Magnetoelectric effect

Transcription

1 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.

2 Kazalo 1 Introduction Basic principles Ferromagnetism What makes (anti)ferromagnets (anti)ferromagnetic? Ferroelectrics What makes ferroelectrics ferroelectric? Multiferroism Requirements for magnetoelectric multiferroics Approaches to coexistence Magnetoelectric coupling Frustrated magnets Example material Slovenian contribution Applications Conclusions References

3 Introduction 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]). 2

4 Basic principles 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 T c that does not have a macroscopic magnetic moment (paramagnetic phase) to a low-temperature phase below T c (T c 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, B s. When the field is reduced to zero, the induction decreases to B r, know as remanent field. The reverse field, H c, 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. 3

5 Basic principles 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 T N. 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 T N. Typical aniferromagnets are Mn0 (T N = 116K), MnS (T N = 160K) and FeO (T N = 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]). 4

6 Basic principles 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, T c, 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 ψ a r 1 and ψ b 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 χ S S = 0 a spatial state of wave function must be symmetric, in case of symmetric triplet state χ T S = 1 a spatial state of wave function must be antisymmetric. Therefore, we can write the wave function for the siglet case ψ S and the triplet case ψ T as ψ S = ψ T = 1 2 ψ a r 1 ψ b r 2 + ψ a r 2 ψ b r 1 χ S, 1 2 ψ a r 1 ψ b r 2 ψ a r 2 ψ b r 1 χ T. (1.1) The energies of the two possible states are E S = ψ S H ψ S dr 1 dr 2, E T = ψ T H ψ T dr 1 dr 2. (1.2) The energy difference E S and E T enables us to define a spin part of the Hamiltonian as H spin = 2JS 1 S 2, (1.3) This Hamiltonian is called Heisenberg exchange interaction, where J = E S E T 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. 5

7 Basic principles For description of extended lattice the Heisenberg model applies: H = 2 J i,j S i S j i>j (1.5) Where J i,j is the exchange constant between the i th and j th 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, ABO 3, 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 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 shortrange 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 lowtemperature 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 d 0 lowest unoccupied state. Typical representative compound is BaTiO 3. A simple charge count gives Ba 2+ Ti 4+ O 3 2. Evidently, Ti 4+ ion is in a d 0 state. 6

8 Basic principles 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 d 0 while magnetic order require d n, 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 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. 7

9 Basic principles 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]) 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 Chemistry: d 0 band Ferroelectric materials have a formal charge corresponding to the d 0 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. 8

10 Approaches to coexistence 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 2 ε 0ε ij E i E j μ 0μ ij H i H j + α ij E i H j + β ijk 2 E ih j H k + γ ijk 2 H ie j E k +. (1.6) The first term on the right hand side describes the contribution resulting from the electrical response to an electric field, where ε ij (T) is relative permittivity. The second term is the magnetic equivalent of the first term, where μ ij (T) is relative permeability. The third term describes linear magnetoelctric coupling via α ij (T). Other terms represent higher-order magnetoelectric coupling coefficients. The magnetoelectric effect can be established in the form P i H j or M i E j by differentiating F. One obtains P i H j = α ij H j + β ijk 2 H j H k +, M i E j = α ij E i + γ ijk 2 E j E k +. (1.7) Term α ij 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 Cr 2 O 3, BiMnO 3, BiFeO 3. Unfortunately, the magnetoelectric effect is usually too small to be practically applicable as term α ij is limited by the relation α ij 2 ε ii μ jj. (1.8) 9

11 Approaches to coexistence 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 Qx n + e 2 sin Qx n ], (1.9) where e 1 and e 2 are two orthogonal unit vectors and wavevector Q is given by 10

12 Approaches to coexistence cos Q 2 = J 4J. (1.10) Like any other magnetic ordering, the magnetic spiral (Figure 7) spontaneously breaks timereversal 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 TbMnO 3 is an example of described mechanism. The spin structure is a sinusoidal antiferromagnetic ordering of the Mn 3+ moments that takes place below T N 41K. A rough sketch of the magnetic order on Mn moments is illustrated in Figure 9. With further decreasing of temperature below T N 27K the ferroelectric phase with spontaneous polarization emerges. The polarization along c axis at ~10K is about C m 2 which is still rather small compared with that of conventional ferroelectrics (~2, C m 2 at 296K in BaTiO 3 ). 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 TbMnO 3 is shown in Figure

13 Approaches to coexistence 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 TbMnO 3. Magnetic fields are applied along the b axis (reproduced from [3]). Figure 11: Temperature versus magnetic field phase diagram for TbMnO 3 for magnetic field applied along the b axis. The shaded areas show magnetic field hysteresis regions (reproduced from [3]). 12

14 Example material Slovenian contribution 4 Example material Slovenian contribution Research field of multiferroic materials is fast evolving and discoveries of compounds such as TbMnO 3 ignited extensive research activities. I will present the magnetic and ferroelectric properties of the FeTe 2 O 5 Br 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 Fe 4 O clusters linked by [Te 4 O 10 Br 2 ] 6 units. Clusters contain two crystallographically non-equivalent Fe 3+ ions (labeled as Fe1 and Fe2 in Figure 12). The exchange interaction between Fe1 and Fe2 moments, J 1, is antiferromagnetic and interaction between Fe1 and Fe1 moments, J 2, is ferromagnetic. The structure of the iron tetrameter suggests competition between J 1 and J 2. Furthermore, Te 4+ cations possess active lone-pare electrons (pared electrons not used in chemical bonding). All intercluster magnetic exchange paths go through Te 4+ sites, thus providing intimate link between the Fe 3+ magnetic moments and Te 4+ lone pairs. Figure 12: Fe 4 O cluster with denoted J 1 and J 2 exchange interactions (reproduced from [7]). Figure 13: Crystal structure of the FeTe 2 O 5 Br system (reproduced from [7]). 13

15 Example material Slovenian contribution 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, in temperature dependence at T T T N = 10,6K C (Figure 14). Neutron diffraction measurements revealed magnetic ordering, suddenly emerging at T N (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 Fe 3+ magnetic moment amplitudes S i, k = S 0 cos Qr i + ψ k. (1.11) Where S 0 is the amplitude and ψ k is the phase of the modulation associated with different Fe 3+ 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 Fe 3+ ions and arrows represent magnetic moments. 14

16 Example material Slovenian contribution 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 C 0 is proportional to ε(t). The FE transition is strongly field dependent, ambiguously proving a coupling between magnetic and polar order in FeTe 2 O 5 Br. Figure 16: Electric polarization hysteresis loop (reproduced from [7]). We discuss the possible origin of the multiferroic behavior in FeTe 2 O 5 Br. 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 Te 4+ ions will take place, leading to the polarization of the Te 4+ lone pair electrons. It is important to stress that all inter-cluster exchange interactions go through Te 4+ ions, thus providing a natural way to couple magnetic and ferroelectric order. The compound FeTe 2 O 5 Br offers a new type of magnetic structure and new type of magnetoelectric coupling. The unique interaction between magnetic frustration and Te 4+ lone pairs bridging magnetic inter-cluster exchange appears vital for the occurrence of magnetoelectric effect. Figure 17: Coupling between magnetic and polar order in FeTe 2 O 5 Br. Parameter C C 0 is proportional to ε(t) and shows great field dependence in all directions (reproduced from [7]). 15

17 Applications 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. 16

18 Conclusions 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 FeTe 2 O 5 Br was characterized as frustrated system where magnetic structure is incommensurate. The origin of ferroelectricity was attributed to the shift of Te 4+ 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. 17

19 References 7 References [1] Hill, N. A.: Why are there so few magnetoelectric materials?, J. Phys. Chem. B 104, (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, (2003) [4] Cheong S.-W. and Mostovoy M.: Multiferroics: A magnetic twist for ferroelectricity, Nature Materials 6, (2007) [5] Eerenstein W., Mathur N. D. and Scott J. F.: Multiferroic and magnetoelectric materials, Nature 442, (2006) [6] Wang K.F., Liu J.-M. and Ren Z.F.: Multiferroicity: the coupling between magnetic and polarization orders, Advances in Physics 58, (2009) [7] Pregelj, M. et al: Spin amplitude modulation driven magnetoelectric coupling in the new multiferroic FeTe 2 O 5 Br, Physical review letters 103, (2009) [8] Ramesh, R.: Emerging routes to multiferroics, Nature 461, (2009) [9] Blundell, Stephen: Magnetism in Condensted Matter. (Oxford university press, New York, 2001) [10] Jackson, John David: Classical Electrodynamics 3 rd ed. (J. Wiley, New York, 1999) 18