Anharmonic lattice interactions in improper ferroelectrics for multiferroic design.

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Anharmonic lattice interactions in improper ferroelectrics for multiferroic design

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 283202 (18pp)

doi:10.1088/0953-8984/27/28/283202

Topical Review

Anharmonic lattice interactions in improper ferroelectrics for multiferroic design Joshua Young1, Alessandro Stroppa2, Silvia Picozzi2 and James M Rondinelli3 1

Department of Materials Science and Engineering, Drexel University, Philadelphia, PA 19104, USA Consiglio Nazionale delle Ricerche—CNR-SPIN, L’Aquila, Italy 3 Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA 2

E-mail: [email protected], [email protected] and [email protected] Received 23 March 2015, revised 20 May 2015 Accepted for publication 22 May 2015 Published 30 June 2015 Abstract

The design and discovery of new multiferroics, or materials that display both ferroelectricity and long-range magnetic order, is of fundamental importance for new electronic technologies based on low-power consumption. Far too often, however, the mechanisms causing these properties to arise are incompatible or occur at ordering temperatures below room temperature. One design strategy which has gained considerable interest is to begin with a magnetic material, and find novel ways to induce a spontaneous electric polarization within the structure. To this end, anharmonic interactions coupling multiple lattice modes have been used to lift inversion symmetry in magnetic dielectrics. Here we provide an overview of the microscopic mechanisms by which various types of cooperative atomic displacements result in ferroelectricity through anharmonic multi-mode coupling, as well as the types of materials most conducive to these lattice instabilities. The review includes a description of the origins of the displacive modes, a classification of possible non-polar lattice modes, as well as how their coupling can produce spontaneous polarizations. We then survey the recent improper ferroelectric literature, and describe how the materials discussed fall within a proposed classification scheme, offering new directions for the theoretical design of magnetic ferroelectrics. Finally, we offer prospects for the future discovery of new magnetic improper ferroelectrics, as well as detail remaining challenges and open questions facing this exciting new field. Keywords: ferroelectrics, improper phase transitions, anharmonic interactions, multiferroics, pseudo-Jahn–Teller (Some figures may appear in colour only in the online journal)

1. Introduction

and widespread integration, including low storage density and high cost [12]. Creating an understanding of how electric polarizations can arise in materials and using this insight to design for function from the atomic scale and up, rather than searching for previously unknown ferroelectrics in structure types known to support ferroelectricity, will accelerate the development process. Knowledge about the microscopic underpinnings of these materials can also allow for further

Ferroelectric materials have recently garnered renewed attention as potential candidates for use in low-power highefficiency electronics [1–4] and solar cells [5, 6], while multiferroics have been proposed for new types of solid-state memory and other novel logic devices [7–11]. There are several challenges to overcome to facilitate industrial production 0953-8984/15/283202+18$33.00

1

© 2015 IOP Publishing Ltd Printed in the UK

Topical Review

J. Phys.: Condens. Matter 27 (2015) 283202

inducing polar displacements and ferroelectric polarizations in the presence of ions with strong magnetic spin or orbital moments are highly desired. To remedy the incompatibilities, new routes to induce electric polarizations in magnetic materials have been sought [19]. One step towards reaching this goal is the recent division of multiferroics into two categories: Type I, in which a material simply displays ferroelectricity and magnetic ordering, or Type II, in which one ferroic property is a consequence of the other [20]. Most proposals now follow the Type I protocol, and rely on choosing various atomic species, e.g. those that are magnetically active (or inert) and ferroelectrically inactive (or active) ions, and then occupying different sites in the lattice with those atoms. Because atoms with a stereoactive lone pair are also considered to exhibit the SOJT effect [21], replacing Ba 2 + with Bi 3 + and Ti 4 + with Fe 3 + in the perovskite structure of BaTiO3 allows the A-sites to collectively displace and generate a ferroelectric polarization while the Fe spins freely interact and order magnetically. The mutual coexisting antiferromagnetic and ferroelectric order in BiFeO3, with a Néel ordering temperature above room temperature, is responsible for its intense popularity in the multiferroics community. Within this setting, a new direction has emerged in the theoretical design of improper magnetic ferroelectrics (IMF). In pioneering experiments on ultra-short period diamagnetic PbTiO3/SrTiO3 superlattices [22], Bousquet et al showed that an additional contribution to the electric polarization arises through anharmonic lattice–lattice interactions that couple multiple normal modes together [23]. Spawned by this observation is the proposal that the electric polarization (and removal of inversion symmetry) may be effectively described as an ‘improper’ displacive ferroelectric transition [24, 25] rather than being due to improper coupling to electronic or orbital degrees of freedom [26–28]. These differences will be described in detail below. Shortly afterwards it was proposed that rather than providing an additional contribution to the total polarization, such non-polar lattice modes could be primarily responsible for the polarization and completely induce it. This form of ferroelectricity in artificial oxide heterostructures also established the potential for the rational design of new magnetic ferroelectrics; one could theoretically predict the behavior of a superlattice by understanding the lattice dynamical properties of the constituent oxides interleaved to form it and evaluate the propensity to an ‘improper’ transition; then as done previously, substitute magnetic cations into the structure. Of course the structure topology and ions must be selected to favor the necessary non-linear coupling between multiple non-polar lattice modes that cooperatively lift inversion symmetry and favor polar displacements [29–31]. Because the origin of the electric polarization in such systems is not electrostatic or due to SOJT-activity, but rather ‘geometric’ distortions, the polarization can be obtained independent of the magnetic ordering [32]. The ‘coexistence challenge’ may then be reformulated into one of inducing electric polarizations in chemistries already known to exhibit long-range magnetic order using non-linear interactions between soft phonons that broadly

advancements, such as coupling between multiple ferroic degrees of freedom, including electric, magnetic, and elastic properties. By determining the atomic scale structural features leading to spontaneous polarizations, it is possible to formulate new models and strategies for the rapid identification of novel ferroelectric or multiferroic materials. Ferroelectric materials rely on broken inversion symmetry at the unit cell level, which allows for the appearance of a macroscopic electric dipole moment per unit volume. This usually arises from the condensation of a polar phonon or soft normal mode of the lattice upon cooling from a centrosymmetric structure. In practice, this is achieved in crystalline oxides, i.e. the materials of topical interest here, by incorporating metal cations into the lattice which have a tendency to displace in a polar manner owing to hybridization with the surrounding anions through the second-order Jahn–Teller (SOJT) effect [13, 14]. However, this effect is almost exclusively limited to metals with a d 0 electronic configuration, such as the Ti 4 + cations in the complex perovskite oxide BaTiO3. Here, the fact that the 3d orbital of Ti is formally empty makes it the lowest unoccupied molecular orbital (LUMO). Ti off-centering is then driven by an increased overlap of the LUMO with the highest occupied molecular orbital (HOMO), i.e. the oxygen 2p states, leading to enhanced covalency of the Ti–O bond. On the other hand, magnetic order in complex oxides, sulfides, and fluorides requires cation chemistries with partially filled d or f electron shells that favor local moment formation as specified by Hund’s rule for intra-atomic exchange. In addition, the local moments at different sites in the crystal are required to interact, otherwise collective magnetic order that frequently appears (vanishes) upon cooling (heating) would not arise. Possible interactions between spins occur through direct exchange (overlap of the wavefunctions due to spatial proximity), superexchange (coupling of spins through spin-dependent hybridization of orbitals with ligands), or via conduction electrons (RKKY-type of interactions) [15, 16]. However, a majority of magnetic oxides do not undergo ferroelectric transitions owing to incompatibilities between the chemical bond formation required to stabilize the polar cation displacements (or competition with antidistortive modes) and the open valence shells required for local moment formation. The requirement of a d 0 valence for ferroelectricity is thus in direct competition with the appearance of spontaneous magnetic ordering in these materials [17]. While there have been several proposed ways of circumventing this electronic constraint, they are all dependent on the specific chemistry of the constituent atomic species. Recent investigations have shown that it is in fact possible for atoms with non-zero d n configurations to undergo proper ferroelectric displacements [18]. This effect arises through vibronic coupling among the ground and excited electronic states, but only if they have the same spin multiplicity and are of opposite parity. Such restrictions limit magnetic cations that undergo ferroelectric displacements to d-orbital occupancies with 3 ⩽ n ⩽ 7, and then only for specific spin states within those configurations. Owing to the fact that magnetism is rarer in 4d and 5d transition metals, the applicability of this approach is mainly limited to 3d transition metals; thus additional mechanisms capable of 2

Topical Review


different types of structural phase transitions [38]. The three types include: (i) reconstructive, (ii) displacive, and (iii) order/ disorder. In a reconstructive transition, the first-coordination chemical bonds are broken and reconstructed leading to discontinuous enthalpy and volume changes. From a thermodynamic point-of-view, the transition exhibits first-order character, and is characterized by the coexistence of multiple phases at equilibrium, as well as hysteresis and metastability. Displacive transitions involve the breaking and reconstruction of second-coordination chemical bonds, while the chemical connectivity of the primary bonds (i.e. the first-coordination neighbor shell) are preserved. These transitions are often classified as having second-order thermodynamic character, and lead to small or vanishing enthalpy and volume changes. In an order/disorder phase transition, the structural difference is related to different chemical occupation of the same crystallographic sites. These transitions are thermodynamically of second or higher order, and are classified into two types: rotational and substitutional. A rotational order/disorder transformation has some characteristics of a displacive transformation in that groups of tightly bound atoms in an ordered structure rotate relative to the rest of the structure and so induce disorder. Substitutional transformations are commonly found in metals, alloys, and multi-anion ceramics where interchanging the position among atoms in a random fashion induces disorder. Typically in displacive and order/disorder phase transitions the space group symmetries of the two phases show a group/subgroup relationship, where the low symmetry phase G approaches the high symmetry G 0 phase continuously. To describe these phase transitions quantitatively, one can introduce an ‘order parameter’ η as a quantity that captures the reduction in symmetry accompanying the phase transition and usually has a clear microscopic meaning, e.g. collective ionic motion or site-distribution changes. It may also be single or multi-component (this will be covered in more detail in subsequent sections). When the transition is driven by temperature changes, the space group of the low temperature G phase is usually a subgroup of the high symmetry G 0 high temperature phase (G ⊂ G 0). Because ferroelectricity is associated with a crystallographic phase transformation from a centrosymmetric non-polar lattice to a non-centrosymmetric polar lattice, this change at the critical temperature Tc is always accompanied by a change of crystal symmetry manifesting in a nonzero value of η. In transitions driven by pressure changes, it is not as easy to predict which one of the low or high pressure phases is more symmetric, although in most cases the high pressure phase is less symmetric. The concept of symmetry-determined functional properties at ferroic phase transitions excels in describing displacive (and order/disorder) phase transitions for which the aforementioned order parameter becomes the critical property to optimize. Within this context it is possible to divide known ferroelectrics into a variety of categories based on the symmetry of the displacement pattern providing the energetic stability for the polar crystal structure. Traditionally, the focus has been on the nature of the phase transition (i.e. whether the transition is continuous (second-order) or discontinuous

describe rotational motions of polyhedral units in solid-state materials [33, 34]. In this review, we focus on establishing a language— atomic scale taxonomy—which accounts for the variety of ways in which polar displacements may result from coupling one or more (non-polar) lattice degrees of freedom to a polar IR-active phonon mode and are fully compatible with magnetic ions. Improper mechanisms offer the best opportunity for the application of this taxonomy, as the polar displacements resulting in an electric polarization arise as a consequence of other, non-polar displacement patterns. We note that rather than focusing on the explicit nature of the phase transition, we explore how the atomic distortions lead to the loss of inversion symmetry—such understanding, formulated within a classification framework, can fuel the design of new materials. For example, once the necessary crystal features are established, the design of new materials can be accomplished by selecting chemistries susceptible to the desired displacement patterns. The main objective is to present recent theoretical solutions to the coexistence challenge based on anharmonic lattice interactions, while simultaneously classifying ‘geometric’ improper ferroelectrics, i.e. compounds where the electric polarization arises from atomic displacement patterns, as opposed to ‘electronic’ improper ferroelectrics with polarizations induced by mechanisms such as spin [35], charge [36], or orbital [37] ordering. In section 2 we briefly describe the essential concepts of displacive phase transitions and lattice dynamics, highlighting the importance of anharmonic coupling of multiple normal modes to the thermodynamic crystal potential energy. Section 3 describes candidate lattice instabilities that repeatedly appear in this review and in the classification scheme described in section 4, drawing distinctions between the electronic or steric origins for each type. We then describe various theoretical proposals for such (magnetic) ferroelectrics, illustrating how this scheme allows for the identification of potentially understudied structural families and compositions for exploration as candidate multiferroics through targeted materials design. In section 5 we discuss additional opportunities for designing functional multiferroics and open challenges faced by the new classes of ferroelectrics where the polarization is not the primary order parameter. 2. Ferroelectric phase transitions According to Ehrenfest’s classification, the thermodynamic order of the phase transition is the smallest order of the derivatives of the Gibbs free energy G (where G = U − TS + PV , and where the derivatives are with respect to T or P) which are discontinuous across the transition. In particular, the first-order derivatives of entropy (S) and volume (V) are discontinuous, leading to enthalpy and volume jumps: dG = −S dT + V dP. In second-order phase transitions, the second-order derivatives, i.e. heat capacity (Cp) and compressibility ( β), are discontinuous, whereas the first-order derivatives are continuous. The classification system proposed by Buerger provides for a useful atomistic, although not rigid, distinction between 3

Topical Review


(first-order)), as well as whether the driving force, e.g. the soft phonon mode leading to this transition, occurs at the zonecenter (proper) or a zone-boundary (improper). We explain this nomenclature next, but note that semantics over these classifications often hinder the design and discovery of new ferroelectrics. Understanding the set of atomic distortions has implications on the underlying microscopic mechanism for inversion symmetry lifting and it also provides useful criteria when selecting the chemistry of a compound for experimentation. Elucidating design rules based on structural factors from which the driving force can subsequently be gleaned and then be applied across a wide variety of material families is arguably more useful. It is this approach we introduce in section 4 and apply in this review.

(b)

(a)

Decreasing T

P=0 T > TC T < TC

P

0

Figure 1. The energetics of a paraelectric to ferroelectric phase transition for a (a) proper and (b) improper ferroelectric system.

Improper ferroelectricity occurs when the polar mode is stable (‘hard’) in the high symmetry phase, but the softening of the non-polar mode(s) drives it to be energetically favorable through an anharmonic interaction; in this case, the polar mode is the secondary order parameter [24, 25]. If more than one mode is coupled to the polar mode, the material is known as a ‘hybrid’ improper ferroelectric [32]. However, there is a caveat to this definition in that some ferroelectric materials exist beyond this simple soft-mode paradigm. The Aurivillius compound Bi4Ti3O12, for example, displays anharmonic coupling of multiple modes which condense simultaneously at the transition temperature in an ‘avalanche’ transition [23]. While some sort of improper transition clearly occurs here, the term hybrid improper is difficult to apply as the unstable modes are not required to condense together, and more than one critical temperature would be expected. Understanding how a non-polar lattice mode can ‘trigger’ or induce polar displacements requires a discussion of anharmonic interactions in these materials, which we review shortly. First we examine the microscopic mechanisms differentiating these types of ferroelectrics in more detail, as well as the mathematical formalism describing the phase transitions.

2.1. Microscopic mechanisms and soft mode theory

All ferroelectrics are characterized as displaying a spontaneous, switchable electric polarization in their ground state. Because ferroelectricity often arises through a phase transition from a paraelectric phase, it is well-described by Landau theory. This framework characterizes phase transitions by the order parameter η which is zero in the high symmetry phase (G 0) and non-zero in the low symmetry phase (G). Although the spontaneous polarization P is traditionally used in Landau analysis of ferroelectric transitions, we wish to distinguish between the types of displacement patterns resulting in a polarization; in this case P does not provide enough information to allow this differentiation. As such we select the collective atomic distortions that lead to the loss of inversion symmetry as the order parameter η. The thermodynamic free energy of the system can be expanded as a Taylor series F(η). Below some critical temperature, TC, the energy minimum of this expression shifts from zero to a non-zero value of the order parameter across the phase transition. However, the order parameter does not always have to be η in a ferroelectric transition; the polar displacements described by η can also arise as a result of some other general distortion Q, which alone need not lift inversion symmetry. In this case η becomes a secondary order parameter which only appears in the presence of Q. This allows for a natural division of ferroelectrics based on whether the polar atomic displacement pattern η is the primary order parameter (‘proper’) or a secondary order parameter (‘improper’). Despite this division, the spontaneous polarization in both proper and improper geometric ferroelectrics results from a specific arrangement of ions in the crystal structure appearing below TC; a description of how this occurs is provided by the soft mode theory of displacive phase transitions [39, 40]. In the high temperature G 0 phase, there are one or more phonon modes whose frequencies decrease (‘soften’) upon cooling of the material, and eventually approach zero at TC. At this point the material undergoes a phase transition to the low symmetry ferroelectric phase, and the patterns of atomic displacements described by the soft phonon mode(s) become frozen in. The material is a proper ferroelectric if the soft mode consists of polar atomic displacements, making the polar mode the primary order parameter driving the phase transition.

2.2. Proper ferroelectrics

In proper ferroelectric transitions, the primary order parameter has the exact same symmetry properties as the pattern of atomic displacements giving rise to the polarization; the simplest second-order phase transition theory model that captures it is α (T ) 2 β 4 F(η) = η + η +… , (1) 2 4

where the temperature-dependent coefficient α changes sign as the polarization becomes non-zero. Across the critical temperature, the distortion leading to a polarization condenses, resulting in an energy gain that leads to a free energy minimum for finite η (figure 1(a)). There is also a minimum at −η corresponding to the same displacement pattern but in opposite sense, giving rise to the characteristic switching ability of ferroelectrics from +P to −P. In perovskite oxide ferroelectrics, like BaTiO3, the softening of an IR-active zone-center phonon produces the polarization. At the atomic scale, the Ti atoms cooperatively displace 4

Topical Review


phonon. The generic Landau functional which describes such transitions takes the form α (T ) 2 β 4 α ′ 2 F(Q, η) = Q + Q + η 2 4 2 β′ + η4 + ξη m ⋅ Q n + … , 4

(2)

where now, a macroscopic polarization is induced by the coupling term ξη m ⋅ Q n when the temperature-dependent α coefficient changes sign. Because the low-order powers of the coupled order parameters are commonly the most important, the coupling term can be truncated [50] to m = 1 and 1 ⩽ n ⩽ 4, as well as any biquadratic terms. Consideration of the order of coupling between Q and η then leads to a natural division among displacement patterns active in an improper ferroelectric. What is Q? Symmetry arguments indicate that Q could be represented as the amplitude of a zone-boundary phonon; the Q mode will then occur at or within the Brillouin zone boundary (k ≠ 0) of the centrosymmetric unit cell and thus is always a multicomponent order parameter (or may represent a combination of different modes). However, routine discovery, let alone design, of improper ferroelectrics which have a suitable coupling between zone-boundary lattice instabilities (that modify the translational symmetry) and polar phonons (which reduce point symmetry) is challenging. Unlike their conventional (proper) counterparts which have well understood stabilization mechanisms for the polar phonon—dpσ -bond formation and/ or stereo-chemical lone-pair activity—tailoring anharmonic interactions to have a soft Q-related phonon coupled to η need not have an obvious chemical prescription. Despite this complexity, however, there exists a larger class of non-polar modes capable of producing a polarization through anharmonic interactions which we review shortly. Determining how to induce this coupling between one or two zone-boundary modes and a set of polar displacements by selecting suitable cations is key to designing new magnetic ferroelectrics. However, we should make note that some forms of order parameter coupling that, while possible, are not examined and fall outside the scope of this Topical Review. First, the presence of a bilinear coupling of the form η Q (m = n = 1) implies that the ferroelectric transition could occur without a change in the size of the unit cell, in contrast to the zoneboundary origin of Q proposed previously. While this has been observed in magnetic or order-disorder driven ferroelectrics (such as the spiral phase of TbMnO3 [51, 52] or Ca9Fe(PO4)7 [53]), we do not consider these ‘pseudo-proper’ ferroelectrics here. Furthermore, if the lowest order coupling is a (negative) biquadratic η2Q 2 term, the ferroelectric transition is known as ‘triggered’ [54]; here, η is not induced solely by Q, but rather Q can be activated, and hence ‘triggered’, by some other instability (for examples and a more detailed treatment of triggered phase transitions, see [55–57]). As the review is not intended to be exhaustive, we have selected to omit discussion of such compounds. The energetics of a paralectric to improper ferroelectric transition are shown in figure 1(b). Here, the energy minima

Figure 2. Out-of-center distortions of transition metal cations in octahedral coordinations occur along the [0 0 1], [1 1 0] or [1 1 1] directions and reduce the point symmetry to C 4v, C 2v and C 3v, yielding acentric polyhedra.

out-of-center of their O6 cages with an accompanied displacement of oxygen anions, stabilized by an energy lowering hybridization of the d 0 transition metal (Ti 4 +) with the oxygen p-states as discussed previously. As a function of decreasing temperature, BaTiO3 undergoes a series of phase transitions from centrosymmetric cubic to polar tetragonal, orthorhombic, and rhombohedral, with the Ti displacements giving polarizations along the [0 0 1], [1 1 0], and [1 1 1] crystallographic directions, respectively (figure 2) [41, 42]. Cations containing a stereo-active lone pair (e.g. Pb 2 + or Bi 3 +) also allow for distortions of this nature. For example, replacing Ba with Pb in PbTiO3 results in an enhancement of the ferroelectric polarization due to the off-centering of the lone-pair active Pb cations supplementing the Ti displacements. Additionally, this lone-pair mechanism produces the extremely large polarization in BiFeO3, with a hybridization of bismuth 6p and oxygen 2p states driving the Bi atoms off-center. The fact that this mechanism is limited to specific electronic configurations, however, severely limits the possible chemistries of ferroelectric compounds, motivating the search for other means to lift inversion symmetry. Nonetheless, this knowledge has subsequently been used to produce dramatic enhancements in heteroepitaxial (proper) ferroelectricity [43–45] and spin-phonon coupling [46] by linking the macroscopic electric polarization with tetragonal lattice distortions that soften the polar displacements in coherent thin films. Finally, this is not to say that proper geometric ferroelectrics without SOJT-ions do not exist; the family of BaMF4 fluorides (M = Mn, Fe, Co, Ni), for example, develop a polarization via a zone-center mode consisting of fluorine polyhedral rotations occurring alongside polar displacements of Ba A-sites with minimal if any changes in chemical bonding [47]. In contrast to the ‘geometric’ ferroelectrics presented here, however, the instability in these proper fluorides is largely driven by ionic size effects [47–49]. Nevertheless, it is much more likely to find non-polar distortions as a zone-boundary phonon mode, hence giving rise to an improper ferroelectric. 2.3. Improper ferroelectrics

In contrast to proper ferroelectrics, improper ferroelectrics exhibit a transition with an order parameter Q that describes some non-polar (in the sense that it does not alone produce a spontaneous electric polarization) atomic distortion pattern; it thus has a physical meaning other than that of a polar IR-active 5

Topical Review


shifts to η, losing the double well potential characteristic of a proper transition. This is because the spontaneous polarization is often dynamically stable with respect to the centrosymmetric structure, and will only arise as a result of some other distortion. This then means that in order to switch the minima to −η, the entire sense of the non-polar distortion Q must be reversed. Finally, the presence of a coupling term in the free energy is not sufficient to declare a material an improper ferroelectric; the energy gain from having both of the distortions present must be greater than either individually. 2.4. Hybrid improper ferroelectrics

Recent reports have identified a new type of rotationallydriven mechanism capable of producing polarizations, known as hybrid improper ferroelectricity [32]. In such materials, equation (2) is extended to include multiple, equally important primary Q order parameters. Polarizations arise in these materials from a symmetry allowed trilinear coupling term in the free energy, given by, for example, F = γQ1Q2η, where Q1 and Q2 are octahedral rotation modes of different symmetry. Microscopically, the non-polar lattice modes serve to remove inversion centers present in the paraelectric structure, and are often rotations or other structural displacements which distort the polyhedral networks in the crystal structure. An important note related to the nature of the transition is that it also depends on the critical temperature at which the two non-polar modes responsible for the appearance of P occur. The two modes can condense at the same temperature (an ‘avalanche’ transition [58]), or they can condense at different temperatures (a ‘staggered’ transition [59]). While an avalanche transition would be a true hybrid improper ferroelectric phase transition, a staggered transition may not necessarily be one. When the first mode condenses, a change of the symmetry or size of the unit cell could mix the remaining mode together with the polar mode at the zone-center of the new phase; the onset of this new mode would actually be a proper ferroelectric transition following the previously discussed definition. However, because this is somewhat different than the proper ferroelectric transition in BaTiO3, such materials have previously been termed ‘weakly polar’ [60, 61]. While in this review we make no distinction between a hybrid improper or weakly polar phase transition when discussing ferroelectricity arising from a trilinear coupling of modes, the existence of this phenomenon is important to understand during the experimental investigations of these materials as it imposes constraints on the temperature-dependent dielectric response.

Figure 3. Depiction of (a) proper rotations and (b) pseudo-rotations in a three dimensional network of corner connected MO6 octahedra. A transition metal (blue/green) sits in the center of the octahedra, and is coordinated by six oxygen atoms (red).

coordination of a metal (M) cation by a varying number of oxygen or halide anions (X), giving rise to the distinct basic building blocks present across various structures. These then take on a number of different regular geometries, including MX3 trigonal planes, MX4 tetrahedra, MX5 trigonal bipyramids, and MX6 octahedra; the materials we discuss here will contain one or more of these units. Furthermore, the connectivity of the building blocks should also be considered. The polyhedral units can be connected via corners, edges, or faces, as well as completely disconnected; the connectivity can also extend through one, two, or three dimensions. The ferroelectric transitions in the materials we classify and discuss in the next section are then driven by the cooperative distortions, or generalized rotations, of these networks. The rotations of these polyhedral networks can take a variety of forms depending on the shape and connectivity, but in general can be divided into either regular or pseudo-rotations. A regular rotation (or proper rotation, not to be confused with a proper phase transition) is one which cooperatively rotates the connected polyhedra while retaining their shape; it can occur in different ways depending on how the various units making up the network are arranged relative to one another. One of the most common examples of these are the in-phase or out-of-phase rotations of MO6 octahedra in perovskite oxides often assigned to a Glazer tilt pattern [62] (figure 3(a)). Here, the initial and final state (left and right structures, figure 3(a)) and each intermediate point are connected by a rotation to the reference state (middle, figure 3(a)) without breaking chemical connectivity. A pseudo-rotational mode describes a somewhat more complex ‘interchange’ of bonds which gives the appearance of rotated polyhedra, rather than a rigid shift of the bonds as in proper rotations. This changing of the chemical connectivity is the familiar Berry pseudo-rotation mechanism [63], which most famously occurs in the PF5 molecule. Within the classification presented here, the primary type of pseudo-rotation considered is the first-order Jahn–Teller distortion; here, an

3. Abilities of lattice instabilities Crystalline materials’ structures can be thought of as continuous networks of polyhedral building blocks, such as octahedra in ABO3 perovskites or tetrahedra in diamond lattices. Because rotations of these units are ubiquitous in solid state materials, especially oxides, we use them as the basis in the subsequent geometric classification of ferroelectrics. The shapes of these polyhedra are most often formed by the 6

Topical Review


Figure 4. Geometric improper ferroelectrics can be divided into Type I or II, based on whether the polarization is coupled to one or two zone-boundary modes, respectively. Additionally, these modes can be classified as proper (A) or pseudo (B) rotational modes.

Figure 5. The typical structure of a M3B7O13X boracite. A corner connected BO4 and BO3 network (a) exists alongside connected MO4 X2 polyhedra (b).

Table 1. Table detailing the types of modes present in each type of

geometric improper ferroelectric.

4.2. Type A

Type

Mode 1

Mode 2

IA IB IIAA

Rotation Pseudo-rotation Rotation

— — Rotation

IIAB IIBB

Rotation Pseudo-rotation

Pseudo-rotation Pseudo-rotation

Type A compounds contain a single proper rotational mode which couples to a polar mode and stabilizes the ferroelectric state. As an example of how geometric improper ferroelectricity can easily coexist with magnetism, we first consider the family of boracites, which have the general formula M3B7O13X; here M is a divalent metal and X is a halogen. The structure consists of a corner-connected network of BO4 tetrahedra and BO3 sheets (figure 5(a)), with highly distorted MO4X2 octahedra forming a second network along the [1 1 0] direction (figure 5(b)). This is one of the oldest and most expansive family of ferroelectrics, with the mineral boracite (Mg3B7O13 Cl) first discovered in the late 1700s [64], and piezoelectricity found in the late 1800s [65]. Indeed, its low temperature crystal structure was determined to be polar Pca21 in the 1950s [66]. Eventually, the compound Ni3B7O13Cl was shown to be ferroelectric [67], and shortly after Ni3B7O13I (which exhibits weak ferromagnetic ordering on the Ni sublattice) became the first identified magnetic ferroelectric [68, 69]. Since then, the members of this family have expanded to encompass a range of chemistries, including M = Mg, Cd, Cr, Mn, Fe, Co, Ni, Cu, and Zn and X = F, Cl, Br, I or even OH or NO3 (see [70–72] and references therein). These compounds all display a paraelectric (non-polar ¯ m cubic phase at high temperature, and undergo achiral) F 43 a phase transition to orthorhombic polar Pca21; certain members also undergo further phase transformations to monoclinic Pb and rhombohedral R3c [73]. The microstructure of these materials is often complex, with many phases displaying twinning and ferroelectric domains [74, 75]. Nonetheless, the primary order parameter driving the transition is clear; at the atomic scale it consists of a rotation of the B-O framework, which is described as a zone-boundary mode given by the irreducible representation X5 [24, 76]. This is coupled to an IR-active Γ4 mode, which produces a spontaneous polarization via displacements of the M atoms along the [0 0 1] direction— the magnitude of which can range from 0.002 to 5 μC cm −2 [77]. Finally, the boracites display several other interesting properties, including strong magnetoelectric coupling (due to the ferroelectricity and magnetism arising from the same atoms) and ferroelastic effects [78]. The hexagonal manganates form another family of magnetic ferroelectrics. The prototypical member, YMnO3, consists

exchange of the short and long bonds and distortion of the anion environment gives the appearance of rotated polyhedra (figure 3(b)). We introduce this distinction because the atomic displacements induced by the proper and pseudo-rotations affect the lattice in unique ways and originate from different underlying structural chemistry mechanisms. We propose that such understanding will facilitate the realization of new multiferroics; for example, this may be the case in the less well-studied class of ferroelectrics in which the spontaneous polarization is induced by pseudo-rotations. 4. Theoretical design of new multiferroics 4.1. Classification scheme

Many geometric improper ferroelectrics may be classified by dividing the materials into two variants based on the number and symmetry of modes that lead to inversion symmetry lifting; that is, the modes active in the transition and appearing in the free energy expansion. The scheme is depicted in figure 4. Type I improper ferroelectrics have one primary mode coupled to a polar IR-mode, whereas Type II distinguishes improper ferroelectrics with two active Q modes linked to a polar mode. A literal ‘A’ or a ‘B’ is then appended to specify whether the microscopic mode(s) involved in describing the symmetry change are rotational or pseudo-rotational, respectively. This classification method results in five categories of improper ferroelectrics that are summarized in table 1. In the next sections, we show how previously identified improper ferroelectrics can be classified in this way. Note that we have not intended to be completely exhaustive, but rather choose illustrative examples of how one can apply the classification scheme to understand the atomistic displacements responsible for lifting inversion symmetry. 7

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Figure 7. The R2(MoO4)3 structure consists of disconnected MoO4

tetrahedra (gray) interspersed with R atoms (purple, R = Tb, Gd, ¯ 1m structure (a) undergoes a transition Sm). The high symmetry P 42 to the ferroelectric Pba2 phase (b) upon condensation of M2M4 and Γ3 modes via a Type A improper mechanism.

Figure 6. The structure of YMnO3 consists of disconnected planes of MnO5 polyhedra (purple) alternating with layers of Y atoms (blue). The ferroelectric transition from P 63/mmc (a) to P 63cm (b) occurs via a zone-boundary K3 mode describing the crumpling of MnF5 planes and non-polar Y displacements coupling to a polar Γ−2 mode.

(by substituting larger rare-earth atoms such as Ce, Pr, and Nd) and epitaxial strain [98]. We also mention that there are several isomorphic halides that may be classified as Type A owing to the same displacement modes found in the hexagonal transition metal oxides, including TlFeBr3, RbMnBr3, TlCoCl3, and KNiCl3 [99, 100]. Finally, hexagonal BaMnO3 also undergoes the same P 63/mmc to P 63cm transition, despite having a different crystal structure (face sharing rather than corner-sharing polyhedra) [101]; indeed, first-principles calculations have shown that the same K3 and Γ−2 modes are responsible for this transition [102]. This microscopic understanding of the hexagonal manganate structure has led to the design of novel ferroelectric materials, both theoretically (RGaO3 and RInO3) [103] and experimentally (YbFeO3) [104]. We next turn our attention to the rare-earth molybdates with chemical formula R2(MoO4)3 (where R = Tb, Gd, or Sm) [105–107], which also exhibit ferroelectricity of this nature [108]. In the paraelectric phase, these oxides exhibit the non¯ 1m space group, and consist of polar non-centrosymmetric P 42 R atoms surrounded by completely disconnected MoO4 tetrahedra (figure 7(a)). Upon condensation of an unstable phonon mode at the M-point (described as M2M4 with respect to the ¯ 1m structure), which manifests as cooperative rotacentric P 42 tions of the MoO4 units, the structures reduce to orthorhombic Pba2 (figure 7(b)), with a corresponding appearance of a spontaneous polarization along the c axis (described by the polar mode Γ3) [100]. The polarization arises via displacements of the Gd cations and MoO4 tetrahedral lattice; the fact that the anti-parallel motions of the Gd atoms mostly cancel, however, means the polarization is only ∼0.2 μC cm −2 [109]. The rare earth atoms also order antiferromagnetically at extremely low temperatures (TN = 0.3 K for Gd and 0.4 K for Tb) [110], suggesting a coupling should exist between the polarization and magnetism; interestingly, a nonlinear magnetoelectric effect was indeed found in these compounds [107, 110], even at temperatures far above TN (∼300 K) [111, 108]. This symmetry analysis has shown that these compounds can also be considered Type A improper ferroelectrics despite having a completely different crystal structure than the boracites

of layers of corner-connected MnO5 bipyramids and exhibits antiferromagnetic ordering of the Mn atoms below 75 K [79, 80]. Upon transition from centrosymmetric P 63/mmc (figure 6(a)) to polar P 63cm (figure 6(b)) at 1260 K, a spontaneous polarization appears owing to a unit cell doubling tilting of MnO5 polyhedra (given by irrep K3) and a ‘crumpling’ of YO planes (irrep Γ−2 ) [81–83]. This ferroelectric phase transition has been extensively studied experimentally, and its exact nature is quite complex and debated [84–88]. First-principles calculations have shown that while the K3 mode is soft and the Γ−2 mode is hard in the phonon band structure of centric P 63/mmc YMnO3, a coupling of the two modes in the free energy expansion of the form F = ... + α1QK33Q Γ−2 + βQK23Q Γ2−2 results in a shift of the Γ−2 mode to a non-zero equilibrium value upon condensation of K3 [89]. Based on the scheme presented here, YMnO3 is then classified as a Type A compound. Although YMnO3 is the prototype of the RMnO3 hexagonal manganates, many of the other members display Type A improper ferroelectricity resulting from the same set of atomic distortions, including R = Ho, Er, Tm, Yb, and Lu [90, 91], albeit with different magnitude. The isostructural hexagonal ferrites (RFeO3) would also be classified as Type A improper ferroelectrics within this scheme. Several members of this family have been synthesized experimentally, such as R = Eu, Er, Tm, Yb, and Lu [92]. Of these, LuFeO3 holds particular promise due to the fact that that the Fe spins magnetically order at a much higher temperature than any of the hexagonal manganites (TN = 440 K), with a net magnetization appearing at 120 K due to canting [93–95]. Recent results have thrown these reported magnetic temperatures into question, however, with the antiferromagnetic ordering only found up to 155 K [96]. Additionally, recent first-principles studies have shown that the hexagonal ferrates display a bulk linear magnetoelectric effect [97], and that it is theoretically possible to increase the polarization by up to 60% through both chemical pressure 8

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Γ2 mode, producing a small net polarization of 0.08 μC cm −2 from an anharmonic interaction [124, 127, 128]. Substitution of 3d transition metals which can easily accommodate tetrahedral coordination, such as Cr, could provide routes towards inducing magnetic ordering in this family. A similar series of transitions occurs in the mineral family of sulfate langbeinites, which have the general formula A2 B2(SO4)3 and consist of a network of disconnected SO4 tetrahedra. Most of the compounds, including the specific mineral langbeinite [K2Mn2(SO4)3], undergo a phase transition from a high temperature P213 phase to a low temperature P212121 phase, both of which are non-polar space groups [129, 130]. However, several members, such as Tl2Cd2(SO4)3 [131], Rb2Cd2(SO4)3 [132], and (NH4)2Cd2(SO4)3 [133, 134], transition to the intermediate monoclinic P21, then triclinic P1 structures, before the P212121 ground state. Note that both intermediate phases are also ferroelectric. As with the stuffed tridymites, rotations of the SO4 tetrahedra described by M-point phonons induce the ferroelectric phase transition. Finally, the compound RbH3(SeO3)2 undergoes a similar P212121 to P21 ferroelectric transition [135]. The structure consists of SeO3 units connected through O-H-O bridges. At 158 K, a phonon mode describing twisting of these units condenses at the Z-point, driving the ferroelectric phase transition [136]. Like the previous two families, the spontaneous polarization is also small in this material, with a magnitude of only 13 nC cm −2 [137]. Interestingly, the Rb member of this family is the only improper ferroelectric; the Li member is a proper ferroelectric, while the Na member is always centrosymmetric. From the analysis to here, we conclude that ferroelectricity induced by cooperative rotations of anion polyhedra via anharmonic interactions is ubiquitous across different structure types and chemistries. It is a promising route towards the discovery of new multiferroic and magnetoelectric materials. Furthermore, knowledge of the atomistic mechanisms responsible for the appearance of the spontaneous polarization can help guide discovery of new compounds (such as by chemical substitution in some of the described structure types) from both an experimental and theoretical approach.

Figure 8. The prototypical A2 BX 4 structure consists of chains of A-sites (purple) and disconnected BX 4 tetrahedra (green). These compounds undergo a phase transition from high-symmetry Pnma (a) to ferroelectric Pna21 (b) via a Σ 2 mode coupling to a polar Γ−4 mode resulting in a spontaneous polarization along c.

or hexagonal manganates. We hypothesize that complete or partial chemical substitution of Mo with a 3d row transition metal element is a possible direction for discovering new multiferroics or enhancing the existing magnetic properties. To the best of our knowledge, no full or partial chemical substitutions have been attempted. A fourth well-known family that can be classified as Type A improper ferroelectrics are those with the chemical formula A2 BX 4, the structure of which is illustrated in figure 8. The prototypical member of this family is K2SeO4. Like Gd2(MoO4)3, the crystal structure consists of disconnected SeO4 tetrahedra; however, the ratio of A-site atoms to tetrahedra per unit cell is greater than in the molybdates. These compounds are also unique in that the members undergo a phase transition from paraelectric Pnma to ferroelectric Pna21 via an intermediate incommensurate phase [112–114] driven by a non-polar zone-boundary mode Σ 2 (describing out-ofphase rotations of BO4 tetrahedra) [115], which is coupled to polar displacements described by Γ−4 [116, 117]. The A2 BX 4 family of improper ferroelectrics includes compounds with vastly different chemistries, including oxides (K2SeO4), halides (Rb2ZnCl4, Rb2ZnBr4, K2ZnCl4), and molecular solids ([N(CH4)3]2ZnCl4, (NH4)2BeF4) [118–122]. Most of the compounds discussed so far undergo a ferroelectric transition into a polar structure from a paraelectric phase containing inversion symmetry. However, this is not a required condition for the appearance of an improper polarization; the paraelectric structure may also be non-polar noncentrosymmetric. For example, the stuffed tridymite BaAl2O4 displays a high symmetry P6322 paraelectric phase [123–125], i.e. a space group that is both non-polar and lacking inversion symmetry [14]. This material consists of corner-connected AlO4 tetrahedra—the SiO2 tridymite structure—with Ba cations located in the space between tetrahedra. At 396 K, a soft phonon mode describing tilting of the AlO4 tetrahedra condenses at the M-point (irrep M2), driving it to the P63 ferroelectric phase [126]. The spontaneous polarization in this structure occurs due to coupling of the M2 mode to a polar

4.3. Type B

The Type B compounds are similar in nature to Type A in that one unstable mode drives the symmetry lowering transition; rather than being a rotational mode, however, it is pseudorotational. Our search reveals that it is difficult to locate materials which contain only one pseudo-rotational mode coupled to a polar mode that induces an electric polarization. We suppose this occurs for a variety of reasons. Although a first-order Jahn–Teller (FOJT) effect is in principle ‘simple’ to achieve through selection of the appropriate transition metal chemistry, this type of this distortion alone is insufficient to lift inversion symmetry in a three-dimensional crystal with corner-connected octahedra [138, 139]. Additionally, although Berry pseudo-rotations are relatively common in organic compounds [140] and molecules such as PF5 [141] or pentacoordinate silicates [142], they are extremely rare, if they occur at all, in crystalline solids with polyhedral networks. 9

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Although our search of the literature revealed no Type B ferroelectrics, we offer some suggestions as to how to locate or purposefully engineer polar phases of this nature for multiferroic design. First, control over the topology of a material could be utilized in combination with the FOJT effect to produce a polar structure. For example, if polyhedra exhibiting FOJT distortions were ordered in such a way that their relative positions cooperatively lift inversion, ferroelectricity could be induced despite the site symmetry of the metal center within the polyhedra itself maintaining inversion symmetry. Second, crystalline solids containing organic networks (such as metal-organic frameworks discussed in section 4.5) could be a promising avenue for Type B ferroelectrics induced by Berry pseudo-rotations as they should also be compatible with magnetically ordered cations. Finally, compounds which contain molecular rotations are also viable candidate materials. The compound lawsonite (CaAl2Si2O7(OH)2⋅H2O), for example, displays ‘crumpling’ of the H2O and OH molecules (described by Y2− mode and driven by steric instabilities) in addition to ferroelectricity [143]; the polarization, however, is actually induced via an order-disorder mechanism. Despite the rarity of these distortions in crystalline solids, their discovery and investigation could open a whole new area of geometric improper ferroelectrics.

Figure 9. Mode decomposition of an ABi2B2O9 (a) paraelectric to

(b) ferroelectric phase transition. In the n = 2 Aurivillius structure, distortions of the Bi2O2 layer couple to out-of-phase rotations of the BO6 octahedra as well as polar displacements to produce a spontaneous polarization. The atoms are colored as follows: A (green), Bi (purple), B (brown), O (red).

a Bi2O2 layer is inserted between every n ABO3 perovskite blocks [147]. The polar displacements in this material are driven by only one rotational mode (irrep X3−), rather than two as in the Ruddlesden–Popper or perovskite families. Interestingly, the rotation mode condenses simultaneously with the Eu polar mode via a proposed avalanche transition, with the appearance of a displacement pattern that describes the shifting of the atoms in the Bi2O2 layers (irrep X2+) [58]. This is quite unique. Rather than two non-polar modes inducing a polarization, the polar mode acts at the primary order parameter with the zone-boundary mode serving as the secondary order parameter; while this does not make the Aurivillius family a Type AA improper ferroelectric in the strictest sense of our definition, we still find it prudent to include this discussion. It provides an example to illustrate that the form of the coupling requires the condensation of multiple modes; however, knowing the form of the coupling alone is insufficient to identify the displacement mode active in the transition. That understanding comes from a lattice dynamical calculation or through careful experimentation, e.g. a temperature-dependent Raman study [148]. Figure 9 shows how these distortions take SrBi2Nb2O9 from its high symmetry I 4/mmm phase (figure 9(a)) to its ferroelectric A21am phase (figure 9(b)). Other members of the family also display ferroelectricity, such as n = 1 Bi2WO6 [149] and n = 3 Bi4Ti3O12 [150]. In order to understand the features characteristic of the Type AA ferroelectrics within this scheme that can facilitate multiferroic design, we now examine how this phenomenon arises in Ruddlesden–Popper and perovskite compounds in detail. The Ruddlesden–Popper structure has the general formula (AO)( ABO3)n; it consists of ABO3 units arranged along the [0 0 1] crystallographic axis with an extra AO layer inserted between every n units [151, 152]. The n = 1 and n = 2 members of this family are illustrated in figures 10(a) and (b). When n → ∞, the ABO3 perovskite results (figure 10(c)). Because

4.4. Type AA

In Type AA improper ferroelectrics, the polarization arises as a result of two rotational modes. This mechanism was first predicted by density functional theory calculations in the Aurivillius compound SrBi2Nb2O9, where it was found that the ferroelectric A21am ground state is a result of a coupling between three displacive modes: an octahedral tilting mode, a mode consisting of oxygen movements in the Bi2O2 layers, and a polar mode [144]. This result was followed by the discovery of ferroelectricity in 1:1 layered artificial superlattices of the perovskite oxides PbTiO3/SrTiO3, in which an in-phase and an out-of-phase rotational mode couple to a polar mode to produce an additional contribution to the total spontaneous polarization [22]. Further first-principles investigations revealed that, similarly, two rotational modes can also lead to a ferroelectric polarization in Ruddlesden–Popper phases via coupling to a polar mode [32]; it was here that this phenomenon was first dubbed ‘hybrid improper’. Since then, this type of ferroelectricity has been predicted and found in a wide variety of material families, most predominately layered oxides [145]. Because Type AA ferroelectricity is driven primarily by rigid rotations of anion polyhedra, substitutions of magnetically active ions can be made on the cation sites without destroying the spontaneous polarization—we describe this procedure in detail below. However, we first examine the aforementioned ferroelectric SrBi2Nb2O9 to show that while a trilinear coupling may exist in a material, the question of which distortions drive the phase transition is not always immediately apparent, requiring detailed experimental and theoretical study. SrBi2Nb2O9 is an n = 2 member of the Aurivillius family [146] with general formula (Bi2O2)( An − 1Bn O3n + 1); here, 10

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Figure 10. Crystal structure of an (AO)(ABO3)n Ruddlesden–

Popper phase with (a) n = 1, (b) n = 2, and (c) n = ∞. When n = ∞, the ABO3 perovskite structure is obtained. Both Ruddlesden–Popper and perovskite oxides consist of a corner connected network of 6-fold anion (red) coordinated B-sites (purple) with A-site cations (blue) residing in the interstices.

Figure 12. The A-site displacements responsible for the

spontaneous polarization in an (a) n = 2 Ruddlesden–Popper and (b) layered perovskite oxide phase.

In the case of perovskites, however, odd-period ABO3/A′BO3 superlattices with layered A-site cation ordering along the [0 0 1]-direction is also required to lift the necessary symmetries. Note that ultrashort period 1/1 superlattices are equivalent to ( A, A′)B2O6 perovskites. The absence of such additional chemical ordering would prohibit the lattice anharmonicities from lifting inversion symmetry [31]. (Other ordering directions for the A and B cations can also yield the same effect, as noted below.) In the n = even Ruddlesden–Popper compounds, this ‘topological’ requirement is satisfied by the inserted AO layer and spatially disconnected two-dimensional perovskite sheets. The paraelectric Ruddlesden–Popper or layered perovskite structure (figure 11(a)) is then reduced to its ferroelectric ground state (figure 11(b)) via this trilinear coupling. In both cases, the materials exhibit anti-aligned A cation displacements; the spontaneous polarization then arises as a consequence of two (chemically) distinct A-sites, preventing a complete cancellation of the oppositely aligned dipoles (figure 12). The fact that the ‘design rules’ summarizing the distortions and cation ordering requirements for lifting inversion and activating an electric polarization in Ruddlesden–Popper and perovskite structures are essentially chemistry-independent [31, 33, 154] opens a large discovery space for the creation of new multiferroics. Additionally, cation ordering along alternative directions in perovskites allows different Glazer rotation patterns to lift inversion; tilts consisting of either two modes (such as a+b−b−) or one mode (such as a0b−b−) can induce improper ferroelectricity [155]. This atomistic understanding has led to a host of new Type AA ferroelectric Ruddlesden– Popper and perovskite compounds. For example, in addition to perovskite oxides, the layered perovskite fluorides (Na,Rb) Hg2F6 and (K,Rb)Hg2F6 have been predicted to display a spontaneous polarization arising from the anharmonic lattice interaction coupling two rotational modes [156]; while fluorides have long been known to be ferroelectric [157], it is very often due to a proper mechanism (such as in BaMF4

Figure 11. Mode decomposition of the (a) paraelectric to (b)

ferroelectric structures in the (AO)(ABO3)2 Ruddlesden–Popper (top) and layered ( A, A′)B2O6 perovskite oxides (bottom). In both cases, in-phase and out-of-phase rotations of the BO6 octahedral network couple to A-site polar displacement through a trilinear term, producing an electric polarization.

these structures are closely related, it should be no surprise that the ‘improper’ polarizations in both compounds can arise from similar structural distortions. As mentioned previously, one requirement is the presence of both in-phase and out-ofphase rotations of the BO6 octahedral network (an a+b−b− tilt pattern in Glazer notation); the atomic displacement mode producing in-phase rotations transforms like the irreducible representation X2+ or M3+ for the Ruddlesden–Popper or perovskite structure, respectively, and that for out-of-phase rotations is given by X3− or R4+ (figure 11). These modes then may couple anharmonically to the polar A-site displacement mode Γ−5 of the cation ordered perovskite (or equivalently to the nominally antipolar X5+ mode of the unordered centrosymmetric ABO3 structure [153]). 11

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calculations predict the presence of a rotationally-induced hybrid improper spontaneous polarization of 16 μC cm −2 in the P21 ground state [30]. Additionally, it was recently shown that substitution of smaller rare earth cations on the A′-site can significantly increase the polarization while retaining the B-site magnetic order [176]. Research on these compounds is still ongoing, however, as experimental measurements on polycrystalline samples of (Na,La)(Mn,W)O6 and (Na,Nd) (Mn,W)O6 have failed to detect the electric polarization [177]. Although the previous discussion has focused exclusively on Ruddlesden–Popper and perovskite compounds, other structural families may be classified as Type AA ferroelectrics. The Dion-Jacobson compounds (general formula A[ An′− 1Bn O3n + 1]) have also recently garnered attention as hybrid improper ferroelectrics, as several members are known to be polar, such as CsBiNb2O7 and CsNdNb2O7 [178, 179]; advances in ion-exchange synthesis has also led to the identification of many new phases [180]. While the polarization is also driven by in-phase and out-of-phase rotations (described by the irreps M2+/M3+ and M5−, respectively) as in the perovskites, the A-site atomic displacements comprising the polar mode (Γ−5 ) are ferro- rather than anti-distortive, resulting in much larger electric polarizations. The n = 2 compounds CsBiNb2O7, RbBiNb2O7, and RbNdNb2O7, for example, have been predicted to show spontaneous polarizations of 40, 36, and 24 μC cm −2, respectively [181–183]. Additionally, the main contribution to the polarization in this family is from the B-site, but is driven by A-site bonding preferences (which provide a somewhat large contribution as well) [184]. Recent experimental results have confirmed the presence of ferroelectricity and piezoelectricity in polar RbBiNb2O7 and CsBiNb2O7 [182, 185], while theoretical studies have shown the potential for enhancement of the piezoelectric response under strain [186]. We summarize this section by noting that layered oxides (and halides) provide an open ‘sand box’ for the discovery of new Type AA (‘hybrid improper’) ferroelectrics with magnetic cations. Although the spontaneous polarizations are produced via different atomic mechanisms, they broadly rely on the anharmonic coupling of two non-polar distortion modes to a polar mode of the crystal.

compounds [47]). It is also important to mention that these criteria can be circumvented in some cases when non-polar ABO3 perovskites are placed under sufficient epitaxial strain, for example in CaTiO3 [158]. Density functional theory (DFT) calculations provide an especially fruitful route to predict new materials without inversion symmetry, as they allow for high throughput testing of a large numbers of chemistries and the evaluation of relative phase stability from the calculation of formation energies. Recent studies to this end, for example, have predicted several new ferroelectrics in the Ruddlesden–Popper and perovskite families [153, 159, 160]. Additionally, because the polarization is A-site driven, B-site substitution with magnetic cations makes it possible to retain ferroelectricity and achieve the magnetic order needed for multiferroism. Consider the compound Ca3Ti2O7, which naturally exists as a polar n = 2 Ruddlesden– Popper compound [161] and was predicted to be a Type AA ferroelectric with an electric polarization of 20 μC cm −2 [32]. Recently, this was confirmed experimentally in Sr-doped phases of Ca3Ti2O7, where a spontaneous polarization was measured (although smaller than the predicted value of 8 μ C cm −2) and successfully switched [162]. Antiferromagnetic ordering of the B-sites can be induced by substituting the Ti atoms with Mn to create isostructural Ca3Mn2O7 [163, 164], which is also predicted to display a small spontaneous polarization of 5 μC cm −2 [32, 33] and found to have a large uniaxial negative thermal expansion [165]. Indeed, these theoretical developments led to synthesis of (CaySr1 −y)1.15Tb1.85Fe2O7, an n = 2 Ruddlesden–Popper phase displaying coexisting electric polarization and weak ferromagnetism at room temperature [166]. This same concept may also be applied to the perovskite family by replacing the non-magnetic B-site in the ordered ( A, A′)B2O6 superlattices mentioned previously. For example, first-principles calculations have predicted the coexistence of magnetism and ferroelectricity in a series of (La,Ln)Fe2O6 ordered perovskite superlattices (where Ln = Ce, Nd, Sm, Gd, Dy, Tm, Lu, Y) [167]. These calculations also predict a significant degree of coupling between the two ferroic properties, giving a design strategy for both multiferroicity and magnetoelectricity. Finally, the family of ( A, A′)(B, B′)O6 double perovskites, which spontaneously exhibit layered A-site and rock salt ordered B-sites [168], are ripe for magnetic cation substitution as well as realizing Type AA ferroelectricity. Similar to the scenario with A-site ordering alone, this type of ‘double’ cation order lifts inversion symmetry in combination with an a+b−b− tilt pattern. These compounds most often contain an alkali metal (such as Na or K) and a rare earth cation (such as La, Nd, or Tb) on the A and A′ site, and have been experimentally synthesized with a wide array of different B and B′ cations. While many of the potential B-sites are non-magnetic, such as Mg, Sc, Ti, Te, W, and Nb, double perovskites with magnetically active cations such as Mn, Co, Ni, and Nb have also been experimentally realized [169–174]. The compound (Na,La)(Mn,W)O6, for example, exhibits G-type antiferromagnetic ordering of the Mn 2 + cations below 10 K and the required a+b−b− tilt pattern [175], while first-principles DFT

4.5. Type AB

Materials exhibiting improper ferroelectricity and containing pseudo-rotational modes could also be considered ‘hybrid’. We classify such compounds as Type AB or BB. Here, we will consider pseudo-rotations produced by first-order Jahn–Teller (FOJT) distortions (see section 3). Like the second-order Jahn–Teller (SOJT) effect discussed in section 1, the appearance of this phenomenon depends on the d electron configuration of a metal cation; however, the FOTJ effect can occur in a wider range of d n configurations. Furthermore, while the SOJT effect itself is responsible for the spontaneous polarization in compounds such as BaTiO3, the FOJT effect cannot drive a ferroelectric phase transition alone, as any distortion of this type preserves inversion symmetry. Finally, although the criteria needed for the appearance of the SOJT effect (i.e. 12

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and co-workers showed using data-mining techniques that out of 105 n = 1 compounds known at the time, only two exhibited a non-centrosymmetric space group [154]. Using the crystalchemistry guidelines put forth from the information analyses, first-principles calculations were used to investigate the A-site ordered n = 1 Ruddlesden–Popper phase (La,Sr)MnO4, which is predicted to have a polar Pca21 ground state displaying a ferroelectric polarization of 1.25 μC cm −2 [188]. This transition occurs via coupling of a Jahn–Teller distortion of the Mn 2 + cations (which transforms as the irreducible representation X2+) and MnO6 octahedral rotations (irrep. X3+) with polar displacements of the A-sites. Additionally, the Mn atoms exhibit an A-type antiferromagnetic ordering with a weak-ferromagnetic component, making this material an interesting magnetoelectric multiferroic for experimentation. The wide variety of firstorder Jahn–Teller active cations capable of being substituted on the B-site make this an attractive route to the discovery of more new Type AB ferroelectrics. The second class of materials which can be classified as Type AB ferroelectrics is the family of metal-organic frameworks (MOFs), especially those with the ABX3 perovskite architecture. These compounds consist of both organic and inorganic components, with transition metal centers coordinated by chains of small organic molecules. Interest in these materials began with the synthesis of a series of formates [C(NH2)3]M[(HCOO)3], where M = Mn, Fe, Co, Ni, Cu, or Zn [37]; each member except M = Zn was found to display spin-canted antiferromagnetic ordering. These structures are similar to perovskites, but rather than the A- and X-sites being occupied by a single cation or anion, they are filled with small organic molecules (here A is the guanidine cation [C(NH2)3] +), B is a transition metal, and X is the formate anion ([COOH] −). Out of these six compounds, one (M = Cu) was thought to be a potential multiferroic due to displaying a polar Pna21 structure. Indeed, first-principles calculations predict the presence of a small polarization of 0.37 μC cm −2 arising from displacements of the NH2 groups [189]. Further investigation of an isostructural Cr MOF (also displaying magnetic order and a small polarization of 0.22 μC cm −2) then showed that the polarization arises due to a hybrid improper mechanism [190]. However, the transition from centrosymmetric Pnna (figure 14(a)) to polar Pna21 (figure 14(b)) occurs via a trilinear coupling (mediated by hydrogen bonding [191]) between rotations of the organic A-site and a Jahn–Teller distortion of the transition metal B-site to induce the polar displacements. Interestingly, the antiferrodistortive order is produced by a combination of two Q2 Jahn–Teller distortions on different crystallographic sites; this ‘switching’ of the short and long bonds is analogous to the Berry pseudorotation mechanism operative in PF5. This family of materials offers the potential for a wide variety of novel magnetic ferroelectrics, as they are highly tunable via both cation substitution and ligand engineering. Different organic molecules can substitute on the A- and X-sites, while various transition metal cations can occupy the B site. For example, the compound [(CH3CH2NH3)]Mn[(COOH)3], where the guanidine molecules have been replaced by ethyl-ammonium while maintaining the presence of a first-order Jahn–Teller

Figure 13. Three types of first-order Jahn–Teller distortions that

an ideal MO6 octahedron can undergo. The apical M-O atoms are extended in each case, but the equatorial bonds can either be (a) all elongated and all compressed (also known as a ‘breathing’ distortion), (b) half elongated and half compressed, or (c) 2/3 compressed and 1/3 elongated and vice versa. These are referred to as a Q1, Q2, or Q3 Jahn–Teller distortion, respectively.

a d 0 transition metal) precludes the appearance of a magnetic moment (with a few exceptions), the FOJT effect has no such restriction. By considering materials in which a spontaneous polarization arises from the coupling of a pseudo-rotation such as a Jahn–Teller distortion and a proper displacive rotational mode, the space for investigation of novel ferroelectrics and multiferroics is further extended. This type of distortion is especially common in six-fold coordinated transition metal environments, so we return our attention to perovskite-derived structures. The corner-connected BO6 octahedral network in these structural families can undergo three different types of Jahn– Teller distortions labeled Q1, Q2, and Q3, in which: (1) all B-O bonds can be elongated or compressed (i.e. ‘breathing’ distortion, figure 13(a)), (2) the apical oxygens can be elongated while two equatorial oxygens are elongated and two are compressed (i.e. ‘d-type’ distortion, figure 13(b)), or (3) all equatorial oxygens can be compressed while the apical oxygens are elongated (i.e. ‘a-type’ distortion, figure 13(c)). A symmetry analysis has revealed no combination of the M2+ or R3+ Jahn–Teller modes with octahedral rotations (described by M3+ or R4+) lifts inversion in perovskites [138, 139], similar to the fact that no combination of the rotational modes M3+ with R4+ result in a polar space group. However, cation ordering may be able to provide a route to lift inversion, as in the case of the Type AA hybrid improper ferroelectrics. Although B-site ordering in combination with Jahn–Teller modes is unable to lift inversion [187] in perovskites, a recent theoretical crystallographic investigation has shown that it is possible via A-site ordering in n = 1 Ruddlesden–Popper phases [154]. This is important because while n = 2 Ruddlesden– Popper phases can display Type AA improper ferroelectricity (such as the aforementioned Ca3Mn2O7), polar space groups are exceedingly rare for the n = 1 RP family; Balachandran 13

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Figure 14. Mode decomposition of the paraelectric (a) to ferroelectric (b) phase transition in the [C(NH2)3]Cr[(HCOO)3] metal-organic framework. Here rotations of the A-site molecules and first-order Jahn–Teller distortions of the CrO6 network couple to polar displacements and drive the structure from centrosymmetric to Pnna to polar Pna21.

active transition metal, also displays a Pna21 space group [192]. First-principles calculations have predicted this material to exhibit an electric polarization of 2 μC cm−2 [193]. Additional substitution of the A-site molecule in this compound with CH3CH2NF3 and PH3CH2NF3 was predicted to further enhance the polarization to 5 μC cm−2 and 6 μC cm−2, respectively. Furthermore, non-collinear spin DFT calculations report that that the collinear antiferromagnetic arrangement exhibits a canted spin structure upon transition to the polar phase, giving rise to a weak ferromagnetic moment; the fact that this ferroelectric transition changes the magnetic properties implies these materials should display strong magnetoelectric coupling [189]. This effect has recently been observed experimentally in the Fe MOF [(CH3)2NH2]Fe[(HCOO)3], where it was found that small electric fields could induce a more than 50% change in the magnetization [194]. Additionally, both electric field control of magnetization and magnetic field control over polarization were realized in a Cu MOF [195]. The ABX3 architecture is not necessarily required for metalorganic frameworks to be Type AB ferroelectrics. For example, the compound (C6H5CH2CH2NH3)2CuCl4 was recently found to exhibit ferromagnetic ordering below 18 K and a small ferroelectric polarization below 4 K on the order of 0.025 μC cm−2 [196]. Although the structure contains a network of octahedrally coordinated Cu 2 + atoms corner-connected in the xy plane, they are separated in the z direction by two layers of the phenylethyl amine organic framework. The ferroelectric phase transition in this compound is driven by a tilting or buckling of the CuCl4 complexes, as well as a Jahn–Teller distortion induced by the Cu 2 + cations, which results in a displacement of the N atoms from their high symmetry positions; this is in contrast to the previously discussed MOFs in which one distortion was a rotation of the organic molecules. As with the previous MOFs, these compounds also offer a high degree of tunability. By replacing the phenylethyl amine molecule by ethyl amine to create (C2H5NH3)2CuCl4, the polarization increases to 18 μC cm −2—an increase of three orders of magnitude [197].

Figure 15. Jahn–Teller distortions associated with the (a) MJT and (b) RJT modes.

arises from the coupling of two pseudo-rotational modes. A focus on materials containing Jahn–Teller active cations appears to be a promising way forward for discovering magnetic Type BB ferroelectrics. Indeed, this type of mechanism has now been predicted from first-principles to arise in the layered vanadate perovskites (La,Y)V2O6 and (La,Pr)V2O6 [198]. While a proper rotationally induced polarization (i.e. a Type AA attribute) also exists in these compounds giving magnitudes of 7.89 ( A′ = Y) and 2.94 μC cm −2 ( A′ = Pr), a second trilinear term of the form F ∝ γMJTRJTPz coupling a polar mode to two distinct types of Jahn–Teller distortions (given by the irreducible representations MJT and RJT, figure 15) arises in the free energy expansion of (La,Pr)V2O6. From this, a small additional spontaneous polarization of 0.34 μC cm −2 is generated perpendicular to the first, owing solely to the Jahn–Teller activity of the V 3 + cations. While electronic structure calculations show the potential for the existence of Type BB ferroelectrics, confirmation of this effect experimentally remains elusive. The understanding obtained from this study, however, indicates a new path towards magnetoelectric multiferroics discovery; the fact that this polarization arises solely from electronic mechanisms allows for stronger coupling to the magnetism in these materials, potentially making Type BB ferroelectrics strong magnetoelectrics. As with Type B improper ferroelectrics, exploring materials with either Jahn–Teller active cations or organic frameworks may prove fruitful in finding new Type BB compounds.

4.6. Type BB

As with Type B improper ferroelectrics, Type BB compounds are fairly difficult to locate as the polarization 14

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5. Prospects and challenges

[8, 9, 210, 211]. For example, the polarization switching in Type A YMnO3 requires only modest electric fields [212]. On the other hand, although some Type AA and AB compounds have been synthesized and determined to be polar, switching of the electric polarization has remained elusive [166]. Recently, however, the Type AA ferroelectric polarization in Ca3Ti2O7 has been reversed [162], albeit the saturated polarization is reduced from the predicted DFT values. Interestingly, isostructural Ca3Mn2O7, which is a candidate multiferroic, has not yet been electrically switched despite comparable calculated energy barriers for the probable switching barriers between the RP titanate and manganate phases. Developing a theory for switching in the various improper ferroelectrics is crucial in realizing technologically useful multiferroics.

The scheme presented allows for the identification of unexplored multiferroic families, followed by targeted materials design based on knowledge of the microscopic structural requirements for electric polarizations. Once these necessary crystal features are established, the design of new materials can be accomplished by an understanding of how and why cations displace; the aforementioned vibronic-coupling theory, for example, offers a description of how electronic structure affects the displacements of ions [199]. Based on this or other crystal-chemistry criteria, one can select appropriate compositions favoring the desired displacement patterns [18, 200]. As an example of the potential design space, consider improper ferroelectrics containing a B-type mode (pseudo-rotations). It is well-known that certain compounds have a tendency to undergo Berry pseudo-rotations (such as those containing MF5 units, or those with hydrogen-bonded frameworks), while others have a tendency to undergo first-order Jahn–Teller distortions (those containing atoms with the required d n electronic configuration leading to orbital degeneracy). Therefore, choosing the chemistry of a structure to specifically contain atoms with a tendency towards those distortions provides a method to engineer new magnetic Type B ferroelectrics. The same argument can also be made for ferroelectrics containing A-type modes, and the use of such a method has indeed resulted in the design of new Type AA compounds from first-principles [159]. It is clear that this classification scheme can allow for the rapid first-principles discovery of new electronic materials, as well as offer many compounds for experimental investigation. Although seemingly straightforward, the application of these theoretical design approaches for realistic materials is not without challenge. First, consider the synthesis challenge. Several of the described compounds rely on incorporation of multiple cations in an ordered fashion into the lattice, and an important aspect for many of the predictions on new compounds relies on evaluating the stability of such phases in the targeted structure type. For bulk systems, first-principles based structure-searching algorithms could be incorporated with the theoretical scheme to down-select the ‘best of class’ compounds for theory guided laboratory synthesis [201–205]. In some scenarios, the predicted phases with an ordered cation sublattice are more suitable for growth via non-equilibrium approaches like molecular beam epitaxy (MBE) or pulsedlaser deposition (PLD), which have proven successful in realizing metastable structures using epitaxial strain stabilization (see, [34, 206–209] and references therein). Second, the magnitude of the electric polarization, ferroelectric Curie temperature, and ease of switching provide a functionality challenge. The multi-mode coupling origin for the electric polarization in the described materials begs the question: To what the extent can large polarizations be achieved, and to what extent can the polarization be reversed (i.e. from +P to −P), which is required for assigning the ‘ferroelectric’ designation to these materials? The latter is also critical towards realizing strong magnetoelectric coupling [167], which is sought to be exploited in a variety of electronic applications, including memory, tunnel junction, spintronic, and microwave devices

6. Summary In this Topical Review, we presented a scheme for classifying displacive improper ferroelectric materials by focusing on the atomistic origin for lifting inversion symmetry. By dividing these materials based on the number and type (symmetry) of the interacting atomic displacement modes, we gleaned that there are multiple routes by which to incorporate magnetic cations as a means to multiferroicity. A main observation is that there are many open-shell magnetic cations that can produce centric lattice distortions by themselves, can also produce spontaneous polarizations in combination with other centric lattice modes, such as octahedral tilting. We believe the scheme is also sufficiently flexible so as to encompass multiferroicity arising via electronic ordering (e.g. spin, charge, or orbital order, as in HoMnO3 [26–28]) within this framework by recasting the order parameter in terms of ‘rotation-like’ modes that preserve identical symmetry constraints. From this theoretical approach, it is possible to discern potentially understudied or ‘missing’ multiferroic compounds and then computationally explore the stability of such phases for accelerated experimentation. Acknowledgments We wish to thank N A Benedek, M Perez-Mato, C J Fennie, Ph Ghosez, J Íñiguez, A Mulder, K Rabe and L Bellaiche for useful discussions. JY and JMR were supported by the Army Research Office under Grant No. W911NF-15-1-0017 and the Penn State NSF-MRSEC Center for Nanoscale Science under Grant No. DMR-1420620, respectively. AS and SP would like to acknowledge the Italian Ministry of Research through the project PRIN ‘Interfacce di ossidi: nuove proprieta’ emergenti, multifunzionalita’ e dispositivi per l’elettronica e lenergia (OXIDE)’ and the support of Fondazione Cariplo, Grant No. 2013-0726. References [1] Scott J F 2000 Ferroelectric Memories (New York: Springer) [2] Defay E 2013 Ferroelectric Dielectrics Integrated on Silicon (New York: Wiley) pp 341–78 15

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18

Artificial design for new ferroelectrics using nanosheet-architectonics concept.

Simultaneous resonant x-ray diffraction measurement of polarization inversion and lattice strain in polycrystalline ferroelectrics.

Magnon breakdown in a two dimensional triangular lattice Heisenberg antiferromagnet of multiferroic LuMnO3.

Ferroelectric phase transition in multiferroic Ge1-x Mn x Te driven by local lattice distortions.

Mechanical cloak design by direct lattice transformation.

HIV-Host Interactions: Implications for Vaccine Design.

Super-crystals in composite ferroelectrics.

Protein-lipid interactions in bilayer membranes: a lattice model.

Ferroelectrics: Negative capacitance detected.

Graphene-multiferroic interfaces for spintronics applications.

Tunnel electroresistance through organic ferroelectrics.

Hierarchical dielectric orders in layered ferroelectrics Bi2SiO5.

Design of a low-emittance lattice for 3.5 GeV synchrotron light source.

Thermotropic phase boundaries in classic ferroelectrics.

Improper Complex-Valued Bhattacharyya Distance.

Multiferroic rhodium clusters.

Ferroelectrics under the Synchrotron Light: A Review.

Voltage control of magnetism in multiferroic heterostructures.

Super Stable Ferroelectrics with High Curie Point.

Hyperferroelectrics: proper ferroelectrics with persistent polarization.

LIGAND-RECEPTOR INTERACTIONS AND DRUG DESIGN.

Helical and skyrmion lattice phases in three-dimensional chiral magnets: Effect of anisotropic interactions.

Single-domain multiferroic BiFeO3 films.

Electron-lattice interactions strongly renormalize the charge-transfer energy in the spin-chain cuprate Li2CuO2.