Multiferroic and magnetoelectric materials-Developments and perspectives

Multiferroic (MF) materials with simultaneous magnetic and electric long range order and occasionally, mutual magnetoelectric (ME) coupling, have recently attracted considerable interest. The small linear ME effect has been shown to control spintronic devices very efficiently, e.g. via the classic ME antiferromagnet Cr2O3 using exchange bias. Similar nano-engineering concepts exist also for type-I MF single phase materials, whose magnetic and polar orders have distinct origins like BiFeO3. Strong ME coupling occurs in type-II multiferroics, where ferroelectricity is due to spiral spin order as in TbMnO3. Record high ME response coming close to applicability arises in stress-strain coupled multiphase magnetoelectrics such as PZT/FeBSiC composites. Higher order ME response in disordered systems (“type-III multiferroics”) extends the conventional MF scenario toward ME quantum paraelectric and multiglass materials with polarization-induced control of magnetic exchange, as e.g. in EuTiO3, Sr0.98Mn0.02TiO3, and PbFe0.5Nb0.5O3.


Introduction
A decade ago, Nicola A. Hill posed the provocative question 'Why are there so few magnetic ferroelectrics?' [1].No doubt, she knew the answer, at least for the oxidic perovskites with the chemical formula ABO 3 , where magnetism becomes established via transition metal ions such as Ni 2+ , Fe 3+ , Mn 4+ .They have partially filled d shells.while practically all ferroelectric (FE) perovskites contain transition metal ions with empty d shells, such as Ti 4+ , Ta 5+ , W 6+ .They favor offcentrality due to their ability to form covalent bonds with neigboring oxygen ions.This process is strongly suppressed by real d electrons, which strongly discourage multiferroicity, i.e. the coexistence of magnetic and electric long-range order [2].Nevertheless, many research groups became involved studying the rare situation of coexisting order parameters and their coupling.
In particular, the magnetoelectric (ME) effect, viz. the cross coupling of the order parameters, magnetization M and polarization P, to their conjugate fields, E and H, enjoyed a breathtaking a e-mail : wolfgang.kleemann@uni-due.derevival [3].Clearly, upcoming visions were challenging and promising, e. g., switching magnetism with bare electric fields and thus getting rid of overheating in microelectronic devices [4].Today we encounter a rich variety of multiferroics and magnetoelectrics.More than 400 papers have been published in 2010 in both of these fields, many of them being mutually linked.An updated world of electrically and magnetically polarizable materials is depicted in Fig. 1.Its still growing complexity will be subject to this brief overview.

Fig 1.
The world of electrically and magnetically polarizable materials including bare ferroics, multiferroics, linear magnetoelectrics [5], and dipole-, spin-, and nonlinear ME multi-glasses [6,33] 2 Magnetoelectric effect The linear ME effect was first verified on the rhombohedral antiferromagnet Cr 2 O 3 [7] and theoretically explored by Landau and Lifshitz [8].They found that quite stringent symmetry properties must be fulfilled.Time and spatial inversion symmetry, T and I, respectively, have to be broken, while the combined symmetry operation, TI, must be valid.In this case the free energy density F of the system contains a contribution W = -αH⋅ ⋅ ⋅ ⋅E, which is bilinearly coupled to H and E via the linear ME susceptibility tensor α.In the axial system Cr 2 O 3 this term enables the formation of single antiferromagnetic (AF) domains by so-called 'ME cooling' to below the AF ordering temperature, T N = 308 K, in simultaneously applied parallel and antiparallel magnetic and electric fields, respectively [9] If a system with polar and magnetic properties does not fulfill the above symmetry conditions, it may still be a candidate for higher order ME effects.They emerge systematically from a series expansion of the free energy under Einstein summation [10], Apart from the field-induced terms coupled to bilinear functions E 2 , H 2   [11] via the electric field-induced components of the magnetization It involves external ac and dc electric and magnetic fields, E = E ac cosωt + E dc and H dc , and records the complex first harmonic ac magnetic moment, . Under well-defined protocols involving appropriate field amplitudes and directions along the crystal coordinates, the full variety of susceptibility tensor components can be determined.In case of a polycrystalline sample 00046-p.2

EMM-FM2011
with volume V the response, m', allows determining orientation averaged coupling parameters α, β and δ, m′ = (αE ac + βE ac H dc + γE ac E dc + 2δE ac E dc H dc )(V/µ o ) (3) The linear ME effect is usually very small.E. g., the peak value of the primordial ME material χ ≈ 100 due to strong FM next-nearest neighbor interaction [13].Fig. 2 shows the ME moment m′ of a polycrystalline sample of EuTiO 3 excited at T = 4.5 K with E ac = 8 kV/m under 'ME annealing' [9] in constant E dc = 80 kV/m and descending µ 0 H dc ≤ 1.5 T. As µ 0 H dc → 0, linear behavior with negative slope, δ eff = -2.1×10 -2 sm/VA, indicates a large third-order δ-effect, which is ≈ 200× larger than that of the first explored example of 3 rd order ME coupling, Sr 0.98 Mn 0.02 TiO 3 [6] (see below).
Closer inspection shows [14] that δ eff contains a contributions due to a second-order β-effect, which becomes allowed due to the formation of net electric polarization upon ME annealing.[14] Most surprisingly, however, the initial ME response suddenly changes sign at 0.6 T and develops a sharp peak with 'giant' m′ ≈ 3×10 -9 Am 2 at µ 0 H c = 0.68 T. At this critical field the system undergoes a phase transition from an AF spin-flop to a (saturated) paramagnetic phase.The ME response is taking advantage of the critical fluctuations of the (AF ordered) transverse magnetization components, ±S x , and thus fulfills the prediction [12] in an impressive way.Very probably the peak is due to electric field-induced Dzyaloshinskii-Moriya exchange interaction, which gives rise to near-divergent non-diagonal 3 rd order ME response as ±S x → 0 [14].

00046-p.3 3 Multiferroics
Multiferroics (MFs) are classified either single or multiphase, if the order parameters involved occur in one single compound or in different components of a composite material [15].Since recently [16] one further distinguishes type-I and type-II single phase MFs.Type Quite often type-I MFs reveal high ordering temperatures, but their theory -including the ME coupling -can be very complex.Probably the most popular type-I single phase MF is BiFeO 3 with record high ordering temperatures, AF T N = 643K and FE T c = 1103 K. Despite its large variety of different FE-ferroelastic and AF domains it has ever since been considered a hot favorite for applications in sensorics or spintronics [17,18].
On the other hand, the theory of type-II MFs is symmetry based and straightforward, albeit often quite sophisticated.In most cases the ordering temperatures are very low and the order parameter amplitudes ridiculously small (from an application point of view).E. g., in the orthorhombic perovskite system TbMnO 3 it was found that spiral spin ordering due to Dzyaloshinskii-Moriya exchange interaction breaks both T and I, and net polarization P = γ γ γ γ (r j -r j+1 )×(S j ×S j+1 ) becomes induced as depicted in Fig.
3 [19].Also the sinusoidal modulation of the uniaxial Etype spin structure in orthorhombic HoMnO 3 [20] was shown to produce a sizable improper ferroelectric polarization.Characteristically for type-II MFs, it is due to an electronic mechanism apart from the lattice one [21].Further, in the Ising chain magnetCa 3 CoMnO 6 [22] alternating Co 2+ /Mn 4+ ionic order creates competing nearest neighbor FM and next-nearest neighbor AF exchange interactions.As a consequence, up-up-down-down ('ANNNI-type') spin ordering arises below T N ≈ 16 K.It is accompanied by asymmetric exchange striction, which breaks I and, hence, induces electric polarization below T N [23].Composite MFs are usuallly based on stress-strain coupling between the order parameters of FEpiezoelectric and FM-magnetostrictive components like BaTiO 3 and CoFe 2 O 4 , respectively [24].This pioneering self-assembled ceramic material became famous for its large ME response, α ME (BaTiO 3 /CoFe 2 O 4 ) = dE/dH = 130 mV/cm•Oe, which exceeds that of Cr 2 O 3 [7] and even that of the record holding single phase type-I MF material TbPO 4 [25] by factors of about 180 and 20, respectively.
Meanwhile even higher conversion rates are available, and ME composites are now considered for applications in transducer, filter and sensor devices [26].Record high ME response can be achieved by taking advantage of resonance effects.One possible design is shown in Fig. 4, where amorphous FM METGLAS (= FeBSiC) layers are excited by a longitudinal magnetic ac field and laterally coupled to a periodically poled FE PZT [= Pb(Zr,Ti)O 3 ] piezofiber layer.The conversion factor α ME = 0.8 kV/cm⋅Oe at the resonance frequency f ≈ 2 kHz [27] exceeds that of archetypical

Multiglasses
The nature of glassy states in disordered materials has long been controversially discussed.In the magnetic community generic spin glasses [28] are meanwhile accepted to undergo phase transitions at a static freezing temperature T g (= glass temperature), where they exhibit criticality and originate well-defined order parameters.Widely accepted, albeit still under debate [29], also polar systems may undergo transitions into generic 'dipolar or orientational glass' states [30], which fulfill similar criteria as spin glasses.Hence, it appears quite natural to introduce the term 'multiglass' for a new kind of MF material revealing both polar and spin glass properties as discovered in ceramic solid solutions of Sr 0.98 Mn 0.02 TiO 3 [6].On one hand, the Mn 2+ ions being randomly distributed and offcentered from their Sr 2+ -sites [31] form nanopolar clusters with frustrated dipolar interaction and give rise to a dipolar glass state below e g T ≈ 38 K. On the other hand, frustrated and random Mn 2+ - O 2--Mn 2+ superexchange is at the origin of spin glass formation below T g m ≈ 34 K.It should be noticed that both glassy states have unanimously been confirmed by clear-cut aging and rejuvenation effects in their respective dc susceptibilities [6].Observation of biquadratic (δ-type) ME interactionsee.Eq. (1) -is fully compatible with the low symmetry of the compound and supposed to crucially reinforce the spin glass 'ordering' as schematically depicted in Fig. 5 [32].(pseudospins σ j ,σ j ′,σ j ") and superantiferromagnetic spin clusters (S j ,S j ′,S j ") [32] In the MF perovskite PbFe 0.5 Nb 0.5 O 3 (PFN), with Fe 3+ and Nb 5+ ions randomly distributed at B sites, two different orderings are about to establish -a soft-mode driven FE one as in PbTiO 3 , and a super-exchange driven AF one in the percolating Fe 3+ subspace [33].Owing to the inherent disorder, however, unconventional phases emerge.The polar phase refers to a so-called relaxor ferroelectric below T c e ≈385 K.It results from quenched random electric fields due to the cationic charge disorder and decays into a polar domain state as known from the related prototype compound PbMg 1/3 Nb 2/3 O 3 (PMN) [34].Even more unusual is the coexistence of two magnetic phases both of which fulfill the requirements of the thermodynamic limit.'Infinitely' large numbers of finite-sized Fe 3+ clusters without mutual overlap make up a spin cluster glass (CG) coexisting with an AF phase of exchange coupled Fe 3+ ions.The phase coexistence is compatible with percolation theory.While the AF phase transition at T N ≈ 153 K is permitted on the bond-percolated infinite cluster of super-exchange coupled Fe 3+ spins, the CG transition at T g ≈ 10 K (Fig. 6, inset) is restricted to the complementary space accommodating isolated and small clusters of Fe 3+ ions, where dipolar and super-exchange interaction via oxygen and lead ions warrant glassy bond coherence.On cooling toward the glass transition a finite number of large, but non-percolating spin clusters is undergoing superantiferromagnetic (SAF) blocking as experienced by extra susceptibility response in both magnetization and second-order ME response (E ac = 12.5 kV/m and µ 0 H dc = 0.2 T; β-effect according to Eq. 3) shown in Fig. 6.Single phase multiferroics open possibilities of double action involving two order parameters.E. g., in the 4-bit memory of Gajek et al. [36] a thin film of the MF ferro-electromagnet La 0.1 Bi 0.9 MnO 3 is proposed to serve as a tunnelling layer in a magnetoresistance element showing four different tunnel magneto-and electro resistances (TMR and TER, respectively) when setting the various magnetic and electric states, ±M and ±P.Interestingly, in this case vanishing linear ME coupling between the two order parameters is explicitly welcome.The need of low-temperatures has hitherto impeded any application of this smart idea.

Conclusion
Presently still the only room temperature type-I MF material BiFeO 3 appears trailblazing for future spintronics applications, while the type-II multiferroics are more challenging from a fundamental point of view because of their fascinating interplay between the different orders.New challenges for theory are brought by ME multiglasses and nonlinear ME effects, which are not as small as hitherto presumed.Among the existing device ideas probably the most promising belong to the field of spintronics such as low current MERAM [35] and MF 2x2 logic cells [36], where the pioneering material chromia, Cr 2 O 3 , is a promising candidate toward novel applications [35,37].

Cr 2 O 3 ,
α zz (T ≈ 260 K) ≈ 4 ps/m[7], denotes an average spin-flip rate of merely ≈ 5 × 10 -7 spins/(kV/cm)[9].Much larger effects are expected in the vicinity of the ferroic phase transitions, where suitable components of the χ χ no really existing material even roughly fulfills the condition of two simultaneous ferroic transition.Recently we have proposed a 'second-best' choice for achieving 'giant' ME response, namely the fluctuation regime (large e ii χ ) of a quantum paraelectric material coming close to a FM instability (large m jj χ ).This applies to EuTiO 3 , which is a G-type AF below T N = 5.4 K, where e ii χ ≈ 400 and m jj

Fig. 2 :
Fig.2: ME response of polycrystalline EuTiO 3 at T = 4.5 K under external fields E ac , E dc , and µ 0 H dc (on decreasing from 1.5 T) as indicated.The initial and final slopes, δ and δ+β, respectively, and the critical field ±µ 0 H c of the AF-to-PM phase transitions are marked[14]

Fig. 7 .
Fig. 7. Schematic view of a MERAM cell based on ME Cr 2 O 3 (0001) controlling the magnetization of the Pt/Co/Pt trilayer FM1 via voltages ±V 0 and constant magnetic stray field H 0 of a NdFeB thick film FM2.R ± is the corresponding giant (or tunneling) magnetoresistance along the trilayer FM1/NM[nonmagnetic Cu or MgO]/FM2 [34].