Cation-ordered A{}^{\prime}_{1/2}A{}^{\prime\prime}_{1/2}B{}_{2}X{}_{4} magnetic spinels as magnetoelectrics

Cation-ordered A′1/2A′′1/2B2X4 magnetic spinels as magnetoelectrics

N. V. Ter-Oganessian Institute of Physics, Southern Federal University, 194 Stachki pr., Rostov-on-Don, 344090 Russia
Abstract

We show that 1:1 ordering of A and A cations in AABX magnetic spinels results in appearance of magnetoelectric properties. Possible value of magnetically induced electric polarization is calculated using the recently proposed microscopic model, which takes into account spin-dependent electric dipole moments of magnetic ions located in noncentrosymmetric crystallographic positions. We build phenomenological models of magnetic phase transitions in cation-ordered spinels, which describe ferromagnetic and antiferromagnetic ordering patterns of B cation spins, and calculate the respective magnetoelectric responses. We find that magnetoelectric coefficients diverge at ferromagnetic or weak ferromagnetic phase transitions in ordered spinels.

keywords:
magnetoelectric, spinel, atomic ordering, multiferroic
journal: Journal of Magnetism and Magnetic Materials

1 Introduction

The search for new magnetoelectric materials has intensified in the last decade due to the discovery of whole new classes of magnetoelectrics and promising technological applications Pyatakov and Zvezdin (2012). Multiferroic materials are sought among composite materials consisting of ferroelectric (piezoelectric) and magnetic (piezomagnetic) subsystems, as well as among single phase multiferroics. Single phase magnetoelectrics are usually divided into the so-called type-I and type-II multiferroics Khomskii (2009). In the type-I magnetoelectrics, or ferroelectromagnets, ferroelectricity and magnetic order occur independently and have different sources. Ferroelectromagnets (the prominent example is BiFeO Catalan and Scott (2009)) usually possess high ferroelectric polarization, but the generally large difference between the ferroelectric and magnetic transition temperatures and the different causes of the two orders result in small coupling between them.

In contrast, the type-II magnetoelectrics, in which ferroelectricity occurs as a result of a magnetic phase transition, offer direct coupling between the magnetic and ferroelectric subsystems. Such magnetoelectrics, however, are characterized by much lower transition temperatures (10 – 40 K) and low electric polarization values (usually of the order of 10 – 100). The prominent examples of such magnetoelectrics are rare-earth manganites RMnO (R=Gd, Tb, and Dy) Kimura et al. (2005). Cupric oxide CuO has the highest phase transition temperature (230 K) among magnetoelectrics discovered to date Kimura et al. (2008), which is still very low for practical applications. Therefore, the search for new single phase magnetoelectrics with higher phase transition temperatures is of paramount importance.

The spinel class with the general chemical formula ABX, where X=O, S, Se, or Te, and A and B are metals, is one of the richest structural classes Krupička and Novák (1982); van Stapele (1982). Spinels offer very high temperatures of magnetic phase transitions of the order of 1000 K (e.g. 860 K in FeO, 1020 K in -FeO, and 790 K in CoFeOBrabers (1995) and allow for great flexibility in both cation and anion substitution Krupička and Novák (1982); van Stapele (1982); Brabers (1995). All this makes spinels interesting from the point of view of searching for new materials and tailoring their properties.

The spinel compounds are widely used as constituents of multiferroic composites Vaz et al. (2010), whereas in single phase spinels the magnetoelectric effect (ME) has been found in a limited number of crystals. The ME effect was observed, for example, in CoCrO Yamasaki et al. (2006) and ZnCrSe Murakawa et al. (2008). However, these magnetoelectric spinels are characterized by low magnetic phase transition temperatures (of the order of 20 K) and incommensurately modulated magnetic order of the ferroelectric phase.

In this work we analyze magnetic spinels with the general chemical formula AABX. We show that the chemical ordering of the A and A cations results in the appearance of magnetoelectricity in such spinels.

2 Atomic ordering in A′1/2A′′1/2B2X4 spinels

The high symmetry ABX cubic spinel structure is described by the space group Fdm (). If more than one sort of cations is present in one of the equivalent sublattices, a tendency generally exists to decrease the internal energy by ordering the cations. Such ordering may possess both short and long range characters, depending on the energy gain and thermodynamic history of the crystal. As a general rule one may state that the higher the difference in valences of inequivalent cations the stronger their tendency to order Krupička and Novák (1982). The examples of spinels exhibiting atomic ordering in the A sublattice are LiGaCrO, LiInCrO Okamoto et al. (2013), CuInCrS Kesler et al. (2005), and FeCuCrS Palmer and Greaves (1999); Aminov et al. (2012), whereas the atomic ordering in the B sublattice may be attained in Zn[LiNb]O, Zn[LiSb]O, and Fe[LiFe]O Krupička and Novák (1982).

Various types of cation orderings in spinels (1:1 in the A sublattice, and 1:1 ordering in the B sublattice, 1:3 ordering in the B sublattice, and others) are considered in Haas (1965); Talanov and Shirokov (2014) and possible orders of the order-disorder phase transitions are established. The atomic ordering results in loss of some symmetry elements, which reduces the cubic symmetry. In this work we focus on 1:1 cation ordering of A cations in AABX spinels, which results in every A cation surrounded by four A cations and viceversa. Such ordering leads to reduction of the crystal symmetry to F3m (Haas (1965) locally if only short range ordering is attained or globally if a long range order is established.

The distribution of atoms over the lattice sites measured by the degree of atomic ordering is an important property of multiatomic crystals. The atomic ordering degree depends on the thermodynamic history of the sample or synthesis conditions and frequently can be varied to a large extent. Among such crystals are ordering alloys and multiatomic compounds such as oxides and halogenides. If such crystals undergo structural or magnetic phase transitions, the temperatures of these transitions and the macroscopic properties of the crystals strongly depend on the type and degree of atomic ordering. Such behavior is, for example, ubiquitous in the perovskite class ABO, which offers great potential for ion substitution. For example, disordered and ordered samples of PbScTaO show completely different dielectric behavior Chu et al. (1993), whereas cation ordering in SrFeMoO significantly influences the magnetotransport and magnetic properties Sarma et al. (2000); Sánchez-Soria et al. (2002).

When interpreting the influence of atomic ordering on properties of crystals one usually proceeds with the assumption that the degree of atomic ordering makes quantitative contribution to the thermodynamic potential Wagner et al. (1996). Within the framework of phenomenological theory this approach reduces to the introduction of the dependence on of the coefficients in the thermodynamic potential expansion with respect to the relevant order parameters Bokov (1989). However, it was shown that the influence of atomic ordering can be much more substantial Sakhnenko and Ter-Oganesyan (2003); Sakhnenko and Ter-Oganessian (2005). Namely, at additional contributions to the thermodynamical potential may arise, which are forbidden by symmetry in the disordered case . These contributions manifest themselves especially strong when they include degrees of freedom, which are described by macroscopic tensors. This results in formation of corresponding macroscopic fields during the phase transitions and divergencies in the corresponding susceptibilities Sakhnenko and Ter-Oganesyan (2003).

The 1:1 cation ordering in the A sublattice of AABX spinels is described by the order parameter transforming according to the irreducible representation (IR) GM of the space group Fdm. Denoting by and the number of atoms and , respectively, in one of the sublattices appearing upon atomic ordering, we can define the atomic ordering degree as

 s=NA′−NA′′NA′+NA′′.

Thus, varies from zero for completely disordered crystal to for completely ordered one. Nonzero results in disappearance of center of inversion and lowering of the crystal symmetry to F3m. The emergence of noncentrosymmetric structure upon 1:1 cation ordering in AABX spinels is to be contrasted with 1:1 cation ordering in AABO or ABBO perovskites, where such ordering results in centrosymmetric crystal lattice.

3 Magnetic phase transitions in spinels and atomic ordering

Spinels exhibit a variety of magnetic structures including ferromagnetic (e.g. CuCrZrS Iijima et al. (2003)), ferrimagnetic (e.g. FeCrS Kalvius et al. (2010)), antiferromagnetic (e.g. MgVO Wheeler et al. (2010)), and incommensurate (e.g. CoCrO Yamasaki et al. (2006)), which is explained by the fact that both the A and B sublattices can incorporate magnetic ions. The B ions also form the so-called pyrochlore lattice, which is known to give rise to very strong geometrical frustration effects Takagi and Niitaka (2011).

Detailed representation analysis of possible magnetic structures in spinels is given in Izyumov et al. (1979). Most of the magnetic structures observed in spinels, especially those appearing at high temperatures, are described by the wave vector , i.e. the magnetic unit cell coincides with the crystal cell Oles et al. (1976). Incommensurate magnetic structures are found in some spinels at temperatures below 20 – 50 K and some of them are also shown to be ferroelectric (e.g. CoCrO Yamasaki et al. (2006) and ZnCrSe Murakawa et al. (2008)). Therefore, despite the fact that spinels exhibit high temperature magnetic properties, ME effect in spinels is observed only at rather low temperatures. Here we show that chemical substitution in the A sublattice of spinels with sufficient degree of cationic ordering results in high temperature magnetically ordered phases becoming magnetoelectric.

The magnetic representation for the A and B positions in ABX for is given by Izyumov et al. (1979)

 dAM =GM4+⊕GM5−, dBM =GM2+⊕GM3+⊕2GM4+⊕GM5+, (1)

respectively. The basis functions for IRs entering into the magnetic representations and are given in Izyumov et al. (1979). It has to be noted, that since the spinel structure possesses spatial inversion a magnetic structure described by a single IR with cannot induce electric polarization Kovalev (1973). This is explained by the following. The symmetry of the paramagnetic phase is , where is the space group and is time inversion. When , for any of the IR’s of a unit matrix corresponds to either or . Therefore, upon a phase transition according to this IR one of these symmetry elements preserves in the ordered phase, but non of them allows non-zero electric polarization.

However, a magnetic phase transition with respect to IR GM, which corresponds to appearance of a simple collinear antiferromagnetic ordering of spins of A cations, results in appearance of a linear ME effect. Denoting the antiferromagnetic ordering of A cation spins by the magnetoelectric interaction can be written in the form

 Lx(MyPz+MzPy)+Ly(MzPx+MxPz)+Lz(MxPy+MyPx),

where and are magnetic moment and electric polarization, respectively. (Here and in the following we define the orthogonal , , and axes along the cubic edges.) Therefore, the magnetic structures with antiferromagnetically ordered spins of A cations possess linear ME effect. Such magnetic structures appear, for example, in MnAlO below  K Krimmel et al. (2006), CoO below  K Roth (1964), and CoRhO below  K Suzuki et al. (2007). The linear ME effect in A-site antiferromagnetic spinels has to be demonstrated experimentally yet.

The A-site antiferromagnetic structures in spinels, however, are rarely observed and occur at rather low temperatures. The magnetic phase transitions in spinels, and especially those taking place at high temperatures, more often occur with respect to IRs even under space inversion and entering into . The resulting magnetic structures neither induce electric polarization nor allow ME effect, since they do not break inversion symmetry. However, cation substituted spinels AABX with partial or full ordering of A and A cations will possess magnetoelectric properties.

The magnetic structures in spinels are most often described by IRs GM or GM. The latter describes antiferromagnetic ordering of B cations. IR GM enters into both and and can induce ferromagnetic, ferrimagnetic or weak ferromagnetic structures, which, besides other causes, depends on whether both the A and B ions are magnetic or not. We denote by and the magnetic order parameters transforming according to GM and describing ferromagnetic and antiferromagnetic ordering of B cations, respectively, whereas by the antiferromagnetic order parameter that transforms according to GM. The following ME interactions in spinels can be obtained

 s (Pxfyfz+Pyfzfx+Pzfxfy), (2) s (Pxgygz+Pygzgx+Pzgxgy), (3) s (Px(gyfz+gzfy)+Py(gzfx+gxfz)+Pz(gxfy+gyfx)), (4) s (Pxa1a3+Pya1a2+Pza2a3), (5) s (Px(a1gy−a3gz)+Py(a2gz−a1gx)+Pz(a3gx−a2gy)), (6) s (Px(a1fy−a3fz)+Py(a2fz−a1fx)+Pz(a3fx−a2fy)). (7)

It follows from (2) – (7) that nonzero A cation ordering () in AABX spinels results in the fact that the magnetically ordered states induced by IRs GM or GM possess linear ME effect, whereas all but the phase state induced by them become improper ferroelectric. Therefore, when interpreting the influence of the A cation order on magnetic phase transitions in spinels one has to include the terms (2) – (7) into the expansion of the thermodynamic potential. The ME coefficients are, thus, directly proportional to the degree of atomic ordering .

4 Magnetoelectric coupling

A microscopic model of ME interactions based on local noncentrosymmetric surroundings of magnetic ions was recently suggested Sakhnenko and Ter-Oganessian (2012). In current work we use this model to estimate the ME coefficients in cation-ordered spinels. In cation-disordered spinels with the cubic Fdm structure the A cations are located in noncentrosymmetric tetragonal positions (8a) with local symmetry , whereas the B cations are in positions (16d) with centrosymmetric rhombohedral symmetry . Therefore, according to the microscopic model Sakhnenko and Ter-Oganessian (2012) the A cations and the oxygen ions, whose surrounding is polar with symmetry , can contribute to the ME effect in spinels with the symmetry Fdm.

The ordering of A cations in AABX spinels results in disappearance of the inversion symmetry operation and lowering of the crystal lattice symmetry to F3m. In the tetrahedral structure the atoms A, A, B, and X are located in positions (4a), (4d), (16e), and (16e), respectively Okamoto et al. (2013). Therefore, local symmetry around the B cations becomes polar and the local electric dipole moments of these ions can contribute to the ME effect.

The primitive unit cell of the tetrahedral structure F3m contains four B cations B () located in positions , , , and , respectively. Their respective electric dipole moments induced by local polar surroundings are equal in size and directed parallel to , , , and , respectively, as shown in Fig. 1(a). This ensures absence of macroscopic electric polarization ().

According to the microscopic model of ME interactions suggested earlier Sakhnenko and Ter-Oganessian (2012), the spins of the cations B modify the electric dipole moments due to spin-orbit interaction as schematically shown in Fig. 1(b). This results in spin-dependent electric dipole moments of the cations B, which may lead to nonzero macroscopic electric polarization for certain spin configurations if .

In order to build the microscopic model we closely follow the scheme developed in Sakhnenko and Ter-Oganessian (2012). In the cubic Fdm structure the B cations are located in trigonally distorted oxygen octahedra. The local symmetry splits the triply degenerate low lying electron states into one orbital and two degenerate states. Therefore, for simplicity, as a zeroth order perturbation we consider the state

 H0|0⟩=Ed|0⟩,

where , is the Hamiltonian including the crystal field of symmetry and is the energy level. In order to obtain spin-dependent electric dipole moments, we consider a single cation B, whereas the dipole moments of the remaining B cations can be obtained from B by crystal symmetry operations. The A-site cation ordering reduces the crystal field symmetry around the B cations to , which is treated perturbatively. Compared to the symmetry the polar distortion gives additional contribution to the crystal field

 VCF=scZ, (8)

where is coefficient and is included in order to reflect the fact that only results . In (8) we consider only the lowest powers in crystal field coordinate expansion around the B cation. Here and in the following we define the orthogonal axes

 X =1√2(x−y), Y =1√6(x+y−2z), Z =1√3(x+y+z).

The spin-orbit coupling is given by

 VSO=−λ(→L⋅→S),

where is the angular momentum operator, is the spin, and is the spin-orbit coupling constant. Thus, the perturbed hamiltonian has the form , with .

The perturbation mixes the unperturbed state with other states and for simplicity it is sufficient to consider only the states with the energy , . One can write the perturbed eigenvector in the form

 |ψ⟩=|0⟩+∑αAα|pα⟩,

where are coefficients. The electric dipole moment is given then by

 →d=⟨ψ|e→r|ψ⟩=∑αAα⟨0|e→r|pα⟩+c.c.,

where is the electron charge. Performing the perturbations up to the third order we obtain the electric dipole moment of the B cation as

 dX =d02λ2Δ2SXSZ, dY =d02λ2Δ2SYSZ, dZ =d0+d0λ2Δ2(S2X+S2Y),

with

 →d0=2e∑αVα0⟨0|→r|pα⟩Δ,

where and . Thus, in addition to , which is the electric dipole moment induced by the local polar crystal field, the spin-orbit coupling gives rise to spin-dependent contribution to the electric dipole moment .

Obtaining similarly the remaining electric dipole moments of the cations B, B, and B, we find the macroscopic electric polarization () as

 Px=q(3(a1gy−a3gz)+gzfy+gyfz−2a1a3+3(a3fz−a1fy)−4fyfz), Py=q(3(a2gz−a1gx)+gxfz+gzfx−2a1a2+3(a1fx−a2fz)−4fxfz), (9) Pz=q(3(a3gx−a2gy)+gyfx+gxfy−2a2a3+3(a2fy−a3fx)−4fxfy),

where is the volume of the primitive cell, , and where we used the basis functions given in Izyumov et al. (1979) to rewrite the spins of the cations B in terms of the magnetic order parameters. The electric polarization (9), obtained from microscopic considerations, reflects the ME invariants (2),  (4),  (5), (6), and (7) and can be used to estimate the respective coefficients in the thermodynamic potential expansion. Performing the quantum perturbations to higher orders one can obtain additional contributions to polarization (9), which reflect in particular the existence of invariant (3).

In order to estimate the electric polarization (9) we use the hydrogen-like orbitals and obtain (), where is the Bohr radius and is the charge of the nucleus and core electrons in units of . Taking LiGaCrO as example Okamoto et al. (2013) we obtain  N and  Å. Using ,  eV,  eV, , and we find  Cm and  C/m. The electric polarization can, therefore, take values of the order of  C/m.

5 Phenomenological models

5.1 Ferromagnetic ordering

In this section we study ferromagnetic ordering in cation-ordered spinels, which is described by IR GM. The thermodynamic potential expansion can be written in the form

 Φ=Af2I1+b14I2+b24I21+κIME+Ap2IP−(fxHx+fyHy+fzHz), (10)

where , , , , and are coefficients, is magnetic field, , , , and . Following the usual premise of the phenomenological theory we assume , where is temperature, is the Curie temperature, and is a coefficient independent of . Since the system is far from a proper ferroelectric phase transition we take .

In the paramagnetic and paraelectric phase , , , and the magnetic susceptibility

 χαβ=∂fα∂Hβ,

where , has the form with other components equal to zero. The linear magnetoelectric coefficient

 Λαβ=∂Pα∂Hβ

is also zero. The potential (10) allows two ferromagnetic phases, either of which may occur at : an improper ferroelectric phase with and a paraelectric phase with and .

5.1.1 The phase fx=fy=fz≠0.

In this phase (phase I), which is stable for , the order parameters are given by and

 Px=Py=Pz=−sκApf2 (11)

with

 f2=−AfApAp(b1+3b2)−2s2κ2.

The magnetic susceptibility and magnetoelectric tensors represented in the coordinates are given by

 χXX=χYY=−Ap(b1+3b2)−2s2κ22Af(Apb1+s2κ2),χZZ=−12Af,
 Λ=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝sκ2f(Apb1+s2κ2)000sκ2f(Apb1+s2κ2)000sκfAfAp⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,

respectively. Nondiagonal components of equal zero. Therefore, the ME tensor components are proportional to the degree of atomic ordering and diverge as at as shown in Fig. 2(a) since .

5.1.2 The phase fx=fy=0, fz≠0.

In this phase (phase II), which is stable for , the order parameters are given by , and with

 f2=−Afb1+b2. (12)

The magnetic susceptibility and magnetoelectric tensors represented in the coordinates are given by

 χxx=χyy=Ap(b1+b2)Af(Apb1+s2κ2),χzz=−12Af,

and

 Λ=⎛⎜ ⎜ ⎜⎝0sκf(Apb1+s2κ2)0sκf(Apb1+s2κ2)00000⎞⎟ ⎟ ⎟⎠, (13)

respectively. Nondiagonal components of equal zero. Similar to the previous case the ME tensor components are proportional to the degree of atomic ordering and diverge as at as shown in Fig. 2(a).

5.1.3 Estimation of ME effect.

To make an order of magnitude estimation of the ME coefficient in the paraelectric phase II we proceed in the following way. We assume that the order parameter in (10) represents the magnetic moment of the B cation measured in Bohr magnetons . It is convenient to introduce three constants

 c1 =afb1+b2, c2 =−2Ap(b1+b2)b1Ap+κ2,

and

 c3=κAp.

The first of them determines the magnetic moment in (12) and can be estimated as  . The second constant represents the ratio of susceptibilities , which we estimate as . The third constant can be estimated from (11) using the results of section 4. Assuming that   induces improper polarization of  C/m we obtain  C/(m) (here and in the following we assume ). The phenomenological constant , which is related to the paramagnetic susceptibility as can be estimated as  mK, where we used the molar susceptibility of ferromagnetic spinel CuCrTe with  Kemu/mol-f.u. Suzuyama et al. (2006). The components of the ME tensor (13) can then be expressed in the form

 Λxy=Λyx=c2c3√c12af√Tc−T=−γ√KTc−T,

where  C/(mOe) ps/m, which is about two orders of magnitude smaller than the maximum ME coefficient observed in CrO Wiegelmann et al. (1994). Assuming a dielectric constant of observed in CoCrO Lawes et al. (2006) we obtain  mV/(cmOe), which is comparable to that of some BaTiO-based bulk particulate magnetoelectric composites Vaz et al. (2010).

5.2 Antiferromagnetic ordering described by GM4+

In this section we study antiferromagnetic ordering of B cations described by IR GM. The thermodynamic potential expansion can be written in the form

 Φ=Af2I1+Ag2Ig1+bg14Ig2+bg24I2g1+wJ+κ1IME1+Ap2IP−(fxHx+fyHy+fzHz), (14)

where , , , , and are coefficients, , , , and . As discussed in section 4 the ME interaction (3) appears in (9) only upon perturbations to higher orders and can be considered smaller than the other ME interactions. Therefore, we do not include this term in the expansion (14). Since the system is far from both the pure ferromagnetic and ferroelectric phase transitions we assume and . Minimization of potential (14) shows that the paramagnetic phase loses stability at and experiences a phase transition either to the ferroelectric phase (phase I) with , , and or paraelectric phase (phase II) with , , , , and . Here we consider only the latter phase for simplicity. Minimization of potential (14) yields the order parameters

 fz =−wAfgz, (15) g2z =−AgAf−w2Af(bg1+bg2).

The antiferromagnetic phase transition according to IR GM is a quasiproper ferromagnetic transition since the ferromagnetic moment transforms according to the same IR. The weak ferromagnetic moment arising due to (15) is usually about two orders of magnitudes smaller than the antiferromagnetic one. Therefore, the ratio can be estimated as 0.01 – 0.05 and will be used as a small parameter in expansions.

Similar to the above case of a ferromagnetic phase transition we assume with , where is the Néel temperature. In the paramagnetic phase at the nonzero components of the magnetic susceptibility tensor are given by

 χαα=AgAfAg−w2=1Af+w2A2fag(T−TN),

whereas the magnetoelectric tensor is equal to zero. Therefore, diverges at . However, due to the smallness of the temperature region of high is very narrow.

In the phase II at temperatures below the nonzero components of magnetic susceptibility tensor are given by

 χxx=χxx=−w2(bg1+bg2)A2fagbg1(TN−T),χzz=1Af+w22A2fag(TN−T),

where for and we consider only the first term in their expansions with respect to .

At temperatures close to the magnetoelectric tensor is given by

 Λ=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝02sκ1w2Apbg1A2fgz02sκ1w2Apbg1A2fgz00000⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (16)

i.e. diverges at as as shown in Fig. 2(a), since

 g2z=ag(TN−T)bg1+bg2.

At temperatures significantly lower than we obtain

 Λ=⎛⎜ ⎜ ⎜ ⎜⎝0−gzsκ1AfAp−g2zs2κ210−gzsκ1AfAp−g2zs2κ2100000⎞⎟ ⎟ ⎟ ⎟⎠. (17)

Here we again considered only the leading term in expansion of with respect to .

To estimate the ME effect we introduce two constants

 c1=agbg1+bg2,c2=−agbg1bg1+bg2.

The first of them determines the temperature dependence of the order parameter as , whereas reflects the temperature dependence of magnetic susceptibility as . The ME coefficients in (16) take then the form

 Λxy=Λyx=2√c1sw2κ1c2A2fAp√TN−T.

Similarly to the case of section 5.1.3 we can use the estimates   and C/(m), whereas can be tentatively taken equal to  mK from section 5.1.3 (both in the current case and in section 5.1.3 determine the temperature dependence of magnetic susceptibility, whereas in the current case the fact that the weak ferromagnetism is considered is accounted for by the factor in the expression for and ). Assuming we obtain

 Λxy=Λyx=γ√KTN−T,

where  ps/m, which is about two orders of magnitude smaller than that in section 5.1.

To estimate the ME coefficients in (17) we assume since the system is far from ferroelectric and ferromagnetic phase transitions. Therefore, nonzero coefficients in (17) can be written as . From the paramagnetic susceptibility  /T of Cr spins in LiGaCrO Okamoto et al. (2013) we can estimate as . Taking we obtain the components of ME tensor (17) of the order of 1.3 ps/m.

5.3 Antiferromagnetic ordering described by GM5+

In this section we study antiferromagnetic ordering of B cations described by IR GM. The thermodynamic potential expansion can be written in the form

 Φ=Af2I1+Aa2Ia1+ba14Ia2+ba24I2a1+κ2IME2+κ3IME3+Ap2IP−(fxHx+fyHy+fzHz), (18)

where , , , and are coefficients, , , , and . In this case we again assume and since the system is far from both the pure ferromagnetic and ferroelectric phase transitions. In the paramagnetic phase at the nonzero magnetic susceptibility tensor components are given by , whereas the linear ME tensor is zero.

The thermodynamic potential (18) allows two antiferromagnetic phases, either of which becomes stable for : a ferroelectric phase with , , and (phase I) and a paraelectric phase with , , , and (phase II). Phase I is stable for , whereas phase II for