The spontaneous, uniform orientation of atomic or molecular magnetic moments to generate what is collo­ quially called a magnet (more correctly, a ferromagnet) has been explored for more than 2,500 years. Barely a century ago, it was discovered that spontaneous order­ ing of electric dipole moments can occur as well1. This phenomenon was named ferroelectricity because of the analogies to ferromagnetism, such as the hysteretic switching between two stable states in an external field. Although the technological merits of ferromagnetism and ferroelectricity are quite different, attempts were made to combine them in the same phase of a mat­ erial to create a so­called multiferroic material (BOX 1). Multiferroic materials are interesting mainly for two reasons. On the one hand, they make it possible to exploit the functionalities of both orders; for example, a magnetic bit may be complemented by an electric bit to establish a four­state memory element. On the other hand, a coupling between the ferromagnetic and the ferro electric states might induce novel functionalities not present in either state alone. The control of the magnetic properties by electric fields instead of magnetic fields is an example of the advantages that multiferroic materials can offer. In the reading and writing of a magnetic bit, if a voltage pulse can be used instead of a magnetic­field­ generating electric current, the waste heat and relatively long build­up time associated with electric currents are avoided. Multiferroics may thus lead to faster, smaller, more energy­efficient data­storage technologies.
The field of multiferroics covers aspects ranging from technological applications to abstract problems of fundamental research. In addition, the study of multi­ ferroics increasingly influences neighbouring research
areas, such as complex magnetism and ferroelectric­ ity, oxide heterostructures and interfaces, and also seemingly remote subjects such as cosmology. In this Review, we give an overview of the twists and turns in the development of the diverse field of multiferroics, and we discuss the trends and challenges that will define its future. Readers looking for a more comprehensive or more technical coverage are referred to more extensive general reviews2,3, or to reviews on particular aspects that are highlighted in further sections.
We begin with a brief survey of the early days of multi­ ferroics. The realization that in some important classes of materials, magnetic and electric long­range order compete with each other4 may be regarded as the mile­ stone separating historical from contemporary research in this field. We continue with an overview of mecha­ nisms permitting the coexistence of magnetic and ferro­ electric order, and evaluate their potential for inducing pronounced magneto electric coupling effects (BOX 1). We then scrutinize heterostructures and, in particu­ lar, interfaces that introduce additional functionalities, bringing multi ferroics closer to device applications. Domains and domain walls are also discussed: any type of coupling between magnetic and electric long­range order in a multiferroic material has its roots in the cou­ pling between the magnetic and electric domains. We then have a closer look at the non­equilibrium dynam­ ics of multiferroic materials, because, considering that the focus in multi ferroics is on the manipulation of the magnetic order by electric fields, it is very important to understand the processes and timescales governing the magnetoelectric coupling. Important progress in the understanding of the coupling of magnetic and electric
Department of Materials, ETH Zürich, Vladimir-Prelog-Weg 4, 8093 Zürich, Switzerland.
All authors contributed equally to this work. Correspondence to M.F.  manfred.fiebig@mat.ethz.ch
Article number: 16046 doi:10.1038/natrevmats.2016.46 Published online 5 Jul 2016
The evolution of multiferroics Manfred Fiebig, Thomas Lottermoser, Dennis Meier and Morgan Trassin
Abstract | Materials with a coexistence of magnetic and ferroelectric order — multiferroics — provide an efficient route for the control of magnetism by electric fields. The study of multiferroics dates back to the 1950s, but in recent years, key discoveries in theory, synthesis and characterization techniques have led to a new surge of interest in these materials. Different mechanisms, such as lone-pair, geometric, charge-ordering and spin-driven effects, can support multiferroicity. The general focus of the field is now shifting into neighbouring research areas, as we discuss in this Review. Multiferroic thin-film heterostructures, device architectures, and domain and interface effects are explored. The violation of spatial and inversion symmetry in multiferroic materials is a key feature because it determines their properties. Other aspects, such as the non-equilibrium dynamics of multiferroics, are underrated and should be included in the topics that will define the future of the field.
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S i
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states has often been introduced by symmetry consider­ ations; for this reason, we pay some attention to funda­ mental symmetry properties of multiferroic materials. Many of the unusual properties of these materials are linked to the unconventional symmetry resulting from the simultaneous presence of magnetic and electric long­ range order. In turn, there are compounds displaying the symmetry of multiferroic materials, and the material properties permitted by this symmetry, but no multifer­ roic order. These systems are equally worth exploring, which we cover in the section devoted to research areas that were significantly influenced by the existence of multiferroics, even though multiferroicity in itself has a minor role in them. Finally, we identify major questions and challenges that will continue to influence research in multiferroics. Fundamental aspects, as well as the efforts to obtain working multiferroic devices, are covered in this Review.
Seeding the field Efforts to combine magnetic and ferroelectric order in a single compound were first undertaken in the former Soviet Union. In 1958, Smolenskii and Ioffe5 suggested the introduction of magnetic ions into ferroelectric perovskites to create solid solutions hosting magnetic long­range order without losing the ferroelectric order. However, the most intensely investigated compounds were boracites, such as Ni3B7O13I, in which a pro­ nounced linear magnetoelectric effect with hysteretic switching of multiferroic domains by electric or mag­ netic fields was observed6. The experimental, theoretical and applied achievements of these early days of the field are summarized in REFS 7,8.
Two precursurs of the present boom in multiferroics are noteworthy. First, in 1978, Newnham and co­ workers9 reported that a spiral­like arrangement of magnetic moments in Cr2BeO4 breaks spatial inversion
Box 1 | Important terms and definitions in the field of multiferroics
Ferroic Ferroic materials display long-range order with respect to at least one macroscopic property, and they develop domains that can be switched by a conjugate field108,109, as shown in the figure. Occasionally, the term primary ferroic167 is used to indicate ferromagnetic, ferroelectric, ferroelastic and ferrotoroidic order.
Magnetoelectric Originally, only materials in which a magnetic (electric) field induces a proportional polarization (magnetization) were referred to as linear magnetoelectrics, as seen in the figure (the prefix linear, which is often omitted, distinguishes this effect from higher-order effects with nonlinear relations). Multiferroics with magnetic and electric order do not necessarily permit the linear magnetoelectric effect (for example, hexagonal YMnO3), and not all materials displaying the linear magnetoelectric effect are multiferroic (for example, Cr2O3). Nowadays, the term magnetoelectric usually refers nonspecifically to any type of coupling between magnetic and electric properties.
Multiferroic When introduced10, the term multiferroic referred to materials with a coexistence of two or more primary ferroic orders in the same phase. Its present use is different; it indicates a coexistence of ferroelectric and ferro-, ferri- or antiferromagnetic order in single- or even multiphase materials. The single-phase compounds are sometimes more specifically referred to as magnetoelectric multiferroics in anticipation of a behaviour as seen in the figure. If the magnetic and ferroelectric orders occur independently, the multiferroic material is denoted as type I. If the ferroelectric and magnetic transitions emerge jointly, the multiferroic is of type II. The two types are depicted in the figure. Note that the term multiferroic prevailed from about the year 2000, whereas terms like ferroelectromagnetic were in use before.
Magnetodielectric This term denominates a material in which the dielectric function, ε, changes in a magnetic field. Because the electrical capacitance, C, is proportional to ε, the term magnetocapacitive is also used. In contrast to the linear magnetoelectric effect, which parameterizes a well-defined relation between magnetic and electric fields, the magnetodielectric effect represents a response function for which this relation is ambiguous.
Domains and domain walls Domains are regions with a uniform orientation of the relevant order parameter: for example, the polarization or the magnetization. At least two orientations of the order parameter (domain states) are allowed for any ferroic material; thus, a typical ferroic material consists of multiple domains, each representing one of the allowed domain states. The interfaces between domains are called domain walls. They can have a width ranging from less than 1 nm to more than 100 nm. They denote the region across which the order parameter reorients between adjacent domains.
E, applied electric field; H, applied magnetic field; M, magnetization; P, polarization; Si, spin at site i.
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symmetry, as does the electric polarization that is thereby induced. In their analysis, the authors foreshadowed much of the physics behind a new type of magnetically driven (improper) ferroelectricity that, a generation later, would be exploited to obtain multiferroics with strong magnetoelectric interactions. Second, in 1993, many of the phenomena, systems and concepts at the base of con­ temporary multiferroics research, including the inven­ tion of the term multiferroic itself, were formulated at a conference on magnetolelectric phenomena — even today, its proceedings are a fascinating read10.
In 2000, Spaldin revisited the original idea of Smolenskii and Ioffe and explained the reason why the ferroelectric and magnetic orders obstruct each other in crystals with the versatile perovskite structure that is technologically relevant4. In perovskites, the ferro­ electric state emerges because the electron clouds of neighbouring ions hybridize, which supports off­ centred ions; this type of ferroelectricity is called dis­ placive ferroelectricity and is particularly energetically favourable when the 3d shell is empty. By contrast, magnetic transition­metal ordering requires partially filled 3d shells — the contradiction is obvious. This realization triggered an intensive search for materials in which ferro electricity is driven by other, nondisplacive mechanisms that are compatible with magnetic order or that do not have the perovskite structure. The first breakthrough was the discovery of pronounced mag­ netoelectric interactions in hexagonal (h­) YMnO3 (REF. 11), orthorhombic (o­) TbMnO3 (REF. 12) and TbMn2O5 (REF. 13). In these last two materials, the magnetoelectric interaction originates from non­cen­ trosymmetric spin textures that induce a magnetically controllable electric polarization. Contrary to the earlier work on Cr2BeO4 (REF. 9), these discoveries inspired the concerted action of different communities — materials science, condensed matter physics and materials theory — that resulted in an impressive expansion of the field of multiferroics.
Mechanisms supporting multiferroicity Following the aforementioned study on perovskite multi­ ferroic materials, the search for ferroelectric materials of the nondisplacive type that permit the coexistence of ferroelectric and magnetic order became the main focus of the field. We distinguish four main classes of these materials on the basis of the mechanisms inducing the multiferroicity (FIG. 1). Other reviews present detailed surveys of these classes14,15 and their distinction by diffraction techniques16.
Ferroelectricity may be driven by electronic lone pairs, geometric effects, charge order or magnetism. In the first three classes, the magnetic and ferroelectric orders occur independently, and the multiferroic material is denoted as type I. In the last class, the ferro electric and magnetic transitions emerge jointly, in which case the multiferroic material is type II (BOX 1).
Lone-pair mechanism. The lone­pair mechanism is based on the spatial asymmetry created by the ani­ sotropic distribution of unbonded valence electrons
around the host ion (FIG. 1a). This mechanism is respon­ sible for the room­temperature ferroelectricity observed in BiFeO3. In this material, a pair of Bi3+ valence elec­ trons in the 6s orbital is not involved in sp hybridiza­ tion and creates a local dipole, yielding a spontaneous polarization17 of ~100 μC cm−2 below the Curie temper­ ature18, TC = 1,103 K. A long­range periodic antiferro­ magnetic structure arises below the Néel temperature19, TN = 643 K. Among the lone­pair systems, BiFeO3 is the only room­temperature single­phase multiferroic material. It has large and robust electric polarization and pronounced magnetoelectric coupling.
Geometric ferroelectricity. Space­filling effects and geometric constraints can cause structural instabilities in materials. If such steric effects, rather than bond chem­ istry, lead to ionic shifts that result in the formation of a polar state, the term geometric ferroelectricity can be used (FIG. 1b). For example, in h­RMnO3 (R = Sc, Y, In or Dy–Lu), unit­cell tripling drives the emergence of a ferroelectric order20–22 at TC ≥ 1,200 K with a polarization Ps = 5.6 μC cm−2 (REF. 23), followed by magnetic ordering at TN ≤ 120 K (REF. 24). A similar behaviour is observed in h­LuFeO3 thin films, which exhibit a larger magnetic moment and room­temperature magnetic order25, but magnetoelectric coupling remains to be demonstrated in this material. Another example is BaNiF4, in which an asymmetry between Ba2+ and F− sites leads to a spon­ taneous electric polarization26. Despite its small value (~0.01 μC cm−2), this polarization is of interest because it couples to a weak ferromagnetic moment, which can thus be reversed along with the electric polarization27. Finally, a cooperation between two nonpolar lattice modes drives a ferroelectric polarization in Ca3Mn2O7; this polarization can interact with the canted magnetic moments of the compound28.
Charge ordering. Valence electrons can be distributed non­uniformly around their host ions in the crystal lat­ tice to form a periodic superstructure. For example, it was suggested that the Fe atoms in LuFe2O4 may form a superlattice with an alternating sequence of Fe2+ and Fe3+ ions29. This kind of charge ordering might be the source of an electric polarization and, hence, ferro­ electricity30 (FIG. 1c). LuFe2O4 is a prime candidate for charge­order­driven multiferroicity, but after a decade of research the occurrence of ferroelectricity in this material is still questioned29,31. Mixed manganites, such as Pr1 − xCaxMnO3, were also taken into consideration32 but did not attract broader attention. For now, charge­ order­driven multiferroicity essentially remains at the stage of an interesting concept.
Spin-driven mechanisms. Magnetic order can break inversion symmetry. The interaction of spins and charges may transfer the non­centrosymmetry from the magnetic to the electric lattice, driving the formation of a polar state. These magnetically induced, so­called improper ferroelectric materials represent the ultimate shift away from displacive ferroelectrics, in which mag­ netic ordering is inhibited, towards materials in which
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b Geometric ferroelectricity
c Charge ordering
Rare-earth ion Mn3+
the electric polarization is induced by the magnetic ordering. So far, three main routes for the creation of this kind of multiferroicity have been established (FIG. 1d), as discussed in specific reviews on the topic33–35.
The most intensely discussed mechanism is the so­called inverse Dzyaloshinskii–Moriya (DM) interaction. Whereas in the DM interaction36 a non­ centrosymmetric crystallographic environment pro­ motes an antisymmetric magnetic interaction, in the inverse DM interaction an acentric spin structure drives a non­centrosymmetric displacement of charges37,38. Spin–orbit coupling is crucial for both the DM and inverse DM interaction. The polarization resulting from the inverse DM interaction is essentially determined by the optimization of the spin configuration from the point of view of antisymmetric exchange, expressed by the antisymmetric product Si × Sj of neighbouring spins Si,j. It yields a one­to­one correlation between (antiferro­)magnetic order and electric polarization. Multiferroicity of this type was first found in Cr2BeO4
(REF. 9), o­TbMnO3 (REF. 12) and CaMn7O12; this last material arguably39 exhibits the highest polarization achieved so far with this mechanism (0.3 μC cm−2)40.
In contrast to the DM interaction, the Heisenberg­like exchange striction describes an acentric displacement of charges derived from the optimization of the symmet­ ric spin product Si · Sj (REF. 41). Ferroelectricity gener­ ated by this kind of displacement was first observed in TbMn2O5 (REF. 13). The dominance of the non­relativistic symmetric exchange over the relativistic antisymmet­ ric mechanism (|Si · Sj| > |Si × Sj|) is well represented in o­TbMnO3, in which an order­of­magnitude increase in the polarization is obtained when a cycloidal order (parameterized by Si × Sj) transforms into a collinear antiferromagnetic order (parameterized by Si · Sj) under pressure42. In general, inversion­symmetry­violating magnetic order may occur in several ways; thus, the existence of spin distributions that could promote ferro­ electricity more effectively than the currently known arrangements is possible.
Figure 1 | Mechanisms promoting the coexistence of magnetic and electric long-range order. a | Lone-pair ferroelectricity in BiFeO3. Ferroelectricity originates from two Bi3+ electrons that shift away from the Bi3+ ion and towards the FeO6 octahedra, giving rise to a spontaneous polarization P along the [111] direction. The lone pair is visualized by the isosurface (red) of the electron localization function of ferroelectric BiFeO3. b | Geometrically driven ferroelectricity in hexagonal (h-) RMnO3 emerges from a tilt and deformation of MnO5 bipyramids, which displace the rare-earth ions as indicated by the arrows, leading to a spontaneous polarization along the [001] axis. c | Charge ordering in LuFe2O3 creates alternating layers with Fe2+/Fe3+ ratios of 2:1 and 1:2. This was argued to create a spontaneous polarization between the two layers, which is oriented parallel to the arrow. d | Mechanisms for spin-induced ferroelectricity. Polar displacement is induced by antisymmetric spin exchange interactions (inverse Dzyaloshinskii–Moriya interaction; top panel) as observed, for example, in orthorhombic (o-) RMnO3: the polarization vector is P ∝ eij × (Si × Sj), where eij is the unit vector connecting neighbouring spins and Si,j are the spins at neighbouring sites i and j. Ferroelectricity arises from symmetric spin exchange in Ca3CoMnO6 (REF. 168) shown in the middle panel, with P ∝ Rij(Si · Sj), where Rij denotes the direction along which the magnetostriction occurs. Spin-driven modulations of the chemical bond between magnetic 3d orbitals and ligand 2p orbitals (indicated by grey clouds) yield a spontaneous polarization along the bond direction in delafossites, such as CuFeO2, as expressed by the relation P ∝ (Si · eij)
2eij (bottom panel). Part d is adapted with permission from REF. 33, Institute of Physics, and from REF. 168, American Physical Society.
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Pressure
Finally, in delafossite systems, such as CuMO2 (M = Fe or Cr), a spontaneous polarization P ≤ 0.03 μC cm−2 is induced by a screw­like spin structure with Si × Sj = 0. This polarization is driven by a variation in the metal–ligand hybridization as a function of spin–orbit coupling43.
Comparison between the different mechanisms. The polarization values resulting from these four mech­ anisms are compared in FIG. 2. So far, the lone­pair mechanism has been the most successful in yielding multiferroicity with the right characteristics for device applications; however, BiFeO3 is the only known mate­ rial to be multiferroic because of this mechanism. In contrast, many spin­driven multiferroic materials are being discovered, and robust room­temperature sys­ tems appear to be within reach40,42,44. This hope is fuelled by the general nature of the driving mechanism itself, which leaves ample room for improvement, using chem­ ical doping, pressure effects and strain, towards higher ordering temperatures or polarization values.
Theory and experiment indicate that there are more mechanisms resulting in multiferroicity than the ones dis­ cussed so far. For example, ferroelectrically induced mag­ netic order was predicted for LiNbO3­type structures45. Here, a weak magnetization can occur because a polar state breaks the inversion symmetry between antiferro­ magnetically aligned spins. Some evidence for this kind of multiferroicity was reported46 for FeTiO3, but it requires additional verification, and the coupled magneto electric switching still has to be demonstrated.
Composite multiferroics. So far, the focus has been on systems permitting the coexistence of ferroelectric and magnetic order in a single material. The alternatives are hybrid systems. A return to the historical idea of alloy­ ing ferroelectrics with magnetic ions led to the discovery of new multiferroic materials with a magnetoelectric response at room temperature47, which can become par­ ticularly pronounced near phase boundaries48. Another possibility is to use composite multiferroic materials, in which a ferromagnetic magnetostrictive and a ferro­ electric piezoelectric constituent are merged in a granu­ lar or layered form. In constituent 1, magnetostriction denotes the induction of strain by a magnetic field; this strain is transferred via strain coupling between the constituents to constituent 2, where it is converted into a voltage via the piezoelectric effect49. The result­ ing magneto electric coupling can be 108 times larger than that of single­phase multiferroics50; however, it is observed only at microwave frequencies and in a very limited range of operation conditions. The other possible strategy is to use strain to change the size and isotropy of unit cells to destabilize a centrosymmetric structure in favour of a polar phase that emerges independently of the magnetic order. Multiferroicity in Ba­alloyed bulk SrMnO3 was obtained this way — the strain was exerted chemically through the large size of Ba3+ ions51. However, precise control of the strain value requires thin­film architectures, which are discussed below.
Thin films and heterostructures Oxide thin films can be grown layer by layer with atomic­ scale precision, thus much more accurately and control­ lably than bulk crystals. At first, the obvious goal was the reproduction of multiferroic bulk phases in thin films and their switching in electric fields — this was accomplished in compounds such as o­TbMnO3, h­RMnO3 and BiFeO3
(REFS 17,52,53). An advantage of thin films is that battery voltages applied across them can generate the electric fields required for magnetoelectric phase control. New phases that are specific to thin films and heterostructures were later explored. For multiferroics, this approach was spectacularly successful in the case of BiFeO3 thin films and heterostructures, which were grown in a large range of structural configurations that are inaccessible in bulk crystals. These structures have already been reviewed in detail54–56; in this Review, we focus on a general classifi­ cation of novel material phases and coupling effects that may arise in heterostructures. In particular, the effects of strain, heteroepitaxy and interfaces in thin­film architec­ tures open unprecedented opportunities for the develop­ ment of multiferroic heterostructures and interface states with potential for device applications. The interfaces between different phases can either transfer the interac­ tion between the constituents of the system or have an active role in determining the properties of the material.
Ferromagnetic–multiferroic heterostructures. A major drawback of almost all bulk multiferroics is their antiferro magnetic order. In contrast to ferromagnets, the lack of a macroscopic magnetization makes them techno­ logically difficult to exploit. However, this shortcoming
Figure 2 | Types of single-phase multiferroic materials with their maximum polarization values. Ferroelectricity may be driven by electronic lone pairs, charge ordering, geometry and magnetism. BiFeO3 displays the largest spontaneous polarization (relaxed, Ps ≈ 100 μC cm−2; strained, Ps ≤ 150 μC cm−2)17. The polarization of LuFe2O4 (25 μC cm−2)29 is still controversial. Hexagonal (h-) RMnO3 develops a polarization of about 5.6 μC cm−2 (REF. 23). Among the spin-driven ferroelectrics, symmetric Heisenberg-like exchange striction leads to a larger polarization than antisymmetric Dzyaloshinskii–Moriya exchange. This is reflected by orthorhombic (o-) TbMnO3, which experiences a transition from a spiral order (Ps ≤ 0.1 μC cm−2) to a collinear antiferromagnetic order (Ps ≈ 1 μC cm−2) under pressure41. The largest spin-spiral-driven polarization has been claimed for CaMn7O12 (Ps ≈ 0.3 μC cm−2)40.
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of the antiferromagnetic state may be overcome by cou­ pling it to a ferromagnetic state. One possible solution is to exploit exchange bias — the displacement and harden­ ing of a ferromagnetic hysteresis induced by the presence of an adjacent antiferromagnet. This is the mechanism used in the read–write heads of computer hard disks. Connecting an electrically­tuneable antiferromagnet with a ferromagnetic layer could make it possible to manipulate the magnetic exchange bias with an electric voltage and to electrically shift the ferromagnetic hyster­ esis loop. This mechanism could form the basis of low­ energy magnetoelectric memories57–59. An important step in this direction was the low­temperature demon­ stration of electrically tuneable pinning of ferromagnetic domains, which resulted in a voltage­controlled modi­ fication of the exchange bias in h­YMnO3–Permalloy60. Following predictions of magnetoelectric coupling61 (and possible weak magnetization) in BiFeO3, a modification of the exchange bias, albeit irreversible, was verified at room temperature in BiFeO3–CoFe (REF. 62).
Another possible method to avoid the disadvantages of the antiferromagnetic state is to transduce it into a ferro­ magnetic state by establishing a rigid coupling between the antiferromagnetic order parameter of the multi ferroic constituent and the order parameter of an adjacent ferro­ magnetic constituent. An important breakthrough was the observation of a voltage­induced rotation of the magnetization in a microscopic BiFeO3–CoFe dot63, which eventually led to the demonstration of repeatable magnetization reversal by an electric field at room tem­ perature — the holy grail of research on magnetoelectric multiferroics64.
Aside from their technological appeal, these results are remarkable because they trace the complex magneto­ electric switching behaviour back to the formation of domains. Ultimately, domain formation is the most signif­ icant factor for the response and performance of devices. All recent experiments on low­energy magnetization reversal aim to establish a one­to­one correlation between domains in ferromagnetic and multiferroic materials64,65.
Strain engineering. Another tool for the engineering of multiferroic thin films is epitaxial strain. By selecting the right substrate, a wide range of tensile and compressive strains can be obtained. The associated modification of the lattice constant can induce a transition to new mate­ rial phases or at least modify the existing ones. By using a ferroelectric and, hence, piezoelectric material as a sub­ strate, strain can even be reversibly tuned post­growth within a 0.1% window using an external voltage66.
The atomic arrangement and the strain provided by the crystal lattice of the substrate can support the growth of otherwise unstable multiferroic phases67; it is also used to pattern the distribution of ferroelectric domains and domain states of multiferroics59,68. Strain was used to demonstrate the correlation between ferro­ electric and magnetic spiral order in BiFeO3 (REF. 69), a result that was sought after for a long time. Strain can even induce ferroic order in otherwise non­ferroic compounds: polar order was induced in SrTiO3 (REF. 70) and, coexisting with magnetic order, in EuTiO3 (REF. 71)
and SrMnO3 (REF. 72), whereas ferromagnetism was induced in LuMnO3 (REF. 73). Strain also controls the density and distribution of vacancies, which represent a powerful degree of freedom for the modification of material properties72.
Finally, strain can be used to couple ferroelectric domains to ferromagnetic domains across an inter­ face through magnetostrictive and magnetoelectric coupling74,75 (FIG. 3b). This effect was demonstrated in experiments on CoFe–BaTiO3, which also revealed that the anisotropy and ordering temperature of interfacial magnetism can be controlled by electric fields76. Note that these experiments are different from experiments involving a magnetoelectric response at microwave frequencies in magnetostrictive–piezoelectric compos­ ites49. The first type of experiment depends on epitaxal interfaces as a source of a well­defined local interaction, whereas the second type involves unspecified, rough interfaces that serve as strain mediators.
Interfaces. Multiferroic states can also be realized at the interface between two constituents (FIG. 3c). At the inter­ face, properties not found in the bulk can exist because of low local symmetry, confinement effects, strain gra­ dients and chemical anisotropy. Indeed, it was found that the interface between a ferromagnetic material, Fe or Co, and a ferroelectric material, BaTiO3, can be multi ferroic, and that the multiferroicity is retained within several monolayers around the interface77. The possible relevance for applications of these 2D multi­ ferroic interface states still has to be elaborated. Note that interfaces break space­ but not time­inversion sym­ metry. As a consequence, the magnetization axis at the interface can be reoriented by an electric field, but the direction along this axis remains ambiguous unless a magnetic bias field is applied.
Another way of confining multiferroicity to an inter­ face might be through domain walls; in this case, the interface is represented by the multiferroic wall between different domains within a single material that is in itself not multiferroic (FIG. 3d).
Domains and domain walls Domains and domain walls (BOX 1) are crucial for the control of many material properties, such as coercivity, resistance and/or fatigue. The magnetoelectric coupling of a multiferroic material roots in the coupling between its individual magnetic and ferroelectric domains.
Although advanced functionalities are often based on complex domain architectures64, early investigations focused on symmetry analysis10 and experiments on a sin­ gle domain state of a single­crystal multiferroic mat erial. An example is the study of the reversal of the electric­ order parameter by a magnetic field (or vice versa) via the linear magnetoelectric effect6. Domain patterns were initially imaged by linear optics, and ferroelectric domains were visualized after chemical surface etching or via optical birefringence in materials with simulta­ neous elastic deformation. Ferromagnetic domains were resolved by the magneto­optical Faraday or Kerr effect10. Using these techniques, domains were studied
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c Multiferroic interface d Multiferroic domain wall
FerroelectricMF
Ferromagnet
antiferromagnetic
M
in multiferroic boracites78, BiFeO3 (REF. 79) and h­YMnO3 (REF. 80). The main limitations of the optical approach are the insensitivity to antiferromagnetic order and the limited resolution of, at most, 1 μm.
A seminal advance was the application of nonlinear optics to multiferroics. In addition to providing access to antiferromagnetic domain states, a technique such as second harmonic generation can resolve electric and magnetic domains within the same experiment, making spatial magnetoelectric coupling phenomena directly visible81. This was crucial for resolving the pronounced coupling between ferroelectric and antiferromagnetic domains in h­YMnO3 and to reveal that this coupling does not originate in the domains but in the domain walls11. Similar studies on spin­spiral multiferroics, such as MnWO4 (REFS 82,83) or o­TbMnO3 (REF. 84), revealed the dynamics of multiferroic domains in external fields and on crossing phase transitions.
Scanning­probe and electron­microscopy tech­ niques pushed the resolution of domain imaging to the nanoscale, making even domain walls accessible with atomic­scale resolution. These advances were extremely helpful for the understanding of the complex multiferroic domain structures in BiFeO3 thin films85. The study of h­RMnO3 revealed a particularly interesting example of
domain physics: the trimerizing lattice distortion leads to a protected pattern of improper ferroelectric domains with unique topological properties86,87.
More recently, attention is moving away from the domains and towards their boundaries — the domain walls88. Whereas heterointerfaces separate constituents in artificial multilayers, domain walls can be regarded as natural homointerfaces occurring within a material. At the walls, charge or spin phenomena similar to the ones occurring at heterointerfaces can emerge but with the advantage that domain walls can be created, moved and erased post­growth. As FIG. 4 shows, the domain walls in type I and type II multiferroics are different. In type I multiferroics, magnetic and electric walls can coincide but do not have to because of the independence of magnetic and electric order. In type II multiferroics, the magnetic order induces the electric order; therefore, each ferroelectric domain wall is a magnetic domain wall at the same time and, thus, multiferroic.
The ground­breaking discovery that domain walls in BiFeO3 have a larger conductance than the surrounding bulk89 triggered a broad interest in domain­wall engi­ neering90. Domain walls with increased, attenuated or anisotropic conductance have been investigated in many materials88. These investigations are mainly targeted at
Figure 3 | Multiferroic thin-film architectures. a,b | In 3D transfer multiferroics, multiferroicity is a composite effect. In panel a, the ferroelectric order, P, is coupled to the antiferromagnetic order, m, within a non-composite multiferroic, and the antiferromagnetic order, in turn, is coupled to the ferromagnetic order, M, of an adjacent constituent via magnetic exchange, J. In panel b, the magnetoelectric coupling between a piezoelectric and magnetostrictive constituent is established via strain, σ. c,d | In 2D confined multiferroics, only the interface between two material phases is multiferroic. In panel c, these phases are the permittivity, ε, and permeability, μ, states of different compounds, and multiferroicity emerges as an interface effect. In panel d, the interface is represented by the wall separating different domains (that is, states A+ and A−, represented by the block arrows) within a crystal. E, applied electric field; MF, multiferroic.
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Crystal Crystal
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M P
the ferroelectric state, and truly multiferroic domain walls receive much less attention. Examples of the phe­ nomena observed at the domain walls are the magne­ toelectric coupling or spontaneous magnetization in the domain walls of multiferroic manganites11,91,92. In spin­driven ferroelectrics, examples are the coupled Bloch­ or Néel­like rotation of magnetic chirality and the induced electric polarization83,93. In contrast to the aforementioned multiferroicity emerging at the inter­ faces between different non­multiferroic compounds (FIG. 3c), multiferroicity occurring at the domain walls within a single non­multiferroic compounds (FIG. 3d) has not yet been observed.
Non-equilibrium dynamics In the field of multiferroics, the strong interest in the electric­field control of the magnetic state is contrasted by the small number of investigations on the temporal evolution of such magnetoelectric switching. However, understanding the switching dynamics is essential for judging and improving the speed, repeatability and control of voltage­induced magnetization reversal. For example, the switching may be inherently faster or slower than conventional magnetic­field switching, a factor that needs to be known before the design of a multiferroic memory element can be envisaged.
Early work on non­equilibrium dynamics revealed a fundamentally different temporal evolution for the polar and magnetic order94 in multiferroic GaFeO3. Another study showed that optical excitation of CuO induces a transition from commensurate to incom­ mensurate magnetic order within picoseconds, which probably influences the accompanying ferroelectric polarization95. LuFeO2 shows indications of a dynamic correlation between the magnetic and an assumed polar­ ization­inducing charge order96. The resonant excitation of spin­spiral multiferroics was predicted to promote an all­optical reversal of the multiferroic order parameters97;
related experiments on o­TbMnO3, performed at acces­ sible terahertz amplitudes, identified an ultrafast spin deflection98 of ~4%, far below a 180° reversal. Ultrafast optical pump­probe spectroscopy was used to reveal the coexistence and coupling of antiferro magnetic or ferroelectric and ferromagnetic order in strained o­TbMnO3 films99. The ultrafast reversal of the multifer­ roic order parameters by an electric field pulse in BiFeO3 is debated100,101; electric field experiments on magneti­ cally driven ferroelectrics even revealed that their order­ parameter reversal can be inherently slow102. In a compos­ ite multiferroic, ultrafast manipulation of the polarization via photo­induced magnetostriction was observed103. As these results show, the field is still at the stage of collecting disconnected bits of information on the non­equilibrium dynamics of multiferroics, and a coherent understanding of them has not yet been reached.
Note that these experiments on switching are dis­ tinct from those in which ultrafast processes in multi­ ferroics are investigated, but not in connection with the multiferroic order — photostrain effects in BiFeO3 are one example104.
Violation of inversion symmetry The Neumann principle relates the physical properties of a material to its structural symmetry, stating that the symmetry displayed by the material properties includes the symmetry elements characterizing the structural arrangement of its charges and spins. Many of the remarkable physical properties of multiferroics arise from the breaking of space­ and time­inversion symme­ try caused by the simultaneous presence of magnetic and electric long­range order — the linear magnetoelectric effect is a prototypical example.
The increasing understanding of the role of symme­ try led to the realization that some phenomena origi­ nally associated with the coexistence of magnetic and electric order are actually a consequence of this two­fold
Figure 4 | Domains and domain walls in type I and type II multiferroics. a | In type I multiferroic materials, magnetic and electric order emerge independently; thus, the respective domain patterns do not have to coincide and domain walls are either magnetic (blue) or electric (red) in nature. If the domain patterns coincide, a multiferroic domain wall is formed (orange). The formation of multiferroic domain walls points to a coupling between magnetic and electric order which, however, is not required by symmetry. b | In type II multiferroics, the magnetic order induces the electric order; thus, ferroelectricity emerges jointly with a magnetic phase transition. Because of the interdependence of the electric and magnetic order parameter, all ferroelectric domain walls are also magnetic domain walls and, therefore, multiferroic. FE, ferroelectric; FM, ferromagnetic; M, magnetic order; MF, multiferroic; P, ferroelectric order.
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symmetry violation and can therefore occur in non­ multi ferroics as well. This could considerably expand the range of host systems in which these phenomena are observed. A prime example is Cr2O3, a non­multiferroic material with broken space­ and time­inversion symme­ try, which displays many of the effects discussed in this Review, such as static and optical magnetoelectric effects, ferrotoroidicity and magnetic quasi­monopoles3,105–107.
Ferrotoroidicity. For a long time, the magnetoelectric coupling in multiferroics has been associated with a viola­ tion of time­reversal symmetry by the magnetic order and a violation of spatial inversion symmetry by the electric order. This vision changed with the discovery that in spin­ driven ferroelectrics, it is the magnetic order that breaks inversion symmetry, a realization that intensified the search for a primary ferroic order violating both space­ and time­inversion symmetry. The discovery of such a state would symmetrize and complete the known forms of primary ferroic order, which at the moment include states violating spatial inversion symmetry (ferro electricity), temporal inversion symmetry (ferromagnetism) or neither of the two (ferroelasticity)108,109.
The multipole expansion of the electrodynamic vec­ tor potential A(r) includes a space–time­antisymmetric term, a so­called anapole110 or toroidal moment111. The spontaneous, uniform alignment of toroidal moments would then establish a ferrotoroidic state, with domains that could be hysteretically switched by an adjunct toroi­ dal (that is, space–time­antisymmetric) external field. Most investigations consider toroidal spin arrangements in the unit cell where the toroidal moment, T = ∑i ri × Si, represents a magnetic whirl (ri is the position of spin Si, FIG. 5a)112. By contrast, the existence of atomic toroidal moments remains debated113.
Compounds exhibiting a toroidal magnetic struc­ ture have been identified; however, this does not mean that these compounds are ferrotoroidic. Ferrotoroidicity requires the presence of domains (FIG. 5b) and their pol­ ing in a toroidal field. Both conditions were realized114,115 in LiCoPO4, in which the coupling to the toroidal field was established with perpendicular magnetic and electric fields. The direct generation of a toroidal field remains an open challenge. Finally, ferrotoroidic structures can also be regarded as antiferromagnetic, and ferrotoroidicity is a reasonable concept only if the toroidal moment can act as the driving (proper) order parameter. This latter criterion was exemplified116 in the case of LiFeSi2O6.
Magnetic monopoles. Magnetic monopoles are a con­ cept closely related to toroidal moments. Magnetic monopoles are the counterpart of the electric (charge) monopoles, but they are not allowed in free space, as expressed by the Maxwell equation, ∇B = 0. However, in condensed­matter systems, magnetic quasi­monopoles with ∇H ≠ 0 or ∇M ≠ 0 can exist in both an excited form (such as spin ice117) or as the ground state. The ground­ state monopole is derived as the second­order term in the multipole expansion of the magnetization density, μ(r), in which the magnetic monopole appears along with the toroidal moment and the magnetic quadrupole
in space–time­inversion­symmetry violating systems107. These three terms are associated to three qualitatively different contributions to the linear magnetoelectric effect107,111. Note that the relation between the deriva­ tions of the toroidal moment from μ(r) and A(r) still needs to be established.
Magnetic monopoles, as toroidal moments, can be studied at the level of atoms or unit cells. The first case is theoretically discussed for the LiMPO4 system107 (M = Mn–Ni). A spin arrangement that may be interpreted as a magnetic monopole is sketched in FIG. 5c and was observed in the P63/cm phase of the h­RMnO3 system118. The existence of a ferro­monopolar state is debated107, but it would depend on the existence of a conjugate sca­ lar monopolar field. A solitary magnetic monopole may be conveniently generated by placing a point charge in front of a magnetoelectric medium, which would con­ vert the electric monopole into a submerged­image mag­ netic monopole. Until now, this possibility has only been explored for topological insulators exposed to a magnetic field119, but a much stronger response can be expected for conventional bulk magnetoelectrics.
Figure 5 | Magnetic toroidal moments and monopoles in crystals. a | Hypothetical unit cell with six spins, Si, at positions ri defining a toroidal moment, T = ∑i Si × ri, per unit cell. The spin arrangements of opposite toroidal moments are shown in the two sketches. b | Ferrotoroidic domain structure with uniform arrangements of toroidal moments (represented by the arrows). c | Hypothetical unit cell with a magnetic-monopole-like arrangement of six spins. Spin arrangements of opposite monopole moments are shown in the two sketches.
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From a technological point of view, toroidal and monopolar systems are interesting because of the inher­ ent linear magnetoelectric effect they display107. This effect may be substantially enhanced by combining a space–time­antisymmetric ferrotoroidic state with a ferroelectric and/or ferromagnetic state, similar to how the combination of space–time­antisymmetric magnetic order with ferroelectric order in spin­driven multiferroics can be the source of giant magnetoelectric effects.
Effects at finite frequencies. The violation of space­ and time­inversion symmetries is not a static phenome­ non; it is also realized at finite frequencies. This fact is exploited in composites in which a ferromagnetic mag­ netostrictive and a ferroelectric piezoelectric constituent are mixed, so that an oscillating magnetic field induces an oscillating electric polarization via strain coupling49.
In single­phase multiferroics, attention is focused on two dynamic effects: electromagnons, a magnetoelec­ tric excitation, and nonreciprocal directional dichro­ ism, a ground­state optical response. Both phenomena are manifestations of the two­fold symmetry violation rather than of the multiferroic state. As a consequence, they are also observed in non­multiferroics.
Electromagnons. An electromagnon is a magnetic resonance that breaks inversion symmetry, giving rise to an oscillating electric polarization (FIG. 6). Electromagnons were first reported in the multi­ ferroics o­TbMnO3 and o­GdMnO3 (REF. 120), in which a dielectric resonance influenced by magnetic order and magnetic fields was observed. In o­GdMnO3, this resonance appeared outside the multiferroic phase, which showed from the start that electromagnons are a consequence of the system symmetry rather than of multiferroicity. Initially, in o­TbMnO3, a spin­wave excitation was observed that induces a polarization oscillation via the (symmetric) Heisenberg exchange121 (FIG. 6b). This is remarkable, because the static polari­ zation of o­TbMnO3 is driven by the (antisymmetric) inverse DM interaction (FIG. 6c,d). Later, o­TbMnO3 also revealed a polarization oscillation mediated by the inverse DM interaction122.
Electromagnons have been identified in various compounds, including type I multiferroics123, high­ temperature multiferroics124 and non­multiferroics125,126.
Nonreciprocal directional dichroism. Identical light waves travelling in opposite directions through a mat­ erial in which space­ and time­inversion symmetry are broken are transmitted with different intensity and polarization, an effect called nonreciprocal directional dichroism (NDD). The reversal of this directional dependence by a magnetic field goes under the name of the optical magnetoelectric effect127,128. Spontaneous NDD induced by multiferroic order has been first stud­ ied in GaFeO3, a material exhibiting relative absorp­ tion differences Δα/α of up to 1.6 × 10−3 (REFS 129,130). Giant NDD with Δα/α = 1 was reported for multiferroic Ba2CoGe2O7 in the terahertz regime131. The requirement of space–time antisymmetry means that NDD is not lim­ ited to multi ferroics132; in fact, the observation of the anti­ ferromagnetic analogue of the Kerr and Faraday effect in Cr2O3 constitutes the earliest example of NDD105,133,134. The strongest possible manifestation of NDD — transmission of light through a material in one direc­ tion only — was also observed in a non­multiferroic. Experiments on CuB2O4 in magnetic fields of up to 53 T showed that such unidirectional propagation would develop at about 75 T (REF. 135).
As discussed, space­ and time­inversion symmetry are violated separately in the two constituents of a composite multiferroic. Another way to acquire such spatially sepa­ rated symmetry breaking is to grow trilayer superlattices formed by two dielectrics and one magnetically ordered constituent136. Other systems are patterned media, such as magnetic photonic crystals and toroidal or chiral meta­ materials137,138. Experimentally, patterned media are more
Figure 6 | Electromagnons in cycloidal spin structures. Grey arrows represent the spins, Si, and their collective movement, and circles indicate the corresponding magnetic easy plane — that is, the plane in which the spins lie. The long-range modulated spin structure induces a polarization along the z axis according to the relation P ∝ eij × (Si × Sj), where eij is the unit vector connecting neighbouring spins and Si,j are the spins at neighbouring sites i and j. a | The spin excitation indicated by the red arrows does not induce a polarization change, because the vector product remains constant. This excitation represents a magnon. b | Excitation leading to a variation in the antisymmetric and also the symmetric spin exchange. It leads to a spatial modulation of the polarization amplitude. c | For excitations that involve a rotation of the magnetic easy plane around the y axis, the polarization also rotates around this axis, yet retaining its amplitude. d | If the magnetic easy plane rotates around the z axis, the cycloidal spin arrangement is transformed into helical order. This leads to a spatially uniform modulation of the polarization amplitude. The excitations in parts a–c represent electromagnons. All the spin deflections shown in parts a–d oscillate with time, and so does the associated modulation of polarization. Figure is adapted with permission from REF. 169, American Physical Society.
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successful in demonstrating NDD than other compos­ ite structures: for example, interfaces139 and magnetized chevrons140 or gratings141.
Other ways to obtain pronounced NDD are based on the resonant excitation of magnetoelectrically active states such as magnons142, electromagnons143 or skyrmions144.
Impact on other research fields The field of multiferroics is experiencing a migration into other research disciplines, in which multiferroic materials are studied for properties that are, at best, indirectly related to their multiferroic nature. This trend was already apparent in the discussion on space­ and time­inversion symmetry violation in the previous section.
Multiferroics had a major impact on the field of ferroelectrics. Research in multiferroics restored an awareness of the wealth of mechanisms, in addition to the displacive mechanism, that can promote spon­ taneous polar order. In particular, the importance of improper ferro electric materials was recognized, because they allow the formation of domains and domain walls with properties and functionalities that do not occur if the polarization is the primary order parameter83,88,145.
An increasing interest in oxide electronics146, in which multiferroic materials and their magnetoelec­ tric interactions are an important topic, stimulated impressive progress in the epitaxial growth of oxide films and hetero structures. This opened a path to var­ ious multiferroic functionalities. It allows coupling of the antiferromagnetic state of a multiferroic material to the ferromagnetic state of an adjacent material, a multi ferroic state to be induced in the first place, and the creation of a magnetoelectric coupling at or across interfaces. Heterostructures also facilitate connection of the multi ferroic system to electronic circuitry.
Research in multiferroics motivated the improvement of many characterization techniques. For example, X­ray diffraction experiments were improved to detect atomic shifts at the femtometre scale147; pyroelectric measure­ ments were refined to measure spontaneous polariza­ tions down to 0.1 nC cm−2; nonlinear laser spectroscopy was developed into a tool for imaging the coexistence and interaction of magnetic and ferroelectric domains11,82; and terahertz spectroscopy was advanced to study electromagnons120.
The field of multiferroics has also entered the realm of high­energy physics. Materials with specific mag­ netic and electric properties may allow the detection of a permanent electric dipole moment of the electron with unparalleled accuracy. The required properties can be obtained by replacing Eu in multiferroic EuTiO3 with Ba until the magnetic order is suppressed148. The multiferroic h­RMnO3 compounds are used to scruti­ nize scaling laws related to string formation in the post­ Big­Bang universe86,87 — upon cooling, both systems undergo topologically similar phase transitions. This similarity is used to relate the distribution of the alleged cosmological strings to the distribution of the lines along which ferroelectric domains meet in h­RMnO3.
Multiferroic materials are a part of these research areas because of their predisposition to host ferroelectric states of uncommon origin or topology.
Trends and challenges Despite the advances in the field of multiferroics, some of the current goals are still the same as in the 1960s, and it is safe to assume that they will continue to keep research­ ers busy for a while. These objectives include the quest for new materials with a strong coupling of magnetic and electric properties, which could raise the very limited number149 of known room­temperature multiferroics. In particular, a room­temperature multi ferroic with pro­ nounced and strongly coupled spontaneous magnetiza­ tion and polarization is not yet known. Novel mechanisms driving multiferroicity may be discovered, but the existing ones are far from being fully exploited. For example, there can be many ways to obtain a magnetic order that can stabilize an improper ferroelectric state. This may lead to inherently higher ordering temperatures and polari­ zations compared with those of the existing spin­driven ferroelectrics — predictions and preliminary discoveries are promising42,44. Other, under­represented classes of materials that may host multiferroic states are non­oxide compounds150 and organic materials151.
A multiferroic device in which the magnetization is controlled by an electric field, preferably at low voltages, at room temperature and with ultrafast switching, remains a prime goal152–155. Major accomplishments obtained so far are the repeatable, room­temperature magnetization reversal by an electric field demonstrated in BiFeO3–CoFe heterostructures64 and the realization of a multiferroic four­state memory operated at low temperatures156. These are important steps towards the integration of multi ferroics into devices, but crucial aspects such as the dynamics, reliability and fatigue of these device concepts still have to be optimized in order to develop a competitive technology. In addition to this, alternative routes for the control of magnetism that use spin torque157 or spin­orbit torque158,159 exerted by a spin­polarized electric current have been presented, and any multiferroic device will need to compete with their functionality and performance.
Multiferroic thin films and heterostructures hold the greatest potential for device applications59. First, they may lead to new types of systems combining magnetic and electric long­range order. Multiferroicity may origi­ nate under strain, confinement or gradient effects within one material, as well as at or across the interface between different materials. All of these possibilities represent degrees of freedom that so far have been only marginally explored. Second, heterostructures with strongly cou­ pled ferroelectric and ferromagnetic layers may be used to build a magnetoelectric memristor. In this kind of device, an electric field sets the ferroelectric layer polari­ zation state, which is then transferred to the magnetic layer, where it defines the memristive state through cou­ pling to an adjacent ferromagnetic reference layer. Third, multilayer heterostructures with an integral symmetry that is different from the symmetry of their individual constituents may be assembled. This integral symmetry may even be switchable by, for example, reversing the
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order parameter of only one of the layers constituting the heterostructure. This was conceptually demonstrated160 by layer­selective polarization orientation of trilayer heterostructures such as PbZr0.2Ti0.8O3–La0.7Sr0.3MnO3– PbZr0.2Ti0.8O3. By setting the relative polarization of the outer layers, the integral space­inversion symmetry of the trilayer structure is activated and deactivated on demand. Accordingly, magnetic­field poling of outer magnetically ordered layers could switch time­ reversal symmetry on or off. These symmetry­tunable multilayer assemblies constitute a powerful option to increase the number of systems with interesting and controllable magnetoelectric coupling effects. Fourth, the focus on domain walls will grow stronger, because it is an advantage to work with oxide interfaces that can be created, shifted and annihilated post­growth (in contrast to conventional interfaces that, once grown, cannot be modified). Controllable domain walls could be the key to electric­field­controlled race­track memo­ ries161. An electric field may act on the magnetic domain wall and move it, exploiting the magnetoelectric effect. Alternatively, if the domain wall is multiferroic, the elec­ tric field may act directly on the wall polarization. As in the bulk compounds, these concepts may be extended to non­oxide thin films.
Another element that might have an important role for future applications of multiferroic materials is skyr­ mions — magnetic whirls that were first observed in half­metallic systems162 and that were detected in multi­ ferroic insulators163. In multiferroic insulators, skyrmions are localized and electric currents — the usual tool to manipulate them — cannot flow, but the coexistence with ferroic order and the possibility of establishing magnetoelectric control of the skyrmions are worth further exploration. For example, skyrmions in systems with acentric spin­spiral order (instead of an acentric crystal structure) may provide a route to giant, locally controllable magnetoelectric interactions.
The field of multiferroics also has a prominent role in promoting cooperation between formerly disjoint research disciplines — systems with strong
magnetoelectric interactions are appealing for different fields. Moreover, bringing together two types of order in one material is an inherently interdisciplinary effort. For example, it is no longer sufficient to track the thick­ ness of a thin film during growth; continuous control of its emerging multiple ferroic properties is also required. The combination of deposition techniques with in situ electro­ and magneto­optical spectroscopy could be very effective to this end.
In multiferroic research, dynamical phenomena are still a highly underrated topic. With magnetoelec­ tric switching as one of the declared goals of the field, more attention must be paid to the temporal evolution of these reorientations. One important aspect is the speed of the reversal of the order parameter, which has to occur within picoseconds if memory applications are envisaged. In this respect, all­optical control164 of a multi ferroic state may be a particularly reward­ ing goal. On the other hand, the reversal needs to be highly reproducible at the level of individual domains if magneto electric switches, sensors or transducers are considered. Apart from establishing a one­to­one association between the orientation of a magnetic order parameter and the orientation of an electric field, more complex types of magnetoelectric control may be taken into consideration. If, for example, a multiferroic state is characterized by three or more order parame­ ters, it might be possible to invert an entire distribu­ tion of magnetic domains; the domain pattern remains unchanged but in each individual domain the order parameter is reversed. The inversion of an inhomoge­ neous physical state has a great technological relevance; for example, it is at the basis of the spin­echo effect in nuclear magnetic resonance tomography165 and of active noise reduction166.
In conclusion, even though the field of multiferroics has reached some maturity after more than 50 years of research, it is still far from being full grown. Its twists and turns continuously lead to the exploration of new systems and effects, and it is possible that the most exciting results and discoveries have not yet been realized.
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