A group-theoretical approach to enumerating magnetoelectric and multiferroic couplings in perovskites

Senn, M.S.; Bristowe, N.C.

doi:10.1107/S2053273318007441

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Volume 74| Part 4| July 2018| Pages 308-321

https://doi.org/10.1107/S2053273318007441

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A group-theoretical approach to enumerating magnetoelectric and multiferroic couplings in perovskites

Mark S. Senn ^a ^* and Nicholas C. Bristowe ^b,^c

^aDepartment of Chemistry, University of Warwick, Gibbet Hill, Coventry, CV4 7AL, UK, ^bSchool of Physical Sciences, University of Kent, Canterbury CT2 7NH, UK, and ^cDepartment of Materials, Imperial College London, London SW7 2AZ, UK
^*Correspondence e-mail: m.senn@warwick.ac.uk

Edited by L. Bourgeois, Monash University, Australia (Received 10 January 2018; accepted 17 May 2018; online 5 July 2018)

A group-theoretical approach is used to enumerate the possible couplings between magnetism and ferroelectric polarization in the parent $[Pm{\overline 3}m]$ perovskite structure. It is shown that third-order magnetoelectric coupling terms must always involve magnetic ordering at the A and B sites which either transforms both as R-point or both as X-point time-odd irreducible representations (irreps). For fourth-order couplings it is demonstrated that this criterion may be relaxed allowing couplings involving irreps at X-, M- and R-points which collectively conserve crystal momentum, producing a magnetoelectric effect arising from only B-site magnetic order. In this case, exactly two of the three irreps entering the order parameter must be time-odd irreps and either one or all must be odd with respect to inversion symmetry. It is possible to show that the time-even irreps in this triad must transform as one of: X₁⁺, M_3,5⁻ or R₅⁺, corresponding to A-site cation order, A-site antipolar displacements or anion rocksalt ordering, respectively. This greatly reduces the search space for type-II multiferroic perovskites. Similar arguments are used to demonstrate how weak ferromagnetism may be engineered and a variety of schemes are proposed for coupling this to ferroelectric polarization. The approach is illustrated with density functional theory calculations on magnetoelectric couplings and, by considering the literature, suggestions are given of which avenues of research are likely to be most promising in the design of novel magnetoelectric materials.

Keywords: magnetoelectric couplings; multiferroic couplings; perovskites; improper ferroelectricity; group theory; irrep analysis; anharmonic couplings.

1. Introduction

The classification of distortions in functional materials is an important part of the process of understanding the structure–property relationship. Perovskites (ABX₃) are among the most studied systems, which is in part due to the many functional properties that they exhibit, but also due to their richness in structural distortions and phase transitions. Schemes classifying the ubiquitous rotations and tilts of the quasi-rigid BO₆ octahedra that drive many of these phase transitions in perovskites can be conveniently classified in terms of Glazer notation (Glazer, 1972 ), and other such schemes also exist for classifying distortions in layered perovskite such as Ruddlesden–Poppers (Aleksandrov & Bartolome, 1994 ). While these schemes have enjoyed much success due to their intuitive nature, there are several limitations, in particular that they are not easily generalized to different systems. Even within the perovskite family, with additional symmetry breaking with respect to the ABX₃ aristotype, it is no longer clear how the occurrence of tilts and rotations can be unambiguously described, or indeed how the symmetry lowering implied by the combined orderings can be derived.

More formally, the degrees of freedom in an aristotype `parent' structure, such as the $[Pm{\overline 3}m]$ ABX₃ perovskite, may be defined as transforming as irreducible representations (irreps) of the parent space group (and setting). The irreps for all special positions in reciprocal space have been tabulated by various authors including by Bradley & Cracknell (1972 ), Miller & Love (1967 ), Kovalev (1993 ) and more recently also at non-special k-points (Stokes et al., 2013 ). With knowledge of these irreps, it is possible to compute the isotropy subgroups of the 230 space groups (Stokes & Hatch, 1988 ), which are the subgroups accessible due to the action of an order parameter (OP) transforming as one of these irreps.

Online tools such as ISODISTORT (Campbell et al., 2006 ) and AMPLIMODES on the Bilbao Crystallographic Server (Aroyo et al., 2006 ; Orobengoa et al., 2009 ) allow distorted structures to be easily decomposed in terms of irreps of a parent space group, and it is now possible to superpose up to three irreps with associated independent incommensurate propagation vectors, and derive the possible subgroups and secondary order parameters (SOPs) (Stokes & Campbell, 2017 ). Additionally, these programs now generate outputs that can be directly read by Rietveld and single-crystal refinement programs (Campbell et al., 2007 ; Perez-Mato et al., 2010 ), allowing refinements to be performed in a symmetry-adapted basis and facilitating easy identification of the active order parameters in a given phase transition.

As a result of much of this work, several group-theoretical studies have emerged that have more formally classified distortions in perovskite-related materials. These include group-theoretical analysis of octahedral tilting in perovskites (Howard & Stokes, 1998 , 2005 ; Knight, 2009 ), cation-ordered and Jahn–Teller distortions in perovskites (Howard & Carpenter, 2010 ), ferroelectric perovskites (Stokes et al., 2002 ), anion ordering (Talanov et al., 2016 ), and works on layered Ruddlesden–Poppers (Hatch & Stokes, 1987 ; Hatch et al., 1989 ). One particularly valuable aspect of classifying these distortions in the formal language of irreps is to understand physical phenomena that can arise due to secondary order parameters which feature at linear order in the Landau-style free energy potential. These odd order terms may always adopt a sign such that they act to lower the overall free energy and hence symmetry analysis alone is sufficient to identify their instability. The process of ascertaining these couplings is greatly simplified using the ideas of invariants analysis (Stokes & Hatch, 1991 ; Saxena et al., 1994 ) when constructing the Landau-style free energy expansion about the parent undistorted phase, and online tools for doing this also exist (Hatch & Stokes, 2003 ).

This process is particularly valuable when understanding improper ferroelectricity (Levanyuk & Sannikov, 1974 ) where third-order terms in the free energy expansion are invariably the key to understanding the resulting polarization. This area has enjoyed a renaissance in the form of the recently much discussed `hybrid improper ferroelectric' mechanism [e.g. see Benedek et al. (2015 ) for a recent review]. The powerful use of magnetic superspace groups for describing multiferroic materials has also allowed magnetoelectric couplings to be trivially identified through analysis of secondary order parameters (Perez-Mato et al., 2012 ). Antisymmetric exchange arguments with respect to the parent perovskite structure have also been used to explain the dominant anisotropic terms that control the directions of spin ordering (Khalyavin et al., 2015 ). And of course, the occurrence of weak ferromagnetism (wFM) by the Dzyaloshinsky–Moriya (DM) interaction (Dzyaloshinsky, 1958 ; Moriya, 1960 ) was first originally rationalized based on such symmetry arguments alone (Dzyaloshinsky, 1958).

Using many of the ideas above, and with the aid of the ISODISTORT (Campbell et al., 2006) tool, we seek here to generalize a recipe for inducing magnetoelectricity in the parent $[Pm{\overline 3}m]$ perovskite. These recipes are based on symmetry arguments alone, and we use as the ingredients structural and magnetic degrees of freedom, which we classify in terms of transforming as irreps of the parent space group. Our results clearly show why certain kinds of coupled distortions and magnetic ordering can never lead to ferroelectric or ferromagnetic secondary order parameters, and by considering which orderings and cation arrangements are commonly observed, we are able to identify several promising avenues for further investigation.

The article is arranged as follows. In §2, we first classify the ingredients for symmetry breaking that are at our disposal in terms of irreps of the parent $[Pm{\overline 3}m]$ space group. To keep our results as general as possible, we will also describe cation and anion ordering in terms of irreps, rather than forming new parent space groups. We then proceed to give various recipes for achieving (multi)ferroic orderings as a consequence of different symmetry-breaking distortions. In §3, for completeness we give the recipe for (hybrid) improper ferroelectricity, while in §§4, 5, 6 we discuss magnetoelectric couplings arising due to third- and fourth-order terms in the free energy expansion. As the most useful multiferroics are those that are ferromagnets (rather than antiferromagnets), in §7 we explain how similar ideas can be used to design systems that exhibit wFM. We also consider in this section systems in which either polarization (P) or wFM is supplied as an external order parameter (as a magnetic or electric field) resulting in the development of wFM or P, respectively, in response to the stimuli. Finally, in §8 we put all of our above ideas together and deal with the design of materials that are both wFM and ferroelectric, and have indirect coupling through at least one primary order parameter (POP).

2. Ingredients for symmetry breaking

First we classify the magnetic degrees of freedom at our disposal in terms of irreps of the space group $[Pm{\overline 3}m]$ . We classify all of these in terms of irreps of the parent perovskite structure $[Pm{\overline 3}m]$ with setting A 1a (0, 0, 0); B 1b (½, ½, ½); X 3c (0, ½, ½). We note that reversing the setting of the structure will result in many of the irrep labels changing, in particular at the X- and R-points, irreps labelled as `+' will correspond to another numbered irrep with the `−' sign and vice versa. The origin of this is that the sign part in these irrep labels refers to whether or not parity (with respect to inversion symmetry) is conserved or violated at the origin (0, 0, 0), and hence interchanging the atom at the origin naturally affects the distortions physically being described by a particular representation. The orderings of the magnetic degrees of freedom will ultimately be devised in such a way as to drive secondary order parameters that are related to ferroelectricity. We restrict ourselves here to the basic types of antiferromagnetic ordering which are commonly observed in perovskites. These are often characterized as A, C and G type having one, two and three antiferromagnetic (AFM) nodes, respectively. They may be classified as corresponding to orderings which transform as irreps at the X[0, ½, 0]-, M[½, ½, 0]- and R[½, ½, ½]-points (Fig. 1). It is important to note that magnetic structures such as A_x and A_yz, which correspond to an ordering with propagation vector X[½, 0, 0] with moment along the propagation axis and perpendicular to it, transform as distinct irreps in this analysis, and will imply physically distinct secondary order parameters. This forms the basis of the antisymmetric exchange arguments of Khalyavin et al. (2015) to determine spin (exchange) anisotropy, and this is why this analysis is so powerful in the perovskite structure where the magnetic atoms sit on high-symmetry sites. Fig. 1 gives full details of how the spin arrangements are related to irreps.

Figure 1
Basic AFM magnetic orderings of the perovskite structure with associated irrep labels and illustrated along high-symmetry OPD A sites, B sites and X sites are shown as green, red and blue spheres, respectively. The parent cubic unit cell is shown in pink so as to illustrate the relationship with the new crystallographic axes (grey). All figures are drawn in ISODISTORT.

Next we classify the various structural degrees of freedom within the perovskite structure for inducing symmetry-lowering phase transitions. The ingredients at our disposal are the commonly observed octahedral rotation and tilt modes, Jahn–Teller distortion modes, cation (charge) ordering modes, antipolar modes and strain. These are all listed in Table 1, along with their corresponding labels in the alternative setting [A at (½, ½, ½)]. Some of these degrees of freedom will be accessible via physical control parameters (such as application of epitaxial strain) whilst others only by chemical design (for example, by inclusion of Jahn–Teller active cations). In the analysis, we will also classify cation and anion orderings in the perovskite structure in terms of transforming as irreps of the parent perovskite. For example, rocksalt cation ordering at the B site transforms as R₂^- and A-site layered cation order as X₁⁺. We may even classify the highly distorted cation-ordered A′A₃B₄O₁₂ quadruple perovskite with aristotype $[Im{\overline 3}]$ as having cation orderings transform as M₁⁺ [with three k-actives = ( $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ , 0); (0, $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ ); ( $[{{1} \over {2}}]$ , 0, $[{{1} \over {2}}]$ )] and octahedral rotations that stabilize the A′ square-planar coordination transforming as M₂⁺.

Table 1
Ingredients for symmetry breaking in the perovskite structure, classified in terms of transforming as irreps of the parent perovskite structure, with the A site at the origin (the corresponding irrep labels for the setting with the B site at the origin are given in parentheses)

Ingredient	Γ	X	M	R
Strain	$[\Gamma _{3}^{+}]$ ; $[\Gamma _{5}^{+}]$
Cation order (A)		X₁⁺ (X₃^-)	M₁⁺ (M₄⁺)	R₁⁺ (R₂^-)
Cation order (B)		X₃^- (X₁⁺)	M₄⁺ (M₁⁺)	R₂^- (R₁⁺)
Anion order (O)		X₁⁺ (X₃^-)	M₄⁺ (M₁⁺); M₅^- (M₅^-)	R₅⁺ (R₄^-)
(Anti-)Polar (A)	$[\Gamma _{4}^{-}]$	X₃^- (X₁⁺); X₅^- (X₅⁺)	M₃^- (M₂^-); M₅^- (M₅^-)	R₄^- (R₅⁺)
(Anti-)Polar (B)	$[\Gamma _{4}^{-}]$	X₁⁺ (X₃^-); X₅⁺ (X₅⁺)	M₂^- (M₃^-); M₅^- (M₅^-)	R₅⁺ (R₄^-)
Jahn–Teller modes	$[\Gamma _{3}^{+}]$	X₃^- (X₁⁺)	M₃⁺ (M₂⁺)	R₃^- (R₃⁺)
Octahedral tilt modes			M₂⁺ (M₃⁺)	R₅^- (R₄⁺)
Magnetic order (A)	$[{\rm m}\Gamma _{4}^{+}]$	mX₃⁺ (mX₁^-); mX₅⁺ (mX₅^-)	mM₃⁺ (mM₂⁺); mM₅⁺ (mM₅⁺)	mR₄⁺ (mR₅^-)
Magnetic order (B)	$[{\rm m}\Gamma _{4}^{+}]$	mX₁^- (mX₃⁺); mX₅^- (mX₅⁺)	mM₂⁺ (mM₃⁺); mM₅⁺ (mM₅⁺)	mR₅^- (mR₄⁺)

Finally, the desired property, ferroelectricity, transforms as the polar mode belonging to the irrep $[\Gamma _{4}^{-}]$ . $[\Gamma _{4}^{-}]$ is a three-dimensional irrep; the most general order parameter direction (OPD) associated with this would hence be written as OP(a,b,c), where special directions (a, 0, 0), (a, a, 0) and (a, a, a) correspond to tetragonal, orthorhombic and rhombohedral directions, respectively, for the macroscopic polarization and off-centre displacements of the atoms. For a full discussion of notation relating to OPDs, including cases where multiple irreps enter into the OP, as will become pertinent in future discussion, the reader is directed to Appendix A. Please note that throughout this article we choose to list the full OPD, instead of the space group and setting. The two are equivalent, but we choose the OPD for the sake of brevity, and also due to its descriptive nature with respect to the magnetic and structural orderings that are allowed to occur. We will now discuss the general design principles by which we can combine the aforementioned degrees of freedom to produce $[\Gamma _{4}^{-}]$ as a secondary OP.

3. Recipes for improper ferroelectric couplings

We begin by considering structural irreps (transforming as time-even) alone, and how they may combine to produce improper ferroelectric couplings, before considering couplings with magnetic irreps in the next section. The concept of improper ferroelectricity was first introduced several decades ago by Levanyuk & Sannikov (1974), but recently there has been renewed interest [see reviews (Varignon et al., 2015b ; Benedek et al., 2015; Young et al., 2015 )] after its observation in epitaxially grown layered perovskite systems (Bousquet et al., 2008 ). In light of work that has highlighted the existence of improper ferroelectricity in naturally layered perovskite-like Ruddlesden–Popper systems (Benedek & Fennie, 2011 ), we believe it is also of interest to enumerate all such possible couplings in the aristotypical perovskite structure here, at least to illustrate the idea, introduce the topic and review the literature, before moving on to magnetoelectric couplings.

The general recipe for constructing improper ferroelectric coupling terms in the Landau-style free energy expansion about the parent perovskite structure that we will use is as follows. The principle of invariants analysis (Hatch & Stokes, 2003) means that, at each term in the free energy expansion, crystal momentum and inversion symmetry must be conserved. In the next section we also consider magnetism, when the additional constraint of time reversal symmetry must be conserved.

We seek initially the dominant coupling term, which means that we should consider the lowest-order term in the free energy expansion that is achievable which has linear order in P. We restrict ourselves to coupling terms only of linear order in P since in these cases we can be sure that symmetry analysis can be sufficient to infer the appearance of P, unlike in even orders where calculation of the sign and strength of the coefficients would be necessary. For example, since P transforms as inversion-odd and has zero crystal momentum, the lowest-order term will be third order (ABP), which has been termed hybrid improper ferroelectricity (Bousquet et al., 2008; Benedek & Fennie, 2011; Fukushima et al., 2011 ). Since trilinear terms will always act to lower the free energy, if A and B are unstable, then P will also be present, adopting a sign (direction of polarization) such as to stabilize the overall free energy.

Invariants analysis tells us that:

for P is inversion-odd; [P] = [0, 0, 0].¹

A·B is inversion-odd; [A] + [B] = [0, 0, 0] must be obeyed leading to all quantities being conserved in the trilinear term:

A·B·P is inversion-even; [A] + [B] + [P] = [0, 0, 0] to be true, where [A] represents crystal momentum associated with OP A and A·B is the multiplication of the characters of the irreps associated with the OP A and B.²

One may further convince oneself that A $[\neq]$ B must be true for this condition to be fulfilled for otherwise AB would be inversion-even, meaning that the quadratic linear term A²P is not permissible in the free energy expansion, and so is not a term that can drive an improper coupling.³ In summary we can say that A and B must both be of opposite parity with respect to inversion symmetry and must have equal crystal momentum. We will explore all trilinear couplings possible within the perovskite parent structure for OPs transforming as X-, M- and R-point irreps below.

The above criterion is necessary, but in a few cases not always sufficient to ensure the desired improper ferroelectric coupling. In practice, this may be conveniently checked using `Method 2' of the online tool ISODISTORT where multiple irreps may be superimposed to form the primary OP of the parent perovskite structure. The program then lists all the possible OPDs associated with this, along with the resulting secondary OPs and the space-group symmetry and basis with respect to the parent structure. It is then trivial to identify from either the space group or the list of secondary OPs if an improper ferroelectric coupling will occur.

Any of the following that have atomic displacements that transform collectively as these irreps will feature in a trilinear term with $[\Gamma _{4}^{-}]$ (where $[\oplus]$ represents the direct sum):

$[{\rm X}_{1,2,3}^{+} \, \oplus \, {\rm X}_{3,5}^{-}]$

$[{\rm M}_{1,2,3,4,5}^{+} \, \oplus \, {\rm M}_{2,3,5}^{-}]$

$[{\rm R}_{1,5}^{+} \, \oplus \, {\rm R}_{2,3,4,5}^{-}]$

While many of these may be difficult to achieve in practice, there are several promising candidates. For example, columnar A-site cation order (M₁⁺) with antipolar B-site displacements (M₅^-) can lead to a trilinear term M₁⁺ M₅^- $[\Gamma _{4}^{-}]$ . We believe this could be the cause of the ferroelectric polarization recently reported in high-pressure perovskite CaMnTi₂O₆ (Aimi et al., 2014 ). Indeed, cation or anion ordering at any of the perovskite sites at the M-point along with antipolar distortions at the A or B sites would produce an improper ferroelectric polarization. In-phase tilting (M₂⁺) or the M-point Jahn–Teller mode (M₃⁺) can alternatively be used in conjunction with the antipolar displacements (such as M₅^-) to induce a polarization, which has been recently predicted in the Pmc2₁ phase of several perovskites (Yang et al., 2012 , 2014 ; Varignon et al., 2016 ), and might also be the origin of the (ionic component of the) ferroelectricity in the P2₁nm half-doped manganites (Giovannetti et al., 2009 ; Rodriguez et al., 2005 ).

The commonly observed rocksalt cation ordering at the B site (King & Woodward, 2010 ) along with (R-point) antipolar distortions on the A site will also produce an improper ferroelectric coupling. While the former is commonly observed, controlling the periodicity of the antipolar distortions such as those induced by lone-pair ordering will be challenging. Cation order on the A sites at the R-point (rocksalt) along with octahedral tilt modes would also produce an improper ferroelectric coupling, as recently predicted through first-principles calculations (Young & Rondinelli, 2013 ). However, it should be noted that A-site cation ordering is more commonly found to be in a layered (X-point) arrangement (King & Woodward, 2010). Very recent reports of improper ferroelectricity in the 134 perovskite HgMn₃Mn₄O₁₂ can also be understood with respect to the present symmetry analysis of ABO₃ perovskites (Chen et al., 2018 ). In this case, the atomic displacements associated with the orbital and charge ordering degrees of freedom on the A and B sites transform as irreps of the parent space group $[Pm\bar{3}m]$ R₅⁺ and R₃^-.

A-site cation layering (X₁⁺) in combination with antipolar A-cation motions is indeed sufficient to induce P. Again, whilst the former is fairly common, the latter is only expected to be an unstable lattice distortion for low tolerance factor perovskites (Mulder et al., 2013 ). However it can itself manifest through an improper appearance with two tilting modes (M₂⁺ R₅^- X₅^-), which gives rise to the fourth-order term described below. At the X-point, one other trilinear term has been predicted to play a role in the P2₁ phase of strained CaTiO₃, whereby A- and B-site antipolar (X₅⁺ and X₅^-) motions induce P (Zhou & Rabe, 2013 ).

Fourth-order terms in P should also be considered and may be more promising on account of the extra degree of flexibility allowed in the recipe.⁴ Here, crystal momentum considerations mean that each relevant fourth-order term must take the form:

A·B·C is inversion-odd; [A] + [B] + [C] = [0, 0, 0] must be obeyed leading to all quantities being conserved in the trilinear term:

A·B·C·P is inversion-even; [A] + [B] + [C] + [P] = [0, 0, 0].

One of the most promising fourth-order candidates involves OPs associated with X⁺, M⁺, R⁻ and $[\Gamma _{4}^{-}]$ : for example, A-site layered cation ordering (X₁⁺), octahedral tilt mode (M₂⁺) and octahedral tilt mode (R₅^-). This explains the significance of layering (X₁⁺) in allowing the two octahedral rotation modes to couple together to produce a polarization and has been the most common example of improper ferroelectricity in perovskites as illustrated in both artificially (Bousquet et al., 2008; Rondinelli & Fennie, 2012 ) and naturally layered double perovskites (Fukushima et al., 2011). A similar term, predicted in half-doped titanates (Bristowe et al., 2015 ), includes A-site layered cation ordering (X₁⁺), M-point Jahn–Teller (M₃⁺) and octahedral tilt modes (R₅^-). Other possibilities include A-site striped cation ordering (X₁⁺), tilting (R₅^-) and charge order (M₄⁺), which we believe to be the origin of the improper polarization in SmBaMn₂O₆ (Yamauchi, 2013 ). Alternatively Jahn–Teller induced, M₃⁺ and R₃^-, ferroelectricity has been discussed in A-site striped cation ordered (X₁⁺) rare-earth vanadates (Varignon et al., 2015a ). Perhaps an interesting avenue for future research is to use anion ordering since the X₁⁺ irrep is also made possible by anion vacancy ordering, which for example is sometimes seen in the cobaltates (Karen et al., 2001 ; Vogt et al., 2000 ; Castillo-Martínez et al., 2006 ).

Other chemically and structurally less promising schemes are still worth a mention: X⁻ M⁻ R⁻ $[\Gamma _{4}^{-}]$ , for example, striped order at the A site (X), antipolar order at the B site (M) and rocksalt cation order at the B site (R); X⁻ M⁺ R⁺ $[\Gamma _{4}^{-}]$ , striped B-site cation order (X₃^-), octahedral tilt mode (M₂⁺), antipolar distortion on the B site (R₅⁺); and X⁺ M⁻ R⁺ $[\Gamma _{4}^{-}]$ , A-site striped cation ordering (X₁⁺), antipolar distortions on the B site (M₂^-), anion order (R₅⁺). Finally, we note that the inclusion of organic cations on the A site or organic link molecules on the X site greatly increases the possible number of such improper ferroelectric coupling schemes (Boström et al., 2018 ) and provides a promising route for designing novel functional materials.

4. Recipes for magnetoelectric coupling

We can extend the ideas discussed above for improper ferroelectrics to magnetoelectric couplings including time-odd irreps that describe magnetic order. We seek initially the strongest magnetoelectric coupling term possible: this means that as before we should consider the lowest-order term in the free energy expansion that is achievable. Since P transforms as time-even, inversion-odd and has zero crystal momentum, the lowest-order term involving two zone-boundary irreps will be third order (ABP). Invariants analysis tells us that:

for P is time-even; P is inversion-odd; [P] = [0, 0, 0].

A·B is time-even; A·B is inversion-odd; [A] + [B] = [0, 0, 0] must be obeyed leading to all quantities being conserved in the trilinear term:

A·B·P is time-even; A·B·P is inversion-even; [A] + [B] + [P] = [0, 0, 0].

As we are seeking a magnetoelectric coupling, at least one of A or B must be magnetic, and inspection of the condition that A·B is time-even means that therefore both A and B must transform as a time-odd irrep. One may further convince oneself that A $[\neq]$ B must be true for this condition to be fulfilled for otherwise AB would be inversion-even, meaning that the quadratic linear term A²P is not permissible in the free energy expansion, and so is not a term that can drive an electromagnetic coupling.⁵ Taking everything together we can say that A and B must both be time-odd, of opposite parity with respect to inversion symmetry and must have equal crystal momentum.

As before with the improper ferroelectrics, the list of magnetoelectric trilinear coupling terms (with respect to the perovskite parent structure) will prove to be rather restrictive, and so we will also consider fourth-order terms in the free energy expansion. This would be equivalent to considering trilinear terms of a new parent structure which has one of the many reported subgroups of $[Pm\bar{3}m]$ due to structural distortions or cation orderings, which themselves can be classified as transforming as irreps of $[Pm\bar{3}m]$ . However, from a materials design perspective, it is most convenient to always list these couplings with respect to the aristotypical symmetry.

If we consider couplings at the fourth order we may now construct terms as follows from the three primary OPs (A, B and C):

A·B·C is time-even

A·B·C is inversion-odd

[A] + [B] + [C] = [0, 0, 0].

If we are seeking a magnetoelectric coupling, precisely two of these terms must be time-odd (since P will always be time-even), but the constraint that the sum of these two terms must conserve crystal momentum is now lifted. We will refer to this design approach as `closing the momentum triangle', since now three irreps may be chosen to produce zero crystal momentum transfer.

This gives greater flexibility in the design strategy, but the price of course is that now three primary OPs are required. This means either these must all spontaneously become thermodynamically favourable at the phase transition, or more likely, and as discussed above, the structure will already contain distortions to the parent phase (such as octahedral rotations) which are ubiquitous in the perovskite structure.

Our approach outlined above is similar in spirit in some ways to that used to consider possible magnetoelectric couplings in the incommensurate phase of BaMnF₄ (Fox et al., 1980 ). However, our approach differs in that we perform the Landau-style expansion of the free energy about a hypothetical aristotypical symmetry, rather than the experimentally observed high-temperature phases. The benefit of our approach is that it encodes as much information as possible regarding the crystal momentum and parity of the time-odd and -even OPs into the problem, making it particularly easy to predict magnetoelectric couplings based on symmetry arguments alone, as we demonstrate here.

5. Trilinear magnetoelectric couplings in AFM systems

We start from the criteria derived above which means that we may superpose the following time-odd irreps when constructing the OP:

$[{\rm mR}_{4}^{+} \oplus {\rm mR}_{5}^{-}\semi {\rm mX}_{3}^{+} \oplus {\rm mX}_{1}^{-}\semi {\rm mX}_{5}^{+} \oplus {\rm mX}_{5}^{-}.]$

At the M-point, all possible magnetic orderings transform as mM⁺ and so no magnetoelectric couplings are possible. This finding immediately rules out a large area of search space. Furthermore, magnetic moments on the A-site cations transform always as mX⁺ and mR⁺ and on the B site always as mX⁻ and mR⁻, meaning any such trilinear magnetoelectric coupling mechanism must involve order on both A and B sites simultaneously. We take these three possible couplings in turn now, and consider which are the most physical and if any experimental realizations already exist.

For mX₃⁺ $[\oplus]$ mX₁^-, the OP is six dimensional OP(a;b;c|d;e;f) and the different choices of OPD result in a total of 22 possible isotropy subgroups. Only a subset of these, in which the condition for conserving crystal momentum is satisfied at a linear term in polarization, have broken inversion symmetry. While many of these lead to polar space groups, some only result in piezoelectric couplings. In these cases application of strain (either external or internal from ferroelastic distortions) will produce the desired polar ground state. Those with broken inversion symmetry correspond to OPDs of OP(a;0;0|d;0;0), OP(a;0;a|d;0;d), OP(a; $[\bar{a}]$ ;a|d; $[\bar{d}]$ ;d) (see Fig. 2). Of these only OP(a;0;0|d;0;0) represents a single k-active and collinear solution, and we shall focus on this for the rest of our discussion. The isotropy group is P_c4cc with basis = [(1, 0, 0),(0, 0, 1),(0, −2, 0)] + (0, 0, 0) and SOPs $[\Gamma _{4}^{-}]$ (polar mode) and $[\Gamma _{3}^{+}]$ (tetragonal strain). This OPD corresponds to the magnetic moments aligned parallel to the propagation vector on both A and B sites.

Figure 2
Magnetic structures giving rise to the magnetoelectric effect resulting from the action of the OP(a,b,c $[\mid]$ d,e,f) transforming as mX₃⁺ $[\oplus]$ mX₁^-, shown along the high-symmetry directions OP(a,0,0 $[\mid]$ d,0,0), OP(a,a,0 $[\mid]$ d,d,0), OP(a, $[\bar{a}]$ ,a $[\mid]$ d, $[\bar{d}]$ ,d).

To illustrate that our symmetry arguments can be used to identify improper ferroelectric couplings, we perform the following computational experiment. Density functional theory (DFT) calculations using the VASP code (Kresse & Hafner, 1993 ; Kresse & Furthmüller, 1996 ) (version 5.4.1) were executed on a hypothetical cubic GdFeO₃ structure in which the unit-cell parameter (the only degree of freedom) was fixed at a = 3.65 Å. This contracted unit cell was to ensure that no polar instability existed in the phonon dispersion curve [in the ferromagnetic (FM) state, or with spin–orbit coupling turned off], such that any later appearance of $[\Gamma _{4}^{-}]$ (with spin–orbit coupling turned on) could be identified as arising through improper, rather than proper, ferroelectricity. This is illustrated in Fig. 3 where the polar mode [ $[\Gamma _{4}^{-}]$ OP (0, h, 0)] is condensed with different amplitudes in the FM phase ( $[{\rm m}\Gamma _{4}^{+}]$ ) to give the expected single well potential centred at zero.

Figure 3
Energy versus polar mode [ $[\Gamma _{4}^{-}]$ OP(0,h,0)] magnitude for AFM [mX₃⁺ mX₁^- OP(a,0,0 $[\mid]$ d,0,0)] and FM ( $[{\rm m}\Gamma _{4}^{+}]$ ) ordering. The inset illustrates the linear behaviour around the origin. The amplitude of the $[\Gamma _{4}^{-}]$ mode is determined by summing the displacements of all the atoms in the unit cell and presented as a percentage with respect to the ground-state amplitude of the AFM phase. In both AFM and FM phases the energy shown is with respect to the structure with zero magnitude of $[\Gamma _{4}^{-}]$ .

We used the GGA PBEsol exchange correlation functional (Perdew et al., 2008 ) and PAW pseudopotentials (PBE functional, version 5.2) with the following valence electron configurations: 5s²5p⁶6s²4f⁸ (Gd), 3p⁶4s²3d⁶ (Fe) and 2s²2p⁴ (O). An on-site Coulomb repulsion U (Liechtenstein et al., 1995 ) was taken as 4 ev for the Gd f electrons and 8 ev for the Fe 3d electrons, which further stiffened $[\Gamma _{4}^{-}]$ , whilst keeping the system insulating. A planewave cut-off of 900 eV and a 6 × 6 × 6 k-grid with respect to the cubic cell were employed.

We then repeat these calculations with magnetic moments fixed on the A and B sites that transform according to the irreps mX₃⁺ and mX₁^- [OP(a,0,0|d,0,0)]. As evident in Fig. 3, the potential shifts away from having a minimum at zero (dashed line) prior to the magnetic interactions being switched on to a position where the minimum energy is at a finite value of the polar mode. This linear trend of the energy at the origin (inset Fig. 3) is indicative of an improper ferroelectric coupling term between mX₃⁺, mX₁^- and $[\Gamma _{4}^{-}]$ . We calculated the polarization after full ionic relaxation to be 4.88 µC cm⁻², which we believe to be one of the largest reported amongst spin-induced ferroelectrics, suggesting a strong trilinear coupling with this magnetic order. We compare this number with the purely electronic contribution to the polarization calculated with the ions fixed in the high-symmetry $[Pm\bar{3}m]$ positions, 0.07 µC cm⁻². This suggests the total polarization of 4.88 µC cm⁻² is predominantly of ionic origin, which is also suggested by the reasonably large cation–anion off-centring in the ground-state structure (0.02 Å). Now that we have used these DFT calculations to illustrate our ideas, we will discuss the remaining magnetoelectric couplings based on symmetry arguments alone.

For mX₅⁺ $[\oplus]$ mX₅^-, the OP is now 12-dimensional OP(a,b;c,d;e,f|h,i;g,k;l,m). The representative (single k-active) OPDs which meet the criteria for zero crystal momentum transfer are however of the form OP(a,b;0,0;0,0|h,i;0,0;0,0). We do not consider OPs with multiple k-actives as in general this will always induce SOPs transforming as M- or R-point irreps, which are already covered in our previous analysis. We note here that we are not saying that these will correspond to physically equivalent examples, only that we can be sure that we have already considered the cases where linear terms in polarization will also be present in the free energy expansions. Hence the representative high-symmetry examples given in Fig. 4 are OP(a,a,0,0,0,0|h, $[\bar{h}]$ ,0,0,0,0) and OP(0,a,0,0,0,0| $[\bar{h}]$ ,0,0,0,0,0). We do not explicitly consider mX₃⁺ $[\oplus]$ mX₅^- or mX₅⁺ $[\oplus]$ mX₁^- here as these represent magnetic structures in which the spins on the A site and B site are non-collinear with each other, which we believe to be less physically likely than the remaining examples that we have already discussed.

Figure 4
Magnetic structures giving rise to the magnetoelectric effect resulting from the action of the OP(a,b,c $[\mid]$ d,e,f) transforming as mX₅⁺ $[\oplus]$ mX₅^-, shown along the high-symmetry directions OP(a,a;0,0;0,0 $[\mid]$ d, $[\bar{d}]$ ;0,0;0,0), OP(a,a;0,0;0,0 $[\mid \bar{d}]$ , $[\bar{d}]$ ;0,0;0,0), OP(0,a;0,0;0,0 $[\mid]$ 0,d,0,0;0,0).

For mR₄⁺ $[\oplus]$ mR₅^-, the resulting OP is OP(a,b,c|d,e,f). There are 14 possible OPDs that result in unique space-group, basis and origin combinations with respect to the parent structure. All other possible OPDs correspond to twin domains of these 14 possibilities. Of these 14 OPDs, we consider here three: OP(a,0,0|d,0,0), $[I_{c}\bar{4}c2]$ , basis = [(−1, −1, 0),(1, −1, 0),(0, 0, 2)] + (0, 0, 0); OP(a,a,0|d,d,0), I_cma2, basis = [(1, 0, 1),(1, 0, −1),(0, 2, 0)] + (0, $[{{1} \over {2}}]$ 0, $[-{{1} \over {2}}]$ 0); OP(a,a,a|d,d,d) R_I3c, basis = [(1, 0, −1),(0, −1, 1),(−2, −2, −2)] + (0, 0, 0), which correspond to collinear magnetic structures shown in Fig. 5. Any of the other lower-symmetry collinear magnetic structures may be constructed through linear combinations of these three OPDs. For the polar space groups (I_cma2 and R_I3c) an SOP transforming as $[\Gamma _{4}^{-}]$ is always active. The only other SOPs are strain. A strategy for stabilizing this ground-state structure, therefore, in addition to designing AFM nearest-neighbour interaction in the system, is to epitaxially pre-strain the sample in a manner that stabilizes terms in the free energy that will also occur at the even order.

Figure 5
Collinear magnetic structures resulting from the action of the OP ( $[a,b,c \mid d,e,f]$ ) transforming as mR₄⁺ $[\oplus]$ mR₅^-, shown along the high-symmetry directions OP(a,0,0 $[\mid]$ d,0,0), OP(a,a,0 $[\mid]$ d,d,0), OP(a,a,a $[\mid]$ d,d,d).

$[I_{c}\bar{4}c2]$ (Fig. 5, left) on the other hand, although it has no inversion symmetry, is only piezoelectric. Indeed, it was very recently demonstrated (Zhao et al., 2017 ) from a combination of first-principle calculations and group-theoretical analysis that the rare-earth gadolinium chromates and ferrites with collinear G-type order on A and B sites along the pseudo-cubic axes lead to a piezoelectric space group. Sheer strain along the [110]-type lattice directions was found to be needed to create a polarization through the piezoelectric effect, consistent with the piezoelectric point group. Our analysis shows an alternative route in which polarization emerges directly, provided that the spins align along the orthorhombic or rhombohedral type axes as in the cases discussed above of OP(a,a,0|d,d,0) and OP(a,a,a|d,d,d). We note also here that the possible observation of weak ferroelectric polarization, which is reported in A- and B-site lattices in which sub-lattice moments along 100-type directions are perpendicular to each other (Zhao et al., 2017), may be understood in the framework of the SOP analysis that we have presented above. We find that $[\Gamma _{4}^{-}]$ arises directly as a consequence of this kind of magnetic ordering [OP(a,0,0|,0,0,d)] with the magnetic space group being F_Smm2 {basis = [(0, 2, 0),(0, 0, 2),(2, 0, 0)], origin = (½, ½, 0)}.

An experimental example of where magnetoelectric properties arise from G-type ordering on the A and B sites can be found in the literature for the 134 perovskite LaMn₃Cr₄O₁₂ (Wang et al., 2015 ). This distorted perovskite structure has the additional structural orderings that can be described as M₁⁺(a;a;a) (1:3 cation ordering) and M₂⁺(a;a;a) (octahedral rotation). However, the observed magnetoelectric effect, that only occurs below both the B-site and the A-site ordering temperature, can be understood in terms of our present results by considering only OP mR₄⁺ $[\oplus]$ mR₅^- with OP(a,a,a|d,d,d) (Fig. 5, right), meaning that the magnetoelectric ground-state structure has rhombohedral lattice symmetry and arises solely as a consequence of the magnetic ordering on both sites.

6. Fourth-order magnetoelectric couplings in AFM systems

An exhaustive list of fourth-order couplings in polarization and zone-boundary irreps is given in Table 2. There are naturally a large number of these, and we will restrict our more detailed discussion to those which are the most physically reasonable and likely to produce the strongest couplings at the highest ordering temperatures. Because of this, we will no longer consider magnetic ordering on the A site which in general only supports rare-earth ions or non-magnetic cations. Notable exceptions to this are the perovskite MnVO₃, but where the magnetic ordering temperatures remain low (Markkula et al., 2011 ), and some highly distorted AA′₃B₄O₁₂ quadruple perovskites that we will not discuss here.

Table 2
Closing the `momentum triangle' – the possible fourth-order magnetoelectric coupling terms

Zeroth row and column correspond to two of the four coupling terms which are always time-odd. At the intersection of the rows and columns, a third time-even irrep is given with the fourth term always being P ( $[\Gamma _{4}^{-}]$ ).

	mX₃⁺ and mX₅⁺	mX₁^- and mX₅^-
mM₂⁺	R₂^-; R₃^-; R₄^-; R₅^-	R₁⁺; R₅⁺
mM₃⁺	R₂^-; R₃^-; R₄^-; R₅^-	R₁⁺; R₅⁺
mM₅⁺	R₂^-; R₃^-; R₄^-; R₅^-	R₁⁺; R₅⁺

	mX₃⁺ and mX₅⁺	mX₁^- and mX₅^-
mR₄⁺	M₃^-; M₅^-	M₁⁺; M₂⁺; M₃⁺; M₄⁺
mR₅^-	M₁⁺; M₂⁺; M₃⁺; M₄⁺	M₃^-; M₅^-

	mM₂⁺ and mM₃⁺ and mM₅⁺
mR₄⁺	X₃^-; X₅^-
mR₅^-	X₁⁺

Considering only B-site magnetism we are left with the following time-odd superposition of irreps to consider: mM_2,5⁺ $[\oplus]$ mX_1,5^-; mR₅^- $[\oplus]$ mX_1,5^-; mR₅^- $[\oplus]$ mM_2,5⁺. In order to close the `momentum triangle' these will now be, respectively, superposed with the following time-even irreps: R_1,5⁺, M_3,5^- and X₁⁺, to produce an OP that transforms as time-even, inversion-odd and has a crystal momentum transfer of zero (see Tables 3, 4 and 5). The relevant structural degrees of freedom (Table 1) to consider are, hence, cation/anion order (X₁⁺, R_1,5⁺) and antipolar displacements (M_3,5^-). Notably, octahedral tilts or Jahn–Teller modes do not appear in this list and hence cannot form part of such a design strategy.

Table 3
Structural SOPs of POPs indicated in the table

Polarization $[\Gamma _{4}^{-}]$ is always an SOP.

POP	SOPs
( $[{\rm X}_{1}^{+} \mid {\rm mM}_{2}^{+} \mid {\rm mR}_{5}^{-}]$ )
(a;0;0 $[\mid]$ 0;b;0 $[\mid]$ c,0,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,0,a); $[\Gamma _{5}^{-}]$ (a,0,0); X₂⁺(a;0;0); X₅^-(a,0;0,0;0,0)
(a;0;0 $[\mid]$ 0;b;0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{4}^{-}]$ (0,a,0); X₃^-(a;0;0)
(a;0;0 $[\mid]$ 0;b;0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); X₅^-(a,a;0,0;0,0)
( $[{\rm X}_{1}^{+} \mid {\rm mM}_{5}^{+} \mid {\rm mR}_{5}^{-}]$ )
(a;0;0 $[\mid]$ 0,0;b,0;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (a,0,0); $[\Gamma _{5}^{-}]$ (0,a,0); X₂⁺(a;0;0); X₅^-(0,a;0,0;0,0)
(a;0;0 $[\mid]$ 0,0;b,0;0,0 $[\mid]$ 0,c,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,a,0); $[\Gamma _{5}^{-}]$ (0,0,a); X₂⁺(a;0;0); X₃^-(a;0;0)
(a;0;0 $[\mid]$ 0,0;b, $[\bar{b}]$ ;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0, $[\bar{a}]$ ); $[\Gamma _{5}^{-}]$ (a,a,0); X₅^-(a, $[\bar{a}]$ ;0,0;0,0)
(a;0;0 $[\mid]$ 0,0;b, $[\bar{b}]$ ;0,0 $[\mid \bar{c}]$ ,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (0,a,0); X₃^-(a;0;0)

Table 4
Structural SOPs of POPs indicated in the table

Polarization $[\Gamma _{4}^{-}]$ is always an SOP.

POP	SOPs
( $[{\rm mX}_1^- \mid {\rm M}_3^- \mid {\rm mR}_5^-]$ )
(a;0;0 $[\mid]$ 0;b;0 $[\mid]$ c,0,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (a,0,0); $[\Gamma _{5}^{-}]$ (0,a,0); M₅⁺(0,0;a,0;0,0)
(a;0;0 $[\mid]$ 0;b;0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); M₅⁺(0,0;a, $[\bar{a}]$ ;0,0); M₂^-(0;a;0)
( $[{\rm mX}_5^- \mid {\rm M}_3^- \mid {\rm mR}_5^-]$ )
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0;b;0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); M₅⁺(0,0;a, $[\bar{a}]$ ;0,0); M₂^-(0;a;0)
(0,a;0,0;0,0 $[\mid]$ 0;b;0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (a,0,0); $[\Gamma _{5}^{-}]$ (0,a,0); M₅⁺(0,0;a,0;0,0)
(0,a;0,0;0,0 $[\mid]$ 0;b;0 $[\mid]$ 0,c,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,a,0); $[\Gamma _{5}^{-}]$ (0,0,a); M₁⁺(0;a;0); M₂⁺(0;a;0)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0;b;0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (0,a,0); M₁⁺(0;a;0); M₄⁺(0;a;0); M₂^-(0;a;0)
( $[{\rm mX}_1^- \mid {\rm M}_5^- \mid {\rm mR}_5^-]$ )
(a;0;0 $[\mid]$ 0,0;0,b;0,0 $[\mid]$ c,0,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,a,0); $[\Gamma _{5}^{-}]$ (0,0,a); M₅⁺(0,0;a,0;0,0)
(a;0;0 $[\mid]$ 0,0;0,b;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,0,a); $[\Gamma _{5}^{-}]$ (a,0,0); M₃⁺(0;a;0); M₄⁺(0;a;0)
(a;0;0 $[\mid]$ 0,0;b,b;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); M₁⁺(0;a;0); M₄⁺(0;a;0)
(a;0;0 $[\mid]$ 0,0;b,b;0,0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (0,a,0); M₅⁺(0,0;a, $[\bar{a}]$ ;0,0)
( $[{\rm mX}_5^- \mid {\rm M}_5^- \mid {\rm mR}_5^-]$ )
(0,a;0,0;0,0 $[\mid]$ 0,0;0,b;0,0 $[\mid]$ c,0,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,0,a); $[\Gamma _{5}^{-}]$ (a,0,0); M₃⁺(0;a;0); M₄⁺(0;a;0)
(0,a;0,0;0,0 $[\mid]$ 0,0;0,b;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,a,0); $[\Gamma _{5}^{-}]$ (0,0,a); M₅⁺(0,0;a,0;0,0)
(0,a;0,0;0,0 $[\mid]$ 0,0;0,b;0,0 $[\mid]$ 0,c,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (a,0,0); $[\Gamma _{5}^{-}]$ (0,a,0); M₁⁺(0;a;0); M₂⁺(0;a;0)
(0,a;0,0;0,0 $[\mid]$ 0,0; $[\bar{b}]$ ,0;0,0 $[\mid]$ c,0,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (a,0,0); $[\Gamma _{5}^{-}]$ (0,a,0); M₃⁺(0;a;0); M₄⁺(0;a;0)
(0,a;0,0;0,0 $[\mid]$ 0,0; $[\bar{b}]$ ,0;0,0 $[\mid]$ 0,c,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,0,a); $[\Gamma _{5}^{-}]$ (a,0,0); M₁⁺(0;a;0); M₂⁺(0;a;0)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0;b,b;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (0,a,0); M₅⁺(0,0;a, $[\bar{a}]$ ;0,0)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0;b,b;0,0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); M₁⁺(0;a;0); M₄⁺(0;a;0)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0;b,b;0,0 $[\mid \bar{c}]$ ,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0, $[\bar{a}]$ ); $[\Gamma _{5}^{-}]$ (a,a,0); M₂⁺(0;a;0); M₃⁺(0;a;0)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0; $[\bar{b}]$ ,b;0,0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0, $[\bar{a}]$ ); $[\Gamma _{5}^{-}]$ (a,a,0); M₁⁺(0;a;0); M₄⁺(0;a;0)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0; $[\bar{b}]$ ,b;0,0 $[\mid \bar{c}]$ ,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); M₂⁺(0;a;0); M₃⁺(0;a;0)

Table 5
Structural SOPs of POPs indicated in the table

Polarization $[\Gamma _{4}^{-}]$ is always an SOP.

POP	SOPs
( $[{\rm mX}_1^- \mid {\rm mM}_2^+ \mid {\rm R}_5^+]$ )
(a;0;0 $[\mid]$ 0;b;0 $[\mid]$ c,0,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,0,a); $[\Gamma _{5}^{-}]$ (a,0,0); R₂^-(a); R₃^-(a,b)
(a;0;0 $[\mid]$ 0;b;0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{4}^{-}]$ (0,a,0); R₂^-(a); R₃^-(a, $[\sqrt{3}\bar{a}]$ )
(a;0;0 $[\mid]$ 0;b;0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); R₂^-(a); R₃^-(a, $[\sqrt{3}\bar{a}]$ ); R₄^-(0,0,a)
( $[{\rm mX}_1^- \mid {\rm mM}_5^+ \mid {\rm R}_5^+]$ )
(a;0;0 $[\mid]$ 0,0;b,0;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (a,0,0); $[\Gamma _{5}^{-}]$ (0,a,0); R₄^-(a,0,0); R₅^-(a,0,0)
(a;0;0 $[\mid]$ 0,0;b,0;0,0 $[\mid]$ 0,c,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,a,0); $[\Gamma _{5}^{-}]$ (0,0,a); R₄^-(a,0,0); R₅^-(a,0,0)
(a;0;0 $[\mid]$ 0,0;b, $[\bar{b}]$ ;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0, $[\bar{a}]$ ); $[\Gamma _{5}^{-}]$ (a,a,0); R₄^-(a, $[\bar{a}]$ ,0); R₅^-(a,a,0)
(a;0;0 $[\mid]$ 0,0;b, $[\bar{b}]$ ;0,0 $[\mid \bar{c}]$ ,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (0,a,0); R₄^-(a, $[\bar{a}]$ ,0); R₅^-(a,a,0)
( $[{\rm mX}_5^- \mid {\rm mM}_2^+ \mid {\rm R}_5^+]$ )
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0;b;0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); R₄^-(a,a,0); R₅^-(a, $[\bar{a}]$ ,0)
(0,a;0,0;0,0 $[\mid]$ 0;b;0 $[\mid]$ c,0,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,a,0); $[\Gamma _{5}^{-}]$ (0,0,a); R₄^-(0,a,0); R₅^-(0,a,0)
(0,a;0,0;0,0 $[\mid]$ 0;b;0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,0,a); $[\Gamma _{5}^{-}]$ (a,0,0); R₄^-(0,a,0); R₅^-(0,a,0)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0;b;0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (0,a,0); R₄^-(a,a,0); R₅^-(a, $[\bar{a}]$ ,0)
( $[{\rm mX}_5^- \mid {\rm mM}_5^+ \mid {\rm R}_5^+]$ )
(0,a;0,0;0,0 $[\mid]$ 0,0;b,0;0,0 $[\mid]$ c,0,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (a,0,0); $[\Gamma _{5}^{-}]$ (0,a,0); R₄^-(0,0,a); R₅^-(0,0,a)
(0,a;0,0;0,0 $[\mid]$ 0,0;b,0;0,0 $[\mid]$ 0,c,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,0,a); $[\Gamma _{5}^{-}]$ (a,0,0); R₄^-(0,0,a); R₅^-(0,0,a)
(0,a;0,0;0,0 $[\mid]$ 0,0;0, $[\bar{b}]$ ;0,0 $[\mid]$ c,0,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,0,a); $[\Gamma _{5}^{-}]$ (a,0,0); R₂^-(a); R₃^-(a,b)
(0,a;0,0;0,0 $[\mid]$ 0,0;0, $[\bar{b}]$ ;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (0,a,0); $[\Gamma _{5}^{-}]$ (0,0,a); R₂^-(a); R₃^-(a,b)
(0,a;0,0;0,0 $[\mid]$ 0,0;0, $[\bar{b}]$ ;0,0 $[\mid]$ 0,c,0)	$[\Gamma _{3}^{+}]$ (a,b); $[\Gamma _{4}^{-}]$ (a,0,0); $[\Gamma _{5}^{-}]$ (0,a,0); R₂^-(a); R₃^-(a,b)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0; $[\bar{b}]$ , $[\bar{b}]$ ;0,0 $[\mid]$ 0,0,c)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (0,a,0); R₂^-(a); R₃^-(a, $[\sqrt{3}\bar{a}]$ ); R₄^-(0,0,a)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0;b, $[\bar{b}]$ ;0,0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0, $[\bar{a}]$ ); $[\Gamma _{5}^{-}]$ (a,a,0); R₃^-(a,0.577a); R₅^-(0,0,a)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0;b, $[\bar{b}]$ ;0,0 $[\mid \bar{c}]$ ,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); R₃^-(a,0.577a); R₅^-(0,0,a)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0; $[\bar{b}]$ , $[\bar{b}]$ ;0,0 $[\mid]$ c,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0,a); $[\Gamma _{5}^{-}]$ (a, $[\bar{a}]$ ,0); R₂^-(a); R₃^-(a, $[\sqrt{3}\bar{a}]$ ); R₄^-(0,0,a)
(a, $[\bar{a}]$ ;0,0;0,0 $[\mid]$ 0,0; $[\bar{b}]$ , $[\bar{b}]$ ;0,0 $[\mid \bar{c}]$ ,c,0)	$[\Gamma _{3}^{+}]$ (a, $[\sqrt{3}\bar{a}]$ ); $[\Gamma _{5}^{+}]$ (0,0,a); $[\Gamma _{4}^{-}]$ (a,0, $[\bar{a}]$ ); $[\Gamma _{5}^{-}]$ (a,a,0); R₂^-(a); R₃^-(a, $[\sqrt{3}\bar{a}]$ ); R₄^-(0,0,a)

For X₁⁺ $[\oplus]$ mM_2,5⁺ $[\oplus]$ mR₅^- we give some possible examples of several magnetic structures in Figs. 6 and 7 corresponding to A-site ordered double perovskites with striped type (X₁⁺) arrangements of cations, such as is commonly found experimentally for cations of substantially different sizes (King & Woodward, 2010). Some of these compounds are already known to be improper ferroelectric (Zuo et al., 2017 ) on account of couplings between the layering and octahedral tilt modes, as discussed in the previous section.

Figure 6
Magnetic structures giving rise to the magnetoelectric effect resulting from the action of the OP(a;b;c $[\mid]$ d;e;f $[\mid g,h,i]$ ) transforming as X₁⁺ $[\oplus]$ mM₂⁺ $[\oplus]$ mR₅^-, shown along the high-symmetry directions indicated. A-site cation ordering is indicated by white and black spheres.

Figure 7
Magnetic structures giving rise to the magnetoelectric effect resulting from the action of the OP(a;b;c $[\mid]$ d,e;f,g;h,i $[\mid]$ j,k,l) transforming as X₁⁺ $[\oplus]$ mM₅⁺ $[\oplus]$ mR₅^-, shown along the high-symmetry directions indicated. A-site cation ordering is indicated by white and black spheres.

The possible high-symmetry OPDs for superposed irreps X₁⁺(0, $[{{1} \over {2}}]$ , 0) $[\oplus]$ mM₂⁺( $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ , 0) $[\oplus]$ mR₅^-( $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ ) are:

$[(a\semi\!0\semi\!0 \!\mid\! 0\semi\!b\semi\!0 \!\mid\! c,\!0,\!0)\semi (a\semi\!0\semi\!0 \!\mid\! 0\semi\!b\semi\!0 \!\mid\! 0,0,c)\semi (a\semi\!0\semi\!0 \!\mid\! 0\semi\!b\semi\!0 \!\mid\! c,\!c,\!0).]$

Conservation of crystal momentum criteria that we have imposed here dictates the relative OPD of the X and M components (k-actives). The three structures listed above and shown in Fig. 6 only differ in the OPD with respect to the mR₅^- irrep, producing two non-collinear magnetic structures and one which has a spin-density-wave. In the case of the non-collinear magnetic structures, the direction of P is parallel to both the cation order planes and the magnetic moment canting direction. For the spin-density-wave structure the polarization vector is perpendicular to the cation ordering planes. Spin-density-wave magnetic structures are in general less common, but we note that X-point order of two magnetically active cations (at the B site) with different magnetic moments could be a way to achieve this.

For X₁⁺ $[\oplus]$ mM₅⁺ $[\oplus]$ mR₅^- as mM₅⁺ is a higher-dimensional irrep than M₂⁺, there are now a larger number of OPD possibilities:

$[\eqalign{&(a\semi\!0\semi\!0 \!\mid\! 0,\!0\semi\!b,\!0\semi\!0,\!0 \!\mid\! c,\!0,\!0)\semi \quad (a\semi\!0\semi\!0 \!\mid\! 0,\!0\semi\!b,\!0\semi\!0,\!0 \!\mid\! 0,\!0,\!c)\semi \cr & (a\semi\!0\semi\!0 \!\mid\! 0,\!0\semi\!b,\!0\semi\!0,\!0 \!\mid\! 0,\!c,\!0)\semi \quad (a\semi\!0\semi\!0 \!\mid\! 0,\!0\semi\!b,\!-b\semi\!0,\!0 \!\mid\! 0,\!0,\!c)\semi \cr & (a\semi\!0\semi\!0 \!\mid\! 0,\!0\semi\!b,\!-b\semi\!0,\!0 \!\mid\! c,\!c,\!0)\semi \quad (a\semi\!0\semi\!0 \!\mid\! 0,\!0\semi\!b,\!-b\semi\!0,\!0 \!\mid\! -c,\!c,\!0).}]$

However, this time several of these high-symmetry OPs give rise to piezoelectric but non-polar space groups [(a;0;0|0,0;b,0;0,0|c,0,0) C222 and (a;0;0|0,0;b,−b;0,0|c,c,0) P222₁]. Although not ferroelectric, the inclusion of any further POP either as an internal or external strain field will drive a ferroelectric ground state in these systems. Fig. 7 shows the representative high-symmetry OPD resulting in polar structures. Similarly for the discussion above, P is parallel and perpendicular to cation ordering for constant moment and spin-density-wave magnetic structures, respectively.

For mX_1,5^- $[\oplus]$ mM_2,5⁺ $[\oplus]$ R₅⁺, in which R₅⁺ could correspond to anion order, the cis-ordering of N for O substitution in oxynitride ABO_3−xN_x perovskite (Yang et al., 2011 ) represents an experimental realization of this. For x = 1.5, this would correspond to a checkerboard anion order and hence we consider R₅⁺(a,a,a) (or the closest high-symmetry equivalent OPD) in the following analysis. As a POP transforming as R₅⁺(a,a,a) always has an SOP transforming as R₁⁺(a), this analysis also turns out to be equivalent to looking at rocksalt ordering on the A-site cation, although we note that such ordering is not particularly common. mX₁^-(0, $[{{1} \over {2}}]$ , 0) mM₂⁺( $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ , 0) R₅⁺( $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ ), with an OPD of (a;0;0|0;b;0|c,c,d), corresponds to a spin-density-wave collinear magnetic structure, where P is in the plane of the magnetic moment directions (Fig. 8). The constant-magnitude spin-canted magnetic structures [mX₅^- $[\oplus]$ mM₂⁺ $[\oplus]$ R₅⁺ (a;0;0|0;b;0|c,c,d), Fig. 8] on the other hand lead to polarizations that are found to be both perpendicular and parallel to the magnetic moment alignment.

Figure 8
Magnetic structures giving rise to the magnetoelectric effect resulting from the action of the OP as shown along the high-symmetry directions indicated. Anion ordering is indicated by blue spheres of differing sizes.

We will not consider the remaining possible couplings for mX₁^- $[\oplus]$ mM₂⁺ $[\oplus]$ R₅⁺, mX⁻ $[\oplus]$ mM₅⁺ $[\oplus]$ R₅⁺ and mX₁^- $[\oplus]$ M⁻ $[\oplus]$ mR₅^- explicitly here but they are tabulated in Table 5 and representative figures are given in Figs. 8, 9 and 10.

Figure 9
Magnetic structures giving rise to the magnetoelectric effect resulting from the action of the OP as shown along the high-symmetry directions indicated. Anion ordering is indicated by blue spheres of differing sizes.

Figure 10
Magnetic structures giving rise to the magnetoelectric effect resulting from the action of the OP as shown along the high-symmetry directions indicated.

7. Stabilizing wFM and magnetoelectric effects in non-ferroelectrics

For multiferroics to be useful in data storage applications, it is likely that they will need to have a ferromagnetically ordered and switchable component. Following the design strategy above we wish to engineer a trilinear term in the free energy expansion of the form A B wFM. Since wFM transforms as $[{\rm m}\Gamma _{4}^{+}]$ which is time-odd, parity-even and has crystal momentum of zero, the constraints on A and B are as follows:

[A] = [B]

A·B is parity-even.

Hence, possible trilinear terms with wFM need to involve OPs that transform as:

$[{\rm mR}^- \oplus {\rm R}^- \semi\, {\rm mR}^+ \oplus {\rm R}^+ \semi\, {\rm mM}^+ \oplus {\rm M}^+ \semi\, {\rm mX}^- \oplus {\rm X}^- \semi\, {\rm mX}^+ \oplus {\rm X}^+.]$

Taking mR₅^- (B-site magnetic order) with R₅^- (octahedral tilting) as an example, G-type magnetic ordering on the B sites with moments along the c axis with out-of-phase octahedral rotations leads to the magnetic space group Im′m′a {basis = [(1, 1, 0),(0, 0, 2),(1, −1, 0)] + (0, 0, $[{{1} \over {2}}]$ )} which has $[{\rm m}\Gamma _{4}^{+}]$ (wFM) as an SOP. Indeed, we believe this is the framework by which the theoretically predicted wFM in Gd Cr/Fe perovskites (Zhao et al., 2017; Tokunaga et al., 2009 ) can be easily understood, and is an example in which the B—O—B exchange angle is allowed to deviate from 180° by a symmetry-breaking event allowing spin canting to occur via the Dzyaloshinsky–Moriya interaction.

At the M-point the above analysis can also be applied. C-type B-site magnetic ordering along the [001] axis [mM₂⁺(a;0;0)], with in-phase octahedral tilts perpendicular to this [M₂⁺(a;0;0), a⁰a⁰c⁺], actually leads to a piezomagnetic space group P4/mbm {basis = [(1, −1, 0),(1, 1, 0),(0, 0, 1)] + ( $[{{1} \over {2}}]$ , $[-{{1} \over {2}}]$ , 0)}. Application of an orthorhombic type strain ( $[\Gamma _{5}^{+}]$ ) for example leads to the occurrence of wFM ( $[{\rm m}\Gamma _{4}^{+}]$ ). Similarly, distortions transforming as X₅^- and magnetic moments as mX₁^- will produce wFM {e.g. $[Cm^{\prime}cm^{\prime}]$ , basis = [(1, 0, 1),(1, 0, −1),(0, 2, 0)] + (0, $[{{1} \over {2}}]$ , 0)}.

Another magnetoelectric effect worth considering is where P is induced by the application of an external magnetic field which may be described as transforming as $[{\rm m}\Gamma _{4}^{+}]$ or conversely wFM is induced by the application of an external electric field ( $[\Gamma _{4}^{-}]$ ). For this we must look at terms involving two zone-boundary irreps like [M] + [S] = [0 0 0], where M is time-odd (magnetic) and S is time-even (structural), and M·S is inversion-odd. Application of an electric field ( $[\Gamma _{4}^{-}]$ ) should then give a fourth-order term in the free energy expansion of the form M S P wFM. A realization of this is mX_1,5^- $[\oplus]$ X₁⁺ $[\oplus]$ P, to give wFM. Finally it is worth pointing out that all systems that are both piezoelectric and piezomagnetic will be magnetoelectric, as application of either an external magnetic or electric field will generate a strain field that mediates a coupling between the two phenomena.

8. Putting it all together

The ultimate goal of course is to have a magnetoelectric in which ferromagnetism is coupled to ferroelectricity. To achieve the strongest such coupling, we envisage first a scenario with two trilinear terms in P and wFM, with one co-dependent OP (see Fig. 11). For example: (i) X₁⁺ $[\oplus]$ X₅^- $[\oplus]$ P with mX₁^- $[\oplus]$ X₅^- $[\oplus]$ wFM. Assuming X₁⁺ represents cation order and may not be reversed, then the reversal of the sign of P would necessitate a reversal of X₅^-. This, in turn, would necessitate a switching of the magnetic structure which most likely would proceed via a reversal of the direction of the wFM. (ii) M₅^- $[\oplus]$ M₂⁺ $[\oplus]$ P and mM₅^- $[\oplus]$ M₂⁺ $[\oplus]$ wFM. Taking M₅^- as anion ordering, then a reversal of P would proceed via reversal of the octahedral rotations (M₂⁺) necessitating a reversal of either mM₅^- or wFM, the latter being more likely. (iii) R₁⁺ $[\oplus]$ R₂^- $[\oplus]$ P and mR₅^- $[\oplus]$ R₂^- $[\oplus]$ wFM, taking R₁⁺ as A-site rocksalt cation ordering, a reversal of P would imply a switching of R₂^- which could represent B-site charge ordering

Figure 11
A scheme for including third-order coupling terms in the free energy expansion involving the OP related to weak ferromagnetic spin canting and ferroelectric polarization.

In fact, if one has an AA′ layered double perovskite (X₁⁺) with the common M₂⁺ and R₅^- tilt pattern (Pnma like) no matter if you have C (mM₅⁺), G (mR₅^-) or A (mX₁^-) magnetic ordering (provided the spins are along certain directions), the ground state is ferroelectric and ferromagnetic with an indirect coupling between them. Efforts should hence be focused on preparing A⁺ A $[^{{\prime}3+}]$ and A²⁺ A $[^{{\prime}4+}]$ layered double perovskites with Mn⁴⁺ and Fe³⁺ on the B site, respectively, to achieve the strongest wFM moments and highest ordering temperatures.

Another scheme involving fourth-order couplings gives a greater degree of flexibility. Similar to the above, the idea here is to construct fourth-order terms with wFM ( $[{\rm m}\Gamma _{4}^{+}]$ ) in. As many of the OPs featuring in the wFM term at the fourth order should also feature in the fourth-order term in P. Fig. 12 envisages one such possible coupling scheme by which an extra degree of freedom related to breaking structural symmetry (S₂) is introduced to the magnetoelectric couplings discussed above, and is equivalent to using antisymmetric (DM) arguments to design wFM. The figure shows that it is possible to construct fourth-order terms with at least two OPs in common in both P and wFM terms, i.e. M₁ M₂ S₁ P and M₁ S₁ S₂ wFM. Each fourth-order term must individually conserve crystal momentum, time reversal and inversion symmetry. Hence the polar part, M₁ M₂ S₁, can be selected according to the analysis in the previous section, leaving the wFM part, M₁ S₁ S₂, to be decided on. Since M₁ S₁ are fixed by the polar part, the only decision to be made is the nature of S₂. We require that the crystal momentum of [S₂] equals the sum of the crystal momentum [M₁] + [S₁], and that parity with respect to inversion is equal to the product of the parity of M₁·S₁ (i.e. opposite to that of M₂). For example, with mM_2,5⁺ (M), mR₅^- (M) and X₁⁺ (S₁), S₂ must be either R_1,5⁺ or M_2,3,5^-.

Figure 12
A scheme for including fourth-order coupling terms in the free energy expansion involving the OP related to weak ferromagnetic spin canting and ferroelectric polarization.

Finally, we note that if one predisposes the system to certain distortions, which are implied as SOPs in the above analysis, certain phases may be thermodynamically favoured over others. This is an important part of controlling the relative OPDs which ultimately affect the higher-order couplings that drive magnetoelectric properties. We discuss now a few of the most promising candidates and propose some design strategies based on SOP analysis. SOPs are listed in Tables 3, 4 and 5 for some fourth-order couplings in ferroelectric polarization. Any further distortion to the $[Pm\bar{3}m]$ aristotype that the system is predisposed to, which transforms as irreps in this list, will act to stabilize one particular OPD over another. Or, put another way, at the harmonic order (quadratic phonon modes) all possible OPDs are degenerate in energy.

The most obvious strategy is to pre-strain (transforming as $[\Gamma _{3,5}^{+}]$ ) the system through epitaxial growth. Another strategy is to search through tables, such as Tables 3, 4 and 5, to find irreps that correspond to the most commonly observed distortions in the perovskite phase, such as the octahedral rotations (M₂⁺ and R₅^-), Jahn–Teller distortions in systems with a degeneracy in their d orbitals or indeed polar distortions themselves in d⁰ systems. In the `undistorted' perovskite structure these will correspond to the lowest-lying phonon modes (rigid unit modes in the cases of the octahedral rotation). Any energy penalty paid at the quadratic order will be kept low with respect to the trilinear terms that always act to lower the free energy, and therefore will drive a phase transition. For example, for X₁⁺ $[\oplus]$ mM₂⁺ $[\oplus]$ mR₅^-, SOPs are strain, $[\Gamma _{5}^{-}]$ , and X-point distortions. The OP(a;0;0 $[\mid]$ 0;b;0 $[\mid]$ c,c,0) P_Amc2₁ is the most promising candidate, as the only SOP is antipolar X5− distortions. Therefore, in addition to striped cation ordering, cations which are susceptible to off-centre distortions should be chosen. For mX₁^- $[\oplus]$ M₅^- $[\oplus]$ mR₅^- or mX₁^- $[\oplus]$ mM₂⁺ $[\oplus]$ R₅⁺, B sites with a propensity to undergo charge (M₄⁺ or R₂^-) and orbital order (M₃⁺ or R₃^-) should be chosen. A similar design strategy of selecting a system which is predisposed to certain SOPs may be adopted for stabilizing wFM.

9. Conclusion

Using group-theoretical means, we have enumerated the possible magnetoelectric couplings in the perovskite structure with respect to its aristotypical symmetry $[Pm\bar{3}m]$ . Our enumeration is complete up to the third-order terms for zone-boundary magnetic structures, and for fourth-order terms for B-site magnetism only. Our results show that, for zone-boundary magnetic ordering, only magnetism on both A and B sites transforming either both as X-point or R-point irreps can produce a magnetoelectric coupling at the third order, which is illustrated with first-principles calculations. For magnetism on the B site alone, then only fourth-order terms can produce the desired effect. We propose a design strategy based on POPs consisting of a superposition of three irreps, one each from the X-, M- and R-points, chosen in such a way that crystal momentum is conserved, that two are time-odd and either one or all are inversion-odd. These ideas are extended to a design strategy for weak ferromagnetism, which may then be coupled to the ferroelectric polarization in a similar manner to the recently much discussed hybrid improper ferroelectric Ca₃Mn₂O₇. Without a doubt, predicting and controlling physical properties arising from magnetic order will remain a challenging area for many years to come. However, our systematic enumeration of coupling mechanisms along with secondary order parameters at least provides some direction for how this might ultimately be achieved.

APPENDIX A

Notation used in order parameter directions

We will be forming OPs from up to three different zone-boundary irreps, which we will denote by using the symbol for direct sum ` $[\oplus]$ ', for example M₂⁺ $[\oplus]$ R₅^- represents an OP that has atomic displacements that transform as both irreps. By forming this OP we will effectively be conducting a thought experiment as to what would happen if the parent $[Pm\bar{3}m]$ became spontaneously unstable with respect to atomic displacements (in this case octahedral rotations) transforming as these irreps. However, specifying these irreps alone does not capture how the displacements (or magnetic orderings) combine with respect to each other, and hence the associated isotropy subgroup. To do this we need to describe the full OPD of the OP transforming as the specified irreps and we follow the notation used in ISODISRORT (Campbell et al., 2006). For M₂⁺ $[\oplus]$ R₅^-, an OPD is OP(a;0;0 $[\mid]$ b,0,0), where ` $[\mid]$ ' denotes a division between the OPD parts belonging to M₂⁺ and R₅^-, respectively. Semi-colons `;' denote divisions between different OPDs resulting from the degeneracy of the propagation vector in $[Pm\bar{3}m]$ . At the M-point the possible k-actives are ( $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ , 0);( $[{{1} \over {2}}]$ , 0, $[{{1} \over {2}}]$ );(0, $[{{1} \over {2}}]$ , $[{{1} \over {2}}]$ ), where the order respects the order of appearance in the OPD. Similarly, for an OP transforming as X-point irreps we will need to specify which k-actives of (0, $[{{1} \over {2}}]$ , 0);( $[{{1} \over {2}}]$ , 0, 0);(0, 0, $[{{1} \over {2}}]$ ) are in use. In general we will only form OPs from one k-active per irrep (equivalent to using the small irrep only). However, the notation we give will always reflect the total number of possible k-actives. At the R-point there is only one possible k-active, but in this case the irrep is multi-dimensional and, as in the case of $[\Gamma _{4}^{-}]$ , this is specified through use of commas between the letters. The total dimension of the OP is hence a function of the number of superposed irreps, the degeneracy of the propagation vectors associated with any of the irreps and the dimensionality of the small irreps themselves. All need to be fully specified along with the setting and space group of the parent structure to uniquely identify the isotropy subgroup.

Footnotes

¹In general the inversion symmetry breaking distortion will transform as the polar $[\Gamma _{4}^{-}]$ mode; however sometimes the symmetry breaking will instead be associated with another $[\Gamma^{-}]$ mode which is piezoelectric in nature.

²Strictly speaking, the relevant OPs are vectors whose elements (real or imaginary numbers) reflect the amplitude of the atomic displacements or magnetic moments that transform according to specific irreps. However, since for the purposes of our symmetry analysis, the information we require concerning crystal momentum and parity is encoded in the irrep label (and we are not concerned with amplitude here), we will also label the OPs using this notation.

³A²P terms can be possible in some systems where A transforms as an irrep with imaginary character, but this is not relevant for the zone-boundary irreps of $[Pm{\overline 3}m]$ that we consider here.

⁴Although we consider these formally as fourth-order terms here we emphasize that they might equally be third-order terms of a lower-symmetry perovskite structure that has already undergone some cation ordering or other structural distortion.

⁵A²P terms can be possible in some systems where A transforms as an irrep with imaginary character, but this is not relevant for the zone-boundary irreps of $[Pm\bar{3}m]$ that we consider here.

Acknowledgements

MSS would like to acknowledge Miss Hanna Boström and Dr Angel Arevalo-Lopez for useful discussions. Calculations were performed on the Imperial College London high-performance computing facility.

Funding information

MSS acknowledges the Royal Commission for the Exhibition of 1851 and the Royal Society for fellowships. NCB also acknowledges the Royal Commission for the Exhibition of 1851 for a fellowship and is also supported by an Imperial College Research Fellowship and a Royal Society research grant. We are also grateful to the UK Materials and Molecular Modelling Hub for computational resources, which is partially funded by EPSRC (EP/P020194/1).

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