## Abstract

The simultaneous presence of ferroelectricity and magnetism in multiferroics breaks both spatial inversion and time reversal symmetries at the macroscopic scale, which opens the door to many interesting phenomena and resembles the violation of these symmetries in particle physics. The symmetry breaking in multiferroics occurs spontaneously at phase transitions rather than at the level of fundamental interactions, and thus can be controlled. Moreover, each crystal is a universe in itself with a unique set of symmetries, coupling constants and ordered patterns, which presents plenty of opportunities to find and design materials with strong magnetoelectric coupling.

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## Introduction

The coupling between the electric and magnetic dipoles in multiferroics has little to do with Maxwell’s electrodynamics and is related to virtual charge fluctuations in Mott insulators and Fermi statistics of electrons that result in exchange interactions between spins, although relativistic effects do play an important role in multiferroics. The magnetoelectric coupling results in unusual responses, such as high sensitivity of the electric polarization to applied magnetic fields, and is of great practical importance: the voltage control of magnetism in multiferroic insulators can reduce dissipation losses and make magnetic devices energy-efficient^{1}.

Depending on the origin of electric polarization, multiferroic materials are divided into two groups^{2}: type-I multiferroics, in which ferroelectricity sets in above the magnetic ordering temperature and type-II multiferroics, in which the electric polarization is induced by ordered spins. The spin-induced polarization naturally gives rise to the magnetic control of ferroelectricity, e.g. rotation of the polarization vector and giant dielectric response triggered by an applied magnetic field^{3,4}, and to new optical phenomena, such as the electric excitation of spin waves^{5}. However, the electric polarization in type-II multiferroics is typically low and the magnetic orders that break inversion and induce the polarization usually appear at low temperatures. Type-I multiferroics show higher electric polarizations and magnetic ordering temperatures, but the coupling between the ferroelectricity and magnetism is not necessarily strong and the number of known magnetic ferroelectrics is still small.

Yet, the studies of multiferroics that have a long history and flourished in the last twenty years, provided solutions to some of the above-mentioned problems. Multiferroic hexaferrites, in which electric polarization is induced by spins forming a conical spiral state, order well above room temperature and exhibit strong magnetoelectric responses making it possible to control the electric polarization with low magnetic fields and to flip the magnetization by applying voltage^{6,7,8}. The electric control of magnetization was also achieved in another family of type-II multiferroics, rare earth orthoferrites, where both the electric polarization and weak ferromagnetic moment change sign across antiferromagnetic domain walls^{9,10}. Despite the absence of linear coupling between the magnetization and electric polarization vectors in BiFeO_{3}, the magnetization of this type-I multiferroic can be reversed with the applied electric field^{11}.

Recent studies of multiferroics also led to the discovery of many new materials and spectacular magnetoelectric phenomena. They improved our understanding of microscopic origins of the coupling between ferroelectricity and magnetism, the optical properties of multiferroics and topological defects in these materials. This research field has overlapped with spintronics and skyrmionics, and there are new ideas on how to integrate magnetoelectric and multiferroic materials into memory and data processing devices^{12}.

These are the main themes of the present review. Written by a theorist, it focuses on mechanisms for the magnetoelectric coupling and phenomenological description of multiferroic behaviors. On the other hand, I discuss existing multiferroic materials rather than large number of theoretical predictions. This review does not do justice to many interesting aspects of multiferroics, e.g. history, first-principle studies, multiferroic heterostructures and device applications, which are reviewed elsewhere^{2,13,14,15,16,17,18}.

## Symmetry considerations

The simultaneous breaking of inversion symmetry, *I*, and time reversal symmetry, *T*, can lead to the linear magnetoelectric effect (LME)^{19,20}, and to the magnetically-induced electric polarization in type-II multiferroics. These two phenomena have common microscopic origins, but they require different kinds of magnetic order.

Close to the magnetic transition, the LME is phenomenologically described by the free energy term,

that couples the magnetic order parameter, *L*, to external electric and magnetic fields, **E** and **H**. In addition to being odd under *I*, and *T*, *L* must be translationally invariant, i.e. the magnetic unit cell must coincide with the crystallographic unit cell. The electric polarization induced by an applied magnetic field is given by, \({P}_{i}=-\frac{\partial {f}_{{{{\rm{me}}}}}}{\partial {E}_{i}}={\alpha }_{ij}{H}_{j}\), where *α*_{ij} = *g*_{ij}*L* is the magnetoelectric tensor, and the electrically induced magnetization is \({M}_{i}=-\frac{\partial {f}_{{{{\rm{me}}}}}}{\partial {H}_{i}}={\alpha }_{ji}{E}_{j}\).

By contrast, multiferroics, in which the electric polarization emerges in absence of applied magnetic fields, require either multicomponent magnetic order parameters or several independent magnetic orders, which is a consequence of time-reversal invariance of microscopic interactions. In the E-phase of orthorhombic rare earth manganites, RMnO_{3}, the two kinds of up-up-down-down states of Mn spins (*↑**↑**↓**↓* state and *↓**↑**↑**↓* state) are described by the order parameters, *L*_{1} and *L*_{2}, forming a two-dimensional representation. Under inversion, *L*_{1} ↔ *L*_{2}, which together with other crystal symmetries allows for the magnetoelectric coupling,

that gives rise to the electric polarization along the *a* axis^{21}.

The magnetoelectric coupling,

describes the electric polarization induced by two independent magnetic orders, *L*_{+} and *L*_{−}, one of which is even and another is odd under inversion, \({L}_{\pm }\mathop{\to }\limits^{I}\pm {L}_{\pm }\)^{22}, so that *P*_{i} = *g*_{i}*L*_{+}*L*_{−}, is odd under inversion and even under time reversal, \({L}_{\pm }\mathop{\to }\limits^{T}-{L}_{\pm }\). Since these two orders break different symmetries, they appear at different temperatures. In the rare earth orthoferrite, GdFeO_{3}, an antiferromagnetic spin ordering of Fe ions, which does not break inversion and is weakly ferromagnetic, sets in at *T*_{Fe} = 661 K, whereas Gd spins order at *T*_{Gd} = 2.5 K. The latter ordering breaks inversion symmetry of the perovskite lattice and makes the material multiferroic^{9}. Equations (2) and (3) are typical of improper ferroelectrics^{23}. The time reversal symmetry breaking by a magnetic ordering in type-II multiferroics is, strictly speaking, not required: in the organic charge-transfer complex TTF-BA, the electric polarization is induced by the spin-Peierls ordering – the spin-singlet formation on every second bond of spin-\(\frac{1}{2}\) chains, which together with the on-site charge alternation along the chains (alternating donor and acceptor molecules) breaks inversion symmetry^{24}.

There is a large group of multiferroics, in which the simultaneous presence of two magnetic orders with opposite parities occurs naturally – spiral magnets with a spin modulation,

where **q** is the spiral wave vector and **e**_{1,2} are two orthogonal unit vectors. Here, \(\cos ({{{\bf{q}}}}\cdot {{{\bf{x}}}})\) is even under **x** → − **x** and \(\sin ({{{\bf{q}}}}\cdot {{{\bf{x}}}})\) is odd. In anisotropic magnets, the sinusoidal and cosinusoidal modulations become ordered at somewhat different temperatures, but at low temperatures the length of the ordered spin at all sites is approximately the same: *A*_{1} ≈ *A*_{2} ≈ *S*.

The mechanism responsible for the electric polarization in spiral magnets is the so-called inverse Dzyaloshinskii-Moriya (DM) interaction. Some 60 years ago Igor Dzyaloshinskii noted that an incommensurate spiral modulation of the magnetization, **M**, in non-centrosymmetric magnets can be phenomenologically described by the so-called Lifshitz invariants in free energy, \({M}_{i}\overleftrightarrow{{\partial }_{j}}{M}_{k}={M}_{i}\frac{\partial {M}_{k}}{\partial {x}_{j}}-{M}_{k}\frac{\partial {M}_{i}}{\partial {x}_{j}}\)^{25}, originating from the microscopic DM interactions of relativistic origin that favor non-collinear spins^{26}. In centrosymmetric magnets, Lifshitz invariants are forbidden, as they change sign under inversion, but they can be coupled to the electric field that is also odd under inversion. In particular, the magnetoelectric coupling,

is an invariant for any crystal lattice, because the product of two axial vectors, *M*_{i}*M*_{j}, transforms in the same way as the product of two polar vectors, *P*_{i}∂_{j} under all symmetry operations. The electric polarization induced by a spiral, \({P}_{i}\propto {g}_{ij}{M}_{i}\overleftrightarrow{{\partial }_{j}}{M}_{j},\) is then proportional to the Lifshitz invariant (inverse DM mechanism)^{27,28,29}.

In spiral antiferromagnets, e.g. DyMnO_{3} and TbMnO_{3}^{30}, the magnetization vector, **M**, is replaced by the Néel vector, **L**, that transforms in the same way as **M**, except that it acquires an extra minus sign when a symmetry transformation interchanges the spin-up and spin-down magnetic sublattices. Since this ‘sublattice sign’ is the same for all components of the vector **L,** *L*_{i}*L*_{j} transforms as *M*_{i}*M*_{j} and the inverse DM mechanism also works in antiferromagnets.

The substitution of Eq. (4) into Eq. (5) gives

where \({{{{\bf{e}}}}}_{3}=\left[{{{{\bf{e}}}}}_{1}\times {{{{\bf{e}}}}}_{2}\right]\) is the normal to the spiral plane. A cycloidal spiral with **e**_{3}⊥**q** induces an electric polarization in crystals of any symmetry, which makes the inverse DM mechanism ubiquitous and explains why there are so many spiral multiferroics^{31}.

Low-symmetry crystals allow for more diverse Lifshitz invariants and an electric polarization can also be induced by a helical spiral with **e**_{3}∥**q**^{32}. The delafossite, CuFeO_{2} (space group \(R\bar{3}m\)) allows for an additional magnetoelectric coupling,

A helical spiral with the wave vector along the two-fold symmetry axis, **q**∥**x**, induces the polarization along the spiral wave vector, **P**∥**q**, whereas for **q**∥**y**, the polarization induced by the helical spiral is still parallel to the two-fold symmetry axis and perpendicular to the wave vector.

An interesting new class of multiferroics are quasi-two-dimensional van der Waals magnets^{33}, which remain multiferroic even when thinned down to a single layer^{34,35}. In transition metal dihalides, the *a**b*-layers with triangular spin lattices are stacked along the *c*-axis and periodically modulated states result from the competition between nearest-neighbor ferromagnetic and third-nearest-neighbor antiferromagnetic intralayer exchange interactions. Ferroelectricity in NiBr_{2} and NiI_{2} is induced by in-plane cycloidal spirals through the inverse DM mechanism^{36,37}. On the other hand, MnI_{2} and CoI_{2} (space group \(P\bar{3}m1\)) show a helical spiral ordering. When the magnetic field rotates in the *a**b* plane, the polarization induced by the helical spiral in MnI_{2} rotates twice as fast in the opposite direction^{38}, which can be obtained from Eq. (7), where **M** is replaced by the Néel vector **L** (the interlayer exchange interaction is antiferromagnetic), assuming that the spiral wave vector follows the magnetic field: **q**∥**e**_{3}∥**H**.

## Magnetic frustration

Frustrated magnetism plays an important role in type-II multiferroics: it gives rise to magnetic orders that spontaneously break inversion symmetry and induce electric polarization. The spiral and up-up-down-down spin states, responsible for ferroelectricity in orthorhombic RMnO_{3} compounds, result from competing exchange interactions. The rare earth ions in orthoferrites, such as GdFeO_{3}, are located approximately in the centers of cubes formed by transition metal ions, which to a large extent cancels exchange fields induced by Fe spins on rare earth sites and allows for independent orders of rare earth spins that break inversion symmetry.

In the geometrically frustrated orthorombic rare earth manganites, RMn_{2}O_{5}, Mn spins form two types of zig-zag antiferromagnetic chains^{39}, described by the Néel vectors, **L**_{1} and **L**_{2}, one of which is even under inversion and another is odd^{40}. The interactions between the antiferromagnetic chains are frustrated, since the term \(({{{{\bf{L}}}}}_{1}\cdot {{{{\bf{L}}}}}_{2})\) is incompatible with inversion symmetry of the crystal lattice, but the magnetoelectric coupling, \(-g{E}_{b}({{{{\bf{L}}}}}_{1}\cdot {{{{\bf{L}}}}}_{2})\), is allowed by symmetry. The resulting electric polarization strongly depends on the relative orientation of the two Néel vector vectors. By contrast, the spin energy depends very little on the angle between **L**_{1} and **L**_{2}, which makes possible to rotate one of the Néel vectors unidirectionally through 360° by applying and removing magnetic field and leads to an unusual 4-state polarization hysteresis^{40}.

Since the magnetically-induced electric polarization is small, the presence of manifolds of low-energy spin configurations in frustrated magnets with the magnetoelectric coupling, is crucial for the electric control of magnetism. Multiferroic hexaferrites show exceptionally soft states with a conical spiral ordering of Fe spins. A magnetic field~200 Oe is sufficient to rotate the magnetization **M**∥**e**_{3} and the electric polarization induced by the spiral, results in record-high magnetoelectric responses and the voltage control of magnetization^{7,8}.

Hexaferrites and the high-temperature spiral multiferroic, YBaCuFeO_{5}, show that magnetic frustration does not necessarily make the spin ordering temperature low. Surprisingly, the Cu/Fe chemical disorder in the latter compound increases the stability region of the spiral state up to~400 K^{41}, which can result from long-ranged interactions between spin-structure deformations around impurity bonds^{42}.

## Microscopic mechanisms

The microscopic mechanisms for the magnetoelectric coupling are closely related to exchange interactions between spins, which involve hopping of negatively charged electrons between magnetic sites and virtual states with positively charged holes on ligand sites. The transfer of charge in intermediate states of the spin-exchange process induces electric dipole moments that depend on spin ordering. If the magnetic ordering breaks inversion symmetry, then the electric dipoles induced on different bonds and sites do not cancel each other. In addition, the ‘magnetic’ inversion symmetry breaking leads to polar shifts of ions, which lower the total energy and contribute to the net electric polarization.

Consider a generic spin Hamiltonian,

where the first term is the exchange interaction with the exchange constant \({J}_{ij}^{ab}\) (*a*, *b* = *x*, *y*, *z*) for spins on sites *i* and *j*, the second term is the single-ion anisotropy and the last term is the Zeeman interaction, \({g}_{i}^{ab}\) being the g-tensor. If local symmetry of bonds and sites allows for linear electric field-dependence of the coupling constants^{43}, then the magnetically-induced electric polarization is,

averaged over a magnetically ordered state (*V* is the sample volume). In particular, the bond electric dipole moment induced by the isotropic Heisenberg exchange interaction, \({J}_{ij}^{ab}={J}_{ij}{\delta }^{ab}\) is \({{{{\bf{d}}}}}_{ij}=-\frac{\partial {J}_{ij}}{\partial {{{\bf{E}}}}}({{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j})\) (the so-called symmetric exchange striction mechanism), whereas DM interactions described by the vector, \({D}_{ij}^{a}=\frac{1}{2}{\varepsilon }^{abc}{J}_{ij}^{bc}\) give rise to \({d}_{ij}^{a}=-\frac{\partial {{{{\bf{D}}}}}_{ij}}{\partial {E}^{a}}\cdot [{{{{\bf{S}}}}}_{i}\times {{{{\bf{S}}}}}_{j}]\).

For DM interactions mediated by a single ligand ion, \({{{{\bf{D}}}}}_{ij}\propto [{\hat{{{{\bf{r}}}}}}_{ij}\times {{{\bf{u}}}}]\), where \({\hat{{{{\bf{r}}}}}}_{ij}\) is the unit vector parallel to the line connecting sites *i* and *j* and **u** is the displacement of the ligand ion away from that line^{44}. To first order in the electric field, \({{{{\bf{D}}}}}_{ij}\propto [{\hat{{{{\bf{r}}}}}}_{ij}\times {{{\bf{E}}}}]\) and

This microscopic expression for the inverse DM interaction gives Eqs. (5) and (6) for a slowly varying magnetization. It was first derived in the framework of the spin current theory of magnetoelectric coupling^{45} that relates the electric polarization to the spin current, \({j}_{i}^{a}\): \({P}^{a}\propto {\epsilon }^{abi}{j}_{i}^{b}\). Persistent divergenceless spin currents flow in non-collinear spin states even in equilibrium. In the spiral state Eq. (4), \({j}_{i}^{a}\propto {q}_{i}{e}_{3}^{a}\), which gives Eq. (6). A more general expression was discussed in ref. ^{46}.

The single-ion magnetic anisotropy tensor \({K}_{i}^{ab}\) is sensitive to distortions of ligand cages surrounding magnetic ions. In addition, the hopping of electrons between the ligand and metal sites, which contributes to the crystal field splitting, involves polar virtual states (*d**p*-mechanism)^{47}, all of which leads to an electric dipole moment induced by a single spin. This mechanism is amplified by close energies of electron orbital states, resulting in an unquenched orbital moment that affects both the single-ion anisotropy and magnetic moment. The *E*-dependence of the *g*-tensor was shown to give a large contribution to the LME in LiFePO_{4}^{48} and is likely responsible for the large magnetoelectric effect observed in the ferrimagnetic state of the polar magnet, Fe_{2}Mo_{3}O_{8}^{49}.

The strength of magnetoelectric coupling scales with the strength of spin-spin interaction. *P* = 1 − 2 *μ*C ⋅ cm^{−2} measured in the collinear E-phase of YMnO_{3} thin film and bulk TbMnO_{3} under pressure originates from the isotropic Heisenberg exchange, which is usually the strongest exchange interaction^{50,51}. The largest electric polarization induced by a spin spiral, *P* ~ 0.3 *μ*C ⋅ cm^{−2}, was measured in DyMnO_{3} under an applied magnetic field^{30} and the ferroaxial helimagnet, CaMn_{7}O_{12}^{52}.

## Type-I multiferroics

The advantages of type-I multiferroics are the large electric polarization (of nonmagnetic origin) and high magnetic transition temperatures, as no magnetic frustration is required. Bringing together ferroelectricity and magnetism is a problem, since ferroelectricity usually involves transition metal ions with empty d-shells that are non-magnetic^{53}. An additional challenge on the way to the electric control of magnetism is the coupling between the electric polarization and magnetization in bulk and at domain walls.

Several ways to circumvent the incompatibility between ferroelectricity and magnetism proved to be effective. The covalent bonding between magnetic and ligand ions can lead to a polar lattice distortion, if the competing lattice instabilities are suppressed by strains and chemical substitutions^{54}, which explains ferroelectricity in Sr_{1−x}Ba_{x}MnO_{3} and Sr_{1−x}Ba_{x}Mn_{1−y}Ti_{y}O_{3}^{55,56} strongly coupled to an antiferromagnetic ordering of Mn spins. Ferroelectricity and magnetism coexist in compounds with both magnetic and non-magnetic transition metal ions, such as BiFeO_{3}^{57}, and improper ferroelectrics, e.g. hexagonal manganites, h-RMnO_{3}, in which the electric polarization is a by-product of a periodic lattice distortion^{58,59}.

It was suggested that ferroelectricity can control the weak ferromagnetic moment through a lattice distortion in the so-called hybrid improper ferroelectrics^{60,61}. In the double-layered perovskite, Ca_{3}Mn_{2}O_{7}, the electric polarization is a result of two coexisting periodic lattice distortions with the wave vector **Q** = (1/2, 1/2, 0) – tilts and rotatons of oxygen octahedra, described by the order parameters, respectively, Δ_{1} and Δ_{2}. Although Δ_{1} breaks inversion, it cannot be coupled to the uniform electric field because of the nonzero wave vector of the tilt-modulation, but the product of the two order parameters can:

i.e. the two coexisting lattice distortions induce an electric polarization. In addition, Δ_{1} affects the antiferromagnetic G-type ordering of Mn spins that sets in at a lower temperature. Importantly, the G-type ordering in bi-layered manganites breaks inversion and has the same wave vector **Q**, which allows for the coupling term,

where *L* is the Néel order parameter describing the antiferromagnetic ordering. This coupling term describes the weak ferromagnetic moment, *M* = *λ*Δ_{1}*L*, resulting from distortions of metal-ligand-metal bonds in the ferroelectric state, which modify DM interactions and force spins to cant. The sign change of Δ_{1} under an applied electric field reverses the weak ferromagnetic moment. Although the coexistence of magnetism and a polar lattice distortion in Ca_{3}Mn_{2}O_{7} and the related compounds, Ca_{3}Mn_{2−x}Ti_{x}O_{7} and (Ca_{x}Sr_{1−x})_{1.15}Tb_{1.85}Fe_{2}O_{7}, was established experimentally^{62,63}, the relatively high conductivity of these materials makes it difficult to reverse the electric polarization and magnetization with an applied electric field^{64}.

Although the linear coupling between the electric polarization and weak ferromagnetism in BiFeO_{3} is forbidden by symmetry^{60}, the electric reversal of the magnetization of a ferromagnetic layer deposited on BiFeO_{3} was demonstrated experimentally^{11}. The switching is made possible by a particular two-step domain switching process, resulting in correlated rotations of the electric polarization, Néel and magnetization vectors^{11}.

Strong magnetoelectric responses are observed in ‘type-I magnetoelectrics’ – polar materials showing large changes in the electric polarization in magnetically ordered states, such as Ni_{3}TeO_{6} near the spin-flop transition^{65} and Zn-doped honeycomb magnet, Fe_{2}Mo_{3}O_{8}^{49}. The latter compound shows a close competition between antiferromagnetic and ferrimagnetic phases with different electric polarizations. Remarkably, the antiferromagnetic state intervenes at the reversal of the magnetization of the ferrimagnetic state, resulting in sharp electric polarization peaks^{66}.

## Domain walls in multiferroics

Multiferroic materials show a great variety of domain walls, at which several order parameters change sign or direction, as it happens, for example, in hexagonal manganites, in which the electric polarization is induced by the lattice trimerization^{58}. The phase of order parameter describing the trimerization changes by 60^{∘} at a structural domain wall, which is accompanied by the electric polarization reversal across the wall and a rotation of antiferromagnetically ordered spins of Mn ions near the wall^{67,68}. The presence of 6 structural domains gives rise to intricate patterns of discrete vortices with an alternating electric polarization^{69}. Non-collinear spin textures of magnetic domain walls induce an electric dipole moment, allowing to move the domain walls in non-multiferroic magnets with inhomogeneous electric fields^{70}.

Switching of a ferroic order parameter by an applied field involves propagation of domain walls, e.g. walls separating the domains with the up- and down-magnetization in ferromagnets. The electric switching of magnetism in multiferroic memory devices^{1} hinges on dynamics of domain walls that are both ferroelectric and magnetic.

A remarkable electric control of magnetization was achieved in rare earth orthoferrites^{9,10}, in which the electric polarization appears when both Fe and rare earth spins are ordered and the magnetoelectric coupling is described by Eq. (3). The sign change of *L*_{+}, describing the weakly ferromagnetic ordering of Fe spins, by an applied electric field will reverse the weak ferromagnetic moment, *M* ∝ *L*_{+}. Whether this can be done depends on the electrically driven dynamics of two types of domain walls: (i) the domain wall in the Fe magnetic sublattice (Fe-DW), at which *L*_{+} changes sign and which is both ferroelectric and ferromagnetic domain wall, and (ii) the domain wall in the rare earth sublattice (R-DW), at which *L*_{−} changes sign and which is a purely ferroelectric domain wall. The electric switching of magnetization occurs if the Fe-DW moves under the applied electric field and R-DW does not. Unfortunately, the opposite happens in GdFeO_{3}, because of the low energy cost of the R-DW for relatively isotropic Gd spins. The replacement of Gd by a mixture of Ising-like Tb and Dy ions, which does not change the symmetry of the rare earth spin ordering, slows the R-DWs and allows for the electric field control of magnetization^{9,10}. The patterns of ferromagnetic and ferroelectric domain walls under applied electric and magnetic fields have been recently imaged optically^{71}.

The composite ferroelectric-ferromagnetic domain walls are also relevant for type-I multiferroics, in particular, the hybrid improper ferroelectric, Ca_{3}Mn_{2}O_{7}, with antiferromagnetic and two kinds of structural domain walls^{61}. The electric switching of magnetism would require propagation of only the tilt domain walls, which reverse the electric polarization and flip the weak ferromagnetic moment.

## Electromagnons

Electromagnons are magnons excited by the electric field of the light wave, first observed in the spiral state of GdMnO_{3} and TbMnO_{3}^{5}. The dynamic linear magnetoelectric response is described by two magnetoelectric tensors, \({\alpha }_{ij}^{{{{\rm{em}}}}}\) and \({\alpha }_{ij}^{{{{\rm{me}}}}}\), relating the electric polarization to the oscillating magnetic field and the magnetization to the electric field, respectively. The Onsager reciprocity relation for the linear response functions implies

where *ω* is the light frequency, **H** is the static magnetic field and \({L}^{{\prime} }\) is the time-reversed magnetic order parameter. The reversal of all spins in a spiral state under time reversal can be undone by the shift along the spiral wave vector by half a period of the spiral. The invariance under this combined symmetry operation implies

for an incommensurate spiral state in the absence of an applied magnetic field.

Interestingly, the excitation of magnons with both electric and magnetic components of the light wave can lead to the non-reciprocal directional dichroism – the difference in absorption for opposite light propagation directions, proportional to \({{{\rm{Im}}}}({\alpha }_{ij}^{{{{\rm{em}}}}}+{\alpha }_{ji}^{{{{\rm{me}}}}})\)^{72}. The non-reciprocal directional dichroism requires breaking of both inversion and time reversal symmetries and in the spiral state, it is only observed under an applied magnetic field [see Eqs. (13) and (14)].

The linear coupling between magnons and photons can be obtained from Eq. (9) by the substitution,

where \(\left\langle {{{{\bf{S}}}}}_{i}\right\rangle\) is the average spin in an ordered magnetic state and *δ***S**_{i}(*t*) ∝ *e*^{−iωt} is the spin component oscillating with the frequency *ω* of the light wave that induces electric dipole oscillations^{73}. The electromagnon wave vector equals that of the ordered magnetic state and the lowest-energy electromagnon modes in spiral multiferroics are collective rotations of the spiral plane, which result in oscillations of the electric polarization induced by the inverse DM mechanism that rotates together with the spiral plane^{74}. Much stronger electromagnon peaks are observed at frequencies of optical magnon modes that induce an oscillating electric polarization through the symmetric exchange-striction mechanism^{73,75,76}. The large spectral weight of the electromagnon peaks in the spin spiral state of orthorombic RMn_{2}O_{5} manganites leads (due to Kramers-Kronig relations) to a giant step-like increase of the static dielectric constant at the transition to the spiral state^{75}. The fact that strong electromagnon peaks are observed in non-collinear magnets is not accidental. Since \(\delta {{{{\bf{S}}}}}_{i}(t)\perp \left\langle {{{{\bf{S}}}}}_{i}\right\rangle\), the symmetric exchange striction mechanism for electromagnon absorption only works in non-collinear spin states: the oscillating bond electric dipole, **d**_{ij}(*t*) ∝ (〈**S**_{i}〉 ⋅ *δ***S**_{j}(*t*)) + (*δ***S**_{i}(*t*) ⋅ 〈**S**_{j}〉), is zero for collinear spins.

It was suggested theoretically that electromagnons can be used to reverse optically the rotation direction of spins in multiferroic spirals^{77}. Although a 180°-rotation of the spiral plane requires strong laser fields, a small rotation of the spiral plane in TbMnO_{3} due to the electromagnon absorption was observed^{78}. Electromagnons have also been used for the electrical switching of natural optical activity of the spiral multiferroic, CuO^{79}.

## Magnon control

Spin waves in (anti)ferromagnets can be controlled electrically through the dependence of spin-spin interactions and spin anisotropy on the applied electric field [see Eq. (9)], as was suggested theoretically^{80,81}. The electric field-dependent shifts of magnon frequencies have been observed in the antiferromagnetic spiral state of BiFeO_{3}^{82}. The strong electric field-dependence of the magnetic anisotropy of Fe ions^{83}, makes possible to tune magnon frequencies electrically by as much as 30%. The probability to excite magnons in the multiferroic antiferromagnet, SmFe_{3}(BO_{3})_{4}, was controlled by an applied electric field that affected relative weights of magnetically ordered states with six different orientations of the Néel vector^{84}. The electrical switching of clamped ferroelectric-antiferromagnetic domains in BiFeO_{3} films was used to change the magnon polarization in thermal spin currents detected by the inverse spin Hall effect^{85}. The Néel vector direction in thin flims of the linear magnetoelectric material, Cr_{2}O_{3}, was controlled by voltage near the spin-flop transition, which made possible to change the polarization of thermal magnons and switch on and off the Spin Seebeck effect signal^{86}.

## Skyrmion control

Magnetic skyrmions are string-like topological defects in a uniform ferromagnetic state. Spins in the skyrmion rotate both in the azimuthal and radial directions, making it difficult to unwind them. Small radius and stability resulting from nontrivial topology make skyrmion a good candidate for information carrier in magnetic memory and data processing devices^{87}. By virtue of its nontrivial topology, skyrmion induces a quantized flux of an effective magnetic field that skew-scatters electrons and magnons and results in topological Hall effects^{88}.

Room-temperature skyrmions, interesting for applications, have been found in the family of chiral Co-Zn-Mn alloys^{89}, where they are stabilized by DM interactions favoring spin rotations in several different directions, and in multilayers of ferromagnetic and heavy-metal materials, in which non-collinear magnetism is favored by interfacial DM interactions^{90}. Skyrmions in conducting magnets are moved by electric currents exerting a spin-transfer torque on skyrmions, or by fluxes of angular momentum flowing into a magnet through the interface with a heavy-metal material. The currents required to set skyrmions into motion are rather high and the voltage control of these topological defects in multiferroic insulators can help to reduce energy consumption of magnetic devices.

The magnetically-induced electric polarization in the chiral cubic Mott insulator, CuO_{2}SeO_{3}^{91}, makes the skyrmion crystal state in the electric field applied along the polarization more stable than the competing conical spiral state^{92} and results in large-angle rotations of the skyrmion crystal relative to the crystal lattice^{93,94}. However, piezoelectricity of CuO_{2}SeO_{3} makes it difficult to distinguish effects of the magnetoelectric coupling from those mediated by strains coupled to the electric field. Creation of skyrmions by electric field pulses was demonstrated both in CuO_{2}SeO_{3} and in heterostructures of ferromagnetic and ferroelectric materials^{95,96,97}.

Nanosized skyrmions resulting in the giant Topological Hall and Nernst effects have been recently observed in a number of centrosymmetric compounds^{98,99,100}. In contrast to chiral magnets, the direction of spin rotation in spiral phases of centrosymmetric magnets is arbitrary, which allows for more versatile spin textures, such as the three-dimensional crystal of magnetic hedgehogs and anti-hedgehogs observed in the itinerant magnet, SrFeO_{3}^{101} and new collective degrees of freedom of topological magnetic defects.

Due to the inverse DM mechanism, non-collinear spin textures induce an electric charge density proportional to topological charge density and the skyrmion acquires an out-of-plane electric dipole moment that depends on the skyrmion helicity (the angle between in-plane spin components and the radial direction)^{102}. The oscillations of the skyrmion helicity angle in the chiral CuO_{2}SeO_{3}, excited both electrically and magnetically, were predicted to result in strong nonreciprocal directional dichroism of microwaves^{103}. In frustrated magnets, helicity is a low-energy collective skyrmion mode and an oscillating electric field can excite large-amplitude dynamics of this mode – the precession of spins in the skyrmion. Owing to the coupling between the helicity and translational dynamics, an oscillating electric field can excite rotation of skyrmions relative to each other and a unidirectional motion of skyrmion-antiskyrmion pairs^{102}.

## Summary

To conclude, recent work defied many preconceptions concerning the single-phase multiferroic materials. Room-temperature multiferroics with a magnetically-induced electric polarization have been found and the electric control of magnetism was demonstrated in this class of materials. Low electric polarizations can be compensated by extreme softness of magnetic states in frustrated magnets with low magnetic anisotropy, which can lead to strong magnetoelectric responses. Ferroelectricity and magnetism in type-I multiferroics are not necessarily mutually exclusive and their coexistence can be controlled by strains, chemical substitutions, ions with lone pairs, charge ordering, etc. New ways to couple these two orders have been proposed. Clamping of ferroelectric and magnetic domain walls, important for magnetoelectric switching, was found in both type-I and type-II multiferroics. These developments give hope that robust electric control of magnetism at room temperatures will be achieved soon.

## Data availability

No datasets were generated or analysed during the current study.

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Mostovoy, M. Multiferroics: different routes to magnetoelectric coupling.
*npj Spintronics* **2**, 18 (2024). https://doi.org/10.1038/s44306-024-00021-8

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DOI: https://doi.org/10.1038/s44306-024-00021-8