Department of Physics

Seminar

Magnetoelectric effect

The challenge of coupling magnetism and ferroelectricity

Luka Vidovič

Mentor: prof. dr. Denis Arčon

Ljubljana, december 2009

Abstract

Magnetism and ferroelectricity are widely used in current technology. However, they

tend to be mutually exclusive and interact weakly when coexist. In multiferroic

materials magnetism and ferroelectricity do coexist and their mutual coupling is

described by magnetoelectric effect. Such magnetic ferroelectricity occurs in frustrated

magnets as a result of competing spin interaction. These compounds have great

potential in many areas as tunable multifunctional devices.

1

Kazalo

1 Introduction ........................................................................................................................................................................ 2

2 Basic principles ................................................................................................................................................................. 3

2.1 Ferromagnetism ..................................................................................................................................................... 3

2.1.1 What makes (anti)ferromagnets (anti)ferromagnetic? ........................................................... 5

2.2 Ferroelectrics ........................................................................................................................................................... 6

2.2.1 What makes ferroelectrics ferroelectric? ........................................................................................ 6

2.3 Multiferroism ........................................................................................................................................................... 7

2.3.1 Requirements for magnetoelectric multiferroics ....................................................................... 7

3 Approaches to coexistence .......................................................................................................................................... 9

3.1 Magnetoelectric coupling................................................................................................................................... 9

3.2 Frustrated magnets ............................................................................................................................................ 10

4 Example material – Slovenian contribution .................................................................................................... 13

5 Applications ..................................................................................................................................................................... 16

6 Conclusions ...................................................................................................................................................................... 17

7 References......................................................................................................................................................................... 18

Introduction

2

1 Introduction

Magnetic and ferroelectric materials are present in wide range of modern science and

technology. For example, ferromagnetic materials with switchable magnetization 𝑀 driven by

external magnetic field are indispensable in data-storage industries. On the other hand, the

sensing industry relies heavily on ferroelectric materials with spontaneous polarization 𝑃

reversible upon an external electric field, because most ferroelectrics are ferroelastics or

piezoelectric with spontaneous strain. This allows such materials to be used in application

where elastic energy is converted in electric and vice versa. Additionally, ferroelectric materials

are also used for data storage in random-access memories (FeRAMs).

This paper focuses on a new class of materials known as magnetoelectric multiferroics, which

are simultaneously ferromagnetic and ferroelectric. Such materials have all the potential

applications of ferromagnetic and ferroelectric materials. In addition, a whole new range of

application possibilities emerges if coupling can be achieved. The possibility to control

magnetization and/or polarization by an electric field and/or magnetic field allows additional

degree of freedom in device design. Other applications include multiple state elements and high

performance for spintronics.

So far around 100 compounds that exhibit the magnetoelectric effect have been discovered (Ref.

[6]) and are believed to have favorable properties regarding application. However, the

coexistence of ferroelectricity (electric dipole order) and magnetism (spin order) in single phase

is extremely difficult. Figure 1 shows relationship between multiferroic and magnetoelectric

materials in relation to magnetically/electrically polarizable materials. Ferroelectricity needs

broken spatial-inverse symmetry and invariant-time reverse symmetry. Electric polarization 𝑃

and electric field 𝐸 change their sign upon an inversion of spatial coordinates 𝑟 → −𝑟 but it is

invariant upon time inversion 𝑡 → −𝑡. In contrast magnetization 𝑀 and magnetic field 𝐻 change

their sign upon time reversal and remain invariant upon spatial inversion. Consequently, a

multiferroic system requires simultaneous breaking of both symmetries. As we will show, this is

possible in spin-frustrated systems which always prefer to have spatially inhomogeneous

magnetization. Second, the efficient coupling between two orders in multiferroic system is often

weak. Strong coupling represents the basis for multi-control of two orders, but shows to be

elusive. Thus, challenge of realization of strong coupling is center of researches.

Figure 1: The relationship between multiferroic and magnetoelectric materials (reproduced from [5]).

Basic principles

3

2 Basic principles

2.1 Ferromagnetism

Magnetic materials respond differently at applied magnetic field. Transition metal, such as 𝐹𝑒

and 𝐶𝑜 with their outer 3𝑑 orbitals, are typical representatives of ferromagnetic materials. A

ferromagnetic material undergoes phase transition from a high-temperature phase above 𝑇𝑐 that

does not have a macroscopic magnetic moment (paramagnetic phase) to a low-temperature

phase below 𝑇𝑐 (𝑇𝑐 𝐹𝑒 = 1043𝐾) that has spontaneous magnetization even when external

magnetic field is switched off. The macroscopic magnetization is caused by magnetic dipole

moments of the atoms tending to line up in the direction of magnetic field. Ferromagnets tend to

concentrate magnetic flux density, they have spontaneous magnetization, which leads to their

widespread usage in applications such as, transformer cores, permanent magnets, and

electromagnets.

But macroscopic magnetization is not uniform across the whole material. In ferromagnets,

alignment of atomic dipoles is almost complete over regions called domains. Upon appliance of

magnetic field, 𝐻, the subsequent alignment and reorientation takes place and results in

hysteresis of magnetization and flux density, 𝐵. Ferromagnetic ordering of dipoles is shown in

Figure 3b.

The ferromagnet starts in an unmagnetized state and upon increased magnetic field magnetic

induction rises to the saturation induction, 𝐵𝑠 . When the field is reduced to zero, the induction

decreases to 𝐵𝑟 , know as remanent field. The reverse field, 𝐻𝑐 , required to reduce the induction

to zero is called the coercivity. A typical hysteresis loop is shown in Figure 2.

Figure 2: Hysteresis loop for ferromagnets (reproduced from [1]).

The characteristic of hysteresis loop determines the suitability of ferromagnetic materials for

particular application. For example, more square-shaped hysteresis loop, with two stable

magnetization states, is suitable for magnetic data storage, while a small hysteresis loop that is

easily cycled between states is suitable for transformer core.

Basic principles

4

Other types of magnetic ordering are also possible such as antiferromagnetic (Figure 3c) or

ferrimagnetic (Figure 3d). They both develop from paramagnetic phase (Figure 3a) below

characteristic 𝑇𝑁 . In paramagnetic phase magnetic dipoles are randomly aligned. Materials in

this phase are slightly attracted by a magnetic field but the material does not retain the magnetic

properties when the external field is removed. In antiferromagnetic phase the magnetic

moments of atoms or molecules align in a regular pattern with neighboring spins pointing in

opposite directions. Generally, antiferromagnetic order may exist at sufficiently low

temperatures, vanishing at and above a certain temperature known as the Neel temperature 𝑇𝑁 .

Typical aniferromagnets are 𝑀𝑛0 (𝑇𝑁 = 116𝐾), 𝑀𝑛𝑆 (𝑇𝑁 = 160𝐾) and 𝐹𝑒𝑂 (𝑇𝑁 = 198𝐾). An

antiferromagnetic interaction acts to anti-align neighboring spins. Interaction term 𝐽 between

spins defines nature of magnetic ordering. If 𝐽 > 0 the spins are aligned and order is

ferromagnetic, for anti-aligned spins interaction term 𝐽 < 0 and order is antiferromagnetic.

Ferrimagnets are somewhat like antiferromagnets with the anti-aligned spins. However, some of

the dipole moments are larger than others, so that the material has a net overall magnetic

moment. As a result, ferromagnetic materials, like ferromagnets, tend to concentrate magnetic

flux in their interiors.

Figure 3: Ordering of magnetic dipoles in magnetic materials (reproduced from [1]).

Basic principles

5

2.1.1 What makes (anti)ferromagnets (anti)ferromagnetic?

The spin of an electron results in a magnetic dipole moment. A classical analogy of magnetic

dipole is current loop, created by the spinning ball of charge. In materials with a filled electron

shells the total dipole moment of all the electrons is zero (i.e., the spins are in up/down pairs).

Thus, only atoms with partially filled shells (i.e., unpaired spins) can possess some kind of

magnetic ordering.

Above the critical Curie temperature, 𝑇𝑐 , the dipoles are randomly aligned, but tend to align

parallel to an external magnetic field. This phase is known as paramagnetic. Below the Curie

temperature, the ferromagnetic or antiferromagnetic ordering emerges. The magnetic dipoles

tend to (anti)align spontaneously, without any applied field. This purely quantum-mechanical

effect brings us to an exchange interaction.

Consider a model with two electrons in state ψa r 1 and ψb r 2 , respectivily. The joint wave

function must behave properly under the operation of particle exchange and can be either

symmetric or antisymmetric. However, by the Pauli exclusion principle no two fermions can

occupy the same state. Since electrons have spin 1/2, they are fermions. This means that the

overall wave function of a system must be antisymmetric. In case of antisymmetric singlet state

𝜒𝑆 𝑆 = 0 a spatial state of wave function must be symmetric, in case of symmetric triplet state

𝜒𝑇 𝑆 = 1 a spatial state of wave function must be antisymmetric. Therefore, we can write the

wave function for the siglet case 𝜓𝑆 and the triplet case 𝜓𝑇 as

𝜓𝑆 = 1

2 ψa r 1 ψb r 2 + ψa r 2 ψb r 1 χS ,

𝜓𝑇 = 1

2 ψa r 1 ψb r 2 − ψa r 2 ψb r 1 χT .

(1.1)

The energies of the two possible states are

𝐸𝑆 = 𝜓𝑆∗ℋ 𝜓𝑆 ⅆr 1 ⅆr 2 ,

𝐸𝑇 = 𝜓𝑇∗ ℋ 𝜓𝑇 ⅆr 1 ⅆr 2 .

(1.2)

The energy difference 𝐸𝑆 and 𝐸𝑇 enables us to define a spin part of the Hamiltonian as

ℋ 𝑠𝑝𝑖𝑛 = −2𝐽𝑆 1𝑆 2 , (1.3)

This Hamiltonian is called Heisenberg exchange interaction, where

𝐽 =𝐸𝑆 − 𝐸𝑇

2 . (1.4)

If 𝐽 > 0 the triplet state 𝑆 = 1 is favored – i.e. the ferromagnetic ordering takes place. If 𝐽 < 0 the

singlet state 𝑆 = 0 is favored – i.e. system tends to order in antiferromagnetic state.

Basic principles

6

For description of extended lattice the Heisenberg model applies:

ℋ = −2 𝐽𝑖 ,𝑗𝑆 𝑖𝑆 𝑗𝑖>𝑗

(1.5)

Where 𝐽𝑖 ,𝑗 is the exchange constant between the 𝑖𝑡ℎ and 𝑗𝑡ℎ spin.

2.2 Ferroelectrics

Ferroelectric material undergoes a phase transition from a high-temperature phase that behaves

as an ordinary dielectric to a low-temperature phase that has spontaneous polarization. Many

properties of ferroelectric materials are analogous to ferromagnets but with corresponding

electric parameters. Ferroelectric materials also have domains and show a hysteresis response.

The most widely studied and used ferroelectrics are perovskite-structure oxides, 𝐴𝐵𝑂3 , which

have cubic structure shown in Figure 4. A structure is characterized by small a cation, 𝐵, at the

centre of an octahendron of oxygen anions with large cations, 𝐴, at the unit cell corner. Below

the Curie temperature, there is a structural distortion to a lower-symmetry phase accompanied

by the shift off-centre of small cations. The spontaneous polarization derives largely from

electric dipole moment created by this shift.

2.2.1 What makes ferroelectrics ferroelectric?

The existence or absence of ferroelectricity is determined by a balance between short-range

repulsion, which favor the nonferrelectric symmetry structure, and additional bonding

consideration, which might stabilize the ferroelectric phase. In ferroelectric materials, the short-

range repulsions dominate at high temperature, resulting in the symmetric, unpolarized state.

As the temperature is decreased, the unit cell undergoes a series of phase transitions. In low-

temperature phase the 𝐵 3𝑑 − 𝑂 2𝑝 hybridization is responsible for stabilization of the

ferroelectric distortion along preferred diagonal.

Also significant is the observation that 𝐵 cation is formally in a 𝑑0 lowest unoccupied state.

Typical representative compound is 𝐵𝑎𝑇𝑖𝑂3. A simple charge count gives 𝐵𝑎2+𝑇𝑖4+𝑂32−.

Evidently, 𝑇𝑖4+ ion is in a 𝑑0 state.

Basic principles

7

In contrast, 𝑑-orbital occupancy is a requirement for existence of magnetic moments and

consequently of magnetic ordering. This brings us to a conflicting situation: ferroelectric

materials favor 𝑑0 while magnetic order require 𝑑𝑛 , thus excluding coexistence of both orders.

Figure 4: Cubic perovskite structure. The small B cation (in black) is at the center of an octahedron of

oxygen anions (in gray). The large A cations (white) occupy the unit cell corners (reproduced from [1]).

2.3 Multiferroism

The term multiferroism is used to describe materials where ferroelectricity and ferromagnetism

occur in the same phase. This means that they have the spontaneous magnetization controlled

by the applied magnetic field and the spontaneous polarization controlled by applied electric

field. In addition, the ability to control charges by applied magnetic fields and spins by applied

voltage offers an extra degree of freedom. However, this proved to be a difficult problem, as this

order parameters turn out to be mutually exclusive. Furthermore, simultaneous presence does

not guarantee strong coupling, as microscopic mechanisms of ferroelecticity and magnetism are

quite different.

2.3.1 Requirements for magnetoelectric multiferroics

Multiferroic physical, structural and electronic properties are restricted to those that occur both

in ferromagnetic and ferroelectric materials. We will analyze a range of properties and discuss

how they limit choice of potential materials.

Basic principles

8

2.3.1.1 Symmetry

From the point of view of symmetry consideration, ferroelectricity needs the broken spatial

inverse symmetry while the time symmetry can be invariant. A structural distortion from

symmetry structure is needed for spontaneous polarization to emerge. In contrast, ferromagnets

possess broken time symmetry and spatial symmetry is invariant. Thus multiferroics have

neither symmetry. The time-reversal and spatial-reversal symmetry in ferroic materials is

illustrated in Figure 5.

Figure 5: Time-reversal and spatial-inversion symmetry in ferroics. a) Ferromagnets. The local magnetic

moment m may be represented classically by a charge that dynamically traces an orbit, as indicated by the

arrowheads. A spatial inversion produces no change, but time reversal switches the orbit and thus m. b)

Ferroelectrics. The local dipole moment p may be represented by a positive point charge that lies

asymmetrically within a crystallographic unit cell that has no net charge. There is no net time dependence,

but spatial inversion reverses p. c) Multiferroics that are both ferromagnetic and ferroelectric possess

neither symmetry(reproduced from [5]).

2.3.1.2 Electric properties

By definition, a ferroelectric material must be an insulator. Ferromagnets, on the other hand, are

often metals. One could assume that small number o multiferroics is due to lack of magnetic

insulators. However, there are also very few antiferromagnetic ferroelectrics, despite

antiferromagnets are usually insulators.

2.3.1.3 Chemistry: 𝒅𝟎 band

Ferroelectric materials have a formal charge corresponding to the 𝑑0 electron configuration on

the 𝐵 cation. But if there are no 𝑑 electrons creating localized magnetic moments, then there can

be no magnetic ordering of any kind. It appears however, that in most cases, as soon as the 𝑑

shell of cation is partially occupied, the tendency to shift and remove the centre of symmetry.

Approaches to coexistence

9

3 Approaches to coexistence

3.1 Magnetoelectric coupling

The magnetoelectric effect in a crystal is traditionally described in Landau theory by writing the

free energy 𝐹 of the system in terms of an applied magnetic field 𝐻 and an applied electric field

𝐸 . Using Einstein summation convention 𝐹 can be written as

−𝐹 𝐸, 𝐻 =1

2𝜀0𝜀𝑖𝑗 𝐸𝑖𝐸𝑗 +

1

2𝜇0𝜇𝑖𝑗 𝐻𝑖𝐻𝑗 + 𝛼𝑖𝑗 𝐸𝑖𝐻𝑗 +

𝛽𝑖𝑗𝑘

2𝐸𝑖𝐻𝑗 𝐻𝑘 +

𝛾𝑖𝑗𝑘

2𝐻𝑖𝐸𝑗𝐸𝑘 + ⋯ . (1.6)

The first term on the right hand side describes the contribution resulting from the electrical

response to an electric field, where 𝜀𝑖𝑗 (𝑇) is relative permittivity. The second term is the

magnetic equivalent of the first term, where 𝜇𝑖𝑗 (𝑇) is relative permeability. The third term

describes linear magnetoelctric coupling via 𝛼𝑖𝑗 (𝑇). Other terms represent higher-order

magnetoelectric coupling coefficients.

The magnetoelectric effect can be established in the form 𝑃𝑖 𝐻𝑗 or 𝑀𝑖 𝐸𝑗 by differentiating 𝐹.

One obtains

𝑃𝑖 𝐻𝑗 = 𝛼𝑖𝑗 𝐻𝑗 +𝛽𝑖𝑗𝑘

2𝐻𝑗 𝐻𝑘 + ⋯ ,

𝑀𝑖 𝐸𝑗 = 𝛼𝑖𝑗 𝐸𝑖 +𝛾𝑖𝑗𝑘

2𝐸𝑗𝐸𝑘 + ⋯ .

(1.7)

Term 𝛼𝑖𝑗 is designated as the linear magnetoelectric effect and corresponds to the induction of

polarization by a magnetic field or a magnetization by an electric field. Materials exhibiting ME

effect are 𝐶𝑟2𝑂3 , 𝐵𝑖𝑀𝑛𝑂3, 𝐵𝑖𝐹𝑒𝑂3. Unfortunately, the magnetoelectric effect is usually too small

to be practically applicable as term 𝛼𝑖𝑗 is limited by the relation

𝛼𝑖𝑗

2 ≤ 𝜀𝑖𝑖𝜇𝑗𝑗 . (1.8)

Approaches to coexistence

10

3.2 Frustrated magnets

Recent discovery report coexistence and gigantic coupling of ferroelectricity and

antiferromagnetism in spin frustrated systems. The key questions are how it is possible that

magnetic ordering can induce ferroelectricity and what the role of frustration is.

The importance of the frustration is illustrated in Figure 6. In some lattices it is not possible to

satisfy all exchange interactions and energy can not be minimized. On the square lattice it is

possible to satisfy the requirement of antiparallel ordering. However, on a triangular lattice

things are not so straightforward. If two neighboring spins are placed antiparallel, the third spin

is faced with a dilemma. In any case the one of two neighbors will not have their energy

minimized. As a result the system is frustrated and tends to release this frustration by forming

unusual magnetic order where magnetization is inhomogeneous.

Figure 6: Frustration of spins.

The coupling between electric polarization and magnetization is governed by the symmetries of

these two order parameters. As we already mentioned, the polarization 𝑃 and electric field 𝐸

change sign on the inversion of all coordinates, 𝑟 → −𝑟 , but remain invariant on time reversal,

𝑡 → −𝑡. The magnetization 𝑀 and magnetic field 𝐻 transform precisely the opposite way.

Because of this difference in transformation properties, the linear coupling between 𝑃, 𝐸 and

𝑀, 𝐻 described by Maxwell’s equations is only possible when these vectors vary both in space

and time; spatial derivatives of 𝐸 are proportional to the time derivative of 𝐻 and vice versa.

This is where frustration comes into play. Its role is to induce spatial variation of magnetization.

The period of magnetic states in frustrated systems depends on strengths of competing

interactions and is often incommensurate (out of proportion) with period of crystal lattice. For

example, a spin chain with a ferromagnetic interaction 𝐽′ > 0 between neighbouring spins has

uniform ground state with all spins parallel. An antiferromagnetic next-nearest-neighbour

interaction 𝐽 < 0 frustrates this simple ordering, and when sufficiently strong stabilizes a spiral

magnetic state:

𝑆 𝑛 = 𝑆[𝑒 1 cos 𝑄𝑥𝑛 + 𝑒 2 sin𝑄𝑥𝑛 ] , (1.9)

where 𝑒 1 and 𝑒 2 are two orthogonal unit vectors and wavevector 𝑄 is given by

Approaches to coexistence

11

cos𝑄

2= −𝐽′ 4𝐽 . (1.10)

Like any other magnetic ordering, the magnetic spiral (Figure 7) spontaneously breaks time-

reversal symmetry. In addition it breaks inversion symmetry, because the change of the sign of

all coordinates inverts the direction of the rotation of spins in the spiral. Thus, the symmetry of

the spiral state allows for a simultaneous presence of electric polarization

Figure 7: Frustrated spin chains with the nearest-neighbour FM and next-nearest-neighbour AFM

interactions J and J´ (reproduced from [4]).

Ferroelectricity is induced by lattice relaxation in a

magnetically ordered state. The exchange between

spins of transition metal ions is usually mediated by

ligands, for example oxygen ions, forming bonds

between pair of transition metals. The effect is

shown in Figure 8. Interaction between spins 𝑆 𝑛 and

𝑆 𝑛+1 pushes negative oxygen ions in one direction

perpendicular to the spin chain formed by positive

magnetic ions, thus inducing electric polarization

perpendicular to the chain.

Figure 8: Ferroelectricity induced by the

exchange striction in a magnetic spiral state(reproduced from [4]).

The perovskite manganite 𝑇𝑏𝑀𝑛𝑂3 is an example of described mechanism. The spin structure is

a sinusoidal antiferromagnetic ordering of the 𝑀𝑛3+ moments that takes place below 𝑇𝑁 ≈ 41𝐾 .

A rough sketch of the magnetic order on 𝑀𝑛 moments is illustrated in Figure 9. With further

decreasing of temperature below 𝑇𝑁 ≈ 27𝐾 the ferroelectric phase with spontaneous

polarization emerges. The polarization along 𝑐 axis at ~10𝐾 is about 8 ∗ 10−4 𝐶 𝑚−2 which is

still rather small compared with that of conventional ferroelectrics (~2,6 ∗ 10−2 𝐶 𝑚−2 at 296𝐾

in 𝐵𝑎𝑇𝑖𝑂3). Dependence of polarization upon temperature in magnetic field is revealed in Figure

10. As the magnetic field 𝐵 is applied along the b-axes the magnetic 𝑄 vector changes and the

individual magnetic moments change their direction. As a result, the direction of polarization

changes and becomes zero in c-direction, while increases in a-direction. Temperature versus

magnetic field phase diagram for compound 𝑇𝑏𝑀𝑛𝑂3 is shown in Figure 11.

Approaches to coexistence

12

Figure 9: A rough sketch of the magnetic order on 𝑀𝑛 moments (reproduced from [3]).

Figure 10: electric polarization along the c and a axes (c and d), respectively, at various magnetic fields in single crystals of 𝑇𝑏𝑀𝑛𝑂3 . Magnetic fields are applied along the b axis (reproduced from [3]).

Figure 11: Temperature versus magnetic field phase diagram for 𝑇𝑏𝑀𝑛𝑂3 for magnetic field applied along

the b axis. The shaded areas show magnetic field hysteresis regions (reproduced from [3]).

Example material – Slovenian contribution

13

4 Example material – Slovenian contribution

Research field of multiferroic materials is fast evolving and discoveries of compounds such as

𝑇𝑏𝑀𝑛𝑂3 ignited extensive research activities. I will present the magnetic and ferroelectric

properties of the 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 system, studied by team of Slovene and Swiss researches (Ref. [7]).

The crystal structure (Figure 13) of compound implies both magnetic frustration and reduced

dimensionality where magnetoelectric effect is probable. The system has monoclinic unit cell

and adopts a layered structure. The layers consist of triangularly arranged 𝐹𝑒4𝑂16 20− clusters

linked by [𝑇𝑒4𝑂10𝐵𝑟2]6− units. Clusters contain two crystallographically non-equivalent 𝐹𝑒3+

ions (labeled as 𝐹𝑒1 and 𝐹𝑒2 in Figure 12).

Figure 13: Crystal structure of the 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 system (reproduced from [7]).

The exchange interaction between 𝐹𝑒1 and

𝐹𝑒2 moments, 𝐽1, is antiferromagnetic and

interaction between 𝐹𝑒1 and 𝐹𝑒1 moments, 𝐽2,

is ferromagnetic. The structure of the iron

tetrameter suggests competition between 𝐽1

and 𝐽2. Furthermore, 𝑇𝑒4+ cations possess

active lone-pare electrons (pared electrons

not used in chemical bonding). All inter-

cluster magnetic exchange paths go through

𝑇𝑒4+ sites, thus providing intimate link

between the 𝐹𝑒3+ magnetic moments and

𝑇𝑒4+ lone pairs.

Figure 12: 𝐹𝑒4𝑂16 20− cluster with denoted 𝐽1 and 𝐽2 exchange interactions (reproduced from [7]).

Example material – Slovenian contribution

14

The reduced dimensionality and the magnetic frustration within the iron tetramers suggest

interesting magnetic properties of the compound. Studies of magnetic susceptibility 𝜒 showed a

distinct change from Curie-Weiss law, 𝜒 =𝐶

𝑇−𝑇𝐶, in temperature dependence at 𝑇𝑁 = 10,6𝐾

(Figure 14). Neutron diffraction measurements revealed magnetic ordering, suddenly emerging

at 𝑇𝑁 (inset Figure 14). Magnetic structure is incommensurate along the crystallographic 𝑏-axis

which arises from the amplitude (Figure 15), rather than directional modulation of the magnetic

characteristics of spiral ordering (Figure 7). The structure is characterized by a periodic

modulation of 𝐹𝑒3+ magnetic moment amplitudes

𝑆 𝑖, 𝑘 = 𝑆0 cos 𝑄 𝑟 𝑖 + 𝜓𝑘 . (1.11)

Where 𝑆0is the amplitude and 𝜓𝑘 is the phase of the modulation associated with different 𝐹𝑒3+

magnetic ions within the unit cell.

Figure 14: Magnetic susceptibility and neutron diffraction in inset (reproduced from [7]).

Figure 15: Incommensurate magnetic structure which arises from the amplitude. Red dots

represent 𝐹𝑒3+ ions and arrows represent magnetic moments.

Example material – Slovenian contribution

15

We discuss the possible origin of the multiferroic behavior in 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟. The studied

compound is highly frustrated which is responsible for the complex magnetic ordering. The

ferroelectric order develops simultaneously within the low-temperature magnetic phase. In this

phase the shift of 𝑇𝑒4+ ions will take place, leading to the polarization of the 𝑇𝑒4+ lone pair

electrons. It is important to stress that all inter-cluster exchange interactions go through 𝑇𝑒4+

ions, thus providing a natural way to couple magnetic and ferroelectric order. The compound

𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 offers a new type of magnetic structure and new type of magnetoelectric coupling.

The unique interaction between magnetic frustration and 𝑇𝑒4+ lone pairs bridging magnetic

inter-cluster exchange appears vital for the occurrence of magnetoelectric effect.

Figure 17: Coupling between magnetic and polar order in 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 . Parameter 𝐶–𝐶0 is proportional to

𝜖(𝑇) and shows great field dependence in all directions (reproduced from [7]).

In low-temperature incommensurate

magnetic structure none of the crystal

symmetry elements is preserved. This

opens the possibility for coexistence of

ferroelectric order. The measurements of

electric polarization, 𝑃, confirmed

existence of spontaneous polarization

(Figure 16).

The existence of magnetoelectric effect is

presented in Figure 17 where parameter

𝐶– 𝐶0 is proportional to 𝜖(𝑇). The FE

transition is strongly field dependent,

ambiguously proving a coupling between

magnetic and polar order in 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟.

Figure 16: Electric polarization hysteresis loop (reproduced from [7]).

Applications

16

5 Applications

Most of the research in multiferroics has been curiosity-driven basic research, but there are a

number of fresh ideas for device applications based on multiferroic materials. Multiferroics

combine application favorable properties of magnetic and ferroelectric materials and blend

them together, opening a new window of opportunities for application usage.

One of the most popular ideas is that multiferroic bits may be used to store information in the

magnetization 𝑀 and polarization 𝑃. This type of encoding information in such four-state

memory has recently been demonstrated (Figure 18). Such memory does not require the

coupling between ferroelectricity and magnetism; a strong cross coupling would be even

disastrous. If magnetoelectric coupling is present, device application could be realized where

information is written electrically and read magnetically. This is attractive, given that it would

exploit the best of aspects of ferroelectric random access memory (FeRAM) and magnetic data

storage, while avoiding the problems associated with reading FeRAM and generating the large

local magnetic field needed to write. However, significant materials developments will be

required to develop magnetoelectric materials that could make a real contribution to the data

storage industry.

The direct application of multiferroics is development of a magnetic field sensor. Multiferroics

possess great potential value due sensitivity to both electric and magnetic field. External

magnetic field could be detected and immediately transformed into electric polarization –

voltage. Multiferroics could offer an alternative to modern field sensors, but sensitivity still has

to be investigated.

Figure 18: Schematic representation of four-state memory.

Conclusions

17

6 Conclusions

In this paper we reviewed the challenge of coupling magnetism and ferroelectricity in a

magnetoelectric effect. The magnetic and electric properties of compound have been presented.

By analyzing characteristics of both type of materials we defined necessary attributes for

multiferroic material. However, we found the requirements for ferrroelectricity and magnetism

to be in contradiction.

We demonstrated a way to get around this problem by introducing frustrated systems.

Frustration removes symmetry and induces spin incommensurate spin ordering which makes

ferroelecticity possible. Example material has been introduced and analyzed.

Slovenian contribution in this exciting field of research was also presented. Compound

𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟 was characterized as frustrated system where magnetic structure is

incommensurate. The origin of ferroelectricity was attributed to the shift of 𝑇𝑒4+ ions which are

also vital for occurrence of magnetoelectric effect. By analysis of the characteristics of selected

compound new materials could be engineered and take a step forward in exploiting

magnetoelectric effect.

As multiferroics possess properties of magnets and ferroelectrics their applicational value is

vast. The magetoelectric effecy offers a whole new dimension in application possibilities.

Already, ideas of four-state memory, spintronics and magnetic field sensors are being under

intense development.

References

18

7 References

[1] Hill, N. A.: Why are there so few magnetoelectric materials?, J. Phys. Chem. B 104, 6697-6709

(2000)

[2] Fiebig, M.: Revival of magnetoelectric effect, J. Phys. D: Appl. Phys. 38, R123-R152 (2005)

[3] Kimura, T. et al: Magnetic control of ferroelectric polarization, Nature 426, 55-58 (2003)

[4] Cheong S.-W. and Mostovoy M.: Multiferroics: A magnetic twist for ferroelectricity, Nature

Materials 6, 13 - 20 (2007)

[5] Eerenstein W., Mathur N. D. and Scott J. F.: Multiferroic and magnetoelectric materials, Nature

442, 759-765 (2006)

[6] Wang K.F., Liu J.-M. and Ren Z.F.: Multiferroicity: the coupling between magnetic and

polarization orders, Advances in Physics 58, 321-448 (2009)

[7] Pregelj, M. et al: Spin amplitude modulation driven magnetoelectric coupling in the new

multiferroic 𝐹𝑒𝑇𝑒2𝑂5𝐵𝑟, Physical review letters 103, 147202 (2009)

[8] Ramesh, R.: Emerging routes to multiferroics, Nature 461, 1218-1219 (2009)

[9] Blundell, Stephen: Magnetism in Condensted Matter. (Oxford university press, New York,

2001)

[10] Jackson, John David: Classical Electrodynamics 3rd ed. (J. Wiley, New York, 1999)