International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.5, pp. 139-145

## Section 1.5.8. Magnetoelectric effect

A. S. Borovik-Romanov,a H. Grimmerb,c* and M. Kenzelmannc*

aP. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia,bLabor für Neutronenstreuung, ETH Zurich, Switzerland, and cPaul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
Correspondence e-mail:  hans.grimmer@psi.ch, michel.kenzelmann@psi.ch

### 1.5.8. Magnetoelectric effect

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Curie (1894) stated that materials that develop an electric polarization in a magnetic field or a magnetization in an electric field may exist. This prediction was given a more precise form by Landau & Lifshitz (1957), who considered the invariants in the expansion of the thermodynamic potential up to linear terms in . For materials belonging to certain magnetic point groups, the thermodynamic potential can be written in the formIf (in the absence of a magnetic field) an electric field is applied to a crystal with potential (1.5.8.1), a magnetization will be produced: Conversely, an electric polarization arises at zero electric field if a magnetic field is applied: This phenomenon is called the magnetoelectric effect. A distinction is made between the linear magnetoelectric effect described above and two types of bilinear magnetoelectric effects. These bilinear effects arise if the thermodynamic potential contains terms of the form or . They will be described in Section 1.5.8.2.

#### 1.5.8.1. Linear magnetoelectric effect

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It is obvious that the linear magnetoelectric effect is forbidden for all dia- and paramagnets, as their magnetic groups possess R as a separate element. The effect is also forbidden if the magnetic space group contains translations multiplied by R, because in these cases the point group also possesses R as a separate element. Since is an axial vector that changes sign under R and is a polar vector that is invariant under time inversion, is an axial tensor of second rank, the components of which all change sign under time inversion (R). From relation (1.5.8.1), it follows that a magnetic group which allows the magneto­electric effect cannot possess a centre of symmetry (). However, it can possess it multiplied by R () (see Table 1.5.8.1). There are 21 magnetic point groups that possess a centre of symmetry. The detailed analysis of the properties of the tensor shows that among the remaining 69 point groups there are 11 groups for which the linear magnetoelectric effect is also forbidden. These groups are , , , , = , , , , , and .

 Table 1.5.8.1| top | pdf | The forms of the tensor characterizing the linear magnetoelectric effect
Magnetic crystal classMatrix representation of the property tensor
SchoenfliesHermann–Mauguin

(unique axis y)
(unique axis y)

All remaining 58 magnetic point groups in which the linear magnetoelectric effect is possible are listed in Table 1.5.8.1. The 11 forms of tensors that describe this effect are also listed in this table.3 The orientation of the axes of the Cartesian coordinate system (CCS) with respect to the symmetry axes of the crystal is the same as in Table 1.5.7.1. Alternative orientations of the same point group that give rise to the same form of have been added between square brackets in Table 1.5.8.1. The tensor has the same form for and , and , and ; it also has the same form for and , and , and .

The forms of for frequently encountered orientations of the CCS other than those given in Table 1.5.8.1 are (cf. Rivera, 1994, 2009)

 (1) (unique axis z): (2) (unique axis z): (3) (unique axis x): (4) (unique axis x): (5) : (6) : (7) :

As mentioned above, the components of the linear magnetoelectric tensor change sign under time inversion. The sign of these components is defined by the sign of the antiferromagnetic vector , i.e. by the sign of the 180° domains (S-domains). This is like the behaviour of the piezomagnetic effect and, therefore, everything said above about the role of the domains can be applied to the magnetoelectric effect.

Dzyaloshinskii (1959) proposed the antiferromagnetic Cr2O3 as the first candidate for the observation of the magnetoelectric (ME) effect. He showed that the ME tensor for this compound has three nonzero components: and . The ME effect in Cr2O3 was discovered experimentally by Astrov (1960) on an unoriented crystal. He verified that the effect is linear in the applied electric field. Folen et al. (1961) and later Astrov (1961) performed measurements on oriented crystals and revealed the anisotropy of the ME effect. In these first experiments, the ordinary magnetoelectric effect MEE (the electrically induced magnetization) was investigated by measuring the magnetic moment induced by the applied electric field. Later Rado & Folen (1961) observed the converse effect MEH (the electric polarization induced by the magnetic field). The temperature dependence of the components of the magnetoelectric tensor in Cr2O3 was studied in detail in both laboratories.

In the following years, many compounds that display the linear magnetoelectric effect were discovered. Both the electrically induced and the magnetically induced ME effect were observed. The values of the components of the magnetoelectric tensor range from 10−6 to 10−2 in compounds containing the ions of the iron group and from 10−4 to 10−2 in rare-earth compounds. Cox (1974) collected values of of the known magnetoelectrics. Some are listed in Table 1.5.8.2 together with more recent results. Additional information about the experimental data is presented in six conference proceedings. The first five are given as references [4] to [8] in Fiebig (2005), the sixth in Fiebig & Spaldin (2009).

 Table 1.5.8.2| top | pdf | A list of some magnetoelectrics
 αmax is the maximum observed value of αij expressed in rationalized Gaussian units.
CompoundTN or TC (K)Magnetic point groupαmaxReferences
Fe2TeO6 219 7–9, 70
DyAlO3 3.5 11–13
GdAlO3 4.0 14
TbAlO3 4.0 12, 15–17
TbCoO3 3.3 12, 16, 18
Cr2O3 318 45–49, 70, 71, W162
Nb2Mn4O9 110 52, 53
Nb2Co4O9 27 52, 53
Ta2Mn4O9 104 53
Ta2Co4O9 21 53
LiMnPO4 35 55, 56, 58, 60
LiFePO4 50 57, 58
LiCoPO4 22 54, 55, R161
LiNiPO4 23 54, 55, 61
GdVO4 2.4 70
TbPO4 2.2 see text
DyPO4 3.4 68, 69
HoPO4 1.4 72
Mn3B7O13I 26 C204
Co3B7O13Cl 12 m S204
Co3B7O13Br 17 88C1
Co3B7O13I 38 90C3
Ni3B7O13I 61.5 74, 75, 77–79, 90C2
Ni3B7O13Cl 9 74R2, 91R1
Cu3B7O13Cl 8.4 88R1
FeGaO3 305 84–86
TbOOH 10.0 114
DyOOH 7.2 92, 114
ErOOH 4.1 93, 114
Gd2CuO4 6.5 W161
MnNb2O6 4.4 101, 102
MnGeO3 16 98–100
CoGeO3 31 70
CrTiNdO5 13 70, 89
Numbers refer to references quoted by Cox (1974); codes 88C1, 90C3, 88R1, 90C2, 74R2, 91R1 refer to references quoted by Burzo (1993); and codes W162, R161, C204, S204 and W161 refer to articles in Ferroelectrics, 162, 141, 161, 147, 204, 125, 204, 57 and 161, 133, respectively.

The values of are given in rationalized Gaussian units, where is dimensionless. Some authors follow Dzyaloshinskii (1959) in writing (1.5.8.1) as , where are the non-rationalized Gaussian values of the components of the magnetoelectric tensor. If SI units are used, then (1.5.8.1) becomes . The connections between the values of the tensor components expressed in these three systems are The units of are s m−1. A detailed discussion of the relations between the descriptions of the magnetoelectric effect in different systems of units is given by Rivera (1994).

Most magnetoelectrics are oxides containing magnetic ions. The ions of the iron group are contained in corundum-type oxides [magnetic point group ], triphyllite-type oxides with different magnetic groups belonging to the ortho­rhombic crystallographic structure and other compounds. The rare-earth oxides are represented by the orthorhombic RMO3 structure with R = rare earth, M = Fe3+, Co3+, Al3+ [magnetic point group ], tetragonal zircon-type compounds RMO4 (R = rare earth, M = P, V) [magnetic point group ], monoclinic oxide hydroxides ROOH [magnetic point groups , ] and other compounds. Of particular interest is TbPO4, which has the highest value for a magnetoelectric tensor component, namely 1.1 × 10−2 at 2.2 K, where the point group is 4′/mmm (Rado & Ferrari, 1973; Rado et al., 1984) and 1.7 × 10−2 at 1.5 K, where the point group is 2′/m (Rivera, 2009). There are also some weak ferromagnets and ferrimagnets that exhibit the linear magnetoelectric effect. An example is the weakly ferromagnetic boracite Ni3B7O13I. These orthorhombic compounds will be discussed in Section 1.5.8.3. Another ortho­rhombic magnetoelectric crystal is ferrimagnetic FeGaO3 (Rado, 1964; see Table 1.5.8.2).

It has been shown in experiments with Cr2O3 that in the spin-flop phase becomes zero but an off-diagonal component arises (Popov et al., 1992). Such behaviour is possible if under the spin-flop transition the magnetic point group of Cr2O3 transforms from to . For the latter magnetic point group, the ME tensor possesses only transverse components.

The temperature dependences determined for the ME moduli, and , of Cr2O3 are quite different (see Fig. 1.5.8.1). The temperature dependence of is similar to that of the order parameter (sublattice magnetization ), which can be explained easily, bearing in mind that the magnetoelectric moduli are proportional to the magnitude of the antiferromagnetic vector (). However, to explain the rather complicated temperature dependence of , it becomes necessary to assume that the moduli are proportional to the magnetic susceptibility of the crystal so that (Rado, 1961; Rado & Folen, 1962)where and are new constants of the magnetoelectric effect which do not depend on temperature. Formulas (1.5.8.5) provide a good explanation of the observed temperature dependence of .

 Figure 1.5.8.1 | top | pdf |Temperature dependence of the components and in Cr2O3 (Astrov, 1961). (αSI = 4πα/c s m−1.)

The linear relation between and is also proved by the fact that when studying the ME effect, the domain structure of the sample is revealed. An annealing procedure to prepare a single-domain sample has been developed. To perform this annealing, the sample must be heated well above the Néel temperature and then cooled below in the presence of electric and magnetic fields. The directions of these fields have to agree with the allowed components of the ME tensor. In some compounds, a single-domain state may be obtained by applying simultaneously pulses of both fields to a multidomain sample at temperatures below (see O'Dell, 1970).

It was shown in the previous section that the piezomagnetic effect can be explained phenomenologically as weak ferromagnetism caused by the change of the symmetry produced by deformation of the lattice. The electric field may act indirectly inducing atomic displacements (similar to the displacements under stress) and as in piezomagnetism may cause the rise of a magnetic moment. Such ideas were proposed by Rado (1964) and expanded by White (1974).

The electric field may act directly to change the admixture of orbital states in the electron wavefunctions. As a result of such a direct action, there may be a change of different terms in the microscopic spin Hamiltonian. Correspondingly, the following mechanisms are to be distinguished. Changes in the g-tensor can explain the ME effect in DyPO4 (Rado, 1969). The electric-field-induced changes in single-ion anisotropy may represent the main mechanism of the ME effect in Cr2O3 (Rado, 1962). Two other mechanisms have to be taken into account: changes in the symmetric and antisymmetric exchange interactions. For details and references see the review article of de Alcantara Bonfim & Gehring (1980).

#### 1.5.8.2. Nonlinear magnetoelectric effects

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Along with linear terms in E and H, the thermodynamic potential may also contain invariants of higher order in : From this relation, one obtains the following formulas for the electric polarization and the magnetization :The third term in (1.5.8.7) describes the dependence of the dielectric susceptibility () and, consequently, of the permittivity on the magnetic field. Similarly, the second term in (1.5.8.8) points out that the magnetic susceptibility may contain a term , which depends on the electric field. The tensors and are symmetric in their last two indices. Symmetry imposes on the same restrictions as on the piezoelectric tensor and on the same restrictions as on the piezomagnetic tensor (see Table 1.5.7.1).

Ascher (1968) determined all the magnetic point groups that allow the terms and in the expansion of the thermodynamic potential . These groups are given in Table 1.5.8.3, which has been adapted from a table given by Schmid (1973). It classifies the 122 magnetic point groups according to which types of magnetoelectric effects (, or ) they admit and whether they admit spontaneous electric polarization (E) or spontaneous magnetization (H). It also classifies the 122 point groups according to whether they contain , or , as in a table given by Mercier (1974). Ferromagnets, ferrimagnets and weak ferromagnets have a point group characterized by H (the 31 groups of types 4–7 in Table 1.5.8.3); dia- and paramagnets as well as antiferromagnets with a nontrivial magnetic Bravais lattice have a point group containing (the 32 groups of types 1, 13, 17 and 19 in Table 1.5.8.3). The 59 remaining point groups describe antiferromagnets with a trivial Bravais lattice. The 31 point groups characterized by E, the 32 containing and the 59 remaining ones correspond to a similar classification of crystals according to their electric properties (see Schmid, 1973).

 Table 1.5.8.3| top | pdf | Classification of the 122 magnetic point groups according to magnetoelectric types
TypeInversions in the groupPermitted terms in thermodynamic potentialMagnetic point groupsNumber of magnetic point groups
1 E     EHH   , , , , , , , , , 10 31   49 122
2   E     EHH HEE , 2
3   E   EH EHH HEE , , , , , 6
4   E H EH EHH HEE , , m, , , , , , , , , , 13 31
5     H EH EHH HEE , , , , , 6
6     H   EHH HEE , 2
7   H     HEE , , , , , , , , , 10
8       EH EHH HEE , , , , , , , , , , , , , 14     73
9         EHH HEE , 2
10       EH     432 1   19
11     EH     , , , , , , , , , , , , , , , , 18
12         EHH   1   11
13       EHH   , , , , , , , , , 10
14           HEE 1   11
15         HEE , , , , , , , , , 10
16           , , 3   16
17           1
18           1
19           , , , , , , , , , , 11

Table 1.5.8.3 shows that for the 16 magnetic point groups of types 16–19, any kind of magnetoelectric effect is prohibited. These are the 11 grey point groups that contain all three inversions, the white group , the grey group ( = and the three black–white groups = , = and = .

Among the 58 magnetic point groups that allow the linear magnetoelectric effect, there are 19 that do not allow the nonlinear effects EHH and HEE (types 10 and 11 in Table 1.5.8.3). The remaining 39 groups are compatible with all three effects, EH, EHH and HEE; 19 of these groups describe ferromagnets (including weak ferromagnets) and ferrimagnets (types 4 and 5 in Table 1.5.8.3).

The 21 point groups of types 7, 14 and 15 allow only the magnetoelectric effect . These groups contain , except . The compounds belonging to these groups possess only one tensor of magnetoelectric susceptibility, the tensor of the nonlinear ME effect. The effect is described by

The magnetic point group of ferrimagnetic rare-earth garnets RFe5O12 ( Gd, Y, Dy) is , which is of type 7. Therefore, the rare-earth garnets may show a nonlinear ME effect corresponding to relations (1.5.8.9) and (1.5.8.10). This was observed by O'Dell (1967) by means of a pulsed magnetic field. As mentioned above, this effect may be considered as the dependence of the permittivity on the magnetic field, which was the method used by Cardwell (1969) to investigate this ME effect experimentally. Later Lee et al. (1970) observed the ME effect defined by relation (1.5.8.10). Applying both static electric fields and alternating ones (at a frequency ), they observed an alternating magnetization at both frequencies and . A nonlinear ME effect of the form was also observed in the weakly ferromagnetic orthoferrites TbFeO3 and YbFeO3. Their magnetic point group is .

Moreover, paramagnets that do not possess an inversion centre may show an ME effect if the point group is not . They have one of the 20 grey point groups given as types 1 or 13 in Table 1.5.8.3. Bloembergen (1962) pointed out that all these paramagnets are piezoelectric crystals. He called the ME effect in these substances the paramagnetoelectric (PME) effect. It is defined by the nonzero components of the tensor : The PME effect was discovered by Hou & Bloembergen (1965) in NiSO4·6H2O, which belongs to the crystallographic point group . The only nonvanishing components of the third-rank tensor are ( in matrix notation), so that and (, , ). Both effects were observed: the polarization by applying static () and alternating ( or ) magnetic fields and the magnetization by applying a static magnetic field and an alternating electric field in the plane . As a function of temperature, the PME effect shows a peak at 3.0 K and changes sign at 1.38 K. The coefficient of the PME effect at 4.2 K isThe theory developed by Hou and Bloembergen explains the PME effect by linear variation with the applied electric field of the crystal-field-splitting parameter D of the spin Hamiltonian.

Most white and black–white magnetic point groups that do not contain the inversion (), either by itself or multiplied by , admit all three types of ME effect: the linear () and two higher-order ( and ) effects. There are many magnetically ordered compounds in which the nonlinear ME effect has been observed. Some of them are listed by Schmid (1973); more recent references are given in Schmid (1994a).

In principle, many ME effects of higher order may exist. As an example, let us consider the piezomagnetoelectric effect. This is a combination of piezomagnetism (or piezoelectricity) and the ME effect. The thermodynamic potential must contain invariants of the form

The problem of the piezomagnetoelectric effect was considered by Rado (1962), Lyubimov (1965) and in detail by Grimmer (1992). All 69 white and black–white magnetic point groups that possess neither nor admit the piezomagnetoelectric effect. (These are the groups of types 2–6, 8–12, 14 and 16 in Table 1.5.8.3.) The tensor , which describes the piezomagnetoelectric effect, is a tensor of rank 4, symmetric in the last two indices and invariant under space-time inversion. This effect has not been observed so far (Rivera & Schmid, 1994). Grimmer (1992) analysed in which antiferromagnets it could be observed.

#### 1.5.8.3. Multiferroics4

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Initially, Schmid defined multiferroics as materials with two or three primary ferroics coexisting in the same phase, such as ferromagnetism, ferroelectricity or ferroelasticity (Schmid, 1994b). The term primary ferroics was defined in a thermodynamic classification, distinguishing primary, secondary and tertiary ferroics (Newnham, 1974; Newnham & Cross, 1976). For magnetoelectric multiferroics, however, it has become customary to loosen this definition. Magnetoelectric multiferroics are now considered materials with coexisting magnetic (ferro- or antiferromagnetic) and ferro­electric order. They can be divided into two classes: multiferroics where the origins of ferroelectricity and magnetic order are independent, and multiferroics where ferroelectricity is induced by magnetic or orbital order.

For the case of magnetically commensurate ferromagnetic ferroelectrics, Neronova & Belov (1959) pointed out that there are ten magnetic point groups that admit the simultaneous existence of spontaneous ferroelectric polarization P and magnetic polarization M, which they called ferromagnetoelectrics. Neronova and Belov considered only structures with parallel alignment of P and M (or L). There are three more groups that allow the coexistence of ferroelectric and ferromagnetic order, in which P and M are perpendicular to each other. Shuvalov & Belov (1962) published a list of the 13 magnetic groups that admit the coexistence of ferromagnetic and ferroelectric order. These are the groups of type 4 in Table 1.5.8.3; they are given with more details in Table 1.5.8.4.

 Table 1.5.8.4| top | pdf | List of the magnetic point groups of the ferromagnetoelectrics
Symbol of symmetry groupAllowed direction of
SchoenfliesHermann–MauguinPM
1 Any Any
2
m

Notice that and must be parallel in eight point groups, they may be parallel in and , and they must be perpendicular in , m and (see also Ascher, 1970). The magnetic point groups listed in Table 1.5.8.4 admit not only ferromagnetism (and ferrimagnetism) but the first seven also admit antiferro­magnetism with weak ferromagnetism. Ferroelectric pure antiferromagnets of type IIIa may also exist. They must belong to one of the following eight magnetic point groups (types 2 and 3 in Table 1.5.8.3): C2v = mm2; C4v = 4mm; C4(C2) = 4′; C4v(C2v) = 4′mm′; C3v = 3m; C6v = 6mm; C6(C3) = 6′; C6v(C3v) = 6′mm′. Table 1.5.8.3 shows that the linear magnetoelectric effect is admitted by all ferroelectric ferromagnets and all ferroelectric antiferromagnets of type IIIa except 6′ and 6′mm′.

The first experimental evidence to indicate that complex perovskites may become ferromagnetoelectric was observed by the Smolenskii group (see Smolenskii et al., 1958). They investigated the temperature dependence of the magnetic susceptibility of the ferroelectric perovskites Pb(Mn1/2Nb1/2)O3 and Pb(Fe1/2Nb1/2)O3. The temperature dependence at T > 77 K followed the Curie–Weiss law with a very large antiferromagnetic Weiss constant. Later, Astrov et al. (1968) proved that these compounds undergo a transition into a weakly ferromagnetic state at Néel temperatures TN = 11 and 9 K, respectively.

The single crystals of boracites synthesized by Schmid (1965) raised wide interest as examples of ferromagnetic ferroelectrics. The boracites have the chemical formula M3B7O13X (where M = Cu2+, Ni2+, Co2+, Fe2+, Mn2+, Cr2+ and X = F, Cl, Br, I, OH, ). Many of them are ferroelectrics and weak ferromagnets at low temperatures. This was first shown for Ni3B7O13I (see Ascher et al., 1966). The symmetries of all the boracites are cubic at high temperatures and their magnetic point group is . As the temperature is lowered, most become ferroelectrics with the magnetic point group . At still lower temperatures, the spins of the magnetic ions in the boracites go into an antiferromagnetic state with weak ferromagnetism. For some of the boracites the ferromagnetic/ferroelectric phase belongs to the group mm2′, and for others to mm′2, m′, m or 1. In accordance with Table 1.5.8.4, the spontaneous polarization P is oriented perpendicular to the weak ferromagnetic moment MD for the groups mm2′ and m. Boracites feature a complicated behaviour in external magnetic and electric fields, which depends strongly on the history of the samples. Changing the direction of the electric polarization by an electric field also changes the direction of the ferromagnetic vector (as well as the direction of the antiferromagnetic vector) and vice versa.

As an example, Fig. 1.5.8.2 shows the results of measurements on Ni–I boracite with spontaneous polarization along [001] and spontaneous magnetization initially along []. A magnetic field was applied along [110] and the polarization induced along [001] was measured. If the applied field H was increased beyond 6 kOe (B = 6 kG = 0.6 T), the induced polarization changed sign because the spontaneous magnetization had been reversed. The applied magnetic field was reversed to obtain the rest of the hysteresis loop describing the response.

 Figure 1.5.8.2 | top | pdf |The hysteresis loop of the linear magnetoelectric effect in ferroelectric and weakly ferromagnetic Ni3B7O13I at 46 K (Ascher et al., 1966). H = 1 kOe corresponds to B = 1 kG = 0.1 T.

If the spontaneous polarization is reversed, e.g. by applying an electric field, the spontaneous magnetization will rotate simultaneously by 90° around the polarization axis. Applying magnetic fields as described above will no longer produce a measurable polarization. If, however, the crystal is rotated by 90° around the polarization axis before repeating the experiment, a hysteresis loop similar to Fig. 1.5.8.2, but turned upside down, will be obtained (cf. Schmid, 1967). The similarity of the jumps in the curves of linear magnetostriction (see Fig. 1.5.7.2) and magnetoelectric effect in Ni-I boracite is noteworthy. More details on multiferroic boracites are given in Schmid (1994b).

Materials can be multiferroic when magnetic order occurs in a ferroelectric material. An important example of this type of multiferroic is BiFeO3, where ferroelectricity arises from the lone-pair activity of the Bi ion. Early on, BiFeO3 was shown to be an antiferromagnet below TN = 643 K using neutron scattering (Kiselev et al., 1962; Michel et al., 1969) and magnetic measurements (Smolenskii et al., 1962; see also Venevtsev et al., 1987). BiFeO3 also possesses a spontaneous ferroelectric polarization. The magnetic point group above TN is 3m1′, and it was suggested that below TN, the magnetic point group is 3m. However, it was shown that the magnetic structure is incommensurately spatially modulated (Sosnovska et al., 1982). Ferroelectric monodomain crystals were used to study the relationship between the direction of the ferroelectric polarization and the magnetic structure (Lebeugle et al., 2008). It was found that the easy-axis plane, in which the magnetic moments are ordered, depends on the direction of the ferroelectric polarization. The antiferromagnetic structure can thus be changed by the application of electric fields.

Another important example of multiferroicity where magnetic order appears in a ferroelectric material is YMnO3. Here, ferroelectricity arises from a complex rotation of the oxygen environment of the transition-metal ions (Bertaut et al., 1964). YMnO3 becomes ferroelectric at Tc = 193 K (with paramagnetic point group 6mm1′) and antiferromagnetic at TN = 77 K. The antiferromagnetic ordering was also proved by investigating the Mössbauer effect (Chappert, 1965). The most recent neutron measurements, using neutron polarimetry, suggest that the magnetic space group is P6′3 (Brown & Chatterji, 2006). The symmetries of both antiferromagnetic ferroelectrics, BiFeO3 and YMnO3, do not allow weak ferromagnetism according to Table 1.5.5.2, and indeed experimentally no spontaneous ferromagnetic moment has been found in the bulk.

In another important class of multiferroics, ferroelectricity is generated by spontaneous magnetic order. Some of the first studies of these kinds of materials were done in the late 1970s on Cr2BeO4 (Newnham et al., 1978). It had already been shown that the magnetic structure is a cycloidal spiral at low temperatures (Cox et al., 1969). Pyroelectric measurements showed that Cr2BeO4 is ferroelectric below TN = 28 K, and it was noted that the magnetic structure breaks all symmetry elements of the space groups.

In 2003, it was shown that TbMnO3 features ferroelectricity below Tc = 27 K (Kimura et al., 2003). TbMnO3 adopts antiferromagnetic order below TN = 42 K, and the ferroelectric onset coincides with a second magnetic transition, which was thought to be a lock-in transition to a commensurate structure. No ferromagnetic order is observed in TbMnO3. Neutron diffraction showed that the magnetic structure remains incommensurate at all temperatures, and that the onset of ferroelectricity coincides with the onset of cycloidal magnetic order (Kenzelmann et al., 2005). The cycloidal order is described by two irreducible representations of the group Gk of those elements of mmm that leave the magnetic modulation vector k = (0, q, 0) invariant. Harris introduced a trilinear coupling theory that clarified the relation between the symmetry of the magnetic structure and ferroelectricity (Kenzelmann et al., 2005; Harris, 2007). Ferro­electricity in TbMnO3 directly emerges from the magnetic symmetry breaking that creates a polar axis along which ferroelectric polarization is observed.

In the case of TbMnO3, the trilinear coupling between the magnetic and ferroelectric order takes the formHere Vijk is a coupling term that couples the incommensurate order parameters Mi and Mj with the ferroelectric polarization Pk. Such a coupling term conserves translational symmetry and has to be invariant under all symmetry elements of the space group. If Mi and Mj belong to the same one-dimensional irreducible representation, Pk has to be invariant under all symmetry elements of the space group and, consequently, has to vanish. If, however Mi and Mj belong to two different one-dimensional irreducible representations, this allows for a nonzero ferroelectric polarization Pk. For temperatures between Tc = 27 K and TN = 41 K, the magnetic structure is described only by one irreducible representation, namely Γ3 (as defined in Table 1.5.8.5). As a result, there can be no ferroelectric polarization. Below Tc, however, the magnetic structure is described by two irreducible representations, Γ2 and Γ3. Since the coupling term H has to be invariant under all symmetry elements of the space group, the ferroelectric polarization has to transform as the product of Γ2 and Γ3, which is Γ4. Therefore, the ferroelectric polarization is only allowed along the c axis, as is experimentally observed.

 Table 1.5.8.5| top | pdf | Irreducible representations of the group Gk for TbMnO3 (Kenzelmann et al., 2005)
12ymzmx
Γ1 1 1 1 1
Γ2 1 1 −1 −1
Γ3 1 −1 1 −1
Γ4 1 −1 −1 1

Magnetically induced ferroelectricity is also possible for commensurate structures. Examples include some of the phases in the RMn2O5 series (Hur et al., 2004, Chapon et al., 2006), and the `up-up-down-down' spin ordering in the MnO2 planes (called E-type ordering) in the orthorhombic RMnO3 series (Lorenz et al., 2007), where R = a rare-earth metal. The E-type magnetic structure is a good example of how a single two-dimensional irreducible representation can induce ferroelectricity. There are also magnetically induced ferromagnetic ferroelectrics, where ferroelectricity arises from antiferromagnetic order, and weak ferromagnetism is present due to uniform canting of the magnetic moments. Here examples include CoCr2O4 (Yamaski et al., 2006) and Mn2GeO4 (White et al., 2012).

Since 2003, a growing number of magnetically induced ferroelectrics have been discovered. Reviews of their symmetry properties have been given by Harris (2007) and Radaelli & Chapon (2007). This phenomenon has been observed for various different transition-metal ions, and for very different crystal structures. They all have in common the fact that ferroelectricity emerges with magnetic order or with a change of an already existing magnetic order. Competing magnetic interactions and low-dimensional magnetic topologies appear to be beneficial for magnetically induced ferroelectricity. The size of the ferroelectric polarization is orders of magnitude smaller than observed in BiFeO3 and YMnO3.

### References

Alcantara Bonfim, O. F. de & Gehring, G. A. (1980). Magnetoelectric effect in antiferromagnetic crystals. Adv. Phys. 29, 731–769.
Ascher, E. (1968). Higher-order magneto-electric effects. Philos. Mag. 17, 149–157.
Ascher, E. (1970). The interactions between magnetization and polarization: phenomenological symmetry considerations on boracites. J. Phys. Soc. Jpn, 28 Suppl., 7–14.
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