Definitions

# Multiferroics

Multiferroics have been formally defined as materials that exhibit more than one primary ferroic order parameter simultaneously (i.e. in a single phase) . 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 alloys 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 1960's . In the most general sense the field of multiferroics was born from studies of magnetoelectric systems . 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 and the discovery of strong magnetic and electric coupling in orthorhombic TbMnO3 and TbMn2O5re-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. 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.

### 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 in order 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 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 (eg-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). 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, 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 where the A-site drives the displacement and partially filled “d” shell on the B-site contributes to the magnetism. Examples include BiFeO3, BiMnO3, 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.

## 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

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

$M_i=alpha_\left\{ij\right\}sum E_j + sum beta_\left\{ijk\right\}E_jE_k+ldots$ 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, 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.

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

## 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.

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 magneto resistance (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. 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. One can also explore multiple state memory elements, where data are stored both in the electric and the magnetic polarizations.

## Reviews on Multiferroics

J. van den Brink and D. Khomskii, “Multiferroicity due to charge ordering”, Cond-Mat/0803.2964 (2008)

M. Bibes and A. Barthelemy, “Oxide spintronics”, Cond-Mat/0706.3015 (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)

## References

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