Symmetry Analysis of Magnetoelectric Effects in Perovskite-Based Multiferroics

Abstract

In this article, we performed symmetry analysis of perovskite-based multiferroics: bismuth ferrite (BiFeO₃)-like, orthochromites (RCrO₃), and Ruddlesden–Popper perovskites (Ca₃Mn₂O₇-like), being the typical representatives of multiferroics of the trigonal, orthorhombic, and tetragonal crystal families, and we explored the effect of crystallographic distortions on magnetoelectric properties. We determined the principal order parameters for each of the considered structures and obtained their invariant combinations consistent with the particular symmetry. This approach allowed us to analyze the features of the magnetoelectric effect observed during structural phase transitions in Bi_xR_1−xFeO₃ compounds and to show that the rare-earth sublattice has an impact on the linear magnetoelectric effect allowed by the symmetry of the new structure. It was shown that the magnetoelectric properties of orthochromites are attributed to the couplings between the magnetic and electric dipole moments arising near Cr³⁺ ions due to distortions linked with rotations and deformations of the CrO₆ octahedra. For the first time, such a symmetry consideration was implemented in the analysis of the Ruddlesden–Popper structures, which demonstrates the possibility of realizing the magnetoelectric effect in the Ruddlesden–Popper phases containing magnetically active cations, and allows the estimation of the conditions required for its optimization.

1. Introduction

Multiferroic, multifunctional materials attract enormous research activity due to the prospects of their implementation in science and technology. The collective state parameters attainable in multiferroics (MFs) are preferable for advanced spintronic devices based on the use of combinatorial functions due to their high efficiency, low power consumption, and adaptability [1].

Despite the variety of currently known magnetoelectric materials, perovskite-like structures remain the most researched MFs compounds. The family of perovskites stands out for its diversity due to the intrinsic instability of the cubic parent perovskite phase [2]. Crystallographic distortions owing to the substitution of various cations at A and B positions can lead to the emergence of ferroelectric and MF properties due to the hybridization of ion orbitals, tilt of oxygen octahedra (BO₆), doping of various cations, and other factors [3]. ABO₃ perovskites can produce a variety of MFs differing in the type of (i) ferroelectric ordering (A- or B-driven ferroelectricity), as in the cases of EuTiO₃ [4] and BiFeO₃ [5]; (ii) magnetic ordering (although most perovskite MFs are G-type antiferromagnets (AFM), canted ferromagnetism and a combination of ferromagnetic and AFM phases are also possible) [6,7]; (iii) mechanism of magnetoelectric coupling, which can be associated with magnetoelectrically active d-ions or f-ions (RFeO₃) [8], the Dzyaloshinskii–Moriya interaction (BiFeO₃) [5], spin–orbit coupling (BiMnTe) [9], antiferroelectric orderings (RCrO₃)) [10]; and (iv) temperatures of ferroelectric and magnetic orderings, which remain low enough for most perovskite-based MFs.

Quite recently, the MF community turned to hybrid layered perovskite structures, considering them as prospective candidates for high-temperature MFs. We refer to layered structures composed of an n-layered perovskite with a single-layer spacer in-between. Depending on the kind of single-layer spacer, one can distinguish the Ruddlesden–Popper (RP) A′₂(A_n−1B_nO_3n+1), Dion–Jacobson (DJ) A′(A_n−1B_nO_3n+1), and Aurivillius Bi₂O₂(A_n−1B_nO_3n+1) structures [11,12]. When n = ∞, the layered perovskite transforms into a classical perovskite.

Among layered perovskites, the most widespread and studied compounds are the RP structures. They received their name in honor of the chemists R. Ruddlesden and P. Popper, who, in 1957, synthesized the series of Sr₂TiO₄, Sr₃Ti₂O₇, and Sr₄Ti₃O₁₀ phases, forming a class of complex oxides S_rn+1Ti_nO_3n+1 [13]. Since then, the class of RP structures has expanded significantly, and, at present, it includes various compounds with the general formula A′₂(A_n−1B_nO_3n+1), where A’ = La, Sr, Li, …; A = Sr, Ca, K, …; and B = Ti, Fe, Mn, Nb, …. The advantages of RP-type perovskites are related to long-term chemical stability, the activity of oxygen reduction reaction, and the possibility of realizing nontrivial magnetic and electrical properties [11,12,13,14,15,16,17]. Due to the activity of the reaction with oxygen, RP structures find applications in solid oxide fuel cells [14]. As for magnetoelectricity, interest in layered perovskites is associated with the discoveries of the colossal magnetoresistance effect in the early 2000s in RP structures with magnetoactive cations (Ca₃Mn₂O₇, …) [14,15,16,17,18,19] and ferroelectricity at room temperatures, found in Ca_3−xSr_xTi₂O₇ in 2015 [20] and then in Sr₃M₂O₇-based oxides and other RP phases [21]. Currently, RP structures are considered promising candidates for high-temperature MFs. The crystal structure, instabilities, and structural phase transitions occurring in RP oxides are well known; however, a number of questions concerning ferroelectricity, magnetic ordering, and their coupling is open for discussion.

The focus of this manuscript was the study of the magnetoelectric effect in Bi_1−xR_xFeO₃, RCrO₃, and Ca₃(Ti_1−xMn_x)₂O₇ multiferroics (MFs). We aimed to develop a unified approach to the analysis of the perovskite-like multiferroics based on the symmetry of the material, taking crystallographic distortions as the primary order parameters.

It is valuable to note that the classification of distortions and the search for recipes for effective magnetoelectric couplings in MFs with a perovskite structure is a long-standing and, at the same time, ‘hot’ problem. We refer to several reviews, references therein, and original papers that discussed various classification schemes for perovskite distortions and group-theoretical approaches, including their implementation in online tools (in particular, ones on the Bilbao Crystallographic Server) for identification of active order parameters, and their possible couplings and invariant polynomials [22,23,24,25,26,27,28,29]. However, distortion classification schemes mainly deal with BO6 octahedrons and do not account for polar distortions, which have significant impact on ferroelectric and MF properties; in addition, they also have limitations for specific systems. Despite the power of the software tools, they have not yet been applied to each MF system; for example, the details of the magnetoelectric coupling in MFs with a variable concentration of rare-earth ions have not yet been investigated, and the same statement is also applied to the RP structures.

Thus, in our research, we appealed to group-theoretical analysis, which is an elegant and effective tool for studying the properties of crystals with a complex magnetic structure, to classify the distortive, ferroelectric, and magnetic orderings in several classes of perovskite MFs. For this purpose, we examined typical MF structures with the perovskite parent phase: BiFeO₃, RCrO₃, Ca₃Mn₂O₇, in which small distortions in the crystallographic lattice lead to systems of different symmetry (trigonal, orthorhombic, and tetragonal). Using the methods of theoretical group analysis, we considered how the symmetry manifests itself in their magnetoelectric properties. We determined magnetic and structural modes, classified them according to the irreducible representations of the corresponding symmetry group, and calculated the coefficients of magnetoelectric couplings.

2. Materials and Methods

In this section, we analyze the crystal structure, and ferroelectric and magnetic properties of representatives of the perovskite-based multiferroics and, using the methods of the theoretical group analysis, determine the features of their magnetoelectric properties.

2.1. Multiferroic BiFeO₃

We start our consideration with the well-known multiferroic BiFeO₃ with high temperatures of ferroelectric and magnetic orderings (T_C = 1083 K, T_N = 643 K). BiFeO₃ crystallizes in the nonpolar symmetry group R3c and is a ferroelectric material. The high-symmetry perovskite group is reduced to R3c due to three types of distortions: (i) relative displacement of Bi and Fe ions along <111> axis, (ii) deformations of oxygen octahedrons, and (iii) counterrotation of oxygen octahedrons around Fe ions. The displacements of Bi and Fe ions from their centrosymmetric positions lead to spontaneous electric polarization P_s directed along one of the <111> crystallographic axes [30]. The magnitude of polarization has been the subject of controversy for a while. As was reported in Refs. [5,20,31,32,33,34,35], spontaneous polarization in the BiFeO₃ crystal and films can attain values varying from 6 to 150 μC/cm². Electric polarization is sufficiently low, P_s~6–9 μC/cm², in single crystals and ceramic samples [34,35]; the large values of P_s, around 100 μC/cm², are achieved under epitaxial strain in the films [5,36,37,38]; and they can also be related to the supertetragonal phase induced by the strain.

The rigorous estimation of electronic polarization requires the implementation of quantum mechanics, including electronic structure methods with correspondence to experimentally measurable observables [39]. In the frame of these approaches in Ref. [40], electric polarization in perovskite BiFeO₃ was calculated as a function of percentage distortion from the high-symmetry nonpolar structure to the ground-state R3c structure, which gives the value 95.0 μC cm⁻². However, as was shown in Ref. [30], similar results can be obtained with the use of the point charge model, where the electric polarization is represented through the atomic displacements of Bi³⁺ and O²⁻ ions.

$$P=\frac{4e}{V}\left(3{\zeta }_R-4{\zeta }_{O1}-2{\zeta }_{O3}\right)$$

(1)

where e is the elementary charge, V is the unit cell volume, ζ_R is the displacement of Bi ions corresponding to (i) distortion, and ζ_O1 and ζ_O3 are the displacements of oxygen ions corresponding to (ii)—distortion.

In magnetic relation, BiFeO₃ is weak ferromagnet with G-type antiferromagnetic (AFM) ordering characterized by AFM vector $\mathit{L}=\frac{1}{V}{{\displaystyle \sum _{i=1}^6\left(-1\right)}}^i{\mathit{\mu }}_i$, where μ_i are the magnetic moments of six Fe ions in a unit cell. The canting of spins is related to rotations of oxygen octahedrons FeO6, determined by the vector Ω, the distortions of (iii)—type [41], which have an impact on the weak ferromagnetic vector, determined as

$$\mathit{M}=\frac{V_0{a}^2}{6J}\mathbf{\Omega }\times \mathit{L}+\frac{V_0{a}^2}{6J}{\displaystyle \sum _{n=1}^6\left({\mathit{n}}_n\times {\zeta }_{O_n}\right)}$$

(2)

where V₀ is a constant, a is the lattice parameter, J is the exchange constant, and n_n is the direction vector, oriented along one of the <100> crystallographic axes. Here, we used an assumption that the Heisenberg exchange constant remains unchanged, which is a commonly used approximation for multiferroic BiFeO₃; in particular, the deviation (noncollinearity) of the nearest-neighbor spins is accounted for by the antisymmetric Dzyaloshinskii–Moriya exchange interaction [42,43]. AFM spin arrangement is superimposed with a spatially modulated magnetic structure with a large period λ = (620 ± 20 Å) incommensurate with the lattice parameter [44]. The magnetic moments of Fe ions retain their local mutually antiferromagnetic G-type orientation and rotate along the propagation direction of the modulation in the plane perpendicular to the basal crystal plane.

To explore the magnetoelectric effect (MEE) in BiFeO₃, we appealed to symmetry analysis. Knowledge of crystal structure and symmetry is of primary importance for this approach. To consider the properties of BiFeO₃, the space group $R3\overline{c}$ was taken as a ‘parent’ symmetry phase [45]. $R3\overline{c}$ differs from the space symmetry group R3c of BiFeO₃ single crystals only by the presence of a polar vector P = (0, 0, P_s). The symmetry analysis of the magnetoelectric properties of BiFeO₃ has been performed in Refs. [45,46,47]. Here, we focused our attention on the magnetoelectric tensor α_ij, which links the electric polarization and the magnetic field

$$P_i={\alpha }_{ij}{H}_j$$

(3)

As was shown in Ref. [47] for BiFeO₃ crystals, the tensor α_ij can be expressed through the components of the AFM vector L

$${\alpha }_{ij}^{\mathrm{B}\mathrm{i}\mathrm{F}\mathrm{e}{\mathrm{O}}_3}=\left|\begin{array}{l}-a_1{L}_x a_4{L}_z+a_1{L}_y a_2{L}_y\\ a_1{L}_y-a_4{L}_z a_1{L}_x -a_2{L}_x\\ -a_3{L}_y a_3{L}_x 0\end{array}\right|$$

(4)

As seen from Equations (3) and (4), the symmetry allows the linear magnetoelectric effect; however, due to the presence of an incommensurate space-modulated structure, it is not observed in BiFeO₃. Thus, when the spin cycloidal structure is destroyed, the linear magnetoelectric effect is restored. It can be achieved by applying a strong magnetic field, pressure, or doping BiFeO₃ with rare-earth ions. In recent years, a series of experimental studies has been conducted, indicating the enhancement of the magnetoelectric effect in BiFeO₃ multilayers and composites.

Here, we discuss the replacement of various rare-earth (RE) ions at the A-positions as a way to improve the MF properties of BiFeO₃, a route that was paved in the 1990s [30,45,46,47,48,49,50], and current research confirms its efficiency [6,51]. As was shown in Ref. [51], La-substitution in Bi_1−xLa_xFeO₃ (x = 0.08~0.22) ceramics induces a continuous structural evolution from R3c to Pna2₁ and, finally, the Pbnm phase. To understand the properties of Bi_1−xR_xFeO₃ structures (R = La, Gd, Dy, …), it is reasonable to introduce the AFM vector (l), which characterizes the magnetic ordering of the rare-earth sublattice, and to consider the Pbnm space symmetry group as the parent phase in the Bi_1−xR_xFeO₃ structure, keeping in mind structural phase transitions Pbnm → Pna2₁. The space group Pbnm contains eight irreducible representations, where their matrix representations are given in the columns corresponding to the generating symmetry elements (

Table 1. Irreducible representations of the Pbnm symmetry group.

Гi	$\overline{1}$	2x	2y	2z	Order Parameters, and Magnetic and Electric Fields
Г1	1	1	1	1	Ly
Г2	1	1	−1	−1	Mx	Lz	Hx
Г3	1	−1	1	−1	My		Hy
Г4	1	−1	−1	1	Mz	Lx	Hz
Г5	−1	1	1	1	ly
Г6	−1	1	−1	−1	lx	Pz	Ez
Г7	−1	−1	1	−1		Py	Ey
Г8	−1	−1	−1	1	lz	Px	Ex

). The vector components of magnetic field H, electric polarization P, magnetization M, and antiferromagnetic vectors L and l form the irreducible representations and are placed in the last column in Table 1 according to their transformation properties. Table 1 allows the determination of the transformation properties of AFM vector components, exchange-coupled structures, and magnetoelectric invariants, contributing to the magnetoelectric energy of the Bi_1−xR_xFeO₃ structure.

$$\begin{array}{l}{\mathrm{\Phi }}_{me}^{{\mathrm{B}}_{1-\mathrm{x}}{\mathrm{R}}_{\mathrm{x}}\mathrm{F}\mathrm{e}{\mathrm{O}}_3}=P_z\left({\gamma }_1\left(l_z{M}_y-l_y{M}_z\right)+{\gamma }_2\left(l_x{L}_y-l_y{L}_x\right)\right)+P_x\left({\gamma }_{xyx}\left(l_y{M}_x-l_x{M}_y\right)+{\gamma }_2\left(l_y{L}_z-l_z{L}_y\right)\right)+\\ P_y\left({\gamma }_3{l}_x{M}_x+{\gamma }_2{l}_x{L}_z+{\gamma }_4\left(l_y{M}_y+L_y^2\right)\right)\end{array}$$

(5)

where γ_i are the magnetoelectric coefficients,

$${\gamma }_1={\gamma }_{zzy},{\gamma }_2={\gamma }_{xyz},{\gamma }_3={\gamma }_{yxx},{\gamma }_4={\gamma }_{yyy}$$

The magnetoelectric coupling coefficient for the Bi_1−xR_xFeO₃ structures can be determined as follows

$${\alpha }_{ij}=\left|\begin{array}{l} a_1\left(l_x{L}_x+l_z{L}_z\right) a_{12}{l}_x -a_{13}{l}_y \\ a_{12}{l}_z a_2{l}_y a_{23}{l}_x \\ a_{13}{l}_y a_{32}{l}_z a_1\left(l_x{L}_x+l_z{L}_z\right)\end{array}\right|\approx \left|\begin{array}{l} 0 a_{12}{l}_x -a_{13}{l}_y \\ a_{12}{l}_z a_2{l}_y a_{23}{l}_x\\ a_{13}{l}_y a_{32}{l}_z 0\end{array}\right|$$

(6)

In the case of Bi_1−xR_xFeO₃, rare-earth elements can have a significant impact on magnetoelectric properties. As seen from Equation (6), α_ij depends on the components of the AFM vector of the rare-earth sublattice (for the case of nonzero magnetization on the R ion site), and it is possible to distinguish a linear MEE related to rare-earth elements, and a quadratic MEE emerges due to cross-coupling of the AFM vectors of iron and rare-earth sublattices.

2.2. Magnetoelectric Properties of RCrO₃

In this section, we consider MEE in the rare-earth orthochromites (RCrO₃), focusing on the relation of magnetoelectric properties with crystallographic distortions. The crystal structure of RCrO₃ belongs to the space symmetry group Pbnm/Pnma [52,53,54]. Its orthorombically distorted perovskite unit cell contains 4 RCrO₃ molecules (Figure 1). In addition, 4 Cr³⁺ and 4 R³⁺ magnetic ions (in the case of R = rare-earth ion) are located in the local positions differing by the symmetry of O²⁻ environments. The d-ions (M³⁺) occupy the position 4b, the f-ions (R³⁺) occupy the position 4c, and oxygen ions occupy the positions 4c and 8d (in Wyckoff notation).

Magnetic moments of the d-ions determined by the vectors M_i (i = 1–4) constitute 4 transition metal magnetic sublattices, and the magnetic moments of the f-ions determined by vectors m_i (i = 1–4) constitute 4 rare-earth magnetic sublattices. The combinations between magnetic moments of the d- and the f-sublattices determine magnetic modes of the ‘d’ ions:

$$\mathit{F}={\mathit{M}}_1+{\mathit{M}}_2+{\mathit{M}}_3+{\mathit{M}}_4, \mathit{A}={\mathit{M}}_1-{\mathit{M}}_2-{\mathit{M}}_3+{\mathit{M}}_4$$

$$\mathit{G}={\mathit{M}}_1-{\mathit{M}}_2+{\mathit{M}}_3-{\mathit{M}}_4,\mathit{C}={\mathit{M}}_1+{\mathit{M}}_2-{\mathit{M}}_3-{\mathit{M}}_4$$

(7)

and magnetic modes of the ‘f’- ions:

$$\begin{array}{l}\mathit{f}={\mathit{m}}_1+{\mathit{m}}_2+{\mathit{m}}_3+{\mathit{m}}_4, \mathit{a}={\mathit{m}}_1-{\mathit{m}}_2-{\mathit{m}}_3+{\mathit{m}}_4, \\ \mathit{g}={\mathit{m}}_1-{\mathit{m}}_2+{\mathit{m}}_3-{\mathit{m}}_4, \mathit{c}={\mathit{m}}_1+{\mathit{m}}_2-{\mathit{m}}_3-{\mathit{m}}_4\end{array}$$

(8)

Neutron diffraction measurements showed that RCrO₃ exhibits one of three G-type AFM configurations with weak magnetic components ${\mathrm{\Gamma }}_1\left(A_x,G_y,C_z\right)$, Г₂ (F_x,C_y,G_z), and Г₄ (G_xA_yF_z).

In contrast to BiFeO₃, the ferroelectric properties of RCrO₃ are weak, and the Curie ordering temperatures are low (T_C~130–250 K) [55,56]. However, as recent experiments have shown, during the polishing of samples in an electric field, an electric polarization of the order of 0.06 μC/cm² is induced; moreover, the electric field affects the spin–reorientation phase transitions, which indicates the presence of magnetoelectric coupling in the material [57,58].

As expected, the magnetoelectric behavior is owed to the crystallographic distortions, which are quite different from those in BiFeO_3. In the case of RCrO₃, the principal distortions are related to the antirotation of the CrO6 octahedrons around the [110] direction and the displacements of R ions from A-positions in the perovskite parent phase. Octahedron rotation is described by the axial vectors ω_i (i = 1 ÷ 4) linked to Cr ions, and, as was shown in Ref. [10], they contribute to the axial order parameter.

$${\mathbf{\Omega }}_b={\mathit{\omega }}_1-{\mathit{\omega }}_2+{\mathit{\omega }}_3-{\mathit{\omega }}_4$$

(9)

Polar order parameters related to the displacements of oxygen and rare-earth ions are also allowed. To compose the polar order parameters, we consider the electric dipole moments ${\mathit{d}}_i=q{\mathit{r}}_i$ emerging in the vicinity of the Cr ions, where ${\mathit{r}}_q={\displaystyle \sum _i^{}{q}_i{\mathit{r}}_{qi}^{}}/{\displaystyle \sum _i^{}{q}_i}$ is the position of the electric dipole charge center, q_i are the signed magnitudes of the charges, and r_qi are the radius vectors of the charges in the local reference frame. For the perovskite-like compounds,

$${\mathit{r}}_q=\frac{\left(+\frac{3}{8}e\right)\cdot {\displaystyle \sum _{i=1}^8{\mathit{r}}_R}+\left(-\frac{2}{2}e\right)\cdot {\displaystyle \sum _{i=1}^6{\mathit{r}}_O}}{\left|8\cdot \left(+\frac{3}{8}e\right)+6\cdot \left(-\frac{2}{2}e\right)\right|} \mathit{r}=(x,y,z)$$

(10)

where e is the elementary charge, r_R are the radius vectors of the rare-earth ions, and r_O are the radius vectors of the oxygen ions measured from the Cr³⁺ ion.

As in the case of magnetic moments (7), (8), electric dipoles ${\mathit{d}}_i=q{\mathit{r}}_i$ constitute 4 ferroelectric sublattices, and the combinations between dipole moments determine ferroelectric modes reads.

$$\begin{array}{l}\mathit{P}={\mathit{d}}_1+{\mathit{d}}_2+{\mathit{d}}_3+{\mathit{d}}_4\\ {\mathit{Q}}_2={\mathit{d}}_1-{\mathit{d}}_2-{\mathit{d}}_3+{\mathit{d}}_4\\ {\mathit{Q}}_3={\mathit{d}}_1-{\mathit{d}}_2+{\mathit{d}}_3-{\mathit{d}}_4\\ \mathit{D}={\mathit{d}}_1+{\mathit{d}}_2-{\mathit{d}}_3-{\mathit{d}}_4\end{array}$$

(11)

The arrangement of the dipole moments in RCrO₃ is shown in Figure 1; as can be seen, they form an antiferroelectric structure ordered according to the D-mode. Direct calculation of the dipole moments shows that the value of D is maximum, and the values of P, Q_2,3 are negligible.

The transformation properties of the order parameters introduced above can be found using the irreducible representations of the space symmetry group of the RCrO₃ compounds given in

Table 2. Irreducible representations of the Pnma symmetry group.

Гi	$\overline{1}$	2x	2y	2z	Magnetic OPs, Magnetic Field		Structural OPs, Electric Field
					4b	4c
Г1	1	1	1	1	$A_z,G_x,C_y$	cy	${\mathrm{\Omega }}_{bx}$
Г2	1	1	−1	−1	$F_z,G_y,C_x,H_z$	$f_z,c_x$	${\mathrm{\Omega }}_{by}$
Г3	1	−1	1	−1	$F_x,A_y,C_z,H_x$	$f_x,c_z$
Г4	1	−1	−1	1	$F_y,A_x,G_z,H_y$	fy	${\mathrm{\Omega }}_{bz}$
Г5	−1	1	1	1		$g_z,a_x$	$Q_{2z}$${,}_{ }{Q}_{3x}$, Dz
Г6	−1	1	−1	−1		ay	$P_z,E_z$$, Q_{3y}$
Г7	−1	−1	1	−1		$a_z,g_x$	$P_y,Q_{2x},E_y$,Dx
Г8	−1	−1	−1	1		gy	$P_x,Q_{2y},E_x$,Dy

.

Knowledge of their transformation properties makes it possible to understand how crystallographic distortions manifest themselves in magnetic and ferroelectric orderings. As in the previous subsection, classification of the components of order parameters according to the irreducible representations of the Pnma symmetry group allows us to determine exchange-coupled magnetic structures, the possible electrostatically coupled dipole structures, to compose the invariant combination of order parameters contributing to the thermodynamic potential and to calculate the magnetoelectric coupling coefficient.

P_x = (α_xyxa_y + α_xzyxg_zG_yG_x + α_xxyxa_xG_y G_x)H_x + (α_xxya_x + α_xxyyf_xG_y G_x + α_xzyyc_zG_y G_x + α_xzyg_z)H_y +…

F_x = (α*_xyxa_y + α*_xzyxg_zG_y G_x + α*_xxyxa_xG_y G_x)E_x + (α*_xxya_x + α*_xxyyf_xG_y G_x + α*_xzyyc_zG_yG_x + α*_xzyg_z)E_y +…

(12)

The magnetoelectric coefficient in terms of the components of G and g vectors is written as follows.

$${\alpha }_{ij}^{RCr{O}_3}=\left|\begin{array}{l}{a}_1{\mathit{G}}_x\left({\mathit{g}}_z{\mathit{G}}_y-{\mathit{g}}_y{\mathit{G}}_z\right) a_2{\mathit{g}}_z a_3{\mathit{g}}_x\\ a_2{\mathit{g}}_z a_1{\mathit{G}}_x\left({\mathit{g}}_z{\mathit{G}}_y-{\mathit{g}}_y{\mathit{G}}_z\right) a_4{\mathit{g}}_y\\ a_4{\mathit{g}}_y a_3{\mathit{g}}_x a_2{\mathit{g}}_z\end{array}\right|$$

(13)

At the end of this section, we draw the reader’s attention to the fact that the presented analysis applies to rare-earth orthoferrites RFeO₃. RFeO₃ compounds crystallize in the same space symmetry group Pnma, and several of these crystals exhibit nonlinear magnetoelectric responses [23,25,59]. The magnetoelectric tensor (12) can also be used to describe their magnetoelectric properties.

2.3. Ruddlesden–Popper Structures

Ruddlesden–Popper (RP) structures are attracting considerable attention due to the recently discovered high-temperature ferroelectric properties and the prospects for their potential implementation as multiferroics at room temperature [11]. The presence of ferroelectric properties, the so-called hybrid improper ferroelectricity (HIF), was theoretically predicted in double-layered perovskite compounds Ca₃Mn₂O₇ and Ca₃Ti₂O₇, and then detected experimentally in Ca_3−xSr_xTi₂O₇ single crystals [20], and in Ca₃(Ti_1−xMn_x)₂O₇ ceramics [21].

In this section, we plan to focus on the RP manganite phases Ca₃(Ti_1−xMn_x)₂O₇ as they contain magnetic Mn ions and, as shown in Ref. [21], are room-temperature HIF materials. In the limiting case x = 1, Ca₃(Ti_1−xMn_x)₂O₇ transforms into the RP phase Ca₃Mn₂O₇, which was thoroughly explored due to the colossal magnetoresistance effect found in the (CaO)-(CaMnO₃)_n (n =1, 2, 3, ∞) RP structures in the 2000s [16,17,18,19]. The structures with n = 1 (2D-structure) and n = ∞ (perovskite) are considered as the end members of the RP series, so the double-layered RP structures receive more attention.

The temperature of AFM ordering T_N = 115 K in Ca₃Mn₂O₇ was first experimentally found in Ref. [16], where it was assumed that there is a G-type AFM order. A neutron diffraction study [19] revealed the dominant contribution of the G-type or C-type AFM phase with an AFM spin arrangement within the bi-layer plane. In that work, the possibility of the existence of a weak ferromagnetic (WFM) state in the RP phase Ca₃Mn₂O₇ was also assumed. A further study on Ca₃Mn₂O₇ showed the transition to the AFM state at the temperature T_N = 134 K and a signature of WFM below 100 K [60,61].

The crystal structure, structural phase transitions, and polar phases of Ca₃Ti₂O₇ [62] and Ca₃Mn₂O₇ [19,63] with double-layered RP structures, studied since 1998 [16], have now been proven [21]. Above room temperature (RT), the crystal structure of Ca₃Mn₂O₇ is described by the tetragonal space group I4/mmm, and at RT, by the space group Cmc2₁ [64]. The transition from the I4/mmm to Cmc2₁ phase should occur through an intermediate phase, which can be either Cmcm or Cmca, as follows from the first-principles calculations [46]. The transition into the ferroelectric phase is observed close to T_C = 280 K, at which the transition into the Cmc2₁ phase occurs. As experimental measurements indicate that ferroelectric ordering in Ca₃Mn₂O₇ occurs up to RT, the existence of an intermediate ferroelectric state is expected here.

Structural transformations can arise due to the substitution of magnetic ions at the A-positions (Ca_3−xLa_xMn₂O₇) or ferroelectric ions in the B-positions (Ca₃(Ti_1−xMn_x)₂O₇). The tetragonal-to-orthorhombic transition presumable through the phases I4/mmm → Fmmm → Cmcm → Cmc2₁, occurring in the 200–300 °C range, was observed during the study of Ca_3−xLa_xMn₂O₇ structures [19,64]. Though the existence of I4/mmm and Cmc2₁ phases is confirmed, the presence of ‘intermediate’ phases is still under discussion.

Thus, in further consideration, we can take the tetragonal I4/mmm structure (Ca₃Mn₂O₇) as the parent phase and apply the symmetry analysis, employed in the previous sections, to explore the magnetoelectric properties of the RP structures. According to the data of the synchrotron X-ray diffraction study [19], the Mn ions occupy the 4e positions (in the Wyckoff notation) in the I4/mmm phase, which indicates the presence of 4 magnetic sublattices μ_i (i = 1÷4) with magnetic moments of the same magnitude. Possible combinations of the vectors μ_i give us the basic magnetic vectors.

$$\begin{array}{l}\mathit{F}={\mathit{\mu }}_1+{\mathit{\mu }}_2+{\mathit{\mu }}_3+{\mathit{\mu }}_4\\ \mathit{A}={\mathit{\mu }}_1-{\mathit{\mu }}_2-{\mathit{\mu }}_3+{\mathit{\mu }}_4\\ \mathit{G}={\mathit{\mu }}_1-{\mathit{\mu }}_2+{\mathit{\mu }}_3-{\mathit{\mu }}_4\\ \mathit{C}={\mathit{\mu }}_1+{\mathit{\mu }}_2-{\mathit{\mu }}_3-{\mathit{\mu }}_4\end{array}$$

(14)

The unit cell of Ca₃Mn₂O₇ shown in Figure 2 contains 2 formula units Ca₃Mn₂O₇. The positions of the symmetry elements $I,C_2^z,C_4^z,C_2^y$(generators of I4/mmm space group) in the unit cell are depicted in Figure 2b in accordance with Figure 2a. Note that the local symmetry elements of the 4e{4mm} positions, which are the $4_z$ axis (principal crystal axis) and 4 planes of symmetry m passing through this axis, remain even for magnetic ions, occupying the 4e positions.

In accordance with Figure 2, it is easy to compose permutation transformations of the ions at the 4e positions (

Table 3. Permutation transformations ions in positions 4e {4mm}.

GF	1	2	3	4
$\overline{1}$	2	1	4	3
4z	1	2	3	4
2z	3	4	1	2
2y	2	1	4	3

) and permutation transformations of basis vectors (

Table 4. Permutation transformations of basis vectors.

GF	F	A	G	C
$\overline{1}$	F	−A	−G	C
4z	F	A	G	C
2z	F	−A	G	−C
2y	F	−A	−G	C

).

Using Table 4, we obtain a cipher (Turov indices) for AFM vectors, which explains how the symmetry elements transform magnetic sublattices into each other.

$$\mathit{A}: 4_z(+)2_z^1(-)2_y(-)\overline{1}(-)$$

$$\mathit{G}: 4_z(+)2_z^1(+)2_y(-)\overline{1}(-)$$

$$\mathit{C}: 4_z(+)2_z^1(-)2_y(+)\overline{1}(+)$$

In consistence with the data of experimental studies [16,19], we assume the G-type of AFM ordering. In this case, the values of AFM vectors A and C are taken to be sufficiently small and we can restrict our consideration with two magnetic order parameters instead of 4. Thus, we leave the ferromagnetic vector M = F and the antiferromagnetic vector L = G as the basic magnetic parameters. Then, we classify them together with the polarization vector according to the irreducible representation of the space symmetry group $D_{4h}^{17}$ (

Table 5. Irreducible representations of the I4/mmm symmetry group and basic functions.

Гi	$E$	$2C_{4z}$	$C_{2z}$	$2C_{2y}$	$\overline{1}$	Basic Vectors F, G, P
Г1	1	1	1	1	1
Г2	1	1	1	−1	1	Fz
Г3	1	−1	1	−1	1
Г4	1	−1	1	1	1
Г5	$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$	$\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$	$\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$	$\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$	$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$	$\left(\begin{array}{c}{F}_x\\ F_y\end{array}\right)$
Г6	1	1	1	1	−1	Gz
Г7	1	1	1	−1	−1	Pz
Г8	1	−1	1	−1	−1
Г9	1	−1	1	1	−1
Г10	$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$	$\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$	$\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$	$\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$	$\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$	$\left(\begin{array}{c}{P}_x\\ P_y\end{array}\right)$
${\mathrm{Г}}_{10}^{’}$	$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$	$\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$	$\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$	$\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$	$\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$	$\left(\begin{array}{c}{G}_x\\ G_y\end{array}\right)$

). The first line of Table 5 contains generators of the $D_{4h}^{17}$ group $G=\left\{E,I,C_2^y,C_2^z,C_4^{z+}\right\}$, where the last column contains basic magnetic and ferroelectric functions.

The decomposition of the basic functions into irreducible representations of the symmetry group I4/mmm allows one to obtain information on the properties of a system. It is seen that the ferromagnetic and antiferromagnetic vectors transform according to the different IRs. In this case the transformation of F and L into each other can occur due to external factors, such as the electric field or strain. The combinations $F_z\left(P_x{G}_x+P_y{G}_y\right),G_z\left(F_x{P}_x+F_y{P}_y\right),P_z\left(F_x{G}_x+F_y{P}_z{G}_y\right),F_z{P}_z{G}_z$, transforming on Г₁, are invariants and have an impact on magnetoelectric energy.

$${\mathrm{\Phi }}_{me}={\gamma }_2{F}_z\left(P_x{G}_x+P_y{G}_y\right)+{\gamma }_3{G}_z\left(F_x{P}_x+F_y{P}_y\right)+{\gamma }_4{P}_z\left(F_x{G}_x+F_y{G}_y\right)+{\gamma }_5{F}_z{P}_z{G}_z$$

(15)

where

$${\gamma }_2={\gamma }_{zxx}={\gamma }_{zyy},{\gamma }_3={\gamma }_{xxz}={\gamma }_{yyz},{\gamma }_4={\gamma }_{xzx}={\gamma }_{yzy},{\gamma }_5={\gamma }_{zzz}$$

Using Table 5 we calculate the magnetoelectric coupling tensor.

$${\alpha }_{ij}^{R-P}=\left|\begin{array}{l}{\gamma }_3{G}_z 0 {\gamma }_2{G}_x\\ 0 {\gamma }_3{G}_z {\gamma }_2{G}_y\\ {\gamma }_4{G}_x {\gamma }_4{G}_y {\gamma }_5{G}_z\end{array}\right|$$

(16)

As seen from Equation (16) it linearly depends on the components of antiferromagnetic vector G. This finding allows us to conclude that the linear magnetoelectric effect is allowed by the symmetry of the RP structures.

3. Results and Conclusions

To summarize, we applied symmetry analysis to the magnetic and magnetoelectric properties of multiferroics (MFs) with a perovskite structure. One of the reasons for choosing perovskite-based MFs as the object of this study is related to the structural instability of the initial perovskite phase, which can lead to a variety of structures with different symmetry. This allows us to demonstrate how crystallographic distortions of various types, even if they are small enough, significantly alter the magnetic and ferroelectric orderings and their couplings. As the typical examples, we considered (i) the family of high-temperature multiferroics Bi_xR_1−xFeO₃ recognizable by their magnetoelectric properties; (ii) the rare-earth orthochromites RCrO₃, promising candidates for MFs, compounds with well-known magnetic structures; and (iii) the RP structures containing magnetic cations, such as Ca₃(Ti_1−xMn_x)₂O₇, novel high-temperature MFs, the ferroelectric and magnetic properties of which are still being researched. These structures crystallize into trigonal, orthorhombic, and tetragonal syngonies, respectively, which allows us to trace the relationship between magnetoelectric properties and the symmetry of a structure, and to consider impacts given by crystallographic distortions.

The direct contribution of the distortion into electric polarization and weak ferromagnetism was considered, using BiFeO₃ as an example. We also analyzed the influence of structural transformation of the Bi_xR_1−xFeO₃ family on their magnetoelectric properties. Calculations showed that the linear magnetoelectric effect, suppressed by the spin-modulated structure in pure BiFeO₃, becomes allowed due to the symmetry of the new phase in Bi_xR_1−xFeO₃, where it is attributed mainly to rare earth (R) ions.

In the case of the rare-earth orthochromites RCrO₃, it was shown that the ferroelectric and magnetoelectric properties are due to crystallographic distortions. The displacements of oxygen ions from their positions in the initial perovskite phase results in the emergence of electric dipole moments in the vicinity of the Cr ions, which are coupled with the magnetic moments of the Cr ions. The distortive, ferroelectric, and magnetic order parameters were classified according to irreducible representations of the Pnma symmetry group of RCrO₃, which allows the invariant combinations to be composed between these parameters.

For the first time, a similar symmetry consideration was implemented in the analysis of the Ruddlesden–Popper structures. Taking into account the local symmetry of magnetic ions in the RP unit cell, we introduced the relevant magnetic order parameters and classified them according to the irreducible representation of the I4/mmm symmetry group, which describes the tetragonal symmetry of the RP structures. Analysis exemplified on the Ca₃(Ti_1−xMn_x)₂O₇ compounds demonstrated the possibility of realizing MEE in the RP phases containing magnetically active cations and allowed the estimation of the magnetoelectric contribution to the thermodynamic potential.

In conclusion, this research allowed us to compare the magnetoelectric effects for different crystal systems of perovskites and thus design a more meaningful organization of the desired experiments.