Multiferroicity: the coupling between magnetic and polarization orders

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Multiferroicity: the coupling betweenmagnetic and polarization ordersK.F. Wang a b , J.-M. Liu a b & Z.F. Ren ca Nanjing National Laboratory of Microstructures, NanjingUniversity and Nanjing 210093, China, and School of Physics,South China Normal University, Guangzhou 510006, Chinab International Center for Materials Physics, Chinese Academy ofSciences, Shenyang, Chinac Department of Physics, Boston College, Chestnut Hill, MA 02467,USAVersion of record first published: 30 Jun 2009.

To cite this article: K.F. Wang , J.-M. Liu & Z.F. Ren (2009): Multiferroicity: the coupling betweenmagnetic and polarization orders, Advances in Physics, 58:4, 321-448

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Advances in PhysicsVol. 58, No. 4, July–August 2009, 321–448

Multiferroicity: the coupling between magnetic

and polarization orders

K.F. Wangab, J.-M. Liuab* and Z.F. Renc

aNanjing National Laboratory of Microstructures, Nanjing University,Nanjing 210093, China, and School of Physics, South China Normal University,

Guangzhou 510006, China; bInternational Center for Materials Physics,Chinese Academy of Sciences, Shenyang, China; cDepartment of Physics,

Boston College, Chestnut Hill, MA 02467, USA

(Received 17 October 2008; final version received 26 March 2009)

Multiferroics, defined for those multifunctional materials in which twoor more kinds of fundamental ferroicities coexist, have become one of thehottest topics of condensed matter physics and materials science in recentyears. The coexistence of several order parameters in multiferroics bringsout novel physical phenomena and offers possibilities for new devicefunctions. The revival of research activities on multiferroics is evidenced bysome novel discoveries and concepts, both experimentally and theoretically.In this review, we outline some of the progressive milestones in thisstimulating field, especially for those single-phase multiferroics wheremagnetism and ferroelectricity coexist. First, we highlight the physicalconcepts of multiferroicity and the current challenges to integrate themagnetism and ferroelectricity into a single-phase system. Subsequently,we summarize various strategies used to combine the two types of order.Special attention is paid to three novel mechanisms for multiferroicitygeneration: (1) the ferroelectricity induced by the spin orders such as spiraland E-phase antiferromagnetic spin orders, which break the spatialinversion symmetry; (2) the ferroelectricity originating from the charge-ordered states; and (3) the ferrotoroidic system. Then, we address theelementary excitations such as electromagnons, and the applicationpotentials of multiferroics. Finally, open questions and future researchopportunities are proposed.

Keywords: multiferroicity; ferroelectricity; magnetism; magnetoelectriccoupling; multiferroics; polarization; magnetization; time-reversion sym-metry breaking; spatial-inversion symmetry breaking; helical spin-orderedstate; charge-ordered state; electromagnon; ferrotoroidicity

*Corresponding author. Email: [email protected]

ISSN 0001–8732 print/ISSN 1460–6976 online

� 2009 Taylor & Francis

DOI: 10.1080/00018730902920554

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Contents page

1. Introduction 3232. Magnetoelectric effects and multiferroicity 326

2.1. Magnetoelectric effects 3262.2. Incompatibility between ferroelectricity and magnetism 3312.3. Mechanisms for ferroelectric and magnetic integration 334

3. Approaches to the coexistence of ferroelectricity and magnetism 3353.1. Independent systems 3353.2. Ferroelectricity induced by lone-pair electrons 337

3.2.1. Mechanism for ferroelectricity induced by a lone pair 3373.2.2. Room-temperature multiferroic BiFeO3 340

3.3. Geometric ferroelectricity in hexagonal manganites 3433.3.1. Geometric ferroelectricity and coupling effects in YMnO3 3433.3.2. Magnetic phase control by electric field in HoMnO3 347

3.4. Spiral spin-order-induced multiferroicity 3503.4.1. Symmetry consideration 3503.4.2. Microscopic mechanism 353

3.4.2.1. The inverse DM model 3533.4.2.2. The KNB model 3553.4.2.3. Electric current cancellation model 356

3.4.3. Experimental evidence and materials 3593.4.3.1. One-dimensional spiral spin chain systems 3593.4.3.2. Two-dimensional spiral spin systems 3613.4.3.3. Three-dimensional spiral spin systems 366

3.4.4. Multiferroicity approaching room temperature 3713.4.5. Electric field control of magnetism in spin spiral

multiferroics 3753.5. Ferroelectricity in CO systems 383

3.5.1. Charge frustration in LuFe2O4 3833.5.2. Charge/orbital order in manganites 3863.5.3. Coexistence of site- and bond-centred charge orders 3883.5.4. Charge order and magnetostriction 390

3.6. Ferroelectricity induced by E-type antiferromagnetic order 3953.7. Electric field switched magnetism 399

3.7.1. Symmetry consideration 4003.7.2. Electric polarization induced antiferromagnetism

in BaNiF4 4003.7.3. Electric polarization induced weak ferromagnetism

in FeTiO3 4013.8. Other approaches 403

3.8.1. Ferroelectricity in DyFeO3 4043.8.2. Ferroelectricity induced by A-site disorder 4063.8.3. Possible ferroelectricity in graphene 4063.8.4. Interfacial effects in multilayered structures 407

4. Elementary excitations in multiferroics: electromagnons 4074.1. Theoretical consideration 4084.2. Electromagnons in spiral spin-ordered (Tb/Gd)MnO3 410

322 K.F. Wang et al.

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4.3. Electromagnons in charge-frustrated RMn2O5 4134.4. Spin–phonon coupling in hexagonal YMnO3 4134.5. Cycloidal electromagnons in BiFeO3 414

5. Ferrotoroidic systems 4165.1. Ferrotoroidic order 4185.2. Magnetoelectric effect in ferrotoroidic systems 4215.3. Observation of ferrotoroidic domains 422

6. Potential applications 4256.1. Magnetic field sensors using multiferroics 4256.2. Electric field control of exchange bias by multiferroics 426

6.2.1. Exchange bias in CoFeB/BiFeO3 spin-valve structure 4286.2.2. Exchange bias in Py/YMnO3 spin-valve structure 428

6.3. Multiferroics/semiconductor heterostructures as spin filters 4316.4. Four logical states realized in a tunnelling junction using

multiferroics 4316.5. Negative index materials 433

7. Conclusion and open questions 434Acknowledgements 436References 437

1. Introduction

Magnetic and ferroelectric materials permeate every aspect of modern science andtechnology. For example, ferromagnetic materials with switchable spontaneousmagnetization M driven by an external magnetic field H have been widely usedin data-storage industries. The discovery of the giant magnetoresistance effectsignificantly promoted magnetic memory technology and incorporated it into theeras of magnetoelectronics or spintronics. The fundamental and application issuesassociated with magnetic random-access memories (MRAMs) and related deviceshave been pursued intensively, in order to achieve high-density integration and alsoovercome the large handicap of the relatively high writing energy [1–4]. On the otherhand, the sensing and actuation industry relies heavily on ferroelectric materials withspontaneous polarization P reversible upon an external electric field E, because mostferroelectrics, especially perovskite oxides, are high-performance ferroelastics orpiezoelectrics with spontaneous strain. The coexistence of strain and polarizationallows these materials to be used in broad applications in which elastic energy isconverted into electric energy or vice versa [5]. In addition, there has been continuouseffort along with the use of ferroelectric random-access memories (FeRAMs) [6] asnovel non-volatile and high-speed memory media, and in promoting their perfor-mance as superior to semiconductor flash memories.

As for the trends toward device miniaturization and high-density data storage, anintegration of multifunctions into one material system has become highly desirable.Stemming from the extensive applications of magnetic and ferroelectric materials,it is natural to pursue a new generation of memories and sensing/actuating devicespowered by materials that combine magnetism and ferroelectricity in an effective andintrinsic manner (as shown in Figure 1). The coexistence of several order parameters

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will bring out novel physical phenomena and offers possibilities for new devicefunctions. The multiferroics addressed in this review represent one such type ofmaterial, which do allow opportunities for humans to develop efficient control ofmagnetization or/and polarization by an electric field or/and magnetic field (Figures1 and 2), and to push their multi-implications. The novel prototype devices basedon multiferroic functions may offer particularly high performance for spintronics,

Figure 1. (Colour online) Sketches of ferroelectricity and ferromagnetism integration as wellas the mutual control between them in multiferroics. Favoured multiferroics would offer notonly excellent ferroelectric polarization and ferromagnetic magnetization (polarization–electric field hysteresis and magnetization–magnetic field hysteresis) but also high-qualitypolarization–magnetic field hysteresis and magnetization–electric field hysteresis.(Reproduced with permission from [14]. Copyright � 2006 Elsevier.)

Figure 2. (Colour online) Relationship between ferroelectricity (polarization P and electricfield E), magnetism (magnetization M and magnetic field H), and ferroelasticity (strain " andstress �): their coupling and mutual control in solid or condensed matters represent the coresof multiferroicity. (Reproduced with permission from [16]. Copyright � 2006 AAAS.)


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for example, reading the spin states, and writing the polarization states to reverse thespin states by electric field, to overcome the high writing energy in MRAMs.

Considering that little attention has been paid to multiferroicity until recently,it now offers us the opportunity to explore some important issues which haverarely been reachable. Although ferroelectricity and magnetism have been the focusof condensed matter physics and materials science since their discovery, quite anumber of challenges have emerged in dealing with multiferroicity within theframework of fundamental physics and technological applications. There are, inprinciple, two basic issues to address in order to make multiferroicity physicallyunderstandable. The first is the coexistence of ferroelectricity (electric dipole order)and magnetism (spin order) in one system (hereafter, detail discussions of compositeintegration strategies for the two types of function are excluded, except for asketched introduction given in Section 2.1), since it was once proven extremelydifficult for the two orders to coexist in a single material. Even so, the explorationof the microscopic conditions by which the two orders can coexist intrinsically in onesystem as a non-trivial problem has continued. Second, an efficient coupling betweenthe two orders in a multiferroic system (we always refer to this coupling to themagnetoelectric coupling) seems to be even more important than their coexistence,because such a magnetoelectric coupling represents the basis for multi-control ofthe two orders by either an electric field or magnetic field. Investigations havedemonstrated that a realization of such strong coupling is even more challengingand, thus, the core of recent multiferroic researches.

It should be mentioned here that most multiferroics synthesized so far aretransitional metal oxides with perovskite structures. They are typically stronglycorrelated electronic systems in which the correlations among spins, charges/dipoles,orbitals and lattice/phonons are significant. Therefore, intrinsic integration andstrong coupling between ferroelectricity and magnetism are essentially related to themulti-latitude landscape of interactions between these orders, thus making thephysics of multiferroicity extremely complicated. Nevertheless, it is also clear thatmultiferroicity provides a more extensive platform to explore the novel physicsof strongly correlated electronic systems, in addition to high TC superconductor andcolossal magnetoresistance (CMR) manganites, etc.

Since its discovery a century ago, ferroelectricity, like superconductivity, has beenlinked to the ancient phenomena of magnetism. Attempts to combine the dipole andspin orders into one system started in the 1960s [7,8], and some multiferroics,including boracites (Ni3B7O13I, Cr3B7O13Cl) [8], fluorides (BaMF4,M¼Mn, Fe, Co,Ni) [9,10], magnetite Fe3O4 [11], (Y/Yb)MnO3 [12] and BiFeO3 [13], were identifiedin the following decades. However, such a combination in these multiferroics hasbeen proven to be unexpectedly tough. Moreover, a successful combination of thetwo orders does not necessarily guarantee a strong magnetoelectric coupling andconvenient mutual control between them. Fortunately, recent work along this linehas made substantial progress by discovering/inventing some multiferroics, mainly inthe category of frustrated magnets, which demonstrate the very strong and intrinsicmagnetoelectric coupling. Our theoretical understanding of this breakthrough isattributed to the physical approaches from various length scales/levels.Technologically, growth and synthesis techniques for high-quality single crystalsand thin films become available. All of these are responsible for an upsurge of


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interest in this topic in recent years. In Tables 1 and 2, we have collected several kindsof single-phase multiferroics discovered and investigated recently [14–20].

In this article our intention is to review the state-of-the-art breakthroughs in thisstimulating research field and we have organized the material in the followingmanner. In Section 2, the relationship and differences between the magnetoelectriccoupling and multiferroicity are addressed and the issue why the coexistence ofmagnetism and ferroelectricity is physically unfavoured will be discussed. Section 3is devoted to the theoretical and experimental efforts made so far, by which themagnetism and ferroelectricity were essentially combined and the improperferroelectricity induced by specific magnetic and charge orders was eventuallydemonstrated. The elementary excitations in multiferroics, electromagnons, areclarified in Section 4. In Section 5, we highlight another way to reach strongmagnetoelectric coupling: ferrotoroidical (FTO) systems. The potential applicationsand unsolved problems associated with multiferroicity are proposed in Sections 6and 7.

It should be mentioned that the authors of this article are not in a position tocover every aspect of multiferroicity and its related topics; in fact, such a task is veryhard and is no way our intention here, not only because of the rapid advances inthis field. The conclusion and perspectives are also biased by the authors’ pointof view. We apologize for any glaring omissions with the related work cited here.Any technical deficiencies in this article are, of course, our own.

2. Magnetoelectric effects and multiferroicity

2.1. Magnetoelectric effects

The magnetoelectric effect, in its most general definition, describes the couplingbetween electric and magnetic fields in matter (i.e. induction of magnetization (M )by an electric field (E ) or polarization (P) generated by a magnetic field (H )). In1888, Rontgen observed that a moving dielectric body placed in an electric fieldbecomes magnetized, which was followed by the observation of the reverse effect:polarization generation of a moving dielectric in a magnetic field [21]. Both, however,are not intrinsic effects of matter. In 1894, when considering crystal symmetry, Curiepredicted the possibility of an intrinsic magnetoelectric effect in some crystals [22].Subsequently, Debye coined this kind of effect as a ‘magnetoelectric effect’ [23]. Thefirst successful observation of the magnetoelectric effect was realized in Cr2O3, andthe magnetoelectric coupling coefficient was 4.13 psm�1 (see [24]). Up to now, morethan 100 compounds that exhibit the magnetoelectric effect have been discoveredor synthesized [14–20,25].

Thermodynamically, the magnetoelectric effect can be understood within theLandau theory framework, approached by the expansion of free energy for amagnetoelectric system, i.e.

F ðE,H Þ ¼ F0 � PsiEi �Ms

iHi �1

2"0"ijEiEj �

1

2�0�ijHiHj � �ijEiHj

�1

2�ijkEiHjHk �

1

2�ijkHiEjEk � � � � ,

ð1Þ


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Table

1.A

list

ofmultiferroicsexcludingthose

multiferroicsinducedbyspiralspin

order

(listedin

Table

2).

Compound

Crystalstructure

(space

group)

Magnetic

ions

Mechanism

formultiferroics

Ferroelectric

polarization

Ferroelectric

transition

temperature

Magnetic

transition

temperature

References

RFe 3(BO

3) 4

(R¼Gd,Tb,et

al.)

R32

R3þ,Fe3þ

Ferroelectric-activeBO

3group

Pa�9mC

cm�2

(under

40kOe

magnetic

field)

�38K

�37K

[37,38]

Pb(B

1/2B’ 1/2)O

3

(B¼Fe,Mn,N

i,Co;

B0 ¼

Nb,W

,Ta)

Pm3m

B0

Bionsinducedferroelectricity,

B’ionsinducedmagnetism

�65mC

cm�2

�385K

�143K

[42–45,47]

BiFeO

3R3c

Fe3þ

LonepairatA-site

P[001]�75mC

cm�2�1103K

�643K

[58–84]

BiM

nO

3C2

Mn3þ

LonepairatA-site

�20mC

cm�2

�800K

�100K

[51–57]

Bi(Fe 0

.5Cr 0

.5)O

3�

*Cr3þ

LonepairatA-site

�60mC

cm�2

�*

�*

[90,91]

(Y,Y

b)M

nO

3HexagonalP63cm

Mn3þ

Geometricferroelectricity

�6mC

cm�2

�950K

�77K

[102–106]

HoMnO

3HexagonalP63cm

Mn3þ


�5.6mC

cm�2

�875K

�76K

forMn3þ

�5K

forHo3þ

[116–119]

InMnO

3HexagonalP63cm

Mn3þ


�2mC

cm�2

�500K

�50K

[123,124]

YCrO

3Monoclinic

P21

Cr3þ

Geometricferroelectricity(?)

�2mC

cm�2

�475K

�140K

[125]

Orthorhombic

Y(H

o)M

nO

3

Orthorhombic

Mn3þ

E-typeantiferromagnetism

�100mC

m�2

�28K

�28K

[129,130]

Pr 1�xCaxMnO

3Pnma

Mn3þ,Mn4þ

Siteandbondcentred

charge-order

�4.4mC

cm�2y

�230K

�230K

forcharge

ordered

state

[232–235]

Pr(Sr 0

.1Ca0.9) 2Mn2O

7Am2m

Mn3þ,Mn4þ

Charge/orbitalorder

–*

–*

TCO1�370K

TCO2�315K

[230,231]

LuFe 2O

4R

� 3m

Fe2þ,Fe3þ

Chargefrustration

�26mC

cm�2

�330K

�330K

forcharge

ordered

state

[220–227]

Ca3Co2�xMnxO

7R3c

Co2þ,Mn4þ

Chargeordered

state

plus

magnetostriction

�90mC

m�2

�16.5K

�16K

[242]

RMn2O

5(R¼Y,

Tb,Dy,etc.)

Pbam

Mn3þ,Mn4þ

Chargeordered

state

plus

magnetostriction

�40mC

cm�2

�38K

TN¼43K

TCM¼33K

TIC

M¼24K

[248–281]

(continued

)


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Table

1.Continued.

Compound

Crystalstructure

(space

group)

Magnetic

ions

Mechanism

formultiferroics

Ferroelectric

polarization

Ferroelectric

transition

temperature

Magnetic

transition

temperature

References

(Fe,Mn)TiO

3R3c(high-pressure

phase)

Fe3þ,Mn3þ

Polarizationinducedweak

ferromagnetism

–*

–*

�*

[290]

DyFeO

3Pbnm

Fe3þ,Dy3þ

Magnetostrictionbetween

adjacentantiferromagnetic

DyandFeions

�0.4mC

cm�2

(under

90kOe

magnetic

field)

�3.5K

TNDy�3.5K

TNFe�645K

[296]

*Noexperim

entaldata

available.

yAssumed

from

theim

ageanddata

oftherefined

electrondiffractionmicroscopy.


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Table

2.A

list

ofmultiferroicswithspiralspin-order-inducedferroelectricity.

Compound

Crystalstructure

Magnetic

ions

Spiralspin

wavevectorq

Ferroelectric

temperature

(K)

Spontaneous

polarization(mC

m�2)

References

LiCu2O

2Orthorhombic

(Pnma)

Cu2þ

(0.5,0.174,0)

523

Pc¼4

[130]

LiCuVO

4Orthorhombic

(Pnma)

Cu2þ

(0,0.53,0)

53

Pa¼20

[134,135]

Ni 3V2O

8Orthorhombic

(mmm)

Ni2þ

(0.28,0,0)

3.9–6.3

Pb¼100

[136]

RbFe(MoO

4) 2

Triangular(P

� 3m1)

Fe3þ

(1/3,1/3,0.458)

53.8

Pc¼5.5

[140]

CuCrO

2,AgCrO

2Delafossite(R

� 3m)

Cr3þ

(1/3,1/3,0)

524

30b

[142]

NaCrO

2,LiCrO

2Ordered

sock

salt(R

� 3m)

Cr3þ

(1/3,1/3,0)and

(�2/3,1/3,1/2)

560

Antiferroelectricity

[142]

CuFeO

2Delafossite(R

� 3m)

Fe3þ

(b,b,0)b¼0.2–0.25

511

P¼300(?

c)(H¼6-13T)a

[143]

Cu(Fe,Al/Ga)O

2

Al/Ga¼0.02

Delafossite(R

� 3m)

Fe3þ

?57

P[110]¼50

[144–146]

RMnO

3(R¼Tb,D

y)

Orthorhombic

(Pbnm)

Mn3þ

(0,k,1)k¼0.2–0.39

528

Pc¼500

[147–165]

CoCr 2O

4Cubic

spinel

(m3m)

Cr3þ

(b,b,0)B¼0.63

526

Pc¼2

[181]

AMSi 2O

6(A¼Na,Li;

M¼Fe,Cr)

Monoclinic

(C2/c)

Fe3þ

Cr3þ

?56

Pb¼14

[174]

MnWO

4Monoclinic

(Pc/2)

Mn2þ

(�0.21,0.5,0.46)

7–12.5

Pb¼55

[166]

CuO

Monoclinic

(C2/c)

Cu2þ

(0.506,0,�0.843)

213–230

Pb¼150

[185]

(Ba,Sr)2Zn2Fe 1

2O

22

RhomboheralY-type

hexaferrite

Fe3þ

(0,0,3d)05

d5

1/2

5325

150(H¼1T)a

[183]

Ba2Mg2Fe 1

2O

22

RhomboheralY-type

hexaferrite

Fe3þ

//[001]

5195

P[120]¼80(H¼0.06–4T)a

[184]

ZnCr 2Se 4

Cubic

spinel

Cr3þ

(b,0,0)

520

–a

[179]

Cr 2BeO

4Orthorhombic

Cr3

+(0,0,b)

528

3b

[180]

aAnexternalmagnetic

fieldisneeded

toinduce

thespiralspin

order

andthen

theferroelectricity.

bPolycrystallinesamples.


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where F0 is the ground state free energy, subscripts (i, j, k) refer to the three

components of a variable in spatial coordinates, Ei and Hi the components of the

electric field E and magnetic field H, respectively, Psi and Ms

i are the components

of spontaneous polarization Ps and magnetization Ms, "0 and m0 are the dielectric

and magnetic susceptibilities of vacuum, "ij and �ij are the second-order tensors

of dielectric and magnetic susceptibilities, �ijk and � ijk are the third-order tensor

coefficients and, most importantly, �ij is the components of tensor � which is

designated as the linear magnetoelectric effect and corresponds to the induction

of polarization by a magnetic field or a magnetization by an electric field. The rest

of the terms in the preceding equations correspond to the high-order magnetoelectric

effects parameterized by tensors � and � (see [25]). Then the polarization is

PiðE,H Þ ¼ �@F

@Ei¼ Ps

i þ "0"ijEj þ �ijHj þ1

2�ijkHjHk þ �ijkHiEj þ � � � , ð2Þ

and the magnetization is

MiðE,H Þ ¼ �@F

@Hi¼Ms

i þ �0�ijHj þ �ijEj þ �ijkHjEi þ1

2�ijkEjEk þ � � � : ð3Þ

Unfortunately, the magnetoelectric effect in single-phase compounds is usually

too small to be practically applicable. The breakthrough in terms of the giant

magnetoelectric effect was achieved in composite materials; for example, in the

simplest case the multilayer structures composed of a ferromagnetic piezomagnetic

layer and a ferroelectric piezoelectric layer [25–28]. Other kinds of magnetoelectric

composites including co-sintered granular composites and column-structure compo-

sites were also developed [29–31]. In the composites, the magnetoelectric effect

is generated as a product property of the magnetostrictive and piezoelectric

effects, which is a macroscopic mechanical transfer process. A linear magneto-

electric polarization is induced by a weak a.c. magnetic field imposed onto a d.c.

bias magnetic field. Meanwhile, a magnetoelectric voltage coefficient up to

100V cm�1Oe�1 in the vicinity of electromechanical resonance was reported [25].

These composites are acceptable for practical applications in a number of devices

such as microwave components, magnetic field sensors and magnetic memories. For

example, it was reported recently that the magnetoelectric composites can be used as

probes in scanning probe microscopy to develop a near-field room temperature

scanning magnetic probe microscope [32]. For a complete introduction to the

magnetoelectric effects in composite materials, readers are referred to the review

papers by Fiebig [25] and Nan et al. [26], and hereafter we no longer consider

magnetoelectric composite materials.One way to enhance the magnetoelectric response in single-phase compounds

significantly is to make use of strong internal electromagnetic fields in the

components with large dielectric and magnetic susceptibilities. It is well known

that ferroelectric/ferromagnetic materials have the largest dielectric/magnetic

susceptibility, respectively. Ferroelectrics with ferromagnetism, i.e. ferroelecto-

magnets [33], would be prime candidates for an enhanced magnetoelectric effect.

Consequently, Schmid called materials with two or more primary ferroic order

parameters (ferroelectricity, ferromagnetism and ferroelasticity) ‘multiferroics’ [34].


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It should be mentioned that, except for the coexistence of ferroelectricity andferromagnetism, materials with strong coupling between primary ferroelastic andferromagnetic order parameters, in the class of ferromagnetic martensitic systems,were also synthesized about 10 years ago. For a review of ferroelastic materials, onemay refer to the excellent book of Salje [35]. Since no substantial breakthroughfor ferromagnetic–ferroelastic coupling has been reported, in this article we restrictour attention specifically to single-phase multiferroic compounds exhibiting(anti)ferromagnetism and (anti)ferroelectricity simultaneously.

2.2. Incompatibility between ferroelectricity and magnetism

Given such a definition of multiferroics, the incompatibility between ferroelectricityand magnetism is the first issue we need to address. From the point of view ofsymmetry consideration, ferroelectricity needs the broken spatial inverse symmetrywhile the time reverse symmetry can be invariant. A spontaneous polarization wouldnot appear unless a structure distortion of the high-symmetry paraelectric (PE) phasebreaks the inversion symmetry. The polarization orientation must be different fromthose crystallographic directions that constrain the symmetry of the point group.In contrast, the broken time-reversal symmetry is the prerequisite for magnetism andspin order, while invariant spatial-inverse symmetry applies for most conventionalmagnetic materials in use, but this is not a prerequisite. Among all of the 233Shubnikov magnetic point groups, only 13 point groups, i.e. 1, 2, 20, m, m0, 3, 3m0, 4,4m0m0, m0m20, m0m020, 6 and 6m0m0, allow the simultaneous appearance ofspontaneous polarization and magnetization. This restriction in the crystallographicsymmetry results in the fact that multiferroics are rare in nature. Even so, it is knownthat some compounds belonging to the above 13 point groups do not show anymultiferroicity. Therefore, approaches different from simple symmetry considera-tions are needed.

Most technologically important ferroelectrics such as BaTiO3 and (Pb,Zr)TiO3

are transitional metal oxides with perovskite structure (ABO3). They usually takecubic structure at high temperatures with a small B-site cation at the centre of anoctahedral cage of oxygen ions and a large A-site cation at the unit cell corners [5,6].In parallel, there are a large number of magnetic oxides in a perovskite or aperovskite-like structure. Attempts to search for or synthesize multiferroics havemostly concentrated on this class of compounds. Nevertheless, in spite of there beinghundreds of magnetic oxides and ferroelectric oxides, there is practically no overlapbetween them. This leads to an unfortunate but clear argument that magnetism andferroelectricity tend to exclude each other. This is an issue that has been addressedrepeatedly. So far, the overall picture suggests that all conventional ferroelectricperovskite oxides contain transition metal (TM) ions with a formal configurationd 0, such as Ti4þ, Ta5þ, W6þ, at B-sites (i.e. the TM ions with an empty d-shell).The empty d-shell seems to be a prerequisite for ferroelectricity generation, althoughthis does not mean that all perovskite oxides with empty d-shell TM ions mustexhibit ferroelectricity.

Magnetism, in contrast, requires TM ions at the B-site with partially filled shells(always d- or f-shells), such as Cr3þ, Mn3þ, Fe3þ, because the spins of electrons


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occupying the filled shell completely add to zero and do not participate in magneticordering. The difference in filling the TM ion d-shells at the B-site, which is requiredfor ferroelectricity and magnetism, makes these two ordered states mutuallyexclusive. However, a closer look at this process reveals even more abundantphysics associated with this issue.

Ferroelectrics have spontaneous polarization that can be switched by an electricfield. In particular, they undergo a phase transition from a high-temperature, high-symmetry PE phase that roughly behaves as ordinary dielectrics, into a low-symmetry polarized phase at low temperature accompanied by an off-centre shift ofthe B-site TM ions, as shown in Figure 3 (structurally distorted). In fact, ionic-bondperovskite oxides are always centrosymmetric (therefore, not ferroelectric-favoured).This is because, for centrosymmetric structures, the short-range Coulomb repulsionsbetween electron clouds on adjacent ions are minimized. The ferroelectric stability istherefore determined by a balance between these short-range repulsions favouringthe non-ferroelectric centrosymmetric structure, and additional bonding considera-tions which stabilize the ferroelectric phase.

Currently, two distinctly different chemical mechanisms for stabilizing thedistorted structures in ferroelectric oxides have been proposed in the literature. Infact, both are described as a second-order Jahn–Teller effect. In this section, we onlyaddress one of them: the ligand-field hybridization of a TM cation with itssurrounding anions. Take BaTiO3 as an example. The empty d-states of TM ions,such as Ti4þ in BaTiO3, can be used to establish strong covalency with thesurrounding oxygen anions which soften the Ti–O repulsion [17,36]. It is favourableto shift the TM ions from the centre of O6 octahedra towards one (or three)oxygen(s) to form a strong covalent bond at the expense of weakening the bonds withother oxygen ions, as shown in Figure 4(a). The hybridization matrix element tpd(defined as the overlap between the wave functions of electrons in Ti and O ions)changes to tpd(1þ gu), where u is the distortion and g is the coupling constant. In thelinear approximation, corresponding terms in the energy �(�t2pd=D), where D is thecharge transfer gap, cancel each other [17]. However, the second-order approxima-tion produces an additional energy difference:

�E ffi �ðtpdð1þ guÞÞ2=D� ðtpdð1� guÞÞ2=Dþ 2t2pd=D ¼ �2t2pdðguÞ

2=D, ð4Þ

Figure 3. (Colour online) Lattice structures of the high-temperature PE phase (left) and low-temperature ferroelectric phase (right) of perovskite BaTiO3. In the ferroelectric phase, the B-site Ti ions shift from the centrosymmetric positions, generating a net polarization and thusferroelectricity.


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If the corresponding total energy gain �u2 exceeds the energy loss due to theordinary elastic energy �Bu2/2 of the lattice distortion, such a distortion would beenergetically favourable and the system would become ferroelectric. Referring toFigure 4(b), one observes that only the bonding bands would be occupied (solidarrows) if the TM ion has an empty d-shell, a process that only allows for electronicenergy. If there is an additional d-electron on the corresponding d-orbital (dashedarrow), this electron will occupy an antibonding hybridized state, thus suppressingthe total energy gain. This seems to be one of the factors suppressing the tendencyof magnetic ions to make a distorted shift associated with ferroelectricity [17,36].

Figure 4. (Colour online) (a) Orbital configuration of O–TM–O chain unit (TM is thetransitional metal ion) in perovskite ABO3 cell and (b) the corresponding energy levels.The B-site TM ions with d 0 configuration tend to move toward one of the neighbouringoxygen anions to form a covalent bond.


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Surely, the incompatibility between ferroelectricity and magnetism has evenmore complicated origins than the above model. More realistic ingredients should beincluded in order to understand the suppression of ferroelectricity in systems withmagnetic ions. For example, it has been argued that the breaking of singlet valencestate ððd"p# � d#p"Þ=

ffiffiffi2pÞ by local spin in magnetic ions is responsible for the

incompatibility [17]. This issue still deserves further attention.

2.3. Mechanisms for ferroelectric and magnetic integration

As stated above, ferroelectric perovskite oxides need B-site TM ions with an emptyd-shell to form ligand hybridization with the surrounding anions. This typeof electronic structure likely excludes magnetism. However, not all experimentaland theoretical results support the argument that ferroelectricity and magnetism areabsolutely incompatible, and an integration of them seems to be possible. First,the famous Maxwell equations governing the dynamics of electric field, magneticfield and electric charges, tell us that rather than being two independent phenomena,electric and magnetic fields are intrinsically and tightly coupled to each other.A varying magnetic field produces an electric field, whereas electric current, or acharge motion, generates a magnetic field. Second, the formal equivalence of theequations governing the electrostatics and magnetostatics in polarizable mediaexplains the numerous similarities in the physics of ferroelectricity and ferromagnet-ism, such as their hysteresis behaviour in response to the external field, anomaliesat the critical temperature and domain structures. On the one hand, these couplingphenomena and similarities in terms of the electric dipoles and spins in polarizablemedia imply the potential to integrate ferroelectricity and magnetism into single-phase materials. On the other hand, the hybridization between the B-site cation andanion (i.e. the covalent bond) in ferroelectrics can be seen as the virtual hopping ofelectrons from the oxygen-filled shell to the empty d-shell of the TM ion. In contrast,however, it is the uncompensated spin exchange interaction between adjacentmagnetic ions that induces the long-range spin order and macroscopic magnetiza-tion, where the spin exchange interaction can be mapped into the virtual hoppingof electrons between the adjacent ions. This similarity also hints a possibility tocombine these two orders into one system.

With respect to the roadmaps for integrating ferroelectricity and magnetism,we incipiently address the conceptually simplest situation: to synthesize materialswhich contain separate functional units. Usually, one mixes the non-centrosym-metric units, which may arouse a strong dielectric response and ferroelectricity,together with those units with magnetic ions. An alternative approach refers toperovskite oxides once more, where the A-sites are usually facilitated with cationsof a (ns)2 valence electron configuration, such as Bi3þ, Pb3þ, which favour thestability of ferroelectrically distorted structures. At the same time, the B-sitesare facilitated with magnetic ions providing magnetism. This approach avoids theexclusion rule of ferroelectricity and magnetism at the same sites because, here, theferroelectricity is induced by the ions at the A-sites instead of the same B-site ionsfor magnetism.

Nevertheless, such simple approaches do allow for ferroelectricity and magnetismin one system, but may not necessarily offer strong magnetoelectric coupling,


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partially because the microscopic mechanisms responsible for ferroelectricity andmagnetism are physically very different. The eventual solution to this paradox,if any, is to search for ferroelectricity that is intrinsically generated by special spinorders. This not only enables an effective combination of the two orders but alsothe spontaneous mutual control of them. Fortunately, substantial progress alongthis line has been achieved in recent years, and some novel multiferroics in whichferroelectricity is induced by a geometric distortion and a helical/conical spin order,as well as a charge-ordered (CO) structure, have been synthesized. The details ofthese efforts and results are presented in the next section.

3. Approaches to the coexistence of ferroelectricity and magnetism

3.1. Independent systems

As mentioned above, the conceptually simplest approach is to synthesize multi-ferroics with two structural units functioning separately for the ferroelectricity andmagnetism. The first and most well-known examples are borates, such asGdFe3(BO3)4, which contain ferroelectricity active BO3 groups and magneticions Fe3þ (see [37,38]). In addition to the multiferroicity, these materials exhibitinteresting optical properties. Boracites, such as Ni3B7O13I, are also in this class[8,39]. One can cite many similar compounds, such as Fe3B7O13Cl (see [40]),Mn3B7O13Cl (see [41]) etc., which may exhibit multiferroic behaviours, but note thatthey do not have a perovskite structure.

We address perovskite oxides here. The first route towards perovskitemultiferroics was taken by Russian researchers. They proposed to mix bothmagnetic TM ions with d electrons and ferroelectrically active TM ions with d 0

configurations at the B-sites (i.e. substituting partially the d 0-shell TM ions bymagnetically active 3d ions while keeping the perovskite structure stabilized). It washoped that the magnetic ions and d 0-shell TM ions favour separately a magneticorder and a ferroelectric order, although this may be difficult if the magneticdoping is over-concentrated. The typical (and one of the most studied) compound isPbFe1/2

3þNb1/25þO3 (PFN) in which Nb5þ ions are ferroelectrically active and Fe3þ

ions are magnetic, respectively. While a theoretical prediction of the ferroelectric andantiferromagnetic orders below certain temperatures was given, simultaneousexperiments confirmed the ferroelectric Curie temperature of �385K and the Neelpoint of �143K (see [7,42–44]), noting that the two ordering temperatures arefar from each other. A saturated polarization as high as �65 mCcm�2 in epitaxialPFN thin films was also reported, as shown in Figure 5(a) [45], demonstrating theexcellent ferroelectric property.

The coupling between magnetic order and ferroelectric order in this kind ofmultiferroics is, in most cases, very weak because these two orders originate fromdifferent kinds of ions. The consequent magnetoelectric coupling can be understoodphenomenologically. According to the Ginzburg–Landau–Devonshire theoryinvestigated by Kimura et al. [46], the thermodynamic potential � in a multiferroicsystem can be expressed as

� ¼ �0 þ aP2 þb

2P4 � PEþ a0M2 þ

b0

2M4 �MHþ �P2M2, ð5Þ


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where �0 is the reference potential, a, a0, b, b0 are related coefficients, respectively,and the term �P2M2 is the coupling between P and M (i.e. the magnetoelectriccoupling term).

Surely, a variation of M would influence the ferroelectricity and, eventually, themagnetic transition would result in a change of dielectric constant " / @2�/@P2

around the transition point. Although this response would be quite weak because ofthe very small coefficient �, one can use this response to check the validity of thistheory. As an example, for PFN, the difference in dielectric constant betweenexperimentally measured "(T) and the data extrapolated from the paramagneticregion at temperature T4TN can be denoted as �". By " / @2�/P2, one easilyobtains �"� �M2 (i.e. �" is proportional to the square of magnetization). Yang et al.synthesized high-quality PFN single crystals using a high-temperature flux techniqueand carefully studied the magnetic and dielectric properties as a function oftemperature [47]. Obvious anomalies in the dielectric constant " near the Neel point(�143K) were observed, as shown in Figure 5(c). A linear relationship between �"and M2 in the range of 130–143K was demonstrated, as shown in Figure 5(d),confirming the Ginzburg–Landau–Devonshire theory. This work revealed that theredoes exist magnetoelectric coupling between the ferroelectric order and magneticorder in PFN. Here, the low-temperature magnetic order was approved by the

Figure 5. (Colour online) (a) Ferroelectric P–E loops of Pb(Fe0.5Nb0.5)O3 thin films,(b) ferromagnetic M–H loop of Pb(Fe0.5Nb0.5)O3 single crystal at T¼ 3K, (c) dielectricconstant as a function of temperature for Pb(Fe0.5Nb0.5)O3 single crystal, measured at afrequency of 104Hz, (d) roughly linear behaviour between dielectric variation �" and squaredmagnetization M2 between T¼ 130 and 143K (see the text for details). (Part (a) reproducedwith permission from [45]. Copyright � 2007 American Institute of Physics. Parts (b), (c) and(d) reproduced with permission from [47]. Copyright � 2004 American Physical Society.)


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Mossbauer spectra. In particular, the weak ferromagnetic order, as shown inFigure 5(b), was argued to originate from the magnetoelectric couplinginteraction [47].

In addition to PFN, other multiferroics falling in the category of AB1�xB0xO3,

such as PbFe1/23þTa1/2

5þO3 (see [41]) and PbFe1/23þW1/2

5þO3 (see [48]), weresynthesized. Similar investigations performed on these materials also revealed a weakmagnetoelectric coupling between the ferroelectric and spin orders. Again, it wasshown that the weak magnetoelectric coupling exists because of the different andindependent origins in the two types of orders. We may call these multiferroicsindependent multiferroic materials.

3.2. Ferroelectricity induced by lone-pair electrons

3.2.1. Mechanism for ferroelectricity induced by a lone pair

In addition to the ligand-field hybridization of a B-site TM cation by its surroundinganions, which is responsible for the ferroelectric order, the existence of (ns)2 (lone-pair) ions may also favour breaking the inversion symmetry, thus inducing andstabilizing the ferroelectric order. In general, those ions with two valence electronscan participate in chemical bonds using (sp)-hybridized states such as sp2 or sp3.Nevertheless, this tendency may not be always true and, for some materials, thesetwo electrons may not eventually participate in such bonding. They are called the‘lone-pair’ electrons. The ions Bi3þ and Pb3þ have two valence electrons in an s-orbit,which belong to the lone pairs. The lone-pair state is unstable and will invokea mixing between the (ns)2 ground state and a low-lying (ns)1(np)1 excited state,which eventually leads these ions to break the inversion symmetry [49–51]. This‘stereochemical activity of the lone pair’ helps to stabilize the off-centre distortionand, in turn, the ferroelectricity. In typical ferroelectrics PbTiO3 and Na0.5Bi0.5TiO3,both the lone-pair mechanism and the ligand-field hybridization take effectsimultaneously [49].

The ions with lone-pair electrons, such as Bi3þ and Pb3þ, always locate at A-sitesin an ABO3 perovskite structure. This allows magnetic TM ions to locate at B-sitesso that the incompatibility for TM ions to induce both magnetism andferroelectricity is partially avoided. The typical examples are BiFeO3 and BiMnO3,where the B-site ions contribute to the magnetism and the A-site ions via the lone-pair mechanism lead to the ferroelectricity. In view of the origins for the two typesof orders and magnetoelectric coupling, this approach shows no essential differencefrom the independent multiferroic materials highlighted in Section 3.1.

What is amazing is the intense investigation of BiFeO3 and BiMnO3 all overthe world, which focuses on the enhanced ferromagnetism and ferroelectricity. Thestrong magnetoelectric coupling in the macroscopic sense, such as the mutual controlof ferroelectric domains and antiferromagnetic domains, were revealed by recentexperiments. Therefore, it may be beneficial to devote some effort to addressing thesetwo materials.

In both BiMnO3 and BiFeO3, Bi3þ ions with two electrons in a 6 s orbit (lone

pair) shift away from the centrosymmetric positions with respect to the surroundingoxygen ions, favouring the ferroelectricity. The magnetism is, of course, from Fe3þ


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or Mn3þ ions. BiMnO3 is unique, in which both M and P are reasonably large.In fact, it is one of the very exceptional multiferroics offering both ferroelectric andferromagnetic orders. BiMnO3 has a monoclinic perovskite structure (space groupC2) [52,53], and shows a ferroelectric transition at Tferroelectric� 800K accompaniedby a structure transition shown in Figures 6(a)–(d) with the remnant polarizationof �16 mCcm�2 (see [54–56]), and a ferromagnetic transition at TFM� 110K shownin Figure 6(e) [57], below which the two orders coexist. The electron localizationfunctions (ELFs) obtained by a first principle calculation facilitate a visualizationof the bonding and long pairs in real space which, in turn, approves the ‘lone-pair’mechanism in BiMnO3 (see [51]). In Figure 7(a) is presented the valence ELFsof cubic BiMnO3 projected onto different lattice planes, together with the ELFs ofcubic LaMnO3 for comparison. The blue end of the scale bar represents the statewith nearly no electron localization, while the white end represents completelocalization. It is clearly shown that the ELFs on the Mn–O plane of both cubiccompounds have similar patterns and even a similar spin polarization. However,large differences can be found on the Bi–O plane. The 6s ‘lone pairs’ around Biions are approximately spherical, forming the orange rings of localization. Thisspherically distributed lone pairs form a domain of localization that is reducible andtends to be unstable. In addition, the localization tendency of the lone pairs toform a lobe pattern can be strong enough to drive a structural distortion [48–50].The calculated ELFs for monoclinic BiMnO3 are shown in Figure 7(b). In order to

Figure 6. A summary of experimental results on BiMnO3. (a) X-ray �–2� diffraction spectra atvarious temperatures; (b)–(d) lattice parameters, thermal analysis YG and DTA, andresistivity as a function of temperature, respectively; (e) magnetic M–H hysteresis and (f )hysteresis of magnetodielectric effect D"(m0H )/"(0) against magnetic field at varioustemperatures. (Reproduced with permission from [46]. Copyright � 2003 American PhysicalSociety.)


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Figure 7. (Colour online) (a) Valence electron localization functions projected onto the Bi–Oand Mn–O planes for cubic BiMnO3 (left column) and cubic LaMnO3 (right column). (b)Valence electron localization functions for monoclinic BiMnO3. The blue end of the scale barcorresponds to no electron localization while the white end corresponds to a completelocalization. (Reproduced with permission from [51]. Copyright � 2001 American ChemicalSociety.)


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adapt the traditional lone-pair geometry, the visible regions in the iso-surfacecorrespond to the lobe-like Bi lone pairs allowed by the distorted geometry of themonoclinic structure. Further calculations reveal that the localized lone pair in thedistorted structure is not only composed of the expected Bi 6s and 6p states, but alsoof some contribution from the 2p states on the oxygen ligands [51]. These predictionssuggest that the lone pairs on the Bi ions in BiMnO3 are stereochemically active andare the primary driving force for the highly distorted monoclinic structure and, thus,ferroelectricity in BiMnO3 (see [51]).

The magnetoelectric coupling between the ferroelectricity and magnetismin BiMnO3 would be weak, as argued above and confirmed experimentally. Theobserved dielectric constant shows only a weak anomaly at TFM and is fairlyinsensitive to external magnetic fields. The maximum decrease of dielectric constant" upon a field of 9T appearing around TFM is around 0.6%, as shown inFigure 6(f ) [46].

3.2.2. Room-temperature multiferroic BiFeO3

BiFeO3 is another well-known multiferroic material because it is one of the fewmultiferroics with both ferroelectricity and magnetism above room temperature.The rhombohedrally distorted perovskite structure can be indexed with a¼ b¼c¼ 5.633 A, �¼ �¼ �¼ 59.4� and space group R3c at room temperature, owing tothe shift of Bi ions along the [111] direction and distortion of FeO6 octahedrasurrounding the [111] axis, as shown in Figure 8(a) [58–61]. The electric polarizationprefers to align along the [111] direction, as shown by the arrow. The ferroelectricCurie point is TC� 1103K and the antiferromagnetic Neel point is TN� 643K, whileweak ferromagnetism at room temperature can be observed due to a residualmoment in a canted spin structure [59,60]. The high ferroelectric Curie point usuallyrefers to a large polarization since other typical ferroelectrics with such Curie pointshave a polarization up to about 100 mCcm�2. However, for BiFeO3 single crystals,the measured P along the [001] direction at 77K was 3.5 mCcm�2, indicate a possibleP of only 6.1 mCcm�2 along the [111] direction, as reported in earlier work [62].For polycrystalline samples, the expected value of P should be smaller. The reasonfor this small polarization is possibly due to the high leakage current as a resultof defects and the non-stoichiometry of the test materials.

In fact, this issue has been clarified recently. To overcome this obstacle, recentwork has focused on new synthesizing methods [63–66] and solid solutions of BiFeO3

with other ABO3 ferroelectric materials [67–73]. By improving the method forsingle-crystal growth, high-quality single crystals of BiFeO3 with a polarizationof about 60 mCcm�2 were obtained [63,64], indicating that the [111]-orientedpolarization can reach up to 100 mCcm�2, as shown in Figure 8(b). A new sinteringmethod for polycrystalline ceramics, the so-called liquid-phase rapid sintering, wasdeveloped in the authors’ laboratory with which the volatilization of Bi ions duringthe sintering was essentially avoided [65]. Rapid annealing of pre-sintered BiFeO3

ceramics was also demonstrated in order to enhance the electric property [66]. Theferroelectricity and magnetism can also be significantly enhanced by substitutingBi ions with rare-earth ions such as La3þ and Pr3þ, similarly due to the suppressionof Bi evaporation and mixed valence of Fe ions [67–70].


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While practical applications prefer high-quality BiFeO3 thin films in hetero-epitaxial form, a large amount of effort was devoted to thin films of BiFeO3 (see[74–80]) where the crystal structure is monoclinic rather than rhombohedral as seenin bulk ceramic samples, due to the strain of substrates. Nowadays, high-qualityBiFeO3 epitaxial films with room-temperature polarization as high as 60–80 mCcm�2, which approaches the theoretical value, are available [74,81].Moreover, researches have revealed that the in-plane strains in the thin films coulddrive a rotation of the spontaneous polarization on the (110) plane, while thepolarization magnitude itself remains almost constant, which is responsible for thestrong strain tunablity of the out-of-plane remnant polarization in (001)-orientedBiFeO3 films [81].

Figure 8. (Colour online) (a) Lattice structure of BiFeO3: Bi ion shifting along the [111]direction and the distorted FeO6 octahedra surrounding the [111] axis. Polarization P pointsalong the [111] direction, indicated by the arrow. (b) Measured P–E loop for BiFeO3 singlecrystal. (c), (d) Spin configuration of BiFeO3. The spiral spin propagation wave vector q isalong the [10�1] direction and the polarization is along the [111] direction. These two directionsdefine the ð�12�1Þ cycloidal plane on which the spin rotation proceeds, as shown by the shadedregion in (c) and (d). (Part (b) reproduced with permission from [63]. Copyright � 2007American Institute of Physics. Parts (c) and (d) are reproduced from [207]. Copyright � 2006American Physical Society.)


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BiFeO3 has a complicated magnetic configuration. Neutron scattering experi-

ments have revealed that the antiferromagnetic spin order is not spatially

homogenous but rather a spatially modulated structure [60], manifested by an

incommensurate (ICM) cycloid structure of a wavelength of � 62 nm, as shown

in Figures 8(c) and (d). The spiral spin propagation wave vector q is along the [10�1]

directions and the polarization is along the [111] directions. These two directions

define the ð�12�1Þ cycloidal plane where the spin rotation occurs, as shown by

Figure 8(d) and the shaded region in Figure 8(c). Owing to this feature, the

antiferromagnetic vector is locked within the cycloid, averaged to zero over a scale

of approximately , and responsible for the very weak magnetization of bulk

BiFeO3. It is expected that this cycloid structure may be partially destroyed if the

sample size is as small as the cycloid wavelength (�62 nm), predicting enhanced

magnetization and even weak ferromagnetism in nanoscale BiFeO3 samples. It is this

mechanism that results in the enhanced magnetization in the thin-film sample [74].

Other grain-reducing methods for improving the ferromagnetism of BiFeO3 were

also reported. For example, BiFeO3 nanowires and nanoparticles do show

ferromagnetism [82,83], as shown in Figure 9. Moreover, the optical decomposition

of organic contaminants by a nanopowder of BiFeO3 as a high photocatalyst was

also demonstrated recently [83,84].Based on the same reasons for BiMnO3, one may postulate that the magneto-

electric coupling in BiFeO3 would also be very weak. However, some recent studies

have found that the ferroelectric polarization is closely tied to the ICM cycloid spin

Figure 9. Measured magnetic M–H hysteresis loops of BiFeO3 nanoparticles with differentsizes at T¼ 300K. Open circles denote the bulk sample. Solid circles open up triangles, openrectangles and solid down-triangles denote the samples with grain sizes of 4, 15, 25 and 40 nm,respectively. The inset shows the saturated magnetization Ms (open circles) and the difference(DM, solid circles) between Ms of the nanoparticles and the bulk samples. (Reproduced withpermission from [82]. Copyright � 2007 American Institute of Physics.)


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structure and a significant magnetoelectric effect was observed in BiFeO3. As thissituation is very similar to the ferroelectricity induced by spiral spin order, wecarefully discuss this effect in Section 3.4.5.

In addition to BiMnO3 and BiFeO3, attention has been given to other Bi-containing multiferroics in the same category. For example, bismuth layer-structuredferroelectrics Bi4þnTi3FenO12þ3n (n¼ 1), which is a member of the Aurivillius-typematerials and has a four-layered perovskite structure, is composed of units withnominal composition (Bi3Ti3FeO13)

2� sandwiched between two (Bi2O2)2þ layers

along the c-axis [85]. It has both the ferroelectric and magnetic orders below a certaintemperature. In order to enhance the ferromagnetism and ferroelectricity in BiFeO3,researchers focused on a theoretical prediction [86,87] that Bi2(Fe,Cr)O6 wouldexhibit huge macroscopic magnetization and polarization, due to the ferromagneticsuperexhange interaction between Fe and Cr ions which induces the ferromagneticstate in La2(Fe,Cr)O6 (see [88,89]). However, it is challenging to synthesize materialswith ordered Fe and Cr ions. Meanwhile, compared with pure BiFeO3, sampleswith disordered Fe/Cr configuration showed no significant improvement of themultiferroicity [90,91]. Similarly, Bi2NiMnO6 was also studied carefully, owing to theferromagnetic superexhange interaction between Ni and Mn ions [92–94]. It is alsoworth noting that multiferroic PbVO3 facilitated with another lone-pair ion, Pb2þ,was synthesized recently [95–98], which is very similar to conventional ferroelectricmaterial, PbTiO3. Furthermore, Cu2OSeO3, which is another lone-pair containingmaterial, exhibits the coexistence of piezoelectricity and ferrimagnetism butunfortunately no spontaneous polarization was measured. It exhibits significantmagnetocapacitance effects below the ferromagnetic Curie temperature of approx-imately 60K (see [99,100]). This is because Cu2OSeO3 is metrically cubic downto 10K but the ferrimagnetic ordering reduces the symmetry to rhombohedral R3which excludes the spontaneous ferroelectric lattice distortion. Similar effects werealso observed in SeCuO3 (see [101]).

3.3. Geometric ferroelectricity in hexagonal manganites

For those ferroelectrics addressed in the last two sections, the main driving forcefor the ferroelectric transitions comes from the structural instability toward the polarstate associated with electronic pairing. They were coined as ‘proper’ ferroelectrics.Different from this class of ferroelectrics, some other ferroelectrics have theirpolarization as the by-product of a complex lattice distortion. This class of materials,together with all other ferroelectrics with their polarization originating fromby-product of other order configurations, were coined as ‘improper’ ferroelectrics.Hexagonal manganites RMnO3 with R the rare-earth element (Ho-Lu, or Y),fall into the latter category, and are often cited as typical examples that violate the‘d 0-ness’ rule.

3.3.1. Geometric ferroelectricity and coupling effects in YMnO3

We take YMnO3 as an example [12,102–105]. It is a well-known multiferroic systemwith a ferroelectric Curie temperature Tferroelectric¼ 950K and an antiferromagneticNeel temperature TN¼ 77K. The hexagonal manganites and orthorhombic


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manganites, RMnO3 where R is the relative large ions such as La, Pr, Nd, etc., havevery different crystal structures from those of small R ions, in spite of their similarchemical formulae. The hexagonal structure adopted by YMnO3 and othermanganites with small R ions consists of non-connected layers of MnO5 trigonalbipyramids corner-linked by in-plane oxygen ions (OP), with apical oxygen ions (OT)which form close-packed planes separated by a layer of Y3þ ions. Schematic viewsof the crystal structure are given in Figure 10(a).

The different crystal structures are facilitated with different electronic config-urations. In contrast to conventional perovskites, YMnO3 has its Mn3þ ions notinside the O6 octahedra but coordinated by a five-fold symmetry (i.e. in the centreof O5 trigonal bipyramid). Similarly, R-ions (e.g. Y ions) are not in a 12-fold but a

Figure 10. (Colour online) (a) Lattice structure of ferroelectric YMnO3, with the arrowsindicating the direction of ion shift from the centrosymmetry positions. (b) Electronicconfiguration of Mn ions in the MnO5 pyramid of YMnO3. (Part (a) reproduced withpermission from [106]. Copyright � 2004 Macmillan Publishers Ltd/Nature Materials.)


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7-fold coordination. Consequently, the crystal field level scheme of Mn ions in thesecompounds is different from the usual scheme in an octahedral coordination. Thed-levels are split into two doublets and an upper singlet, instead of a triplet t2g and adoublet eg in orthorhombic perovskites (Figure 10(b)). Therefore, the four d electronsof Mn3þ ions occupy the two lowest doublets, leaving no orbital degeneracy.Consequently, Mn3þ ions in these compounds are not Jahn–Teller active.

Early work in the 1960s established YMnO3 to be ferroelectric with space groupP63cm, and revealed an A-type antiferromagnetic order with non-collinear Mn spinsoriented in a triangular arrangement [12,102]. The ferroelectric polarization arisesfrom an off-centre distortion of Mn ions towards one of the apical oxygen ions.However, careful structural analysis revealed that Mn ions remain very close tothe centre of the oxygen bipyramids and, thus, are definitely not instrumental inproviding the ferroelectricity [106]. The first principle calculation also predictsthat the off-centre distortion of Mn ions is energetically unfavourable. The maindifference between the PE P63/mmc structure and ferroelectric P63cm structureis that all ions in the PE phase are restricted within the planes parallel to the ab plane,whereas in the ferroelectric phase, the mirror planes perpendicular to the hexagonalc-axis are lost, as shown in Figures 10(a) and 11. The structural transition from thecentrosymmetric P63/mmc to the ferroelectric P63cm is mainly facilitated by twotypes of atomic displacements. First, the MnO5 bipyramids buckle, resulting in ashorter c-axis and the OT in-plane ions are shifted towards the two longer Y–OP

bonds. Second, the Y ions vertically shift away from the high-temperature mirrorplane, keeping the constant distance to OT ions. Consequently, one of the two �2.8 AY–OP bond length is reduced down to �2.3 A, and the other is elongated to3.4 A, leading to a net electric polarization [106]. The polarization-dependentX-ray absorption spectroscopy (XAS) at OK and Mn L2,3 edges of YMnO3

demonstrated that the Y 4d states are indeed strongly hybridized with the O 2pstates. This results in large anomalies in the Born effective charges on the off-centredY and O ions [107].

The above picture suggests that the main dipole moments are contributed by theY–O pairs instead of the Mn–O pairs. This is an additional example for the A-siteions-induced ferroelectricity, but the details of the mechanism for such distortionremain puzzling. The demand for close packing is one possible reason. To realize theclose packing, the rigid MnO5 trigonal bipyramids in YMnO3 prefer to tilt and thenlead to the loss of inversion symmetry and the ferroelectricity. Moreover, for ahexagonal RMnO3, a combinatorial approach by structural characterization andelectronic structure calculation, as performed already, seems to devalue the roleof re-hybridization and covalency in driving the ferroelectric transition, which isinstead cooperatively driven by the long-range dipole–dipole interactions and oxygenrotations [106]. Interestingly, the huge Y–OP off-centre displacements are quitedistinct from the small displacements induced by chemical activity available forconventional ferroelectric perovskite oxides, but the induced electric polarizationremains much smaller. Thus, one may argue that this is a completely differentmechanism for ferroelectric distortion [108,109].

The spin configuration of hexagonal YMnO3 is frustrated, which will beaddressed carefully in the next section. The easy plane anisotropy of Mn spinsrestricts the moments strictly on the ab plane, which are thus dominated by the


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strong in-plane antiferromagnetic superexchange interaction. The inter-planeexchange between the Mn spins is two orders of magnitude lower. Therefore,YMnO3 is an excellent example of a quasi-two-dimensional Heisenberg magnet on atriangular lattice with a spin frustration generated by geometric constraint.Accordingly, the Mn spins undergoing long-range order at TN usually developinto a non-collinear configuration with a 120� angle between neighbouring spins[110–115].

Figure 11. Crystal structures of (a) PE phase and (b) ferroelectric phase of YMnO3. Thespheres and pyramids represent Y ions and MnO5 pyramids, respectively. The arrows indicatethe direction of ion shift from the centrosymmetry positions, and the numbers are the bondlengths. (Reproduced with permission from [106]. Copyright � 2004 Macmillan PublishersLtd/Nature Materials.)


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For hexagonal manganites, all theoretical and experimental evidence consistentlyfavours the Y–d 0-ness with re-hybridization being the driving force for theferroelectricity. This stands for a substantial new approach to ferroelectricity.In this framework, the strong coupling between ferroelectric order and magneticorder (magnetoelectric coupling) may be expected because both orders are essentiallyassociated with the lattice structure. For example, Fiebig et al. employed opticalsecond harmonic generation (SHG) to map the coupled magnetic and ferroelectricdomains in YMnO3 (see [113]). In this case, as proposed by the symmetry analysis,YMnO3 has four types of 180� domains denoted by (þP,þl ), (þP,�l ), (�P,�l )and (�P,þl ), respectively, where �P and �l are the independent components of theferroelectric and antiferromagnetic order parameters. Any ferroelectric domainwall will be coupled with an antiferromagnetic domain wall, as shown in Figure 12,thus the sign of the product Pl must be conserved upon crossing a ferroelectricdomain wall [113]. Moreover, a significant anomaly of the dielectric constantin response to the electric field along the ab plane ("ab) can be observed at TN, but noanomaly at TN is available when the electric field is along the c-axis (see [114]). Theseexperiments provide fascinating evidence that supports the strong magnetoelectriccoupling in YMnO3.

3.3.2. Magnetic phase control by electric field in HoMnO3

For RMnO3, such as hexagonal HoMnO3, in addition to the complex Mn spinstructure, usually R3þ ions also carry their own spin (magnetic moment) that isnon-collinear with the Mn spins. The ferroelectric phase of HoMnO3 appears at the

Figure 12. (Colour online) Coupled magnetic and ferroelectric domain structures observed inYMnO3. YMnO3 has four types of 180

� domains denoted by (þP,þl ), (þP,�l ), (�P,�l ) and(�P,þl ), respectively, where �P and �l are the independent components of the ferroelectricand AFM order parameters. (Reproduced with permission from [113]. Copyright � 2002Macmillan Publishers Ltd/Nature.)


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Curie point TC¼ 875K, and possesses P63cm symmetry with a polarizationP¼ 5.6 mCcm�2 (see [116–119]) along the hexagonal c-axis. In addition to theMn3þ ions, Ho3þ ions with f electrons also contribute a non-zero magnetic momentwith the easy axis anisotropy along the c-axis, noting that the Mn3þ spins arerestricted within the basal ab plane due to the anisotropy. The as-induced frustrationfavours four kinds of possible triangular antiferromagnetic configurations, as shownin Figure 13(a), in which the magnetic-ordered states are composed of three magneticsublattices with Mn3þ (3d3) ions at the 6c positions and Ho3þ (4f 10) ions at the 2aand 4b positions, respectively. At low temperatures, the exchange coupling betweenHo3þ and Mn3þ magnetic subsystems becomes strong enough so that additionaldistinct changes of magnetic structure may occur. Below TN� 76K, the Mn spinsfavour the non-collinear antiferromagnetic ordering. The coupling between the Mn

Figure 13. (Colour online) (a) Spin configurations and lattice symmetry of HoMnO3 indifferent temperature ranges with and without electric field. The red arrows represent the Hospins and the yellow arrows for the Mn spins (see the text for details). (b) Dielectric constant asa function of temperature for HoMnO3, indicating three anomalies. (c) Dielectric constant as afunction of temperature for HoMnO3 under different magnetic fields. (Part (a) reproducedwith permission from [119]. Copyright � 2004 Macmillan Publishers Ltd/Nature. Part (c)reproduced with permission from [116]. Copyright � 2004 American Physical Society.)


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spins and Ho spins drives an in-plane rotation of the Mn spins at TSR� 33K.Correspondingly, the Ho spins become magnetically polarized and a smallmagnetization from the antiferromagnetic sublattice was detected and enhancedas temperature fell. In fact, the measured c-axis magnetic susceptibility hasan abrupt decrease at TSR, although the change is small, indicating the onset ofthe antiferromagnetic Ho spin order with magnetic moments aligned along thehexagonal c-axis. At an even lower temperature, THo� 5K, another spinreorientation transition associated with the Ho spins takes place, leading to a low-temperature phase with P63cm magnetic symmetry and a remarkable enhancementof Ho spin moment. This configuration remains antiferromagnetic. The two Ho3þ

sublattices are assumed to be Ising-like ordered along the z(c)-axis, exhibiting theantiferromagnetism or ferri-/ferromagnetism.

It is important to mention that the dielectric property of HoMnO3 is verysensitive in response to the subtle variation of the magnetic order [117,118].The dielectric constant as a function of temperature, "(T), under zero magnetic field,exhibits three distinct anomalies, as shown in Figure 13(b) and (c). At the Neel point,"(T ) shows a clear decrease due to the onset of an antiferromagnetic order with theMn spins. This feature was confirmed in other hexagonal manganites or multi-ferroics and is usually viewed as a symbol of antiferromagnetic ordering. Thetransition into the P63cm magnetic structure at THo� 5.2K is accompanied by asharp increase of "(T ). The most notable anomaly of "(T ) is the sharpest peak atTSR� 32.8K. In addition, the dielectric constant and these anomalies exhibit anevident dependence on the magnetic field. A magnetic field H, imposed along thec-axis, shifts the sharpest peak at TSR toward a lower temperature, and the peak atTHo toward a higher temperature. Eventually, the two peaks develop similar plateausand merge atH� 33 kOe, as shown in Figure 13(c). AboveH� 40 kOe, all anomaliesassociated with "(T ) are suppressed, leaving a small drop at approximately 4K.These additional anomalies indicate the phase complexity and mark the generationof a field-induced reentrant novel phase due to the indirect coupling between theferroelectric and antiferromagnetic orders [117,118].

The most fascinating effect with hexagonal RMnO3 is the magnetic phase controlby an electric field, as demonstrated in HoMnO3 (see [119]). Using an optical SHGtechnique, it was observed that at TN, an external electric field may drive HoMnO3

into a magnetic state different from that under zero electric field, thereby modulatingthe magnetic order of the Mn3þ sublattice, as shown in Figure 13. Moreover,compared with YMnO3, HoMnO3 has an extra magnetic sublattice consisting ofHo3þ ions, which shows an interesting response to electric field. In the presence ofan electric field, the para- or antiferromagnetic state under zero field is convertedinto a ferromagnetic order with strong macroscopic magnetization. The proposedmechanism for this phase control is the microscopic magnetoelectric couplingoriginating from the interplay of the Ho3þ-Mn3þ interactions and ferroelectricdistortion [119]. The large difference in far-infrared spectroscopy regarding theantiferromagnetic resonance splitting of Mn ions between YMnO3 and HoMnO3

demonstrates the ferromagnetic exchange coupling between Mn ions and thesurrounding Ho ions [120]. However, the role of Ho3þ ionic spins in HoMnO3

remains ambitious up to now. For example, the X-ray resonant scatteringexperiments indicated that the magnetic structure of Ho3þ ions remains unchanged


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upon an applied electric field as high as 107Vm�1 (see [121]), which may suggestno contribution of Ho3þ spins to the ferromagnetic state of HoMnO3 under anelectric field.

Similar effects were also identified in other multiferroics in the same category,such as YbMnO3 (see [122]), InMnO3 (see [123,124]) and (Lu/Y)CrO3 (see[125–127]). However, the detailed mechanism of ferroelectricity in these compoundsremains a puzzle. For example, more recently a new concept of ‘local non-centrosymmetry’ in YCrO3 has been proposed to account for the small value ofpolarization observed in spite of the large A-cation off-centring distortion [125,126].It is amazing that these multiferroics may possibly be prepared in a constrainedmanner so that a metastable phase can be maintained using special approaches.For instance, bulk TbMnO3 is of an orthorhombic structure (it is ferroelectric, to beaddressed in next section), but a hexagonal metastable TbMnO3 can be epitaxiallydeposited on an in-plane hexagonal Al2O3 substrate [128]. With respect to the bulkphase, the hexagonal TbMnO3 films may exhibit an around 20 times larger remnantpolarization with the ferroelectric Curie point shifting to approximately 60K.In addition, while an antiferroelectric-like phase and a clear signature of themagnetoelectric coupling were observed in hexagonal TbMnO3 films, the metastableorthorhombic (Ho/Y)MnO3 can be synthesized under high-pressure conditions [129].In the orthorhombic HoMnO3, below the antiferromagnetic Neel point, the Ho spinstilt toward the a-axis from their original alignment (along the c-axis) in thehexagonal phase, and a larger magnetoelectric coupling was detected, probably beingascribed to the E-phase antiferromagnetic order [130], which is carefully discussedin Section 3.6.

3.4. Spiral spin-order-induced multiferroicity

So far, we have reviewed various mechanisms for multiferroicity in several typesof multiferroics. These mechanisms definitely shed light on research on novelmultiferroics. Nevertheless, it should be noted that the perspectives of thesemechanisms are somewhat disappointing. In these multiferroics, the ferroelectricityand magnetism basically originate from different ions or subsystems. In a generaland macroscopic sense, one may not expect a very strong magnetoelectric couplingin these multiferroics. An exception is owed to the ferroelectricity induced directly bythe spin order, meaning that an intrinsic magnetoelectric coupling occurs betweenthe ferroelectric and magnetic order parameters. Keeping this in mind, the primaryproblem is how to overcome the inter-exclusion between ferroelectricity andmagnetism so that any special spin order can induce ferroelectricity.

3.4.1. Symmetry consideration

The inter-exclusion between ferroelectricity and magnetism originates not only fromthe d 0-ness rule, but also from the symmetry restriction of the two types of order.Ferroelectricity needs the broken spatial-inverse symmetry and usually invarianttime-reverse symmetry, in which electric polarization P and electric field E changetheir signs upon an inversion operation of all spatial coordinates r!�r but mayremain invariant upon an operation of time reversal t!�t. In contrast, the broken


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time-reversal symmetry is the prerequisite for magnetism (spin order), in whichmagnetization M and magnetic field H change their signs upon time reversal andmay remain invariant upon spatial inversion. Consequently, a multiferroic systemthat is both ferromagnetic and ferroelectric requires the simultaneous breaking of thespatial-inversion and time-reversal symmetries. The magnetoelectric couplingbetween polarization P and magnetization M is derived from this general symmetryargument [131–133].

First, time reversal t!�t must leave the magnetoelectric coupling invariant.As this operation transforms M!�M, and leaves P invariant, the lowest ordermagnetoelectric coupling term has to be quadratic in M. However, the fourth-orderterm �P2M2 does not contribute to any ferroelectricity because it is compensated bythe energy cost for a polar lattice distortion proportional to �P2, although �P2M2

term may account for the small change in dielectric constant at a magnetictransition (as identified for BiMnO3 etc.) [46]. However, given the case of aspatially inhomogeneous spin configuration (i.e. magnetization M is a functionof spatial coordinates), the above symmetry argument allows for the third-ordermagnetoelectric coupling (i.e. the coupling between a homogeneous polarizationand an inhomogeneous magnetization can be linear in P and contains one gradientof M ) [132].

This simple symmetry argument immediately leads to the following magneto-electric coupling term in the Landau free energy [132,133]:

�MEðrÞ ¼ P � f� � rðM2Þ þ � 0½Mðr �MÞ � ðM � rÞM� þ � � �g, ð6Þ

where r, P and M are vectorized spatial coordinate, polarization and magnetization,and � and � 0 are the coupling coefficients. The first term on the right-hand sideis proportional to the total derivative of the square of magnetization and wouldnot give contribution unless P is assumed to be independent of spatial coordinate r.By including the energy term associated with P, i.e. P2/2e, where e is the dielectricsusceptibility, into the free energy, a minimization of the free energy with respect to Pproduces

P ¼ � 0e½Mðr �MÞ � ðM � rÞM�: ð7Þ

This simple symmetry argument predicts the possible multiferroicity in spin-frustrated systems which always prefer to have spatially inhomogeneous magneti-zation owing to the competing interactions. For example, a one-dimensional spinchain with a ferromagnetic nearest-neighbour interaction J5 0 has a uniformground state with parallel-aligned spins. An additional antiferromagnetic next-nearest-neighbour interaction J 04 0 which meets J 0=j J j4 1=4, (i.e. the Heisenbergmodel H¼�n[J �Sn �Snþ 1þ J 0 �Sn �Snþ 2], where Si is the Heisenberg spin momentat site i referring to a spin chain), frustrates this simple spin order [134], as shownin Figure 14(a). The frustrated ground state is characterized by a spiral spin order(spiral spin-density wave (SDW)) and can be expressed as

Sn ¼ S1e1 cosQ � rþ S2e2 sinQ � r, ð8Þ

where the unit vectors ei (i¼ 1, 2, 3) form an orthogonal basis, e3 is the axis aroundwhich spins rotate and vector Q is given by cos(Q/2)¼�J 0/(4J ). If only S1 or S2 are


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non-zero, this equation describes a sinusoidal SDW, which cannot induce any

ferroelectricity because it is invariant upon the spatial inversion operation r!�r.

Given that S1 and S2 are both non-zero, equation (8) describes a spiral spin order

(spiral SDW) with the spin rotation axis e3. Like any other magnetic order, the spiral

spin order spontaneously breaks the time-reversal symmetry. In addition, it also

breaks the spatial inversion symmetry because the sign reversal of all coordinates

inverts the direction of the spin rotation in the spiral. Therefore, the symmetry of the

spiral spin state allows for a simultaneous presence of multiferroicity. Using equations

(7) and (8), one finds that the average polarization is transverse to both e3 and Q:

�P ¼1

V

Zd 3xP ¼ � 0eS1S2½e3 Q�: ð9Þ

The above simple model can be extended to two- or three-dimensional spin systems.

In general, two or more competing magnetic interactions can induce the spin

Figure 14. (Colour online) (a) Sinusoidal (upper) and spiral (lower) spin order for a one-dimensional spin chain with competing exchange interactions. (b) Geometric spin frustrationin a two-dimensional triangular lattice. The DM interaction in La2CuO4 and RMnO3 areillustrated in (c) and (d). The open arrow in (d) for La2CuO4 denotes the direction of weakferromaguetism and the open arrow in (d) for RMnO3 denotes the direction of as-generatedpolarization. (Reproduced with permission from [19]. Copyright � 2007 Macmillan PublishersLtd/Nature Materials.)


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frustration and the spiral (helical) spin order which, in turn, breaks the spatial-inversion and time-reversal symmetries simultaneously, thus establishing theferroelectric order.

What should be mentioned here is that whether the spiral spin order (spiralSDW) is a prerequisite for generating ferroelectricity remains unclear. It wastheoretically predicted that the acentric dislocated SDW may also drive a ferro-electric polarization [133]. For a SDW order described by M¼M0 cos(qmxþ ’)where qm is the magnetic ordering wave vector and ’ is its phase, the magnetizationM is phase-dislocated with respect to the lattice wave vector. As the spins arecollinear and sinusoidal, a centre of symmetry exists but no directionality isavailable, eventually no ferroelectricity is possible. However, for an acentric SDWsystem, M2 falls behind with respect to polarization P, which is the immediateconsequence of the finite phase difference ’. Thus, M2 has some directionality inrelation to P, which is a sufficient condition for a direct coupling between the twotypes of orders and a macroscopic polarization [133].

Surely, one may expect additional long-range and spatially inhomogeneous spinstructures which can produce non-zero polarization P, following Equation (7). Thisissue remains interesting and deserves further investigation.

3.4.2. Microscopic mechanism

In addition to the symmetry argument disclosed above, a microscopic mechanismresponsible for ferroelectricity in magnetic spiral systems is required. Unfortunately,it was found that such a mechanism is very complex and a clear answer has not yetbeen found. Currently, three theories on the microscopic aspect of magnetoelectriccoupling in magnetic spiral multiferroics have been proposed: the inverseDzyaloshinskii–Moriya (DM) model (exchange striction approach) [135,136], thespin current model (KNB model) [137], and the electric current cancellationmodel [138].

3.4.2.1. The inverse DM model. A plausible microscopic mechanism for ferrroelec-tricity in the spin spiral system is the displacement of oxygen ions driven by theantisymmetric DM interaction [139,140], which is a relativistic correction to theusual superexchange interaction. In fact, it has been a long-standing issue whethera weak (canted) ferromagnetism can be generated by the DM interaction in somecompounds such as La2CuO2. As early as 1957, Dzyaloshiskii pointed out that a‘weak’ ferromagnetism may be possible in antiferromagnetic compounds such asFe2O3 but may not in the isostructural oxide Cr2O3. This prediction was made withinthe framework of symmetry argument. Dzyaloshiskii proposed that an invariantin the free energy expansion of the following form [141]:

EDML ¼ D � ðM LÞ, ð10Þ

where D is the material-specific vector coefficient, M is the magnetization and Lis the antiferromagnetic order parameter (vector), will result in appearance of thesecond-order parameter M at the antiferromagnetic ordering temperature. In otherwords, if the symmetry of a pure antiferromagnetic state is such that the appearance


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of a small magnetization does not lead to further symmetry lowering, anymicroscopic mechanism which favours a non-zero magnetization, even if it israther weak, will lead to M 6¼ 0. A possible microscopic mechanism was proposedsubsequently by Moriya, who pointed out that such an invariant with the requiredform can be realized by an antisymmetric microscopic coupling between twolocalized magnetic moment Si and Sj (see [140]):

EDMij ¼ dij � ðSi SjÞ, ð11Þ

where dij is the prefactor. This invariant term is the so-called DM interaction, anddij is the DM factor.

For a spiral spin-ordered state, the classical low-temperature spin structure canbe described as Si

n ¼ Si0 cosðn� þ �

iÞ, where i¼ (x, y, z). A detailed consideration fortypical multiferroic TbMnO3 was given by Sergienko and Dagotto [135] and isdescribed here. For TbMnO3, S

x0 ¼ S

y0 ¼ Sz

0 ¼ 1:4, �¼ 0.28�, �i is a constant, but notcritical to the physics. Assuming that the positions of Mn ions are fixed and oxygenions may displace from their centre positions, the isotropic superexchange interac-tion of a Mn–O–Mn chain in the x direction (as shown in Figure 14(c)) can bedescribed as

Hex ¼ �Xn

J0 þ1

2J 0jjx

2n þ

1

2J 0?ðy

2n þ z2nÞ

� �ðSn � Snþ1Þ, ð12Þ

where J0, J 0? and J 0k are the exchange constants, and rn¼ (xn, yn, zn) is thedisplacement of oxygen ions located between the Mn spins Sn and Snþ 1. In anorthorhombically distorted structure, the displacement of an oxygen ion can bedescribed as rn¼ (�1)nr0þ �rn, where r0 is a constant and �rn is the additionaldisplacement associated with the ICM structure. Taking into account the elasticenergy Hel ¼ �

Pn ð�x

2n þ �y

2n þ �z

2nÞ=2 associated with the displacement, where � is

the stiffness, the total free energy upon a minimization yields

�zn ¼ ð�1Þn J0?z02�

Xi

S i0fcos � þ cos½ð2nþ 1Þ� þ 2�i�g, ð13Þ

and similar expressions for �xn and �yn can be obtained. Note that this displacementstill cannot induce the ferroelectric polarization because of �nrn¼ 0.

Further consideration has to go to the antisymmetric DM interaction Di(rn) �(SnSnþ 1) which will change its sign under the spatial inversion. For a perovskitestructure, the DM factor Dx(rn)¼ �(0,�zn, yn) and Dy(rn)¼ �(zn, 0,�xn) for the Mn–O–Mn chain along the x and y directions, respectively. The Hamiltonian, dependingon �rn for the Mn–O–Mn chain along the x directions, respectively, can be written as

�HDM ¼Xn

Dxð�rnÞ � ½Sn Snþ1� þHel, ð14Þ

and a minimizing of Equation (12) with respect to �rn (exchange strictive effect) yields

�zn ¼�

�Sx0S

y0 sin � sinð�x � �yÞ,

�x ¼ �y ¼ 0:ð15Þ


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Hence, the DM interaction drives the oxygen ions to shift in one directionperpendicular to the spin chain, thus resulting in an electric polarization, as shownin the lower panel of Figure 14(d). If the spin configuration is collinear, parameter�rn as given by Equation (15) vanishes, i.e. the PE state. For example, in La2CuO4,the weak ferromagnetism (as shown in the upper part of Figure 14(d)) would inducealternative displacement of O atoms and then no ferroelectric polarization. Thissuggests that a non-collinear spin configuration is a necessary ingredient offerroelectricity generation by the DM interaction.

Applying this conceptual picture to a realistic system, such as a perovskitemanganite RMnO3, one can develop a practically applicable microscopic model.Combining the orbitally degenerate double-exchange model together with the DMinteraction, a microscopic Hamiltonian for orthorhombical multiferroic manganitescan be described as [135]

H ¼ �Xia, ��

ta��dþi��diþ a, �� JH

Xi

si � Si þ JAFXia

Si � Siþa

þXia

Dað~r Þ � ½Si Siþa� þHJT þ�12

Xi

ðQ2xi þQ2

yiÞ þ�22

Xi

Xm

Q2mi

!,

ð16Þ

where the first term on the right-hand side accounts for electron hopping (kineticenergy term), the second term is the Hund coupling, the third is an antiferromagneticsuperexchange interaction between neighbour local spins, the fourth term includesthe DM interaction, the fifth refers to the Jahn–Teller term, and the last twoterms come from the ferroelectric phonon modes (the displacement of O atoms).The roles of these terms are summarized in Figure 15. A simulation based on thisHamiltonian revealed the appearance of ICM magnetic ferroelectric phase inducedby ordered oxygen displacement, as shown in Figure 16(a), and the simulatedrelative displacement of oxygen ions (i.e. ferroelectric polarization) is shown inFigure 16(b). This model produces a phase diagram that is in excellent agreementwith experiments [135].

A Monte Carlo simulation on the multiferroic behaviours of a two-dimensionalMnO2 lattice based on this model for multiferroic manganites was reported recently.The simulated ferroelectric polarization induced by the spiral spin ordering and itsresponse to the external magnetic field agree with reported experimental observa-tions [136]. Furthermore, the possible coexistence of clamped ferroelectric domainsand spiral spin domains is predicted in this simulation. In short, it has been arguedthat the DM interaction, competing with other exchange interactions, stabilizes thehelical (spiral) spin order, while the exchange striction effect favours the ferroelectricpolarization.

3.4.2.2. The KNB model. The spin current model to be addressed here was proposedby Katasura, Nagaosa and Balatsky, hence the name KNB model. It serves as thesecond microscopic explanation of multiferroicity in a spiral spin-ordered systemand also refers to manganites [137]. This model is very famous and has been widelyutilized to explain a number of experimentally observed facts due to its clear physicsand simple picture.


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For a spin chain, the spin current from site n to site nþ 1 can be expressed asjn,nþ 1 / SnSnþ 1, which describes the precession of spin Sn in the exchange fieldcreated by spin Snþ 1. The DM interaction leads to the spiral spin configuration andacts as the vector potential or gauge field to the spin current. The induced electricdipole between the site pair is then given by Pn,nþ 1 / rn,nþ 1 jn,nþ 1, where rn,nþ 1 isthe vector pointing to site nþ 1 from site n. Although the model may be over-simplified, it is physically equivalent to the exchange striction approach.

3.4.2.3. Electric current cancellation model. This model stems from fundamentalelectromagnetic principles [138]. The current operator of electrons is defined as thechange in Hamiltonian with respect to the variance of vector potential ofelectromagnetic field, i.e.

J ¼ �c�H=�A, ð17Þ

where A is the vector potential of electromagnetic field and c is the light velocity.In non-relativistic quantum mechanics, the definition of electric current includesthree terms generated from three different physical origins: (1) the contributionof standard momentum; (2) the spin contribution; (3) the contribution of

Figure 15. (Colour online) A schematic illustration of the competing interactions involved inthe Hamiltonian proposed for multiferroic manganites by Sergienko and Dagotto [135]. Themiddle part explains the double-exchange and super-exchange interactions among the Mn 3dorbitals. The lower part shows the phonon modes of oxygen ions, which are coupled to the t2gelectrons of Mn ions by the DM interaction. The upper part shows the modes of the Jahn–Teller distortion.


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spin–orbital coupling. For example, we consider a single electron in a band structure

described by the Hamiltonian

He ¼ð p� eðA=cÞÞ2

2mþ � p� e

A

c

� �� ½� rVðrÞ� � �ðr AÞ � �, ð18Þ

where m* is the effective mass of electrons, � the effective spin–orbital coupling

parameter, �¼ ge/2mc and � is the spin of electrons. In the absence of external

electrodynamic field, i.e. A¼ 0, for a given wave function �(r), the electric current

from the above equation is given by

j ¼ j0 þ �cr ð��Þ þ �eð��Þ rVðrÞ,

j0 ¼ieh

2m½ðr�Þ��ðr�Þ�,

ð19Þ

where h is the Planck constant, and the three terms precisely correspond to the three

physical origins mentioned above. For the magnetization of electrons in the band

with a simple spiral magnetic order, one has

M ¼M0 cosðqx=aÞ, sinðqx=aÞ, 0½ �, ð20Þ

Figure 16. (Colour online) Monte Carlo simulation of multiferroicity based on the Sergienko–Dagotto model Hamiltonian. (a) Simultaneous ferroelectric and magnetic transitionscharacterized by polarization P and AFM structural factor S(�/2,�/2). (b) Spin configurationof the spiral-ordered state and oxygen ion displacement (ferroelectric polarization). Thearrows indicate the direction of the Mn spins and the filled circles represent the oxygen ions.(Reproduced with permission from [135]. Copyright � 2006 American Physical Society.)


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whereM0 is the magnetic moment, q is the spiral wave vector, a is the lattice constantand x is the coordinate. The electric current associated with the magnetization isgiven by

JM ¼ �cr M ¼�cqM0

að0, 0, cos qx=aÞ, ð21Þ

which represents the current along the z direction.In an insulator, the net electric current with such a configuration must be zero,

based on Kohn’s proof of the insulator property. The total electric currentcontributed from j0 in the band also vanishes since the lattice mirror symmetry in thex–y plane is not broken for the non-collinear multiferroics in the absence of anexternal magnetic field. Therefore, the electric current from the magnetic orderingmust be counterbalanced by the electric current induced from the spin–orbitcoupling. This cancellation requirement leads to

�cr Mþ �eMrVðrÞ ¼ 0: ð22Þ

By a simple algebraic modification and averaging over the total space, the aboveequation becomes

�e2

�cEh i ¼

ðM � EÞM

M20

� �þ

M ðr MÞ

M20

� �, ð23Þ

where h. . .i refers to the space averaging and V(r)¼�eE(r). The first term on theright-hand side of Equation (23) usually vanishes when a space averaging for aspatially modulated spin density is made. The total ferroelectric polarization canthen be written as

P ¼"0�c

�e2M ðr MÞ

M20

� �: ð24Þ

It is worth mentioning that the generated polarization P is inversely proportionalto the effective spin–orbital coupling parameter, a very unusual argument.Moreover, one can conclude that there is no contribution to the ferroelectricityfrom the completely filled bands since electrons in a fully filled band do not havea magnetization response. Therefore, the contribution to the ferroelectricity onlycomes from the band which is partially filled, i.e. multiferroics must not be anconventional insulator but an insulator with a partially filled band. The strongelectron–electron coupling or spin–exchange coupling between the electrons on theband and the localized spin moment can cause an insulator with partially filled band.

The significance of this model is presented by a limitation on the ferroelectricpolarization, i.e. the energy gap Dg in the insulator. If there is an internalelectric field E which is spontaneously generated, the electric field must satisfyejE jh=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mDg

p5Dg in order to maintain the validity of the insulator. A semi-

quantitative estimation gives a polarization of only approximately 100 mCm�2 fortypical manganites, a disappointing prediction from the point of view of techno-logical applications.

In spite of different microscopic origins, the three models outlined above givea similar prediction: Pn,nþ1 / rn,nþ1 (SnSnþ1). Furthermore, these models are


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all based on the transverse spiral spin-ordered state in which the spin spiral planecontains the propagation vector of spin modulation. This postulation, in fact, maynot be always true. Some other spiral spin-ordered states, which are not reachableby the three models, can indeed induce ferroelectricity, to be addressed below.In summary, the issue of multiferroicity as generated in spiral spin-ordered systemsremains attractive, thus making a more careful consideration necessary. However,it is now generally accepted that the spin–orbit coupling and the DM interaction doplay important roles.

3.4.3. Experimental evidence and materials

The wealth of evidence that supports spiral spin-order-induced multiferroicityand intrinsic magnetoelectric coupling parallels the current theoretical progress. Wecollect some of the main results below and we show that the theory of spiral spin-order-induced ferroelectricity is in principle an appropriate description although thistheory does not take all of phenomena observed so far into account.

3.4.3.1. One-dimensional spiral spin chain systems. We first deal with the one-dimensional (1D) spin systems. The 1D chain magnet with competingnearest-neighbour ferromagnetic interaction (J) and next-nearest-neighbour anti-ferromagnetic interaction (J 0) will develop its configuration into a frustrated spiralspin order as long as jJ 0=Jj4 1=4 (see [142]), as already theoretically predictedin Figure 14(a). Experimentally, the spin configuration of LiCu2O2 can beapproximately treated as a quasi-1D spin chain system and the crystal structure isshown in Figure 17(a), where magnetic Cu2þ ions are blue and non-magnetic copperions are green with red dots for oxygen ions. The blue bonds constitute the quasi-1Dtriangle spin ladders, with the weaker inter-ladder interaction (J?) than the in-ladderinteractions (J1 and J2). Therefore, each ladder can be viewed as an independent 1Dspin chain, as shown in Figure 17(b). In fact, the picture of a quasi-1D spin spiral isalso physically sound since the equivalent nearest-neighbour exchange interactionsand frustration ratio estimated experimentally for LiCu2O2 are J1¼ 5.8meV andJ2/J1¼ 0.294 1/4 (see [142,143]). Indeed, a non-collinear spiral spin order wasidentified for these quasi-1D spin ladders with a spiral propagation vector (0.5, , 0)and ¼ 0.174 was determined. Consequently, within the theoretical frameworkaddressed above, the ferroelectric polarization along the c-axis (Pc) would beexpected, and was experimentally evident in LiCu2O2, as shown in Figure 17(c).The anomaly of the dielectric constant at the magnetic transition point and thespontaneous Pc below this point, shown in Figure 18, are quite obvious [142].

More exciting is the intrinsic magnetoelectric coupling between the spin orderand ferroelectric order, which is evident in the response of polarization to an externalmagnetic field [142]. The external field along the b-axis drives the rotation of spinswithin the bc plane (Figure 17(c)) toward the ab plane, as shown in Figure 17(d) and,correspondingly, a switch of the polarization orientation from the c-axis to thea-axis, as shown in Figure 18, was observed [142].

Nevertheless, it should be mentioned that not all of the experimental results onLiCu2O2 can be successfully explained by this one-dimensional spin chain model


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[144,145], while similar copper oxide, LiCuVO4, was also identified as a multiferroicmaterial [146,147]. For example, we look at the response of polarization P to anexternal magnetic field. Whatever the magnetic field applies along the b-axis ora-axis, the spiral spin order will be transferred into a parallel aligned configurationwhich would no longer generate any spontaneous polarization, while experimentallythe suppression of polarization along the c-axis is accompanied with the appearanceof polarization along the a-axis, which is not explainable theoretically. Therefore,one may argue that an additional contribution to the polarization generation isinvolved. Furthermore, for LiCu2O2 and LiCuVO4, early neutron scatteringstudies revealed the ICM magnetic structure with a modulation vector(0.5, 0.174, 0), in which the Cu2þ magnetic moment lies in the CuO2 ribbon plane

Figure 17. (Colour online) (a) Crystal structure of LiCu2O2 and (b) its spin-orderedconfiguration with multifold exchange interactions. The blue lines indicate the quasi-1D spinladders consisting of Cu ions. The red spheres represent the oxygen ions and grey spheresdenote the Li ions. Spiral arrangements of the Cu spin ladders and corresponding polarizationunder (c) zero magnetic field and (d) 9.0 T applied along the b-axis. (Reproduced withpermission from [142]. Copyright � 2007 American Physical Society.)


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(i.e. the ab plane) [142]. However, according to the KNB model or the inverse-DMmodel, the spontaneous polarization along the a-axis is associated with the ab-planespin spiral. This is true for LiCuVO4 (see [146]), but unfortunately for LiCu2O2 thepolarization aligns along the c-axis [142]. A possible reason is that the KNB modeland the inverse DM model were formulated for the t2g electron system, while forLiCu2O2 an unpaired spin resides in the eg orbital.

This issue was recently checked carefully by XAS and neutron scattering, anda possible bc-plane spin spiral was proposed [145]. Moreover, experiments revealedthat the ground state of LiCu2O2 has long-range two-dimensional-like ICMmagnetic order rather than being a spin liquid of quantum spin-1/2 chains due tothe large interchain coupling which suppresses quantum fluctuations along the spinchains. The spin coupling along the c-axis is essential for generating electricpolarization [148]. Nevertheless, so far no conclusive understanding has beenreached.

3.4.3.2. Two-dimensional spiral spin systems. An example of the two-dimensional(2D) frustrated spin system is the Kagome staircase Ni3V2O8 which can be viewed as

Figure 18. Measured physical properties of LiCu2O2 as a function of temperature: (a) mag-netic susceptibility along the b-axis and its temperature derivative; (b) dielectric constant alongthe c-axis; (c) polarization along the c-axis and that along the a-axis; (d) under variousmagnetic fields as numbered (in Tesla). (Reproduced with permission from [142]. Copyright �2007 American Physical Society.)


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a quasi-2D spin structure with a frustrated spin order. Similar experiments regardingthe electric polarization together with the spin structure and phase diagram aresummarized in Figure 19 [149–151].

The well known geometrically frustrated spin systems go to those 2D triangularlattices with an antiferromagnetic interaction, as shown in Figure 14(b). While thesecond spin can easily align in antiparallel with the first spin due to the anti-ferromagnetic interaction, the third, however, cannot align in a stable way to the firstand second spins simultaneously, leading to a frustrated spin structure. Surely, realsystems seem far more complicated than this simple picture and the inter-spininteractions can be competitive and entangled. Given the classical Heisenberg spins,the 2D triangular lattice generally favours the 120� spiral spin order at the groundstate. Depending on the sign of anisotropy term H¼D�(Sz

i )2 where Sz

i is the z-axiscomponent of spin Si, the spin spiral is confined parallel (D4 0, easy-plane type) to,or perpendicular (D5 0, easy-axis type) to, the triangular-lattice plane [152].

RbFe(MoO4)2 (RFMO) exhibits the typical easy-plane triangular lattice, whichis described by space group P�3m1 at room temperature. At T0¼ 180K, the symmetryis lowered to P3 by a lattice distortion, as shown in Figure 20(a), in which the

Figure 19. (Colour online) (a) Lattice structure and three spin arrangements in Ni sublatticefor Ni3V2O8. The LTI (low-temperature insulator) phase exhibits a spin spiral structure whichcan induce ferroelectric polarization P along the b-axis, while the spins in the HTI (high-temperature insulator) and CAF (canted antiferromagnetic) phases are collinear. (b) Phasediagram of magnetic field against temperature for Ni3V2O8 under magnetic field along the a-axis and c-axis, respectively. (c) Polarization along the b-axis as a function of temperature andmagnetic field applied along the a-axis and c-axis, respectively. (Reproduced with permissionfrom [151]. Copyright � 2005 American Physical Society.)


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out-of-plane ions lead to two types of triangles: the ‘up triangles’ with a green oxygentetrahedron above the plane and the ‘down triangles’ with a tetrahedron below theplane [153]. For T5T0, RFMO contains perfect Fe3þ triangular lattice planesin which spins S¼ 5/2 are coupled through antiferromagnetic superexchangeinteractions. The magnetism is dominated by the intra-plane interactions of anenergy scale of approximately 1.0meV and the inter-plane interaction of at least25 times weaker [154]. Therefore, RFMO is essentially a XY antiferromagnet on atriangular lattice with a long-range magnetic ordering at TN¼ 3.8K. The magneticground state is shown in Figure 20(b). The magnetic ordering wave vector in thereciprocal lattice units is q¼ (1/3, 1/3, qz) with qz� 0.458 at T5TN under zeromagnetic field. This feature implies the absence of a mirror plane perpendicularto the c-axis, and experimental measurement revealed an electric polarization ofapproximately 5.5 mCcm�2 along the c-axis [153].

However, according to the KBN model or the inverse DM model, the generatedlocal polarization is Pn,nþ 1/ rn,nþ 1 (SnSnþ 1), and thus lies in the basal planefor RFMO. In view of the three-fold rotation axis, the macroscopic polarizationP¼�nPn,nþ 1 vanishes. This means that neither the KNB model nor the inverse DMmodel can explain the origin of ferroelectricity in RFMO.

CuCrO2 with the delafossite structure (as shown in Figure 21(a)) is anothertypical triangular-lattice antiferromagnet with the easy-axis type, and the magnetic

Figure 20. (Colour online) Crystal lattice structure (a) and 120� spin-ordered state (b) intriangular-lattice RbFe(MoO4)2. (Reproduced with permission from [153]. Copyright � 2007American Physical Society.)


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properties are dominated by Cr3þ ions with S¼ 3/2 spin [155]. Recent studiesrevealed the 120� spin structure with the easy-axis anisotropy along the c-axis, inwhich the spin spiral is in the (110) plane and the spins rotate in the planeperpendicular to the wave vector, as shown in Figure 22. Again, the KNB model orthe inverse DM model predicts that only polarization perpendicular to the spin spiralplane (along the [110] directions) is possible and the net polarization vanishesbecause any 120� spin structure produces the same SnSn,nþ 1 for all bondsin the triangular lattice. Nevertheless, experiments revealed a sharp anomaly of

Figure 21. (Colour online) Crystal lattice structures of (a) CuCrO2 with delafossite structureand (b) (Li/Na)CrO2 with ordered rock salt structure. (Reproduced with permission from[155]. Copyright � 2008 American Physical Society.)

Figure 22. (Colour online) (a) Symmetry elements in CuCrO2 with space group R�3m: two-foldrotation axis 2, reflection mirror m, and three-fold rotation axis along the c-axis with inversioncentre. (b) Symmetry elements (left) and a schematic figure (right) of the 120� spin-orderedstructure with (110) spiral plane. (Reproduced with permission from [155]. Copyright � 2008American Physical Society.)


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dielectric constant at TN and a polarization of approximately 20 mCm�2 below TN

(see [155]).In addition to RFMO and CuCrO2, LiCrO2 and NaCrO2 also exhibit the 2D

triangular-lattice structure, but they do not exhibit any electrical polarization over

the whole temperature range since they are probably antiferroelectrics due to a

different sock salt structure, as shown in Figure 21(b) [155]. CuFeO2 is a quasi-2D

example consisting of Cu and Fe triangular layers, as shown in the inset of

Figure 23(a) [156,157]. The complex magnetization behaviour such as five M–H

plateaus was observed, which is in physics attributed to the spin–phonon coupling

[157]. For a magnetic field between 6 and 13T, the ground state will evolve from the

collinear commensurate (CM) order into non-collinear ICM frustrated state. The

non-zero polarization inside this magnetic field range, accompanied with remarkable

dielectric anomalies at the magnetic transition point below 11K, was observed, as

shown in Figure 23. A doping at the Fe sites with non-magnetic ions such as Al3þ

and Ga3þ can also induce the non-collinear ICM spin state and then observable

electric polarization [158,159]. It is revealed that the possible microscopic origin of

the ferroelectricity is the variation in the metal–ligand hybridization with spin–orbit

coupling [160].

Figure 23. (Colour online) A summary of experimental results on CuFeO2: (a) phase diagramof the magnetic field against temperature (the inset in the top right corner is the crystalstructure); (b) a.c. magnetic susceptibility (the inset is the dimensional dilation); (c) dielectricconstant measured in parallel and perpendicular to the c-axis as a function of temperature,respectively; (d) polarization perpendicular to the c-axis as a function of magnetic field atseveral temperatures (from top to bottom: T¼ 2, 7, 9, 10 and 11K). (Reproduced withpermission from [156]. Copyright � 2007 American Physical Society.)


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3.4.3.3. Three-dimensional spiral spin systems. We finally highlight recent investiga-tions on three-dimensional (3D) frustrated spin systems [161–195]. Typical examplesare perovskite manganites Tb(Dy)MnO3 (see [161–180]). We pay special attentionto TbMnO3 which has been investigated extensively. At room temperature, TbMnO3

has an orthorhombically distorted perovskite structure (space group Pbnm), differentfrom antiferromagnetic–ferroelectric hexagonal rare-earth manganites RMnO3

(R¼Ho, Y, etc.). The t32ge1g electronic configuration of the Mn3þ site is identical

to the parent compound of CMR manganites LaMnO3 where the staggeredd3x2�r2=d3y2�r2 orbital order favours the ferromagnetic spin order in the ab-planeand antiferromagnetic order along the c-axis. A replacement of La by smaller ions,such as Tb and Dy, enhances the structural distortion and strengthens the next-nearest-neighbour antiferromagnetic exchange, compared with the nearest-neigh-bour ferromagnetic interaction in the ab-plane. Consequently, the competitionbetween the two types of interactions frustrates the spin configuration within theab-plane and then induces successive magnetic phase transitions at low temperature.Theoretical investigation predicted that the Jahn–Teller distortion, together with therelatively weak next-nearest-neighbour superexchange coupling in perovskite multi-ferroic manganites is shown to be essential for the spiral spin order [161].

At room temperature, the crystal symmetry of TbMnO3 has an inversion centre,and the system is non-polar. Magnetic and neutron scattering experiments showedthat the spin structure of Tb(Dy)MnO3 favours an ICM collinear sinusoidalantiferromagnetic ordering of Mn3þ spins along the b-axis, taking place at TN¼ 41Kwith a wave vector q¼ (0, ks� 0.29, 1) in the Pbnm orthorhombic cell, as shown inFigure 24(a), (b) and (c) [162]. It is easily understood that the collinear sinusoidalantiferromagnetic state is PE and the ferroelectric phase may not appear unless thespin order is spiral or helicoidal-like. The non-zero polarization appears only belowapproximately 30K (Tlock) where an ICM–CM (or lock-in) transition occurs,generating a helicoidal structure with the magnetic modulation wave vector kswhich is nearly temperature-independent and locked at a constant value of about0.28 (see [162]).

It is easily predicted that the generated electric polarization P� e ks where eis the unit vector connecting the neighbouring two spins, is parallel to the c-axis,because vector ks is along the b-axis and the spin helicoidal points to the a-axis.This prediction is consistent with experiments, as shown in Figure 24(d) and (e). Thedielectric constant along the c-axis ("c) exhibits a sharp peak at the lock-in point(Tlock), below which only the polarization along the c-axis is observable under a zeromagnetic field. Further decrease of the temperature leads to the third anomalyof magnetization and specific heat as a function of temperature at approximately7K, at which the Tb3þ spins initiate the long-range ordering with a propagationvector (0,�0.42, 1). Simultaneously, the electric polarization also exhibits a smallanomaly.

TbMnO3 is similar to those improper ferroelectrics mentioned earlier and itspolarization is a secondary order parameter induced by the lattice distortion. As thelattice modulation (distortion) is accompanied with the spin order, the intrinsicmagnetoelectric coupling between the spin and polarization may be expected. In fact,experiments confirmed the realignment of polarization by an external magnetic field.A magnetic field over approximately 5T applied along the b-axis suppresses the


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polarization along the c-axis (Pc) significantly, below a temperature Tflop whichincreases with increasing magnetic field, as shown in Figure 25(c). In contrast, afinite polarization along the a-axis (Pa), is generated, with the onset point perfectlyconsistent with Tflop (as shown in Figure 25(d)). These experiments demonstrateconvincingly the intrinsic magnetoelectric coupling effect characterized by aspontaneous switching of polarization from one alignment to another, as shownin Figures 25(c) and (d). DyMnO3 also exhibits polarization flop from Pkc to Pka byapplied magnetic field. Whereas in TbMnO3 the polarization flop is accompaniedby a sudden change from ICM to CM wave vector modulation, in DyMnO3 the wavevector varies continuously through the flop transition [175].

At the same time, the colossal magnetodielectric effect associated with theremarkable response of dielectric constant along the c-axis and a-axis, respectively,to external magnetic field, is shown in Figures 25(a) and (b). This colossal effect wasargued to be related to the softening of element excitations in these systems, just thesame as the softening of phonons in normal ferroelectrics [175]. However, careful

Figure 24. (a) Spin configuration of the ICM collinear sinusoidal spin-ordered state atT¼ 35K (upper) and the spiral spin-ordered state at 15K (middle: the bc-plane, lower: 3Dview) in TbMnO3. The measured magnetization and specific heat, modulation wave number,dielectric constant and polarization along the a-axis, b-axis and c-axis, respectively, are shownin (b), (c), (d) and (e). (Reproduced with permission from [162]. Copyright � 2003 MacmillanPublishers Ltd/Nature.)


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study of the dielectric spectra of DyMnO3 found that this colossal effect is aphenomenon emerging only below 105–106Hz and the spectrum shape is not theresonance type but the relaxation type, indicating an origin other than the bosonicexcitations [181]. It was postulated that this colossal effect may be attributed to thelocal electric field-driven motion of the multiferroic domain walls between thebc-plane spin cycloid (Pkc) and ab-plane spin cycloid (Pka) domains, as shown inFigure 26. Moreover, this motion exhibits an extremely high relaxation rate of about107 s�1 even at low temperature, indicating that the multiferroic domain wallemerging at the polarization flop is thick rather than the Ising-like thin domainwall identified in conventional ferroelectrics [181].

It should be pointed out that the effect of magnetic field on the electricpolarization is orientation-dependent. This remains to be a non-trivial issue [169].On the one hand, when the magnetic field is applied along the a-axis or the b-axis,both the magnetization and polarization along the c-axis exhibit double metamag-netic transitions, and the polarization (Pc) is drastically suppressed at the secondmetamagnetic transition (�10T for Hka and �4.5T for Hkb). This suppression isdue to the flop of the electric polarization from the c-axis to the a-axis, as shown inFigures 25(c), 27(d) and 27(e), coinciding with a first-order transition to a CM butstill long-wavelength magnetically modulated state (revealed by the magnetizationcurves in Figures 27(a) and (b)), with a propagation vector of (0, 1/4, 1) (see [154]).On the other hand, a magnetic field above approximately 5T applied along the c-axiscauses a single metamagnetic transition, and suppresses the polarization along any

Figure 25. Dielectric constants (a) and (b) and polarizations (c) and (d) along the c-axis anda-axis as functions of temperature under different magnetic fields for TbMnO3. (Reproducedwith permission from [162]. Copyright � 2003 Macmillan Publishers Ltd/Nature.)


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crystallographic orientation, as shown in Figures 27(c) and (f). This effect is relatedto the disappearance of the ICM antiferromagnetic ordering with the (0, 1, 0)magnetic Bragg reflection.

As for the mechanism of the electric polarization flop induced by externalmagnetic field along the a-axis or b-axis, two possible scenarios were proposed. Thefirst and direct scenario is that the field-induced phase with P along the a-axis is also

Figure 26. (Colour online) (a) Various multiferroic domain walls conceivable in DyMnO3. (b)Calculated domain wall structure between the Pkþc and Pkþa domains. Blue and red arrowsrepresent the Mn spins and local polarizations, respectively. The colour gradation representsthe angle of local polarization relative to the a-axis. (Reproduced with permission from [181].Copyright � 2009 American Physical Society.)


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a spiral spin-ordered state, corresponding to the spin rotation from the a-axis to thec-axis. However, recent neutron scattering experiment revealed that the field-inducedmagnetic phase is a non-spiral CM phase with the propagation vector (0, 1/4, 0)(see [168]). This spin-modulation-induced lattice distortion is attributed to theferroelectric order due to the E-type antiferromagnetic, which is discussed againin Section 3.6.

The multiferroicity in systems with spiral spin order was confirmed in severalother perovskite manganites. Figure 28 summaries the phase diagram by plottingtemperature T against the Mn–O–Mn bond angle � which scales the rare-earthionic radius. The shaded region corresponds to the spiral spin order and, thus, themultiferroicity [163]. Those manganites with even smaller rare-earth ions may exhibitgeometrical ferroelectricity, as already discussed in Section 3.3.

At the end of this section, we mention that the multiferroics with spiral spin orderdo show the intrinsic magnetoelectric coupling, as demonstrated by carefulexperiments. However, their ferromagnetism seems to be very weak since essentiallyno spontaneous magnetization is available due to the helical or spiral spin order.An extension of this spiral spin order concept can partially avoid this problem.For example, conical spin state is also a kind of spiral spin order, in which thespontaneous component Sk (homogeneous ferromagnetic part) and spiral compo-nent of the magnetization coexist, as shown in Figure 29(a) [196]. If the spiralcomponent lies in the (e1, e2) plane, S points to the e3-axis, one has the spin momentSn¼S1e1cos(Q � r)þS2e2sin(Q � r)þSke3, where Q is the wave vector and r is thespace coordinate. Chromite spinels, CoCr2O4 (Figure 29(b)) [197–200] do show suchexceptional conical spin structure.

Figure 27. (a)–(c) Magnetization and (d)–(f) magnetic-field-induced changes of polarizationalong the c-axis for TbMnO3, as a function of external magnetic field along the a-axis, b-axisand c-axis, respectively, at various temperatures. The inset of (a) shows a magnified view ofthe high-field region. (Reproduced with permission from [169]. Copyright � 2005 AmericanPhysical Society.)


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In CoCr2O4, Co2þ and Cr3þ ions occupy the tetrahedral (A) and octahedral (B)

sites respectively. Owing to the nearest-neighbour and isotropic antiferromagneticA–B and B–B exchange interaction (JAA and JBB) with JBB=JAA 4 2=3, a conicalstate with the spiral wave vector Q� 0.63 was identified below approximately 27K(see [110]). The ferromagnetic M–H hysteresis and spontaneous polarizationP�Q [001]� [�1, 1, 0], were identified, as seen in Figures 29(c) and (d) [197]. Areversal of external magnetic field could trigger the switching of polarization becauseof the transition of (M, Q) to (�M, �Q), as seen in Figures 29(e) and 30(c). Thisprocess is very quick, which makes it attractive for potential applications. Moreover,there is another magnetic transition at TL� 14K which is a magnetic lock-intransition and has the first-order nature. The spontaneous polarization exhibits adiscontinuous jump and changes its sign without reversal of spin spiral wave vectorQ at this transition temperature. This fact is in contrast to the above discussion, asshown in Figure 30(a) [201]. Below this temperature, although the electricpolarization can be reversed, a reversal of H also induces the 180� flip of Q andthen polarization P, as shown in Figure 30(c) [201].

3.4.4. Multiferroicity approaching room temperature

All of the physics associated with multiferroicity from spiral spin structure illustratesthe fact that the spiral magnetic order often arises from the competing magneticinteractions. These competing interactions usually reduce the ordering temperatureof conventional spin-ordered phase. Hence, it is hardly possible for the spiral spin

Figure 28. Phase diagram of temperature against Mn–O–Mn bonding angle � (correspondingto different rare-earth ionic radii) for manganites RMnO3. The inset shows the wave numbersof spiral spin order for these manganites. (Reproduced with permission from [163]. Copyright� 2004 American Physical Society.)


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order (phase)-induced ferroelectricity to appear above a temperatures of approxi-mately 40K, far below room temperature required for service of most devices.

One of the possible ways to overcome this barrier is to search for those magneticmaterials with very strong competing magnetic interactions, and this effort has beenmarked with some progress recently. In fact, it was once revealed that the magnitudeand sign of the principal super-exchange interaction J in low-dimensional cupratesdepend remarkably on the Cu–O–Cu bond angle ’ (see [202,203]). In cuprateswith ’� 180�, J has an order of magnitude of around 102meV, thus favouringferromagnetic order. Upon decreasing ’, J is monotonically suppressed andeventually becomes negative (favouring ferromagnetic order) at ’� 95�, as shown

Figure 29. (Colour online) Physical properties of CoCr2O4: (a) spin configuration andpolarization of the conical spin-ordered state; (b) crystal lattice, electronic and spin structures.The measured hysteresis loops of magnetization and polarization against external magneticfield at two temperatures are shown in (c) and (d). (e) Switching (reversal) of polarizationinduced by time-dependent magnetic field. The upper part of (e) illustrates the spiral spin andpolarization structures. (Reproduced with permission from [197]. Copyright � 2006 AmericanPhysical Society.)


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in Figure 31(a). Therefore, for those cuprates with � deviating away from 180�, theferromagnetic interaction (J ) competes with the higher-order superexchangeinteractions, often leading to the spiral magnetic order with relatively high orderingtemperature. While LiCu2O2 discussed above is the typical example exhibitingthe spiral magnetic order and simultaneously ferroelectricity below about 25K, therelationship between parameters J and � in cuprates allows us to tune the strength ofthe spiral magnetic order. For example, CuO with C2/c monoclinic crystal structurecan be viewed as a composite of two types of zigzag Cu–O chains running along the[10�1] and [101] directions, respectively, with �¼ 146� and 109�. A Cu–O–Cu angleof 146� seems to be an intermediate value between 95� and 180� and a large magneticsuper-exchange interaction is expected. Strong competition between this super-exchange interaction and the ferromagnetic interaction was identified, resultingin the ICM spiral magnetic order (AF2) which appears over the temperature rangefrom 213 to 230K, as shown in Figure 31(b). This argument was confirmed by aclear ferroelectric polarization measured in this temperature region, as shownin Figure 31(c) [202].

In addition to CuO, hexaferrite Ba0.5Sr1.5Zn2Fe12O22 is another multiferroicsystem offering the spiral magnetic and ferroelectric orders at a relatively hightemperature [204]. Similarly, hexaferrite Ba2Mg2Fe12O22 was also found to exhibitmagnetic field induced ferroelectricity at relatively high temperature, although it

Figure 30. (Colour online) (a) Temperature (T )-dependence of electric polarization (P) alongthe �110

direction and magnetization M along the [001] direction in CoCr2O4 below 30K.

Note how P suddenly switches sign when cooling across 14K without changing signs ofM andQ. (b), (c) H-dependence of M and P at 20 and 10K, respectively. (Reproduced withpermission from [201]. Copyright � 2009 American Physical Society.)


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does not show ferroelectricity under zero magnetic field [205]. The helical spin orderwith propagation vector ks along the [001] direction appears at approximately 200K,and so does the as-induced ferroelectricity, under a very small magnetic field(�30mT). More interesting here is that a magnetic field as small as about 30mT issufficient to stimulate a transverse conical spin structure with respect to the magneticfield direction. In agreement with the inverse DM model and the KNB model, thistransverse conical spin order allows a polarization P to align perpendicular to boththe magnetic field and the propagation vector ks. An oscillating or multidirectionallyrotating field is able to excite the cyclic rotation of polarization P. For example, therotating magnetic field with magnitude from 30mT to 1.0 T, within the plane normal

Figure 31. (Colour online) (a) Relationship between the principal superexchange interaction Jand the Cu–O–Cu bond angle ’ in low-dimensional cuprates. (b) Schematic drawing of theCM collinear (AF1) and ICM non-collinear (AF2) antiferromagnetic spin orders in CuO.(c) Measured polarization as a function of temperature in CuO. (Reproduced with permissionfrom [202]. Copyright � 2008 Macmillan Publishers Ltd/Nature Materials.)


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to the [001] direction, drives polarization P to vary in proportion to sin � where �is the angle defined by the magnetic field and spiral propagation axis, as shown inFigure 32 [205].

3.4.5. Electric field control of magnetism in spin spiral multiferroics

It is now believed that the ferroelectricity in those frustrated magnetic oxidesoriginates from specific frustrated spin configuration, e.g. the spiral spin structure.Therefore, the control of polarization by the magnetic field becomes quite naturaland was demonstrated extensively. Owing to the intrinsic magnetoelectric couplingin those materials, one may also expect an effective control of the magnetization

Figure 32. (Colour online) Measured polarization of Ba2Mg2Fe12O22 as a function of therotation angle � of magnetic field with respect to the [001] direction and the rotation angle � ofmagnetic field with respect to the [120] direction. The magnetic field rotates horizontally(A–D) and vertically (E–H) in the shaded planes shown in A and D. (Reproduced withpermission from [205]. Copyright � 2008 AAAS.)


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by an electric field as the counterpart of the magnetic field control of polarization.Nevertheless, so far no dynamic and macroscopically reliable evidence of thismagnetization control by an electric field has been presented, while the microscopicidentification in some multiferroics was reported only very recently. One example isTbMnO3, in which the magnetization switching associated with the polarizationreversal was observed using spin-polarized neutron scattering [180]. This seems to bethe first evidence to demonstrate that the helical spin order can be modulated by anelectric field or by polarization switching. In Figure 33(a) we show the recorded twosatellite scattering peaks at position (4,�q,L¼ 1) with the neutron spins parallel(mode I") and antiparallel (mode I#), respectively, to the scattering vector in theferroelectric phase. It is seen that the intensities of the two peaks are reversed upona switching of the polarization between the two states along the �c-axis. Thissuggests that the spin helicity, in clockwise or counterclockwise mode, can becontrolled by reversing the polarization, as schematically shown in Figure 33(b)[180]. Although this effect is very weak and the spin helicity reversal might not berealized at a temperature above TC, a reversible magnetoelectric coupling betweenmagnetization and polarization in TbMnO3 was identified.

Furthermore, this effect was recently demonstrated also in BiFeO3. Although theferroelectricity in BiFeO3 is commonly believed to originate from the lone-pairelectrons of Bi ions and the magnetoelectric coupling should be weak, it was pointedout that the Fe3þ ions are ordered antiferromagnetically (G-type) and their momentalignment constitutes a cycloid with a period of approximately 62 nm (see [206]).Owing to the rhombohedral symmetry, there are three equivalent propagationvectors for the cycloidal rotation: k1¼ [�, 0,��], k2¼ [0, �,��], and k3¼ [��, �, 0] with�� 0.0045. This allows one to argue about the possibility that such a cycloidal spin

Figure 33. (Colour online) Reversal of electric polarization Pc and spiral spin order inducedby external electric field along the c-axis in TbMnO3. (a) Scattering intensities and (b) spinconfigurations of the spin-order states with different polarization Pc. See the text for details.(Reproduced with permission from [180]. Copyright � 2007 American Physical Society.)


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structure may also contribute to the ferroelectricity in BiFeO3, or one may expectintrinsic coupling between the ferroelectricity and the cycloidal spin structure. This

idea was indirectly confirmed experimentally by a successful observation of theintimate link between the cycloidal magnetic structure of Fe3þ ions and thepolarization vector [206–209].

The coupling between ferroelectric domains and antiferromagnetic domains in

BiFeO3 provides direct evidence of the above argument. Experimentally, piezo-forcemicroscopy (PFM) or in-plane piezo-force microscopy (IPPFM) allows researchersto observe the ferroelectric domains under different electric fields [210,211], while

X-ray photoemission electron microscopy (PEEM) can be used to monitor theantiferromagnetic domains simultaneously. High-resolution images of both theantiferromagnetic and ferroelectric domains in (001)-oriented BiFeO3 films were

obtained. As mentioned previously, the spontaneous polarization of BiFeO3 directsalong the [111]-axis, enabling eight equivalent orientations along the four cubicdiagonals. This geometry thus allows for the 180�, 109� and 71� domain switching

driven by appropriate electric field, as shown in Figure 34. Figures 35(c) and (d)show the PFM images of the BiFeO3 film before and after the electric field poling,

Figure 34. (Colour online) Four equivalent electric polarization directions of BiFeO3 crystal.The numbers in each figure indicate the reversal angles relative to the polarization along the[111] direction. The shaded planes represent the AFM plane perpendicular to the spiral spinplanes. (Reproduced with permission from [208]. Copyright � 2006 Macmillan PublishersLtd/Nature.)


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respectively, and 109� domain switching (regions 1 and 2) was identified in additionto the 180� and 71� domain switching (regions 3 and 4). It can be seen that themultidomain state consists of stripe regions with two different polarizationdirections. The PEEM images of the same regions (Figures 35(a) and (b)) clearlyindicate the reverse contrast in regions 1 and 2 upon the electric field poling [208].These results demonstrate the switching of the antiferromagnetic order from the

Figure 35. (Colour online) PEEM images of BiFeO3 film (a) before and (b) after electricpoling as indicated by arrows; and IPPFM images of the same area (c) before and (d) after theelectric poling, noting the 109� ferroelectric domain switching (regions 1 and 2) and 180� and71� domain switching (regions 3 and 4). (e) PFM image of the same area with polarizationlabelled. (Reproduced with permission from [208]. Copyright � 2006 Macmillan PublishersLtd/Nature.)


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orange plane to the green plane (Figure 34(a)) due to the 109� polarization switching.The neutron scattering on single crystal of BiFeO3 also revealed that the intensitiesaround the (1/2, �1/2, 1/2) Bragg position in the P111 and P1-11 domains (as shownby the lower half and upper half of the pattern in Figure 36(a), respectively) aredifferent, which implies that the 71� domain switching by an electric field along the[010] direction brings the rotation of the Fe spiral spin plane and then inducesthe flop of the antiferromagnetic sublattices [206], as shown in Figure 36(b). Theseexperiments unveiled the coupling between M and P at atomic level, although noglobal linear magnetoelectric effect exists because of no net magnetization (hMi¼ 0).

These phenomena illustrated above can be understood as following. Lebeugleet al. [206] pointed out that a coupling energy term EDM¼ (P eij)(SiSj) should beincluded into the total energy, owing to the DM interaction. This coupling energy

Figure 36. (Colour online) (a) Neutron scattering intensity in the adjacent P111 (lower half)and P1-11 (upper half) domains in BiFeO3 single crystal. (b) Schematic drawing of the planes ofspin rotations and cycloids k1 vector for the two polarization domains with the domain wall(light grey plane). (Reproduced with permission from [206]. Copyright � 2008 AmericanPhysical Society.)


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favours the canting of Fe3þ spins, which exactly compensates the loss of theexchange energy. Moreover, this coupling energy is zero if polarization P isperpendicular to the local spin moments and maximum if it lies on the cycloidrotation plane. This picture explains reasonably the flop of antiferromagneticdomains associated with the switching of ferroelectric domains in BiFeO3 thinfilms [206].

Another experiment on the dynamics of ferroelectric domain switching in BiFeO3

films also provides evidence of the coupling between the ferroelectric andantiferromagnetic domains. A well-known fact is that the ferroelectric domains inconventional ferroelectric films are smooth, stripe-like, and the domain width growsin proportion to the square root of the film thickness, i.e. the so-called Landau–Lifshitz–Kittel (LLK) scaling law [212]. However, qualitatively different behavioursof the ferroelectric domains from the LLK scaling in very thin BiFeO3 filmswere observed. First, the domain walls are not straight, but irregular in shape,characterized by a roughness exponent of approximately 0.5–0.6 and an in-planefractal Hausdorff dimension Hk � 1:4� 0:1. The average domain size appears todepart from the LLK square root dependence on the film thickness, but scaleswith an exponent of 0.59� 0.08 (see [209]). Second, the ferroelectric domains aresignificantly larger in size than those in other ferroelectric films of the samethickness, but closer to magnetic domains in typical magnetic materials. This impliesthat the ferroelectric domains are coupled with the antiferromagnetic domains inBiFeO3 films. The magnons coupled with polarization (electromagnons) observedin BiFeO3, as is emphasized in Section 4, also map the coupling between theferroelectric order and cycloidal spin order in BiFeO3 (see [207]).

In fact, dealing with the roadmaps to control the ordering state of a multiferroicsystem, two types of approach are possible: phase control or domain control.In the first case, an external field is used to trigger a phase transition betweentwo fundamentally different phases. Although this approach cannot be realizedin BiFeO3, the coupling between the ferroelectric domains and antiferromagneticdomains provides the second approach, i.e. external field triggers a transitionbetween two equivalent but macroscopically distinguishable domain states. Thisapproach together with the exchange interaction at the interfaces, makes the electric-field control of magnetism possible. An excellent example developed recently by Chuet al. comes to BiFeO3 again [213]. For a Ta/Co0.9Fe0.1/BiFeO3/SrRuO3/SrTiO3(001)heterostructure, a 10 10 mm2 region in the BFO layer was upward with a �21Vbiased electric voltage, as shown by the red square in Figure 37(a). Subsequently,a 5 5 mm2 smaller area inside this region was downward poled with a þ12V biasedvoltage, as shown by the green square. The magnetic domains in the CoFe layerexhibit two distinct regions, as we can see from the PEEM images in Figure 37(b).These regions are the in-plane ferromagnetic domains aligned horizontally from leftto right (black) and vertically from down to up (grey). The formation of the twotypes of domains is due to two switching events by rotation of the polarizationprojection on the (001) plane: the 70� in-plane switching and the 109� out-of-planeswitching, which gives rise to the corresponding rotation of the antiferromagneticorder in BiFeO3, as discussed in the beginning of this section [206–209]. The rotationof the antiferromagnetic order drives the reversal of magnetism of the CoFe layervia the exchange bias effect on the BiFeO3–CoFe interface [213]. This approach to


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control the ferromagnetism by an electric field can be utilized in dynamic switchingdevices, i.e. back-switching the ferroelectric domains in BiFeO3 and then theferromagnetic domains in CoFe layer to the initial states by an opposite electric field.These investigations sketch a possible magnetoelectric random-access memory

Figure 37. (Colour online) (a) In-plane PFM image showing the ferroelectric domainstructure of BiFeO3 with a large (10mm, red square) and small (5 mm, green square) electricallyswitched region. (b) Corresponding XMCD-PEEM image for the CoFe film grown on theelectrically written BiFeO3 film. (c) Schematic drawings of the two adjacent domains in[001]-oriented BFO. (Reproduced with permission from [213]. Copyright � 2008 MacmillanPublishers Ltd/Nature Materials.)


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(MeRAM) element (as shown in Figure 38) [214]. The binary information is storedby the magnetization of the bottom ferromagnetic layer (blue), which is read bymonitoring the resistance of the magnetic trilayer and written by applying a voltageacross the multiferroic ferroelectric/antiferromagnetic layer (green). If the magne-tization of the bottom ferromagnetic layer is coupled to the ferroelectric/antiferromagnetic layer, a reversal of the polarization P in the multiferroic layerstimulates the switching of the magnetic configuration in the trilayer from theparallel alignment to the antiparallel alignment, and thus the resistance from the lowstate RP to the high state RAP (see [214]).

There are some other 3D frustrated oxides, such as MnWO4 (see [182–190]) andpyroxenes (NaFeSi2O6 and LiCrSi2O6) [191], which were found to be spiral spin-order-induced multiferroics. In fact, not only those oxides but also some thiospinelcompounds with frustrated spin order such as Cd(Hg)Cr2S4, also exhibit multi-ferroicity. In addition, the magnetoelectric coupling and a colossal magnetodielectriceffect were observed in these multiferroics [192–196]. For convenience to readers,we collected the main physical properties of those so-far investigated multiferroicsof spiral spin order and induced ferroelectricity and present them in Table 2.

Although there has been extensive research on this kind of multiferroicsand several comprehensive models have been developed, so far no quantitativeunderstanding of the multiferroicity in spiral spin-ordered materials has been madeavailable. For example, first principle calculations on LiCu2O2 (see [144]) andTbMnO3 (see [215,216]) predict that all of these models are inadequate. Carefulcalculation on TbMnO3 reveals that both the electronic and lattice effects have

Figure 38. A possible multiferroic random-access memory element using antiferromagneticmultiferroic materials. (Reproduced with permission from [214]. Copyright � 2008 MacmillanPublishers Ltd/Nature Materials.)


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contribution to the electric polarization and the latter can be even dominant [215].It is surprising to be disclosed that the displacements of Mn3þ and Tb3þ ionsare generally larger than those of O2� ions and have a significant contribution to thepolarization [216], which is not possible in the framework of the current theories.

3.5. Ferroelectricity in CO systems

In parallel to the development of multiferroics of spiral (helical) spin order, anothertype of multiferroics, i.e. CO multiferroics, has also been receiving attention. Forconventional ferroelectrics and all of the multiferroics addressed above, theferroelectricity originates from the relative displacement between anions and cationsas well as the lattice distortion associated with the second-order Jahn–Teller effect.However, an alternative mechanism, electronic ferroelectricity [217], was proposedrecently, in which the electric dipole originates from the electronic correlation ratherthan the covalency. This would offer an attractive possibility for novel ferroelec-tricity that could be controlled by the charge, spin and orbital degrees of freedomof the electron.

In many narrowband metal oxides with strong electronic correlations, chargecarriers may become localized at low temperature and form a periodic structure(i.e. CO state). The often cited example is magnetite Fe3O4, which undergoes ametal–insulator transition at approximately 125K (the Verwey transition) with arather complex iron charge order pattern [218]. It is expected that a non-symmetriccharge order may induce electric polarization. Another of the well-studied COmaterials are manganites [219]. When LaMnO3 (or related compounds in which thecharge of Mn ions is formally 3þ) and CaMnO3 (in which the Mn charge is formally4þ) are alloyed, the resulting arrangement of Mn3þ and Mn4þ ions can be ordered ina particular case, as shown in Figure 39. Moreover, electrons around the atoms theyoccupy may have several choices among their energetically equivalent (or degenerate)electronic orbitals. This orbital degree of freedom allows for a manifold of possibleelectronic states to be chosen. For example, the Mn ions can occupy either of the twod-orbitals. However, these choices are not independent, and the charge distributionaround these ions is distorted with adjacent oxygen ligands which would bedislodged once a valence electron localizes in a definite Mn d-orbital. Eventually,a spontaneously ordered pattern of the occupied orbitals throughout the crystallattice (i.e. orbital-ordered state) is yielded.

3.5.1. Charge frustration in LuFe2O4

The ferroelectricity associated with a CO state was first demonstrated in a mixedvalence oxide, LuFe2O4 (see [220–227]). At room temperature, LuFe2O4 has ahexagonal layered structure (space group R�3m, a¼ 3.44 A, c¼ 25.28 A) in whichall Fe sites are crystallographically the same. The crystal structure consists of analternative stacking of triangular lattices of rare-earth elements, irons and oxygens.The Fe2O4 layers and Lu3þ ion layers stack alternatively with three Fe2O4 layers perunit cell. Each Fe2O4 layer is made up of two triangular sheets of corner-sharingFeO5 trigonal bipyramids.


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In LuFe2O4, an equal number of Fe2þ and Fe3þ ions coexists on the same siteof the triangular lattice. With respect to the average Fe2.5þ valence, Fe2þ and Fe3þ

ions are considered to be facilitated with an excess and a deficient half electron,respectively. The Coulomb preference for pairing oppositely signed charges Fe2þ andFe3þ causes the degeneracy in the lowest energy for charge configuration in thetriangular lattice, and then the charge-ordered state. The charge order pattern ofalternating Fe2þ : Fe3þ layers with ratios of 2 : 1 and 1 : 2, appearing at a temperatureas high as around 370K, is shown in Figure 40(a). This postulated CO structureallows the presence of a local electric polarization, because the centres of Fe2þ ionsand Fe3þ ions do not coincide in the unit cell of the superstructure, as highlightedby the arrow in Figure 40(c). The high-resolution transmission electron microscopyimage shown in Figure 40(b) [222,226] is consistent with this pattern. An electricpolarization as high as approximately 26 mCcm�2 was measured using thepyroelectric current method [220]. In response to temperature variation, significantdecaying of the polarization occurs at approximately 250K, the magnetic transitionpoint, and at approximately 330K where the CO superstructure of Fe2þ and Fe3þ

disappears, as displayed in Figure 40(d) [220].LuFe2O4 also exhibits remarkable magnetoelectric coupling effect. For example,

that remarkable response of the dielectric constant to a small magnetic field atroom temperature was given by a change of 25% upon a field of approximately1 kOe (see [227]). Moreover, Fe2þ onsite crystal field excitations are sensitive to the

Figure 39. (Colour online) CO–OO structure of Pr0.5Ca0.5MnO3 at low temperature.(Reproduced with permission from [219]. Copyright � 2000 AAAS.)


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monoclinic distortion which can be driven by temperature/magnetic field. Thedistortion further splits the three groups of Fe 3d level of D3d symmetry, and thena large magneto-optical effect was observed [228].

Nevertheless, the first principle calculation, in combination with Monte Carlosimulation, reveals another CO state in connection with the Fe2O4 layers ofLuFe2O4, consisting of Fe2þ chains alternating with Fe3þ chains in each triangularsheet. This state has almost the same stability as the CO state discussed above,although it does not favour ferroelectricity and is not the ground state. The chargefluctuations associated with the inter-conversion between the two different types ofCO states could be remarkable because they are very similar in energy. In this sense,LuFe2O4 can be viewed as a phase-separated system in terms of the CO state,

Figure 40. (Colour online) Atomic configuration of the charge-order state of LuFe2O4 (a) onthe ab-plane and (c) in 3D space (c). The red arrow in (c) indicates the direction ofpolarization. (b) Transmission electron diffraction pattern of LuFe2O4 along the ½1�10�direction at T¼ 20K. (d) Electric polarization of LuFe2O4 as a function of temperature undertwo different cooling field modes. (Parts (a) and (b) reproduced from [222]. Copyright � 2007American Physical Society. Part (c) reproduced from [19]. Copyright � 2007 MacmillanPublishers Ltd/Nature Materials. Part (d) reproduced with permission from [220]. Copyright� 2005 Macmillan Publishers Ltd/Nature.)


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consisting of two types of CO domains separated by domain boundaries. The giantdielectric constant of LuFe2O4, as observed, may be ascribed to this kind of CO statefluctuations. Given an external magnetic field, the Zeeman energy may preferentiallystabilize one of the two CO states because the two states most likely have differenttotal spin moments. This explains why the dielectric response and polarization willbe weakened by a magnetic field, which suppresses the charge fluctuations [221].Experimental results also indicate that the charge fluctuations have an onset pointwell below the CO temperature [228].

Surely, there are still enormous disputes and further investigations of the detailsof the proposed multiferroic origin in LuFe2O4 are required. For example, someresults of X-ray scattering experiments revealed an ICM charge order withpropagation vector close to (1/3, 1/3, 3/2) below 320K, which contains polar Fe/Odouble layers with antiferroelectric stacking [229].

3.5.2. Charge/orbital order in manganites

A charge-ordered state is also often observed in manganites RxA1–xMnO3.In addition, the charge/orbital-ordered (CO–OO) state is highly favoured inRuddlesden–Popper series manganites [219]. For example, Pr(Sr0.1Ca0.9)2Mn2O7 iscomposed of bilayers of MnO6 octahedra, exhibiting two CO–OO phases: high-temperature phase (CO1) and low-temperature phase (CO2). The space groups areAmam (with a¼ 5.410, b¼ 5.462, c¼ 19.277 A at 405K) for the charge-disorderedphase (T4TCO1), Pbnm (with a¼ 5.412, b¼ 10.921, c¼ 19.234 A at 330K) for theCO1 phase (TCO15T5TCO2) and Am2m (with a¼ 10.812, b¼ 5.475, c¼ 19.203 Aat 295K) for the CO2 phase (T5TCO2). The charge/orbital configurations for thethree phases are shown in Figure 41. From the synchrotron X-ray oscillationphotography, it was found that with respect to the CO1 phase, the orbital stripes andzigzag chains rotate by 90� when T falls down to TCO2 [230,231]. Above TCO1, eachMnO6 octahedron tilts towards the b-axis, as shown in Figure 41(a). Within a singlebilayer unit, pairs of tilted MnO6 octahedra on the upper and lower layers line upwith the shared O2� shifting towards the þb and �b directions. Such a situationcauses the alternation of the Mn–O–Mn bond in the MnO plane along the b-axis.In the adjacent bilayer, the arrangement of the bond alternation shifts by (0, 1/2, 0).For the low-temperature phases, the structural modulation accompanied by theCO–OO is superimposed onto this structure.

For simplicity and without losing the essence of the charge polarization problem,we take into account the charge ordering in the assumed Amam orthorhombic lattice.For the charge order transition, as shown in Figures 41(b) (CO1) and 41(c) (CO2),the checkerboard pattern of the CO state is superimposed onto the bond alternationpattern. Consequently, the charge polarization appears along the b-axis in eachbilayer. In the CO1 phase, however, the CO pattern stacks along the c-axis witha shift by (1/2, 0, 0) with respect to the next bilayer, as shown in Figure 41(b),facilitating the inter-bilayer coupling of the polarization antiferroelectrically innature. At TCO2, on the other hand, the rotation of the orbital stripes is accompaniedby the rearrangement of Mn3þ and Mn4þ ions and, thus, the CO stacking pattern,as shown in Figure 41(c). In the CO2 phase, the charge-order sequence stacks alongthe c-axis with a shift by (0, 1/2, 0) with respect to the next bilayer, coinciding with


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the stacking of the bond alternation. Therefore, the polarization of each bilayeralong the b-axis is excessive, forming a charge-polarized state below TCO2. In fact,optical SHG signals clearly demonstrate the breaking of the space-inversionsymmetry. However, the direct detection of the electrical polarization by, forexample, a pyroelectric current measurement, is hard to perform because of the lowresistivity around TCO2 (see [230]).

Figure 41. (Colour online) Synchrotron X-ray oscillation photographs (upper row), andschematic CO–OO configurations (middle row) as well as lattice structures (lower row) ofPr(Sr0.1Ca0.9)2Mn2O7 at three different temperatures. (Reproduced with permission from[230]. Copyright � 2006 Macmillan Publishers Ltd/Nature Materials.)


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3.5.3. Coexistence of site- and bond-centred charge orders

There has been active debate over the validity of the conventional CO picture

in which the 3þ and 4þ Mn ions orbitally order in a checkerboard arrangement

with the so-called CE-type antiferromagnetism (Efremov et al. referred to this state

as the site-centred order, as shown in Figures 39 and 42(a)). An alternative model

of ferromagnetic Mn–Mn dimers (the bond-ordering model of Efremov et al., see

Figure 42. (Colour online) (a) Site-centred and (b) bond-centred CO–OO phases as well as (c)the superposition of the two ordered phases for mixed-valence manganites. The green circlesrepresent the Mn ions, the blue circles for the rare-earth ions and the red circles for the oxygenions. The arrow indicates the direction of polarization P. (d) Predicted phase diagramof Pr1�xCaxMnO3. Abbreviations FM, C, CE and A represent the ferromagnetic, C-type,CE-type and A-type antiferromagnetic phases, respectively. The yellow region is predicted toexhibit ferroelectricity. (Reproduced with permission from [232,233]. Copyright � 2004Macmillan Publishers Ltd/Nature Materials.)


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Figure 42(b)) was proposed for La0.5Ca0.5MnO3 and likely identified forLa0.6Ca0.4MnO3 (see [232]). Recently, Efremov et al. proposed an intermediatestate, a kind of superposition of these two different charge-ordered patterns, andpredicted the local dipole moments that add up to a macroscopic ferroelectricpolarization (Figure 42(c)) [232,233]. In La0.5Ca0.5MnO3, adjacent dipole momentspoint to opposite directions so that there is no overall electric polarization due tothe moment cancellation. However, given a composition away from 50% CaMnO3,the cancellation should not be complete and a net polarization could enter, as shownin the calculated phase diagram (Figure 42(d)) [232].

So far, no direct experimental evidence with ferroelectricity in such charge-ordered manganites has been reported due to the high conductivity and possiblyother unknown reasons, while indirect characterization of the ferroelectricity wasreported recently [234,235]. A so-called electric field gradient (EFG) tensor viahyperfine techniques was developed to map the whole compositional range ofPr1–xCaxMnO3 and a new phase transition occurring at a temperature betweenthe CO point and antiferromagnetic Neel point was evidenced [234]. Although thistransition can be detected in all samples with CO state, the critical temperature forthe transition is suppressed upon the shift of the composition away from x¼ 0.5.The principal EFG component VZZ characterizing the local PE susceptibility showsa sharp increase in the vicinity of this new transition due to the polar atomicvibrations. Therefore, this new transition gives a hint of the local spontaneouspolarization below the CO transition point. The refined electron diffractionmicroscopy data also provide indirect evidence for the electric polarization inPr0.68Ca0.32MnO3 (see [235]). The results revealed that the Zener polaron orderstructure is non-centrosymmetric and the relative displacements of the bound cation–anion charge pairs create permanent electric dipoles, resulting in a net permanentpolarization Pa¼ 4.4mCcm�2. This polarization is much larger than those multi-ferroic manganites with spiral spin orders and thus allows for potential applicationsof the charge-ordered manganites. Nevertheless, the related experiments are verylimited and definitely, the ferroelectricity in charge-ordered ABO3 manganites seemsto be a hot issue for further careful study.

Another possible and intriguing multiferroic material, benefiting from thecoexistence of bond-centred and site-centred CO states, is magnetite, Fe3O4, whichundergoes a metal–insulator transition at approximately 125K (the Verweytransition) with a rather complex CO pattern of Fe2þ and Fe3þ ions [218]. Fe3O4

crystallizes in an inverted cubic spinel structure with two distinct iron positions. Theiron B sites locate inside the oxygen octahedra and contain two-thirds of the totaliron ions, with equal numbers of Fe2þ and Fe3þ ions. These sites by themselves forma pyrochlore lattice, consisting of a network of corner-sharing tetrahedra. The iron Asites contain the other one-third of the Fe ions and are considered to be irrelevantfor the CO state. The originally proposed charge-order pattern consists of analternation of Fe2þ and Fe3þ ions at the B-sites in the x–y planes and was shown tobe too simple in later reports. The difficulty in determining the CO structure in Fe3O4

is related to the strong frustration of simple biparticle ordering on a pyrochlorestructure and details of the CO pattern remains to be an issue [236–239].

Alternatively, a much earlier report claimed that Fe3O4 in the insulating statebelow the Verwey temperature is ferroelectric and the electric polarization points to


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the b-direction [236]. The polarization leads to the formation of a ferroelectricdomain structure which can be explained only by assuming a triclinic structure.Although the real microscopic origin of the ferroelectricity in magnetite remains tobe unveiled, the most probable one is the coexistence of site-centred and bond-centred CO states [237–239]. In the proposed structure and charge pattern shownin Figure 43, there exist strong fluctuations of the Fe–Fe distance (the bond length)and the site occupancy of Fe2þ and Fe3þ ions. The variation of the Fe–Fe bondlength, in addition to the alternative occupancy of Fe2þ and Fe3þ ions along theh110i Fe chain on the x–y planes, results in the mixed bond- and site-centred COchains. Such a configuration would give a non-zero contribution to the electricalpolarization.

3.5.4. Charge order and magnetostriction

The spiral magnetic order with active antisymmetric exchange coupling and the COstate are not the only possible ways towards the magnetism-induced ferroelectricity.It has been postulated that the exchange striction associated with symmetricsuperexchange coupling plus charge-ordered state can also generate ferroelectricity.The ground magnetic order of the one-dimensional Ising spin chain with thecompeting nearest-neighbour ferromagnetic interaction (JF) and next-nearest-neighbour antiferromagnetic interaction (JAF) is of the up–up–down–down (""##)type if jJAF=JFj4 1=2 (see [240]), as shown in Figure 44. If the magnetic ions alignalternatively along the chain, the exchange striction associated with the symmetricsuperexchange interaction shortens the bonds between the parallel spins, whilestretches those between the antiparallel spins. Ultimately, the inversion symmetryis broken and an electronic polarization yields along the chain direction [241].

Figure 43. (Colour online) Structure and polarization of CO Fe3O4. In the xy chains the Fe2þ

and Fe3þ ions (filled and open circles) align alternatively, and simultaneously there is analternation of short and long Fe–Fe bonds (the black arrows indicate the direction of Fe-ionshift and red arrows indicate the direction of polarization).


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Ca3Co2O6 is a typical Ising spin chain system, and a half-doping of this compoundby Mn at the Co site produces the novel compound Ca3Co2�xMnxO6, where theCo2þ and Mn4þ ions tend to be located in the centre of oxygen cages of face-sharedtrigonal prisms and octahedra aligned alternatively along the c-axis. This is becauseMn ions have a strong tendency to avoid the trigonal prismatic oxygen coordination.At x¼ 1, all of the Co ions are located in the trigonal prismatic sites and all of theMn ions occupy the octahedral sites, leading to the CO state associated with theIsing spin chain in Ca3CoMnO6. This configuration would generate electronicferroelectricity. In fact, a clear ferroelectric polarization was observed at 16.5K, theonset point for the magnetic order, which is signified by a broad peak in the magneticsusceptibility [242].

Ca3Co2�xMnxO6 and the undoped compound Ca3Co2O6 are famous for theirsuccessive metamagnetic transitions and the magnetization plateaus under amagnetic field [243–247]. In agreement with the magnetization plateaus, thedielectric constant of Ca3Co2O6 also shows plateaus [246]. For multiferroicCa3Co2�xMnxO6 (x� 0.96), magnetization and neutron-scattering measurementsrevealed successive metamagnetic transitions from the ""## spin configurationunder zero field to the """# state and then the """" state [247]. Inversion symmetrybroken in the ""## state is restored in the """# state, resulting in the disappearanceof the spontaneous polarization [247].

In addition to Ca3Co2�xMnxO6, manganites RMn2O5 as multiferroics weresupposed to follow a similar mechanism for ferroelectricity generation. RMn2O5

with R¼Ho, Tb, Dy, Y and Er etc, represents another kind of CO manganitesin addition to CO ABO3-type manganites [248–281]. They exhibit very complexmagnetic and ferroelectric phase transitions upon temperature variation. At roomtemperature, TbMn2O5 belongs to the orthorhombic space group of Pbam, hostingMn3þ (S¼ 2) ions surrounded by oxygen pyramids and Mn4þ (S¼ 3/2) ionssurrounded by oxygen octahedra, as shown in Figure 45(a). The magnetic structureof RMn2O5 is extremely complicated and determined with multi-manifold exchange

Figure 44. (Colour online) (a) One-dimensional chain with alternating charges (CO state) andup–up–down–down spin structure. (b) Magnetostriction effect, which shortens the ferromag-netic bonds and generates a ferroelectric polarization.


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interactions, as shown in Figure 45(b). Along the c-axis, the Mn spins are arrangedin the five-spin loop: Mn4þ–Mn3þ–Mn3þ–Mn4þ–Mn3þ. The nearest-neighbourmagnetic coupling in the loop is of the antiferromagnetic type, favouring antiparallelalignment of the neighbouring spins. However, because of the odd number of spinsin one loop, a perfect antiparallel spin configuration cannot be possible, eventuallyleading to the frustrated complex magnetic structure [248].

Also, upon temperature fluctuation and external stimuli, RMn2O5 exhibitscomplex magnetic transitions. From Figure 46(a), it is seen that TbMn2O5 showsseveral magnetic and ferroelectric phase transitions accompanied by the appearanceof electric polarization and dielectric anomalies along the b-axis. Starting froman ICM antiferromagnetic ordering at TN¼ 43K with a propagation vector(�0.50, 0, 0.30), the spin configuration locks into a CM antiferromagnetic state atTCM¼ 33K with propagation vector (0.50, 0, 0.25). The dielectric constants alsoexhibit anomalies at these magnetic transitions, as shown in Figures 46(a) and (b).Spontaneous polarization arises at a temperature T¼Tferroelectric� 38K between TN

and TCM, as shown in Figure 46(d). As the temperature drops down to TICM¼ 24K,the ICM configuration re-enters, with a sudden decrease of the polarization anda jump of the vector to (0.48, 0, 0.32). The polarization increases again withtemperature decreasing down to about 10K, as shown in Figure 46(d) [248].

Figure 45. (a) Crystal structures of TbMn2O5 on the ab-plane (left) and a(b)–c-plane (right).Five types of magnetic exchange interactions are denoted by J1, J2, J3, J4 and J5, respectively.(b) Spin (solid arrows) configuration and crystal distortion (electric polarization, open arrows)of TbMn2O5. (Reproduced with permission from [249]. Copyright � 2006 American PhysicalSociety.)


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Interestingly, experiments revealed that TbMn2O5 belongs to the space groupPbam, which meets the spatial-inversion symmetry and thus would excludeferroelectricity because of the lack of spatial-inversion breaking. While the factthat TbMn2O5 develops a spontaneous polarization is still puzzling, it was suspectedthat the symmetry group should be Pb21m which allows for ferroelectricity, but nodirect evidence has been presented [252,253]. It was also postulated that the CO stateplus the CM magnetic order is responsible for the polarization, where the Mn spinconfiguration in the CM phase is composed of antiferromagnetic zigzag chains alongthe a-axis. Half of the Mn3þ–Mn4þ spin pairs across the neighbouring zigzags alignin an approximately antiparallel manner, whereas the other half favours, more orless, the parallel alignment. The exchange striction effect drives a shift of ions (mostlyMn3þ ions inside the pyramids) in a way that optimizes the spin–exchange energy:those ions with antiparallel spins are pulled close to each other, whereas those with

Figure 46. (a) Temperature dependence of magnetic susceptibilities and dielectric constantsalong the a-axis, b-axis and c-axis, respectively, as well as specific heat for TbMn2O5.(b) Dielectric constant along the b-axis as a direction of magnetic field along the a-axis.(c), (d) Polarization along the b-axis as a function of magnetic field along the a-axis.(Reproduced with permission from [248]. Copyright � 2004 Macmillan Publishers Ltd/Nature.)


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parallel spins move away from each other. This leads to a distorted pattern labelled

by the open black arrows in Figure 45(b), which breaks the inversion symmetry and

induces a net polarization along the b-axis. For the ICM magnetic phase, the

magnetization of Mn ions in each zigzag chain is modulated along the a-axis and

the spins in every other chain are rotated slightly toward the b-axis. It should be

mentioned here that rare-earth Tb3þ ions also have magnetic moments and will

exhibit a non-collinear magnetic order at a low temperature of approximately 10K.

The net distortion associated with the Tb spins is even larger than that associated

with the Mn spins. Therefore, a polarization enhancement with temperature

decreasing down to approximately 10K was observed, as shown in Figure 46(d).

A magnetic fieldH applied along the a-axis will force alignment of the Tb spins along

the a-axis but leave the Mn spins nearly unchanged. This realignment of Tb spins

makes the associated lattice distortion disappear and causes a rotation of the net

polarization by 180�, as shown in Figures 46(c) and (d); note that this rotation is very

quick (as shown in the inset of Figure 46(c)) and may be utilized for memory

applications [248].Similar but slightly different multiferroic effects were observed in other RMn2O5

systems for R¼Tm, Er, Dy, Ho, Gd and Y (see [249–282]). For example, the CM

ferroelectric phase in TbMn2O5 is replaced by an ICM ferroelectric phase in

ErMn2O5 and TmMn2O3 (see [281]). The complex behaviour of the electric

polarization, especially the anomaly of polarization at the low-temperature CM–

ICM transition, remains unclear. Research into YMn2O5, which excludes the effects

of magnetic moments at the Tb ions, postulated that there are two ferroelectric

phases owing to the complex spin structure. The spiral spin orders ensues both in the

ac and bc planes, noting that the up–up–down–down order in the ab plane was

described above. Both types of orders can induce ferroelectric polarization,

according to the KNB model and the magnetostriction mechanism, corresponding

to the intermediate-temperature CM ferroelectric phase and low-temperature ICM

ferroelectric phase, as shown in Figure 47 [282]. It is possible that both the

mechanisms play important roles in these complex systems.For RMn2O5, the magnetoelectric coupling between magnetism and ferroelec-

tricity can be even more fascinating. For example, the dielectric response of

TbMn2O5 (as shown in Figure 46(b)) and DyMn2O5 to a magnetic field can be very

large: approximately 109% at 3K upon a field of around 7T (see [250]). This

extraordinary magnetodielectric effect seems to originate from the high sensitivity

of the ICM spin state to external perturbations. A manipulation of the magnetic

structure by electric field was also observed in this type of multiferroic. For

ErMn2O5, which shows its magnetic and ferroelectric transitions very similar to

TbMn2O5, a static electrical field may significantly enhance the magnetic scattering

intensity. The reason may be that an electric field stabilizes the ferroelectric phase,

which pushes the spin configuration into the CM magnetic phase by modulating the

direction of magnetic moment via the magnetoelectric coupling. The X-ray scattering

intensity I as a function of the applied electric field at approximately 38.5K shows

the butterfly-type hysteresis, which is also evidence of the manipulation/switching

of magnetic structure by the electric field [251].


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3.6. Ferroelectricity induced by E-type antiferromagnetic order

It is well established that the time-inverse symmetry imposes rather strict conditions

for possible magnetic orders that can induce ferroelectricity: the magnetic structure

must have enough low symmetry in order for the lattice to develop a polar axis.

As a consequence, the spin configuration should usually have complicated non-

collinear structures, including spiral and ICM structures. The non-collinear magnetic

structures can be stabilized by either competing interactions (frustration) or

anisotropies generated by spin–orbital coupling, which usually lead to reduced

transition temperatures and weak order parameters. In turn, there is a type of

so-called collinear multiferroics, which are rare so far but may be more promising

since they are less prone to the obstacles mentioned above. One type of unusual

collinear multiferroics come from those with E-type antiferromagnetic order

(E-phase). In the perovskite manganite family RMnO3 (space group Pbnm), the

E-phase was first reported in orthorhombic HoMnO3, with the spin configuration

shown in Figure 48(a), which was considered as an example to demonstrate the

collinear E-phase-induced ferroelectricity [129,283,284].We first come to look at a simple model associated with the E-phase and

understand how the ferroelectricity is generated. In the E-phase, the parallel Mn

Figure 47. (Colour online) Magnetic structure of the low-temperature ICM ferroelectric phaseof YMn2O5. (a), (b) and (c) represent the ab-, ac- and bc-planes, respectively. (d) Polarizationinduced by the magnetic striction mechanism in the ab-plane. (Reproduced with permissionfrom [282]. Copyright � 2008 American Physical Society.)


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spins form zigzag chains in the ab-plane, with the chain link equal to the nearest-neighbour Mn–Mn distance. The neighbouring zigzag chains along the b-axisare antiparallel and the ab-planes are stacked antiferromagnetically along the c-axis.The symmetric coordinates corresponding to the E-phase can be defined as [283]

E1 ¼ S1 þ S2 � S3 � S4 � S5 � S6 þ S7 þ S8, ð25aÞ

E2 ¼ S1 � S2 � S3 þ S4 � S5 þ S6 þ S7 � S8, ð25bÞ

where Si is the spin of the ith Mn atom in the magnetic unit cell, as shownin Figure 48(a). Considering that the Mn spins in HoMnO3 point along the b-axis,

Figure 48. (Colour online) (a) Spin structures of two AFM E-phases in perovskite HoMnO3.The arrows on the Mn atoms (blue spheres) denote the directions of their spins, and thedirection of polarization is indicated by the black arrows. The red spheres denote the O atoms.(b) In-plane ferroelectric configuration of AFM E-phase (E1) HoMnO3. The small red spheresdenote the O atoms. The bigger spheres represent the Mn atoms, and the regions shaded byblue and pink colour denote the AFM coupled spin zigzag chains. The green and yellowarrows represent the directions of ionic displacement of Mn (left) and O (right) respectively,and the resultant polarization is denoted by the thick arrow at the bottom. (c) The ac-planecharge density isosurface plot in the energy region between �8 eV and 0 eV (0 eV is the top ofthe valence band) for the relaxed structure of AFM E-phase (E1) HoMnO3 by first principlecalculations. (Part (a) reproduced with permission from [283]. Copyright � 2006 AmericanPhysical Society. Parts (b) and (c) reproduced with permission from [284]. Copyright � 2007American Physical Society.)


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we can only consider the b-components of E1,2, denoted by E1,2. Coordinates E1 and

E2 span an irreducible representation of the space group Pbnm corresponding

to k¼ (0, 1/2, 0). Taking into account the polarization P as a polar vector, the

Landau potential corresponding to the E-phase can be defined as

F ¼ aðE21 þ E2

2Þ þ b1ðE21 þ E2

2Þ2þ b2E

21E

22cðE

21 � E2

2ÞPa

þ dðE21 � E2

2ÞE1E2Pb þ1

2P2,

ð26Þ

where is the dielectric susceptibility of the PE phase and other coefficients are

the phenomenological parameters of the Landau theory. Minimizing F with respect

to P yields

Pa ¼ �cðE21 � E2

2Þ,

Pb ¼ �dðE21 � E2

2ÞE1E2,

Pc ¼ 0,

ð27Þ

where Pi is the component of P along the i-axis. Equation (27) shows us that the four

domains in the E-phase space, i.e. (�E1, 0) and (0, �E2), are all multiferroic, with

polarization P pointing to the a-axis but its sign depending on the relative balance

between coordinates E1 and E2.To understand the microscopic mechanism for this E-phase-induced ferroelec-

tricity, one has to take into account the orbitally degenerate double exchange with egelectrons per Mn3þ ion. The Hamiltonian can be written as [283]

H ¼ �Xia��

Ci, iþatia��dþi�diþa, � þ JAF

Xia

Si � Siþa þ Xi

ðQ1i�i þQ2i�xi þQ3i�ziÞ

þ1

2

Xim

�mQ2mi, ð28Þ

where di� is the annihilation operator for the eg electrons on two orbitals

�¼ �(x2� y2), and �(3z2 – r2), a is the direction of the link connecting the two

nearest-neighbour Mn sites, and Si the classical unit spin of t2g electrons of Mn sites,

Cij the double-exchange factor arising due to the large Hund’s coupling that projects

out the eg electrons with spin antiparallel to Si, and Qmi represents the classical

adiabatic phonon modes.An explicit solution to this model seems challenging. Orthorhombic perovskite

manganites have a GdFeO3-like distorted lattice with the Mn–O–Mn bond angle

(’ia) deviating from 180�. In order to include the initial structural buckling distortion

in orthorhombic perovskites, the dependence of hopping parameter tia�� on ’ia must

be considered explicitly, and the classical adiabatic phonon modes also must be

defined such that the elastic energy term is minimal for ’ia¼’05 180�, as shown

in Figure 49(a). For example, ’0 is approximately 144� for HoMnO3.Monte Carlo simulation of this model Hamiltonian manifests the crucial role

of the double exchange in the formation of the ferroelectric state. Owing to the factor

Ci,iþa, electron hopping between Mn ions with opposite t2g spins is prohibited. The

displacements of the corresponding oxygen ions perpendicular to the Mn–Mn bond


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(these displacements are not Jahn–Teller active) depend only on the elastic energy,favouring a small angle ’0. In contrast, the hopping along the ferromagnetic zigzagchains is usually allowed and the hopping energy is minimal for ’0¼ 180�. Therefore,the displacements of the oxygen ions are eventually determined by the competitionbetween the hopping energy and elastic energy. The resultant optimized angle �should satisfy condition ’05 ’5 180�. Since angle ’ only depends on the bondnature (ferromagnetic or antiferromagnetic), the oxygen displacements for all zigzagchains have the same direction even though the neighbouring chains have oppositespin alignment, as shown in Figures 48(b) and 49(b). This leads to the overallcoherent displacements of the O ions with respect to the Mn sublattice, i.e.ferroelectricity, as shown in right-hand side of Figure 49(c). Clearly, these coherentdisplacements, however, do not exist if the Mn spin alignment is disordered as shownin the left-hand side of Figure 49(c).

In fact, it was revealed by Monte Carlo simulation that the magnitude ofpolarization P does depend on ’0. It would be zero if ’0¼ 180� since both thehopping energy and elastic term are optimal at ’ia¼’0¼ 180�, as shown inFigure 49(d). In summary, it is understood that the symmetry of zigzag spin chainsin the E-phase orthorhombic perovskites with buckling distorted oxygen octahedraallows for the formation of a polar axis along the a-axis, i.e. spontaneouspolarization along this direction.

The first principle calculation proved that the inequivalence of the in-planeMn–O–Mn configurations for parallel and antiparallel spins is an efficient

Figure 49. (Colour online) Monte Carlo simulation of the AFM E-phase inducedpolarization. (a) Starting configuration of a Mn–O–Mn bond. (b) A Monte Carlo snapshotof the E-phase at T¼ 0.001. The arrows on the Mn ions denote their spin and theferromagnetic zigzag chains are shown by solid red lines. (c) Local arrangement of the Mn–O–Mn bonds with (left) disordered Mn spins and (right) opposite Mn spin chains. The arrowsindicate the oxygen displacements, open and crossed circles denote the direction of Mn spins.(d) Dependence of polarization on the starting Mn–O–Mn angle �0. (Reproduced withpermission from [283]. Copyright � 2006 American Physical Society.)


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mechanism for driving a considerable ferroelectric polarization [284]. Moreover,in addition to the polar ionic displacement mechanism, a larger portion of theferroelectric polarization was found to arise owing to the quantum-mechanicaleffects of electron orbital polarization [284]. Figure 48(c) shows the charge densityisosurface in the ac-plane in the energy region between �8 eV and 0 eV (0 eV is thetop of the valent band) for the relaxed structure of the antiferromagnetic E-phase(E1) HoMnO3 with centrosymmetric atom arrangement by first principle calcula-tions. From the charge density distribution, in addition to the expected checker-board-like orbital ordering, two kinds of inequivalent oxygen ions with differentcharge distribution are confirmed due to the energy range of hybridized Mn eg andO p orbitals. The result of this polar charge distribution would be a polarizationwhose direction and quality are same as that induced by ionic displacement, and isdue to symmetry breaking by the antiferromagnetic-E ordering [284].

To end this section, we look at experiments on orthorhombic HoMnO3. It shouldbe mentioned that HoMnO3 is of hexagonal structure at normal ambient pressure.However, orthorhombic HoMnO3 was successfully synthesized by high-pressuresintering. It does exhibit the E-phase below approximately 26K, and consequently amacroscopic polarization along the orthorhombic a-axis [129,285–287], confirmingthe theoretical prediction. However, the rapidly enhanced polarization below 15Kwas argued to be related to the non-collinear spin order of Ho3þ ions rather than theE-phase.

The E-phase and associated ferroelectricity generation were used to explain thepolarization flop at the critical field HC along the c-axis in TbMnO3, intensivelycoinciding with the first-order transition to a CM magnetic phase with propagationvector (0, 1/4, 0) (see [170]). Another possible case is nickelates which also exhibit anE-phase consisting of zigzag spin chains with different directions in the ab-plane anddifferent stacking modes along the c-axis (þ þ � �). The Landau theory analysispredicts a polarization along the b-axis in such E-phase nickelates [283].

3.7. Electric field switched magnetism

In Section 3.4.5, we have already discussed the electric field control of magnetismin spiral multiferroic materials. In spite of this, in general it is difficult to realize sucha control, in particular a switching of the magnetization state. So far, no electricfield switching of a magnetization between a pair of 180� equivalent states has beendemonstrated. One possible reason might be that the electronic polarization appearsas a second-order parameter coupled to the primary order parameter (magnetiza-tion) in those multiferroics with magnetism-induced ferroelectricity. It is generallybelieved that a realization of such a switching can be easy if the magnetization isa second-order parameter coupled to polarization as the primary order parameter,i.e. the ferroelectricity-induced magnetism [288,289], while this argument seems tobe challenging in principle. Even so, it should be noted that most multiferroicsaddressed so far have zero or weak macroscopic magnetization because of theantiferromagnetic nature of the spin configuration, while a large magnetizationwill be one of the prerequisites for practical applications. These motivations makethe idea of ferroelectrically induced ferromagnetism very attractive although no


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substantial progress along this line has been accomplished. We discuss this issuein this section.

3.7.1. Symmetry consideration

According to the discussion in Section 3.4.2, the antisymmetric microscopic couplingbetween two localized magnetic moments, i.e. the DM interaction is maximized whenthe two magnetic moments form a 90� angle, or more accurately, when dij, Si and Sj

form a left-handed coordinate system for determinant jdijj4 0 in Equation (11).However, in these compounds under consideration, the Heisenberg-type interactionEHij ¼ Jij � ðSi � SjÞ is usually much stronger than the DM interaction. For Jij¼ Jji, the

Heisenberg interaction favours an angle of either 0 or 180� between Si and Sj,therefore, the presence of the DM interaction can only lead to a small cantingof these interactive moments, corresponding to a weak macroscopic magnetization,i.e. weak ferromagnetism.

Now we discuss the DM factor dij. If the midpoint between the interactivemoments is an inversion centre, dij is identically zero. For conventional ferroelectricsof interest today, a small polar structural distortion away from a centrosymmetricPE structure exists. If the midpoint between two neighbouring magnetic ions in thePE structure is an inversion centre, this symmetry will be broken by the ferroelectricdistortion, which actually ‘switches on’ the DM interaction (a non-zero dij) betweenthe two ions, i.e. switches on a non-zero magnetization. This criterion was coinedas the structural–chemical criterion, and the material-specific parameter D definedin Equation (11) can be identified by the polarization P. In summary, a ferroelectricdistortion can generate a weak magnetization when the phenomenological invariant

EDML ¼ P � ðM LÞ ð29Þ

is allowed in the free energy of an antiferromagnetic–PE phase. Consequently, at theferroelectric transition point, once polarization P becomes nonzero, the system gainsan energy EDML by simultaneously generating a non-zero M. Moreover, if it ispossible to reverse polarization P using an electric field without varying the directionof vector L (in Equation (29)), magnetization M will certainly reverse in order tominimize the total free energy. Therefore, the DM interaction (i.e. invariant termEquation (11)) allows a possibility for electric-field-induced switching of magnetiza-tion. Note that some other symmetry operation associated with the ferroelectrictransition or electric-field-induced sequences would result in dij¼ 0, whichunfortunately will prevent such a macroscopic magnetization from appearing[263]. In the following two sections, we illustrate some examples to demonstrate suchan effect.

3.7.2. Electric polarization induced antiferromagnetism in BaNiF4

The first example is BaNiF4, which was theoretically predicted to exhibitferroelectrically induced magnetic order [289]. BaNiF4 is a representative of theisostructural family of barium fluorides of chemical formula BaMF4 with M¼Mn,Fe, Co, Ni, Zn or Mg, etc. They crystallize in the base-centred orthorhombicstructure with space group Cmc21. The magnetic unit cell is doubled in comparison


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with the chemical unit cell, and contains four magnetic Ni ions arranged in sheetsperpendicular to the b-axis. The cations within each sheet form a puckeredrectangular grid, with the magnetic moments of neighbouring cations alignedantiferromagnetically but all parallel to the b-axis. The coupling between differentsheets is weak.

Nevertheless, first principle calculation including the spin–orbit coupling predictsthat the collinear spin configuration with all spins aligned along the b-axis is unstableand the moments assume a non-collinear configuration where all spins are slightlytilted toward the �c-axis. Although this issue remains to be further clarifiedexperimentally, the weak spin canting can be explained by the DM interaction.The magnetic space group of BaNiF4 does not allow the occurrence of weakferromagnetism, nevertheless a non-zero DM factor between the magnetic nearestneighbours along the c-axis, dc, is available, in spite of such DM interaction alongthe a-axis vanishes. The canting due to the DM interaction generates a weakantiferromagnetic order parameter Lc¼ s1þ s2� s3� s4 in addition to experimentallyobserved (primary) antiferromagnetic order parameter Lab¼ s1� s2� s3þ s4.Following the theory of the DM interaction described above, the DM couplingbetween order parameters Lab and Lc on the macroscopic level can be written as

EDMmacro ¼ D � ðLab LcÞ: ð30Þ

Surely, an inclusion of this DM term in the free energy allows control of the Lc bythe electric field. In fact, computation shows that no canting of the magneticmoments in the non-polar structure is possible and the resultant magnetic ordercorresponds to a collinear structure, as shown in Figure 50(a). For a polar distortingstructure, the magnetic order becomes non-linear, as shown in Figure 50(b). Whenpolarization P is reversed in the calculation, clear orientation dependence of Lc on Pis obtained, if the order parameter Lab, is fixed, as shown in Figure 50(c). This doesindicate a reversal of Lc upon a reversal of P driven by electric field [289].

3.7.3. Electric polarization-induced weak ferromagnetism in FeTiO3

The second example is FeTiO3. Before discussing this system, we look at BiFeO3

first, which was described carefully in the earlier sections, because BiFeO3 is astarting example for designing multiferroics with ferroelectrically induced weakferromagnetism.

In PE BiFeO3 with space group R3c, Bi ions occupy the A sites with the Wyckoffposition 2a of local site symmetry 32 (as shown in Figure 51(a)) whereas magneticFe ions occupy the B-sites with the Wyckoff position 2b of inversion symmetry.The Fe spins order ferromagnetically within the antiferromagnetically coupled (111)planes of magnetic easy axis perpendicular to the [111] direction. Although in the PEphase of BiFeO3, the symmetry operator I transforms each Fe sublattice onto itself,i.e. IL¼ I(S1�S2)¼L, as shown in Figure 51(b), in this case, the invariant EPLM

is forbidden by the symmetry (the PE point group is 20/m0 or 2/m for which weakferromagnetism is allowed). In other words, the midpoints between the magnetic sitesare not the inversion centres, as shown in Figure 51(b). First principle calculationreveals that for BiFeO3, the sign of vector coefficient D defined in Equation (27)


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Figure 51. (a) Crystal structure and symmetry elements of PE BiFeO3 with space group R�3c.(b) Spin structure and symmetry elements of BiFeO3. (c) Spin structure and symmetryelements of FeTiO3. (Reproduced with permission from [288]. Copyright � 2008 AmericanPhysical Society.)

Figure 50. (a) Collinear magnetic structure of BaNiF4 extracted from experimentalobservations. (b) Canting spin-ordered structure, i.e. weak magnetic order obtained fromfirst-principle calculations including the spin–orbit coupling. (c) Reversal of polarizationin (b) leads to a reversal of the canted magnetic moments and thus to a reversal of vector Lc.(Reproduced with permission from [289]. Copyright � 2006 American Physical Society.)


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is independent of the polar distortion, instead it is determined by a rotational(non-polar) distortion of the oxygen octahedral network [290].

The situation would be entirely different if we place the magnetic ions on theA sites which are ordered similarly so that the magnetic criterion is still satisfied,as shown in Figure 51(c). This corresponds to the A-site magnetism, and one hasIL¼�L, i.e. the midpoints between the magnetic sites are the inversion centres.Placing a ferroelectrically active ion such as Ti4þ on the B site would then satisfy thestructural–chemical criterion. Although the magnetic point group in the PE phasebecomes 2/m0 (20/m) in which weak ferromagnetism is also forbidden, a ferroelectricdistortion by design, via the term EDLM, would favour a lower symmetry m0 (m), thusallowing the weak ferromagnetism, as shown in Figure 51(c). The high-pressuremetastable phase of FeTiO3 and MnTiO3 [290–293] with space group R3c meetsthe criteria above, and provides the possibility of realizing the electric-field switchingof magnetization.

In fact, first principle calculation along this line is quite optimistic and direct.For FeTiO3, a PE phonon of symmetry type A2u can be polarized along the [111]direction in the R�3c!R3c transition. One highly unstable mode in the A2u phononsconsisting of displacements of the Fe ions and Ti ions against oxygen was alsopredicted, which is similar to other R3c ferroelectrics such as BiFeO3 and LiNdO3.A spontaneous polarization as large as 80–100 mCcm�2, together with a ferroelectrictransition point as high as 1500–2000K, was estimated. More exciting is the chiralitychange of the S–O–S bonds (as shown in Figure 52) associated with the variationof polarization P in orientation, was revealed in the calculation [288].

Although BaNiF4 and FeTiO3 were predicted to be multiferroics with ferro-electrically induced magnetism, so far no experimental evidence has been madeavailable owing to the challenges faced in sample synthesis. High-quality samplesand experimental verification of these predictions are urgently needed, so that asubstantial step towards practical control and switching of magnetism by an electricfield can be made.

3.8. Other approaches

Before ending this long section, we make some remarks on other strategies ofintegrating the two functions, ferroelectricity and magnetism, into one

Figure 52. (Colour online) Chiral nature of the S1–O–S2 bonds of FeTiO3 in the ferroelectricphase with polarization P up (left), polarization P down (right) and in the PE phase (middle).(Reproduced with permission from [288]. Copyright � 2008 American Physical Society.)


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single-phase compound. The unveiled physics may shed light on the design andsynthesis of novel multiferroics.

3.8.1. Ferroelectricity in DyFeO3

It has been known that in orthorhombic DyAlO3 there exists a large linearmagnetoelectric component [294,295]. However, the Al ions are diamagnetic at theground state, and then these materials do not show spontaneous polarization P andmagnetization M. Substitution of a magnetic ion at the B-site may produce themultiferroic state. Recently, researchers found the magnetic-field-induced ferroelec-tricity in orthorhombic DyFeO3 (see [296]). The magnetic structures of DyFeO3

are shown in Figure 53. Below Tr� 37K, the Fe spins align antiferromagneticallyin configuration AxGyCz where the G-type and A-type components of Fe spins aredirect towards the b-axis and a-axis respectively, while the C-type components arealong the c-axis. Upon further cooling, magnetization M shows another anomalyat T

DyN � 4K, corresponding to the antiferromagnetic ordering of Dy moments in the

GxAy configuration. Moreover, below Tr, a magnetic field H4HFer along the c-axis

causes the configuration change to GxAyFz so as to produce a weak ferromagneticcomponent along the c-axis. Under H¼ 30 kOe along the c-axis, a large P only along

Figure 53. (Colour online) Magnetic structures of DyFeO3 below TFeN under a magnetic field

(along the c-axis)H5HFer ((a) and (b)) andH4HFe

r ((c) and (d)). In (b) and (d), the magneticstructures of Dy ions are different from (a) and (c), and then a reversed polarization appears in(d). (Reproduced with permission from [296]. Copyright � 2008 American Physical Society.)


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the c-axis below TDyN was observed, as shown in Figure 54(a). Moreover, with

increasing H, the polarization P as a normal linear magnetoelectric componentincreases monotonically from zero, but shows an anomaly atH ¼ HFe

r � 24 kOe. Theextrapolated value of P from the data within the region of H4HFe

r backforward toH¼ 0 is non-zero, as shown by the dashed line in Figure 54(b). This demonstrates theexistence of spontaneous polarization P. In fact, the multiferroic state can be furtherconfirmed by the PE hysteresis, as shown in Figures 54(c) and (d) [296].

The orientation relationship between P and M (PkM) in DyFeO3, which isdifferent from the relationship in spiral magnets such as DyMnO3, together with thedisappearance of P at TDy

N and the anomaly of P at HFer , suggests that the mechanism

for ferroelectricity in DyFeO3 is different from the inverse DM interaction anddepends on both the magnetic structures of Dy and Fe ions. It is postulated that theexchange striction between those adjacent Fe3þ and Dy3þ layers with the interlayerantiferromagnetic interaction (see Figures 53(c) and (d)) results in the multiferroicphase. The ferromagnetic sheets formed by Fe and Dy ions stack along the c-axis.For the Ay component along the b-axis, the spins on the Fe layer become parallel tothe moments on one of the nearest-neighbour Dy layers and antiparallel to thosemoments on another nearest-neighbour layer. As a result, the Dy layers shoulddisplace cooperatively toward the Fe layers with antiparallel spins, via the exchangestriction. Then the polarization along the c-axis appears [296].

Figure 54. (a) Temperature dependence of polarization of DyFeO3 along the a-axis, b-axis,c-axis under magnetic field of 30 kOe (4HFe

r ) along the c-axis. The dotted line shows thepolarization along the c-axis under a magnetic field of 500Oe (5HFe

r ). (b) Magnetic field(along the c-axis) dependence of the residual polarization obtained by P–E loops (filled circles)and the displacement current measurement (solid line) at T¼ 3K. The dashed line is theextrapolated polarization curve in the regions of H4HFe

r towards H¼ 0. (c), (d) Magneticfield (along the c-axis) dependence of the P–E loops measured under Hkc and Ekcconfigurations with different frequencies by a Sawyer–Tower bridge. (Reproduced withpermission from [296]. Copyright � 2008 American Physical Society.)


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3.8.2. Ferroelectricity induced by A-site disorder

Perovskite lattice instabilities are often described by the tolerance factort ¼ ðrA þ rOÞ=

ffiffiffi2pðrB þ rOÞ, where rA, rB and rO are the A-site, B-site and O ionic

radii, respectively. Conventional ferroelectric materials such as BaTiO3 often havet4 1, indicating that the B-site ion is too small for ideal cubic structure. Assisted bythe covalent hybridization with O ions, B-site ions deviate from the centre positionsand then cause ferroelectric polarization. The ferroelectrics with a lone-pairmechanism, as stated in Section 3.2, have t5 1 and the ferroelectricity is from off-centring of the A-site ions. Without Pb or Bi, perovskite structures of t5 1 generallyhave tilted BO6 octahedra instead of the A-site off-centring. Unfortunately, themagnetic perovskite materials often have t5 1 and tilted BO6 octahedra becausethose ions with d electrons are generally larger than d 0 ions, and then notferroelectrically active.

However, first principle calculations suggest that the octahedral tilting isprevented in KNbO3–LiNbO3 alloys with the average tolerance factor significantlysmaller than one, because K ions and smaller Li ions are distributed randomly in thelattice, which is coined as an A-site disorder. The ferroelectricity appears to originatefrom the large off-centring of Li ions, contributing significantly to the differencebetween the tetragonal and rhombohedral ferroelectric states and yielding atetragonal ground state even without strain coupling [297].

Based on above discussion, it is predicted that (La,Lu)MnNiO6 with t5 1exhibits polar-type lattice distortion [298]. This polar behaviour arises from thefrustration of the octahedral tilting instabilities due to the mixture of A-site cationsof different sizes and the fact that the coherence length for the A-site off-centringis shorter than that for the tilting instabilities [298]. On the other hand, Mn3þ andNi3þ ions can occupy the B-sites in an order form, resulting in the double perovskitestructure. The superexhange interaction between Mn3þ and Ni3þ is ferromagnetic[86,87]. Owing to these mechanisms, (La,Lu)MnNiO6 may exhibit large ferroelectricpolarization and ferromagnetism simultaneously. However, again it is difficult tosynthesize (La,Lu)MnNiO6 because of the phase separation and competing phases,which often occur for perovskite oxide materials with mismatching A-site species.Therefore, so far no experimental evidence with this A-site disorder inducedmultiferroicity has been made available.

3.8.3. Possible ferroelectricity in graphene

In addition to those approaches in transitional metal oxides substantially addressedabove, some other approaches may be also utilized to synthesize novel multiferroicmaterials. For example, an electronic phase with coexisting magnetic and ferro-electric orders in graphene ribbons with zigzag edges is predicted [299–303]. Thephysics lies in that the coherence of the Bardeen–Cooper–Schrieffer (BCS) wavefunction for electron–hole pairs in the edge bands, available in each spin channel,is related to the spin-resolved electric polarization [299]. Although the totalpolarization may vanish due to the internal phase locking of the BCS state, strongmagnetoelectric effects are expectable. By placing the graphene between twoferromagnetic dielectric materials, theoretical analysis predicts that the magneticinteraction at the interface affects the graphene band structure and leads to an


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effective exchange bias between the magnetic layers, which is highly dependenton the electronic properties (particularly on the position of the electrochemicalpotential, i.e. the Fermi level) of the graphene layer. Therefore, an external electricfield (the gate bias) can modulate the exchange bias [304].

3.8.4. Interfacial effects in multilayered structures

Interfacial effects, which are different from the macroscopic mechanical transferprocess, can be exemplified in multiferroic superlattices, and then significantmagnetoelectric effects can be expected. For example, first principle calculationpredicts that, in the ferroelectric/ferromagnetic multilayers such as the Fe/BaTiO3

structure, the bond fluctuation on the ferroelectric/ferromagnetic interface willmodulate the interfacial magnetization upon the polarization reversal due to theinterface bonding sensitive to the atomic displacements on the interface [305,306].Similar effects are also predicted by first principle calculation in the Fe3O4/BaTiO3

oxide superlattice [307]. The effects of the charge imbalance and strain as well asoxygen vacancies on the interfaces of superlattice, may play important roles [308].Moreover, first-principle calculation claims that even for the Fe (001), Ni (001) andCo (0001) films, an external electric field can induce remarkable changes in thesurface magnetization and surface magnetocystalline anisotropy, originatingfrom spin-dependent screening of the electric field at the metal surface, as shownin Figure 55 (see also [309]). However, these effects still need an experimentaldemonstration. Another approach to multiferroics is the use of so-called tricolourmultilayered oxides structures. Tricolour multilayered structure (i.e. ABCABC . . . )without ferroelectric layer, where at least one layer or one interfacial layer should beferromagnetic, such as the LaAlO3/La0.6Sr0.4MnO3/SrTiO3 structure, exhibitsmultiferroicity on the ferromagnetic interface. The details of the tricolourmultilayered structure can refer to recent literature [310–314].

4. Elementary excitation in multiferroics: electromagnons

For condensed matters, it is well established that any spontaneous breaking ofsymmetry will induce novel elemental excitation [315]. For conventional ferro-electrics, a displacive structural phase transition is associated with one of thetransverse optical (TO) phonons softening with its frequency, corresponding tothe square root of the inverse order parameter susceptibility (0), i.e. !2 / 1/(0)(see [5]). Here, a soft polar phonon directly couples to the divergent dielectricsusceptibility and broken spatial-inversion symmetry. Spin waves (i.e. magnons) arethe characteristic excitations of the magnetic structure. It is expected that thesimultaneous breaking of the spatial-reversion symmetry and time-inversion sym-metry and, thus, the strong coupling between the magnetic and lattice degreesof freedom can lead to complex excitations. In this setting, the character associatedwith the soft mode is less obvious since the multiferroic order does not arise frompure structural degrees of freedom but from their complex interplay with magnetism.Thus, the collective excitation directly reflecting the inverse DM mechanism is therotation mode of the spiral plane that is driven by electric field, and a consequencefundamentally different from ferroelectric and spin excitation exists: electro-active


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magnons, or electromagnons (i.e. the spin waves that can be excited by an a.c.electric field). This kind of elemental excitation due to the magneto-dielectricinteraction was predicted theoretically more than 30 years ago [316], but no otherexperimental observations have been made until very recently [317–336].

4.1. Theoretical consideration

We first outline the theoretical framework of electromagnons, developed recently[317]. From the KNB theory, the spin supercurrent in non-collinear magnets, js /

Figure 55. Difference between the spin densities for a 15-monolayer-thick Fe film with andwithout external electric field (E ) of 1V A�1, i.e. D�¼ �(E)� �(0). (Reproduced withpermission from [309]. Copyright � 2008 American Physical Society.)


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SiSj, leads to the electric polarization defined by P / eij js, with eij the unit vectorconnecting two sites i and j. An effective Hamiltonian describing the couplingbetween spins and atomic displacement ui may take the following form:

H ¼ H1 þH2 þH3 þH4,

H1 ¼ �Xm, n

JðRm � RnÞSm � Sn,

H2 ¼ �Xm

ðum ezÞ � ðSm Smþ1Þ,

H3 ¼Xm

�

2u2m þ

1

2mP2m

� �,

H4 ¼Xm

DðSymÞ

2,

ð31Þ

where H1 denotes the Heisenberg interaction with Ri and Si the coordinates and spinmoment of site i; the spin–lattice interaction H2 stems from the relativistic spin–orbitinteraction and corresponds to the DM interaction once the static displacementhumi is non-zero and the inversion symmetry is broken, um is regarded as the lowest-frequency representative coordinate relevant to polarization P, i.e. the TO phonons,P¼ e*um with a Born charge e*; in the term H3, k and m are the spring constant andeffective mass of um; the term H4 deals with the easy-plane spin anisotropy withanisotropic factor D.

This Hamiltonian allows a helical spin ordering with decreasing temperature,corresponding to the softening and condensation of the spin bosons. The phononmode ux does not show any frequency softening, but the spontaneous polarization isrealized through the hybridization of ux with the spin bosons. One may assumethat the spins are on the easy plane, i.e. Sz

n ¼ S cosðQRn þ �Þ, Sxn ¼ S sinðQRn þ �Þ,

Syn ¼ 0, where Q is the spiral wave number and � is the phase angle. Also, the

equations of motion for spins and displacements can be derived out from theHamiltonian. Considering the lowest-temperature region with spin order andconverting the lattice into a rotating local coordinate system ( , �, �) and momentumspace (q), one has the equations of motion:

_S�q ¼ �AðqÞS q,

_uq ¼ pq=m,

_S q ¼ BðqÞS�q � iS2 eiQa � e�iQa

2iei�uq�Q þ

Q!�Q

�!��

� �� ,

_pq ¼ ��uq þ iSeiQa � eiðq�QÞa

2iei�S�q�Q þ

Q!�Q

�!��

� �� ,

ð32Þ

with

AðqÞ ¼ 2S2JðQÞ � JðQþ qÞ � JðQ� qÞ

2þ22S2

�sin2

qa

2

� � sin2ðQaÞ

� �,

BðqÞ ¼ 2S JðQÞ � JðqÞ þ2S2

�sin2ðQaÞ þ K

� �:

ð33Þ


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From the equations of motion, one can evidently see the coupling between the spinwave modes and electric polarization. Here, S� and S are the canonical variablesand form a harmonic oscillator at each q in the rotate frame. The spin wave at q iscoupled with the phonon u at q�Q, or uq is coupled to Sn at q�Q. The uniformlattice displacement uyo is coupled to e�i�S�Q � ei�S��Q, which corresponds to therotation of both the spin plane and the direction of polarization along the z-axis.This mode is the Goldstone mode with frequency !¼ 0 if K¼ 0. The spin wave modeat q¼ 0 corresponds to the sliding mode, i.e. spiral phason.

The dynamic dielectric function can be obtained by the retarded Green functionand it has the poles at !�, given by

!2� ¼

1

2!20 þ !

2p �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið!2

0 þ !2pÞ

2� 4AðQÞK!2

0

q� �,

!0 ¼

ffiffiffiffik

m

r,

!p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAðQÞBðQÞ

p,

ð34Þ

where !0 is the frequency for the original phonon and !p is the frequency of themode e�i�S�Q � ei�S��Q. In the limit of � !2

0, one can see from the above equationthat there are two modes contributing to the dielectric function. One is the phononmode with frequency !þ�!0 which is high and does not show any softening. Thedielectric function is most likely contributed from the other mode, i.e. the z-axisrotation mode (spin wave mode) at !� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiAðQÞK

p, which is hybridized with the

polarization mode uyo.The theoretical study on the elementary excitation based on the symmetry

analysis and the Landau theory also gives similar results [318]. The above discussioncombined with a realistic estimation of material parameters allows one to calculatethe frequency of the collective mode, i.e. the electromagnon. As well establishedalready, ferroelectricity induced by the spin order is usually observable in ICMspiral/helical spin-ordered systems. This special spin order can be suppressed by anexternal magnetic field, thus the corresponding electromagnon can be wiped out.As a consequence, a significant response of the reflection spectrum, ranging from d.c.up to the terahertz frequency range, to an external magnetic field can be expected.Along this line, dielectric spectroscopy under an external magnetic field can be aroadmap to reveal electromagnons in multiferroics. In fact, preliminary experimentsto disclose this electromagnon excitation, reported recently, were quite successfuland good agreement between experimental observations and theoretical predictionswas found, as we show below.

4.2. Electromagnons in spiral spin-ordered (Tb/Gd)MnO3

For multiferroics, the frequency dependence of the dielectric response usually showsa broad relaxation-like excitation. The characteristic frequency for GdMnO3 is�0¼ 23(�3) cm�1 and for TbMnO3 it is v0¼ 20(�3) cm�1, as shown in Figure 56.The dielectric response of this excitation increases with decreasing temperatureand becomes saturated once the low-temperature spiral magnetic phase enters.


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Upon a magnetic field, both the imaginary and real parts of the dielectric constantwill be suppressed. In particular, the excitation will be suppressed when the a.c.component of the electric field e is rotated from eka to ekb, given a constant magneticfield. However, it remains unchanged when the a.c. component of the magneticfield h is rotated from hkc to hkb, as shown in the inset of Figure 56(c) (seealso [319]). The significant sensitivity of the excitation to the a.c. electric fielddemonstrates the strong coupling between the magnetic and lattice degrees offreedom, reflecting the close correlation of the spin structure and electric polarizationand thus providing possible evidence for electromagnons in multiferroics [319].

These experimental results are also quantitatively consistent with theoreticalpredictions. For TbMnO3, from the spin wave dispersion data observed in theneutron scattering and electron spin resonance (ESR) spectroscopy, the exchangecoupling J1 was estimated to be 8SJ1¼ 2.4meV and the spin–lattice coupling was� 1.0meV A�1. The Born charge was assumed to be 16e where e is the bareunit charge. Then the evaluated frequency of the collective mode in TbMnO3 is!�� 10 cm�1, which is of the same order with experimental data (�20 cm�1).

Figure 56. (Colour online) Dielectric spectra of GdMnO3 and TbMnO3 at differenttemperatures under various combinations of electric and magnetic fields. (Reproduced withpermission from [319]. Copyright � 2006 Macmillan Publishers Ltd/Nature Physics.)


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Inelastic neutron scattering (INS) is the most powerful technique for finding themagnetic excitation in spin systems [320]. The INS dispersion relations for spin waveexcitations in TbMnO3 along the a-axis and c-axis of the Pbnm lattice in PEsinusoidal phase and in the ferroelectric spiral phase, respectively, are presented inFigures 57(a) and (b). Clearly, the three low-lying magnons are revealed, as shownin Figure 57(c), in which the lowest-energy mode is the sliding mode of the spiral.The other two modes at 1.1 and 2.5meV correspond to the rotations of the spiralrotation plane, as shown in Figure 57(d). The latter two modes are coupled with theelectric polarization and the outcome is in perfect agreement with the infrared spec-troscopy result. This is a hybridized phonon–magnon excitation (i.e. electromagnon).

It should be mentioned that some other methods such as the far-infraredspectroscopy were recently utilized for probing electromagnons in spiral

Figure 57. (Colour online) Dispersion relations of spin-wave excitations in (a) PE and (b)ferroelectric phases of TbMnO3, respectively. The dashed lines are the dispersion relations ofLaMnO3 for comparison. (c) Spectra of element excitations in PE and ferroelectric phases ofTbMnO3. (d) Three magnons in the ferroelectric spiral spin-order phase of TbMnO3.(Reproduced with permission from [320]. Copyright � 2007 American Physical Society.)


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multiferroics, such as in Eu1�xYxMnO3 (see [321,322]) and GdMnO3 (see [323]),which also demonstrate the existence of the spin (magnon)–lattice (phonon) couplingand electromagnons in perovskite RMnO3. It is worth of noting that the elementaryexcitation in RMnO3 remains ambiguous although the experimental results are alsoquantitatively consistent with theoretical predictions. According to the inverse DMmechanism, the k¼ 0 magnon mode responding to the rotation mode of the spiralplane should be active for the E-vector perpendicular to the spin spiral plane. Then,the polarization selection rule for the electromagnon, i.e. the absorption band inEu1�xYxMnO3 with Pka, should be E!

kc. However, the absorption band with E!ka

was observed in Eu1�xYxMnO3. This discrepancy appeals for further research. Forexample, the wide range optical spectra on Eu1�xYxMnO3 revealed that the possiblecandidate of origin for this absorption band is the two-magnon excitation drivenby the electric field [324].

4.3. Electromagnons in charge-frustrated RMn2O5

Additional evidence on electromagnons comes from the far-infrared transmissionspectra for YMn2O5 and TbMn2O5. TbMn2O5 (YMn2O5) favours the paramagnetic/PE state at T4 41(45)K, the CM magnetic order and ferroelectric order at24(20)K5T5 38(41)K, and the ICM magnetic order and ferroelectric order below24(20)K. The far-infrared transmission data revealed a clear electromagnonexcitation feature below the lowest phonon centred at approximately 97 cm�1 andthe strongest absorption near 10 cm�1: at 7.9 cm�1 for YMn2O5 and 9.6 cm�1 forTbMn2O5, as shown in Figure 58 (see also [325]).

4.4. Spin–phonon coupling in hexagonal YMnO3

Although the ferroelectric ordering and magnetic ordering in hexagonal RMnO3 isnot concomitant, there exists a strong interplay between the two order parameters,as discussed in Section 3.4. It is reasonable to postulate that the spin–phononcoupling in hexagonal RMnO3 is strong. Looking at such a coupling in YMnO3,characterized by thermal conductivity, it was observed that the thermal conductivityexhibits an isotropic suppression in the cooperative paramagnetic state, followedby a sudden increase upon the magnetic ordering. This unprecedented behaviourwithout any associated structural distortion is probably the consequence of a strongdynamic coupling between the acoustic phonons and low-energy spin fluctuationsin geometrically frustrated magnets [326]. Some other experiments, such as thermalexpansion [327], Raman scattering [328], and ultrasonic measurement [329] alsorevealed the existence of a giant spin–lattice (phonon) coupling.

Also, such a coupling can be probed by INS, which plays an important role in thestudy of the elementary excitation in YMnO3 (see [330]). Figure 59(a) shows themagnetic structure of YMnO3 and the dashed lines in Figure 59(b) plot the magnondispersion of three modes along the a*-direction (as shown in Figure 59(a); the OTOP

tilting direction involved in the ferroelectric distortion) measured by neutronscattering (symbols). The dispersions of the transverse phonon mode mainlypolarized along the c-axis in the ferroelectric phase with propagation along the


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a*-axis, obtained at 200K (triangles) and 18K (circles), are shown in Figure 59(b),together with the optical phonon mode (squares) and three magnon modes (dashedline). It is evident that a gap in the phonon dispersion opens at q0� 0.185 anda crossing of the 200K phonon dispersion with the magnon mode 2 arises atqacross� 0.3. Moreover, the gap opens mainly below TN, indicating its coupling withthe magnetic subsystem. Figure 59(c) shows the nuclear dynamical structure factorrevealing the phonon-like component of the hybrid excitation, where a jump fromthe lower to the upper mode is observed, providing a natural interpretation of theexperimentally observed gap. These data reveal a strong coupling between spins andphonons and possible electromagnons, i.e. the hybridization between the two typesof elementary excitation in hexagonal manganites [330].

4.5. Cycloidal electromagnons in BiFeO3

BiFeO3 is similar to YMnO3 in the sense that the ferroelectricity and magnetismoriginate from different ions. However, as illustrated in Section 3.4.4, theferroelectricity in BiFeO3 is closely related to the cycloidal antiferromagneticorder, implying an intimate relationship between the electric polarization and spinwave excitations (magnons), i.e. the electromagnons [207,331–333]. BiFeO3 exhibitsa G-type antiferromagnetic order which is subjected to a long-range modulationassociated with a cycloidal spiral of periodicity approximately 62 nm. The spiralpropagates along the ½10�1� direction with the spin rotation within the ð�12�1Þ plane,as shown in Figure 8(c) and (d) (see also [207]).

Recently, low-energy Raman scattering spectroscopy was used to unveil themagnon spectra of BiFeO3 (see [207]). Although no phonons below 50 cm�1 are

Figure 58. (Colour online) Far-infrared optical transmission spectra for (a) YMn2O5 and (b)TbMn2O5 at different temperatures under various combinations of electric and magneticfields. (Reproduced with permission from [325]. Copyright � 2007 American PhysicalSociety.)


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Figure 59. (Colour online) (a) Schematic of the spin order and atomic positions in YMnO3.The squares represent oxygen ions, and the circles indicate Mn ions. (b) Dispersion of phononsand magnons in YMnO3. The dashed lines indicate the measured magnon dispersions alongthe a*-axis in (a). Triangles and circles represent the phonon dispersions obtained at T¼ 200and 18K, respectively. The squares indicate the optical phonon mode. The gap in the phonondispersion opens at q0� 0.185, and the crossing of the 200K phonon dispersion with themagnon mode arises at qcross� 0.3. (c) Nuclear dynamical structure factor calculated as afunction of the wave vector along (q, 0, 6) and energy. A jump from the lower mode to theupper mode, which results in an experimentally observed gap, occurs. (Reproduced withpermission from [330]. Copyright � 2007 American Physical Society.)


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expected, several peaks in the Raman spectra were observed. The two configurations

with parallel polarization and crossed polarization on the (010) plane produced

spectra with two distinct sets of peaks, as shown in Figure 60(a). The two sets of

peaks, respectively corresponding to the two species of spin wave excitations lying in

(cyclone modes) and out of (extra-cyclone modes) the cycloidal plane, exhibit

distinctive dispersive energy curves that depend on their coupling to the electric

polarization, as shown in Figure 60(b). The antiferromagnetic magnon zone folding,

induced by the periodicity of the cycloidal spin order, leads to a very simple

expression for the energy level structure of the cyclone mode. This cyclone mode

remains gapless, as expected from the antiferromagnetic ordering, but a gap is

expected for the extra-cyclone mode due to the pinning of the cycloidal plane by the

polarization. The experimental results do fit this picture and an extra-cyclone mode

with gap was unambiguously assigned, demonstrating the cycloidal electromagnons,

as shown in Figure 60(b).The elementary excitations in multiferroics will significantly affect the physical

properties, which reveals a new possibility for applications. For example, the

magnetic sublattice precession is coherently excited by picosecond thermal modifi-

cation of the exchange energy during detection of the magnetic resonance mode in

multiferroic Ba0.6Sr1.4Zn2Fe12O22 using time-domain pump-probe reflectance

spectroscopy. This excitation induces the modulation of the material’s dielectric

tensor and then a dynamic magnetoelectric effect [334].In addition to those examples cited above, more experiments did reveal the strong

spin–phonon (lattice) coupling in other multiferroics, such as the 2D triangular

CuFeO2 (see [335]). These experiments unveiled the existence of electromagnons in

a broad category of materials. However, a comprehensive understanding of their

origins, conceptual pictures and dynamics, seems far from sufficient. One key point

is whether the origin of electromagnon excitations is the DM exchange interaction.

Some works pointed out that the electromagnon excitation in multiferroic

orthorhombic RMnO3 should result from the Heisenberg coupling between spins

despite the fact that the polarization arises from the much weaker DM exchange

interaction [336].

5. Ferrotoroidic systems

In practice, ferromagnetism, ferroelectricity and ferroelasticity are widely utilized in

modern technology. The three functions are always called fundamental ferroicity.

One common character for these functions is the domain structure associated with

the spontaneous magnetization, polarization and elastic transform, respectively.

These domains are the key units for data memory. For example, ferromagnetic

domains are memory units in computer hard disks and ferroelectric domains are

found in FeRAMs. Recently, the fourth ferroicity, ferrotoroidicity, was proposed as

being one of the fundamental ferroicities, and consequently the fourth kind of

ferroics, ferrotoroidics, was addressed [337–348].


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Figure 60. (Colour online) (a) Raman spectra of spin excitations in BiFeO3. The equally andnon-equally spaced modes at low frequencies correspond to the � and � cycloidal modesselected out using parallel (||) and crossed (œ) polarizations. The inset shows the superpositionof these two kinds of modes on another sample. (b) � (circles) and � (squares) cycloidal modefrequency as a function of the mode index n, respectively. (Reproduced with permission from[207]. Copyright � 2007 American Physical Society.)


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5.1. Ferrotoroidic order

As is well known, a magnetic toroidic moment is generated by a vortex of a magneticmoment, such as atomic spins or orbital currents, which can be represented by a

time-odd polar (or ‘axiopolar’) vector T ¼ 12 �iri Si, where ri and Si are the ith

magnetic moment and its positional vector, respectively [337–339]. This toroidicvector T changes its sign upon both time-inversion and space-inversion operations

and is generally associated with a ‘circular’ or ‘ring-like’ arrangement of spins. Theconcept of a magnetic toroidic moment can be sketched by a ring-shape torus with

an even number of current windings which exhibit a magnetic toroidic moment T(the green arrow in Figure 61(a)) perpendicular to the ring plane. In a magnetictoroidic system, it is possible to induce a magnetization M by an electric field E and

a polarization P by a magnetic filed H, which is one of the reasons why muchattention has been paid to magnetic toroidic systems. For example, in the systemshown in Figure 61(a), a magnetic field along the ring plane drives a congregation of

the current loops in one direction and, eventually, an electric polarization along thisdirection appears, as shown in Figure 61(b).

A system in which the toroidic moments are aligned spontaneously in a

cooperative way is coined as a ferrotoroidic system. The macroscopic vector T of thissystem can also be used as the order parameter for various d.c./optical magneto-electric phenomena, which describe the genuinely electronic couplings between an

electric field and a magnetic field. For details, the toroidic moment T describes thecoupling between polarization P and magnetization M and one can easily deriveout T / PM for multiferroics of ferroelectric and ferromagnetic orders. However,

it should be mentioned here that a non-zero macroscopic T does not necessarilyrequire the coexistence of P and M. For example, GaFeO3 is a prototypical

ferrotoroidic system, as shown in Figure 62(a) and (b) (see [340–343]). It ispyroelectric in nature with the built-in electronic polarization along the b-axis in theorthorhombic cell, and its spontaneous magnetization stems from the ferromagnetic

arrangement of Fe spins. However, the displacements of two Fe-ion sites areopposite, as if it was antiferroelectric. In this case, a macroscopic toroidal moment

is present but its magnitude is larger than PM. This is also one of the reasonswhy antiferromagnetics or antiferroelectrics are categorized into the componentsof multiferroics. On the other hand, the difference between ferrotoroidics and

Figure 61. (Colour online) Magnetic toroidic moment in a simple system: (a) a ring-shapedtorus with an even number of current windings exhibits a toroidic moment T (the greenarrow); (b) a magnetic field along the ring plane induces the congregation of the current loopsin one direction and eventually an electric polarization along this direction. (Reproduced withpermission from [348]. Copyright � 2007 Macmillan Publishers Ltd/Nature.)


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multiferroics is disputed since the definition for each of the two types of ferroicsremains unclear. For example, typical ferrotoroidic GaFeO3 has been regarded as atypical multiferroic or magnetoelectric material [341].

In fact, any physical system can be characterized by its behaviour upon spatialand temporal reversals. Ferromagnetics and ferroelectrics correspond to the systemswhose order parameters change their sign upon the temporal and spatial reversal,respectively. For a ferroelastic system, no such change occurs under the tworeversals, as shown in Figure 63. It is apparent that the three fundamental ferroicorders correspond to three of the four parity-group representations and the residualshould be assigned as the ferrotoroidic order which changes sign under both the tworeversals. This is the reason to coin ferrotoroidics as the fourth type of fundamentalferroics and the relationship between ferrotoroidics and multiferroics can behighlighted. The multiferroics are spatial and time asymmetric because of thecoexistence of two order parameters: one violating the spatial-reversal symmetry andthe other breaking the temporal-reversal symmetry.

Figure 62. (Colour online) (a) Crystal lattice structure and (b) schematic magnetic toroidicmoments of GaFeO3; (c) four kinds of magnetic-optical effects. (Reproduced with permissionfrom [14]. Copyright � 2007 Elsevier.)


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It is well known that ferroelasticity is always related to ferroelectricity, andsimilarly ferrotoroidicity is intrinsically linked to antiferromagnetism because of itsvortex nature. Figure 64 shows four simple and typical antiferromagnetic systems:Figures 64(a) and (b) have equal and opposite toroidal moments and the anti-ferromagnetic arrangement in Figure 64(c) also has a toroidal moment, while thearrangement in Figure 64(d) does not.

In addition GaFeO3, LiCoPO4 and LiNiPO4 also are the typical ferrotoroidics[344–348]. LiCoPO4 has a olivine crystal structure with mmm symmetry in a param-agnetic state. The Co2þ ions are located at coordinates such as (1/4 þ ", 1/4, ��)

Figure 63. (Colour online) Relations between the ferroic orders and the space-/time-reversal.(Reproduced with permission from [348]. Copyright � 2007 Macmillan Publishers Ltd/Nature.)

Figure 64. (Colour online) Possible antiferromagnetic spin orders: (a) and (b) have equal andopposite toroidal moments, and the antiferromagnetic arrangement in (c) also has a toroidalmoment, while the arrangement shown in (d) does not.


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where " and � are small displacements allowed by the mmm symmetry, as shown inFigure 65(a). At 21.8K, the Co2þ ions order in a compensated antiferromagneticconfiguration with spins along the y-axis while the symmetry changes to mmm0.Moreover, recent neutron scattering data revealed a rotation of the spins by ’¼ 4.6�

away from the y-axis and a reduction of symmetry down to 20 with x as the twofoldaxis. This magnetic order is not of the helical type and all magnetic momentsorder antiparallely with Snk(0, cos’, sin’), contributing to the weak magnetism alongthe y-axis.

LiCoPO4 exhibits a FTO order in the x–z plane, as shown in Figure 65(b). Thespin part of the toroidical moment is described by T ¼ 1

2

Pn rn Sn with rn the radius

vector and Sn the spin of the nth magnetic ions, taking the centre of the unit cell asthe origin. Note that only the components of Sn that are oriented perpendicularlyto rn contribute to T, as shown by the green arrows in Figure 65(b). Clearly, thecontribution of the spins in ions 1 and 3 are in contrast to the contribution of thespins in ions 2 and 4. However, the clockwise contribution from ions 1 and 3 is largerthan the anticlockwise contribution from ions 2 and 4 because r1,34 r2,4, leadingto a residual toroidical moment Ty perpendicular to the x–y plane. Any sign reversalof either Sx or ’ will result in the reversal of order parameters of antiferromagnetismand ferrotoroidicity (�l and �T). In fact, recent experiments using resonant X-rayscattering demonstrated the existence of FTO moment in this system, noting thatLiNiPO4 is very similar to LiCoPO4, although the spins in LiNiPO4 are aligned alongthe z-axis.

5.2. Magnetoelectric effect in ferrotoroidic systems

The toroidic moment T (i.e. the coupling between P and M) can cause someinteresting optical magnetoelectric effects. One of them originates from thepolarization component induced by optical magnetic field as an analogue of themagnetoelectric coupling in the optical frequency. The normal Faraday (or Kerr)

Figure 65. (Colour online) Arrangements of spins of Co2þ ions on (a) the yz-plane and (b) thexz-plane for the ground state of LiCoPO4. The solid and open circles represent the Co ions atx� 3/4 and x� 1/4 positions, respectively. The grey arrows are the spins of Co ions.(Reproduced with permission from [348]. Copyright � 2007 Macmillan Publishers Ltd/Nature.)


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rotation, as shown in Figure 62(c), stems from the dichroism or birefringencewith respect to the right-hand or left-hand circularly polarized light. The opticalmagnetoelectric effect refers to the dichroism/birefringence with respect to the lightpropagation vector, irrespective of the light polarization, as shown in Figure 62(c).Another important feature of the optical magnetoelectric effect is the second-ordernon-linear optical activity. Owing to the presence of the toroidal moment T, thesecond harmonic (SH) light with polarization in parallel to T can be generated(Figure 62(c)) in addition to the ordinary SH light polarized along the P direction.Eventually, the incident light polarized along the T direction can generate the SHcomponents polarized along the P and T directions, respectively. This T-induced SHcomponent may reverse its phase upon the magnetization reversal. Consequently,the polarization of the SH light can rotate depending on the magnetization directionor, equivalently, on the toroidic moment direction. This non-linear Kerr rotationcan be used to sensitively probe the toroidic moment or the breaking of the inversionsymmetry.

Both LiCoPO4 and LiNiPO4 exhibit very large magnetoelectric coupling and thelow-temperature symmetry of the magnetic ground state allows the existence of alinear magnetoelectric effect [344,345]. For example, in LiNiPO4, the magneto-electric tensor � has two non-zero components �xz and �zx (see [345]); correspond-ingly for LiCoPO4 subscript z should be replaced by y. Figure 66 shows the electricpolarization along the z-axis under an external magnetic field along the x-axis belowthe magnetic transition point around 20K. It is evident that a relatively largemagnetic field along the x-axis can induce a large polarization along the z-axis.More exciting is that the relationship between the polarization along the z-axis andmagnetic field along the x-axis exhibits a butterfly loop around the magnetictransition point and this loop disappears at a lower temperature. It is well knownthat the butterfly loop always corresponds to the appearance of spontaneousmoments, as in ferromagnetics and ferroelectrics, and this phenomenon demon-strates the existence of the macroscopic and spontaneous toroidical moments.

5.3. Observation of ferrotoroidic domains

The domain structure and wall also apply to ferrotoroidics, although the spin orderis essentially antiferromagnetic. A FTO system can exhibit a FTO domain that isindependent of an antiferromagnetic domain because of the different symmetries inthese systems. Take LiCoPO4 as an example again. The antiferromagnetic orderingreduces the symmetry from mmm to mmm0 and the number of symmetry operationsfrom 16 to 8, corresponding to two antiferromagnetic domains (�l). The spinrotation around x reduces the symmetry to 20 and number of symmetry operationsto two, corresponding to two FTO domains (�T) [348].

The SHG appears to be a powerful tool in detecting domain structure inferrotoroidics. For the first-order approximation, sign reversal of the orderparameter O will induce the reversal of SHG susceptibility (O). This means a180� phase shift of the SHG light from opposite domains, which allows one toidentify the domain structure. Given the fact that different ferroic orders correspondto different symmetries and then (O), it is possible to image different domains


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coexisting by polarization analysis. This approach was demonstrated recently inLiCoPO4 using the SHG technique, where the FTO domains were successfullyimaged, providing direct evidence for ferrotoroidicity as a kind of fundamentalferroicity. Figure 67(a) shows the zzz image obtained at 2.25 eV for a nearly singleantiferromagnetic domain in LiCoPO4 (100) single crystal, where the singleantiferromagnetic domain with a single antiferromagnetic domain wall at thelower left, shown by the dark line (the black patch in the centre of the sample isdamaged), is mapped.

The images using SHG light from yyz and zyy components exhibit completelydifferent patterns. Figure 67(b) gives the images using SHG light from yyzþzyy.Extra patterns with bright or dark areas are observed in the single antiferromagnetic

Figure 66. Relations between electric polarization along the z-direction and magnetic fieldalong the x-direction at different temperatures adjacent to the magnetic transition point ofLiNiPO4. (Reproduced with permission from [346]. Copyright � 2000 American PhysicalSociety.)


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domain region, indicating the existence of other ferroic domain structures than theantiferromagnetic domain. Moreover, a rotation of the detected SHG polarizationaround x by 90� (i.e. transform from yyz þ zyy to yyz�zyy) leads to a reversal ofthe brightness of all regions (as shown in Figure 67(c)). This is because the rotationchanges the sign of the zyy contribution which inverts the interference and, thus,the contrast between the zyy and yyz contributions. This reversal is possible onlyif the SHG contributions responsible for the interference stem from independentsources such as the antiferromagnetic and FTO domains. The extra domain structurewas regarded as the FTO domain, as sketched in Figure 67(d). It is noted that thereare three kinds of domain in this sample. With respect to the largest domain

Figure 67. (Colour online) Images of a single antiferromagnetic domain in LiCoPO4 (100)single crystal, obtained using SHG light at 10K. (a), (b) and (c) are the images by SHG lightfrom zzz, yyzþzyy and yyz�zyy at 2.25 eV. (d) The three kinds of domains in this sample,and their relations to the largest domain (AFM, þl; FTO, þT: shown as ‘þ þ’ in the figure),the red domains have (þl, �T ) and the blue domain has (�l, �T ). The black patch in thecentre of sample in all figures is a damage defect. (Reproduced with permission from [348].Copyright � 2007 Macmillan Publishers Ltd/Nature.)


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(antiferromagnetic, þl; FTO, þT: labelled as ‘þþ’ in Figure 67), the red domainshave (þl, �T) and the blue domain has (�l, �T) (see [348]).

Although the existence of ferrotoroidic order was demonstrated by experimentalidentification of FTO domains and other relevant evidence, several important andconfusing questions on ferrotoroidicity do exist. One of them is the exact microscopicdefinition of the ferrotoroidic moment, as done for ferroelectric polarization andmagnetization. While it was claimed that the ‘toroidization’ represents the toroidalmoment per unit cell volume, the periodic boundary condition in the bulk periodiccase leads to a multivaluedness of the toroidization and only the toroidizationdifferences are observable quantities [349]. Based on the concept of a Berry phase,it was presented that a geometric characterization of the ferrotoroidic moment,in terms of a set of Abelian Berry phases, provides a computational method tomeasure the ferrotoroidic moment [350]. So far, no well-accepted exact definitionof the ferrotoroidic moment has been proposed.

6. Potential applications

Multiferroics, or ferrotoroidics, simultaneously exhibit ferroelectricity and magne-tism and provide alternative ways to encode and store data using both electricpolarization and magnetization. Even more exciting is the mutual control betweenthe electric polarization and magnetization due to the strong magnetoelectriccoupling between them in multiferroics. Consequently, huge potential applicationsin the sensor industry, spintronics and so on, are stimulated and expected.

6.1. Magnetic field sensors using multiferroics

The easiest, and most direct, application of multiferroics is to utilize the sensitivityof electric polarization (voltage) to an external magnetic field, for the developmentof a magnetic field sensor, as shown in Figure 68(a). A prototype read head usingmultiferroic materials is shown in Figure 68(b). Even more attractive is the reversedprocess of the order parameter (i.e. the control of magnetization by an externalelectric field or electric polarization). For example, Multiferroics can provide a novelmeans for modulating the phase and amplitude of millimetre wavelength signalspassing through a fin-line waveguide. The fin-line is a rectangular waveguide loadedwith a slab of dielectric material at the centre of the waveguide. Conventional meansfor the control of magnetic parameters control implies cumbersome electromagnets.Magnetoelectric materials provide the possibility of tuning magnetic parameters byvoltage. Applying a voltage across the slab results in a shift in the absorption line forthe multiferroic material, thus allowing the modulation of the phase and amplitudeof the propagating wave with the electric field.

Unfortunately, magnetization switching by electric field/polarization seems to bevery difficult or insignificant. On the other hand, almost all of present multiferroicmaterials are antiferromagnetic and exhibit a small macroscopic magnetic moment.So it is challenging to detect the tiny influence of an external electric field onmagnetization and the change of electric polarization directly.


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Given the fact that a ferromagnetic layer can be pinned by its antiferromagneticneighbour, and that most multiferroics are antiferromagnetic, it is possible to utilizethis pinning effect to monitor magnetization switching by electric field/polarization[20]. To do so, a soft ferromagnetic layer can be deposited onto an antiferromagneticmultiferroic film, as shown in Figure 69. Utilizing the magnetoelectric couplingof the multiferroic film, one applies an external electric field to modulate themagnetization of the multiferroic film, and eventually switch the magnetization ofthe soft ferromagnetic layer due to the magnetic pinning. In this way themagnetoelectric process can be realized as a read out operation of information.Following this roadmap, a NiFe alloy film deposited on (0001) epitaxial YMnO3 filmwas reported and the magnetic pinning and exchange bias in this structure wasconfirmed [351,352].

6.2. Electric field control of the exchange bias by multiferroics

Utilizing multiferroics to control the transport behaviours of spin-valve structuresrepresents a promising direction towards the potential applications of multi-ferroicity. We first briefly present the physics of the exchange bias effect associatedwith the spin-valve structure, which is simplified as a bilayer structure consisting

Figure 68. (Colour online) (a) Multiferroic materials as a probe of the magnetic field. Themiddle layer (the white layer) is multiferroic, and the upper and lower layers (grey layers) areferromagnetic metals. An external magnetic field will induce the electric polarizationperpendicular to the magnetic field direction, and then a voltage. (b) The read-head deviceusing the probe in (a). The blue layer is the magnetic media (magnetic disk) and the blackarrows in it indicate two opposite bits.


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of a ferromagnetic layer in contact with an antiferromagnetic layer, and then discusshow to couple multiferroics with this structure.

There are two general manifestations of exchange interactions that have beenobserved in the interface between the ferromagnetic layer and an antiferromagneticlayer. The first is an exchange bias of the magnetic hysteresis as a consequence ofpinned uncompensated spins on the interface, which is where the practical interestin the conventional antiferromagnetic layer in spin-valve structures lies [353]. Theexchange bias manifests itself by a shift of the hysteresis along the magnetic field axisfor the ferromagnetic layer. The second is an enhancement of the coercivity of theferromagnetic layer as a consequence of enhanced spin viscosity or the spin drag effect.

Within a simple model on the exchange bias effect, the exchange fieldHE dependson the interface coupling Jeb¼ JexSFSAF/a

2, where Jex is the exchange parameter,SF and SAF are the moments of the interfacial spins in the ferromagnetic/antiferromagnetic layers, respectively, a is the unit cell parameter of the anti-ferromagnetic layer. Here HE also depends on magnetization M and thickness tFof the ferromagnetic layer, the anisotropic factor KAF and thickness tAF of theantiferromagnetic layer. These dependences can be formulated as [353]

HE ¼�Jeb�0MtF

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

J2eb4K2

AFt2AF

s¼ H1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

1

4<2

r, ð35Þ

where H1 is the effective field and <¼KAFtAF/Jeb is the normalized factor.However, this model has a long-standing discrepancy with experimental observation,while the random field model proposed by Malozemoff [354] and the concept ofmultidomain structure with the antiferromagnetic layer give relatively betterconsistency with experiments. In the case of < 1, HE is then given by [353,354]

HE ¼ H1 ¼ � JexSAFSF

�0MtFaL, ð36Þ

where L is the domain size of the antiferromagnetic layer and the prefactordepending on the domain shape and average number z of the frustrated interactionpaths for each uncompensated interfacial spin.

Figure 69. (Colour online) Schematic of a soft ferromagnetic layer deposited on multiferroicantiferromagnetic film. The external electric field induces a variation in magnetization of theantiferromagnetic multiferroic film, and eventually results in the reversal of magnetization inthe soft ferromagnetic layer due to the magnetic pinning effects. (Reproduced with permissionfrom [20]. Copyright � 2007 Macmillan Publishers Ltd/Nature Materials.)


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If we replace the antiferromagnetic layer with a multiferroic layer, such asBiFeO3 which is of ferroelectric order and antiferromagnetic order and theantiferromagnetic domains are cross-coupled with ferroelectric domains, as discussedin Section 3.4.5, a multiferroic spin-valve structure is developed. As discussed inSection 3.4.5, an external electric field will drive motion and/or switching of theferroelectric domains, and thus modulate the coupled antiferromagnetic domains.In this case, the exchange bias effect can be controlled by means of an electricfield instead of the magnetic field in conventional spintronics. This approach thusallows the possibility to modulate/switch the magnetization of the ferromagneticlayer in the spin-valve structure. As shown below, recent experiments havedemonstrated the applicability of this approach.

6.2.1. Exchange bias in the CoFeB/BiFeO3 spin-valve structure

Related experiments have focused on the exchange bias effect for a ferromagneticCoFeB layer at 300K deposited on an adjacent antiferromagnetic BiFeO3 film[355,356]. Figure 70(a) shows the hysteresis loops of different CoFeB/BiFeO3

structures and significant exchange bias was observed. Microscopically, X-rayPEEM and piezoresponse force microscopy were utilized to map the antiferromag-netic domains and ferroelectric domains of BiFeO3. A linear variation of theexchange field with the inverse antiferromagnetic domain size was evaluated, whichshowed excellent agreement with the theoretical predictions (equation (36)), asshown in Figure 70(b). Simultaneously, a fitting of the experimental data gives ¼ 3.2 which gives a hint of the existence of uncompensated spins on theferromagnetic/antiferromagnetic interfaces.

Regarding the magnetic moment on the interface, polarized neutron reflectivity(PNR) investigations have revealed that an interface layer of 2.0� 0.5 nm inthickness carries a magnetic moment of 1.0� 0.5 mB/f.u. However, within theframework of the Malozemoff model, the interfacial moment due to the pinneduncompensated spins is mpin

s ¼ 2SAF=aL � 0:32�B nm�2, only 1% of the measured

moment by PNR. This indicates that majority of the uncompensated spins on theinterface are non-pinned and they can rotate with the spins in the CoFeB layer,resulting in a significantly enhanced coercivity. This means that the coercivityand magnetization of the ferromagnetic layer in the spin-valve structure can bemanipulated by controlling the number or density of non-pinned uncompensatedspins on the interface, while the latter can be modulated by the effective interfacialanisotropy or antiferromagnetic domain size of the BiFeO3 layer. As pointed outabove, the antiferromagnetic domain size of the BiFeO3 layer depends on itspolarization or the electric field applied on it [356]. This is the strategy for the electricfield modulated exchange bias in the CoFeB/BiFeO3 spin-valve structure.

6.2.2. Exchange bias in the Py/YMnO3 spin-valve structure

In addition to the CoFeB/BiFeO3 spin-valve structure reviewed above, a similarexperiment on a Cr2O3/ferromagnetic alloy bilayer structure, in which Cr2O3 is amagnetoelectric compound rather than a multiferroic compound, was also reported[357]. However, the detected signal was tiny, while a significant effect observed


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in Pt/YMnO3/Py, as shown in the inset of Figure 71(b), was reported recently.In this structure, YMnO3 is the pinning layer and Py is the soft ferromagnetic alloy[358]. Figure 71(a) plots the magnetic hysteresis (M–H loops) measured underdifferent electric fields at T¼ 2K. The loop shift from the origin point indicates anexchange-bias field of approximately 60Oe under zero electric field (Ve¼ 0), notingthat the magnetization and exchange-bias field depend on temperature. Uponapplying an electric field across the YMnO3 layer, the shift of the M–H loopgradually disappears, indicating suppression of the exchange-bias field andcoercivity. At Ve¼ 1.2V, the loop becomes asymmetric and narrow. Moreover, the

Figure 70. (Colour online) Exchange bias in the CoFe/BiFeO3 system. (a) Magnetic fielddependence of magnetization of CoFeB/BiFeO3/SrTiO3(001) multilayer (upper left), CoFeB/BiFeO3/SrTiO3(111) (upper right), CoFeB/BiFeO3/La0.7Sr0.3MnO3/SrTiO3(001) (lower left),and CoFeB/BiFeO3/La0.7Sr0.3MnO3/SrTiO3(111) (lower right). (b) Dependence of theexchange field on the inverse of the domain size for BiFeO3 films. LFE and LAF representthe sizes of the FE and AF domains. (c) Thickness dependence of exchange field for CoFeB/BiFeO3 grown on SrTiO3 (001). (Reproduced with permission from [355]. Copyright � 2008American Physical Society.)


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electric-field-induced magnetization reversal was also realized in this structure, whichis evident by the decrease in magnetization with an increasing electric field from zerountil Ve¼ 0.4V, at which the magnetization changes its sign (i.e. switching), asshown Figure 71(b). Unfortunately, this process is irreversible and no back-switchingof the magnetization to the initial M4 0 state upon decreasing of the electric fieldfrom the maximum value was observed.

The transport behaviour of the Pt/YMnO3/Py structure modulated by anexternal electric field is shown in Figure 72, where the anisotropic magnetoresistance(AMR) at 5K under various electric fields are presented with R the resistivity and�a the angle between measuring magnetic field Ha and electric current J (�a¼ 0corresponds to JkHa). The increasing electric field Ve results in an additional R(�a)minimum at around 270� because the electric field mimics the effect of theincreasing temperature/magnetic field, and then reduces the uniaxial exchange-bias-based energy barrier [358]. These results reveal a genuine electric field effecton the exchange bias in the YMnO3/Py heterostructure and may be utilized inspintronics.

Figure 71. (Colour online) Measured (a) M–H loops and (b) magnetization of the exchange-biased Pt/YMnO3/Py structure under different external electric fields Ve. The inset in (a)shows the relation between magnetization and temperature, and the inset in (b) is themultilayered structure. (Reproduced with permission from [358]. Copyright � 2006 AmericanPhysical Society.)


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6.3. Multiferroic/semiconductor structures as spin filters

Multiferroic/semiconductor heterostructures are attractive owing to some noveleffects. In fact, much effort has been directed towards synthesizing and characteriz-ing YMnO3 thin films as potential gate dielectrics for semiconductor devices[359–361]. The most widely studied system is a YMnO3/GaN heterostructurebecause YMnO3 and GaN both have hexagonal symmetry [359]. So far, however,less attention has been paid to the role of the heterostructure interface. Firstprinciple calculation predicts different band offsets at the interface betweenantiferromagnetically ordered YMnO3 and GaN for the spin-up and spin-downstates. This behaviour is due to the interface-induced spin splitting of the valenceband. The energy barrier depends on the relative orientation of the electricpolarization with respect to the polarization direction of the GaN substrate,suggesting an opportunity to create a magnetic tunnel junction in this hetero-structure [362,363].

6.4. Four logical states realized in a tunnelling junction using multiferroics

FeRAMs represent a typical device for ferroelectric applications in recent years,favoured by 5 ns access speed and 64 MB memory density. The disadvantage ofFeRAMs is the destructive read and reset operation. By comparison, MRAMs havebeen lagging far behind FeRAMs, mainly because of the slow and high-power read/write operation. Multiferroics offer a possibility to combine the advantages ofFeRAMs and MRAMs in order to compete with electrically erasable programmableread-only memories (EEPROMs). Recently, Fert and his group developed a novelmagnetic tunnelling junction (MTJ) in which multiferroic La0.1Bi0.9MnO3 (LBMO)

Figure 72. (Colour online) AMR of Pt/YMnO3/Py structure measured at T¼ 5K underdifferent electric fields, where �a is the angle between magnetic field Ha and electric current J(�a¼ 0� corresponds to JkHa). (Reproduced with permission from [358]. Copyright � 2006American Physical Society.)


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was used as the insulating barrier, and ferromagnetic half-metal La2/3Sr1/3MnO3

(LSMO) and Au were used as the bottom and top electrodes, respectively[364,365]. The structure and energy level of this new MTJ are sketched inFigure 73. The ferroelectricity and ferromagnetism of the as-prepared ultra thinLa0.1Bi0.9MnO3 film down to 2 nm in thickness were identified. This MTJ exhibitsnormal tunnelling magnetoresistance (TMR) effect (i.e. the resistance is low whenthe magnetization of bottom electrode La2/3Sr1/3MnO3 is aligned with that ofLa0.1Bi0.9MnO3, and higher when their magnetizations are antiparallel), as shown inFigure 74.

In addition to the normal TMR, one may expect a modulation of resistanceby the ferroelectricity of the La0.1Bi0.9MnO3 film (i.e. the electroresistance effect).The bias-voltage dependence on the current for two different bias sweep directions(as shown by the arrow in Figure 75(a)) exhibit significant hysteresis (i.e. thetunnelling current is smaller when the voltage is swept from þ2V to �2V). Theelectric field has a huge effect on the TMR value, which is evident by the highresistance at a þ2V voltage than rather at a �2V voltage. Consequently, it ispossible to obtain four different resistance states at a low bias voltage in this TMRstructure by combining the TMR and electroresistance effect, as shown inFigure 75(d). This prototype device allows for an encoding of quaternary

Figure 73. (Colour online) Structure and energy landscape of a new magnetic tunnellingjunction (MTJ) in which multiferroic La0.1Bi0.9MnO3 (LBMO) was used as the insulatingbarrier and ferromagnetic half-metal La2/3Sr1/3MnO3 (LSMO) and Au were used as thebottom and top electrodes, respectively. (Reproduced with permission from [364]. Copyright� 2007 Macmillan Publishers Ltd/Nature Materials.)


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information by both ferromagnetic and ferroelectric order parameters, and a non-destructive reading by the resistance measurement [364]. This paves the way for novelreconfigurable logic spintronics architectures and an electrically controlled readoutin quantum computing schemes using the spin filter effects [365].

6.5. Negative index materials

One other application, among many, is associated with negative index materials(NIMs). Materials that simultaneously display negative permittivity and perme-ability, often referred to as NIMs, have been presently receiving special attentionbecause the interaction of such materials with electromagnetic radiations can bedescribed by a negative index of refraction [366]. To date, experimental realization ofa negative index has only been gained in metamaterials composed of high-frequencyelectrical and magnetic resonant reactive circuits that interact in the microwave band[366]. A lot of effort has also been directed to the far-infrared band. Using an idealmodel in which both ferromagnetic and ferroelectric resonances are available, anegative index of refraction in the terahertz region using a finite difference methodin the time domain (FDTD) was predicted [367]. These results favour the capabilityof the mechanical phase in a multiferroic material to control the phase between theelectric field E and magnetic field H, and thus manipulates the direction of powerpropagation that identifies multiferroics as a possible source for a negative indexof refraction.

Figure 74. (Colour online) Measured TMR effects in a La2/3Sr1/3MnO3/La0.1Bi0.9MnO3/Aumultilayer structure. The red curve represents the resistivity and the black curve represents themagnetoresistance ratio. (Reproduced with permission from [364]. Copyright � 2007Macmillan Publishers Ltd/Nature Materials.)


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7. Conclusion and open questions

In summary, because of the promising application potentials of magnetoelectric

coupling and mutual control between two or more fundamental ferroic order

parameters in data memories/storages and their significance in condensed matter andmaterials sciences, multiferroic and FTO materials have attracted a lot of attention

from physicists and material scientists. Several breakthroughs and milestones have

been accomplished due to this upsurge in interest. We conclude this state-of-the-art

review with a highlight of some important challenges that remain unresolved.Comprehensive approaches to them are needed in order to advance this active and

exciting field of multiferroicity:

(a) For BiFeO3, one of the rare room-temperature multiferroics, the relationshipbetween the spontaneous polarization and ICM cycloid spin order needs

Figure 75. (Colour online) (a) Influence of the external electric field on the tunnelling currentin La2/3Sr1/3MnO3/La0.1Bi0.9MnO3/Au junctions. The arrows denote the sequence for electricfield application. (b) Dependence of the tunnelling electroresistance effect (TER) andtunnelling magnetoresistance (TMR) on external electric field Vdc. (c) Measured TMR uponan electric bias of þ2V and �2V. (d)–(g) Four states of resistance in the junction.(Reproduced with permission from [364]. Copyright � 2009 Macmillan Publishers Ltd/NatureMaterials.)


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further study. Is the ‘lone-pair’ mechanism sufficient to account for thepolarization in BiFeO3? Can the room-temperature multiferroicity of BiFeO3

provide some clues to search for novel room-temperature multiferroics?What is the physical mechanism for the strong coupling between theferroelectric polarization and ICM spin order? These problems will shed lighton the discovery of novel room-temperature multiferroics and their practicalapplications.

(b) The mechanism of ferroelectricity in hexagonal manganites remains unclear.For hexagonal manganites, disputes on the relatively large ferroelectricpolarization are active, and the polarizations originating respectively fromthe electronic orbitals and lattice distortion need more clarification. Howclosely is the ferroelectricity in YMnO3 linked with the frustrated triangularspin lattice? Moreover, the mechanism for electric field control of themagnetic phase in HoMnO3 and the nature of the Ho3þ magnetism remainconfusing.

(c) Although several microscopic models have been proposed to explain theferroelectricity in spiral spin-ordered multiferroics, they are far fromsufficient to illustrate all of those abundant phenomena observed experi-mentally, in particular in the quantitative sense. The ferroelectricity in the egsystems such as LiCu2O2 is still a controversial issue, and the multiferroicityassociated with either the easy-plane-type or easy-axis-type 120� spiral spinorder in triangular lattices is not fully understood even in a qualitative sense.

(d) Special and continuous attention has to be paid to mechanisms withwhich the ferroelectricity and ferromagnetism can be integrated effectively,in particular for charge-ordered multiferroics. So far no quantitative theoryon the ferroelectricity in LuFe2O4 is available, and the predictedferroelectricity in Pr1�xCaxMnO3 and Pr(Sr0.1Ca0.9)2Mn2O7 has not yetbeen confirmed by direct experimental evidence. For RMn2O5, a fullunderstanding of the ferroelectricity origin seems to be extremelychallenging.

(e) Ferroelectricity in antiferromagnetic E-phase and weak ferromagnetisminduced by ferroelectricity remain to be theoretical concepts and no reliableexperimental evidence is available. The antiferromagnetic E-phase-inducedferroelectricity in orthorhombic TbMnO3 or YMnO3 remains unclear andneeds further clarification. High-quality samples of non-centrosymmetricMnTiO3 and FeTiO3 have not yet been made available even by high-pressuresynthesis. High-quality materials for experimental and theoretical investiga-tions are necessary.

(f) Complex elementary excitations in multiferroic materials have yet to beexplored. New elemental excitations, electromagnons, are expected and havebeen confirmed by preliminary experiments. However, a comprehensiveunderstanding of their origins, conceptual pictures and dynamics, is stilllacking. So far no practical prediction of these element excitations, in termsof their potential applications, has been given.

(g) Although quite a number of multiferroics have been synthesized andcharacterized, almost all of them exhibit either small/net spontaneousmagnetization or electric polarization. The observed electric polarization


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in multiferroic manganites is nearly two orders of magnitude less than typicalferroelectrics, which is too small to be practically applicable. The magneticorder state in multiferroic and FTO systems is usually of antiferromagnetictype. Moreover, the temperature for the coexistence of ferroelectricity andmagnetism and, thus, the mutual control between them, remains very low,although recent work has revealed that CuO seems to be a multiferroic withthe ferroelectric Curie point as high as 220K. These issues essentially hindermultiferroics from practical applications at room temperature.

(h) Basically, the magnetoelectric coupling and mutual control betweenferroelectricity (polarization) and magnetism (magnetization) for mostmultiferroics remain weak. Although the mutual control has been identifiedin some multiferroic systems, few of them show the reversal of polarizationupon a magnetic field reversal, which is very useful in practical applications.Moreover, the inverse process (i.e. the magnetization switching driven byelectric field/polarization) also seems to be difficult. The major challenge is tosearch for novel materials and mechanisms to realize the effective mutualcontrol between these ferroic order parameters.

(i) Owing to the advanced techniques for materials synthesis and fabrication,the objects of modern condensed matter physics and material sciences havebeen extended to artificial structures, such as nanoscale quantum dots/wires/wells and superlattices, etc. The domain/interface engineering has been inrapid development. Novel multiferroics stemming from new mechanismsfor the magnetoelectric coupling/mutual control between these ferroic orderparameters can be fabricated with artificial designs. The physics and novelgiant effect associated with these new artificial structures given thecoexistence of two or more ferroic orders, is very promising for futureinvestigations.

(j) Our understanding of FTO systems is still quite preliminary. Up to now,there has been no unified and clear definition of the macroscopic toroidicalmoment in FTO systems. The relationship between ferrotoroidicity andmultiferroicity remains unclear and should be clarified in future.

(k) Practical applications of multiferroic and FTO materials seem to bechallenging, although some possible prototype devices, in storages, sensors,spintronics and other fields, have been proposed. The mutual controlbetween the ferroic order parameters, and also some additional effects (e.g.,the control of exchange bias by electric field), deserve extensive explorationin the future.

Acknowledgements

The invaluable support of Professor N. B. Ming and Professor D. Y. Xing in NanjingUniversity is gratefully acknowledged. We appreciate the stimulating discussions with Dr C.W. Nan, Dr X. G. Li and Dr X. M. Chen. This work is supported by the National NaturalScience Foundation of China (50832002, 50601013), the National Key Projects for BasicResearches of China (2009CB623303, 2009CB929501, 2006CB921802), the 111 Project ofMOE of China (B07026), DOE DE-FG02-00ER45805 (ZFR), and DOE DE-FG02-087ER46516 (ZFR).


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Multiferroicity: the coupling between magnetic and polarization orders