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

International Tables for Crystallography (2006). Vol. D, ch. 1.5, pp. 137-142

Section 1.5.8. Magnetoelectric effect

A. S. Borovik-Romanova and H. Grimmerb*

aP. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and bLabor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH-5234 Villigen PSI, Switzerland
Correspondence e-mail:  hans.grimmer@psi.ch

1.5.8. Magnetoelectric effect

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Curie (1894[link]) 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[link]), who considered the invariants in the expansion of the thermodynamic potential up to linear terms in [H_i]. For materials belonging to certain magnetic point groups, the thermodynamic potential [\Phi] can be written in the form[\Phi = \Phi _{0} - \alpha _{ij}E_{i}H_{j}. \eqno(1.5.8.1)]If (in the absence of a magnetic field) an electric field [\bf E] is applied to a crystal with potential (1.5.8.1)[link], a magnetization will be produced: [M_{j} = -{{\partial{\Phi}}\over{\partial{H_{j}}}} = \alpha_{ij}E_{i}. \eqno(1.5.8.2)]Conversely, an electric polarization [\bf P] arises at zero electric field if a magnetic field is applied: [P_{i} = -{{\partial{\Phi}}\over{\partial{E_{i}}}} = \alpha_{ij}H_{j}. \eqno(1.5.8.3)]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 [E_{i}H_{j}H_{k}] or [H_{i}E_{j}E_{k}]. They will be described in Section 1.5.8.2.[link]

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 [\bf H] is an axial vector that changes sign under R and [\bf E] is a polar vector that is invariant under time inversion, [\alpha_{ij}] is an axial tensor of second rank, the components of which all change sign under time inversion (R). From relation (1.5.8.1)[link], it follows that a magnetic group which allows the magneto­electric effect cannot possess a centre of symmetry ([C_{i} = \bar{1}]). However, it can possess it multiplied by R ([C_{i}R = \bar{1}']) (see Table 1.5.8.1[link]). There are 21 magnetic point groups that possess a centre of symmetry. The detailed analysis of the properties of the tensor [\alpha_{ij}] shows that among the remaining 69 point groups there are 11 groups for which the linear magnetoelectric effect is also forbidden. These groups are [{\bi C}_{3h} =\bar{6}, {\bi C}_6({\bi C}_3)=6', {\bi C}_{6h}({\bi C}_{3h})=] [6'/m, {\bi D}_{3h}=\bar{6}m2, {\bi D}_{3h}({\bi C}_{3h})=\bar{6}m'2', {\bi D}_{6h}({\bi D}_{3h})=6'/mmm'], [{\bi D}_6({\bi D}_3)=6'22', {\bi C}_{6v}({\bi C}_{3v})=6'm'm,{\bi T}_d=\bar{4}3m,{\bi O}({\bi T})=4'32'] and [{\bi O}_h({\bi T}_d)=] [m'\bar{3}'m].

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 [\alpha_{ij}]
SchoenfliesHermann–Mauguin
[{\bi C}_{1}] [1] [\left[\matrix { \alpha_{11} & \alpha_{12} & \alpha_{13}\cr \alpha_{21} & \alpha_{22} & \alpha_{23}\cr \alpha_{31} & \alpha_{32} & \alpha_{33} }\right]]
[{\bi C}_{i}({\bi C}_{1})] [{\bar 1}']
   
[{\bi C}_{2}] [2\,(=121)] [\left[\matrix { \alpha_{11} & 0 & \alpha_{13}\cr 0 & \alpha_{22} & 0 \cr \alpha_{31} & 0 & \alpha_{33} }\right]]
[{\bi C}_{s}({\bi C}_{1})] [m'\,(=1m'1)]
[{\bi C}_{2h}({\bi C}_{2})] [2/m'\,(=1\,2/m'\,1)]
  (unique axis y)
[{\bi C}_{s} ] [m\, (=1m1)] [\left[\matrix { 0 & \alpha_{12} & 0 \cr \alpha_{21} & 0 & \alpha_{23}\cr 0 & \alpha_{32} & 0 }\right]]
[{\bi C}_{2}({\bi C}_{1})] [2'\,(=12'1)]
[{\bi C}_{2h}({\bi C}_{s})] [2'/m\,(=1\,2'/m\,1)]
  (unique axis y)
[{\bi D}_{2} ] [222] [\left[\matrix { \alpha_{11} & 0 & 0 \cr 0 & \alpha_{22} & 0 \cr 0 & 0 & \alpha_{33} }\right]]
[{\bi C}_{2v}({\bi C}_{2})] [m'm'2\,[2m'm',\,m'2m']]
[{\bi D}_{2h}({\bi D}_{2})] [m'm'm']
[{\bi C}_{2v}] [mm2] [\left[\matrix { 0 & \alpha_{12} & 0 \cr \alpha_{21} & 0 & 0 \cr 0 & 0 & 0 }\right]]
[{\bi D}_{2}({\bi C}_{2})] [2'2'2]
[{\bi C}_{2v}({\bi C}_{s})] [2'mm'\,[m2'm']]
[{\bi D}_{2h}({\bi C}_{2v})] [mmm']
[{\bi C}_{4},\, {\bi S}_{4}({\bi C}_{2}),\, {\bi C}_{4h}({\bi C}_{4})] [4,\, {\bar 4}',\,4/m' ] [\left[\matrix { \alpha_{11}& \alpha_{12} & 0 \cr - \alpha_{12} & \alpha_{11} & 0 \cr 0 & 0 & \alpha_{33} }\right]]
[ {\bi C}_{3},\, {\bi S}_{6}({\bi C}_{3})] [3,\, {\bar 3}' ]
[{\bi C}_{6},\, {\bi C}_{3h}({\bi C}_{3}),\, {\bi C}_{6h}({\bi C}_{6})] [6,\, {\bar 6}',\,6/m']
[{\bi S}_{4} ] [{\bar 4}] [\left[\matrix { \alpha_{11}& \alpha_{12} & 0 \cr \alpha_{12}&- \alpha_{11} & 0 \cr 0 & 0 & 0 }\right]]
[{\bi C}_{4}({\bi C}_{2})] [4']
[{\bi C}_{4h}({\bi S}_{4})] [4'/m']
[{\bi D}_{4},\, {\bi C}_{4v}({\bi C}_{4})] [422,\,4m'm' ] [\left[\matrix { \alpha_{11} & 0 & 0 \cr 0 & \alpha_{11} & 0 \cr 0 & 0 & \alpha_{33} }\right]]
[{\bi D}_{2d}({\bi D}_{2}),\, {\bi D}_{4h}({\bi D}_{4})] [{\bar 4}'2m'\, [{\bar 4}'m'2],\,4/m'm'm' ]
[{\bi D}_{3},\, {\bi C}_{3v}({\bi C}_{3}),\, {\bi D}_{3d}({\bi D}_{3}) ] [32,\,3m',\, {\bar 3}'m' ]
[{\bi D}_{6},\, {\bi C}_{6v}({\bi C}_{6})] [622,\,6m'm']
[{\bi D}_{3h}({\bi D}_{3}),\, {\bi D}_{6h}({\bi D}_{6})] [{\bar 6}'m'2\, [{\bar 6}'2m'],\, 6/m'm'm']
[{\bi C}_{4v},\, {\bi D}_{4}({\bi C}_{4})] [4mm,\,42'2' ] [\left[\matrix { 0 & \alpha_{12} & 0 \cr - \alpha_{12} & 0 & 0 \cr 0 & 0 & 0 }\right]]
[{\bi D}_{2d}({\bi C}_{2v}),\, {\bi D}_{4h}({\bi C}_{4v})] [{\bar 4}'2'm\, [{\bar 4}'m2'],\,4/m'mm]
[{\bi C}_{3v},\, {\bi D}_{3}({\bi C}_{3}),\, {\bi D}_{3d}({\bi C}_{3v}) ] [3m,\,32',\, {\bar 3}'m]
[{\bi C}_{6v},\, {\bi D}_{6}({\bi C}_{6}) ] [6mm,\,62'2' ]
[{\bi D}_{3h}({\bi C}_{3v}),\, {\bi D}_{6h}({\bi C}_{6v})] [{\bar 6}'m2'\, [{\bar 6}'2'm],\,6/m'mm]
[{\bi D}_{2d},\, {\bi D}_{2d}({\bi S}_{4})] [{\bar 4}2m,\, {\bar 4}m'2'] [\left[\matrix { \alpha_{11} & 0 & 0 \cr 0 &- \alpha_{11} & 0 \cr 0 & 0 & 0 }\right]]
[{\bi D}_{4}({\bi D}_{2}),\, {\bi C}_{4v}({\bi C}_{2v})] [4'22',\,4'm'm]
[{\bi D}_{4h}({\bi D}_{2d})] [4'/m'm'm]
[{\bi T},\, {\bi T}_{h}({\bi T})] [23,\,m' {\bar 3}' ] [\left[\matrix { \alpha_{11} & 0 & 0 \cr 0 & \alpha_{11} & 0 \cr 0 & 0 & \alpha_{11} }\right]]
[{\bi O},\, {\bi T}_{d}({\bi T}),\, {\bi O}_{h}({\bi O})] [432,\, {\bar 4}'3m',\,m' {\bar 3}'m']

All remaining 58 magnetic point groups in which the linear magnetoelectric effect is possible are listed in Table 1.5.8.1[link]. 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[link]. Alternative orientations of the same point group that give rise to the same form of [\alpha_{ij}] have been added between square brackets in Table 1.5.8.1[link]. The tensor has the same form for [32 \;(=321)] and [312], [3m'1] and [31m'], [\bar{3}'m'1] and [\bar{3}'1m']; it also has the same form for [3m1] and [31m], [32'1] and [312'], [\bar{3}'m1] and [\bar{3}'1m].

The forms of [\alpha_{ij}] for frequently encountered orientations of the CCS other than those given in Table 1.5.8.1[link] are (cf. Rivera, 1994[link])

  • (1) [112, 11m', 11\,2/m'] (unique axis z):[\left[\matrix{\alpha_{11} &\alpha_{12} & 0 \cr \alpha_{21} &\alpha_{22} & 0 \cr 0 & 0 &\alpha_{33}}\right]\semi]

  • (2) [11m, 112', 11\,2'/m] (unique axis z):[\left[\matrix{0 & 0 &\alpha_{13} \cr 0 & 0 &\alpha_{23} \cr \alpha_{31} &\alpha_{32} & 0 }\right]\semi]

  • (3) [2mm, 22'2', m'm2'\,[m'2'm], m'mm]:[\left[\matrix{0 & 0 & 0 \cr 0 & 0 &\alpha_{23} \cr 0 &\alpha_{32} & 0}\right]\semi]

  • (4) [m2m, 2'22', mm'2'\,[2'm'm], mm'm]:[\left[\matrix{0 & 0 &\alpha_{13} \cr 0 & 0 & 0 \cr \alpha_{31} & 0 & 0}\right]\semi]

  • (5) [\bar 4m2, \bar 42'm', 4'2'2, 4'mm', 4'/m'mm']:[\left[\matrix{0 &\alpha_{12} & 0 \cr \alpha_{12}& 0 & 0 \cr 0 & 0 & 0 }\right].]

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 [\bf L], 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[link]) 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: [\alpha_{11} = \alpha_{22}] and [\alpha_{33}]. The ME effect in Cr2O3 was discovered experimentally by Astrov (1960[link]) on an unoriented crystal. He verified that the effect is linear in the applied electric field. Folen et al. (1961[link]) and later Astrov (1961[link]) performed measurements on oriented crystals and revealed the anisotropy of the ME effect. In the first experiments, the ordinary magnetoelectric effect MEE (the electrically induced magnetization) was investigated. This means the magnetic moment induced by the applied electric field was measured. Later Rado & Folen (1961[link]) 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[link]) collected values of [\alpha_{\rm max}] of the known magnetoelectrics. Some are listed in Table 1.5.8.2[link] together with more recent results. Additional information about the experimental data is presented in three conference proceedings (Freeman & Schmid, 1975[link]; Schmid et al., 1994[link]; Bichurin, 1997[link]).

Table 1.5.8.2| top | pdf |
A list of some magnetoelectrics

CompoundTN or TC (K)Magnetic point groupMaximum [\alpha_{\rm obs}]References
Fe2TeO6 219 [4/{m}'{m}'{m}'] [3\times10^{-5}] 7–9, 70
DyAlO3 3.5 [{m}'{m}'{m}'] [2\times10^{-3}] 11–13
GdAlO3 4.0 [{m}'{m}'{m}'] [1\times10^{-4}] 14
TbAlO3 4.0 [{m}'{m}'{m}'] [1\times10^{-3}] 12, 15–17
TbCoO3 3.3 [mm{m}'] [3\times10^{-5}] 12, 16, 18
Cr2O3 318 [{\bar 3}'{m}'] [1\times10^{-4}] 45–49, 70, 71, W162
Nb2Mn4O9 110 [{\bar 3}'{m}'] [2\times10^{-6}] 52, 53
Nb2Co4O9 27 [{\bar 3}'{m}'] [2\times10^{-5}] 52, 53
Ta2Mn4O9 104 [{\bar 3}'{m}'] [1\times10^{-5}] 53
Ta2Co4O9 21 [{\bar 3}'{m}'] [1\times10^{-4}] 53
LiMnPO4 35 [{m}'{m}'{m}'] [2\times10^{-5}] 55, 56, 58, 60
LiFePO4 50 [mm{m}'] [1\times10^{-4}] 57, 58
LiCoPO4 22 [mm{m}'] [7\times10^{-4}] 54, 55, R161
LiNiPO4 23 [mm{m}'] [4\times10^{-5}] 54, 55, 61
GdVO4 2.4 [{4}'/{m}'{m}'m] [3\times10^{-4}] 70
TbPO4 2.2 [{4}'/{m}'{m}'m] [1\times10^{-2}] 66, 67
DyPO4 3.4 [{4}'/{m}'{m}'m] [1\times10^{-3}] 68, 69
HoPO4 1.4 [{4}'/{m}'{m}'m] [2\times10^{-4}] 72
Mn3B7O13I 26 [{m}'m{2}'] [2\times10^{-6}] C204
Co3B7O13Cl 12 m [3\times10^{-4}] S204
Co3B7O13Br 17 [{m}'m{2}'] [2\times10^{-3}] 88C1
Co3B7O13I 38 [{m}'m{2}'] [1\times10^{-3}] 90C3
Ni3B7O13I 61.5 [{m}'] [2\times10^{-4}] 74, 75, 77–79, 90C2
Ni3B7O13Cl 9 [{m}'m{2}'] [3\times10^{-4}] 74R2, 91R1
Cu3B7O13Cl 8.4 [{m}'{m}'2] [3\times10^{-6}] 88R1
FeGaO3 305 [{m}'m{2}'] [4\times10^{-4}] 84–86
TbOOH 10.0 [2/{m}'] [4\times10^{-4}] 114
DyOOH 7.2 [2/{m}'] [1\times10^{-4}] 92, 114
ErOOH 4.1 [{2}'/m] [5\times10^{-4}] 93, 114
Gd2CuO4 6.5 [mm{m}'] [1\times10^{-4}] W161
MnNb2O6 4.4 [mm{m}'] [3\times10^{-6}] 101, 102
MnGeO3 16 [mm{m}'] [2\times10^{-6}] 98–100
CoGeO3 31 [mm{m}'] [1\times10^{-4}] 70
CrTiNdO5 13 [mm{m}'] [1\times10^{-5}] 70, 89
Numbers refer to references quoted by Cox (1974[link]); codes 88C1, 90C3, 88R1, 90C2, 74R2, 91R1 refer to references quoted by Burzo (1993[link]); 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 [\alpha_{ij}] are given in rationalized Gaussian units, where [\alpha_{ij}] is dimensionless. Some authors follow Dzyaloshinskii (1959[link]) in writing (1.5.8.1)[link] as [\Phi=\Phi_0-(\alpha^{\prime}_{ij}/4\pi)E_iH_j], where [\alpha^{\prime}_{ij}] are the non-rationalized Gaussian values of the components of the magnetoelectric tensor. If SI units are used, then (1.5.8.1)[link] becomes [\Phi=\Phi_0-\alpha^{\rm SI}_{ij}E_iH_j]. The connections between the values of a tensor component expressed in these three systems are [4\pi\alpha_{ij} = \alpha^{\prime}_{ij} = 3\times 10^{8}\alpha^{\rm SI}_{ij}. \eqno(1.5.8.4)]

The units of [\alpha^{\rm SI}_{ij}] 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[link]).

Most magnetoelectrics are oxides containing magnetic ions. The ions of the iron group are contained in corundum-type oxides [magnetic point group [{\bi D}_{3d}({\bi D}_{3}) = \bar{3}'m']], triphyllite-type oxides with different magnetic groups belonging to the ortho­rhombic crystallographic structure [{\bi D}_{2h} = mmm] 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 [{\bi D}_{2h}({\bi D}_{2}) = m'm'm']], tetragonal zircon-type compounds RMO4 (R = rare earth, M = P, V) [magnetic point group [{\bi D}_{4h}({\bi D}_{2d})=4'/m'm'm]], monoclinic oxide hydroxides ROOH [magnetic point groups [{\bi C}_{2h}({\bi C}_2) = 2/m'], [{\bi C}_{2h}({\bi C}_s) = 2'/m]] and other compounds. Of particular interest is TbPO4, which has the highest value of the magnetoelectric tensor components, 1.2 × 10−2 (Rado & Ferrari, 1973[link]; Rado et al., 1984[link]). 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[link]. Another ortho­rhombic magnetoelectric crystal is ferrimagnetic FeGaO3 (Rado, 1964[link]; see Table 1.5.8.2[link]).

It has been shown in experiments on Cr2O3 that in the spin-flop phase [\alpha_{ \parallel }] becomes zero but a non-diagonal component [\alpha_{xz}] arises (Popov et al., 1992[link]). Such behaviour is possible if under the spin-flop transition the magnetic point group of Cr2O3 transforms from [{\bi D}_{3d}({\bi D}_{3}) = \bar{3}'m'] to [{\bi C}_{2h}({\bi C}_{s}) = 112'/m]. For the latter magnetic point group, the ME tensor possesses only transverse components.

The temperature dependences determined for the ME moduli, [\alpha_{ \parallel }] and [\alpha_{\perp}], in Cr2O3 are quite different (see Fig. 1.5.8.1[link]). The temperature dependence of [\alpha _{\perp}] is similar to that of the order parameter (sublattice magnetization [M_{0}]), which can be explained easily, bearing in mind that the magnetoelectric moduli are proportional to the magnitude of the antiferromagnetic vector ([\alpha \propto L_{z} = 2M_{0}]). However, to explain the rather complicated temperature dependence of [\alpha_{ \parallel }] it becomes necessary to assume that the moduli [\alpha] are proportional to the magnetic susceptibility of the crystal so that (Rado, 1961[link]; Rado & Folen, 1962[link])[\alpha_{ \parallel } = a_{ \parallel }\chi_{ \parallel }L_{z}, \quad\alpha_{\perp} = a_{\perp}\chi_{\perp}L_{z}, \eqno(1.5.8.5)]where [a_{ \parallel }] and [a_{\perp}] are new constants of the magnetoelectric effect which do not depend on temperature. Formulas (1.5.8.5)[link] provide a good explanation of the observed temperature dependence of [\alpha].

[Figure 1.5.8.1]

Figure 1.5.8.1 | top | pdf |

Temperature dependence of the components [\alpha_{ \parallel }] and [\alpha_{\perp}] in Cr2O3 (Astrov, 1961[link]).

The linear relation between [\alpha] and [L_{z} = 2M_{0}] 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 [T_{N}] 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 simultaneous pulses of both fields to a multidomain sample at temperatures below [T_{N}] (see O'Dell, 1970[link]).

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[link]) and expanded by White (1974[link]).

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 g-tensor can explain the ME effect in DyPO4 (Rado, 1969[link]). The electric-field-induced changes in single-ion anisotropy may represent the main mechanism of the ME effect in Cr2O3 (Rado, 1962[link]). 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[link]).

1.5.8.2. Nonlinear magnetoelectric effects

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Along with linear terms in E and H, the thermodynamic potential [\Phi] may also contain invariants of higher order in [E_{k}, H_{i}]: [\Phi = \Phi_0 - \alpha_{ik}E_i H_k - {\textstyle{{1}\over{2}}}\beta_{ijk}E_i H_j H_k - {\textstyle{{1}\over{2}}}\gamma_{ijk}H_i E_j E_k. \eqno(1.5.8.6)]From this relation, one obtains the following formulas for the electric polarization [P_{i}] and the magnetization [M_{i}]:[\eqalignno{P_i &= \alpha_{ik}H_k + \textstyle{{1}\over{2}}\beta_{ijk}H_j H_k + \gamma_{jik}H_j E_k, &(1.5.8.7)\cr M_i &= \alpha_{ki}E_k + \beta_{jik}E_j H_k + \textstyle{{1}\over{2}}\gamma_{ijk}E_j E_k. &(1.5.8.8)}%fd1.5.8.8]The third term in (1.5.8.7)[link] describes the dependence of the dielectric susceptibility ([\chi^{e}_{ik} = P_{i}/E_{k}]) and, consequently, of the dielectric permittivity [\varepsilon_{ik}], on the magnetic field. Similarly, the second term in (1.5.8.8)[link] points out that the magnetic susceptibility [\chi^{m}] may depend on the electric field ([\delta\chi^{m}_{ik} = \beta_{jik}E_{j}]). The tensors [\beta_{ijk}] and [\gamma_{ijk}] are symmetric in their last two indices. Symmetry imposes on [\beta_{ijk}] the same restrictions as on the piezoelectric tensor and on [\gamma_{ijk}] the same restrictions as on the piezomagnetic tensor (see Table 1.5.7.1[link]).

Ascher (1968[link]) determined all the magnetic point groups that allow the terms [EHH] and [HEE] in the expansion of the thermodynamic potential [\Phi]. These groups are given in Table 1.5.8.3[link], which has been adapted from a table given by Schmid (1973[link]). It classifies the 122 magnetic point groups according to which types of magnetoelectric effects ([EH], [EHH] or [HEE]) 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 [\bar{1}], [1'] or [\bar{1}'], as in a table given by Mercier (1974[link]). 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[link]); dia- and paramagnets as well as antiferromagnets with a nontrivial magnetic Bravais lattice have a point group containing [1'] (the 32 groups of types 1, 13, 17 and 19 in Table 1.5.8.3[link]). The 59 remaining point groups describe antiferromagnets with a trivial Bravais lattice. The 31 point groups characterized by E, the 32 containing [\bar{1}] and the 59 remaining ones correspond to a similar classification of crystals according to their electric properties (see Schmid, 1973[link]).

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 [1'] E     EHH   [1'], [21'], [m1'], [mm21'], [41'], [4mm1'], [31'], [3m1'], [61'], [6mm1'] 10 31   49 122
2   E     EHH HEE [6'], [6'mm'] 2  
3   E   EH EHH HEE [mm2], [4mm], [4'], [4'mm'], [3m], [6mm] 6  
4   E H EH EHH HEE [1], [2], m, [2'], [m'], [m'm2'], [m'm'2], [4], [4m'm'], [3], [3m'], [6], [6m'm'] 13 31
5     H EH EHH HEE [2'2'2], [42'2'], [{\bar 4}], [{\bar 4} 2'm'], [32'], [62'2'] 6  
6     H   EHH HEE [{\bar 6}], [{\bar 6} m'2'] 2  
7 [{\bar 1}]   H     HEE [{\bar 1}], [2/m], [2'/m'], [m'm'm], [4/m], [4/mm'm'], [{\bar 3}], [{\bar 3} m'], [6/m], [6/mm'm'] 10  
8       EH EHH HEE [222], [{\bar 4}'], [422], [{\bar 4}2m], [4'22'], [{\bar 4}'2m'], [{\bar 4}'2'm], [32], [{\bar 6}'], [622], [{\bar 6}'m'2], [{\bar 6}'m2'], [23], [{\bar 4}'3m'] 14     73
9         EHH HEE [{\bar 6} m2], [6'22'] 2    
10       EH     432 1   19
11 [{\bar 1}']     EH     [{\bar 1}'], [2/m'], [2'/m], [mmm'], [m'm'm'], [4/m',] [4'/m'], [4/m'm'm'], [4/m'mm], [4'/m'm'm], [{\bar 3}'], [{\bar 3}'m'], [{\bar 3}'m], [6/m'], [6/m'm'm'], [6/m'mm], [m'{\bar 3}'], [m'{\bar 3}'m'] 18  
12         EHH   [{\bar 4}3m] 1   11
13 [1']       EHH   [2221'], [{\bar 4}1'], [4221'], [{\bar 4}2m1'], [321'], [{\bar 6}1'], [6221'], [{\bar 6}m21'], [231'], [{\bar 4}3m1'] 10  
14           HEE [4'32'] 1   11
15 [{\bar 1}]         HEE [mmm], [4'/m], [4/mmm], [4'/mmm'], [{\bar 3} m], [6'/m'], [6/mmm], [6'/m'm'm], [m{\bar 3}], [m{\bar 3} m'] 10  
16 [{\bar 1}']           [6'/m], [6'/mmm'], [m'{\bar 3}'m] 3   16
17 [1']           [4321'] 1  
18 [{\bar 1}]           [m{\bar 3} m] 1  
19 [{\bar 1}, \,1', \,{\bar 1}']           [{\bar 1}1'], [2/m1'], [mmm1'], [4/m1'], [4/mmm1'], [{\bar 3}1'], [{\bar 3} m1'], [6/m1'], [6/mmm1'], [m{\bar 3}1'], [m{\bar 3} m1'] 11  

Table 1.5.8.3[link] 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 [{\bi O}_h = m\bar{3}m], the grey group ([{\bi O}+R{\bi O})=4321'] and the three black–white groups [{\bi C}_{6h}({\bi C}_{3h}) = 6'/m], [{\bi D}_{6h}({\bi D}_{3h}) = 6'/mmm'] and [{\bi O}_h({\bi T}_d) = m'\bar{3}'m].

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[link]). 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[link]).

The 21 point groups of types 7, 14 and 15 allow only the magnetoelectric effect [HEE]. These groups contain [{\bi C}_i = \bar{1}], except [4'32']. The compounds belonging to these groups possess only one tensor of magnetoelectric susceptibility, the tensor [\gamma_{ijk}] of the nonlinear ME effect. The effect is described by [\eqalignno{P_{i} &= \gamma_{jik}H_{j}E_{k}, &(1.5.8.9)\cr M_{i} &= \textstyle{{1}\over{2}}\gamma_{ijk}E_{j}E_{k}. &(1.5.8.10)}%fd1.5.8.10]

The magnetic point group of ferrimagnetic rare-earth garnets RFe5O12 ([R =] Gd, Y, Dy) is [{\bi D}_{3d}({\bi S}_{6})=\bar{3}m'], which is of type 7. Therefore, the rare-earth garnets may show a nonlinear ME effect corresponding to relations (1.5.8.9)[link] and (1.5.8.10)[link]. This was observed by O'Dell (1967[link]) 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[link]) to investigate this ME effect experimentally. Later Lee et al. (1970[link]) observed the ME effect defined by relation (1.5.8.10)[link]. Applying both static electric fields and alternating ones (at a frequency [\omega]), they observed an alternating magnetization at both frequencies [\omega] and [2\omega]. A nonlinear ME effect of the form [HEE] was also observed in the weakly ferromagnetic orthoferrites TbFeO3 and YbFeO3. Their magnetic point group is [{\bi D}_{2h}({\bi C}_{2h})=m'm'm].

Moreover, paramagnets that do not possess an inversion centre [{\bi C}_i = \bar{1}] may show an ME effect if the point group is not [4321']. They have one of the 20 grey point groups given as types 1 or 13 in Table 1.5.8.3[link]. Bloembergen (1962[link]) 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 [\beta_{ijk}]: [\eqalignno{P_{i} &= \textstyle{{1}\over{2}}\beta_{ijk}H_{j}H_{k}, &(1.5.8.11)\cr M_{i} &= \beta_{jik}E_{j}H_{k}. &(1.5.8.12)}%fd1.5.8.12]The PME effect was discovered by Hou & Bloembergen (1965[link]) in NiSO4·6H2O, which belongs to the crystallographic point group [{\bi D}_{4}=422]. The only nonvanishing components of the third-rank tensor are [\beta_{xyz}] [= \beta_{xzy}] [= -\beta_{yzx}] [= -\beta_{yxz}] [= \beta] ([\beta_{14} = -\beta_{25} = 2\beta] in matrix notation), so that [{\bf P} = \beta(H_yH_z, -H_xH_z, 0)] and [{\bf M} = \beta]([-E_yH_z], [E_xH_z], [E_xH_y-E_yH_x]). Both effects were observed: the polarization [\bf P] by applying static ([H_{z}]) and alternating ([H_{x}] or [H_{y}]) magnetic fields and the magnetization [\bf M] by applying a static magnetic field [H_{z}] and an alternating electric field in the plane [xy]. 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 is[\beta (4.2 \;{\rm K}) = 2.2 \times 10^{-9}\hbox{ cgs units.} \eqno(1.5.8.13)]The 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 ([{\bi C}_{i}=\bar{1}]), either by itself or multiplied by [R = 1'], admit all three types of ME effect: the linear ([EH]) and two higher-order ([EHH] and [HEE]) effects. There are many magnetically ordered compounds in which the nonlinear ME effect has been observed. Some of them are listed by Schmid (1973[link]); more recent references are given in Schmid (1994a[link]).

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 [\Phi] must contain invariants of the form[\Phi = \Phi_{0} - \pi_{ijk\ell}E_{i}H_{j}T_{k\ell}. \eqno(1.5.8.14)]

The problem of the piezomagnetoelectric effect was considered by Rado (1962[link]), Lyubimov (1965[link]) and recently in detail by Grimmer (1992[link]). All 69 white and black–white magnetic point groups that possess neither [{\bi C}_{i}=\bar{1}] nor [R = 1'] admit the piezomagnetoelectric effect. (These are the groups of types 2–6, 8–12, 14 and 16 in Table 1.5.8.3[link].) The tensor [\pi_{ijk\ell}] that 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[link]). Grimmer (1992[link]) analyses in which antiferromagnets it could be observed.

1.5.8.3. Ferromagnetic and antiferromagnetic ferroelectrics

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Neronova & Belov (1959[link]) pointed out that there are ten magnetic point groups that admit the simultaneous existence of spontaneous dielectric polarization [\bf P] and magnetic polarization [\bf M]. Materials with such a complicated ordered structure are called ferromagnetoelectrics. Neronova and Belov considered only structures with parallel alignment of [\bf P] and [\bf M] (or [\bf L]). There are three more groups that allow the coexistence of ferroelectric and ferromagnetic order, in which [\bf P] and [\bf M] are perpendicular to each other. Shuvalov & Belov (1962[link]) 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[link]; they are given with more details in Table 1.5.8.4[link].

Table 1.5.8.4| top | pdf |
List of the magnetic point groups of the ferromagnetoelectrics

Symbol of symmetry group Allowed direction of
SchoenfliesHermann–MauguinPM
[{\bi C}_1] 1 Any Any
[{\bi C}_2] 2 [ \parallel 2] [\parallel 2]
[{\bi C}_2({\bi C}_1)] [2'] [ \parallel 2'] [\perp 2']
[{\bi C}_s={\bi C}_{1h}] m [ \parallel m] [\perp m]
[{\bi C}_s({\bi C}_1)] [m'] [ \parallel m'] [ \parallel m']
[{\bi C}_{2v}({\bi C}_2)] [m'm'2] [ \parallel 2] [ \parallel 2]
[{\bi C}_{2v}({\bi C}_s)] [m'm2'] [ \parallel 2'] [\perp m ]
[{\bi C}_4] [4] [ \parallel 4] [ \parallel 4]
[{\bi C}_{4v}({\bi C}_4)] [4m'm'] [ \parallel 4] [ \parallel 4]
[{\bi C}_3] [3] [ \parallel 3] [ \parallel 3]
[{\bi C}_{3v}({\bi C}_3)] [3m'] [ \parallel 3] [ \parallel 3]
[{\bi C}_6] [6] [ \parallel 6] [ \parallel 6]
[{\bi C}_{6v}({\bi C}_6)] [6m'm'] [ \parallel 6] [ \parallel 6]

Notice that [\bf P] and [\bf M] must be parallel in eight point groups, they may be parallel in [1] and [m'], and they must be perpendicular in [2'], m and [m'm2'] (see also Ascher, 1970[link]). The magnetic point groups listed in Table 1.5.8.4[link] 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[link]): [{\bi C}_{4}({\bi C}_{2}) = 4']; [{\bi C}_{4v}({\bi C}_{2v}) = 4'mm']; [{\bi C}_{6}({\bi C}_{3}) = 6']; [{\bi C}_{6v}({\bi C}_{3v}) = 6'mm';] [{\bi C}_{2v} = mm2]; [{\bi C}_{4v} = 4mm]; [{\bi C}_{3v} = 3m]; [{\bf C}_{6v} = 6mm].

The first experimental evidence to indicate that complex perovskites may become ferromagnetoelectric was observed by the Smolenskii group (see Smolenskii et al., 1958[link]). 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[link]) proved that these compounds undergo a transition into a weakly ferromagnetic state at temperatures TN = 11 and 9 K, respectively.

BiFeO3 is an antiferromagnet below TN = 643 K. This was proved by neutron scattering (Kiselev et al., 1962[link]; Michel et al., 1969[link]) and magnetic measurements (Smolenskii et al., 1962[link]; see also Venevtsev et al., 1987[link]). BiFeO3 also possesses a spontaneous electric polarization. The magnetic point group above TN is 3m1′ and below it should have been 3m (Kiselev et al., 1962[link]), but in reality it possesses an antiferromagnetic spatially modulated spin structure (Sosnovska et al., 1982[link]). Another ferroelectric antiferromagnet, YMnO3, was found by Bertaut et al. (1964[link]). It becomes ferroelectric at Tc = 913 K (with paramagnetic point group 6mm1′) and antiferromagnetic at TN = 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[link]). The symmetries of both antiferromagnetic ferroelectrics described above do not allow weak ferromagnetism according to Table 1.5.5.2[link], and, experimentally, a spontaneous ferromagnetic moment has not been observed so far.

Since Schmid (1965[link]) developed a technique for growing single crystals of boracites, these compounds have become the most interesting ferromagnetoelectrics. The boracites have the chemical formula M3B7O13X (where M = Cu2+, Ni2+, Co2+, Fe2+, Mn2+, Cr2+ and X = F, Cl, Br, I, OH, [\hbox{NO}_3^-]). Many of them are ferroelectrics and weak ferromagnets at low temperatures. This was first shown for Ni3B7O13I (see Ascher et al., 1966[link]). The symmetries of all the boracites are cubic at high temperatures and their magnetic point group is [\bar{4}3m1']. As the temperature is lowered, most become ferroelectrics with the magnetic point group [mm21']. 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 [m'm2'] and for others to [m'm'2], [m'], m or 1. In accordance with Table 1.5.8.4[link], the spontaneous polarization [\bf P] is oriented perpendicular to the weak ferromagnetic moment [{\bf M}_{D}] for the groups [m'm2'] 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[link] shows the results of measurements on Ni–I boracite with spontaneous polarization along [001] and spontaneous magnetization initially along [[\bar{1}\bar{1}0]]. 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 [\hbox{ME}_{ \parallel }] response was obtained.

[Figure 1.5.8.2]

Figure 1.5.8.2 | top | pdf |

The hysteresis loop in the linear magnetoelectric effect in ferromagnetoelectric Ni3B7O13I at 46 K (Ascher et al., 1966[link]).

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[link] but turned upside down will be obtained (cf. Schmid, 1967[link]).

The similarity of the jumps in the curves of linear magneto­striction (see Fig. 1.5.7.2[link]) 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[link]).

The ferromagnetoelectrics appear as type 4 and the ferroelectric antiferromagnets of type IIIa as types 2 and 3 in Table 1.5.8.3[link]. The table shows that the linear magnetoelectric effect is admitted by all ferromagnetoelectrics and all ferroelectric antiferromagnets of type IIIa, except those that belong to the two point groups [{\bi C}_{6}({\bi C}_{3}) = 6'] and [{\bi C}_{6v}({\bi C}_{3v}) = 6'mm'].

Concluding Section 1.5.8[link], 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[link]; Bichurin, 1997[link], 2002[link]).

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