory shift with the opposite polarity, however, this shift would be much smaller. To be specific, let us focus on the concrete example CuMoP$_{2}$S$_{6}$ monolayer in the following discussion. In this material, there exists a large shift of Cu from the mid plane. However, the ground state of monolayer CuMoP$_{2}$S$_{6}$ is a stripe type antiferroelectric state in the No. 4 space group. As shown in Figs. 2(a) and 2(b), the Cu ions at neighboring rows undergo shift in opposite directions. The antiferroelectric ground state has no net electric polarization. The similar antiferroelectric ground state has been reported in several MTP materials, such as bulk/layered CuCrP$_{2}$S$_{6}$ and CuInP$_{2}$S(Se)$_{6}$.[21,25,27,29] The ferroelectric state, in which all Cu ions shift in the same direction, corresponds to a metastable state of CuMoP$_{2}$S$_{6}$ ML. In this state, the shift of Cu is about 1.43 Å from the mid plane and the calculated electric dipole is $P=41.28$ $e$Å. Now the space group changes to No. 143, which is quite different from the AFE configuration. Note the instability of the MTP towards AFE and FE states can also be inferred from the phonon spectrum of high symmetry phase (see the SI and Figs. S2 and S3) and the polarization is constrained to be along the out-of-plane ($z$) direction due to the local $C_{3z}$ rotational symmetry. In FE structure, the $C_{3}$ rotation symmetry constrains all Mo sites with equivalent coordination. While in AFE, due to the striped configuration, this $C_{3}$ symmetry is broken, meanwhile, there emerges a glide mirror $G_{y} = (-x, y+1/2, 0)$. As already shown in Figs. 2(c) and 2(f), the Cu$^{+}$ ions prefer low symmetry structure and shift spontaneously from octahedral center toward three S atoms (dashed triangles) of one side forming CuS$_{3}$ tetrahedron. In AFE phase, the CuS$_{3}$ tetrahedron locates on opposite sides alternatively [CuS$^{\rm (u)}_{3}$, CuS$^{\rm (d)}_{3}$]. As the MoS$_{6}$ octahedrons are titled, Mo$^{3+}$ ions connect both up and down S-sublayers [S$^{\rm (u)}$ and S$^{\rm (d)}$] by $\sim$$180^\circ\!$ S$^{\rm (u)}$–Mo–S$^{\rm (d)}$ geometry [Fig. 2(c)]. On the other hand, the CuS$_{3}$ tetrahedron locates on the same side in the FE phase and now, Mo$^{3+}$ in MoS$_{6}$ octahedron connects only one S-sublayer by $\sim$$90^\circ\!$ S$^{\rm (u)}$–Mo–S$^{\rm (u)}$ [or S$^{\rm (d)}$–Mo–S$^{\rm (d)}]$ geometry [Fig. 2(f)]. This difference may result in the difference in exchange interactions between AFE and FE. Next, we turn to the magnetism. In CuMoP$_{2}$S$_{6}$, the magnetism comes mainly from the magnetic moments of the Mo$^{3+}$ ions, which can be clearly visualized from the spin density plot in Figs. 3(a) and 3(c). For the trivalent Mo$^{3+}$ state, there are three remaining electrons occupying the $d$ shell. In the octahedral crystal field from the surrounding S ions, the Mo $d$ orbitals split into the lower $t_{\rm 2g}$ and higher $e_{\rm g}$ orbitals. The three electrons will occupy these $t_{\rm 2g}$ orbitals, and according to Hund's rule, they will have parallel spins, leading to a local magnetic moment of 3$\mu_{_{\scriptstyle \rm B}}$. This picture is confirmed by our first-principles calculations. For example, it is manifested by the spin-polarized $t_{\rm 2g}$ orbitals in the projected band plots in Figs. 3(b) and 3(d). Also, the calculation finds a local moment $\sim$$3\mu_{_{\scriptstyle \rm B}}$ for Mo$^{3+}$ ions.
cpl-38-7-077501-fig3.png
Fig. 3. (a) The structure of s-AFE CuMoP$_{2}$S$_{6}$ monolayer with yellow (blue) color isosurface represents spin-up (spin-down) charge density. (b) The projected band structure and DOS of s-AFE state. (c) The structure of FE CuMoP$_{2}$S$_{6}$ monolayer with spin charge density. (d) The projected band structure and DOS of FE state. The $e_{\rm g}$ and $t_{\rm 2g}$ orbital characters are labeled by blue and red colors, respectively. Note that only spin-up bands are shown here for simplicity. The weight centers of $e_{\rm g}$ are indicated by blue dashed lines and obviously in s-AFE, the $e_{\rm g}$ states are broadened in energy resulting in the increase ($\delta$) of exchange gap between $t_{\rm 2g}$ and $e_{\rm g}$.
Remarkably, the ordering of these magnetic moments is strongly coupled to the electric polarization. In the AFE ground state, the system favors a striped AFM (s-AFM) among several AFM configurations,[30] which conforms to the pattern of the electric ordering [see Fig. 3(a)]; whereas for the FE state, the system prefers the ferromagnetism [see Fig. 3(c)]. Thus, the transition between the two electric states represents a phase transition between (AFE + AFM) and (FE + FM) orderings. The extremely strong coupling between electric and magnetic orderings revealed here distinguishes the current system from conventional type-I multiferroic materials. In bulk type-I multiferroics such as BiFeO$_{3}$, the ferroelectricity can tilt the collinear magnetism and form certain long period magnet spirals.[30] In comparison, here the electric polarization even reverses the type of magnetic coupling from AFM to FM. The observation is consistent with the general tendency that couplings between occupied $t_{\rm 2g}$-derived states and unoccupied $e_{\rm g}$ states are enhanced with reduced dimension. Regarding the magnetic exchange interactions, in 2D, they are mediated by virtual hopping processes confined within the 2D layer, therefore the interaction will have more sensitive dependence on the structural configurations. Hence, the electric ordering transition between FE and AFE corresponds to magnetic ordering transition between FM and AFM. The well-known Goodenough–Kanamori–Anderson (GKA) rules summarize the dependence of magnetic super-exchange interactions in insulators on some typical lattice configurations.[31–33] Here, the lattice structure is much more complicated and there are multiple exchange paths between the local moments at Mo$^{3+}$ ions. Although it is difficult to give a simple geometric picture like the cases in GKA rules, the first-principles results do indicate that the magnetic super-exchange is strongly affected by the polarization of the Cu$^{1+}$ ions in the 2D layer. This AFE-to-FE transition affects local structure of MoS$_{6}$ octahedrons (Table S2) and crystal field of Mo-$d$ orbital. Although the band widths of occupied $t_{\rm 2g}$ orbitals for both states are similar, the unoccupied $e_{\rm g}$ orbitals of magnetic Mo$^{3+}$ ions at FE state are much narrower in energy than the $e_{\rm g}$ orbitals at AFE state [Figs. 3(b) and 3(d), also see the HSE06 results in Fig. S5] and the split gap between $t_{\rm 2g}$ and $e_{\rm g}$ orbitals is decreased from AFE to FE, also proved by the on-site $d$ levels based on the maximally localized Wannier functions (MLWFs)[34] shown in the SI for details. Thus, exchange virtual gap between the lower $t_{\rm 2g}$ and higher $e_{\rm g}$ orbitals of magnetic ions is reduced by the transition from AFE to FE, which promotes the transition from AFM to FM. To have a quantitative estimation of the magnetic interaction, we adopt the following classical spin model: $$ H_{\mathrm{spin}}=J_{1}\sum_{i,j\in \mathrm{intra}} {{\boldsymbol S}_{i}\cdot {\boldsymbol S}_{j}} +J_{2}\sum_{i,j\in \mathrm{inter}} {{\boldsymbol S}_{i}\cdot {\boldsymbol S}_{j}} +K\sum_i \left( S_{i}^{z} \right)^{2}, $$ where ${\boldsymbol S}_{i}$ is the normalized local spin at site $i$, the first and second terms are the exchange interaction between neighboring spins within the same stripe (intra-stripe) and neighboring stripe (inter-stripe), and the third term represents the magnetic anisotropy. The model parameters, including the magnetic exchange couplings $J_{1}$, $J_{2}$, and the anisotropy strength $K$, can be extracted from the first-principles calculations via the energy mapping approach. Here, $J_{1}$ and $J_{2}$ should equal in the ferroelectric state, and they differ in the antiferroelectric state. Importantly, one observes that while the intra-stripe exchange coupling $J_{1}$ remains to be FM and is only slightly affected by the electric polarization, the inter-stripe coupling $J_{2}$ switches sign between the two states: $J_{2}(J_{1})= -0.25$ meV being FM in the ferroelectric state, and $J_{2}= 0.16$ meV being AFM in the AFE state (see Table 1). By Monte Carlo (MC) simulation with anisotropic magnetic energy (see details of anisotropy strength $K$ in the SI Table S4), the critical temperature of magnetic ordering is estimated to be $\sim $15 K ($\sim $5 K) for FM (AFM) (see Fig. S7).
Table 1. Polarization-dependent effective-exchange interaction $J_{2}$ in Cu$M$P$_{2}X_{6}$ ($M$ = Cr, Mo or W; $X$ = S or Se).
$d ^{{3}}$ (meV) Cr–S Mo–S W–S Cr–Se Mo–Se W–Se
$J_{2}$ FE $-1.51$ $-0.25$ 3.29 $-1.53$ 0.21 5.54
AFE $-1.08$ 0.16 3.82 $-0.97$ 0.62 8.51
cpl-38-7-077501-fig4.png
Fig. 4. (a) Energy contour by moving two Cu atoms in the parallel and antiparallel directions. (b) Calculated CINEB barrier between FE and AFE. During the electric transition, the magnetic phase varies from FM to AFM. (c) and (d) Calculated transmission spectrum of CuMoP$_{2}$S$_{6}$ sandwiched by two 2D metallic electrodes under two ordering states (FE + FM and AFE + AFM) with red and blue lines indicating spin-up and spin-down channels, respectively. The insets show the $I$–$V$ curves of the two spin channels.
Since the polarization is along the out-of-plane direction, this ferroelectric state can be achieved from the ground state by applying a gate electric field. Indeed, we have checked that the system remains insulating during the transition between the AFE and FE states. To investigate the transition barrier between the (AFE + AFM) and (FE + FM) states, in Fig. 4(a), we plot the free-energy contours for monolayer CuMoP$_{2}$S$_{6}$ in the parameter space of the shifts of Cu ions in neighboring stripes. The two minimum points A and A$'$ correspond to the (AFE + AFM) ground states. Meanwhile, the two local minima B and B$'$ give the (FE + FM) metastable states with opposite electric polarizations. We apply the climbing image nudged elastic band method (CINEB)[35] to determine the transition barrier between the A (AFE + AFM) and B (FE + FM) states. The estimated activation energy barrier from A to B is 74.1 meV/f.u. and it is 159.2 meV/f.u. from B to A [Fig. 4(b)]. These values are comparable to the traditional ferroelectric materials such as PbTiO$_{3}$ ($\sim $200 meV)[36,37] and the 2D ferroelectric In$_{2}$Se$_{3}$ (66 meV),[5] indicating that an applied gate field can be readily to switch monolayer CuMoP$_{2}$S$_{6}$ from the antiferroelectric ground state to two ferroelectric states (with $\pm P$). The above analysis can be extended to other 2D materials Cu$M$P$_{2}$S(Se)$_{6}$ in the family, with trivalent magnetic ions $M^{3+}$, such as Cr and W. The corresponding results are presented in Table 1 and the SI. One can observe that when varying from 3$d$ ion Cr$^{3+}$ to 5$d$ ion W$^{3+}$, the enlarged orbital radius tends to enhance the AFM exchange coupling. This is manifested by the observation that CuCrP$_{2}$S$_{6}$ and CuCrP$_{2}$Se$_{6}$ always favor FM ordering, whereas CuWP$_{2}$S$_{6}$ and CuWP$_{2}$Se$_{6}$ always favor the AFM ordering, for both AFE and FE states, although there is obvious change in the inter-stripe exchange coupling strengths ($J_{2}$) between the two states. The anion group also affects the exchange interaction. One finds that the change from more localized S-3$p$ to relatively delocalized Se-4$p$ orbitals promote the AFM exchange interaction. Although for these other materials, the AFE and FE states may not necessarily correspond to different magnetic orderings, the electric polarization does strongly affect the magnetic interaction strengths (Table 1). This also suggests that it should be possible to achieve different magnetic orderings via external perturbations, such as lattice strain. Indeed, we find that a small $2\%$ uniaxial strain is enough to turn the meta-stable state of CuMoP$_{2}$Se$_{6}$ ML from (FE + AFM) to (FE + FM) ordering and the ground state is still (AFE + AFM) ordering, thus it becomes a 2D multiferroic, qualitatively the same as CuMoP$_{2}$S$_{6}$ ML (see Fig. S8). To elucidate the effects of two ordering states (AFE + AFM and FE + FM) on electron transmission, a tunneling structure with CuMoP$_{2}$S$_{6}$ ML sandwiched by 2D metallic electrodes (ReS$_{2}$, see Fig. S9) is constructed. As shown in Figs. 4(c) and 4(d), the transmissions of two spin channels are nearly degenerated for AFE + AFM state. However, the transmission of FE + FM state is largely spin-dependent above and below fermi level, enabling electric control of spin current [inset of Fig. 4(d)]. Although the magnetic orderings remain for other 2D materials Cu$M$P$_{2}$S(Se)$_{6}$, the electric ordering does strongly affect the magnetic interaction strengths and the spin-dependent transmissions of them are also different for the two electric ordering states (Fig. S10). Therefore, Cu$M$P$_{2}$S(Se)$_{6}$ ML provides an ideal platform to the pursuit of controlling magnetism using electric gate at atomic scale. In summary, we have demonstrated a magnetoelectric effect with logical operation between electric and magnetic orderings and established that the experimentally fabricated MTPs could host such multiferroic properties with out-of-plane polarization (e.g., ML CuMoP$_{2}$S$_{6}$ or stained CuMoP$_{2}$Se$_{6}$). They enjoys the unique property that a gate field can control both its electric polarization and its magnetic ordering. This advantage is of paramount importance for constructing functional devices. For example, it enables magnetic storage that can be controlled purely by gate field without currents. It also enables the magnetic information to be read out through the measurement of electric polarization. References The evolution of multiferroicsObservation of Ferroelastic and Ferroelectric Domains in AgNbO 3 Single CrystalClassifying multiferroics: Mechanisms and effectsMagnetoelectricity in multiferroics: a theoretical perspectivePrediction of intrinsic two-dimensional ferroelectrics in In2Se3 and other III2-VI3 van der Waals materialsTwo-dimensional ferroelectricity and switchable spin-textures in ultra-thin elemental Te multilayersLayer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limitDiscovery of intrinsic ferromagnetism in two-dimensional van der Waals crystalsElemental Ferroelectricity and Antiferroelectricity in Group-V MonolayerPrediction of Intrinsic Ferromagnetic Ferroelectricity in a Transition-Metal Halide MonolayerTwo-dimensional hyperferroelectric metals: A different route to ferromagnetic-ferroelectric multiferroicsMagnetoelectric Coupling in Multiferroic Bilayer VS 2 Multiferroicity in atomic van der Waals heterostructuresBinary Compound Bilayer and Multilayer with Vertical Polarizations: Two-Dimensional Ferroelectrics, Multiferroics, and NanogeneratorsElectric-Field Switching of Magnetic Topological Charge in Type-I MultiferroicsTwo-dimensional multiferroic semiconductors with coexisting ferroelectricity and ferromagnetismType-II Multiferroic Hf 2 VC 2 F 2 MXene Monolayer with High Transition TemperatureDesign of Two-Dimensional Multiferroics with Direct Polarization-Magnetization CouplingInversion of Ferrimagnetic Magnetization by Ferroelectric Switching via a Novel Magnetoelectric CouplingElectrical Control of Magnetic Phase Transition in a Type-I Multiferroic Double-Metal Trihalide MonolayerRoom-temperature ferroelectricity in CuInP2S6 ultrathin flakesMetal Thio- and Selenophosphates as Multifunctional van der Waals Layered MaterialsLow-Temperature Copper Ordering in the Layered Thiophosphate CuVP2S6: A Time-of-Flight Neutron Powder Diffraction StudyCopper sublattice ordering in layered CuMP2Se6 (M=In, Cr)Off-plane polarization ordering in metal chalcogen diphosphates from bulk to monolayerFerrielectric ordering in lamellar CuInP 2 S 6 Two-dimensional ferromagnetism and driven ferroelectricity in van der Waals CuCrP 2 S 6Off-center atomic displacements in zinc-blende semiconductorStrain engineering to facilitate the occurrence of 2D ferroelectricity in CuInP 2 S 6 monolayerBiFeO 3 Thin Films: A Playground for Exploring Electric-Field Control of MultifunctionalitiesAntiferromagnetism. Theory of Superexchange InteractionTheory of the Role of Covalence in the Perovskite-Type Manganites [ La ,   M ( II ) ] Mn O 3 Superexchange interaction and symmetry properties of electron orbitalsLigand-field helical luminescence in a 2D ferromagnetic insulatorA generalized solid-state nudged elastic band methodOrigin of ferroelectricity in perovskite oxidesA New Model of Ferroelectric Phase Transition with Neglectable Tunneling Effect
[1] Fiebig M, Lottermoser T, Meier D, and Trassin M 2016 Nat. Rev. Mater. 1 16046
[2] Zhao W, Fu Z, Deng J, Li S, Han Y, Li M R, Wang X, and Hong J 2021 Chin. Phys. Lett. 38 037701
[3] Khomskii D 2009 Physics 2 20
[4] Dong S, Xiang H, and Dagotto E 2019 Natl. Sci. Rev. 6 629
[5] Ding W, Zhu J, Wang Z, Gao Y, Xiao D, Gu Y, Zhang Z, and Zhu W 2017 Nat. Commun. 8 14956
[6] Wang Y, Xiao C, Chen M, Hua C, Zou J, Wu C, Jiang J, Yang S A, Lu Y, and Ji W 2018 Mater. Horiz. 5 521
[7] Huang B, Clark G, Navarro-Moratalla E, Klein D R, Cheng R, Seyler K L, Zhong D, Schmidgall E, McGuire M A, Cobden D H, Yao W, Xiao D, Jarillo-Herrero P, and Xu X 2017 Nature 546 270
[8] Gong C, Li L, Li Z, Ji H, Stern A, Xia Y, Cao T, Bao W, Wang C, Wang Y, Qiu Z Q, Cava R J, Louie S G, Xia J, and Zhang X 2017 Nature 546 265
[9] Xiao C, Wang F, Yang S A, Lu Y, Feng Y, and Zhang S 2018 Adv. Funct. Mater. 28 1707383
[10] Huang C, Du Y, Wu H, Xiang H, Deng K, and Kan E 2018 Phys. Rev. Lett. 120 147601
[11] Luo W, Xu K, and Xiang H 2017 Phys. Rev. B 96 235415
[12] Liu X, Pyatakov A P, and Ren W 2020 Phys. Rev. Lett. 125 247601
[13] Gong C, Kim E M, Wang Y, Lee G, and Zhang X 2019 Nat. Commun. 10 2657
[14] Li L and Wu M 2017 ACS Nano 11 6382
[15] Xu C, Chen P, Tan H, Yang Y, Xiang H, and Bellaiche L 2020 Phys. Rev. Lett. 125 37203
[16] Qi J, Wang H, Chen X, and Qian X 2018 Appl. Phys. Lett. 113 043102
[17] Zhang J J, Lin L, Zhang Y, Wu M, Yakobson B I, and Dong S 2018 J. Am. Chem. Soc. 140 9768
[18] Zhang J, Shen X, Wang Y, Ji C, Zhou Y, Wang J, Huang F, and Lu X 2020 Phys. Rev. Lett. 125 017601
[19] Weng Y, Lin L, Dagotto E, and Dong S 2016 Phys. Rev. Lett. 117 037601
[20] Xu M, Huang C, Li Y, Liu S, Zhong X, Jena P, Kan E, and Wang Y 2020 Phys. Rev. Lett. 124 067602
[21] Liu F, You L, Seyler K L, Li X, Yu P, Lin J, Wang X, Zhou J, Wang H, He H, Pantelides S T, Zhou W, Sharma P, Xu X, Ajayan P M, Wang J, and Liu Z 2016 Nat. Commun. 7 12357
[22] Susner M A, Chyasnavichyus M, McGuire M A, Ganesh P, and Maksymovych P 2017 Adv. Mater. 29 1602852
[23] Burr G, Durand E, Evain M, and Brec R 1993 J. Solid State Chem. 103 514
[24] Bourdon X, Maisonneuve V, Cajipe V B, Payen C, and Fischer J E 1999 J. Alloys Compd. 283 122
[25] Song W, Fei R, and Yang L 2017 Phys. Rev. B 96 235420
[26] Maisonneuve V, Cajipe V, Simon A, Von Der M R, and Ravez J 1997 Phys. Rev. B 56 10860
[27] Lai Y, Song Z, Wan Y, Xue M, Wang C, Ye Y, Dai L, Zhang Z, Yang W, Du H, and Yang J 2019 Nanoscale 11 5163
[28] Wei S H, Zhang S B, and Zunger A 1993 Phys. Rev. Lett. 70 1639
[29] Sun Z Z, Xun W, Jiang L, Zhong J L, and Wu Y Z 2019 J. Phys. D 52 465302
[30] Yang J C, He Q, Yu P, and Chu Y H 2015 Annu. Rev. Mater. Res. 45 249
[31] Anderson P W 1950 Phys. Rev. 79 350
[32] Goodenough J B 1955 Phys. Rev. 100 564
[33] Kanamori J 1959 J. Phys. Chem. Solids 10 87
[34] Seyler K L, Zhong D, Klein D R, Gao S, Zhang X, Huang B, Navarro-Moratalla E, Yang L, Cobden D H, McGuire M A, Yao W, Xiao D, Jarillo-Herrero P, and Xu X 2018 Nat. Phys. 14 277
[35] Sheppard D, Xiao P, Chemelewski W, Johnson D D, and Henkelman G 2012 J. Chem. Phys. 136 074103
[36] Cohen R E 1992 Nature 358 136
[37] Yin H M, Zhou H W, and Huang Y N 2019 Chin. Phys. Lett. 36 070501