Multiferroics

Multiferroics have been formally defined as materials that exhibit more than one primary ferroic order parameter simultaneously (i.e. in a single phase).^[1] The four basic primary ferroic order parameters are ferromagnetism, ferroelectricity, ferroelasticity and ferrotoroidicity, the latter still being under debate. However, many researchers in the field consider materials as multiferroics only if they exhibit coupling between the order parameters. On the other hand, the definition of multiferroics can be expanded as to include non-primary order parameters, such as antiferromagnetism or ferrimagnetism.

Typical multiferroics belong to the group of the perovskite transition metal oxides, and include rare-earth manganites and -ferrites (e.g. TbMnO₃, HoMn₂O₅, LuFe₂O₄). Other examples are the bismuth compounds BiFeO₃ and BiMnO₃, and non-oxides such as BaNiF₄ and spinel chalcogenides, e.g. ZnCr₂Se₄. These alloys show rich phase diagrams combining different ferroic orders in separate phases. Apart from single phase multiferroics, composites and heterostructures exhibiting more than one ferroic order parameter are studied extensively. Some examples include magnetic thin films on piezoelectric PMN-PT substrates and Metglass/PVDF/Metglass trilayer structures. Besides scientific interest in their physical properties, multiferroics have potential for applications as actuators, switches, magnetic field sensors or new types of electronic memory devices.

History

The term multiferroic was first used by H. Schmid in 1994. His definition referred to multiferroics as single phase materials which simultaneously possess two or more primary ferroic properties. Today the term multiferroic has been expanded to include materials which exhibit any type of long range magnetic ordering, spontaneous electric polarization, and/or ferroelasticity. Working under this expanded definition the history of magnetoelectric multiferroics can be traced back to the 1960s.^[2] In the most general sense the field of multiferroics was born from studies of magnetoelectric systems.^[3] After an initial burst of interest, research remained static until early 2000 (see figure). In 2003 the discovery of large ferroelectric polarization in epitaxially grown thin films of BiFeO₃^[4] and the discovery of strong magnetic and electric coupling in orthorhombic TbMnO₃^[5] and TbMn₂O₅^[6] re-stimulated activity in the field of multiferroics.

Symmetry

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

	Space Invariant	Space Variant
Time Invariant	Ferroelastic	Ferroelectric
Time Variant	Ferromagnetic	Ferrotoroidic

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

In general, a variety of mechanisms can cause lowering of symmetry resulting in multiferroicity as described below.

Types of multiferroics

Charge ordered

Mechanism:

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

Examples:

One prominent example for a charge ordered multiferroic is LuFe₂O₄, which shows improper ferroelectricity below 330 K.^[7] The arrangements of the electrons arise from the charge frustration on a triangular lattice with the mixed valence state of Fe²⁺ and Fe³⁺ ions. Ferrimagnetic behavior occurs below 240 K.

In addition, charge ordered ferroelectricity is suggested in Fe₃O₄ and (Pr,Ca)MnO₃.^[8]

Geometrically frustrated multiferroics

Geometric frustrated multiferroicity is related to a structural phase transition at high temperature. Several compounds belong to this important class of multiferroics: K₂SeO₄, Cs₂CdI₄, hexagonal RMnO₃. These systems are proto-typical multiferroics which can be understood by competition between local interactions on several ion sites. For example, in hexagonal manganites h-RMnO₃ (R=Ho-Lu, Y), the ferroelectric polarization at high temperature is correlated to lattice distortions through off-centering of ions. Geometric frustration gives rise to novel spin arrangements at low temperature: The spins order in a variety of non-collinear, e.g. (in-plane) triangular or Kagomé structures to relieve the geometric frustration. The coexistence of ferroelectric and magnetic order occurs together with a strong coupling between two disparate order parameters.

The mechanism of the ferroelectric ordering in hexagonal RMnO₃ is still questionable in the scientific community and must be understood before a comprehensive picture of multiferroic phenomena in spin frustrated systems can be built. It is still matter of debate whether the geometric distortion is the origin of the electric polarization or whether the off-centering of Mn ions also contributes to the polarization.

Physical properties of geometric multiferroics are dominated by the behavior of the d-shell electrons (e.g.-orbitals) and of the rare earth elements with an unfilled f-shell. Hexagonal manganites show the largest deviation from perovskite structure due to the small size of rare-earth ion. Although geometrically frustrated multiferroics exhibit a simple chemistry, they provide a unique set of physical properties, such as rich phase diagrams or multiple frustrations. The strong coupling between ferroelectric and magnetic orders is represented by an anomaly in the static dielectric constant at magnetic phase transitions. Geometric frustrated ferroelectrics are prime candidates for device memory applications.

Magnetically driven ferroelectricity

Magnetically driven multiferroics are insulating materials, mostly oxides, in which macroscopic electric polarization is induced by magnetic long-range order. A necessary but not sufficient condition for the appearance of spontaneous electric polarization is the absence of inversion symmetry. In these materials inversion symmetry is broken by magnetic ordering. Such a symmetry breaking often occurs in so-called frustrated magnets, where competing interactions between spins favor unconventional magnetic orders. The microscopic mechanisms of magnetically induced ferroelectricity involve the polarization of electronic orbitals and relative displacement of ions in response to magnetic ordering.

Many multiferroics show the cycloidal spiral ordering, in which spins rotate around an axis perpendicular to the propagation vector of the spiral. The induced electric polarization is orthogonal to the propagation vector and lies in the spiral plane. An abrupt change of the spiral plane induced by magnetic field results in the corresponding rotation of the polarization vector. In DyMnO3 this transition is accompanied by the 600% increase of dielectric constant (the giant magnetocapacitance effect^[9]). The microscopic mechanism of magnetoelectric coupling in spiral multiferroics involves spin-orbit coupling.

E-type Antiferromagnet (I.e. ortho-HoMnO₃): In the presence of strong uniaxial anisotropy, as in the ANNNI model,^[10] competing interaction can stabilize a *periodic collinear spin arrangement of the up-up-down-down* type. Such a spin modulation commensurate with the structural or charge modulation can induce electric polarization via exchange striction mechanism that does not require spin-orbit coupling.

Lone pair multiferroics

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

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

Strain induced multiferroics

Thin film strategy also enables achievement of interfacial multiferroic coupling through a mechanical channel in heterostructures consisting of a magnetoelastic and a piezoelectric component. This type of heterostructure is composed of an epitaxial magnetoelastic thin film grown on a piezoelectric substrate. (Ref 1, 2, 3)For this system, application of a magnetic field will induce a change in the dimension of the magnetoelastic film. This process, called magnetostriction, will alter residual strain conditions in the magnetoelastic film, which can be transferred through the interface to the piezoelectric substrate. Consequently a polarization is introduced in the substrate through the piezoelectric process. (Ref 1, 2, 3) The overall effect is that the polarization of the ferroelectric substrate is manipulated by an application of a magnetic field, which is the desired magnetoelectric effect. In this case, the interface plays an important role in mediating the responses from one component to another, realizing the magnetoelectric coupling. For an efficient coupling, a high-quality interface with optimal strain state is desired. In light of this interest, advanced deposition techniques have been applied to synthesize these types of thin film heterostructures. Recently molecular beam epitaxy has been demonstrated to be capable of depositing structures consisting of piezoelectric and magnetostrictive components. Materials systems studied included cobalt ferrite, magnetite, SrTiO3, BaTiO3, PMNT (ref 4).

1.R. V. Chopdekar and Y. Suzuki, "Magnetoelectric coupling in epitaxial CoFe2O4 on BaTiO3," Applied Physics Letters, vol. 89, Oct 2006.

2.H. F. Tian, T. L. Qu, L. B. Luo, J. J. Yang, S. M. Guo, H. Y. Zhang, Y. G. Zhao, and J. Q. Li, "Strain induced magnetoelectric coupling between magnetite and BaTiO3," Applied Physics Letters, vol. 92, Feb 2008

3.C. Thiele, K. Dorr, O. Bilani, J. Rodel, and L. Schultz, "Influence of strain on the magnetization and magnetoelectric effect in La(0.7)A(0.3)MnO(3)/PMN-PT(001) (A=Sr,Ca)," Physical Review B, vol. 75, Feb 2007

4. S. Xie, J. Cheng, B. W. Wessels, and V. P. Dravid, "Interfacial structure and chemistry of epitaxial CoFe2O4 thin films on SrTiO3 and MgO substrates," Applied Physics Letters, vol. 93, pp. 181901-181903, Nov 2008.

Magnetoelectric effect

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

$P_i=\sum \alpha_{ij}H_j + \sum \beta_{ijk}H_jH_k+\ldots$

$M_i=\sum \alpha_{ij}E_j + \sum \beta_{ijk}E_jE_k+\ldots$

where P is the electric polarization, M the magnetization, E and H the electric and magnetic field, and α and β are the linear and nonlinear ME susceptibilities. The effect can be observed in single phase and composite materials. Some examples of single phase magnetoelectrics are Cr₂O₃,^[14] and multiferroic materials which show a coupling between the magnetic and electric order parameters. Composite magnetoelectrics are combinations of magnetostrictive and electrostrictive materials, such as ferromagnetic and piezoelectric materials. The size of the effect depends on the microscopic mechanism. In single phase magnetoelectrics the effect can be due to the coupling of magnetic and electric orders as observed in some multiferroics. In composite materials the effect originates from interface coupling effects, such as strain. Some of the promising applications of the ME effect are sensitive detection of magnetic fields, advanced logic devices and tunable microwave filters.^[14]

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

Flexomagnetoelectric effect

Magnetically driven ferroelectricity is also caused by inhomogeneous ^[15] magnetoelectric interaction. This effect appears due to the coupling between inhomogeneous order parameters. It was also called as flexomagnetoelectric effect.^[16] Usually it is describing using the Lifshitz invariant (i.e. single-constant coupling term).^[17] It was shown that in general case of cubic hexoctahedral crystal the four phenomenological constants approach is correct.^[18] The flexomagnetoelectric effect appears in spiral multiferroics ^[19] or micromagnetic structures like domain walls^[20] and magnetic vortexes.^[21] Ferroelectricity developed from micromagnetic structure can appear in any magnetic material even in centrosymmetric one.^[22] Building of symmetry classification of domain walls leads to determination of the type of electric polarization rotation in volume of any magnetic domain wall. Existing symmetry classification ^[23] of magnetic domain walls was applied for predictions of electric polarization spatial distribution in their volumes.^[24][25] The predictions for almost all symmetry groups conform with phenomenology in which inhomogeneous magnetization couples with homogeneous polarization. The total synergy between symmetry and phenomenology theory appears if energy terms with electrical polarization spatial derivatives are taking into account.^[26]

Applications

Multiferroic composite structures in bulk form are explored for high-sensitivity ac magnetic field sensors and electrically tunable microwave devices such as filters, oscillators and phase shifters (in which the ferri-, ferro- or antiferro-magnetic resonance is tuned electrically instead of magnetically.^[14]

In multiferroic thin films, the coupled magnetic and ferroelectric order parameters can be exploited for developing magnetoelectronic devices. These include novel spintronic devices such as tunnel magnetoresistance (TMR) sensors and spin valves with electric field tunable functions. A typical TMR device consists of two layers of ferromagnetic materials separated by a thin tunnel barrier (~2 nm) made of a multiferroic thin film.^[27] In such a device, spin transport across the barrier can be electrically tuned. In another configuration, a multiferroic layer can be used as the exchange bias pinning layer. If the antiferromagnetic spin orientations in the multiferroic pinning layer can be electrically tuned, then magnetoresistance of the device can be controlled by the applied electric field.^[28] One can also explore multiple state memory elements, where data are stored both in the electric and the magnetic polarizations.

See also

• Ferroics
• Ferromagnetism
• Ferroelectricity
• Ferroelasticity
• Ferrotoroidicity
• Magnetoelectric effect
• Photon Induced Electric Field Poling

Reviews on Multiferroics

Y. Tokura and S. Seki, "Multiferroics with Spiral Spin Orders", Adv. Mater. 22, 1554 (2010).

K. F. Wang, J. M. Liu, and Z. F. Ren, "Multiferroicity: the coupling between magnetic and polarization orders", Cond-Mat/0908.0662 (2009); Adv. Phys. 58, 321 (2009)

J. van den Brink and D. Khomskii, “Multiferroicity due to charge ordering”, Cond-Mat/0803.2964 (2008); Journal of Physics: Condensed Matter 20, 434217 (2008).

T. Kimura, "Spiral magnets as Magnetoelectrics", Annu. Rev. Mater. Res. 37, 387-413 (2007)

M. Bibes and A. Barthelemy, “Oxide spintronics”, IEEE Trans. Electron. Dev. 54, 1003 (2007).

S.-W. Cheong and M. Mostovoy, “Multiferroics: a magnetic twist for ferroelectricity”, Nature Materials 6, 13 (2007).

R. Ramesh and N. A. Spaldin, “Multiferroics: progress and prospects in thin films”, Nature Materials 6, 21 (2007).

Y. Tokura, “Multiferroics – toward strong coupling between magnetization and polarization in a solid”, J. Mag. Mag. Mat. 310, 1145 (2007)

C. N. R. Rao and C. R. Serrao, “New routes to multiferroics”, J. Mat. Chem. 17, 4931 (2007)

W. Eerenstein, N. D. Mathur, and J. F. Scott, "Multiferroic and magnetoelectric materials“, Nature 442, 759 (2006)

D. I. Khomskii, “Multiferroics: Different ways to combine magnetism and ferroelectricity”, J. Mag. Mag. Mat. 306, 1 (2006)

Y. Tokura, "Multiferroics as Quantum Electromagnets”, Science 312, 1481 (2006)

N. A. Spaldin and M. Fiebig, “The Renaissance of Magnetoelectric Multiferroics”, Science 309, 391 (2005)

M. Fiebig, “Revival of the magnetoelectric effect”, J. Phys. D: Appl. Phys. 38, R123 (2005)

W. Prellier, M. P. Singh, and P. Murugavel, “The single-phase multiferroic oxides: from bulk to thin film”, J. Phys.: Condens. Matter 17, R803 (2005)

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