# Multiferroics

Multiferroics have been formally defined as materials that exhibit more than one primary ferroic order parameter simultaneously (i.e. in a single phase),[1] and many researchers in the field consider materials to be multiferroics only if they exhibit coupling between primary order parameters.[citation needed] However, the definition of multiferroics can be expanded to include non-primary order parameters, such as antiferromagnetism or ferrimagnetism.[citation needed]

The four basic primary ferroic order parameters are

The last is a topic of some debate, as there was no evidence for switching ferrotoroidicity until recently.[citation needed]

Many multiferroics are transition metal oxides with perovskite crystal structure, and include rare-earth manganites and -ferrites (e.g. TbMnO3, HoMn2O5, LuFe2O4 and recently, "PZTFT",[2][3]). Other examples are the bismuth compounds BiFeO3 and BiMnO3, non-perovskite oxide LiCu2O2,[4] and non-oxides such as BaNiF4 and spinel chalcogenides, e.g. ZnCr2Se4. These alloys show rich phase diagrams combining different ferroic orders in separate phases.

Apart from single phase multiferroics, composites and heterostructures exhibiting more than one ferroic order parameter are studied extensively. Some examples include magnetic thin films on piezoelectric PMN-PT substrates and Metglass/PVDF/Metglass trilayer structures.[citation needed]

Besides scientific interest in their physical properties, multiferroics have potential for applications as actuators, switches, magnetic field sensors or new types of electronic memory devices.[citation needed]

## History

Multiferroics history use of terms magnetoelectric and multiferroic

The term multiferroic was first used by H. Schmid in 1994. His definition referred to multiferroics as single phase materials which simultaneously possess two or more primary ferroic properties. Today the term multiferroic has been expanded to include materials which exhibit any type of long range magnetic ordering, spontaneous electric polarization, and/or ferroelasticity. Working under this expanded definition the history of magnetoelectric multiferroics can be traced back to the 1960s.[5] In the most general sense the field of multiferroics was born from studies of magnetoelectric systems.[6] After an initial burst of interest, research remained static until early 2000 (see figure). In 2003 the discovery of large ferroelectric polarization in epitaxially grown thin films of BiFeO3[7] and the discovery of strong magnetic and electric coupling in orthorhombic TbMnO3[8] and TbMn2O5[9] re-stimulated activity in the field of multiferroics.

## Symmetry

Each multiferroic property is closely linked to symmetry. The primary ferroic properties (see table) can be characterized by their behavior under space and time inversion. Space inversion for example will reverse the direction of polarization P while leaving the magnetization M invariant. Time reversal, in turn, will change the sign of M, while the sign of P remains invariant.

 Space invariant Space variant Time invariant Ferroelastic Ferroelectric Time variant Ferromagnetic Ferrotoroidic

Magnetoelectric multiferroics require simultaneous violation of space and time inversion symmetry. In BiFeO3, for example, off-centering of ions gives rise to an electric polarization, while at a lower temperature additional magnetic ordering breaks time-reversal symmetry.

## Mechanisms for ferroelectricity in multiferroics

A necessary but not sufficient condition for the appearance of spontaneous electric polarization is the absence of inversion symmetry. We can distinguish between proper and improper ferroelectric (FE). The difference lies in the driving force (the primary order parameter) that leads to ferroelectricity: In the case of proper FE, the primary order parameter is the ferroelectric distortion. One example of proper FE is BaTiO3 where a covalent bonding between the transition metal and the oxygen happens to allow a polar state. In usual perovskite-based ferroelectrics like BaTiO3, the ferroelectric distortion occurs due to the displacement of B-site cation (Ti) with respect to the oxygen octahedral cage. Here the transition metal ion (Ti in BaTiO3 ) requires an empty “d” shell since the ferroelectric displacement occurs due to the hopping of electrons between Ti “d” and O “p” atoms.

For technological applications, it is highly desirable to combine ferroelectric and ferromagnetic order within one material, but it has become clear that usual displacive ferroelectric order, e.g. like in BaTiO3, cannot coexist together with magnetic order. Whereas the latter requires at least partially filled d-shells (e.g.-orbitals) providing a non-zero magnetic moment, usual displacive ferroelectricity requires empty d-shells.[10] Therefore, new ferroelectric driving mechanisms must be present for electric and magnetic ferroic order to occur simultaneously. A variety of driving mechanism is described below.

### Lone pair multiferroics

In usual perovskite-based ferroelectrics like BaTiO3, the ferroelectric distortion occurs due to the displacement of B-site cation (Ti) with respect to the oxygen octahedral cage. Here the transition metal ion (Ti in BaTiO3 ) requires an empty "d" shell since the ferroelectric displacement occurs due to the hopping of electrons between Ti "d" and O p atoms. This normally excludes any net magnetic moment because magnetism requires partially filled "d" shells. However, partially filled "d" shell on the B-site reduces the tendency of perovskites to display ferroelectricity.

In order for the coexistence of magnetism and ferroelectricity (multiferroic), one possible mechanism is lone-pair driven[10] where the A-site drives the displacement and partially filled “d” shell on the B-site contributes to the magnetism. Examples include BiFeO3,[11][12] BiMnO3,[13] PbVO3. In the above materials, the A-site cation (Bi3+, Pb2+) has a stereochemically active 6s2 lone-pair which causes the Bi 6p (empty) orbital to come closer in energy to the O 2p orbitals. This leads to hybridization between the Bi 6p and O 2p orbitals and drives the off-centering of the cation towards the neighboring anion resulting in ferroelectricity.

### Improper geometric ferroelectricity

In improper geometric ferroelectrics a structural phase transition at high temperatures causes the appearance of ferroelectricity. A prototypical compound is the family of hexagonal rare earth manganites (h-RMnO3 with R=Ho-Lu, Y), showing a structural phase transition at around 1300 K providing the necessary symmetry lowering by tilting of the MnO5 bipyramids. This allows for an electrostatically driven corrugation of the R-ion layers. The value of the induced polarization of several µC/cm² is comparable to that of usual displacive ferroelectrics. The multiferroic phase is entered only at cryogenic temperatures when antiferromagnetic order due to spin frustration arises.[14][15] Thus only weak, indirect coupling between two disparate order parameters can occur. The exact microscopic mechanism of the ferroelectric ordering in hexagonal RMnO3 is still questionable in the scientific community, i.e. it is still matter of debate whether only the corrugation of R-ions is the origin of the electric polarization or whether an off-centering of Mn ions also contributes to the polarization.

### Charge ordering

A possible origin for a multiferroic state is charge ordering. Such an order can occur in a compound containing ions of mixed valence and with geometrical or magnetic frustration. These ions form a polar arrangement, causing improper ferroelectricity (i.e. no ionic displacement). If magnetic ions are present, a coexisting magnetic order can be established and may be coupled to ferroelectricity

One prominent example for a charge ordered multiferroic is LuFe2O4, which shows improper ferroelectricity below 330 K.[16] The arrangements of the electrons arise from the charge frustration on a triangular lattice with the mixed valence state of Fe2+ and Fe3+ ions. Ferrimagnetic behavior occurs below 240 K.

In addition, charge ordered ferroelectricity is suggested in Fe3O4 and (Pr,Ca)MnO3.[17]

### Magnetically driven ferroelectricity

Magnetically driven multiferroics[18] are insulating materials, mostly oxides, in which macroscopic electric polarization is induced by magnetic long-range order. A necessary but not sufficient condition for the appearance of spontaneous electric polarization is the absence of inversion symmetry. We can distinguish between proper and improper ferroelectricity (FE). The difference lies in the driving force (the primary order parameter) that lead to ferroelectricity: In the case of proper FE, the primary order parameter is the ferroelectric distortion. One example of proper FE is BaTiO3 where a covalent bonding between the transition metal and the oxygen happens to allow a polar state.

In the case of improper FE, the primary order parameter is not the ferroelectric distortion but another type of phase change, like magnetic ordering or a structural change. The FE distortion is a secondary order parameter in the sense that it is driven by the presence of other order parameters. One example of improper FE is when the inversion symmetry of a crystal is broken by magnetic structure like spiral magnetic ordering. This is the spin-driven ferroelectric. The microscopic mechanism of magnetoelectric (ME) coupling in spiral multiferroics involves spin-orbit coupling. The polarization is smaller than the one of proper FE. ME coupling is very strong because ferroelectricity is driven by magnetic order and do not exist without the latter. That means that any change in the magnetic order will affect the ferroelectricity.

### Optical properties

Optical properties of multiferroics.

### List of materials

critical temperature
Crystal TC [K] Type
ZnCr2Se4
BaNiF4
PbVO3 lone pair
BiMnO3 lone pair
BiFeO3 1143 lone pair
LuFe2O4 charge order
HoMn2O5 39[19]
h-YMnO3 1270 geometric
K2SeO4 geometric
Cs2CdI4 geometric
TbMnO3 27 spin spiral
Ni3V2O8 6.5[20]
MnWO4 13.5[21] spin spiral
CuO 230 spin spiral

## Domains

Schematic picture of the four possible domain states of a ferroelectric ferromagnetic material. Both, the polarization (electric dipole indicated by charges) and the magnetization (red arrow), may have two orientations. The domains are separated by different types of domain walls, classified by the order parameter that is changed throughout the wall.

Like any ferroic material, a multiferroic system is fragmented into domains. A domain is a spatially extended region with a constant direction and phase of its order parameters. Neighbouring domains are separated by transition regions called domain walls.

### Properties of multiferroic domains

In contrast to materials with a single ferroic order, domains in multiferroics have additional properties and functionalities. For instance, they are characterized by an assembly of at least two order parameters.[22] The order parameters may be independent (typical yet not mandatory for a split-order-parameter multiferroic) or coupled (mandatory for a joint-order-parameter multiferroic).

Many outstanding properties that distinguish domains in multiferroics from those in materials with a single ferroic order are consequences of the coupling between the order parameters.

• The coupling can lead to patterns with a distribution and/or topology of domains that is exclusive to multiferroics.
• The order-parameter coupling is usually homogeneous across a domain, i.e., gradient effects are negligible.
• In some cases the averaged net value of the order parameter for a domain pattern is more relevant for the coupling than the value of the order parameter of an individual domain.[23]

These issues lead to novel functionalities which explain the current interest in these materials.

### Properties of multiferroic domain walls

Domain walls are spatially extended regions of transition mediating the transfer of the order parameter from one domain to another. In comparison to the domains the domain walls are not homogeneous and they can have a lower symmetry. This may modify the properties of a multiferroic and the coupling of its order parameters. Multiferroic domain walls may display particular static[24] and dynamic[25] properties.

Static properties refer to stationary walls. They can result from

• The reduced dimensionality
• The finite width of the wall
• The different symmetry of the wall
• The inherent chemical, electronic, or order-parameter inhomogeneity within the walls and the resulting gradient effects.[26]

Dynamic properties refer to moving walls. In a magnetic ferroelectric, the magnetoelectric interaction is, at its roots, usually synonymous to the movement of the multiferroic domain walls. Because of the order-parameter coupling this may reflect characteristic features of both, ferroelectric and magnetic domain wall movement.

## Magnetoelectric effect

The magnetoelectric (ME) effect is the phenomenon of inducing magnetic (electric) polarization by applying an external electric (magnetic) field. The effects can be linear or/and non-linear with respect to the external fields. In general, this effect depends on temperature. The effect can be expressed in the following form

${\displaystyle P_{i}=\sum \alpha _{ij}H_{j}+\sum \beta _{ijk}H_{j}H_{k}+\ldots }$

${\displaystyle M_{i}=\sum \alpha _{ij}E_{j}+\sum \beta _{ijk}E_{j}E_{k}+\ldots }$

where P is the electric polarization, M the magnetization, E and H the electric and magnetic field, and α and β are the linear and nonlinear ME susceptibilities. The effect can be observed in single phase and composite materials. Some examples of single phase magnetoelectrics are Cr2O3,[27] and multiferroic materials which show a coupling between the magnetic and electric order parameters. Composite magnetoelectrics are combinations of magnetostrictive and electrostrictive materials, such as ferromagnetic and piezoelectric materials. The size of the effect depends on the microscopic mechanism. In single phase magnetoelectrics the effect can be due to the coupling of magnetic and electric orders as observed in some multiferroics. In composite materials the effect originates from interface coupling effects, such as strain. Some of the promising applications of the ME effect are sensitive detection of magnetic fields, advanced logic devices and tunable microwave filters.[27]

The SI-Unit of α is [s/m] which can be converted to the practical unit [V/(cm Oe)] by [s/m]=1.1 x10−11 εr [V/(cm Oe)]. For the CGS unit, [unitless] = 3 x 108 [s/m]/(4 x π)

### Strain driven magnetoelectric heterostructured effect

Thin film strategy also enables achievement of interfacial multiferroic coupling through a mechanical channel in heterostructures consisting of a magnetoelastic and a piezoelectric component.[28] This type of heterostructure is composed of an epitaxial magnetoelastic thin film grown on a piezoelectric substrate. For this system, application of a magnetic field will induce a change in the dimension of the magnetoelastic film. This process, called magnetostriction, will alter residual strain conditions in the magnetoelastic film, which can be transferred through the interface to the piezoelectric substrate. Consequently, a polarization is introduced in the substrate through the piezoelectric process. The overall effect is that the polarization of the ferroelectric substrate is manipulated by an application of a magnetic field, which is the desired magnetoelectric effect. In this case, the interface plays an important role in mediating the responses from one component to another, realizing the magnetoelectric coupling.[29] For an efficient coupling, a high-quality interface with optimal strain state is desired. In light of this interest, advanced deposition techniques have been applied to synthesize these types of thin film heterostructures. Molecular beam epitaxy has been demonstrated to be capable of depositing structures consisting of piezoelectric and magnetostrictive components. Materials systems studied included cobalt ferrite, magnetite, SrTiO3, BaTiO3, PMNT.[30][31][32]

## Flexomagnetoelectric effect

Magnetically driven ferroelectricity is also caused by inhomogeneous[33] magnetoelectric interaction. This effect appears due to the coupling between inhomogeneous order parameters. It was also called as flexomagnetoelectric effect.[34] Usually it is describing using the Lifshitz invariant (i.e. single-constant coupling term).[35] It was shown that in general case of cubic hexoctahedral crystal the four phenomenological constants approach is correct.[36] The flexomagnetoelectric effect appears in spiral multiferroics[37] or micromagnetic structures like domain walls[38] and magnetic vortexes.[39][40] Ferroelectricity developed from micromagnetic structure can appear in any magnetic material even in centrosymmetric one.[41] Building of symmetry classification of domain walls leads to determination of the type of electric polarization rotation in volume of any magnetic domain wall. Existing symmetry classification[42] of magnetic domain walls was applied for predictions of electric polarization spatial distribution in their volumes.[43][44] The predictions for almost all symmetry groups conform with phenomenological theory in which inhomogeneous magnetization couples with homogeneous polarization. The total synergy between symmetry and phenomenological theory appears if energy terms with electrical polarization spatial derivatives are taking into account.[45]

## Synthesis

Multiferroics properties can appear in a large variety of materials. Therefore, several routes for conventional material fabrication are being applied. Popular techniques within the multiferroic community are: solid state synthesis.,[46] hydrothermal synthesis, sol-gel processing, vacuum based deposition, and floating zone.

However some types of multiferroics require specific processing conditions within certain techniques. For instance:

• Vacuum based deposition (for instance: MBE, PLD) for thin film deposition to exploit certain advantages that may come with 2-dimensional layered structures such as: strain mediated multiferroics, heterostructures, anisotropy.
• High pressure solid state synthesis to stabilize metastable or highly distorted structures as for example lone pair multiferroics like Bi based multiferroics due to their low melting point.

## Dynamics

The study of dynamics in multiferroic systems is concerned with understanding the time evolution of the coupling between various ferroic orders, in particular under external applied fields. Current research in this field is motivated both by the promise of new types of application reliant on the coupled nature of the dynamics, and the search for new physics lying at the heart of the fundamental understanding of the elementary MF excitations.

An increasing number of studies of MF dynamics are concerned with the coupling between electric and magnetic order parameters in the so-called magnetoelectric (ME) multiferroics. In this class of materials, the leading research is exploring, both theoretically and experimentally, the fundamental limits (e.g. intrinsic coupling velocity, coupling strength, materials synthesis) of the dynamical ME coupling and how these may be both reached and exploited for the development of new technologies.

At the heart of the proposed technologies based on ME coupling are switching processes, which describe the manipulation of the material's macroscopic magnetic properties with electric field and vice versa. Much of the physics of these processes is described by the dynamics of domains and domain walls. An important goal for current research is the minimization of the switching time, from fractions of a second ('quasi'-static regime), towards the nanosecond range and faster, the latter being the typical time scale needed for modern electronics, such as next generation memory devices.

Ultrafast processes operating at picosecond, femtosecond, and even attosecond scale are both driven by, and studied using, optical methods that are at the front line of modern science. The physics underpinning the observations at these short time scales is governed by non-equilibrium dynamics, and usually makes use of resonant processes. One demonstration of ultrafast processes is the switching from collinear antiferromagnetic state to spiral antiferromagnetic state in CuO under excitation by 40 fs 800 nm laser pulse.[47] A second example shows the possibility for the direct control of spin waves with THz radiation on antiferromagnetic NiO.[48] These are promising demonstrations of how the switching of electric and magnetic properties in multiferroics, mediated by the mixed character of the magnetoelectric dynamics, may lead to ultrafast data processing, communication and quantum computing devices.

Current research into MF dynamics aims to address various open questions; the practical realisation and demonstration of ultra-high speed domain switching, the development of further new applications based on tunable dynamics, e.g. frequency dependence of dielectric properties, the fundamental understanding of the mixed character of the excitations (e.g. in the ME case, mixed phonon-magnon modes – 'electromagnons'), and the potential discovery of new physics associated with the MF coupling.

## Applications

Multiferroic composite structures in bulk form are explored for high-sensitivity ac magnetic field sensors and electrically tunable microwave devices such as filters, oscillators and phase shifters (in which the ferri-, ferro- or antiferro-magnetic resonance is tuned electrically instead of magnetically).[27]

In multiferroic thin films, the coupled magnetic and ferroelectric order parameters can be exploited for developing magnetoelectronic devices. These include novel spintronic devices such as tunnel magnetoresistance (TMR) sensors and spin valves with electric field tunable functions. A typical TMR device consists of two layers of ferromagnetic materials separated by a thin tunnel barrier (~2 nm) made of a multiferroic thin film.[49] In such a device, spin transport across the barrier can be electrically tuned. In another configuration, a multiferroic layer can be used as the exchange bias pinning layer. If the antiferromagnetic spin orientations in the multiferroic pinning layer can be electrically tuned, then magnetoresistance of the device can be controlled by the applied electric field.[50] One can also explore multiple state memory elements, where data are stored both in the electric and the magnetic polarizations.

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