 1. Introduction
 2. Ingredients for symmetry breaking
 3. Recipes for improper ferroelectric couplings
 4. Recipes for magnetoelectric coupling
 5. Trilinear magnetoelectric couplings in AFM systems
 6. Fourthorder magnetoelectric couplings in AFM systems
 7. Stabilizing wFM and magnetoelectric effects in nonferroelectrics
 8. Putting it all together
 9. Conclusion
 References
 1. Introduction
 2. Ingredients for symmetry breaking
 3. Recipes for improper ferroelectric couplings
 4. Recipes for magnetoelectric coupling
 5. Trilinear magnetoelectric couplings in AFM systems
 6. Fourthorder magnetoelectric couplings in AFM systems
 7. Stabilizing wFM and magnetoelectric effects in nonferroelectrics
 8. Putting it all together
 9. Conclusion
 References
research papers
A grouptheoretical approach to enumerating magnetoelectric and multiferroic couplings in perovskites
^{a}Department of Chemistry, University of Warwick, Gibbet Hill, Coventry, CV4 7AL, UK, ^{b}School of Physical Sciences, University of Kent, Canterbury CT2 7NH, UK, and ^{c}Department of Materials, Imperial College London, London SW7 2AZ, UK
^{*}Correspondence email: m.senn@warwick.ac.uk
A grouptheoretical approach is used to enumerate the possible couplings between magnetism and ferroelectric polarization in the parent perovskite structure. It is shown that thirdorder magnetoelectric coupling terms must always involve magnetic ordering at the A and B sites which either transforms both as Rpoint or both as Xpoint timeodd irreducible representations (irreps). For fourthorder couplings it is demonstrated that this criterion may be relaxed allowing couplings involving irreps at X, M and Rpoints which collectively conserve crystal momentum, producing a magnetoelectric effect arising from only Bsite magnetic order. In this case, exactly two of the three irreps entering the must be timeodd irreps and either one or all must be odd with respect to inversion symmetry. It is possible to show that the timeeven irreps in this triad must transform as one of: X_{1}^{+}, M_{3,5}^{−} or R_{5}^{+}, corresponding to Asite cation order, Asite antipolar displacements or anion rocksalt ordering, respectively. This greatly reduces the search space for typeII 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 (ABX_{3}) 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 quasirigid BO_{6} 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 ABX_{3} 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 ABX_{3} perovskite, may be defined as transforming as irreducible representations (irreps) of the parent (and setting). The irreps for all special positions in have been tabulated by various authors including by Bradley & Cracknell (1972), Miller & Love (1967), Kovalev (1993) and more recently also at nonspecial kpoints (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 (OP) transforming as one of these irreps.
in an `parent' structure, such as theOnline 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 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 singlecrystal programs (Campbell et al., 2007; PerezMato et al., 2010), allowing refinements to be performed in a symmetryadapted basis and facilitating easy identification of the active order parameters in a given phase transition.
As a result of much of this work, several grouptheoretical studies have emerged that have more formally classified distortions in perovskiterelated materials. These include grouptheoretical analysis of octahedral tilting in perovskites (Howard & Stokes, 1998, 2005; Knight, 2009), cationordered 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 Landaustyle 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 Landaustyle 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 thirdorder 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 groups for describing multiferroic materials has also allowed magnetoelectric couplings to be trivially identified through analysis of secondary order parameters (PerezMato 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 perovskite. These recipes are based on symmetry arguments alone, and we use as the ingredients structural and magnetic which we classify in terms of transforming as irreps of the parent 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 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 symmetrybreaking 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 fourthorder terms in the free energy expansion. As the most useful multiferroics are those that are (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 (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 (POP).
2. Ingredients for symmetry breaking
First we classify the magnetic 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 Rpoints, 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 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 highsymmetry sites. Fig. 1 gives full details of how the spin arrangements are related to irreps.
at our disposal in terms of irreps of the . We classify all of these in terms of irreps of the parent perovskite structure with settingNext we classify the various structural , along with their corresponding labels in the alternative setting [A at (½, ½, ½)]. Some of these 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 R_{2}^{} and Asite layered cation order as X_{1}^{+}. We may even classify the highly distorted cationordered A′A_{3}B_{4}O_{12} quadruple perovskite with as having cation orderings transform as M_{1}^{+} [with three kactives = (, , 0); (0, , ); (, 0, )] and octahedral rotations that stabilize the A′ squareplanar coordination transforming as M_{2}^{+}.
within the perovskite structure for inducing symmetrylowering 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

Finally, the desired property, ferroelectricity, transforms as the polar mode belonging to the irrep . is a threedimensional irrep; the most general 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 offcentre 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 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 to produce as a secondary OP.
direction (OPD) associated with this would hence be written as OP(3. Recipes for improper ferroelectric couplings
We begin by considering structural irreps (transforming as timeeven) 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 perovskitelike 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 Landaustyle 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 lowestorder 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 inversionodd and has zero crystal momentum, the lowestorder 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 inversionodd; [P] = [0, 0, 0].^{1}
A·B is inversionodd; [A] + [B] = [0, 0, 0] must be obeyed leading to all quantities being conserved in the trilinear term:
A·B·P is inversioneven; [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 B must be true for this condition to be fulfilled for otherwise AB would be inversioneven, meaning that the quadratic linear term A^{2}P 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 Rpoint 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 spacegroup symmetry and basis with respect to the parent structure. It is then trivial to identify from either the 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 (where represents the direct sum):
While many of these may be difficult to achieve in practice, there are several promising candidates. For example, columnar Asite cation order (M_{1}^{+}) with antipolar Bsite displacements (M_{5}^{}) can lead to a trilinear term M_{1}^{+} M_{5}^{} . We believe this could be the cause of the ferroelectric polarization recently reported in highpressure perovskite CaMnTi_{2}O_{6} (Aimi et al., 2014). Indeed, cation or anion ordering at any of the perovskite sites at the Mpoint along with antipolar distortions at the A or B sites would produce an improper ferroelectric polarization. Inphase tilting (M_{2}^{+}) or the Mpoint Jahn–Teller mode (M_{3}^{+}) can alternatively be used in conjunction with the antipolar displacements (such as M_{5}^{}) to induce a polarization, which has been recently predicted in the Pmc2_{1} 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 P2_{1}nm halfdoped manganites (Giovannetti et al., 2009; Rodriguez et al., 2005).
The commonly observed rocksalt cation ordering at the B site (King & Woodward, 2010) along with (Rpoint) 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 lonepair ordering will be challenging. Cation order on the A sites at the Rpoint (rocksalt) along with octahedral tilt modes would also produce an improper ferroelectric coupling, as recently predicted through firstprinciples calculations (Young & Rondinelli, 2013). However, it should be noted that Asite cation ordering is more commonly found to be in a layered (Xpoint) arrangement (King & Woodward, 2010). Very recent reports of improper ferroelectricity in the 134 perovskite HgMn_{3}Mn_{4}O_{12} can also be understood with respect to the present symmetry analysis of ABO_{3} perovskites (Chen et al., 2018). In this case, the atomic displacements associated with the orbital and charge ordering on the A and B sites transform as irreps of the parent R_{5}^{+} and R_{3}^{}.
Asite cation layering (X_{1}^{+}) in combination with antipolar Acation motions is indeed sufficient to induce P. Again, whilst the former is fairly common, the latter is only expected to be an unstable for low tolerance factor perovskites (Mulder et al., 2013). However it can itself manifest through an improper appearance with two tilting modes (M_{2}^{+} R_{5}^{} X_{5}^{}), which gives rise to the fourthorder term described below. At the Xpoint, one other trilinear term has been predicted to play a role in the P2_{1} phase of strained CaTiO_{3}, whereby A and Bsite antipolar (X_{5}^{+} and X_{5}^{}) motions induce P (Zhou & Rabe, 2013).
Fourthorder 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 fourthorder term must take the form:
A·B·C is inversionodd; [A] + [B] + [C] = [0, 0, 0] must be obeyed leading to all quantities being conserved in the trilinear term:
A·B·C·P is inversioneven; [A] + [B] + [C] + [P] = [0, 0, 0].
One of the most promising fourthorder candidates involves OPs associated with X^{+}, M^{+}, R^{−} and : for example, Asite layered cation ordering (X_{1}^{+}), octahedral tilt mode (M_{2}^{+}) and octahedral tilt mode (R_{5}^{}). This explains the significance of layering (X_{1}^{+}) 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 halfdoped titanates (Bristowe et al., 2015), includes Asite layered cation ordering (X_{1}^{+}), Mpoint Jahn–Teller (M_{3}^{+}) and octahedral tilt modes (R_{5}^{}). Other possibilities include Asite striped cation ordering (X_{1}^{+}), tilting (R_{5}^{}) and charge order (M_{4}^{+}), which we believe to be the origin of the improper polarization in SmBaMn_{2}O_{6} (Yamauchi, 2013). Alternatively Jahn–Teller induced, M_{3}^{+} and R_{3}^{}, ferroelectricity has been discussed in Asite striped cation ordered (X_{1}^{+}) rareearth vanadates (Varignon et al., 2015a). Perhaps an interesting avenue for future research is to use anion ordering since the X_{1}^{+} 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; CastilloMartínez et al., 2006).
Other chemically and structurally less promising schemes are still worth a mention: X^{−} M^{−} R^{−} , 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^{+} , striped Bsite cation order (X_{3}^{}), octahedral tilt mode (M_{2}^{+}), antipolar distortion on the B site (R_{5}^{+}); and X^{+} M^{−} R^{+} , Asite striped cation ordering (X_{1}^{+}), antipolar distortions on the B site (M_{2}^{}), anion order (R_{5}^{+}). 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
to magnetoelectric couplings including timeodd irreps that describe magnetic order. We seek initially the strongest magnetoelectric coupling term possible: this means that as before we should consider the lowestorder term in the free energy expansion that is achievable. Since P transforms as timeeven, inversionodd and has zero crystal momentum, the lowestorder term involving two zoneboundary irreps will be third order (ABP). Invariants analysis tells us that:for P is timeeven; P is inversionodd; [P] = [0, 0, 0].
A·B is timeeven; A·B is inversionodd; [A] + [B] = [0, 0, 0] must be obeyed leading to all quantities being conserved in the trilinear term:
A·B·P is timeeven; A·B·P is inversioneven; [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 timeeven means that therefore both A and B must transform as a timeodd irrep. One may further convince oneself that A B must be true for this condition to be fulfilled for otherwise AB would be inversioneven, meaning that the quadratic linear term A^{2}P 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 timeodd, of opposite parity with respect to inversion symmetry and must have equal crystal momentum.
As before with the improper
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 fourthorder 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 due to structural distortions or cation orderings, which themselves can be classified as transforming as irreps of . 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 timeeven
A·B·C is inversionodd
[A] + [B] + [C] = [0, 0, 0].
If we are seeking a magnetoelectric coupling, precisely two of these terms must be timeodd (since P will always be timeeven), 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
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 BaMnF_{4} (Fox et al., 1980). However, our approach differs in that we perform the Landaustyle expansion of the free energy about a hypothetical aristotypical symmetry, rather than the experimentally observed hightemperature phases. The benefit of our approach is that it encodes as much information as possible regarding the crystal momentum and parity of the timeodd 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 timeodd irreps when constructing the OP:
At the Mpoint, 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 Asite 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 mX_{3}^{+} mX_{1}^{}, the OP is six dimensional OP(a;b;cd;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;0d;0;0), OP(a;0;ad;0;d), OP(a;;ad;;d) (see Fig. 2). Of these only OP(a;0;0d;0;0) represents a single kactive and collinear solution, and we shall focus on this for the rest of our discussion. The isotropy group is P_{c}4cc with basis = [(1, 0, 0),(0, 0, 1),(0, −2, 0)] + (0, 0, 0) and SOPs (polar mode) and (tetragonal strain). This OPD corresponds to the magnetic moments aligned parallel to the propagation vector on both A and B sites.
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 GdFeO_{3} structure in which the unitcell parameter (the only degree of freedom) was fixed at a = 3.65 Å. This contracted 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 (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 [ OP (0, h, 0)] is condensed with different amplitudes in the FM phase () to give the expected single well potential centred at zero.
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: 5s^{2}5p^{6}6s^{2}4f^{8} (Gd), 3p^{6}4s^{2}3d^{6} (Fe) and 2s^{2}2p^{4} (O). An onsite 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 , whilst keeping the system insulating. A planewave cutoff of 900 eV and a 6 × 6 × 6 kgrid 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 mX_{3}^{+} and mX_{1}^{} [OP(a,0,0d,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 mX_{3}^{+}, mX_{1}^{} and . 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 spininduced 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 highsymmetry 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 offcentring in the groundstate 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 mX_{5}^{+} mX_{5}^{}, the OP is now 12dimensional OP(a,b;c,d;e,fh,i;g,k;l,m). The representative (single kactive) OPDs which meet the criteria for zero crystal momentum transfer are however of the form OP(a,b;0,0;0,0h,i;0,0;0,0). We do not consider OPs with multiple kactives as in general this will always induce SOPs transforming as M or Rpoint 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 highsymmetry examples given in Fig. 4 are OP(a,a,0,0,0,0h,,0,0,0,0) and OP(0,a,0,0,0,0,0,0,0,0,0). We do not explicitly consider mX_{3}^{+} mX_{5}^{} or mX_{5}^{+} mX_{1}^{} here as these represent magnetic structures in which the spins on the A site and B site are noncollinear with each other, which we believe to be less physically likely than the remaining examples that we have already discussed.
For mR_{4}^{+} mR_{5}^{}, the resulting OP is OP(a,b,cd,e,f). There are 14 possible OPDs that result in unique spacegroup, 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,0,0d,0,0), , basis = [(−1, −1, 0),(1, −1, 0),(0, 0, 2)] + (0, 0, 0); OP(a,a,0d,d,0), I_{c}ma2, basis = [(1, 0, 1),(1, 0, −1),(0, 2, 0)] + (0, 0, 0); OP(a,a,ad,d,d) R_{I}3c, 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 lowersymmetry collinear magnetic structures may be constructed through linear combinations of these three OPDs. For the polar space groups (I_{c}ma2 and R_{I}3c) an SOP transforming as is always active. The only other SOPs are strain. A strategy for stabilizing this groundstate structure, therefore, in addition to designing AFM nearestneighbour interaction in the system, is to epitaxially prestrain the sample in a manner that stabilizes terms in the free energy that will also occur at the even order.
(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 firstprinciple calculations and grouptheoretical analysis that the rareearth gadolinium chromates and ferrites with collinear Gtype order on A and B sites along the pseudocubic axes lead to a piezoelectric 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 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,0d,d,0) and OP(a,a,ad,d,d). We note also here that the possible observation of weak ferroelectric polarization, which is reported in A and Bsite lattices in which sublattice moments along 100type 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 arises directly as a consequence of this kind of magnetic ordering [OP(a,0,0,0,0,d)] with the magnetic being F_{S}mm2 {basis = [(0, 2, 0),(0, 0, 2),(2, 0, 0)], origin = (½, ½, 0)}.
An experimental example of where magnetoelectric properties arise from Gtype ordering on the A and B sites can be found in the literature for the 134 perovskite LaMn_{3}Cr_{4}O_{12} (Wang et al., 2015). This distorted perovskite structure has the additional structural orderings that can be described as M_{1}^{+}(a;a;a) (1:3 cation ordering) and M_{2}^{+}(a;a;a) (octahedral rotation). However, the observed magnetoelectric effect, that only occurs below both the Bsite and the Asite ordering temperature, can be understood in terms of our present results by considering only OP mR_{4}^{+} mR_{5}^{} with OP(a,a,ad,d,d) (Fig. 5, right), meaning that the magnetoelectric groundstate structure has rhombohedral lattice symmetry and arises solely as a consequence of the magnetic ordering on both sites.
6. Fourthorder magnetoelectric couplings in AFM systems
An exhaustive list of fourthorder couplings in polarization and zoneboundary 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 rareearth ions or nonmagnetic cations. Notable exceptions to this are the perovskite MnVO_{3}, but where the magnetic ordering temperatures remain low (Markkula et al., 2011), and some highly distorted AA′_{3}B_{4}O_{12} quadruple perovskites that we will not discuss here.

Considering only Bsite magnetism we are left with the following timeodd superposition of irreps to consider: mM_{2,5}^{+} mX_{1,5}^{}; mR_{5}^{} mX_{1,5}^{}; mR_{5}^{} mM_{2,5}^{+}. In order to close the `momentum triangle' these will now be, respectively, superposed with the following timeeven irreps: R_{1,5}^{+}, M_{3,5}^{} and X_{1}^{+}, to produce an OP that transforms as timeeven, inversionodd and has a crystal momentum transfer of zero (see Tables 3, 4 and 5). The relevant structural (Table 1) to consider are, hence, cation/anion order (X_{1}^{+}, 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.



For X_{1}^{+} mM_{2,5}^{+} mR_{5}^{} we give some possible examples of several magnetic structures in Figs. 6 and 7 corresponding to Asite ordered double perovskites with striped type (X_{1}^{+}) 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.
The possible highsymmetry OPDs for superposed irreps X_{1}^{+}(0, , 0) mM_{2}^{+}(, , 0) mR_{5}^{}(, , ) are:
Conservation of crystal momentum criteria that we have imposed here dictates the relative OPD of the X and M components (kactives). The three structures listed above and shown in Fig. 6 only differ in the OPD with respect to the mR_{5}^{} irrep, producing two noncollinear magnetic structures and one which has a spindensitywave. In the case of the noncollinear magnetic structures, the direction of P is parallel to both the cation order planes and the canting direction. For the spindensitywave structure the polarization vector is perpendicular to the cation ordering planes. Spindensitywave magnetic structures are in general less common, but we note that Xpoint order of two magnetically active cations (at the B site) with different magnetic moments could be a way to achieve this.
For X_{1}^{+} mM_{5}^{+} mR_{5}^{} as mM_{5}^{+} is a higherdimensional irrep than M_{2}^{+}, there are now a larger number of OPD possibilities:
However, this time several of these highsymmetry OPs give rise to piezoelectric but nonpolar space groups [(a;0;00,0;b,0;0,0c,0,0) C222 and (a;0;00,0;b,−b;0,0c,c,0) P222_{1}]. 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 highsymmetry OPD resulting in polar structures. Similarly for the discussion above, P is parallel and perpendicular to cation ordering for constant moment and spindensitywave magnetic structures, respectively.
For mX_{1,5}^{} mM_{2,5}^{+} R_{5}^{+}, in which R_{5}^{+} could correspond to anion order, the cisordering of N for O substitution in oxynitride ABO_{3−x}N_{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 R_{5}^{+}(a,a,a) (or the closest highsymmetry equivalent OPD) in the following analysis. As a POP transforming as R_{5}^{+}(a,a,a) always has an SOP transforming as R_{1}^{+}(a), this analysis also turns out to be equivalent to looking at rocksalt ordering on the Asite cation, although we note that such ordering is not particularly common. mX_{1}^{}(0, , 0) mM_{2}^{+}(, , 0) R_{5}^{+}(, , ), with an OPD of (a;0;00;b;0c,c,d), corresponds to a spindensitywave collinear magnetic structure, where P is in the plane of the directions (Fig. 8). The constantmagnitude spincanted magnetic structures [mX_{5}^{} mM_{2}^{+} R_{5}^{+} (a;0;00;b;0c,c,d), Fig. 8] on the other hand lead to polarizations that are found to be both perpendicular and parallel to the alignment.
We will not consider the remaining possible couplings for mX_{1}^{} mM_{2}^{+} R_{5}^{+}, mX^{−} mM_{5}^{+} R_{5}^{+} and mX_{1}^{} M^{−} mR_{5}^{} explicitly here but they are tabulated in Table 5 and representative figures are given in Figs. 8, 9 and 10.
7. Stabilizing wFM and magnetoelectric effects in nonferroelectrics
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 which is timeodd, parityeven and has crystal momentum of zero, the constraints on A and B are as follows:
[A] = [B]
A·B is parityeven.
Hence, possible trilinear terms with wFM need to involve OPs that transform as:
Taking mR_{5}^{} (Bsite magnetic order) with R_{5}^{} (octahedral tilting) as an example, Gtype magnetic ordering on the B sites with moments along the c axis with outofphase octahedral rotations leads to the magnetic Im′m′a {basis = [(1, 1, 0),(0, 0, 2),(1, −1, 0)] + (0, 0, )} which has (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 symmetrybreaking event allowing spin canting to occur via the Dzyaloshinsky–Moriya interaction.
At the Mpoint the above analysis can also be applied. Ctype Bsite magnetic ordering along the [001] axis [mM_{2}^{+}(a;0;0)], with inphase octahedral tilts perpendicular to this [M_{2}^{+}(a;0;0), a^{0}a^{0}c^{+}], actually leads to a piezomagnetic P4/mbm {basis = [(1, −1, 0),(1, 1, 0),(0, 0, 1)] + (, , 0)}. Application of an orthorhombic type strain () for example leads to the occurrence of wFM (). Similarly, distortions transforming as X_{5}^{} and magnetic moments as mX_{1}^{} will produce wFM {e.g. , basis = [(1, 0, 1),(1, 0, −1),(0, 2, 0)] + (0, , 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 or conversely wFM is induced by the application of an external electric field (). For this we must look at terms involving two zoneboundary irreps like [M] + [S] = [0 0 0], where M is timeodd (magnetic) and S is timeeven (structural), and M·S is inversionodd. Application of an electric field () should then give a fourthorder term in the free energy expansion of the form M S P wFM. A realization of this is mX_{1,5}^{} X_{1}^{+} 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 codependent OP (see Fig. 11). For example: (i) X_{1}^{+} X_{5}^{} P with mX_{1}^{} X_{5}^{} wFM. Assuming X_{1}^{+} represents cation order and may not be reversed, then the reversal of the sign of P would necessitate a reversal of X_{5}^{}. 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) M_{5}^{} M_{2}^{+} P and mM_{5}^{} M_{2}^{+} wFM. Taking M_{5}^{} as anion ordering, then a reversal of P would proceed via reversal of the octahedral rotations (M_{2}^{+}) necessitating a reversal of either mM_{5}^{} or wFM, the latter being more likely. (iii) R_{1}^{+} R_{2}^{} P and mR_{5}^{} R_{2}^{} wFM, taking R_{1}^{+} as Asite rocksalt cation ordering, a reversal of P would imply a switching of R_{2}^{} which could represent Bsite charge ordering
In fact, if one has an AA′ layered double perovskite (X_{1}^{+}) with the common M_{2}^{+} and R_{5}^{} tilt pattern (Pnma like) no matter if you have C (mM_{5}^{+}), G (mR_{5}^{}) or A (mX_{1}^{}) 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 and A^{2+} A layered double perovskites with Mn^{4+} and Fe^{3+} on the B site, respectively, to achieve the strongest wFM moments and highest ordering temperatures.
Another scheme involving fourthorder couplings gives a greater degree of flexibility. Similar to the above, the idea here is to construct fourthorder terms with wFM () in. As many of the OPs featuring in the wFM term at the fourth order should also feature in the fourthorder term in P. Fig. 12 envisages one such possible coupling scheme by which an extra degree of freedom related to breaking structural symmetry (S_{2}) 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 fourthorder terms with at least two OPs in common in both P and wFM terms, i.e. M_{1} M_{2} S_{1} P and M_{1} S_{1} S_{2} wFM. Each fourthorder term must individually conserve crystal momentum, time reversal and inversion symmetry. Hence the polar part, M_{1} M_{2} S_{1}, can be selected according to the analysis in the previous section, leaving the wFM part, M_{1} S_{1} S_{2}, to be decided on. Since M_{1} S_{1} are fixed by the polar part, the only decision to be made is the nature of S_{2}. We require that the crystal momentum of [S_{2}] equals the sum of the crystal momentum [M_{1}] + [S_{1}], and that parity with respect to inversion is equal to the product of the parity of M_{1}·S_{1} (i.e. opposite to that of M_{2}). For example, with mM_{2,5}^{+} (M), mR_{5}^{} (M) and X_{1}^{+} (S_{1}), S_{2} must be either R_{1,5}^{+} or M_{2,3,5}^{}.
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 higherorder 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 fourthorder couplings in ferroelectric polarization. Any further distortion to the 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 prestrain (transforming as ) 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 (M_{2}^{+} and R_{5}^{}), Jahn–Teller distortions in systems with a degeneracy in their d orbitals or indeed polar distortions themselves in d^{0} systems. In the `undistorted' perovskite structure these will correspond to the lowestlying 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 For example, for X_{1}^{+} mM_{2}^{+} mR_{5}^{}, SOPs are strain, , and Xpoint distortions. The OP(a;0;00;b;0c,c,0) P_{A}mc2_{1} 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 offcentre distortions should be chosen. For mX_{1}^{} M_{5}^{} mR_{5}^{} or mX_{1}^{} mM_{2}^{+} R_{5}^{+}, B sites with a propensity to undergo charge (M_{4}^{+} or R_{2}^{}) and orbital order (M_{3}^{+} or R_{3}^{}) 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 grouptheoretical means, we have enumerated the possible magnetoelectric couplings in the perovskite structure with respect to its aristotypical symmetry . Our enumeration is complete up to the thirdorder terms for zoneboundary magnetic structures, and for fourthorder terms for Bsite magnetism only. Our results show that, for zoneboundary magnetic ordering, only magnetism on both A and B sites transforming either both as Xpoint or Rpoint irreps can produce a magnetoelectric coupling at the third order, which is illustrated with firstprinciples calculations. For magnetism on the B site alone, then only fourthorder 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 Rpoints, chosen in such a way that crystal momentum is conserved, that two are timeodd and either one or all are inversionodd. 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 Ca_{3}Mn_{2}O_{7}. 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
directionsWe will be forming OPs from up to three different zoneboundary irreps, which we will denote by using the symbol for direct sum `', for example M_{2}^{+} R_{5}^{} 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 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 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 M_{2}^{+} R_{5}^{}, an OPD is OP(a;0;0b,0,0), where `' denotes a division between the OPD parts belonging to M_{2}^{+} and R_{5}^{}, respectively. Semicolons `;' denote divisions between different OPDs resulting from the degeneracy of the propagation vector in . At the Mpoint the possible kactives are (, , 0);(, 0, );(0, , ), where the order respects the order of appearance in the OPD. Similarly, for an OP transforming as Xpoint irreps we will need to specify which kactives of (0, , 0);(, 0, 0);(0, 0, ) are in use. In general we will only form OPs from one kactive per irrep (equivalent to using the small irrep only). However, the notation we give will always reflect the total number of possible kactives. At the Rpoint there is only one possible kactive, but in this case the irrep is multidimensional and, as in the case of , 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 of the parent structure to uniquely identify the isotropy subgroup.
Footnotes
^{1}In general the inversion symmetry breaking distortion will transform as the polar mode; however sometimes the symmetry breaking will instead be associated with another mode which is piezoelectric in nature.
^{2}Strictly 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.
^{3}A^{2}P terms can be possible in some systems where A transforms as an irrep with imaginary character, but this is not relevant for the zoneboundary irreps of that we consider here.
^{4}Although we consider these formally as fourthorder terms here we emphasize that they might equally be thirdorder terms of a lowersymmetry perovskite structure that has already undergone some cation ordering or other structural distortion.
^{5}A^{2}P terms can be possible in some systems where A transforms as an irrep with imaginary character, but this is not relevant for the zoneboundary irreps of that we consider here.
Acknowledgements
MSS would like to acknowledge Miss Hanna Boström and Dr Angel ArevaloLopez for useful discussions. Calculations were performed on the Imperial College London highperformance 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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