In the last two decades, significant progress has been made in combining ferroelectricity and magnetism in the same material. Usually, magnetism and ferroelectricity were thought of as independent phenomena. However, the discovery of spin-induced ferroelectricity with a strong coupling between magnetism and electric polarization is intriguing. Whether they interact non-trivially is determined by the symmetries of the crystal lattice and the appearing magnetic structure, which, in turn, depends on the nature of magnetic ions, their exchange, and anisotropic interactions, the presence of frustration, etc. Several structural families of compounds exhibiting magnetoelectric (ME) or multiferroic properties have already been reported. This paper gives an account of the richness of structure, magnetism, magnetoelectric and multiferroic properties of spinels. After presenting symmetry aspects of the crystal and magnetic structures of spinels related to these phenomena, we give several examples of magnetoelectric and multiferroic spinels explaining the relations between magnetism and electric polarization and discuss their future perspectives.

It is well known that electricity and magnetism are interrelated, but they are generally studied independently in materials. However, some magnetic insulators exhibit a magnetoelectric effect consisting in a linear coupling of electric and magnetic fields in them.1 The first material predicted to show a magnetoelectric effect was the corundum type antiferromagnet Cr2O3 with Néel temperature TN=307 K.2,3 Below TN, an applied magnetic field or electric field induces an electric polarization P or magnetization M, respectively, which varies linearly with the magnetic field (Pi=αikHk) or electric field (Mk=αikEi), where αik is the linear magnetoelectric coefficient. Cr2O3 exhibits a magnetoelectric coefficient, αzz=4.31 ps/m around 263 K. Following this, several other materials were found to exhibit a linear magnetoelectric effect. All these materials require broken time-reversal symmetry and spatial inversion. Therefore, knowledge of crystal and magnetic structures is essential to find magnetoelectric materials.

There exists another class of magnetoelectric materials called multiferroics, where magnetism and ferroelectricity not only coexist but the ferroelectric polarization appears without applying an external magnetic field also. These materials can be further classified into type-I and type-II multiferroics.4 In type-I multiferroics, electric polarization and magnetic ordering occur independently. For example, the ferroelectric transition in BiFeO3 takes place at 1103 K, and the antiferromagnetic ordering temperature is 643 K.5 Thus, due to the large difference between the temperatures of magnetic and ferroelectric phase transitions, the intrinsic coupling between electric polarization and magnetism is weak. On the other hand, the ferroelectricity in type-II multiferroics is induced by magnetic ordering below the magnetic ordering temperature, which implies a strong coupling between the two orders. Thus, type-II multiferroics can be more promising for technological applications. Recently, polar (pyroelectric) magnets have been suggested to constitute another class of multiferroics. In these materials, the polar distortion arises from chemical ordering, and magnetic ordering induces a change in polarization that may or may not be switched by the electric field.6 

In type-II multiferroics, the spin structure that induces ferroelectric polarization is decided by the nature of the magnetic interactions, which depends on various factors such as the magnetic cations, and their connectivity in the lattice structure and the local or global symmetry. Most of the type-II multiferroics show incommensurate spiral magnetic structures resulting from a weak to moderate magnetic frustration; however, commensurate or even collinear spin structures can also break inversion symmetry and induce ferroelectricity.7 

Several microscopic mechanisms, such as inverse Dzyaloshinskii–Moriya (DM) interaction, exchange striction, and spin dependent p-d hybridization, have been proposed for spin-induced ferroelectricity in various structural families of materials.8 Therefore, the investigation of magnetic materials with different crystal structures is important for understanding the origin of ferroelectricity and developing multiferroics with a strong coupling at room temperature. Several review articles on multiferroic and magnetoelectric materials have been published in the literature recently, including the discussion of microscopic mechanisms of the coupling of magnetism and ferroelectricity, and possible applications.8–13 

Type-II multiferroics with various structural families have been reported in the literature. The first material to show magnetic field control of electric polarization was TbMnO3 with an orthorhombic perovskite structure.14 Here, the distorted and vortex shared MnO6 octahedra give rise to competing ferromagnetic and antiferromagnetic interactions, which result in a noncollinear cycloidal magnetic structure. The spontaneous polarization at the cycloidal ordering temperature (TN=28 K) arises through the inverse DM interaction.15–17 Interestingly, as the magnetic interactions are varied through the size of the rare-earth cation, a collinear E-type magnetic structure induces a large polarization below TN=26 K in the orthorhombic HoMnO3 stabilized under high pressure.18 On the other hand, the family of orthorhombic oxides RMn2O5 (R = Tb, Ho, and Dy) with the space group Pbam consists of edge shared Mn4+O6 forming a ribbon along the c axis and Mn3+O5 pyramids that connect with the former by corner-sharing.19,20 Such a lattice geometry of magnetic ions results in competing magnetic exchange interactions, which give rise to complex incommensurate or commensurate magnetic structures at low temperatures. These magnetic structures are reported to be polar and thus these materials exhibit spontaneous electric polarization below the magnetic ordering temperatures.

The AMO2 (A = Cu, Ag; M = Cr, Fe) system with the delafossite structure having the centrosymmetric space group (R3¯m) exhibits a simple proper screw spin structure that breaks inversion symmetry and induces ferroelectricity.21–23 In this structure, the edge shared MO6 octahedra form layers connected through linearly coordinated A-cations along the c-axis. More importantly, each element (A/M/O) forms a triangular lattice stacked along the c-axis. Since magnetic ions in a triangular lattice experience frustration, these compounds show a proper screw spin structure either at applied or zero magnetic fields. According to the inverse DM model, the proper screw spin structure cannot induce polarization because it involves (Si×Sj)q, where q is the magnetic propagation vector. However, symmetry-based analysis and several microscopic mechanisms support the origin of P in this system.24–26 Many other compounds with very different structural types, such as, e.g., Ni3V2O8, MnWO4, BaYFeO4, CuO, and hexaferrites, are reported to exhibit multiferroic properties.27–30 In a number of type-I and type-II multiferroics, charge ordering has been suggested to play a role in inducing ferroelectricity.31 These materials include doped manganites, magnetite, and LuFe2O4. However, still there is no consensus on the observation of ferroelectricity or the mechanism of ferroelectricity.

In this article, we focus on the structure, magnetism, and magnetoelectric and multiferroic behavior of spinels and their future perspectives. The spinel crystallographic class is arguably as diverse as the perovskite class. The spinel structure can accommodate transition metals, which in many cases, results in various types of magnetic orderings at high temperatures. However, unlike perovskite ferroelectrics, intrinsic ferroelectricity in spinels has not been demonstrated experimentally except for the suggestion of off-centering of B-site cation along [111] direction and a recent report on soft mode driven ferroelectricity in CdCr2S4.32,33 On the other hand, diverse magnetic properties make spinels potentially perspective for new discoveries of magnetoelectric and multiferroic compounds, which we discuss in this paper.

Spinel represents a family of compounds having the general formula AB2X4 (X=O, S, Se, Te). Most of the spinels crystallize in the cubic structure with the space group Fd3¯m, similar to that of the mineral spinel, MgAl2O4. A polyhedral view of the crystal structure of the spinel is shown in Fig. 1(a). In this structure, the anions form a cubic-closed packed arrangement, while the B3+-cations occupy one-half of the 32 available octahedral sites and the A2+-cations occupy 1/8th of the tetrahedral sites. The materials with this kind of cation arrangement are called normal spinels. In inverse spinels, one-half of the B-cations occupy the tetrahedral site and the remaining B-cations and the A-cations occupy the B-sites. The choice of normal vs inverse spinel is decided by the crystal field stabilization energy of the cations occupying the A- and B-sites. In some materials, an arbitrary mixing of these ions also occurs in both these sites.

FIG. 1.

(a) Crystal structure of cubic spinel (Fd3¯m). (b) Polyhedral view of the tetrahedral arrangement of A-site cations with their surrounding ligands showing the diamond lattice. (c) Polyhedral view of B-cations showing the pyrochlore lattice.

FIG. 1.

(a) Crystal structure of cubic spinel (Fd3¯m). (b) Polyhedral view of the tetrahedral arrangement of A-site cations with their surrounding ligands showing the diamond lattice. (c) Polyhedral view of B-cations showing the pyrochlore lattice.

Close modal

Spinels exhibit a wide variety of magnetism because of their complex structure and the ability to accommodate various magnetic and non-magnetic cations in both A and B sublattices. Therefore, it is important to look at various coupling pathways of magnetic ions in different sublattices. In this respect, it is useful to decompose the spinel structure into individual interpenetrating A and B sublattices with their oxygen polyhedra, as shown in Figs. 1(b) and 1(c), respectively. The tetrahedrally coordinated A-cations form the diamond lattice [Fig. 2(a)], whereas the B-cations constitute the pyrochlore lattice [Fig. 2(b)], which is a three-dimensional network of corner-sharing tetrahedra. The triangle-based nature of the B-sublattice results in a strong geometrical frustration of spins in the pyrochlore lattice.34,35 In turn, if next-nearest-neighbor interactions are strong enough, the diamond lattice is also frustrated for a certain range of the next-nearest- JAANNN to nearest-neighbor JAANN interaction ratio.36 

FIG. 2.

(a) Diamond and (b) pyrochlore lattices corresponding to A- and B-sites in spinels. JAANN and JAANNN denotes nearest- and next-nearest-neighbor interactions in the diamond lattice, respectively.

FIG. 2.

(a) Diamond and (b) pyrochlore lattices corresponding to A- and B-sites in spinels. JAANN and JAANNN denotes nearest- and next-nearest-neighbor interactions in the diamond lattice, respectively.

Close modal

High inherent frustration in many spinels with nonmagnetic cations in either of the sublattices A or B is reflected in high values of the frustration parameter f=|ΘCW|/TN, where ΘCW is the Curie–Weiss temperature and TN is the temperature of the onset of long-range magnetic ordering. In the case of magnetic B and nonmagnetic A cations, e.g., in the compounds AB2O4 with A = Mg, Zn, Cd, Hg and B = V, Cr, Mn, Fe, |ΘCW| reaches 850 K in ZnV2O4, whereas the frustration parameter can be as high as 31.2 in ZnCr2O4, which reflects strong B-B spin interactions.37 In the case of magnetic A and nonmagnetic B cations, e.g., A = Mn, Fe, Co and B = Al, Sc, Ga, Rh, the Curie–Weiss temperature is much lower and |ΘCW| reaches only 143 K for MnAl2O4, whereas the value of f can be of the order of 20 and the frustration can still be high. When magnetic ions occupy A and B-sites, they frequently undergo ferrimagnetic ordering at high temperatures, as in the ferrimagnet Fe3O4.

Néel first described the ferrimagnetic properties of spinels by introducing two sublattices with antiparallel magnetic moments and employing the Weiss molecular field theory.38 The model allowed explaining the deviation of the paramagnetic susceptibility from the Curie–Weiss law and anomalous temperature dependence of the magnetization such as, for example, the presence of the compensation point. In the original model, Néel introduced two sublattices, namely, the tetrahedral A and octahedral B sublattices, and exchange interactions JAA and JBB within sublattices and JAB between them. In the case of antiferromagnetic (ferromagnetic) JAB and ferromagnetic JAA and JBB interactions, a ferrimagnetic (ferromagnetic) ordering is the ground state. However, if the exchange interaction is antiferromagnetic and sufficiently high within one of the sublattices, the model predicts that the respective sublattice is unsaturated at low temperatures. In case both JAA and JBB are antiferromagnetic and sufficiently high, the model does not predict magnetic ordering. The reason for such behavior is in the assumption of only two sublattices, whereas the antiferromagnetic interactions JAA or JBB encourage further splitting of the tetrahedral and octahedral sublattices. In other words, the Néel model is applicable in the case of dominance of the intersublattice interaction JAB.

By allowing for further subdivision of the A- and B-sublattices and considering noncollinear magnetic moments, Yafet and Kittel have found that the ground state may consist of a triangular arrangement of the spins or have antiferromagnetic ordering in the A- and B-sublattices.39 Moreover, their model allowed for several phase transitions from one type of order to another even with temperature-independent interactions. Further works of Kaplan et al. have shown that the Yafet–Kittel (YK) triangular ordering can be stable in a class of tetragonally distorted spinels; however, it does not minimize the exchange energy in cubic spinels.40–42 Indeed, the models of Néel and Yafet–Kittel is restricted to the case of k=0, i.e., without multiplication of the primitive cell, whereas allowing for k0 may further minimize the exchange energy. A rigorous analysis of the ground state problem with the help of the generalized Luttinger–Tisza method reveals the appearance of ferrimagnetic spirals with k[110].43 

Thus, the presence in the primitive cell of many atomic positions, which can be occupied by various magnetic species and competing exchange interactions between them results in the richness of magnetic properties and magnetically ordered states. The appearance of multiferroic or magnetoelectric properties in spinels, as in any other crystal, is essentially a question of symmetry. Therefore, in this respect, the symmetry analysis of the possible magnetic structures in spinels is indispensable.

In the k=0 case, i.e., for magnetic orderings that do not result in the multiplication of the unit cell, the magnetic representations dMA and dMB for the A and B positions, respectively, split into the following sums of irreducible representations (IRs):44 

dMA=Γ4+Γ5,
(1)
dMB=Γ2+Γ3+2Γ4+Γ5+.
(2)

It can be found that Γ4+ describes the ferromagnetic ordering of spins in both sublattices. Since the mutual orientation of the magnetization of sublattices is not determined by symmetry, their arrangement can also be ferrimagnetic. Furthermore, various types of antiferromagnetic orderings of spins are described by all IR’s entering into dMA and dMB.44 Note that IR Γ4+ also describes an antiferromagnetic ordering of spins in the B-sublattice, since it enters into dMB twice.

Since the occurrence of multiferroic or magnetoelectric properties requires the breaking of spatial inversion symmetry, it follows from the analysis of (1) and (2) that, in the case of k=0 magnetic structures, only an antiferromagnetic ordering of spins in the A-sublattice breaks inversion symmetry, which is described by Γ5. Thus, the other IR’s in (1) and (2) do not induce magnetoelectric coupling, because they are even under space inversion operation. On this basis, the linear ME effect in A-site antiferromagnetic spinels was theoretically predicted and experimentally found.45,46

In spinels, multiferroic or magnetoelectric properties of magnetic structures with k0 can appear due to peculiar features of the Fd3¯m space group. Indeed, it can be found that any magnetic order parameter with k0 induces improper ferroelectric phases.47,48 It has also been shown that any order parameter allowing Lifshitz invariants, i.e., invariants linear in spatial derivatives, induces improper ferroelectric phases.49 The space group Fd3¯m has only three non-parametric Brillouin zone points, namely, the X, L, and W points, and all IR’s in these points allow Lifshitz invariants either of second or fourth order with respect to the order parameter. An example of such fourth-order invariant allowed in the L point is given in Ref. 50. This fact means that (i) magnetic instabilities in these Brillouin zone points can be prone to incommensurate modulations and (ii) improper ferroelectric magnetic structures can be expected. Magnetic structures with modulations that do not belong to the Brillouin zone’s special points naturally involve Lifshitz invariants and, consequently, improper ferroelectric phases. Therefore, magnetic instabilities in spinels with k0 can result in cascades of phase transitions between incommensurate, commensurate, and improper ferroelectric phases. The occurrence of a particular phase depends primarily on the exchange constants and anisotropic interactions.

In this section, we present few examples of linear magnetoelectric spinel oxides (Fd3¯m) containing magnetic ions only at the A-site with k=0 magnetic structures and discuss the possibility of magnetoelectric effect in cation-ordered B-site magnetic spinels with the F4¯3m structure. We also present second-order magnetoelectric effect in B-site magnetic chalcogenide spinels, which exhibit incommensurate helical magnetic structures.

A family of A-site antiferromagnetic spinels such as MnB2O4 (B=Ga, Al), Co3O4, and CoAl2O4 crystallizes in the normal spinel structure with the space group Fd3¯m. In this structure, the magnetic ions (Mn2+ and Co2+) occupy the A-site with local symmetry Td. A combined study on polycrystalline and single-crystal samples of MnGa2O4 has demonstrated a linear magnetoelectric effect below TN=33 K.46 Due to small inversion or anti-site disorder between the tetrahedral and octahedral sites, the magnetization increases below TN, as shown in Fig. 3(a). However, as seen from the inset, the lambda anomaly in the temperature dependence of heat capacity confirms the long-range antiferromagnetic ordering of Mn-moments. At zero applied magnetic field, no dielectric anomaly appears at the magnetic order; however, an anomaly appears and its magnitude increases with the magnetic field [Fig. 3(b)] typical of linear magnetoelectric material. Similarly, no polarization occurs in zero fields but appears under magnetic fields. The proportionality between the polarization and the applied magnetic fields confirms the linear magnetoelectric effect (α=0.17 ps/m at E=15 kV/cm), as seen in Fig. 3(c). On the phenomenological level, one can understand the appearance of a linear magnetoelectric effect below TN in the following way. Denoting the antiferromagnetic ordering of A cation spins by the order parameter (Lx,Ly,Lz), which transforms according to IR Γ5 [cf. Eq. (1)], the magnetoelectric interaction can be written in the form

Lx(MyPz+MzPy)+Ly(MzPx+MxPz)+Lz(MxPy+MyPx),
(3)

where M and P are magnetization and electric polarization, respectively. Therefore, the appearance of non-zero L below TN results in the linear coupling between the magnetization and electric polarization in the thermodynamic potential expansion.

FIG. 3.

(a) Magnetization vs temperature in MnGa2O4. Inset shows C/T vs T. (b) Temperature and field dependence of dielectric constant showing the appearance of the field-induced anomaly at TN. (c) Variation of field-induced electric polarization as a function of temperature. (d) Variation of P[011] under applied magnetic along H[111], H[100], and H[011]. Reproduced with permission from Saha et al., Phys. Rev. B 94, 014428 (2016). Copyright 2016 American Physical Society.

FIG. 3.

(a) Magnetization vs temperature in MnGa2O4. Inset shows C/T vs T. (b) Temperature and field dependence of dielectric constant showing the appearance of the field-induced anomaly at TN. (c) Variation of field-induced electric polarization as a function of temperature. (d) Variation of P[011] under applied magnetic along H[111], H[100], and H[011]. Reproduced with permission from Saha et al., Phys. Rev. B 94, 014428 (2016). Copyright 2016 American Physical Society.

Close modal

The available neutron diffraction data on MnGa2O4 do not determine the magnetic easy axis. However, the results of the magnetic field induced polarization measurements on a single-crystal sample Fig. 3(d) and the phenomenological theory suggested that the easy axis of MnGa2O4 is [111]. It has been shown theoretically that the microscopic mechanism responsible for the linear magnetoelectric effect is the single-ion effects of Mn2+ ions located at the A-site of the spinel. Similarly, MnAl2O4 has also been shown to exhibit a linear magnetoelectric effect.

Unlike MnGa2O4, there is no anti-site disorder in Co3O4 because of the low spin t2g6eg0 state of Co3+ ions, located at the B-sites. Co3O4 experiences a typical antiferromagnetic transition, as displayed in Fig. 4(a). The temperature and field dependence of the real part of dielectric constant at 100 kHz and the electric polarization, as obtained from pyroelectric measurements, P(H) variations, as shown in Figs. 4(b)4(d), demonstrate the linear magnetoelectric effect in Co3O4.46 

FIG. 4.

(a) Temperature dependence magnetization of Co3O4. Inset shows CP vs T. (b) Temperature and field dependence of dielectric constant showing the appearance of the field-induced anomaly at TN. (c) Variation of field-induced electric polarization as a function of temperature. (d) The linear variation of P with the applied magnetic field. Reproduced with permission from Saha et al., Phys. Rev. B 94, 014428 (2016). Copyright 2016 American Physical Society.

FIG. 4.

(a) Temperature dependence magnetization of Co3O4. Inset shows CP vs T. (b) Temperature and field dependence of dielectric constant showing the appearance of the field-induced anomaly at TN. (c) Variation of field-induced electric polarization as a function of temperature. (d) The linear variation of P with the applied magnetic field. Reproduced with permission from Saha et al., Phys. Rev. B 94, 014428 (2016). Copyright 2016 American Physical Society.

Close modal

In CoAl2O4, the anti-site disorder becomes significant and can be controlled by sample preparation conditions.51 More importantly, this compound has received much attention because its magnetic ground state has been a controversial issue. The ratio of the two competing exchange interactions (JAANNN/JAANN) in the diamond lattice, namely, the nearest-neighbor interactions (JAANN) between the two interpenetrating fcc lattices and the next-nearest-neighbor interaction between the spins within each sublattice (JAANNN) determines the magnetic ground state. It has been shown that in the range 0JAANNN/JAANN1/8, the ground state is a collinear antiferromagnet, whereas for JAANNN/JAANN>1/8, it is a degenerate spiral spin liquid state.36 In CoAl2O4, the ratio JAANNN/JAANN0.109 is close to the boundary JAANNN/JAANN=1/8 separating the two extreme ground states and, thus, there are several reports of contradicting results. A careful magnetoelectric measurement and Monte Carlo simulations support that the ground state of this compound with a low anti-site disorder Co0.95Al0.05[Al1.95Co0.05]O4 is a long-range collinear antiferromagnet. However, with increasing anti-site disorder, the systems move to a spin-glass state. Figure 5(a) shows the antiferromagnetic transition in the sample with the low anti-site disorder. The antiferromagnetic order parameter calculated using the Monte Carlo method is shown in Fig. 5(b). As seen in Figs. 5(c) and 5(d), the temperature and field dependence of the dielectric constant and polarization data confirm the linear magnetoelectric effect.

FIG. 5.

(a) Temperature dependence magnetization in Co1xAlxAl2xCoxO4 (x=0.5). Inset shows CP vs T. (b) Temperature dependence of the antiferromagnetic order parameter L (c) Temperature and field dependence of dielectric constant showing the appearance of the field-induced anomaly at TN. (d) Variation of field-induced electric polarization as a function of temperature. The inset shows the linear variation of P with the applied magnetic field for various x. Reproduced with permission from Ghara et al., Phys. Rev. B 95, 094404 (2017). Copyright 2017 American Physical Society.

FIG. 5.

(a) Temperature dependence magnetization in Co1xAlxAl2xCoxO4 (x=0.5). Inset shows CP vs T. (b) Temperature dependence of the antiferromagnetic order parameter L (c) Temperature and field dependence of dielectric constant showing the appearance of the field-induced anomaly at TN. (d) Variation of field-induced electric polarization as a function of temperature. The inset shows the linear variation of P with the applied magnetic field for various x. Reproduced with permission from Ghara et al., Phys. Rev. B 95, 094404 (2017). Copyright 2017 American Physical Society.

Close modal

The spinels with nonmagnetic A-cations and magnetic B-cations display various physical phenomena due to the strongly frustrated nature of the pyrochlore lattice. For example, the ZnCr2O4 spinel has a dramatic frustration parameter f=31.2 but develops a long-range antiferromagnetic order below TN=12.5 K. This ordering becomes possible because it is associated with a structural phase transition at the same temperature, which suppresses the geometrical frustration and lifts the spin degeneracy.52–54 The other representative spinel, ZnV2O4, displays a very high Curie–Weiss temperature |ΘCW|850 K and an antiferromagnetic phase transition at TN=40 K, which is preceded by an orbital ordering phase transition at TS=50 K.55 

The resulting combination of magnetic ordering and the crystal structure does not induce multiferroic or magnetoelectric properties. It was suggested that 1:1 chemical ordering of A-site cations occurring, e.g., in LiGaCr4O8, can help induce the coupling between magnetism and ferroelectricity because such atomic ordering results in the loss of inversion symmetry and the crystal symmetry becomes F4¯3m.45 Furthermore, the B-site magnetic ions become located in a polar local environment 3m, which, thus, can provide an opportunity to induce magnetoelectric coupling due to single-ion anisotropy. The cation-ordered phase without inversion symmetry does not suffer from the problem of even and odd IR’s under inversion operation discussed in Sec. II. Therefore, any k=0 magnetic ordering described by three-dimensional IR’s can induce magnetoelectric effects.45 Furthermore, the F4¯3m structure allows a second-order magnetoelectric coupling of the form

PxMyMz+PyMzMx+PzMxMy;

however, the respective magnetoelectric coefficient is small without appropriate magnetic ordering.

Several attempts have been made to find magnetoelectric coupling or multiferroic properties in the A-site ordered spinels LiMCr4O8 (M = Ga, In, and Fe), which, however, failed as these compounds exhibit k0 magnetic structures with, apparently, trivial coupling to electric polarization.46,56 However, these materials exhibit interesting magnetic and dielectric properties due to the breathing pyrochlore lattice resulting from the chemical ordering of cations of different ionic sizes.

The ZnCr2Se4 spinel demonstrates strong ferromagnetic spin interactions reflected in a positive Curie–Weiss temperature ΘCW=90 K but develops a proper screw spin structure below TN=21 K with propagation vector k(0,0,0.44) as shown in Fig. 6.57–59 Below TN application of the magnetic field is assumed to convert the magnetic structure into a conical one, which for certain field directions results in the appearance of electric polarization that is proportional to the square of the magnetic field in the low-H region, i.e., PH2, as shown in Fig. 7.60,61 The quadratic magnetic field dependence of P at low fields suggests that the proper screw magnetic structure breaks inversion symmetry but does not induce electric polarization so that the crystal becomes essentially a piezoelectric. Assuming the magnetic instability in the commensurate kc=(0,0,1/2) point of the Brillouin zone, the proper screw magnetic structure is described by a spatially modulated order parameter {ni} (i=112) transforming according to IR Δ5. The modulation is induced by the respective Lifshitz invariants, whereas the ME interaction can be written in the form

(n12n22+n32n42n92+n102n112+n122)MyMzPx+(n52+n62n72+n82+n92n102+n112n122)MxMzPy+(n12+n22n32+n42+n52n62+n72n82)MxMyPz.

Therefore, below TN non-zero components of {ni} appear, and electric polarization can be induced by an external magnetic field applied at an angle to the axes, which is indeed observed experimentally. At fields higher than 6 T, the Cr spins align along the field, which results in the disappearance of the conical structure and the respective vanishing of polarization, as shown in Fig. 7(c).

FIG. 6.

Magnetic structure of ZnCr2Se4. Only the Cr atoms are shown for clarity. Reproduced with permission from Yokaichiya et al., Phys. Rev. B 79, 064423 (2009). Copyright 2009 American Physical Society.

FIG. 6.

Magnetic structure of ZnCr2Se4. Only the Cr atoms are shown for clarity. Reproduced with permission from Yokaichiya et al., Phys. Rev. B 79, 064423 (2009). Copyright 2009 American Physical Society.

Close modal
FIG. 7.

Magnetic field dependences of (a) magnetization, (b) polarization current, and (c) electric polarization of ZnCr2Se4 for various directions of magnetic field H. Reproduced with permission from Murakawa et al., J. Phys. Soc. Jpn. 77, 043709 (2008). Copyright 2008 Journal of the Physical Society of Japan.

FIG. 7.

Magnetic field dependences of (a) magnetization, (b) polarization current, and (c) electric polarization of ZnCr2Se4 for various directions of magnetic field H. Reproduced with permission from Murakawa et al., J. Phys. Soc. Jpn. 77, 043709 (2008). Copyright 2008 Journal of the Physical Society of Japan.

Close modal

Several synchrotron and neutron diffraction studies suggested lowering crystal symmetry below TN by spin-lattice distortion to either the I41/amd or Fddd space groups.62,63 These space groups are centrosymmetric and do not allow magnetic field-induced electric polarization. Therefore, the presence of the ME effect in ZnCr2Se4 below TN calls for a reinterpretation of the diffraction data in noncentrosymmetric (piezoelectric) crystal structures, although the structural distortion is minimal.

The ZnCr2S4 spinel demonstrates two magnetic phase transitions at TN1=15 K and TN2=8 K.58,64 At TN2<T<TN1, the magnetic structure is characterized by the propagation vector k1(0,0,0.787) and is similar to that observed in ZnCr2Se4. Upon cooling, the intensities of k1 reflections decrease and below TN2 two commensurate propagation vectors k2(1/2,1/2,0) and k3(1,1/2,0) appear. The crystal structures in these two magnetic phases were identified as I41/amd and Imma, respectively. Given the similarity between the magnetic structure of ZnCr2Se4 and that in ZnCr2S4 at TN2<T<TN1, one can expect magnetic field-induced polarization similar to ZnCr2Se4. Considering a higher structural distortion in ZnCr2S4 accompanying the magnetic phase transitions, a higher ME effect can be expected in this spinel compound. As discussed in Sec. II, the magnetic structure below TN2 characterized by k2 and k3 can also potentially show multiferroic or ME properties. Furthermore, one can note the spinel HgCr2S4 also showing a magnetic structure similar to that of ZnCr2Se4 and interesting dielectric properties.65,66

In this section, we present examples of type-II multiferroic spinels, in which electric polarization is induced by either a modulated or commensurate magnetic structure, as well as examples of type-I multiferroic spinels, in which electric polarization appears independently of the magnetic ordering.

The CoCr2O4 spinel undergoes three magnetic phase transitions at Tc=93 K, Ts=26 K, and Tlockin=15 K.67 The first transition results in the appearance of a ferrimagnetic phase at Ts<T<Tc, whereas a ferrimagnetic incommensurate spiral structure appears at Tlockin<T<Ts as shown in Fig. 8(a). The ferrimagnetic spiral, which is in line with the predictions of Lyons et al.43, is characterized by the propagation vector k=(0.63,0.63,0) that locks-in to a commensurate value (2/3,2/3,0) below Tlockin.67–69 The ferrimagnetic magnetic moment is directed along [001], which also defines the cone axis, whereas the components of spins in the (001) plane order incommensurately at Tlockin<T<Ts.

FIG. 8.

(a) Spiral spin ordering pattern for CoCr2O4. (b) Schematic illustration of spin order-induced electric polarization. Reproduced with permission from Kim et al., Appl. Phys. Lett. 94, 042505 (2009). Copyright 2009 AIP Publishing LLC.

FIG. 8.

(a) Spiral spin ordering pattern for CoCr2O4. (b) Schematic illustration of spin order-induced electric polarization. Reproduced with permission from Kim et al., Appl. Phys. Lett. 94, 042505 (2009). Copyright 2009 AIP Publishing LLC.

Close modal

The development of the ferrimagnetic spiral below Ts=26 K is accompanied by the appearance of electric polarization directed along [1¯10], i.e., normal both to the magnetization easy axis and the spiral propagation vector (Fig. 9).70 At the lock-in phase transition at Tlockin, the electric polarization shows a tiny anomaly. Interestingly, the electric polarization can be reversed by reversing the ferrimagnetic moment under a magnetic field applied along [001].70 

FIG. 9.

The temperature dependence of (a) magnetization and specific heat C/T, and (b) dielectric constant ε and electric polarization along y[1¯10]. Reproduced with permission from Yamasaki et al., Phys. Rev. Lett. 96, 207204 (2006). Copyright 2006 American Physical Society.

FIG. 9.

The temperature dependence of (a) magnetization and specific heat C/T, and (b) dielectric constant ε and electric polarization along y[1¯10]. Reproduced with permission from Yamasaki et al., Phys. Rev. Lett. 96, 207204 (2006). Copyright 2006 American Physical Society.

Close modal

On the phenomenological level of theory, the multiferroic behavior observed in CoCr2O4 can be understood as follows. The modulated components of the ferrimagnetic spiral are described by IR’s Σ3 and Σ4 in the (2/3,2/3,0) point of the Brillouin zone and the respective order parameters can be denoted by {ni} and {mi} (i=112). Taking into account only two arms of the 12-armed star, namely, (2/3,2/3,0) and (2/3,2/3,0), one can write the ME interaction in the form

IME=(n1m2n2m1)(PxPy)=r1r2(PxPy)sin(ϕ2ϕ1),
(4)

where we have used n1=r1cosϕ1, n2=r1sinϕ1, m1=r2cosϕ2, and m2=r2sinϕ2. Therefore, this ME interaction implies that at Ts the condensation of r1 and r2 with Δϕ=ϕ2ϕ1=±π/2, which describe, respectively, the components of spins parallel to [11¯0] and [110], induces electric polarization along [1¯10].

The reversal of electric polarization observed upon reversing of the ferrimagnetic component can be due to the change of the phase difference Δϕ from π/2 to π/2 or vice versa. However, the mechanism of this change is not clear at the moment, i.e., it is not clear what interactions of the order parameters with the magnetic field lead to such a difference. Several reports of Monte Carlo studies of the magnetic field influence on the magnetic structure and electric polarization in CoCr2O4, from our point of view, do not pay sufficient attention to this problem.71,72 Furthermore, an unexpected reversal of electric polarization across the lock-in phase transition at Tlockin is found,73 whereas the magnetic structure below Tlockin is characterized by both commensurate ordering with the propagation vector (2/3,2/3,0) and an incommensurate component.69,73 The latter implies that the ferrimagnetic spiral model of Lyons et al.43 may not be the actual ground state in CoCr2O4.68 

MnCr2O4 is among the spinel compounds that show ferrimagnetic spiral ordering. In this case, the Curie temperature is Tc=51 K, while the ferrimagnetic spiral order with propagation vector (0.626,0.626,0) sets in below Ts=16 K.74 Similar to CoCr2O4, the onset of ferrimagnetic spiral ordering is accompanied by the appearance of electric polarization and a dielectric anomaly.75,76 However, it appears that since the phase transition temperatures, the magnetic propagation vector, and the multiferroic properties vary between different reports and are highly sensitive to the sample quality and degree of cation inversion, the magnetic ground state of MnCr2O4 is still under debate.75,77–79

In the GeCu2O4 spinel, a Jahn–Teller active Cu2+ ion located at the B-site is responsible for the strong tetragonal distortion to the I41/amd structure. This spinel shows interesting magnetic properties developing a complex long-range antiferromagnetic ordering at TN=33 K.80 It consists of slabs of a pair of layers with orthogonal chains along a and b axis in the ab plane along the c axis. In each chain, the spin moments are aligned in ↑↑↓↓ configuration. The reported magnetic structure is consistent with the magnetic point group 4¯2m1, which breaks both space inversion and time-reversal symmetries allowing linear magnetoelectric effect. Surprisingly, as shown in Fig. 10, this compound exhibits spontaneous polarization below TN indicating multiferroicity.81,82 With respect to the tetragonal I41/amd structure the magnetic propagation vector is (1/2,1/2,1/2) and the magnetic order parameters in this point of the Brillouin zone allow improper ferroelectric phases. Therefore, the discovered type-II multiferroicity in GeCu2O4 warrants further investigation of the magnetic structure to understand the origin of spontaneous electric polarization.

FIG. 10.

(a) Temperature dependent dielectric constant measured under zero field and 8 T at 50 kHz in GeCu2O4. (b) Switchable spontaneous polarization and the inset shows the field-dependent polarization. Reproduced with permission from Yanda et al., Solid State Commun. 272, 53 (2018). Copyright 2018 Elsevier.

FIG. 10.

(a) Temperature dependent dielectric constant measured under zero field and 8 T at 50 kHz in GeCu2O4. (b) Switchable spontaneous polarization and the inset shows the field-dependent polarization. Reproduced with permission from Yanda et al., Solid State Commun. 272, 53 (2018). Copyright 2018 Elsevier.

Close modal

LiCuVO4 is an inverse spinel and crystallizes in the centrosymmetric orthorhombic structure (sp. gr. Imma), in which the non-magnetic V5+ ions are in tetrahedral positions, whereas Li+ and magnetic Cu2+ occupy the octahedral sites in an ordered way. This compound with S=1/2 spin-chains and frustrated nearest-neighbor ferromagnetic and next-nearest neighbor antiferromagnetic interactions received much attention as a low-dimensional quantum spin system. Because of the frustration, it undergoes a helical spin ordering at TN2.4 K in the ab plane with the propagation vector kinc=(0,0.532,0).83 The magnetic phase transition is accompanied by the appearance of electric polarization along the a-direction.84 The spin rotation plane can be reoriented by an applied magnetic field, which is accompanied by switching of electric polarization with strong changes in its absolute value.85 Phenomenologically, the modulated magnetic order and the spin-order-induced ferroelectricity can be described similar to other type-II multiferroics by considering the instability in the (0,1/2,0) point of the Brillouin zone and taking into account the Lifshitz invariants.86 The studies of LiCuVO4 in magnetic fields reveal complex phase diagrams, which include collinear modulated and spin-nematic phases.85,87–89 Recently, the magnetic field–temperature phase diagram has been found to extend to temperatures above TN as shown in Fig. 11.90 The newly observed phase (III) has been ascribed to the so-called vector-chiral (VC) phase, in which the long-range spin order is absent while the vector chirality correlations—the vector product of adjacent spins Si×Sj—survive.91 The VC phase is characterized by non-zero electric polarization, which is, however, about one order of magnitude smaller than in the zero field phase below TN. The interpretation of the observed behavior of LiCuVO4 provided by Ruff et al.90 is plausible assuming that the vector chirality-like terms in the Hamiltonian are often responsible for the induction of electric polarization in multiferroics; however, we would like to suggest another possible interpretation here. Indeed, on the phenomenological level, the applied magnetic field can nontrivially couple to magnetic order with modulation vector (0,1/3,0) by terms linear in the magnetic field and of third order with respect to the magnetic order parameter (i.e., HL3-type terms). The magnons with kinc soften at TN;92 however, a sufficiently high magnetic field above a critical field Hc can result in a first-order phase transition above TN to a phase with modulation vector (0,1/3,0). Such a scenario also explains the absence of electric polarization at low magnetic fields. Neutron diffraction measurements in presence of a magnetic field are needed in order to clarify the presence of long-range magnetic ordering in the VC phase.

FIG. 11.

HT phase diagram of LiCuVO4 obtained according to various measurement methods. Reproduced from Ruff et al., npj Quantum Mater. 4, 24 (2019). Copyright 2019 Nature Research.

FIG. 11.

HT phase diagram of LiCuVO4 obtained according to various measurement methods. Reproduced from Ruff et al., npj Quantum Mater. 4, 24 (2019). Copyright 2019 Nature Research.

Close modal

MnCr2S4 presents an interesting case bearing an analogy to the A-site antiferromagnetic spinels Co3O4, MnGa2O4, and MnAl2O4 discussed above, which are linear magnetoelectrics. Below, Tc=65 K MnCr2S4 exhibits a ferrimagnetic ordering with antiparallel magnetic moments of the Cr and Mn sublattices. Upon cooling, the Mn spins remain not fully saturated, which results in the ordering of their transverse components in a triangular Yafet–Kittel type structure at TYK=5 K as shown in Figs. 12(a) and 12(b).93,94 The YK order is accompanied by electric polarization below TYK at zero applied magnetic field (Fig. 13), which classifies this spinel compound as multiferroic.94 The multiferroic properties of MnCr2S4 can be understood as coming from the interplay of ferrimagnetic ordering occurring below Tc and described by IR Γ4+ and (partial) antiferromagnetic ordering of Mn spins below TYK described by IR Γ5, which couple together with electric polarization in a trilinear interaction equation (3). This mechanism is to be compared and contrasted to A-site antiferromagnetic spinels that are linear ME materials, in which Γ5 appears below TN, whereas an external magnetic field induces macroscopic magnetization.

FIG. 12.

(a) and (b) Schematic illustration of the Yafet–Kittel type ordering in MnCr2S4. (c) Schematic evolution of the spin structure with increasing external magnetic field. Reproduced with permission from Ruff et al., Phys. Rev. B 100, 014404 (2019). Copyright 2019 American Physical Society.

FIG. 12.

(a) and (b) Schematic illustration of the Yafet–Kittel type ordering in MnCr2S4. (c) Schematic evolution of the spin structure with increasing external magnetic field. Reproduced with permission from Ruff et al., Phys. Rev. B 100, 014404 (2019). Copyright 2019 American Physical Society.

Close modal
FIG. 13.

The temperature (a) and magnetic field (b) dependence of electric polarization in MnCr2S4. Reproduced with permission from Ruff et al., Phys. Rev. B 100, 014404 (2019). Copyright 2019 American Physical Society.

FIG. 13.

The temperature (a) and magnetic field (b) dependence of electric polarization in MnCr2S4. Reproduced with permission from Ruff et al., Phys. Rev. B 100, 014404 (2019). Copyright 2019 American Physical Society.

Close modal

Ultrahigh magnetic field magnetization measurements up to 110 T reveal that the magnetic structure in MnCr2S4 undergoes a sequence of field-induced phase transitions between magnetic phases with differently ordered Mn and Cr spins as shown in Fig. 12(c).94,95 The experimental phase diagram is shown in Fig. 14. The respective phase transitions are reflected in the complex magnetic field dependence of electric polarization (Fig. 13), which can serve as a playground for testing the microscopic theories behind the magnetoelectric coupling.

FIG. 14.

Temperature–magnetic field phase diagram of MnCr2S4. Reproduced with permission from Miyata et al., Phys. Rev. B 101, 054432 (2020). Copyright 2020 American Physical Society.

FIG. 14.

Temperature–magnetic field phase diagram of MnCr2S4. Reproduced with permission from Miyata et al., Phys. Rev. B 101, 054432 (2020). Copyright 2020 American Physical Society.

Close modal

Lodestones, which consist mainly of magnetite, Fe3O4, have been of practical importance since more than 2000 years. Studies reveal now that magnetite is arguably the first multiferroic known to mankind, and together with göthite, α-FeOOH, they may represent the most abundant in nature multiferroic and linear magnetoelectric crystals, respectively.96,97 Magnetite is ferrimagnetic below 858 K and experiences the metal-insulator Verwey transition at TV=120 K, which is a charge ordering transition.98–100 Careful structural analyses in several works have disproved the initial Verwey model and showed that the charge ordering scheme and the coupled atomic displacements are due to the instability in the Δ-point of the Brillouin zone.101,102 Although in some works a centrosymmetric crystal structure of Fe3O4 below TV was proposed,103,104 many authors suggest a polar structure with the Cc symmetry.102,105,106 The latter is also corroborated by several density functional theory calculation reports and is in accordance with electric polarization loops and magnetoelectric properties determined below TV.107–112 Despite the fact that electric conductivity decreases by two orders of magnitude at TV, it is still rather high, which makes the electric polarization and magnetoelectric measurements difficult. Furthermore, electric polarization is, apparently, induced as a secondary order parameter upon the condensation of IR Δ5, which describes the transformation of the primary order parameter. Therefore, the improper nature of electric polarization results in asymmetric P-E loops; however, the value of polarization is quite high reaching 2 μC/cm2.

A broad class of spinels, known as lacunar spinels, with the general chemical formula AB4X8 (A = Ga, Ge; B = V, Mo, Nb, Ta; X = S, Se) has attracted a lot of interest recently due to the richness of physical phenomena including, for example, pressure-induced superconductivity,113 large negative magnetoresistance,114 and metal-to-insulator transitions.115 The lacunar spinel compounds can be thought of as usual spinel with one-half of tetrahedral A-site cations removed, i.e., A1/21/2B2X4, which results in 1:1 ordering of the vacancies and the remaining A-cations, thus lowering the symmetry to F4¯3m similar to the case described in Sec. III B.

Lacunar spinels can undergo structural phase transitions that result in the appearance of ferroelectricity. The GeV4S8 compound at TS=30 K experiences a Jahn–Teller distortion of the V4-clusters reducing their symmetry from 4¯3m to mm2, which results in a ferroelectric crystal structure described by space group Imm2.116,117 Similar orbital ordering, however, to a rhombohedral structure was recently identified in GaV4S8 at TS=44 K.118 Thus, both compounds experience a first-order structural phase transition to a ferroelectric structure with quite large electric polarization of the order of 1 μC/cm2. Upon further cooling, both compounds show magnetic ordering at TC=13 K (GaV4S8) and TN=13 K (GeV4S8) into ferromagnetic cycloidal and antiferromagnetic states, respectively, which renders these compounds type-I multiferroics.118,119 Besides the compounds mentioned above, several other lacunar spinels, including GaMo4S8, GaMo4Se8, and GeV4Se8, have been shown to exhibit orbital order-induced ferroelectricity and subsequent magnetic ordering at lower temperatures.120–122 

What makes the lacunar spinels even more impressive is discovering the skyrmion lattice (SkL) phases.118,123 Skyrmions—topologically protected whirls of spins—have high future potential for applications in magnetic storage or logic devices and to date have been observed in many materials ranging from metals, to semi-metals, to semiconductors and insulators.124–126 Lacunar spinels, which have a noncentrosymmetric crystal structure and experience the frequent occurrence of modulated ferromagnetic ordering, have become a perfect host for skyrmion physics, although at rather low temperatures. A typical temperature–magnetic field phase diagram is shown in Fig. 15, in which one can discern the cycloidal magnetic phase, the low-temperature ferromagnetic phase, and the SkL phase in GaV4S8. All magnetic phases have an excess polarization ΔP of the order of 10–100 μC/m2 over the orbitally ordered phase, and the temperature and magnetic field dependencies of ΔP show step-like features at the boundaries between the phases. This allowed Ruff et al.118 to distinguish four different ferroelectric states corresponding to the above mentioned phases, which are, thus, characterized by their own polarization. Compared to Cu2OSeO3, another insulating skyrmion compound with the noncentrosymmetric space group P213, the magnetic order-induced excess polarization in GaV4S8 is about two orders of magnitude larger.127 

FIG. 15.

The temperature–magnetic field phase diagram of GaV4S8. Reproduced from Ruff et al., Sci. Adv. 1, e1500916 (2015).

FIG. 15.

The temperature–magnetic field phase diagram of GaV4S8. Reproduced from Ruff et al., Sci. Adv. 1, e1500916 (2015).

Close modal

The spinel structure presents a versatile platform for the creation of magnetic materials. It can accommodate various magnetic ions in two crystallographically different positions and a set of different anions, which results in a great diversity of chemical compositions. The symmetry analysis of possible magnetic structures in spinels with the Fd3¯m structure performed in Sec. II allows making the following conclusions.

The magnetic structures without multiplication of the primitive cell, i.e., with k=0, can induce or non-trivially couple to electric polarization only when an antiferromagnetic ordering of A-cation spins is present because only the corresponding IR Γ5 breaks inversion symmetry. No k=0 spin ordering of B cations alone can induce magnetoelectric or multiferroic properties. This is exemplified by the linear magnetoelectric case of antiferromagnetic A-site spinels discussed in Sec. III A as well as by the multiferroic MnCr2S4 (Sec. IV D), in which the antiferromagnetic ordering of some components of Mn spins occurs at lower temperatures, in addition, to k=0 ordering of Cr spins. Therefore, the antiferromagnetic exchange coupling JAA between nearest neighbors in the A-sublattice can promote their antiferromagnetic order and, respectively, magnetoelectric properties.

The situation with magnetic structures characterized by a magnetic unit cell that does not coincide with the primitive cell, i.e., which have k0, is more favorable for the sought phenomena, because in the Fd3¯m structure, any magnetic order parameter with k0 can in principle induce improper ferroelectric phases. The phenomenon, which, from the symmetry grounds, accompanies the induction of improper ferroelectric phases, is the common occurrence of modulated magnetic phases. As the k0 magnetic order parameters in the Fd3¯m structure are usually multidimensional, they generally induce many magnetic phases. However, only some of them are improper ferroelectric or noncentrosymmetric (the latter case is exemplified by the ZnCr2Se4 compound discussed in Sec. III C). The k0 magnetic structures appear due to competing exchange interactions, whereas anisotropic interactions also play an important role in selecting the specific magnetic phase out of those that can be induced by the given order parameter. Therefore, direct search or targeted design of spinel compounds with magnetic interactions that lead to k0 magnetic structures, and, thus, potentially to symmetry-determined coupling to electric polarization, can be a fruitful strategy.

From our point of view, significant progress in the search for new multiferroic or magnetoelectric compounds within spinels can also be made by selecting appropriate compositions that result in atomic ordering. Although the spinel compounds, in contrast to the perovskites, lack intrinsic ferroelectric instabilities, the combined action of atomic ordering and structural distortions can result in sizable electric polarization, as exemplified by the case of lacunar spinels discussed in Sec. IV F. One of the simplest atomic orderings in spinels, namely, the 1:1 ordering of A-cations, as mentioned in Sec. III B, results in breaking of inversion symmetry—a necessary condition for the coupling of magnetism and electricity. Therefore, one can say that atomic ordering can induce the coupling of magnetism and ferroelectricity by symmetry, which are otherwise uncoupled in a disordered crystal.

Besides the aforementioned 1:1 ordering of A-cations, other and more complex types of atomic orderings are possible and common in spinels, and their group-theoretical, structural, and thermodynamic studies have been performed over the decades by Talanov and co-workers.128–132 A recent study of possible atomic orderings in both cationic sublattices A and B as well as in the anion sublattice X lists several hundred different phases induced by separate or simultaneous ordering in the sublattices. Of course, most of them are elusive; however, some are experimentally observed.

LiFe5O8 can exist in a disordered form with a random distribution of cations, as well as in an ordered form that can be attained by annealing at suitable temperatures.133,134 In the ordered structure described by enantiomorphic space groups P4132 or P4332, the Li atom located in the B-sublattice experiences a 1:3 ordering with the Fe atoms. LiFe5O8 develops a magnetic order at a very high temperature (Tc905 K),133 but according to our knowledge, the precise magnetic structure is not known. The magnetoelectric effect was recently confirmed in this spinel below room temperature, where the resistivity becomes sufficiently high to allow the measurements.135 Despite the absence of inversion symmetry in the P4132 or P4332 structure, these crystal structures do not allow neither piezoelectricity or second-order magnetoelectric effect (i.e., the magnetoelectric interactions of the form PM2 are forbidden). However, a higher-order magnetoelectric coupling (PM4) is possible, explaining the observed phenomenon. Several other spinels, e.g., Mn4+-containing AM0.5Mn1.5O4 (A=Li, Cu; M=Ni, Mg), that show ferro- or ferrimagnetism and can be prepared in the cation-ordered P4332 structure, can show magnetoelectric phenomena similar to the case of LiFe5O8.136 

An interesting example is provided by the LiZn0.5Mn1.5O4 spinel, which possesses an Fd3¯m structure when quenched from above 750 °C, a P213 structure for slowly cooled samples, and possibly a P4332 structure when quenched from 600 °C.137 The P213 structure is characterized by the simultaneous ordering of Li and Zn in the A-sublattice and Li and Mn in the B-sublattice, so that the chemical formula can be written as (Li0.5Zn0.5)Td[Li0.5Mn1.5]OhO4.130 Given the facts that in the ordered form, this compound has a piezoelectric crystal structure and experiences a ferromagnetic phase transition at Tc=21 K with high spontaneous magnetization, one can expect a linear magnetoelectric behavior below Tc that is made possible by atomic ordering, which results in the noncentrosymmetric lattice.138 

Above, we discussed the interaction of atomic ordering and magnetism, resulting in magnetoelectric or multiferroic properties. However, the other promising strategy is to combine atomic ordering with structural distortions, e.g., orbital ordering, which can result in a polar structure as found in lacunar spinels discussed in Sec. IV F.

The authors are thankful to Rana Saha, Somnath Ghara, and Premakumar Yanda for helping with the article. A.S. acknowledges the International Center for Materials Science (ICMS) and Sheikh Saqr Laboratory (SSL) for various experimental facilities. S.A. also acknowledges BRICS research project, Department of Science and Technology, Government of India, for a research grant [Sl. No. DST/IMRCD/BRICS/PilotCall2/EMPMM/ 2018(g)]. N.V.T.-O. acknowledges financial support from the Russian Foundation for Basic Research under Grant No. 18-52-80028 (BRICS STI Framework Programme).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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