International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 1.5, pp. 137142
Section 1.5.8. Magnetoelectric effect^{a}P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and ^{b}Labor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH5234 Villigen PSI, Switzerland 
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.
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 .

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.^{4} 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)
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 (Sdomains). 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 Cr_{2}O_{3} 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 Cr_{2}O_{3} 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 the first experiments, the ordinary magnetoelectric effect ME_{E} (the electrically induced magnetization) was investigated. This means the magnetic moment induced by the applied electric field was measured. Later Rado & Folen (1961) observed the converse effect ME_{H} (the electric polarization induced by the magnetic field). The temperature dependence of the components of the magnetoelectric tensor in Cr_{2}O_{3} 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 rareearth 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 three conference proceedings (Freeman & Schmid, 1975; Schmid et al., 1994; Bichurin, 1997).
^{†}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 nonrationalized 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 a tensor component 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 corundumtype oxides [magnetic point group ], triphyllitetype oxides with different magnetic groups belonging to the orthorhombic crystallographic structure and other compounds. The rareearth oxides are represented by the orthorhombic RMO_{3} structure with R = rare earth, M = Fe^{3+}, Co^{3+}, Al^{3+} [magnetic point group ], tetragonal zircontype compounds RMO_{4} (R = rare earth, M = P, V) [magnetic point group ], monoclinic oxide hydroxides ROOH [magnetic point groups , ] and other compounds. Of particular interest is TbPO_{4}, which has the highest value of the magnetoelectric tensor components, 1.2 × 10^{−2} (Rado & Ferrari, 1973; Rado et al., 1984). There are also some weak ferromagnets and ferrimagnets that exhibit the linear magnetoelectric effect. An example is the weakly ferromagnetic boracite Ni_{3}B_{7}O_{13}I. These orthorhombic compounds will be discussed in Section 1.5.8.3. Another orthorhombic magnetoelectric crystal is ferrimagnetic FeGaO_{3} (Rado, 1964; see Table 1.5.8.2).
It has been shown in experiments on Cr_{2}O_{3} that in the spinflop phase becomes zero but a nondiagonal component arises (Popov et al., 1992). Such behaviour is possible if under the spinflop transition the magnetic point group of Cr_{2}O_{3} 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 , in Cr_{2}O_{3} 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 .
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 singledomain 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 singledomain state may be obtained by applying simultaneous 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 displacement (similar to the displacement 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 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 gtensor can explain the ME effect in DyPO_{4} (Rado, 1969). The electricfieldinduced changes in singleion anisotropy may represent the main mechanism of the ME effect in Cr_{2}O_{3} (Rado, 1962). Two other mechanisms have to be taken into account: changes in symmetric and antisymmetric exchange. For details and references see the review article of de Alcantara Bonfim & Gehring (1980).
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 dielectric permittivity , on the magnetic field. Similarly, the second term in (1.5.8.8) points out that the magnetic susceptibility may depend 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 dielectric 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 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 rareearth garnets RFe_{5}O_{12} ( Gd, Y, Dy) is , which is of type 7. Therefore, the rareearth 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 dielectric 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 TbFeO_{3} and YbFeO_{3}. 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 NiSO_{4}·6H_{2}O, which belongs to the crystallographic point group . The only nonvanishing components of the thirdrank 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 crystalfieldsplitting 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 higherorder ( 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 recently 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 that describes the piezomagnetoelectric effect is a tensor of rank 4, symmetric in the last two indices and invariant under spacetime inversion. This effect has not been observed so far (Rivera & Schmid, 1994). Grimmer (1992) analyses in which antiferromagnets it could be observed.
Neronova & Belov (1959) pointed out that there are ten magnetic point groups that admit the simultaneous existence of spontaneous dielectric polarization and magnetic polarization . Materials with such a complicated ordered structure are called ferromagnetoelectrics. Neronova and Belov considered only structures with parallel alignment of and (or ). There are three more groups that allow the coexistence of ferroelectric and ferromagnetic order, in which and are perpendicular to each other. Shuvalov & Belov (1962) published a list of the 13 magnetic point groups that admit ferromagnetoelectric 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.

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 III^{a} 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): ; ; ; ; ; ; .
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(Mn_{1/2}Nb_{1/2})O_{3} and Pb(Fe_{1/2}Nb_{1/2})O_{3}. 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 temperatures T_{N} = 11 and 9 K, respectively.
BiFeO_{3} is an antiferromagnet below T_{N} = 643 K. This was proved by neutron scattering (Kiselev et al., 1962; Michel et al., 1969) and magnetic measurements (Smolenskii et al., 1962; see also Venevtsev et al., 1987). BiFeO_{3} also possesses a spontaneous electric polarization. The magnetic point group above T_{N} is 3m1′ and below it should have been 3m (Kiselev et al., 1962), but in reality it possesses an antiferromagnetic spatially modulated spin structure (Sosnovska et al., 1982). Another ferroelectric antiferromagnet, YMnO_{3}, was found by Bertaut et al. (1964). It becomes ferroelectric at T_{c} = 913 K (with paramagnetic point group 6mm1′) and antiferromagnetic at T_{N} = 77 K. Below this temperature, its magnetic point group is 6′mm′. The antiferromagnetic ordering was also proved by investigating the Mössbauer effect (Chappert, 1965). The symmetries of both antiferromagnetic ferroelectrics described above do not allow weak ferromagnetism according to Table 1.5.5.2, and, experimentally, a spontaneous ferromagnetic moment has not been observed so far.
Since Schmid (1965) developed a technique for growing single crystals of boracites, these compounds have become the most interesting ferromagnetoelectrics. The boracites have the chemical formula M_{3}B_{7}O_{13}X (where M = Cu^{2+}, Ni^{2+}, Co^{2+}, Fe^{2+}, Mn^{2+}, Cr^{2+} and X = F^{−}, Cl^{−}, Br^{−}, I^{−}, OH^{−}, ). Many of them are ferroelectrics and weak ferromagnets at low temperatures. This was first shown for Ni_{3}B_{7}O_{13}I (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 the ferromagnetoelectric phase belongs to the group and for others to , , m or 1. In accordance with Table 1.5.8.4, the spontaneous polarization is oriented perpendicular to the weak ferromagnetic moment for the groups and m. There results a complicated behaviour of boracites in external magnetic and electric fields. It 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 was increased beyond 6 kOe, the induced polarization changed sign because the spontaneous magnetization had been reversed. On reversing the applied magnetic field, the rest of the hysteresis loop describing the response was obtained.

The hysteresis loop in the linear magnetoelectric effect in ferromagnetoelectric Ni_{3}B_{7}O_{13}I at 46 K (Ascher et al., 1966). 
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 about the present state of investigation of the ferromagnetoelectrics are presented in the review article of Schmid (1994b).
The ferromagnetoelectrics appear as type 4 and the ferroelectric antiferromagnets of type III^{a} as types 2 and 3 in Table 1.5.8.3. The table shows that the linear magnetoelectric effect is admitted by all ferromagnetoelectrics and all ferroelectric antiferromagnets of type III^{a}, except those that belong to the two point groups and .
Concluding Section 1.5.8, it is worth noting that the magnetoelectric effect is still actively investigated. Recent results in this field can be found in papers presented at the 1993 and 1996 conferences devoted to this subject (see Schmid et al., 1994; Bichurin, 1997, 2002).
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