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Physics and Applications of Bismuth Ferrite
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By Gustau Catalan* and James F. Scott*

BiFeO3 is perhaps the only material that is both magnetic and a strong

ferroelectric at room temperature. As a result, it has had an impact on the field

of multiferroics that is comparable to that of yttrium barium copper oxide

(YBCO) on superconductors, with hundreds of publications devoted to it in

the past few years. In this Review, we try to summarize both the basic physics

and unresolved aspects of BiFeO3 (which are still being discovered with

several new phase transitions reported in the past few months) and device

applications, which center on spintronics and memory devices that can be

addressed both electrically and magnetically.

1. Introduction

1.1. History

The basic idea that crystals could be simultaneously ferromag-netic and ferroelectric probably originated with Pierre Curie inthe 19th century.[1] After switching was discovered in ferroelectricRochelle Salt by Valasek in 1920[2] there was a rash of supposeddiscoveries of magnetoelectric properties by Perrier,[3,4] butunfortunately in materials such as Ni in which they are nowunderstood to be impossible. A history of this period of solid-statephysics is given in O’Dell’s text.[5] True magnetoelectricity –defined as a linear term in the free energy G(P,M,T)¼aij Pi Mj –,where P is the polarization and M is the magnetization was firstunderstood theoretically by Dzyaloshinskii[6] with special predic-tions being made for Cr2O3 and discovered experimentally in thatmaterial by Astrov.[7] However this material is paraelectric andantiferromagnetic, making microelectronics applications imprac-tical. The more interesting case of ferromagnetic ferroelectricswaited for some years until the work of Schmid on boracites.[8]

The boracites are also impractical materials for device applica-tions: they have low symmetry with large unit cells and grow inneedle shapes; more importantly, they exhibit magnetoelectricityonly at extremely low temperatures. Meanwhile Smolenskii’sgroup in Leningrad pioneered[9] the study of bismuth ferrite,BiFeO3, but they found that they could not grow single crystalsand that ceramic specimens were too highly conducting (probablycaused by oxygen vacancies and mixed Fe valences) to be used inapplications.[10] They tried to address the conductivity problem by

[*] Dr. G. Catalan, Prof. J. F. ScottDepartment of Earth SciencesUniversity of CambridgeDowning Street, Cambridge CB2 3EQ (United Kingdom)E-mail: [email protected]; [email protected]

DOI: 10.1002/adma.200802849

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doping other ions into both the A and Bsites of the lattice, but no practical deviceswere obtained.

Reviews of the general study of magne-toelectricity appeared by Schmid in 1994[11]

and more recently by Fiebig[12] and byEerenstein et al.[13] The current interest inbismuth ferrite was stimulated primarily bya 2003 paper from Ramesh’s group,[14]

which showed that it had unexpectedlylarge remnant polarization, Pr, 15 timeslarger than previously seen in bulk, togetherwith very large ferromagnetism of ca. 1.0Bohr magneton (mB) per unit cell. Single

crystals grown more recently in France in 2006–7 have confirmedthe large value of the polarization first observed in the films,showing also that it is intrinsic;[15-19] on the other hand, theintrinsic magnetization of thin films is now thought to be nearzero[20, 21] – ca. 0.02 magnetons/cell – and possible reasons fordiscrepancies between the magnetization values encountered inthe literature are discussed later in this review. At any rate, the2003 Science paper has proved enormously stimulating, and hasinspired both new fundamental physics and exciting deviceapplications.

1.2. The Hypothesis of Spaldin (Hill)

In parallel with the specific investigation of bismuth ferrite andrelated compounds has been a more general approach to the ideaof multiferroics. Nicola Hill (now Spaldin) has asked[22] why thereare so fewmaterials that are magnetic and ferroelectric; implicitlylimiting her discussion to transition-metal oxides, especiallyperovskites, she observed that the ferroelectrics (e.g., titanates)have B-site ions with d8 electrons,[23] whereas the magnetsrequire dj electrons with j different from zero. This line ofthinking has also proved very stimulating, although it is helpful toremind ourselves that there are many potential multiferroics thatare not oxides, as further discussed in Section 1.3.

On the other hand, oxide perovskites do not all have the samemechanism of ferroelectricity: the center Ti ion plays the key rolein BaTiO3 but the lone-pair Pb ion is dominant in PbTiO3.

[24]

Indeed, this seems to be the case in BiFeO3, where thepolarization is mostly caused by the lone pair (s2 orbital) ofBiþ3, so that the polarization comes mostly from the A site whilethe magnetization comes from the B site (Fe3þ); this same ideahas led Spaldin and co-workers to propose a host of otherperovskites with possible A-site ferroelectricity and B-sitemagnetism, such as Bi(Cr,Fe)O3 and BiMnO3. Her work hasalso been instrumental in triggering the quest for other ways ofachieving coexistence of ferroelectricity andmagnetism in oxides;

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Gustau Catalan studiedPhysics at the University ofBarcelona and earned hisPhD at the Queen’sUniversity of Belfast,followed by postdoctoralappointments in CSIC(Spain) and the ZernikeInstitute for AdvancedMaterials (Holland). He iscurrently a Senior ResearchAssociate in the FerroicsLaboratory of the Universityof Cambridge, which hejoined in 2005. His areas of

interest include the properties of perovskite oxides and thestudy of how small size affects the phase transitions andfunctionality of thin films and nanocrystals.

James F. Scottwas educatedat Harvard (B.A., Physics1963) and Ohio StateUniversity (Ph.D., Physics1966). After six years in theQuantum ElectronicsResearch Department atBell Labs he was appointedprofessor of Physics at Univ.Colorado (Boulder). He wasDean of Science andProfessor of Physics foreight years in Australia(UNSW, Sydney, and RMIT,Melbourne) and has been

Professor of Ferroics in the Earth Sciences Department at theUniversity of Cambridge since 1999.

Figure 1. Compositional phase diagram of BiFeO3. Reproduced withpermission from [49]. Copyright 2008, American Physical Society.

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among the new findings are ferroelectricity induced by spiral spinorder,[25,26] magnetic exchange striction,[27] or charge order.[28]

Based on these ideas, new oxides have been predicted to bemultiferroics, some of which await direct experimental verifica-tion.[26,29,30] In addition, there are materials in which theferroelectricity itself causes spin canting[31,32] and these shouldbe explored further.

1.3. Oxide vs. Fluoride Multiferroics

There are hundreds of different crystals with ferroelectrictransition temperature, Tc, above room temperature at atmo-spheric pressure, with approximately a log-normal distribution oftransition temperatures.[33] Many of these materials are notoxides, yet almost all work on multiferroics has emphasizedoxides. This is convenient as they are easy to grow, particularly inthin-film form, and researchers in high-Tc superconductors,colossal magnetoresistance manganites, and ceramics all study

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oxides. However, there is no a priori reason to expect that they willhave the most interesting multiferroic physics or will make thebest devices. The exclusive emphasis on oxides seems unwise,and magnetoelectric fluorides should probably receive moreattention in this context.

Abrahams has given a list of magnetic materials that areprobably ferroelectric.[34–36] His criterion was the existence of astructure that has very small acentric displacements of ions, withthe assumption that ferroelectric switching is likely in suchlattices. Most of these predicted ferroelectrics are indeed oxides,but some are fluorides.[37] BaMnF4, for example, is the earliestknown example of a material with spin canting induced byferroelectricity,[28] the physics being the same as in theferroelectrically induced local spin canting in BiFeO3, discussedlater in this Review. K3Fe5F15

[38–40] is a ferroelectric ferrimagnetwith 2 Feþ3 ions and 3 Feþ2 ions per unit cell, adding to a weak netferromagnetic moment. A particularly interesting multiferroic isSr3(FeF6)2, which merits further study.[41] An additional family ofmultiferroics that seems promising is Pb5Cr3F19.

[42,43] There aremany other fluoride multiferroics, and (NH4)3FeF6 has receivedcareful study.[44,45] These systems include both pure fluorides andoxyfluorides,[45] with the latter including Bi2TiO4F2; Ravez hasgiven a review encompassing both fluorides and oxyfluorides,[46]

and Nenert and Palstra[29] have also recently reviewed otherpossible multiferroic fluorides.

2. The Phase Diagram of BiFeO3

2.1. Phase Decomposition and Impurities

The phase diagram for the system Bi2O3/Fe2O3 has been mappedout[47,48,49] and is shown in Figure 1. BiFeO3 is usually preparedfrom equal parts of Bi2O3 and Fe2O3, and under hightemperatures it can decompose back into these starting materials,as shown in Equation 1

2BiFeO3 ! Fe2O3 þ Bi2O3 (1)

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Figure 2. Impurities have a strong effect on functional properties. In thisfigure, the remnant magnetization, M, of BiFeO3 thin films grown inoxygen-deficient conditions is shown to be directly correlated to theamount of g-Fe2O3 parasitic phase as extracted from X-ray diffraction(XRD) peaks. Figure courtesy of Manuel Bibes (Thales-CNRS).

Bismuth ferrite is very prone to show parasitic phases that tendto nucleate at grain boundaries and impurities.[50] It has beenargued that BiFeO3 is in fact metastable in air, with opticallyvisible impurity spots appearing well below the meltingtemperature.[49,51,52] Impurities and oxygen vacancies are alsoimportant for thin films, because they are known to artificiallyenhance the remnant magnetization[19,21] (Fig. 2). Minimizingthem requires very careful tuning of growth parameters,particularly oxygen pressure.[21]

At room temperature under applied fields of ca. 200 kVcm�1

(typical switching voltages across thin films), BiFeO3 decom-poses, yielding magnetite Fe3O4 as a by-product.[53] This wassomewhat surprising and is thought to occur via the followingreaction

6BiFeO3 ! 2Fe3O4 þ 3Bi2O3 þ O (2)

The magnetite phase was unambiguously identified by meansof micro-Raman studies; the Raman spectra of Fe3O4 is quitedistinct and unlike those of Fe2O3. However, the Bi2O3 was notdetected, possibly because it is a well-known glass-formingcompound, or perhaps because of its evaporation during thermaldecomposition. Bi2O3 melts at a temperature slightly above800 8C.[49] Similar phenomena occur in the electrical stressing oflead zirconate titanate (PZT),[54] which decomposes into rutileTiO2 (but not anatase), and both a-PbO and b-PbO (litharge andmassicot). In both bismuth ferrite and PZT these observationsraise concerns about the lifetime of ferroelectric memories madefrom them. In the case of BiFeO3, this decompositionmechanism also provides another possible explanation for theappearance of remnant magnetization in thin films: it may comefrom localized spots of magnetite in the sample.

We also point out that Bi2O3 and Pt are known to react easilyand exothermically with each-other.[55,56] Bismuth has a lowmelting temperature of 270 8C, at which it readily forms aneutectic alloy with Bi2Pt and, at higher temperatures, other Bi–Ptalloys are also formed.[57] Therefore, it is probably best NOT to usePt electrodes when probing the high-temperature properties ofbismuth-based materials,[58] including bismuth ferrite. We alsonote that most BiFeO3 ceramics and single crystals are made in

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platinum crucibles which, given the high mutual reactivity of Biand Pt, may not be the most suitable receptacle.

The above discussion underlines what is currently one of themost important difficulties in implementing BiFeO3 for practicalapplications, namely, its compositional instability, with associatedfickleness of functional behavior. Addressing this problem isessential if BiFeO3 is to succeed as a technologically relevantmaterial.

2.2. Crystal Structure

The room-temperature phase of BiFeO3 is classed as rhombohe-dral (point group R3c).[59] The perovskite-type unit cell has alattice parameter, arh, of 3.965 A and a rhombohedral angle,arh, of ca. 89.3–89.48 at room temperature,[60,61] with ferroelectricpolarization along [111]pseudocubic.

[61] The unit cell can also bedescribed in a hexagonal frame of reference, with the hexagonalc-axis parallel to the diagonals of the perovskite cube, i.e.,[001]hexagonal jj [111]pseudocubic. The hexagonal lattice parametersare ahex¼ 5.58 A and chex¼ 13.90 A.[60–62] The coefficient ofthermal expansion is neither completely linear nor isotro-pic,[62–64] and reported values[62,63] differ notably, ranging fromca. 6.5� 10�6 to ca. 13� 10�6 K�1.

A very important structural parameter is the rotation angle ofthe oxygen octahedra. This angle would be 08 for a cubicperovskite with perfectly matched ionic sizes. A measure of howwell the ions fit into a perovskite unit cell is the ratio ðrBi þ r0Þ=l,where r is the ionic radius of the respective ion and l is the lengthof the octahedral edge. This is completely analogous to thecommonly used Goldschmid tolerance factor,[65] which is definedas t ¼ rBi þ rOð Þ

� ffiffiffi2

pðrFe þ rOÞ. For BiFeO3 we obtain t¼ 0.88

using the ionic radii of Shannon,[66] with Biþ3 in eightfoldcoordination (the value for 12-fold coordination is not reported)and Feþ3 in sixfold coordination and high spin. When this ratio issmaller than one, the oxygen octahedra must buckle in order to fitinto a cell that is too small. For BiFeO3, v is ca. 11–148 around thepolar [111] axis,[59,61,67] with the directly related Fe–O–Fe angle,u¼ ca. 154–1568.[61,64] The Fe–O–Fe angle is important because itcontrols both the magnetic exchange and orbital overlap betweenFe and O, and as such it determines the magnetic orderingtemperature and the conductivity, as will be discussed in latersections

2.3. Symmetry of the High-Temperature b and g Phases

At approximately 825 8C there is a first-order transition to ahigh-temperature b phase that is accompanied by a suddenvolume contraction.[49,68] The transition is also accompanied by apeak in the dielectric constant;[68,69] this has been taken as anindication of a ferroelectric–paraelectric transition, althoughdielectric peaks can also occur in ferroelectric–ferroelectrictransitions, such as the orthorhombic–rhombohedral transitionin the archetypal perovskite ferroelectric BaTiO3 (which is alsofirst order). Nevertheless, although there is disagreement aboutthe exact symmetry of the b phase above 825 8C, most reportsagree that it is centrosymmetric,[70–75] so it is probably a safe bet

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that the a–b transition at TC¼ 825 8C is indeed the ferroelec-tric–paraelectric transition.

Palai et al.[49] propose that the symmetry of the b phase isorthorhombic, although their data does not allow establishing theexact space group with certainty. Some authors have argued thatthe b phase may be tetragonal or pseudotetragonal,[67,71] but thatis impossible, since the domain structure rules out a tetragonalsymmetry and the perovskite a,b,c lattice constants are each quitedifferent.[49,72] It was also proposed that this phasemay instead bemonoclinic;[71,72] the measured monoclinic angle was never-theless initially quoted as 90o within experimental error,[71] so thatthe b phase was in effect ‘‘metrically orthorhombic’’ (i.e., theangles may be 908, but internal ion positions in each unit cell donot satisfy orthorhombic constraints). More recently, however,Haumont et al. have quoted a monoclinic angle of 90.018.[72]

On the other hand, our group does not see in our specimensthe extra XRD lines used to infer monoclinic structure. Also, thedomains studied optically do not reveal the many extra wallorientations that would exist if the symmetry were monoclinicinstead of orthorhombic. Furthermore, we find that the b–gphase transition to the cubic metallic phase encountered at1204K (atmospheric pressure) seems second-order, and acubic–monoclinic second-order phase transition would violatethe principle of maximal subgroup.[76] This is an additionalargument in favor of the b phase being orthorhombic:cubic–orthorhombic transitions are allowed to be second orderand still satisfy the maximal subgroup criterion.

A very recent work has added more ‘‘fuel to the fire’’ regardingthis problem. Selbach et al.[73] claim that the paraelectric b phasemay neither be orthorhombic nor monoclinic, but rhombohedral(space groupR3c). It is, however, hard to reconcile this claim with

Figure 3. Specific-heat measurements by a) Kaczmarek et al. [124] and b) Palathat the transition to the orthorhombic b phase (815 8C in Kaczmarek’s measuours) may proceed via an intermediate phase which nucleates at ca. 25–35 8C bthe monoclinic phase reported by Haumont and co-workers, or else it may becoexistence between rhombohedral and orthorhombic. c) At temperatures(110) diffraction peaks appear unsplit (the small split in the diffractogramradiation), indicating that the g phase is cubic. Reproduced with permiCopyright 1974, Elsevier (a). Reproduced with permission from [49]. CopyrighPhysics Society (b).


the splitting of the pseudocubic lattice parameters observed byother groups,[49,71,72,75] or with the symmetry of the domain wallsin this phase. Perhaps, more importantly, if both the a and b

phases belong to the same crystal class (rhombohedral), then thetransition cannot be ferroelastic, which would appear to contra-dict the observed change in the ferroelastic domain configurationat this transition.[49]

Part of the disagreements between all the above works canperhaps be justified by the fact that X-rays are not particularlysensitive to the position of the oxygen ions as a result of the lowelectronic density of O compared with the Bi and Fe ions. In thisrespect, neutron diffraction is a far more helpful technique.High-temperature neutron-diffraction experiments have recentlybeen undertaken by Arnold et al.,[74] who show that the b phase isorthorhombic Pbnm, which is the same non-polar orthorhombicsymmetry of the GdFeO3 orthoferrite family. In retrospect, ofcourse, this seems quite obvious: once the distinctive feature ofBiFeO3 (its ferroelectric polarization) is removed, one may expectthis material to be just like all the other perovskite orthoferrites,i.e., it should be orthorhombic. This is, of course, just an ad hocargument, but apart from the neutron diffraction and our XRDexperiments, there may be additional indirect support for anorthorhombic b phase based on chemical-doping experimentswith ions other than Bi, as discussed in the next Section. It isworth mentioning as well that the neutron-diffraction experi-ments of Arnold et al.[74] suggest phase coexistence in the b phaseand, indeed, optical viewgraphs at high temperature do show acoexistence of rhombohedral and orthorhombic domains.[49] Thisis consistent with the strongly first-order nature of this transition,and perhaps the mixture of rhombohedral and orthorhombicphases could also explain why the b phase may seem

i et al. [49] suggestrement, 825 8C inelow. This may bea region of phaseabove 930 8C theis due to a1/a2

ssion from [124].t 2008, American

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monoclinic in some experiments.[71,72] Onefinal note relevant to this issue is thatdifferential thermal analysis (DTA) measure-ments (Fig. 3) show a smaller anomaly ca.30 8C below the transition to the orthothombicb phase. This suggests the existence of anintermediate phase, which may either be themonoclinic phase seen by Haumont andco-workers, or a region of phase coexistenceas seen by Arnold et al. and us.

As for the symmetry of the highest-temperature g phase, Redfern’s XRD data[75]

show that the most intense Bragg peak,(110)pseudocubic, has a large splitting at roomtemperature and atmospheric pressure, but isunsplit (resolution 0.058 in 2u) in the g phaseabove 931 8C (Fig. 3), with the proposedsymmetry for the cubic phase beingPm3m.[49,75] Unfortunately, BiFeO3 is veryunstable at the high temperature of the b–gtransition and it rapidly decomposes intoparasitic phases such as Bi2Fe4O9 or Fe2O3

(Bi2O3 becomes a liquid at that temperature soit does not show up in the diffraction scans).Accordingly, measurements of the b–g transi-tion at 930 8C have to be performed on veryhigh-quality samples (preferably singlecrystals) and using very fast heating/cooling

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Figure 4. Properties of Bi1-xLaxFeO3 as a function of La doping. There areindications of at least three phase transitions as x increases. Reproducedwith permission from [68]. Copyright 1974, Wiley.

ramps; the different measurement protocols used by differentgroups mean that not all of them have been able to measure thecubic phase at high temperature. On the other hand, a tendencytowards cubic symmetry has also been reported for BiFeO3 as afunction of decreasing grain size,[77] which indirectly supports theconclusion that the highest-symmetry phase is cubic.

2.4. Phase Transitions with Pressure or with La Doping

The diffraction peaks are also unsplit in Redfern’s measurementsat room temperature and high pressure above 47GPa (althoughthe experimental resolution is low, with 0.588 peak broadening);this suggests the same cubic symmetry at high pressure asobserved at high temperature, in agreement with the early phasediagram of Scott et al.[84] Gavriliuk et al. report a rhombohedralstructure instead,[78] in agreement with ab initio simulations.[79]

The experimental resolution precludes a completely unambig-uous answer at this stage, as very small splittings may have been


masked by broadening in our measurements or, conversely, lackof hydrostatic equilibriummay cause peak asymmetry that can bewrongly interpreted as peak splitting in Gavriliuk’s. As a sidecomment, we note that the question of whether materials tendtowards cubic symmetry with high pressure is, surprisingly,unresolved even for simple elements.[80]

Recently, Pashkin et al. [81] have reported additional phasetransitions at room temperature at ca. 3–5 and 7.5–10GPa. Thereported pressure-induced transition near 10GPa is to anorthorhombic Pnma (Pbnm) state. This could also be theorthorhombic symmetry for the high-temperature b phaseproposed by Palai et al.[49] and Arnold et al.[74] While theselow-pressure transitions have not been confirmed by high-pressure studies in the USA,[82] or Russia,[82,83] the transition tothe orthothombic b-phase near 10 GPa has been recentlyreproduced by Redfern et al.[75] It thus seems that therhombohedral-orthorhombic-cubic sequence of phase transitionsis the same as a function of pressure as it is as a function oftemperature.

An alternative way of inducing ‘‘pressure’’ in a crystal is bychemical substitution of an ion for another of the same valence butdifferent size–what is sometimes called ‘‘chemical pressure’’. Themost common isovalent substitute in BiFeO3 is La3þ for Biþ3.However, interpretation of the effects of La doping in terms ofchemical pressure is not straightforward because La3þ has almostexactly the same ionic radius as Biþ3 (1.16 and 1.17 A, respectively[66]).Furthermore, the lone-pair orbital of Bi3þ (6s2) is stereochemicallyactive and responsible for the ferroelectric distortion; distortionsinduced by La doping are therefore more likely to be caused by theturning off of the lone-pair activity (i.e., the turning off of the ferro-electricity) than to direct differences in ionic size.

The first phase diagram for Bi1�xLaxFeO3 was published byPolomska et al.,[68,85] who looked at the dielectric constant andvolume expansion as a function of La doping concentration(Fig. 4). In their study, there is a first-order transition with sharpvolume contraction for x¼ ca. 0.2 (Fig. 4) and several othertransitions, the last one of which is at x¼ ca. 0.75 to theorthorhombic Pnma (centric) phase of pure LaFeO3, also reportedfor pure BiFeO3 above ca. 10GPa

[81] at room temperature or above825 8C at ambient pressure.[74] The nature of the intermediatebridging phases, however, is unclear. Gabbasova et al.[86] andZalesskii et al.[87] claim a noncentric orthorhombic phase (C222)for 0.2< x< 0.6 that could also be associated with the high-temperature b phase of pure BiFeO3 – the sudden volumecontraction at x¼ 0.2 is in this context very reminiscent of thevolume contraction observed at the temperature-induced a–btransition (825 8C). At any rate, the exact nature and even thenumber of structural phase transitions as a function of La dopingis still an open question.[63,86–88]

2.5. Other Anomalies above Room Temperature: Phase

Transitions vs. Defects

Krainik et al.[51] measured the GHz dielectric constant andthermal expansion of BiFeO3 between room temperature and900 8C. They found small anomalies at 130, 200, 280, 370, 460,600, 670, 740, and 845 8C. However, the authors themselvesmention that the samples are ‘‘almost phase pure’’, which is


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Figure 6. Sketch of a possible phase diagram as a function of pressure andtemperature. Solid points are experimental data, the lines are only a visualguide. The ground state is rhombohedral, and the b phase is orthorhombic.The reported monoclinic phase transition [71,72,81] has not been con-firmed, and may actually be a coexistence of rhombohedral and orthor-hombic (not unusual for a strongly first-order phase transition). Pressure isknown to increase TNeel in orthoferrites at a rate of 4.0–7.5 KGPa�1 [204].Accordingly, we expect that TNeel will rise slowly with hydrostatic pressureup until the triple point is reached, but there is no direct experimentalevidence of this; above the triple point, the pressure-induced meta-l–insulator (MI) transition (TMI) delocalizes the electrons and induces aPauli paramagnetic state, so that themagnetic-ordering temperature, TN, isforced to track down TMI The metallic state at high pressures and lowtemperatures has been claimed by Gavriliuk et al. [78] and Gonzalez-Vazquez and Iniguez [79] to be rhombohedral, while the data of Redfernet al. is consistent with cubic[49,75]. This schematic phase diagram doesnot include any of the new magnetic phase transitions observed at lowtemperatures and discussed in later Sections.

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another way of saying that they are not pure; accordingly, some oftheir anomalies could be due to parasitic phases and defects (seeSection 2.1), particularly since many have never been reproducedin later measurements. Nonetheless, some of these anomalies areclearly correlated with known phase transitions: the 845 8C peakis almost certainly the a–b phase transition, whereas the anomalyat 370 8C, also reported by Polomska,[68] is caused by magneto-electric coupling to the antiferromagnetic Neel temperature(TNeel). With the exception of the anomaly at the Neeltemperature, none of the other possible phase transitions showsup in the refractive index as a function of temperature.[89]

The most intriguing of these ‘‘ghost’’ transitions is perhaps theanomaly in both dielectric constant and thermal expansionreported by Polomska near 185 8C or 458K.[68,85] It is possible thatthe phase transition reported at 458K and ambient pressure couldbe the same as that observed at room temperature and ca.4GPa.[81] Both sides of this phase boundary were first reported byPashkin et al. as rhombohedral[81a] and, hence, the transition—ifit were a transition –would not be ferroelastic.[90] Other ferro-electrics, such as nickel iodine boracite, have had isomorphictransitions proposed for them which do not change symmetry,[91]

so this is not out of the question. On the other hand, the transitionat 458K is not universally observed: our own single-crystaldielectric measurements do not show any clear feature aroundthat temperature. Nor is there a clear signature of that transitionin the phonon behavior,[49,70] all of which argues in favor of anextrinsic origin of the anomalies. Having said that, the 458Kdielectric anomaly has been reported by other groups,[93] and alsoappears in electrical resistivity measurements in our laboratory(Fig. 5). This could, of course, still be related to impurities ratherthan being intrinsic; comparison between two-probe andfour-probe measurements, for example, shows the anomaly tobe much stronger in the former, suggesting contact- resistanceeffects, although the derivative of the four-probe resistivity doesstill show a peak near 185 8C. However, other reports of resistivitydo not show any anomaly near ca. 185 8C.[94,95]

While the coincidence of our resistive anomaly withPolomska’s (whose impedance and dilatometry measurementswere in a completely different set of samples) is tantalizing, at thispoint the evidence for and against an intrinsic origin seems to besplit down the middle, so this possible transition certainly merits

Figure 5. Resistance as a function of temperature of BiFeO3 single crystafour-probe (right) measurements. The sharp anomaly at 185 oC/458K in the twvery washed-out in the four-probe resistivity, although the derivative (inset) doexactly the same temperature. The four-probe measurements were performedUniversity of Cambridge.


further careful studies in order to unambiguously establish itsnature.

Based on the pressure/doping effects and the known behaviorof the magnetic transition in orthoferrites, we propose aschematic temperature–pressure phase diagram for BiFeO3

(Fig. 6). At present this is but an informed guess, with enormousgaps in real experimental data, so we very much encourage thecareful exploration of this map.

ls: two-probe (left) ando-probe measurement ises show a peak at almostby Julia Herrero-Albillos,

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3. Conductivity, Bandgap,and Metal–Insulator (MI)Transition

3.1. Resistivity of BiFeO3

The dc resistivity of good-qualitybulk samples of BiFeO3 exceeds1010Ohmcm.[49,95] As temperatureincreases, the resistivity decreases aswould be expected from any wide-bandgap semiconductor. Around the TN(370 8C) there is no change in the absolutevalue of resistivity, but Arrhenius plotsshow a change in slope (Fig. 7), with the

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Figure 7. Arrhenius plot of the two-probe resistivity of a single crystal,showing a change of slope at the Neel temperature.

activation energy of the charge carriers decreasing fromca. 1.3 to ca.0.6 eVas thematerial is heated above TN. Resistive anomalies at TNhave also been reported by Selbach et al.[77] This indicates thatmagnetic ordering affects the conductivity bandgap, increasing it inthe antiferromagnetic phase, which is consistent with ab initiocalculations.[92] The correlation between bandgap and magneticordering suggests that BiFeO3 could be magnetoresistive. Indirectevidence of this exists from the dielectric measurements of Kambaet al.[96] and direct measurements are currently underway in ourlaboratory.

At even higher temperatures there are further resistiveanomalies correlated with the a–b (rhombohedral–orthorhombic)transition, the b–g (orthorhombic–cubic) transition and, finally,the decomposition temperature.[49] Specifically, the resistivitydecreases (but remains semiconducting) at the a–b transition[77]

and the slope of the resistivity as a function of temperaturechanges sign[48] at the b–g transition, which is consistent with ametal–insulator (MI) transition, as discussed below.

3.2. Bandgap and MI Transition

Reported values for the optical bandgap of BiFeO3 at roomtemperature range from ca. 2.3 to ca. 2.8 eV.[49,95,97–100] According

Figure 8. Optical bandgap of BiFeO3 as a function of pressure and temperature. Figuresreproduced with permission from [83] and [49], respectively. Copyright 2007, Materials ResearchSociety and 2008, American Physics Society, respectively.

to some authors, this bandgap is direct,[98,99]

although other reports suggest also thepresence of an indirect bandgap roughly0.4–1.0 eV smaller than the direct one.[95,97]

Ab initio calculations using screenedexchange formalism show that bismuth ferriteis a semiconductor with a room-temperaturegap of ca. 2.8 eV.[49,100] The valence-bandmaximum is at the R-point corner of theBrillouin zone, whereas the conduction-bandminimum is at the center, G, so that the gap isindirect. However, the calculated valence bandin the rhombohedral state is in fact almostflat[100] so that BiFeO3 should in practicebehave as a direct-bandgap semiconductor atroom temperature. The same calculations,however, show an evolution towards indirect


bandgap as BiFeO3 goes from rhombohedral to orthorhombic tocubic, with the indirect bandgap decreasing markedly at each ofthese transitions. Note that the screened exchange band structurecalculation gave good results for the bandgap versus temperature,in comparison with experiment, but it was not a total energycalculation and hence does not assess stability of phases.

As temperature increases, Palai et al.[49] have indeed measuredthat the optical bandgap decreases and goes to zero abruptly at theg -phase, signaling a temperature-driven MI transition. MItransitions are of particular interest in solid-state physics, andare often studied as a function of pressure as well as temperature.An MI phase transition has indeed been observed in bismuthferrite at room temperature and at a pressure of ca.50GPa,[78,82,83] as well as at 1204K at a pressure of 1 atm.[49]

As far as we know, MI transitions have not been reported inperovskite ferrites other than BiFeO3.

The evidence for the MI transition at the orthorhombic–cubictransition near TMI (¼ ca.1204K) is first, that the optical bandgapgoes to zero at that temperature[49] or at room temperature andhigh pressure[78,82,83] (Fig. 8); second, that the magnetismdisappears;[78,101] third, that the temperature derivative ofresistivity changes sign;[49,78] and fourth, that the reflectivityincreases abruptly.[49,78] As the temperature rises, the deviationfrom cubic structure decreases and the experimentally measuredgap[49] decreases to ca. 1.6 eV by 500 8C (723K, still in therhombohedral phase). At 1204K the structure becomes cubic viaa second-order transition,[49,75] and the conduction-band mini-mum now overlaps the valence-band maximum. Thus asemimetal is formed, as in elemental Bi or graphite. Althoughthe behavior is now metallic, the material is not strictly aconventional metal with a half-filled band.

The pressure-induced MI transition is correlated with the lossof magnetism.[101] A possible interpretation of this is that the MItransition triggers the magnetic one by delocalizing the magneticelectrons. However, a different interpretation has been proposedby Gavriliuk et al.,[78] who think instead that the order ofprecedence is different, i.e., the magnetic transition induces theMI change rather than the other way round. These authorssuggest that the MI transition may be Mott-type. This means thatthe bandgap is caused by electron– electron Coulombic repulsion(the Hubbard parameter,U), and that there is a critical value ofUwhich can be reached with either temperature or pressure.[104,105]


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Figure 9. Polarization of BiFeO3: bulk single crystal (top) and epitaxial thinfilm (bottom). Figures reproduced with permission from [15] and [14],respectively. Copyright 2007, American Institute of Physics (top) and 2003,AAAS (bottom).

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According to Gavriliuk et al., the change in U would be due to achange in the spin configuration from the ground-state high spin(the five d-shell electrons occupying one each of the t2g and eglevels, giving a total magnetic moment, S¼ 5/2[102,103]) to lowspin (no electrons in the eg level and S¼ 1/2), with the weakermagnetic interactions in the low-spin state being consistent withthe observed paramagnetism.[101] While there is no directexperimental evidence for this, ab initio calculations do agreewith a low-spin configuration at high pressures and lowtemperatures.[79]

The mechanism proposed by Gavriliuk et al. for thepressure-driven MI transition is unlikely to work for thetemperature-driven one. For one thing, BiFeO3 is magneticallydisordered both above and below TMI. Furthermore, thetransition we observe appears to be second order, which violatesone of Mott’s principal requirements.[104] We think instead thatthe MI transition is triggered by the structural change, ahypothesis supported by the screened-exchange model,[49] whichshows that BiFeO3 is metallic in only the cubic phase. BiFeO3 isviewed as a charge-transfer insulator, with the bandgap controlledby the orbital overlap between the O 2p and the Fe 3d levels.[49,100]

The overlap integral in turn depends on the Fe–O–Fe exchangeangle; accordingly, the observed straightening of the bond anglewith increasing temperature[64,75] would result in the observeddecrease of bandgap with increasing temperature. This mechan-ism is in fact completely analogous to that of the MI transition inthe perovskite nickelates[106–108] (parenthetically we note thatperovskite nickelates are also thought to be multiferroic[25,26,29]),with the main difference being that, whereas in the nickelates thebond angle is tuned by the ionic size, in BiFeO3 the Fe–O–Febond angle is controlled by the ferroelectric distortion.[92,100,109]

The correlation between orbital overlap and bandgap is also veryrelevant for the local properties – including conductivity – of thedomain walls, as further discussed in Section 8.

It seems strange to have two different mechanisms dependingon whether the MI transition is induced by pressure ortemperature. On the other hand, the high-pressure symmetryis reported as rhombohedral[78,79] (although some experimentssuggest cubic, as mentioned in Section 2) whereas the hightemperature one is cubic.[49,75] So, structurally at least, the twophases may indeed be different and different physical mechan-isms for the MI transition could therefore be expected. However,we suggest that there could be a ‘‘third way’’, reconciling aspectsof the two models. Here we note that the low-spin ionic radius ofFe3þ is rLS¼ 0.55 A, whereas that of the high-spin one isrHS¼ 0.645 A;[66] the smaller low-spin radius is, of course, thereason why pressure can induce the transition to low spin in thefirst place. The smaller size of the low-spin Fe must necessarilyresult in a shrinking of the oxygen octahedron around it, meaningthat the Fe–O–Fe bond angle can straighten: the Goldschmidtolerance factor is 0.88 for the high-spin configuration, and 0.93for the low-spin one, with the octahedral rotation angle beingcloser to 0 as it approaches 1. Thus, while the idea of thepressure-induced MI transition associated with a change to lowspin may be correct, we also believe that the change in bandgapmay not itself be caused by a reduction in the Mott–Hubbardelectron–electron repulsion, but by the low-spin-induced straigh-tening of the Fe–O–Fe bond angle.


4. Ferroelectricity

4.1. Bulk

The ferroelectric polarization of bulk bismuth ferrite is alongthe diagonals of the perovskite unit cell ([111]pseudocubic/[001]hexagonal). Early measurements of bulk ferroelectricity inthe 1960s and 1970s yielded only small values of the polarization.However, the small value of Pr (ca. 6mCcm�2) reported by Teagueet al.[110] for single crystals was viewed by those authors as limitedby lack of saturation, and they remarked, presciently, that ‘‘. . .theactual polarization of BiFeO3 is an order of magnitude higherthan we have measured.’’. It took more than 30 years before theywere proved right by measurements on high-quality thin films,[14]

single crystals,[15,16] and ceramics.[111]

The unprecedented large polarization of the thin films wasinitially thought[14] to be due to strain enhancement, but this is nolonger the case: good single crystals were eventually grown[15–19]

with Pr values very similar (Fig. 9) to those of the films: ca.60mCcm�2 normal to (001) and, therefore, approximately100mCcm�2 along [111]pseudocubic, and high polarization wasalso found in ceramics.[111] Ab initio calculations also agree withthe statement that the polarization of bulk BiFeO3 is intrinsicallyhigh[92,109] (ca. 90–100mCcm�2) and relatively insensitive tostrain.[109]

4.2. Thin Films and Strain Effects

In addition to having excellent ferroelectric properties asdiscussed above, thin films of BiFeO3 often have differentcrystallographic structures than single crystals do. Freestanding


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Figure 10. The absolute value of the ferroelectric polarization in thin films(c) is essentially independent from in-plane compression of the films (a,b).Reproduced with permission from [114]. Copyright 2008, American Insti-tute of Physics.

dendritic films were prepared as early as the mid 1980s by HansSchmid and others in Geneva, and these are like single crystals. Inparticular, their crystal class at ambient temperatures isrhombohedral. However, when bismuth ferrite is epitaxiallygrown as a thin film onto, for example, an SrTiO3 [001] substrate,the resulting morphology is monoclinic, where the symmetry-lowering distortion arises from in-plane contraction and out-of-plane elongation as a result of lattice mismatch between filmand substrate. This has been characterized by several groups.[112]

An as yet unresolved issue is whether there is a further changein symmetry, from monoclinic to tetragonal, as film thickness isreduced. XRD and Raman spectroscopy data suggest that epitaxialBiFeO3 grown on SrTiO3 becomes tetragonal[112,113] below acritical thickness of ca. 100 nm. On the other hand, piezo-response atomic force microscopy (PFM) studies performed onultrathin films[115] still show eight polarization variants, con-sistent with polarization oriented along the diagonals, as expectedfrom a monoclinic structure, rather than the two variantsexpected from an in-plane-compressed tetragonal phase. Whilethis discrepancy is still unresolved, we note that both observationsare not necessarily incompatible, as the ultrathin films couldperhaps be ‘‘metrically tetragonal’’ but with the actual point groupbeingmonoclinic. That is, while the external shape of the unit cellmay be tetragonal, the internal degrees of freedom responsiblefor the polarization might remain monoclinic. Such a decouplingbetween crystal class and internal symmetry has been previouslyreported in other epitaxially strained perovskite thin films.[116,117]

The in-plane compression was initially thought[14,112] toenhance the polarization, a natural assumption given the strongeffect of strain on the ferroelectricity of other perovskitefilms.[118,119] As discussed in the previous section, however, thisis now known not to be the case. Direct experimental proof of thesmall sensitivity of the polarization to the strain state was recentlypublished by Kim et al.,[114] who show that the polarization ofepitaxial BiFeO3 stays constant even as the epitaxial strain isrelaxed with increasing film thickness (Fig. 10). A newer studyhas also been published by Jang et al.[120] looking closely at therelationship between strain and polarization in BiFeO3; theseauthors confirm that the spontaneous polarization does notchange in absolute magnitude but can be rotated out-of-planethrough the monoclinic symmetry plane.[120]

The reason for the relatively small sensitivity of BiFeO3 toepitaxial strain is that its piezoelectric constant, which linksstrain to polarization, is also relatively low (between15–60 pmV�1[14,15,111,121] compared with 100–1000 pmV�1 forother perovskite ferroelectrics). The piezoelectric constant ofproper ferroelectrics/improper ferroelastics with a centrosym-metric paraphase can itself be linked to a more fundamentalparameter: the electrostrictive coefficient, Q. This relates thestrain, s, to the square of the polarization, s¼QP2. The effectivepiezoelectric coefficient is defined as the derivative of the strainwith respect to the electric field

deff33 � @s

@E¼ @s

@P

@P

@E¼ 2QP" (3)

where e is the dielectric constant. Using the experimentally

measured values of d33[14,111,121] we obtain from Equation 3 that


the effective electrostrictive coefficients are Qeff33¼ ca.

1–4� 10�2m4C�2, with the lower limit being in good agreement

with experimental estimates[121]. The full electrostrictive tensor

has in fact been calculated by Zhang et al.,[122] who give values

of Q1111¼ 0.032m4C�2, Q1122¼�0.016m4C�2, and Q1212¼0.01m4C�2, all of which are in the range estimated here from

reported piezoelectric constants. It is interesting to note that these

electrostrictive values are similar to those of BaTiO3 or

SrTiO3,[123] and yet the strain effect on the ferroelectricity of

BaTiO3 and SrTiO3 is much bigger. In other words, although the

piezoelectric coefficient of BiFeO3 is small, its electrostrictive

coefficient is not. The reason for this seemingly surprising result

is likely to be the small dielectric constant (see Section 5), which

affects piezoelectricity as given in Equation 3. Possible reasons for

the smallness of the dielectric constant are discussed in the next

section.

5. Dielectric Properties

5.1. Dielectric Constant from Radio Frequency to Optical

Frequency

The GHz dielectric constant of BiFeO3 at room temperature iser¼ ca. 30.[68,51,96,124] It peaks at the rhombohedral–orthorhombictransition (825–840 8C), possibly – though not necessarily – dueto a ferroelectric–paraelectric transition. This dielectric constant


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is small compared with those of typical perovskite ferroelectricssuch as BaTiO3, (Ba, Sr)TiO3 and Pb(Zr,Ti)O3 (PZT), which, asargued in the previous section, is the reason why the associatedpiezoelectric coefficient is also smaller. The mean refractiveindex, n, of BiFeO3 is[89] ca. 2.62, so the optical frequencydielectric constant can be estimated as er¼ n2¼ ca. 6.86. This isonly an average value, however; BiFeO3 is in fact stronglybirefringent with Dn¼ ca. 0.34[89] meaning that the dielectricconstant at optical frequencies is very anisotropic.

Although ca. 30 can be regarded as the intrinsic dielectricconstant of this compound at radiofrequencies, the impedancemeasurements in parallel-plate capacitors often yield highervalues: between 50 and 300 depending on sample morphology,orientation, and frequency range. This is because at thefrequencies typically accessible by impedance analyzers(100Hz to 1MHz), domain-wall motion and space-chargecontributions can be important and add to the measuredpermittivity. While the intrinsic value er¼ ca. 30 may seemsmall for a ferroelectric, this value is not unreasonable. For onething, the ferroelectric Curie temperature of BiFeO3 is very high,meaning that at room temperature the ferroelectric polarization isalready saturated and, thus, small electric fields will barely affect it(the dielectric constant is essentially a measure of polarizability).Furthermore, this is a strongly first-order transition to start with,so again there is little phonon softening and thus the dielectricconstant predicted by the frequency-shift according to theLyddane–Sachs–Teller relationship can also be expected to bevery low. Finally, and this is just a hypothesis, it may be thatperovskite ferroelectrics in which the polarization comes from theA site (e.g., PbTiO3 and BiFeO3) have intrinsically lower dielectricconstants than those where polarization comes from the B site(e.g., BaTiO3). Experimentally this certainly seems to be the case,but at present we know of no satisfactory explanation for this fact,if indeed it is more than just a coincidence.

At low frequencies or at high temperatures, colossal dielectricconstants have also been reported[96,125] and these are clearlydue to finite conductivity leading to Maxwell–Wagner (M–W)behavior.[96,125-127] The temperature at which the M–W effects setin depends on the sample conductivity; for some samples thiseffect happens at temperatures as low as 200K;[96] in our ownsingle crystal and ceramics the finite resistivity effects typicallyappear above room temperature, enabling a more confidentanalysis of intrinsic dielectric effects.

Figure 11. Anomalies in the relative dielectric constant (�), possibly due to coupling to magnetic(or magnetoelastic) transitions at low temperature. The anomalies do not seem to affect thedielectric loss (tan d). Adapted from [142], with permission. Copyright 2008, Insitute of Physics.

5.2. Dielectric Anomalies at Magnetic

Transitions

Because bismuth ferrite is piezoelectric at alltemperatures below 1100K, any magnetoelas-tic phenomena at its magnetic-phase transi-tions are apt to create responses in thedielectric response. These are shown inFigure 11. The subtle low-temperature anoma-lies at 200 and at 50 K coincide with thetemperatures where magnetic, magneto-opticand elastic anomalies have been seen, asdiscussed in the next section. None of thedielectric anomalies is strong and, curiously,none seems to affect the dielectric loss. Their


weakness shows that they do not correspond to ferroelectric phasetransitions, but arise instead fromweak coupling to another orderparameter, most likely magnetic.

Additional dielectric and conductivity anomalies arereported[68] at TNeel¼ 643K (370 8C), clearly related to magneto-electric coupling, and magnetodielectric coupling is alsoresponsible for the reported anomaly in the birefringence ofBiFeO3 at TNeel.

[89] Another is reported at the heretoforemysterious transition at 458K (185 8C), although this dielectricanomaly may itself be an artifact caused by the change inresistivity.[125–128]

6. Magnetism

6.1. Magnetic Symmetry and Spin Cycloid

The local short-range magnetic ordering of BiFeO3 is G-typeantiferromagnet, that is, each Feþ3 spin is surrounded by sixantiparallel spins on the nearest Fe neighbors. The spins are infact not perfectly antiparallel, as there is a weak canting momentcaused by the local magnetoelectric coupling to the polarization(see next section). Superimposed on this canting, however, is alsoa long-range superstructure consisting of an incommensuratespin cycloid of the antiferromagnetically ordered sublattices. Thecycloid has a very long repeat distance of ca. 62–64 nm, and apropagation vector along the [110] direction.[129,130] The magneticeasy plane (the plane within which the spins rotate) is defined bythe propagation vector and the polarization vector (Fig. 12). Themagnetic Neel temperature is ca. 643K (370 8C) and the exponentcharacterizing the sublattice magnetization as a function oftemperature, b, is known to be approximately 0.43 frombirefringence[89] and 0.37 from Mossbauer hyperfine split-tings.[102] Other critical exponents are discussed in theliterature.[131,132]

The cycloidal model of spin ordering in bismuth ferrite wasfirst proposed by Sosnowska et al. (1982),[130] whose group hasmade a number of detailed studies via XRD, neutron scattering,Mossbauer measurements, etc.[130–135] However, in recent yearsZalesskii and co-workers[136–138] have proposed that the simplecycloid is distorted at low temperatures. However, no publisheddata from either group indicate the phase-transition temperaturewhere the spin reorientation transition should occur.


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Figure 12. Schematic representation of the spin cycloid. The canted anti-ferromagnetic spins (blue and green arrows) give rise to a net magneticmoment (purple arrows) that is spacially averaged out to zero due to thecycloidal rotation. The spins are contained within the plane defined by thepolarization vector (red) and the cycloidal propagation vector (black).Figure reproduced with permission from [129]. Copyright 2008, AmericanPhysical Society.

It is also worth noting that in some single-crystal monodomainsamples the cycloid propagates along only one of the threesymmetry-equivalent<110> directions. This unique propagationdirection is suggestive of a magnetic symmetry-lowering effect(from rhombohedral to monoclinic), as emphasized by Lebeugleet al.[129] and Schmid.[139].

In 2007–2008 two groups reported evidence for furthermagnetic phase transitions at 140 and 200K. Cazayous et al. firstreported a transition at 140K[141] and, independently, Singh et al.found that one and an apparently stronger one at 200K.[103] Apossible origin of these transitions is discussed below togetherwith evidence for spin-glass behavior.[142,143]

6.2. Spin Reorientation in Orthoferrites

In the magnetic orthoferrites (e.g., ErFeO3) there are phasetransitions within the antiferromagnetic phase at which thesublattice spin orientations rotate. These occur at temperatures(90 and 103K in ErFeO3) far below the Neel temperature(TN¼ 633K in ErFeO3) and, hence, have nothing to do with loss

Figure 13. (Left) Intensity of magnon peaks in the Raman spectra as a function of temperature.These show clear phase transitions at ~140 K and ~200 K, the origin of which is as yet unclear butwhich is tentatively attributed to spin reorientations. (Above right) Magnon linewidth narrowingshows "critical slowing down" of spin fluctuations near 140 K, proof that the cross sectiondivergence cannot come from impurities. (Below right) Preliminary electron paramagneticresonance measurements show clear anomalies also at 140 and 200 K. Figures courtesy of M.Singh (Puerto Rico) and Pavle Cevc (Ljubliana).

of magnetic order. Generally these transitionsoccur in pairs: at the upper temperature thespins begin to rotate out of plane; and at thelower temperature, the rotation is complete sothat the spins are now 908 from their originaldirections, perpendicular to the plane. Thesephenomena are well understood in orthofer-rites[144] and Raman spectroscopy of magnonsnear the reorientation temperatures showsfrequency dips, cross-section enhancements,and linewidth narrowing.[145] The dip infrequency would be 100% (to zero) if therewere no magnetoelastic behavior, but actuallyreaches 50% in ErFeO3

[145] and only 5% inBiFeO3.

[103] The cross-section divergences areshown for bismuth ferrite in Figure 13. Thelinewidth narrowing for the magnons goesfrom 3.5 to <1.9 cm�1 (resolution-limited) atthe reorientation transition temperatures. Alsoshown in Figure 13 is the EPR susceptibility,which shows discontinuities at both 140and 200K; these anomalies confirm theinterpretation of these two temperatures asthose of magnetic-phase transitions. Note,


however, that the phenomena at 140K are different from thoseat 200K.

The orthoferrites are, as the name implies, orthorhombic,whereas BiFeO3 is crystallographically rhombohedral; however,its local magnetic structure is monoclinic,[146] with a monoclinicangle very near 908, which justifies approximating the spinstructure as orthorhombic, as in the orthoferrites. We note alsothat the Fe–O–Fe exchange angle (ca.1568), octahedral rotationangle (ca. 128) and Neel temperature (ca.640K) are all in thesame range as those of the rare-earth orthoferrites.[147] On thisbasis, one may hypothesize that the magnon anomaliesobserved at 140 and 200K may be indicative of spin reorientationin BiFeO3 analogous to that observed in orthoferrites such asErFeO3.

On the other hand, the spin reorientation in orthoferrites suchas ErFeO3 is thought to be brought about by the magneticinfluence of the rare-earth ions.[148] Clearly this cannot be the casein BiFeO3 as bismuth is not magnetic. So, again, whether or notthe phase transitions at 140 and 200K are indeed due to spinreorientation remains uncertain.

6.3. Spin-Glasslike Behavior

The evidence for spin-glass (or, at least, nonergodic) behavior inBiFeO3 is

[143,149] first that there is a large difference between itsfield-cooled (FC) and zero-field-cooled (ZFC) magnetizationbelow ca. 240 K (Fig. 14) (weaker FC effects were also reportedby Pradhan et al.[150] and Nakamura et al.[151]); second, that thereis a cusp at ca. 50 K in the magnetic susceptibility;[143] and third,that the temperature of the cusp in magnetic ac susceptibilityappears to be dependent upon the frequency of the magneticfield.[143]

When magnetic spins are subjected to competing forces andgeometric constrains, frustration can result in a chaotic glassy


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Figure 14. Field-cooled (FC) vs. zero-field-cooled (ZFC) magnetization insingle crystals of BiFeO3 for different magnetic fields (H). The strongdifference between the two is consistent with a spin-glass state. Figurecourtesy of M. Singh, University of Puerto Rico.

Figure 15. Almeida–Thouless fit of the irreversibility temperature deter-mined from the FC vs. ZFC data in Figure 14.

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state. The original spin-glass model of Kirkpatrick andSherrington[152] gave detailed predictions for such systemswithin a mean-field theory. The spin glasses studied experimen-tally are centric; that is, their spatially averaged structure has aninversion center. More recent work has generally appliedIsing-model statistics to such spin glasses, but bismuth ferritewould be a rare (perhaps unique) case of a spin glass that isferroelectric and hence non-centrosymmetric (acentric). AsFisher and Hertz have emphasized in their text,[153] no publishedtheories apply to acentric spin glasses, and Isingmodels definitelycannot apply to them.

Spin glasses are characterized by the frequency dependence ofthe peak in their magnetic susceptibilities. As the frequency isincreased, the peak in the ac magnetic susceptibility moves tohigher temperatures. If we call Tf the temperature at which the acsusceptibility has a maximum for a measurement frequency f, andTSG the extrapolated value of Tf at f = 0, then the spin-relaxationtime, t, varies as:

tðTf Þ ¼ a½TSG=ðTf � TSGÞ�zn (4)

where a is a constant independent of T and the exponent zn is a

characteristic spin-glass critical exponent describing the slowing

down of spin fluctuations near TSG. Although zn¼ 7–9 for Ising

models, there are other systems known to have 1< zn< 2, as in

the present case; La0.5Mn0.5FeO3 is a good example of such a

nonstandard spin glass,[154] with zn¼ 1.0.When the susceptibility

data are analyzed quantitatively, a spin-glass freezing temperature

of 29.4 K is estimated, and the critical exponent zn¼ 1.4� 0.2 that

characterizes the relaxation dynamics. This critical exponent zn is7–9 for Isingmodels, but 2 inmean-field theory.[152] This suggests

that the spin glass in BiFeO3 may be mean field; this would be

reasonable in view of the strong elastic effects manifest near the

magnetic transition temperatures, as discussed in the next


section. We remind readers that strain is always unscreened and

therefore very long-range and generally mean-field: if the

magnetic-order parameter is coupled to strain, the mean-field

nature of the strain might induce mean-field behavior in the

magnetic state.It is also useful to consider the absolute magnitude of the shift

in Tf with f. This is usually defined by a dimensionless sensitivityparameter, K

K ¼ DTf=ðTf D log f Þ (5)

Typically in a superparamagnetic crystal, 0.01<K< 0.1;whereas in a conventional spin glass, 0.001<K< 0.01. InBiFeO3, Singh et al. [143] found K¼ 0.014, which is at the marginbetween the two cases, and hence they infer that bismuth ferrite isnot a conventional spin glass. Further, they note that themagnitude of the ac magnetic susceptibility increases withfrequency; which is not reasonable for any glass, since glassystates are always less responsive as frequency increases. Theysuggest that this might be due to electrical or mechanicalresonances in the kHz regime, butmore work clearly is warrantedto clarify these issues, including what is the origin of the glassystate and whether or not it is an intrinsic feature as opposed to adefect-related phenomenon. An unpublished report was given onglassy behavior in single crystals of BiFeO3 very recently whichmay be helpful in this regard. Shvartsman et al. confirmre-entrant non-ergodic behavior of the low-field magnetization atlow temperature, but they exclude a generic spin-glass phase,since only cumulative relaxation is found after isothermal agingbelow Tg instead of classic hole burning and rejuvenation.[155]

Other technical details need attention, such as the possiblepresence in bismuth ferrite of the Almeida–Thouless (AT)transition line, which describes the stability of a spin glass at finitetemperatures and magnetic fields.[156] Such an AT line is shownas a function of magnetic field in Figure 15 for BiFeO3.

[149]

Although the physical interpretation is not yet clear, it is worthnoticing that the extrapolation temperature of the AT line isca.140K, which is one of the two critical temperatures of theelectromagnon spectra. It does not, however, coincide with thefreezing temperature extracted from the ac-susceptibility analysis.


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It is not easy to prove the existence of a spin-glass stateexperimentally: superparamagnets can exhibit AT lines, pinneddomains can exhibit aging and rejuvenation, and relaxors exhibitfrequency dependent susceptibilities; so unambiguous evidencewill require many different kinds of measurement.

6.4. Low-Temperature Ferromagnetism?

As explained earlier, BiFeO3 is antiferromagnetic at roomtemperature, with the weak local canting moment beingcompletely cancelled by the averaging out effect of the cycloid.However, there are several reports, including hysteresis mea-surements in single crystals (Fig. 16) suggesting that at very lowtemperatures there could be a weakly ferromagnetic state.[157,158]

It is important to confirm whether or not this is intrinsic because,although the net magnetic moment is minuscule (ca.10�6mB perFe), it would have important consequences regarding magneticsymmetry and, thus, also whether or not the linear magneticcoupling is allowed. The existence of ferromagnetism at very lowtemperatures would also reflect an underlying competitionbetween antiferromagnetic and ferromagnetic interactions,which, of course, would be consistent with the spin-glass statein the intermediate temperature range.

On the other hand, the observation of ferromagnetic hysteresisat low temperatures is not universal andmay be explained by evena very small concentration of impurities; Lebeugle et al., forexample, note that just 1mol% of paramagnetic Fe3þ (probablydue to the presence of Bi25FeO39) can account for all thelow-temperature magnetic enhancement in their single crystals,and that removing such impurities with HNO3 removes virtuallyall traces of ferromagnetism in their samples.[15]

6.5. Elastic Anomalies at Magnetic Transitions

Redfern et al. have used dynamic mechanical analysis (anoscillating three-point bending measurement) to estimate theelastic constant of BiFeO3 ceramics below room temperature. [142]

Their reported results show an anomaly between 200–250K and

Figure 16. Magnetization of BiFeO3 single crystals at low temperatures.Figure courtesy of M. Singh, University of Puerto Rico.


perhaps another one near 140K. The exact temperature ofthe anomaly at ca. 225K depends strongly on the frequency of theappliedmechanical stress. This frequency dependence can be dueto a thermally activated defect state, with an Arrhenius-typebehavior, or else it is an indication of some glassy underlyingprocess. While the mechanical data alone does not allowelucidation of the answer, it is worth pointing out that between200 and 250K there are also indications of a magnetic transition,the nature of which is still unclear but possibly related to aspin-glass state (see Section 6.3). It is therefore possible that theelastic anomaly is due to magnetoelastic (magnetostrictive)coupling to a glassy magnetic transition.

We have also measured the mechanical response of BiFeO3

above room temperature and several broad peaks are apparent.These are rather puzzling, since no structural transition has beenreported between room temperature and the Neel temperature. Itis again possible that these mechanical anomalies are caused bydefect states, but theymay also be real: as discussed in Section 2, alarge number of ‘‘ghost transitions’’, which are awaitingclarification, have been reported for BiFeO3.

Resonant ultrasound spectroscopy has also been deployed tocharacterize the elastic behavior of BiFeO3. Preliminary low-temperature data (Fig. 17) shows a very clear transition in theregion 30–60K, where elastic attenuation leads to the disap-pearance and re-entrance of the elastic resonances. This massiveattenuation is suggestive of a highly dissipative state. Given thatthe magnetic measurements suggest a spin-glass state in thistemperature region, the elastic measurements are consistent withcoupling between elasticity and the spin glass, lending support toa magnetoelastic mean-field character for the transition.

7. Magnetoelectric Coupling

7.1. Magnetoelectric Coupling and Spin Cycloid

The existence of a spin cycloid averages out any linearmagnetoelectric (ME) coupling between polarization (P) and

Figure 17. Resonant ultrasound spectroscopy of a BiFeO3 ceramic. All theresonant peaks disappear in the low-temperature region between ca. 50and 30 K, a temperature range in which the ac magnetic susceptibility anddielectric constant have also been reported to show broad anomalies.Other elastic anomalies can also be seen at higher temperatures. Figurecourtesy of Julia Herrero-Albillos and Michael Carpenter, University ofCambridge.


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magnetization (M). Any macroscopic magnetoelectric couplingmust therefore be higher order (quadratic). Indeed, up tomagnetic fields of several Tesla the magnetically inducedpolarization is found to be proportional to the square of themagnetic field (Fig. 18a). The full magnetoelectric tensor was firstcharacterized by Tabares-Munoz et al.,[159] and is given by (inhexagonal coordinate axis with P3 parallel to the spontaneouspolarization)

P1 ¼ b111ðH21 �H2

2Þ þ b113H1H3 (6)

P2 ¼ b113H2H3 � 2b111H1H2 (7)

P3 ¼ b311ðH21 �H2

2Þ þ b333H23 (8)

with experimentally measured coefficients b111¼ 5.0�10�19 s A�1, b113¼ 8.1� 10�19 s A�1, b311¼ 0.3� 10�19 s A�1,

and b333¼ 2.1� 10�19 s A�1.[159]

Above a certain critical field, however, the magnetoelectricpolarization markedly changes, signaling a change in the spinconfiguration. Ismailzade et al.[158] first reported spin flop at acritical field, HC, of only 5 kOe, (1Oe¼ 10�4 T) but this wasalmost certainly an artifact, since no subsequent measure-ments[159,161,162] were able to reproduce it. The real critical fieldappears to be much higher, at ca. 20 T[161,162] (Fig. 18). Above thiscritical value, the magnetoelectric polarization changes sign andbecomes linearly dependent onmagnetic field (Fig. 18, left). Sincethe linear magnetoelectric effect is forbidden by the cycloid, itsonset signals that the cycloid has been destroyed by the highmagnetic field. A second observation is that above the critical fieldfor the cycloid destruction (or spin flop), the field-inducedmagnetization jumps to a higher value. Linear extrapolation ofthis field-induced magnetization to zero-field yields a ‘‘remnant’’magnetization ca.0.3 emug�1 (Fig. 18, right).

The theory behind these effects is subtle. The local (short-range) magnetic symmetry of BiFeO3 is such that, if it werecentrosymmetric (paraelectric), it would be a perfect G-type

Figure 18. Magnetoelectric effect in BiFeO3 (left): at low fields, P is proportionaME coupling). Above BC ¼ 20 T, P is linearly dependent on H instead. Sinceforbidden in the presence of a cycloid, the cycloid is destroyed above 20 T. Notthe magnetically induced polarization is very small (ca.105 times smaller thapolarization). Once the cycloid is destroyed, the small canted magnetic mo(right). Extrapolation of the magnetization to zero field yields a small net ma0.3 emug�1. Figures adapted from [161], with permission. Copyright 2006, E


antiferromagnet with no net magnetic moment. However, theferroelectric polarization breaks the center of symmetry andinduces a small canting of the spins via the Dzyaloshinskii–Moriya interaction. This canting results in the very smallmagnetization of 0.3 emug�1.[32,162] Ferroelectrically inducedcanting magnetism is neither new nor unique to BiFeO3, as it wasalready reported for BaMnF4 in the 1970s.[28] What is specialabout BiFeO3 is that, in addition to this canting, there is also aferroelectrically induced spin cycloid that averages out the localcanted magnetism. This cycloid appears because polarization canalso couple to gradients of magnetization, thereby inducing aninhomogeneous spin configuration (the spin cycloid).[161,162] Thisis the converse effect of the ferroelectric polarization induced bymagnetic spirals.[25,26]

Although the cycloid averages out the macroscopic cantingmoment, this is locally still present at the unit-cell level. Highmagnetic fields can destroy the cycloid, thereby recovering thecanted state and its associated remnant magnetization (Fig. 18b)and, in this state, the linear magnetoelectric is allowed (Fig. 18a),so both effects in Figure 18 are consistent with each other. Thespin cycloid can also be destroyed by doping[164] or by epitaxialstrain,[165] so fully strained epitaxial thin films can in principledisplay a weak remnant magnetization, although not as big asinitially reported.[20,21]

7.2. Ferroelectric Control of Magnetism

Recent experimental works have explored in more detail therelationship between the ferroelectric polarization and magneticsymmetry. With the exception of the work by Kubel andSchmid,[61] early bulk research was performed on samples whichwere mostly either polycrystalline or polydomain single crystalsand, hence, some subtle directional effects were averaged out.Groups at Saclay[129] and Rutgers,[140] however, have managed tomake ferroelectric monodomain crystals of BiFeO3 by growingthem at temperatures below the ferroelectric transition, and havedeployed very high-resolution neutron diffraction to elucidate the

l toH2, (quadraticthe linear ME is

e that, in any case,n the ferroelectricment is recoveredgnetization of ca.lsevier.

H & Co. KGaA, Weinhe

relationship between ferroelectricity and anti-ferromagnetism in this compound.

Specifically, they have shown that themagnetic moments rotate within the planedefined by the polarization (P// [111]pseudocubic)and the cycloid propagation vector (k//[1,0,�1]pseudocubic) (Fig. 19). This has profoundconsequences, for if the direction of thepolarization is changed, so too will themagnetic easy plane: indeed, by applying avoltage and switching the polarization by 718,both Lebeugle et al.[129] and Lee et al.[140] wereable to show that the magnetic easy planeswere rotated. Importantly too, the magneticeasy plane can be switched only if thepolarization changes direction, but not if itmerely changes polarity; 1808 switching of thepolarization should not affect the magneticorientation.

The recent observations in single crystalsare closely related to a previous investigation

im Adv. Mater. 2009, 21, 2463–2485

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8.2. Domain Size

Figure 19. Relationship between the magnetic easy plane containing thespins, the vector of ferroelectric polarization, and the vector of cycloidpropagation. Rotating the polarization by 718 (i.e., switching only one of thecomponents of the polarization) results in a change of the magnetic easyplane, meaning that sublattice magnetization can be switched by anapplied voltage. Reproduced with permission from [129]. Copyright2008, American Physical Society.

on thin films reported by Zhao et al.[166] The thin films had nospin cycloid (due to either strain or reduced thickness), andinstead they had the homogeneous G-type antiferromagnetism(with a slight canting), with the magnetic easy plane perpendi-cular to the ferroelectric polarization. Using a combination ofpiezoelectric-force microscopy and X-ray photoelectron micro-scopy, these authors have managed to visualize simultaneouslythe ferroelectric and the antiferromagnetic domains, establishingthat both types of domains are completely correlated with eachother. Again, switching the polarization by an angle other than1808 (in rhombohedral symmetry, the polar vectors can beswitched by 718 and 1098 as well as 1808, as discussed in Section8) changes the magnetic easy plane. This electrically inducedswitching of the magnetic easy plane is of seminal importance,because it opens the possibility of magnetoelectric devices basedon the voltage control of magnetization (see Section 9).

Figure 20. Schematic of the three types of ferroelectric domain wallsseparating domains with one, two, or all three components of the polar-ization switched. The domain walls are labeled according to the anglebetween the polarization vectors on either side. Note that in this simplifiedpicture the 718 and 1808 walls are not in their most stable configurations,since the head-to-head polarization perpendicular to the wall would lead tolarge electric fields at the interface.

8. Domains and Domain Walls

Research on domains and domain walls has intensified recentlybecause i) the behavior of domains is directly responsible forswitching characteristics (switching of polarization takes placethrough nucleation and growth of domains) and ii) domain sizescales with sample size, so thin films can have very small domainsand, therefore, a high volume density of domain walls. BiFeO3

displays new domain-wall related phenomena of its own whichmake this subject particularly fascinating.

8.1. Domain Walls in Rhombohedral Symmetry

In rhombohedral BiFeO3 the ferroelectric polarization can pointalong any of the four diagonals of the perovskite unit cell, with two


antiparallel polarities for each direction: hence there are eightdifferent polar domains in BiFeO3. Separating adjacent domains,there are three possible types of ferroelectric domain wall, whichare usually labeled according to the angle formed between thepolarization vectors on either side of the wall. When only onecomponent of the diagonal polarization is reversed (say, onedomain has [111] orientation and the adjacent one is [11�1]), thenthe polar vectors form an angle of approximately 718 and thedomain wall that separates the two polarizations is called a 718wall. When two polar components are reversed, it is a 1098 wall,and when all three components of the polarization are reversed itis a 1808 wall. This is schematically depicted in Figure 20.Minimization of electrostatic and elastic fields imposes con-straints on the orientation of the walls; typically, 718 walls areparallel to {110} planes, 1098 walls are in {100} planes, and 1808walls should be on planes containing the [111] polar vector.[167] Ofcourse, nonequilibrium walls can be in different planes and mayeven be very irregular in shape.

8.2.1. Stripe Domains

Since the work of Landau and Lifshitz in 1935,[168] and later Kittelin 1946,[169] it is understood that domain size scales as the squareroot of film thickness. While their arguments were initiallyproposed for ferromagnetic films, they were later extended toferroelectric materials[170] and ferroelastics,[171] andmore recentlyalso to multiferroics[172] and nanostructures other than thinfilms.[173] The basic argument is that domain size results from thecompetition between a surface energy (demagnetization, depo-larization, strain) which is directly proportional to domain width,w, and a domain-wall energy that is proportional to the numberdensity of walls and, hence, inversely proportional to w. Theenergy of the walls is also proportional to their size, which scalesas the film thickness, d. Minimization of these components leadsto the familiar expression w ¼ A

ffiffiffid

p, where A is a constant. This

scaling has been experimentally verified for the ferroelectricstripe domains of BiFeO3 films,[174] but not for ultrathin films,which have a very different domain morphology (Fig. 21).


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Figure 21. Ferroelectric stripe domains (left) and fractal domains (right) inepitaxial thin films of BiFeO3. The latter tend to appear in very thin filmsonly. (PFM images of stripe domains and fractal domains courtesy of Y. H.Chu and M. Bibes, respectively).

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8.2.2. Fractal Domains

For ultrathin films the domains in BiFeO3 are no longer stripedand instead form irregularly shaped mosaic structures(Fig. 20).[115,175] The critical thickness depends on substrateand electrodes, among other things. The domain walls are rough,and are well described by a fractal Hausdorff dimension, h¼ ca.2.5 (for a perfectly smooth domain wall it should be two). TheHausdorff dimension describes the scaling between the area ofthe domains and their perimeters; since the domain size resultsfrom a competition between domain energy (proportional to area)and the wall energy (proportional to perimeter), it is reasonable toexpect that Kittel’s law will need to be modified in order toincorporate this ‘‘fractality’’. The modified equation turns out tobe[175]

w ¼ A0dh?3�hk (9)

whereA0 is a constant and h?, hjj are theHausdorff dimensions of

the domain wall in the directions perpendicular and parallel to the

plane of the film, respectively. This becomes the standard Kittel’s

law when the walls are perfectly smooth (h?¼ hjj ¼ 1).The reason for the thickness-induced transition to fractal

morphology remains unknown, but we note that irregular wallsare elastically costly, so they cannot be the equilibriumconfiguration in a perfect crystal. Two necessary ingredientsfor their appearance must be a random distribution of pinningdefects and a low crystal anisotropy (so that the wall can deformwithout much elastic cost). In this respect, the possiblethickness transition to a tetragonal symmetry for ultrathin filmsof BiFeO3

[113] would indeed facilitate the latter, since tetragonalc-axis domains face no elastic constraints (other than surfaceminimization) on the in-plane orientation of the 1808 walls.

Figure 22. The suppression of the ferroelectrically induced in-plane con-traction (Q13 is negative) leads to a lattice expansion in the directionperpendicular to the wall, which is accommodated by straightening of thebond angle. This results in increased orbital overlap and higher conductivity.

8.3. Domain-Wall Conductivity

Domain walls have their own local symmetry and, hence, alsotheir own properties. In the case of BiFeO3, this includesenhanced local conductivity. Ramesh and co-workers haverecently reported that certain domain walls of BiFeO3 are muchmore conductive than the domains themselves.[176] Furthermore,


the conductivity of the walls is directly related to the type ofdomains they separate. Thus, 1808 walls are the most conductive,followed by 1098walls and, finally, the 718walls, which, in fact, donot have any measurable transport enhancement. The authorsargue that there are at least two reasons for the enhancedconductivity of the walls. First, the polarization normal to thedomain wall is observed not to be constant across it; thisgenerates an electrostatic depolarization field that may attractcharge carriers. Second, the electronic bandgap is considerablyreduced for the 1808 and 1098 domain walls.

Aplausible explanation for the bandgap decrease has to do withthe local distortion of the Fe–O–Fe bond angle, which, asdiscussed in Section 3, controls the orbital overlap. In the middleof the domains, the octahedra are quite buckled (and hencethe gap is big). If the unit cell expands, the buckling angle canbecome straighter, thereby increasing the orbital overlap andreducing the bandgap. The local suppression of polarization atthe domain walls leads to precisely such a volume expansion viathe cancelling of the spontaneous strain (Fig. 22) and, hence, thelocal straightening of the bond angles at the walls reduces the gapin much the same way as temperature or pressure would.

This model is consistent with the correlation between the typeof wall and its conductivity. The absolute value of the polarizationis smallest in themiddle of the 1808walls (Pwall¼ 0), intermediatein the 1098 walls (Pwall¼P0/

ffiffiffi3

p¼ ca. 0.57P0), and maximum in

the 718 walls (Pwall¼P0

ffiffi2

pffiffi3

p ca.0.82P0); therefore, the volumechange (and the associated change in conductivity) will bebiggest for the 1808 walls, intermediate in the 1098 walls, andsmallest in the 718 walls, in agreement with the observedsequence of wall conductivities.

8.4. Domain-Wall Magnetization: Privratska–Janovec and

Daraktchiev Theories

Privatska and Janovec[177] observed that magnetoelectric couplingcould lead to the appearance of net magnetization in themiddle ofantiferromagnetic domain walls. Specifically, they showed thatthis effect is allowed for R3c space groups, which is the symmetryof BiFeO3, but their group-symmetry arguments do not allow anyquantitative estimate. Later, Fiebig and co-workers analyzedmagnetoelectric coupling in the walls of multiferroics such asYMnO3 and HoMnO3.

[178,179]

Daraktchiev et al. have proposed a thermodynamic (Land-au-type) model[172] with the aim of quantitatively estimatingwhether the walls of BiFeO3 can be magnetic and, if so, to whatextent they might contribute to the observed enhancement of


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Figure 24. Ferromagnetic hysteresis due to uncompensated surface spins

magnetization in ultrathin films. Their starting point is a verybasic two-parameter expansion with biquadratic coupling (whichis always symmetry-allowed) between net polarization and netmagnetization

DG ¼ a

2P2 þ b

4P4 þ kðrPÞ2 þ a

2M2 þ b

4M4

þ lðrMÞ2 þ g

2P2M2 (10)

Analysis of the phase space of this thermodynamic potentialshows that it is possible for net magnetization to appear in themiddle of ferroelectric walls even when the domains themselvesare not ferromagnetic (Fig. 23). This, however, is presently justonly a ‘‘toy model’’ which does not take into account the exactsymmetry of BiFeO3, so it cannot yet quantitatively estimate howmuch domain walls can contribute to the magnetization ofBiFeO3. The exact theory of magnetoelectric coupling at thedomain walls of BiFeO3 remains to be formulated.

in BiFeO3 nanocrystals. Reproduced with permission from [180]. Copyright2007, American Chemical Society.

9. Bismuth Ferrite Nanotubes, Nanowires,Nanocrystals

There is a fast-growing body of research devoted to themanufacture and characterization of complex nanoscopic shapesother than thin films. These 3D nanostructures generally havetheir own distinctive size effects, and multiferroic BiFeO3 is noexception. For example, nanocrystals of BiFeO3 show enhancedmagnetization and superparamagnetism correlated with decreas-ing diameter[180] (Fig. 24). Similar size-induced magnetism hasalso been reported for BiFeO3 nanowires[181] and nanopow-ders.[125] This is thought to be due to the large fraction ofuncompensated spins from the surfaces of the nanocrystals, aneffect that is well known from classic antiferromagnets such asNiO.[182]

Prof. Wong at SUNY Stony Brook reported crystalline BiFeO3

nanotubes in 2004.[183] These tubes were 240–300 nm in diameterand as much as 50mm long. Prepared via a sol-gel technique and

Figure 23. Domain-wall profile in a hypothetical multiferroic with biquadraticoupling between polarization andmagnetization. Note that the domain wallsmoment, even though the domains themselves do not.


a porous alumina (AAO) template technique, they werepolycrystalline with some amorphous content. Those authorsremoved the alumina template completely by immersion inNaOH, meaning that the tubes were left out in a pile which washard to characterize electrically. Zhang et al.[184] used porousalumina templates instead, managing to make ordered arrays ofstanding BiFeO3 nanotubes and measure their piezoelectrichysteresis loops. This proved that they were indeed ferroelectric.

Such nanotubes may be of considerable interest in terms ofnew theories relating to them.[185] Using a Landau-type thermo-dynamic analysis, the phase diagrams of magnetoelectricnanotubes as a function of radius have been calculated. Thesmall diameter of the tubes affects the ferroelectric criticaltemperature of the ferroelectric state as a result of the stressproduced by the curvature; when the ferroelectric criticaltemperature approaches the magnetic one, the magnetoelectriccoupling can be enhanced by several orders of magnitude

c magnetoelectrichave net magnetic

mbH & Co. KGaA, Wein

(Fig. 25). The theoretical predictions regardingmagnetoelectric enhancement still awaitexperimental confirmation.

10. Device Applications

10.1. Ferroelectricity and Piezoelectricity

Being a room-temperature multiferroic,BiFeO3 is an obvious candidate for applica-tions. Interestingly, however, the first applica-tion that may reach the market might not usethe multiferroic properties of BiFeO3 at all.The remnant polarization of BiFeO3 is verylarge, 100mCcm�2 along the polar [111]direction. To put this into context, this is thebiggest switchable polarization of any perovs-kite ferroelectric, and is roughly twice as big as

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Figure 25. Magnetoelectric coefficient as a function of nanotube (R12> 0)and nanowire (R12< 0) radius. The magnetoelectric coefficient increasesas the external radius decreases. Reproduced with permission from [185].Copyright 2008, American Physics Society.

Figure 26. Enhanced piezoelectric coefficient, d33, of thin films at themorphotropic phase boundary between pure BiFeO3 and SmFeO3. Repro-duced with permission from [190]. Copyright 2008 American Institute ofPhysics.

Figure 27. a) Experimental setup used by Takahashi et al. [191] formeasuring the THz emission of BiFeO3. (LT: low temperature; LSAT:LaAlO3–Sr2AlTaO6). b) Experimentally measured radiation and c) theFourier components of the amplitude of the measured THz radiationswitch when the ferroelectric polarization is switched, indicating thepossibility of using THz-based optical methods for reading ferroelectricmemories. Copyright 2006, American Physics Society.

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the polarization of the most widely used material in ferroelectricmemories, PZT. Moreover, unlike PZT, BiFeO3 is a lead-freematerial, a bonus regarding health and safety. It is therefore notsurprising that manufacturers of ferroelectric memories such asFujitsu are considering BiFeO3 as the potential active material intheir next generation of ferroelectric memory devices.[186] Forsuch an application to ever come into fruition, however,important obstacles must be removed, such as: i) the higherconductivity (and thus also dielectric losses) of BiFeO3 relative toPZT, ii) its tendency to fatigue,[187] and iii) the fact that it appearsto thermally decompose at voltages quite close to the coercivevoltage.[53]

A second potential application unrelated to magnetoelectricproperties is piezoelectricity. The piezoelectric coefficient of pureBiFeO3 is actually quite small, as argued in Section 4.2. However,its rhombohedral ground state means that mixing it with atetragonal ferroelectric such as PbTiO3 leads to a morphotropicphase boundary (MPB) at a composition of 30% mol PbTiO3.

[188]

This is important because MPBs are commonly thought to be thekey behind the large piezoelectric coefficients of PZT andrelaxors,[189] so the MPB of BiFeO3–PTmight lead to equally largepiezoelectric constants. A newer MPB with enhanced piezo-electric coefficients has been reported[190] for a solid solution ofrhombohedral BiFeO3 with orthorhombic SmFeO3 (Fig. 26); it islikely that other solid solutions between BiFeO3 and any of theother orthoferrites should also display similar MPBs. In thiscontext, it is surprising that the widely studied solid solutionbetween BiFeO3 and LaFeO3 does not appear to have yet beenpiezoelectrically characterized; perhaps this is due to its highconductivity.

10.2. Terahertz Radiation

Another possible application which does not make use of themagnetoelectric properties of BiFeO3 is its reported emissionof THz radiation. Takahashi et al.[191] reported that, when hit with


a femtosecond laser pulse, BiFeO3 films emit THz radiation,which is currently of great interest for many applications rangingfrom telecommunications to security.[192]

Furthermore, the authors note that the THz radiation iscompletely correlated with the poling state of the films (Fig. 27);accordingly, THz emission could provide an ultrafast (picosecondresponse time) and non-destructive method for ferroelectric


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Figure 29. The magnetization of a small island of ferromagnetic Co–Fe isswitched when a voltage is applied to the underlying BiFeO3 (white andblack correspond to different magnetic polarities). Figure reproduced withpermission from [200]. Copyright 2008, Nature Publishing Group.

Figure 30. Tunneling magnetoresistance (TMR) of BiFeO3 sandwichedbetween (La,Sr)MnO3 and Co. Reproduced with permission from [193].Copyright 2008, Institute of Physics.

memory readout. As an additional advantage, at such highfrequencies the response is insensitive to leakage, whichautomatically gets rid of one of the major obstacles forimplementing BiFeO3 as a ferroelectric memory material.

10.3. Spintronics

But the real drivers behind most of the applied research onBiFeO3 aremagnetoelectric and spintronic applications.[193] Chiefamong these would be memories that can be written using avoltage and read using a magnetic field. Using a voltage forwriting has three advantages: i) this can be implemented in asolid-state circuit without mobile parts, ii) it has a low-energyrequirement, and iii) the voltage requirements automatically scaledown with thickness. Reading the memory magnetically, on theother hand, has the advantage that it is a non-destructive readoutprocess, unlike direct ferroelectric reading, which requiresswitching the polarization in order to read it.

For such memories to actually work, the magnetic statetherefore must be a) electrically switchable and b) magneticallyreadable. As discussed in the previous section, the first conditionis met by BiFeO3, because the easy plane of its antiferromagneticdomains is correlated with the polar direction, and rotating theferroelectric polarization results in a rotation of the sublatticemagnetization,[129,140,166] i.e., the magnetic state of the samplecan be changed by a voltage. On the other hand, the secondcondition is not directly met, because antiferromagnetic (or, atbest, weakly canted antiferromagnetic) domains cannot be easilyread.

An elegant solution to the problem of reading antiferromag-netic states consists in using the mechanism known as exchangebias. Crudely, exchange bias is the magnetic interaction betweenthe spins at the uppermost layer of an antiferromagnet and a thinferromagnetic layer attached to it. The exchange bias modifies thehysteresis loops of the ferromagnetic layer, either offsetting orwidening them.[194] What is relevant here is that voltage-inducedchanges to the underlying antiferromagnetic domains will resultin changes to the ferromagnetic hysteresis of the upper layer,which can then be read by conventional mechanisms. Theimplementation of this concept for Cr2O3 (which is magneto-electric but not ferroelectric) was first done by Borisov et al.,[195]

and the first investigation with an actual multiferroic (YMnO3)was done by Laukhin et al.[196]

The race to implement this idea using BiFeO3 (which has theadvantage over YMnO3 that it works at room temperature) hasbeen on for a while, and has been punctuated by severalimportant milestones, such as the observation of exchange bias inthin ferromagnetic layers grown on BiFeO3

[197,198] (Fig. 28), the

Figure 28. Exchange-biased magnetic hysteresis loops of thin Co grown onfigures correspond to measurements along the four in-plane orientations [10[�100]. Reproduced with permission from [198]. Copyright 2006, American In


correlation between exchange bias and ferroelectric domains,[199]

the observation that the antiferromagnetic domains can beswitched by a voltage,[166] and, most recently, the final proof-of-concept that the exchange-biased ferromagnetic layer canindeed be switched by a voltage[200] (Fig. 29).

A second line of work uses BiFeO3 as a barrier layer inspintronics. Sandwiching BiFeO3 between two ferromagneticmetals results in tunneling magnetoresistance[193,198] (Fig. 30).For this, the only requirement is that the BiFeO3 layer bereasonably insulating down to tunneling thicknesses. However,an extra ingredient provided by BiFeO3 is the fact that it alsoremains a robust and switchable ferroelectric down to a thicknessof 2 nm,[201] and thus it could in principle be used as anelectrically switchable tunnel junction, whereby the ferroelectricstate controls the magnetic state of the thin ferromagneticelectrodes, thus modifying the tunneling magnetoresistance. Asimilar concept using a ferromagnetic multiferroic (La–BMO)was indeed demonstrated by Gajek et al.,[202] who showed that thetunneling resistivity could be controlled both by electric andmagnetic fields, giving rise to a four-state memory device. Thevoltage-dependent barrier characteristics of BiFeO3 have not yetbeen established.

The above developments show that, at least in principle, it is

BiFeO3. The four0], [010], [0�10],stitute of Physics.

mbH & Co. KGaA, We

now possible to develop an MERAM (Magneto-electric Random Access Memory) based onBiFeO3. A schematic of such a device has beenproposed by Bibes and Barthelemy,[203] and isreproduced in Figure 31.

11. Closing Remarks

It is difficult to write a review on a topic that isso popular and that is continuing to develop at

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Figure 31. MERAM based on exchange-bias coupling between a multi-ferroic that is ferroelectric and antiferromagnetic (FE-AFM, green layer),and a thin ferromagnetic electrode (blue). A tunneling barrier layer betweenthe two top ferromagnetic layers provides the two resistive states. Inter-estingly, BiFeO3 could act not only as the magnetoelectric active layer, butalso as the tunneling barrier. Reproduced with permission from [203].Copyright 2008, Nature Publishing Group.

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such a rapid rate. So, rather than just summarizing what is knownso far about BiFeO3, we have chosen to focus our attention hereon four basic issues which we believe are still open: 1) What arethe high-temperature phases and in particular what is the natureof the MI transition (at high temperature? at high pressure? Arethey the same? Are they Mott-like? is the ‘‘Polomska transition’’ at458 K extrinsic?). 2) What are the low-temperature magneticphases? (Is there a spin glass? If so, is it Ising-like? long-range?mean-field? How does it couple to strain? Is Sosnowska’smagnetic structure correct or Zalesskii’s or neither? Is there anAlmeida–Thouless AT line? Are there two magnetic transitions at140 and 201K or four? Is the spin-glass onset at 230 or 50 K? Is theVolgel–Fulcher glass-freezing temperature 29 K? Is it a ferro-magnet below ca. 10K? Are thesemagnetic transitions intrinsic?).3) What are the magnetoelectric properties near room tempera-ture? (Is there a linear magnetoelectric term? quadratic? Is ituseful? for spin filters and spintronics? for memories?). 4) Whatare the intrinsic properties of the domain walls? (Are theyferromagnetic? If so, how much? Do they affect the functionalproperties of thin films? What is the mechanism for enhancedconductivity? Is it intrinsic or extrinsic? How will that affect theperformance of thin-film devices with a high density of domains?)There is an urgent need for many more studies focusing on thephase diagram and the dynamics. Very little yet is known aboutswitching processes. As a final, concrete example: If we don’tknow the magnetic space group or even point-group symmetry ofthis material below room temperature in what could be severalmagnetic phases plus a glassy phase, how can we even decidewhether a linear magnetoelectric coupling is allowed orforbidden?

We cannot provide definitive answers to most of thesequestions. For some readers that may be a disappointment:Why write a 15 000-word review if you don’t know the answers?Instead, our aim has been to provide a fair view of the pertinentworks from different sources, stating as clearly as possible whatwe think the questions are, together with evidence for and against


certain models. We emphasize that the study of BiFeO3 is NOTallwrapped up. So this is not a Bible about bismuth ferriteproperties; rather, it should be read as an inspiration for furtherstudies. We hope the reader finds it useful as such.

Acknowledgements

During the elaboration of this review we have had many fruitful exchangesthat have enhanced our understanding of this material. We wouldparticularly like to thank Michel Viret, John Robertson, Jorge Iniguez, andHans Schmid for their insights. There are also a number of unpublishedresults that have been kindly made available to us for this review; we areindebted to Simon Redfern, Julia Herrero-Albillos, Hong Jiawang, MichaelCarpenter, Manooj Singh, Ram Katiyar, Manuel Bibes, Brahim Dhkil, JensKreisel, Finlay Morrison, Eddie Chu, Ramamoorthy Ramesh, MarenDaraktchiev, Maria Polomska, and Pavle Cevc for sharing their workwith us.

Received: September 25, 2008

Published online: May 4, 2009

Note added in proof: After submission of this paper, Choi et al. reported[205] a polarization-controlled diode effect and large photovoltaic currentsin BiFeO3, opening the way for novel device applications that combine boththe ferroelectric and semiconducting properties of this material.

[1] P. Curie, J. Physique 1894, 3, 393.

[2] J. Valasek, Phys. Rev. 1920, 15, 537.

[3] A. Perrier, A. J. Staring, Arch. Sci. Phys. Nat. (Geneva) 1922, 4, 373.

[4] A. Perrier, A. J. Staring, Arch. Sci. Phys. Nat. (Geneva) 1923, 5, 333.

[5] T. H. O’Dell, The Electrodynamics of Magneto-electric Media, North-

Holland, Amsterdam 1970.

[6] a) I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 1959, 37, 88. b) Sov. Phys. —

JETP 1959, 10, 628.

[7] a) D. N. Astrov, Zh. Eksp. Teor. Fiz. 1960, 38, 984. b) Sov. Phys. — JETP

1960, 11, 708.

[8] E. Ascher, H. Rieder, H. Schmid, H. Stossel, J. Appl. Phys. 1966, 37, 1404.

[9] a) G. A. Smolensky, V. A. Isupov, A. I. Agronovskaya, Sov. Phys. — Solid

State 1959, 1, 150. b) G. A. Smolenskii, A. I. Agranovskaya, S. N. Popov,

V. A. Isupov, Zh. Tekh. Fiz. 1958, 28, 2152. c) A. Smolenskii, A. I.

Agranovskaya, S. N. Popov, V. A. Isupov, Sov. Phys. — Tech. Phys.

1958, 3, 1981.

[10] a) G. A. Smolenskii, I. E. Chupis, Usp. Fiz. Nauk 1982, 137, 415. b) G. A.

Smolenskii, I. E. Chupis, Sov. Phys. Usp. 1982, 25, 475. c) G. A. Smolenskii,

I. E. Chupis, Sov. Phys. — Solid State 1962, 4, 807.

[11] H. Schmid, Ferroelectrics 1994, 162, 665.

[12] M. Fiebig, J. Phys. D 2005, 38, R123.

[13] W. Eerenstein, N. D. Mathur, J. F. Scott, Nature 2006, 442, 759.

[14] J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D.

Viehland, V. Vaithyanathan, D. G. Schlom, U. V. Waghmare, N. A. Spaldin,

K. M. Rabe, M. Wuttig, R. Ramesh, Science 2003, 299, 1719.

[15] a) D. Lebeugle, D. Colson, A. Forget, M. Viret, P. Bonville, J. F. Marucco, S.

Fusil, Phys. Rev. B 2007, 76, 024 116. b) D. Lebeugle, D. Colson, A. Forget,

M. Viret, P. Bonville, J. F. Marucco, S. Fusil, Appl. Phys. Lett. 2007, 91,

022 907.

[16] D. Lebeugle, ‘Groupement de Recherche sur les Nouveaux Etats Electro-

niques de la Matiere: GdR NEEM’; Gif sur Yvette, November 2006.

[17] R. P. S. M. Lobo, R. L. Moreira, D. Lebeugle, D. Colson, Phys. Rev. B 2007,

76, 172 105.

[18] M. Cazayous, D. Malka, D. Lebeugle, D. Colson, Appl. Phys. Lett. 2007, 91,

071910.

[19] D. Lebeugle, Ph.D. Thesis Saclay Institute of Matter and Radiation, 2007.


REVIE

W

www.advmat.de

[20] W. Eerenstein, F. D. Morrison, J. Dho, M. G. Blamire, J. F. Scott, N. D.

Mathur, Science 2005, 307, 1203a.

[21] H. Bea, M. Bibes, A. Barthelemy, K. Bouzehouane, E. Jacquet, A. Khodan,

J.-P. Contour, Appl. Phys. Lett. 2005, 87, 072508.

[22] N. A. Hill, J. Phys. Chem. 2000, B104, 6694.

[23] A. Filippetti, N. A. Hill, Phys. Rev. B 2002, 65, 195120.

[24] a) R. E. Cohen, Nature 1992, 358, 136. b) Z. Wu, R. E. Cohen, Phys. Rev.

Lett. 2005, 95, 9 037 601.

[25] M. Mostovoy, Phys. Rev. Lett. 2006, 96, 067601.

[26] S.-W. Cheong, M. Mostovoy, Nature Materials 2007, 6, 13.

[27] I. A. Sergienko, C. Sen, E. Dagotto, Phys. Rev. Lett. 2006, 97, 227204.

[28] D. V. Efremov, J. van den Brink, D. I. Khomskii, Nature Materials 2004, 3,

853.

[29] G. Nenert, T. T. M. Palstra, J. Phys.: Condens. Matter 2007, 19, 406213.

[30] G. Catalan, Phase Transitions 2008, 81, 729.

[31] D. L. Fox, J. F. Scott, J. Phys. C: Sol. St. Phys. 1977, 10, L329.

[32] C. J. Fennie, Phys. Rev. Lett. 2008, 100, 167203. C. Ederer, C. J. Fennie,

J. Phys. Cond. Mat. 2008, 20, 434219.

[33] J. F. Scott, Ferroelectrics 1995, 166, 95.

[34] S. C. Abrahams, Acta Cryst. B 2007, 63, 257; 2003, 59, 811; 2003, 59, 541;

2002, 58, 34; 2002, 58, 316; 2000, 56, 793; 1996, 52, 790.

[35] S. C. Abrahams, K. Mirsky, R. M. Nielson, Acta Cryst. B 1996, 52, 806.

[36] S. C. Abrahams, Ferroelectrics 1990, 104, 37; Acta Cryst. B 1999, 55, 494;

1989, 45, 228; 1988, 44, 585.

[37] S. C. Abrahams, J. Ravez, Ferroelectrics 1992, 135, 21.

[38] J. Ravez, S. C. Abrahams, R. de Pape, J. Appl. Phys. 1989, 65, 3987.

[39] Y. Calage, S. C. Abrahams, J. Ravez, J. Appl. Phys. 1990, 67, 430.

[40] R. Blinc, G. Tavcar, B. Zemva, D. Hanzel, P. Cevc, C. Filipic, A. Levstik, Z.

Jaglicic, Z. Trontelj, N. Dalal, V. Ramachandran, S. Nellutla, J. F. Scott, J.

Appl. Phys. 2008, 103, 074114.

[41] E. Kroumova, M. I. Aroyo, J. M. Perez-Mato, R. Hundt, Acta Cryst. B 2001,

57, 599.

[42] S. Sarraute, J. Ravez, R. VonderMuhll, G. Bravic, R. S. Feigelson, S. C.

Abrahams, Acta Cryst. B 1996, 52, 72.

[43] S. C. Abrahams, J. Albertsson, C. Svensson, Acta Cryst. B 1990, 46, 497.

[44] a) M. Lorient, R. VonderMuhll, J. Ravez, A. Tressaud, Solid State.

Commun. 1980, 36, 383. b) M. Lorient, R. VonderMuhll, J. Ravez, A.

Tressaud, Solid State. Commun. 1981, 40, 847.

[45] a) A. Tressauld, J. Ravez, M. Lorient, J. Fluorine Chem. 1982, 21, 30. b) A.

Tressauld, J. Ravez, M. Lorient, Rev. Chim. Minerale 1982, 19, 128.

[46] J. Ravez, J. Physique III 1997, 7, 1129.

[47] E. J. Speranskaya, V. M. Skorikov, E. Ya. Kode, V. A. Terektova, Bull. Acad.

Sci. U. S. S. R. 1965, 5, 873.

[48] M. I. Morozov, N. A. Lomanova, V. V. Gusarov, Russ. J. Gen. Chem. 2003,

73, 1680.

[49] R. Palai, R. S. Katiyar, H. Schmid, P. Tissot, S. J. Clark, J. Robertson, S. A. T.

Redfern, G. Catalan, J. F. Scott, Phys. Rev. B 2008, 77, 014110.

[50] M. Valant, A.-K. Axelsson, N. Alford, Chem. Mater. 2007, 19, 5431.

[51] N. N. Krainik, N. P. Khuchua, V. V. Zhdanova, V. A. Evseev, Sov. Phys. —

Solid State 1966, 8, 654.

[52] H. Schmid: private communication.

[53] X. J. Lou, C. X. Yang, T. A. Tang, Y. Y. Lin, M. Zhang, J. F. Scott, Appl. Phys.

Lett. 2007, 90, 262908.

[54] X. J. Lou, X. J. M. Zhang, M. S. A. T. Redfern, J. F. Scott, Phys. Rev. Lett.

2006, 97, 177601.

[55] G. Garton, S. H. Smith, B. M. Wanklyn, J. Cryst. Growth 1971, 13, 588.

[56] D. Elwell, A. W. Morris, B. W. Neate, J. Cryst. Growth 1972, 16, 67.

[57] H. Okamoto, J. Phase Equilib. 1991, 12, 207.

[58] T. Atsuki, N. Soyama, T. Yonezawa, K. Ogi, Jpn. J. Appl. Phys. 1995, 34,

5906.

[59] J. M. Moreau, C. Michel, R. Gerson, W. J. James, J. Phys. Chem. Solids

1971, 32, 1315.

[60] a) V.S Filip’ev, I.P Smol’yaninov, E.G Fesenko, I.I Belyaev, Kristallografiya

1960, 5, 958. b) Sov. Phys. — Crystallogr. 1960, 5, 913.


[61] F. Kubel, H. Schmid, Acta Cryst. B 1990, 46, 698.

[62] J. D. Bucci, B. K. Robertson, W. J. James, J. Appl. Cryst. 1972, 5, 187.

[63] J. R. Chen, W. L. Wang, J.-B. Li, G. H. Rao, J. Alloys Compd. 2008, 459,

66.

[64] A. Palewicz, R. Przeniosło, I. Sosnowska, A. W. Hewat, Acta Cryst. 2007,

B63, 537.

[65] V. M. Goldschmidt, Naturwissenschaften 1926, 14, 477.

[66] R. D. Shannon, Acta Cryst. A 1976, 32, 751.

[67] H. D. Megaw, C. N. W. Darlington, Acta Cryst. A 1975, 31, 161.

[68] M. Polomska, W. Kaczmarek, Z. Pajak, Phys. Stat. Sol. 1974, 23, 567.

[69] N. N. Krainik, N. P. Khuchua, A. A. Bierezhnoi, A. G. Tutov, A. J.

Cherkashtschenko, Izv. Akad. Nauk ser. Fiz. 1965, 29, 1026.

[70] a) R. Haumont, J. Kreisel, P. Bouvier, F. Hippert, Phase Trans. 2006, 79,

1043. b) R. Haumont, J. Kreisel, P. Bouvier, F. Hippert, Phys. Rev. B 2006,

73, 132 101.

[71] I. A. Kornev, S. Lisenkov, R. Haumont, B. Dkhil, L. Bellaiche, Phys. Rev.

Lett. 2007, 99, 227602.

[72] R. Haumont, I. A. Kornev, S. Lisenkov, L. Bellaiche, J. Kreisel, B. Dkhil,

Phys. Rev. B 2008, 78, 134108.

[73] S. M. Selbach, T. Tybell, M.-A. Einarsrud, T. Grande, Adv. Mater. 2008, 20,

3692.

[74] D. C. Arnold, K. S. Knight, F. D. Morrison, P. Lightfoot, Phys. Rev. Lett.

2009, 102, 027602.

[75] S. A. T. Redfern, J. N. Walsh, S. M. Clark, G. Catalan, J. F. Scott,

arXiv:0901.3748, 2009.

[76] a) E. Ascher, J. Phys. C: Solid State Phys. 1977, 10, 1365. b) M. V. Jaric, D.

Mukamel, Nucl. Phys. 1990, B336, 475.

[77] S. M. Selbach, T. Tybell, M.-A. Einarsrud, T. Grande, Chem. Mater. 2007,

19, 6478.

[78] A. G. Gavriliuk, V. V. Struzhkin, I. S. Lyubutin, S. G. Ovchinnikov, M. Y. Hu,

P. Chow, Phys. Rev. B 2008, 77, 155 112.

[79] O. E. Gonzalez-Vazquez, J. Iniguez, Phys Rev. B 2009, 79, 064102.

[80] P. Karen, D. Legut, M. Friak, M. Sob, Phys. Today 2008, 61, 10.

[81] a) A. Pashkin, K. Rabia, S. Frank, C. A. Kuntscher, R. Haumont, R.

Saint-Martin, J. Kreisel, arXiv:0712.0736v1, 2007.b) R. Haumont, P. Bouvier,

A. Pashkin, K. Rabia, S. Frank, B. Dkhil, W. A. Crichton, C. A. Kuntscher, J.

Kreisel, arXiv:0811.0047, 2008.

[82] a) A. G. Gavriliuk, I. S. Lyubiutin, V. V. Strukin, JETP Lett. 2007, 86, 532.

b) A. G. Gavriliuk, I. S. Lyubiutin, V. V. Strukin, JETP Lett. 2007, 86, 197.

c) A. G. Gavriliuk, I. S. Lyubiutin, V. V. Strukin, Mater. High Press. 2007,

987, 147.

[83] A. G. Gavriliuk, V. Struzhkin, I. S. Lyubutin, I. A. Trojan, Mi, Y. Hu, P. Chow,

Mater. Res. Soc. Symp. Proc. 2007, 987, 05.

[84] J. F. Scott, R. Palai, A. Kumar, M. K. Singh, N. M. Murari, N. K. Karan, R. S.

Katiyar, J. Am. Ceram. Soc. 2008, 91, 1762.

[85] M. Polomska, W. Kaczmarek, Acta Phys. Polonica A 1974, 45, 199.

[86] Z. V. Gabbasova, M. D. Kuz’min, A. K. Zvezdin, I. S. Dubenko, V. A.

Murashov, D. N. Rakov, I. B. Krynetsky, Phys. Lett. A 1991, l58, 491.

[87] A. V. Zalesskii, A. A. Frolov, T. A. Khimich, A. A. Bush, Phys. Sol. St. 2003,

45, 141.

[88] G. L. Yuan, S. W. Or, H. L. W. Chan, J. Phys. D: Appl. Phys. 2007, 40,

1196.

[89] J.-P. Rivera, H. Schmid, Ferroelectrics 1997, 204, 23.

[90] J. C. Toledano, Annal. Telecomm. 1974, 29, 249.

[91] J. Holakovsky, F. Smutny, J. Phys. C: Solid State 1978, 11, L611.

[92] P. Ravindran, R. Vidya, A. Kjekshus, H. Fjellvag, O. Eriksson, Phys. Rev. B

2006, 74, 224412.

[93] T. T. Carvalho, European Summer School of Multiferroics, Girona, 2008.

[94] J. Rymarczyk, D. Machura, J. Ilczuk, Eur. Phys. J. Special Topics 2008, 154,

191.

[95] T. P. Gujar, V. R. Shinde, C. D. Lokhande, Mater. Chem. Phys. 2007, 103,

142.

[96] S. Kamba, D. Nuzhnyy, M. Savinov, J. Sebek, J. Petzelt, J. Prokleska, R.

Haumont, J. Kreisel, Phys. Rev. B 2007, 75, 024403.


REVIE

W

www.advmat.de

2484

[97] V. Fruth, E. Tenea, M. Gartner, M. Anastasescu, D. Berger, R. Ramer, M.

Zaharescu, J. Eur. Ceram. Soc. 2007, 27, 937.

[98] J. F. Ihlefeld, N. J. Podraza, Z. K Liu, R. C. Rai, X. Xu, T. Heeg, Y. B. Chen,

J. Li, R. W. Collins, J. L. Musfeldt, X. Q. Pan, J. Schubert, R. Ramesh, D. G.

Schlom, Appl. Phys. Lett. 2008, 92, 142908.

[99] Y. Xu, M. Shen, Mat. Lett. 2008, 62, 3600.

[100] S. J. Clark, J. Robertson, Appl. Phys. Lett. 2007, 90, 132903.

[101] A. G. Gavriliuk, V. V. Struzhkin, I. S. Lyubutin, M. Y. Hu, H. K. Mao, JETP

Lett. 2005, 82, 224.

[102] C. Blaauw, F. van der Woude, J. Phys. C: Solid State Phys. 1973, 6, 1422.

[103] M. K. Singh, R. S. Katiyar, J. F. Scott, J. Phys. Condens. Mat. 2008, 20,

252203.

[104] N. F. Mott, Rev. Mod. Phys. 1968, 40, 677.

[105] P. Schlottmann, Phys. Rev. B 1992, 45, 5784.

[106] J. B. Torrance, P. Lacorre, A. I. Nazzal, E. J. Ansaldo, C. Niedermayer, Phys.

Rev. B 1992, 45, 8209.

[107] P. C. Canfield, J. D. Thompson, S. W. Cheong, L. W. Rupp, Phys. Rev. B

1993, 47, 12357.

[108] M. Medarde, J. Mesot, P. Lacorre, S. Rosenkranz, P. Fischer, K. Gobrecht,

Phys. Rev. B 1995, 52, 9248.

[109] C. Ederer, N. A. Spaldin, Phys. Rev. Lett. 2005, 95, 257601.

[110] J. R. Teague, R. Gerson, W. J. James, Solid State Commun. 1970, 8,

1073.

[111] V. V. Shvartsman, W. Kleemann, R. Haumont, J. Kreisel, Appl. Phys. Lett.

2007, 90, 172115.

[112] a) J. Li, J. Wang, M. Wuttig, R. Ramesh, N. Wang, B. Ruette, A. P. Pyatakov,

A. K. Zvezdin, D. Viehland, Appl. Phys. Lett. 2004, 84, 5261. b) G. Xu,

H. Hiraka, G. Shirane, J. Li, J. Wang, D. Viehland, Appl. Phys. Lett. 2005,

86, 182905. c) S Keisuke, K. Saito, A. Ulyanenkov, V. Grossmann, H. Ress,

L. Bruegemann, H. Ohta, T. Kurosawa, S. Ueki, H. Funakubo,

Jpn. J. Appl. Phys. 2006, 45, 7311. d) G. Y. Xu, J. F. Li, D. Viehland, Appl.

Phys. Lett. 2006, 89, 222901.

[113] H. Bea, M. Bibes, S. Petit, J. Kreisel, A. Barthelemy, Philos. Mag. Lett. 2007,

87, 165.

[114] D. H. Kim, H. N. Lee, M. D. Biegalski, H. M. Christen, Appl. Phys. Lett.

2008, 92, 012911.

[115] Y. H. Chu, T. Zhao, M. P. Cruz, Q. Zhan, P. L. Yang, L. W. Martin, M.

Huijben, C. H. Yang, F. Zavaliche, H. Zheng, R. Ramesh, Appl. Phys. Lett.

2007, 90, 252906.

[116] F. He, B. O. Wells, S. M. Shapiro, Phys. Rev. Lett. 2005, 94, 176101.

[117] G. Catalan, A. Janssens, G. Rispens, S. Csiszar, O. Seeck, G. Rijnders,

D. H. Blank, B. Noheda, Phys. Rev. Lett. 2006, 96, 127602.

[118] K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker,

P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, L.-Q. Chen, D. G. Schlom,

C. B. Eom, Science 2004, 306, 1005.

[119] J. H. Haeni, P. Irvin, W. Chang, R. Uecker, P. Reiche, Y. L. Li, S. Choudhury,

W. Tian, M. E. Hawley, B. Craigo, A. K. Tagantsev, X. Q. Pan, S. K. Streiffer,

L. Q. Chen, S. W. Kirchoefer, J. Levy, D. G. Schlom,Nature (London) 2004,

430, 758.

[120] H. W. Jang, S. H. Baek, D. Ortiz, C. M. Folkman, R. R. Das, Y. H. Chu, P.

Shafer, J. X. Zhang, S. Choudhury, V. Vaithyanathan, Y. B. Chen, D. A.

Felker, M. D. Biegalski, M. S. Rzchowski, X. Q. Pan, D. G. Schlom, L. Q.

Chen, R. Ramesh, C. B. Eom, Phys. Rev. Lett. 2008, 101, 107602.

[121] C. W. Bark, S. Ryu, Y. M. Koo, H. M. Jang, H. S. Youn, Appl. Phys. Lett.

2007, 90, 022902.

[122] J. X. Zhang, Y. L. Li, Y. Wang, Z. K. Liu, L. Q. Chen, Y. H. Chu, F. Zavaliche,

R. Ramesh, J. Appl. Phys. 2007, 101, 114105.

[123] J. Zhao, A. E. Glazounov, Q. M. Zhang, B. Toby, Appl. Phys. Lett. 1998, 72,

1048.

[124] W. Kaczmarek, Z. Pajak, M. Polomska, Solid State Commun. 1975, 17,

807.

[125] R. Mazumder, S. Ghosh, P. Mondal, D. Bhattacharya, S. Dasgupta, N.

Das, A. Sen, A. K. Tyagi, M. Sivakumar, T. Takami, H. Ikuta, J. Appl. Phys.

2006, 100, 033908.


[126] G. Catalan, J. F. Scott, Nature 2007, 448, E4–E5.

[127] G. Catalan, Appl. Phys. Lett. 2006, 88, 102902.

[128] J. F. Scott, JETP Lett. 1989, 49, 233.

[129] D. Lebeugle, D. Colson, A. Forget, M. Viret, A. M. Bataille, A. Gukasov,

Phys. Rev. Lett. 2008, 100, 227602.

[130] I. Sosnovska, T. Peterlin-Neumaier, E. Steichele, J. Phys. C 1982, 15, 4835.

[131] J. F. Scott, M. K. Singh, R. S. Katiyar, J. Phys.: Condens. Matter 2008, 20,

322203.

[132] J. F. Scott, M. K. Singh, R. S. Katiyar, J. Phys.: Condens. Matter 2008, 20,

425223.

[133] A. Palewicz, T. Szumiata, R. Przeniosło, I. Sosnowska, I. Margiolaki, Solid

State Commun. 2006, 140, 359.

[134] R. Przeniosło, M. Regulski, I. Sosnowska, J. Phys. Soc. Jpn. 2006, 75,

084 718.

[135] R. Przeniosło, A. Palewicz, M. Regulski, I. Sosnowska, R. M. Ibberson,

K. S. Knight, J. Phys.: Condens. Matter 2006, 18, 2069.

[136] A. A. Bush, A. A. Gippius, A. V. Zalesskii, E. N. Morozova, JETP Lett. 2003,

78, 389.

[137] A. V. Zalesskii, A. A. Frolov, A. K. Zvezdin, A. A. Gippius, E. N. Morozova,

D. F. Khozeev, A. A. Bush, V. S. Pokatilov, JETP 2002, 95, 101.

[138] A. V. Zalesskii, A. K. Zvezdin, A. A. Frolov, A. A. Bush, JETP Lett. 2000, 71,

465.

[139] H. Schmid, J. Phys.: Condens. Matter 2008, 20, 434 201.

[140] S. Lee, W. Ratcliff, S.-W. Cheong, V. Kiryukhin, Appl. Phys. Lett. 2008, 92,

192 906.

[141] M. Cazayous, Y. Gallais, A. Sacuto, R. de Sousa, D. Lebeugle, D. Colson,

Phys. Rev. Lett. 2008, 101, 037 601.

[142] S. A. T. Redfern, C. Wang, J. W. Hong, G. Catalan, J. F. Scott, J. Phys.: Cond.

Mat. 2008, 20, 452205.

[143] M. K. Singh, W. Prelier, M. P. Singh, R. S. Katiyar, J. F. Scott, Phys. Rev. B

2008, 77, 144403.

[144] R. M. White, R. J. Nemanich, Conyers. Herring, Phys. Rev. B 1982, 25,

1822. S. Venugopalan, M. Dutta, A. K. Ramdas, J. P. Remeika, Phys. Rev. B

1985, 31, 1490.

[145] N. Koshizuka, S. Ushioda, Phys. Rev. B 1980, 22, 5394.

[146] C. Ederer, N. A. Spaldin, Phys. Rev. B 2005, 71, 060401R.

[147] A. Bombik, B. Lesniewska, J. Mayer, A. W. Pacyna, J. of Mag. Mag. Mat.

2003, 257, 206.

[148] T. Yamaguchi, J. Phys. Chem. Solids. 1974, 35, 479.

[149] M. K. Singh, R. S. Katiyar, W. Prelier, J. F. Scott, J. Phys.: Condens. Matter

2009, 21, 042202.

[150] A. K. Pradhan, Kai Zhang, D. Hunter, J. B. Dadson, G. B. Loiutts,

P. Bhattacharya, R. Katiyar, J. Zhang, D. J. Sellmyer J. Appl. Phys. 2005,

97, 093903.

[151] S. Nakamura, S. Soeya, N. Ikeda, M. Tanaka, J. Appl. Phys. 1993, 74, 5652.

[152] S. Kirkpatrick, D. Sherrington, Phys. Rev. B 1978, 17, 4384.

[153] K. H. Fischer, J. Hertz, Spin Glasses, Cambridge Univ. Press, Cambridge

1991.

[154] K. De, M. Thakur, A. Manna, S. Giri, J. Appl. Phys. 2006, 99, 013 908.

[155] V. V. Shvartsman, R. Haumont, W. Kleemann, unpublished.

[156] J. R. L. de Almeida, D. J. Thouless, J. Phys. A 1978, 11, 983.

[157] S. T. Zhang, M. H. Lu, D. Wu, Y. F. Chen, N. B.Ming, Appl. Phys. Lett. 2005,

87, 262907.

[158] H. Naganuma, S. Okamura, J. Appl. Phys. 2007, 101, 09103.

[159] C. Tabares-Munoz, J.-P. Rivera, A. Bezinges, A. Monnier, H. Schmid, Jpn. J.

Appl. Phys. 1985, 24–2, 1051.

[160] I. H. Ismailzade, R. M. Ismailov, A. I. Alekperov, F. M. Salaev, Phys. Stat.

Sol. A 1980, 57, 99.

[161] a) Y. F. Popov, A. K. Zvezdin, G. P. Vorobev, A. M. Kadomtseva, V. A.

Murashev, D. N. Rakov, JETP Lett. 1993, 57, 69. b) A. K. Zvezdin, A. M.

Kadomtseva, S. S. Krotov, A. P. Pyatakov, Y. F. Popov, G. P. Vorob’ev,

J. Magn. Magn. Mater. 2006, 300, 224.

[162] A. M. Kadomtseva, A. K. Zvezdin, Y. F. Popov, A. P. Pyatakov, G. P.

Vorob’ev, JETP Lett. 2004, 79, 571.


REVIE

W

www.advmat.de

[163] I. Sosnowska, A. K. Zvezdin, J. Magn. Magn. Mater. 1995, 140, 167.

[164] N. Wang, J. Cheng, A. Pyatakov, A. K. Zvezdin, J. F. Li, L. E. Cross, D.

Viehland, Phys. Rev. B 2005, 72, 104434.

[165] F. Bai, J. Wang, M. Wuttig, J. F. Li, N. Wang, A. P. Pyatakov, A. K. Zvezdin,

L. E. Cross, D. Viehland, Appl. Phys. Lett. 2005, 86, 32511.

[166] T. Zhao, A. Scholl, F. Zavaliche, K. Lee, M. Barry, A. Doran, M. P. Cruz,

Y. H. Chu, C. Ederer, N. A. Spaldin, R. R. Das, D. M. Kim, S. H. Baek, C. B.

Eom, R. Ramesh, Nat. Mater. 2006, 5, 823.

[167] S. K. Streiffer, C. B. Parker, A. E. Romanov, M. J. Lefevre, L. Zhao, J. S.

Speck, W. Pompe, C. M. Foster, G. R. Bai, J. Appl. Phys. 1998, 83, 2742.

[168] L. D. Landau, E. M. Lifshitz, Phys. J. Sowjetunion 1935, 8, 153.

[169] C. Kittel, Phys. Rev. 1946, 70, 965.

[170] T. Mitsui, J. Furuichi, Phys. Rev. 1953, 90, 193.

[171] A. L. Roytburd, Phys. Stat. Sol. A 1976, 37, 329.

[172] M. Daraktchiev, G. Catalan, J. F. Scott, Ferroelectrics 2008, 375, 122.

[173] G. Catalan, A. Schilling, J. F. Scott, J. M. Gregg, J. Phys.: Cond. Mat. 2007,

19, 132201.

[174] Y. B. Chen, M. B. Katz, X. Q. Pan, R. R. Das, D. M. Kim, S. H. Baek, C. B.

Eom, Appl. Phys. Lett. 2007, 90, 072907.

[175] G. Catalan, H. Bea, S. Fusil, M. Bibes, P. Paruch, A. Barthelemy, J. F. Scott,

Phys. Rev. Lett. 2008, 100, 027602.

[176] J. Seidel, L. W. Martin, Q. He, Q. Zhan, Y.-H. Chu, A. Rother, M. E.

Hawkridge, P. Maksymovych, P. Yu, M. Gajek, N. Balke, S. V. Kalinin, S.

Gemming, H. Lichte, F. Wang, G. Catalan, J. F. Scott, N. A. Spaldin, J.

Orenstein, R. Ramesh, Nat. Mater. 2009, 8, 229.

[177] J. Privratska, V. Janovec, Ferroelectrics 1997, 204, 321.

[178] A. Goltsev, R. Pisarev, T. Lottermoser, M. Fiebig, Phys. Rev. Lett. 2003, 90,

177204.

[179] T. Lottermoser, M. Fiebig, Phys. Rev. B 2004, 70, 220407R.

[180] T.-J. Park, G. C. Papaefthymiou, A. J. Viescas, A. R. Moodenbaugh, S. S.

Wong, Nano Lett. 2006, 7, 766.

[181] F. Gao, Y. Yuan, K. F. Wang, X. Y. Chen, F. Chen, J.-M. Liu, Z. F. Ren, Appl.

Phys. Lett. 2006, 89, 102506.

[182] a) J. T. Richardson, W. O. Milligan, Phys. Rev. 1956, 102, 1289. b) J. T.

Richardson, D. I. Yiagas, B. Turk, K. Forster, M. V. Twigg, J. Appl. Phys.

1991, 70, 6977.

[183] T. J. Park, Y. B. Mao, S. S. Wong, Chem. Commun. 2004, 2708.

[184] X. Y. Zhang, C. W. Lai, X. Zhao, D. Y. Wang, J. Y. Dai, Appl. Phys. Lett. 2005,

87, 143102.

[185] M. D. Glinchuk, E. A. Eliseev, A. N. Morozovska, R. Blinc, Phys. Rev. B

2008, 77, 024106.


[186] Fujitsu website: http://www.fujitsu.com/my/news/pr/fmal20060808.

html (accessed January 2009).

[187] H. W. Jang, S. H. Baek, D. Ortiz, C. M. Folkman, C. B. Eom, Y. H. Chu, P.

Shafer, R. Ramesh, V. Vaithyanathan, D. G. Schlom, Appl. Phys. Lett. 2008,

92, 062910.

[188] D. I. Woodward, I. M. Reaney, R. E. Eitel, C. A. Randall, J. Appl. Phys. 2003,

94, 3313.

[189] a) B. Noheda, D. E. Cox, G. Shirane, J. A. Gonzalo, L. E. Cross, S. E. Park,

Appl. Phys. Lett. 1999, 74, 2059. b) R. Guo, L. E. Cross, S.-E. Park, B.

Noheda, D. E. Cox, G. Shirane, Phys. Rev. Lett. 2000, 84, 5423.

[190] S. Fujino, M. Murakami, V. Anbusathaiah, S.-H. Lim, V. Nagarajan, C. J.

Fennie, M. Wuttig, L. Salamanca-Riba, I. Takeuchi, Appl. Phys. Lett. 2008,

92, 202904.

[191] K. Takahashi, N. Kida, M. Tonouchi, Phys. Rev. Lett. 2006, 96, 117402.

[192] V. Ryzhii, J. Phys.: Condens. Matter 2008, 20, 380301.

[193] H. Bea, M. Gajek, M. Bibes, A. Barthelemy, J. Phys.: Cond. Mat. 2008, 20,

434231.

[194] J. Nogues, I. K. Schuller, J. Magn. Magn. Mat. 1999, 192, 203.

[195] P. Borisov, A. Hochstrat, X. Chen, W. Kleemann, C. H. Binek, Phys. Rev.

Lett. 2005, 94, 117203.

[196] V. Laukhin, V. Skumryev, X. Martı, D. Hrabovsky, F. Sanchez, M. V.

Garcıa-Cuenca, C. Ferrater, M. Varela, U. Luders, J. F. Bobo, J. Fontcuberta,

Phys. Rev. Lett. 2006, 97, 227201.

[197] J. Dho, X. Qi, H. Kim, J. L. MacManus-Driscoll, M. G. Blamire, Adv. Mater.

2006, 18, 1445.

[198] H. Bea, M. Bibes, S. Cherifi, F. Nolting, B. Warot-Fonrose, S. Fusil, G.

Herranz, C. Deranlot, E. Jacquet, K. Bouzehouane, A. Barthelemy, Appl.

Phys. Lett. 2006, 89, 242114.

[199] H. Bea, M. Bibes, F. Ott, B. Dupe, X.-H. Zhu, S. Petit, S. Fusil, C. Deranlot,

K. Bouzehouane, A. Barthelemy, Phys. Rev. Lett. 2008, 100, 017204.

[200] Y.-H. Chu, L. W. Martin, M. B. Holcomb, M. Gajek, S.-J. Han, Q. He, N.

Balke, C.-H. Yang, D. Lee, W. Hu, Q. Zhan, P.-L. Yang, A. Fraile-Rodrıguez,

A. Scholl, S. X. Wang, R. Ramesh, Nat. Mater. 2008, 7, 478.

[201] H. Bea, S. Fusil, K. Bouzehouane, M. Bibes, M. Sirena, G. Herranz, E.

Jacquet, J.-P. Contour, A. Barthelemy, Jpn. J. Appl. Phys. 2006, L187, 45.

[202] M. Gajek, M. Bibes, S. Fusil, K. Bouzehouane, J. Fontcuberta, A. Barthel-

emy, A. Fert, Nat. Mater. 2007, 6, 296.

[203] M. Bibes, A. Barthelemy, Nat. Mater. 2008, 7, 425.

[204] N. A. Halasa, G. DePasquali, H. G. Drickamer, Phys. Rev. B 1974, 10, 154.

[205] T. Choi, S. Lee, Y. J. Choi, V. Kiryukhin, S.-W. Cheong, Sciencexpress,

DOI: 10.1126/science.1168636 (2009).


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