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Self-organization into ferroelectric andantiferroelectric crystals via the interplay betweenparticle shape and dipolar interactionKyohei Takaea,1 and Hajime Tanakaa,1

aDepartment of Fundamental Engineering, Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan

Edited by Ren-Gen Xiong, Nanchang University, Nanchang, China, and accepted by Editorial Board Member Thomas E. Mallouk August 16, 2018 (receivedfor review May 25, 2018)

Ferroelectricity and antiferroelectricity are widely seen in varioustypes of condensed matter and are of technological significancenot only due to their electrical switchability but also due tointriguing cross-coupling effects such as electro-mechanical andelectro-caloric effects. The control of the two types of dipolarorder has practically been made by changing the ionic radius ofa constituent atom or externally applying strain for inorganiccrystals and by changing the shape of a molecule for organiccrystals. However, the basic physical principle behind such con-trollability involving crystal–lattice organization is still unknown.On the basis of a physical picture that a competition of dipolarorder with another type of order is essential to understand thisphenomenon, here we develop a simple model system com-posed of spheroid-like particles with a permanent dipole, whichmay capture an essence of this important structural transition inorganic systems. In this model, we reveal that energetic frustra-tion between the two types of anisotropic interactions, dipolarand steric interactions, is a key to control not only the phasetransition but also the coupling between polarization and strain.Our finding provides a fundamental physical principle for self-organization to a crystal with desired dipolar order and realizationof large electro-mechanical effects.

dipolar crystal | ferroelectricity | antiferroelectricity |structural phase transition | mechanical switching

Ferroelectricity and antiferroelectricity are manifestations oflong-range dipolar ordering with and without macroscopic

polarization, respectively. These phases are widely observed invarious materials, such as inorganic oxides (1), organic molec-ular solids (2), polymers (3), liquid crystals (4), metal–organicframeworks (5), and supramolecular systems (6). Phase tran-sitions between paraelectric, ferroelectric, and antiferroelectricphases are often accompanied by a large change in spontaneouspolarization, dielectric permittivity, crystalline structure (latticestrain), and entropy. Because of these intriguing characteris-tics, they have attracted considerable attention from condensedmatter physics (7–10), chemistry (6, 11, 12), and technologicalapplications including nonvolatile memories, electro-mechanicalactuators, and electro-caloric refrigerators (13–22).

Typical examples of ferroelectric–antiferroelectric phasetransitions in inorganic substances are Pb(Zr,Ti)O3 (1) and(Bi,RE)FeO3 (14, 23), where RE stands for a rare earth metalatom. In these substances, the ionic radius ratio (or the Gold-schmidt tolerance factor) is a key factor controlling the phasebehavior (23). Interestingly, similar effects arising from the ionicradii of constituent atoms have also been reported for the mag-netic analogue, i.e., the emergence of antiferromagnetic phase.It is worth noting that, in some cases, colossal magnetoresistancecan be induced by using phase transitions between ferromag-netic and antiferromagnetic phases (24). Another known methodto control antiferroelectric transformation is to apply a biax-ial epitaxial stress on thin films (14–16). In an epitaxial thinfilm of PbZrO3, for example, strain generated by a lattice mis-

fit to the substrate is known to be a crucial factor determiningthe crystalline structure and polarization ordering (25, 26), sug-gesting the importance of electro-mechanical coupling in thephase transition. Examples of antiferroelectric phases can alsobe seen in molecular crystals (2, 6, 27), organic–inorganic hybridperovskites (28), and liquid crystals (4) such as benzimidazolesorganic crystals (29). For a 1:1 adduct of iodanilic/chloranilicacid with 5,5′-dimethyl-2,2′-bipyridine (55DMBP), for exam-ple, ferroelectricity and antiferroelectricity emerge for iodanilicand chloranilic acids, respectively (11, 30). It was also shownthat the ferroelectric transition in the former case vanishesat high pressure, while another anomaly in dielectric per-mittivity remains. Quite recently, furthermore, ferroelectricitywas observed in columnar supramolecular crystals of benzene-1,3,5-triamides (BTAs) by controlling steric intermolecularinteractions (12).

In all of the above examples, the type of dipolar order is con-trolled by a certain controlling parameter. In particular, in theorganic substances, a key controlling factor is thought to be amolecular shape, although there is no unified physical under-standing on how to stabilize each phase by such modifications.Although the key structural factor responsible for the emer-gence of each of ferroelectric and antiferroelectric ordering hasbeen investigated for individual systems (25, 26, 31–38), the

Significance

Controlling ferroelectricity and antiferroelectricity of a crys-talline solid is one of the central issues in material scienceand technological applications because the dielectric, elec-tromechanical, and thermoelectric properties crucially dependon the type of dipolar order. However, still missing is a sim-ple particle-based physical model that shows self-organizationinto a lattice structure of desired dipolar order in a controlledmanner. Here we elucidate the importance of competitionbetween anisotropic steric and dipolar interactions in suchself-organization. We also reveal that the interplay betweenthe two types of anisotropy with different origins is a key tocross-coupling properties of solids such as mechanical switch-ability of ferroelectric and antiferroelectric phases. These find-ings will provide a useful guide for designing materials withdipolar order.

Author contributions: K.T. and H.T. designed research; K.T. performed research; K.T. andH.T. analyzed data; and K.T. and H.T. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. R.-G.X. is a guest editor invited by the EditorialBoard.

Published under the PNAS license.1 To whom correspondence may be addressed. Email: tanaka@iis.u-tokyo.ac.jp ortakae@iis.u-tokyo.ac.jp.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1809004115/-/DCSupplemental.

Published online September 17, 2018.

www.pnas.org/cgi/doi/10.1073/pnas.1809004115 PNAS | October 2, 2018 | vol. 115 | no. 40 | 9917–9922

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physical origin of the structural phase transition accompanyinga crystal–lattice change between the two types of polarizationordering and its coupling to mechanics remain elusive. Further-more, it is unclear whether there are any common underlyingphysics behind ferro-to-antiferro ordering seen in a wide class ofmaterials.

As a first step toward the unified understanding of these phe-nomena, we propose a simple candidate mechanism, which maybe relevant for the organic systems: competition between stericand dipolar interactions. To realize this by a simple particle-based model, we construct a system consisting of spheroid-like Lennard-Jones particles with a point dipole, whose shapeanisotropy (i.e., the aspect ratio characterized by an anisotropyparameter η) is introduced as a key physical parameter to controlthe short-range repulsive interaction (Materials and Methods).So far there have been many particle-based modelings of fer-roelectric ordering (39–44), but no reports on antiferroelectricordering and phase controllability. Our model may not be uni-versal [e.g., not directly applied to perovskite-type solids where apoint dipole moment cannot be well defined even approximately(45)], but captures essential physics behind self-organization ofdipoles into two types of long-range dipolar order into differentcrystalline lattices: competing orderings. Fig. 1A schematicallyshows a key idea of our model: The electric interaction betweenpoint dipoles is intrinsically anisotropic, and its sign depends onthe particle arrangement (46). Thus, the particle arrangementfavored by steric repulsions is not necessarily favored by dipo-lar interactions. Such frustration is more significant for particleswith larger anisotropy η. Thus, the increase of η destabilizes theferroelectric crystalline lattice structure and results in a struc-tural transition to an antiferroelectric phase with a differentlattice structure (Fig. 1 A–D). In relation to this, it is worth not-ing that even in simple dipolar systems an interesting hystereticresponse is observed under a specially designed arrangement ofdipoles (47). In this article, we show a simple physical principle,by which we are able to control the tendency toward ferroelectricor antiferroelectric ordering, and discuss its relevance to materialdesign.

ResultsPhase Diagram. We show the T − η phase diagram of our modelin Fig. 1B, where η represents the degree of shape anisotropyand η= 0 corresponds to the spherical dipoles (43) (Materi-als and Methods and SI Appendix, A). We fix the pressure ata high value so that the crystalline state is stable for a broadtemperature range. For η close to zero, there are three stablephases: the equilibrium liquid, paraelectric crystal (plastic crys-tal characterized by long-range translational order but withoutorientational order), and ferroelectric crystal (Fig. 1C), fromhigh to low temperature. Here the paraelectric–ferroelectricphase transition is of second order (SI Appendix, B). The crys-talline structure of the paraelectric phase is either face-centeredcubic (Fm 3m) or hexagonal closed packed (P63/mmc) withalmost equal probability; upon transition to the ferroelectricphase, it transforms into a rhombohedral (R3m) or hexagonal(P63mc) structure by dipole alignment accompanying the asso-ciated change of (111) interlayer spacing. With increasing η,the paraelectric–ferroelectric transition temperature decreasessince the ferroelectric attractive interaction decreases (Fig. 1A).Eventually, an antiferroelectric phase with a monoclinic (P2/m)structure (Fig. 1D) is formed from the ferroelectric phase viaa strongly first-order transition (see SI Appendix, C for thedetails of the crystalline structure). At η= 1.4, the ferroelectric–antiferroelectric phase transition temperature coincides withthat of the paraelectric–ferroelectric phase transition, indicatingthe presence of a triple point. Interestingly, a strong electrostric-tive coupling for larger η changes the nature of the (metastable)paraelectric–ferroelectric transformation from second order to

T

Ferroelectric

Antiferroelectric

ParaelectricLiquid

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

A

antiferroelectric order

ferroelectric order

++

DC

B

E z

yx -z

-y-x

-x x

y

-y

-x x

-y

y

diagonal view top view bottom view

Fig. 1. Phase behavior. (A) Schematic representation of electro-mechanicalcoupling in our model. Since the dipolar interaction changes its signdepending on the particle arrangement, the ferroelectric/antiferroelectricordering crucially depends on the aspect ratio of the spheroids and theapplied strain. (B) Equilibrium (η, T)-phase diagram of our model (see Mate-rials and Methods for the calculations). The solid and dashed lines arethe phase boundaries between equilibrium phases and those betweenmetastable states, respectively. The phase transitions are first order for theantiferroelectric–ferroelectric/paraelectric transitions (black line) and thecrystal–liquid phase transition (blue line). The transition between the stableferroelectric and the paraelectric phase for η < 1.4 (solid red line) is sec-ond order, whereas the one between their metastable phases for η≥ 1.4(dashed red line) is weakly first order due to the electrostrictive couplingbetween polarization and strain (SI Appendix, B). (C) A ferroelectric phaseobserved for η= 1.0 and T = 0.6. The color denotes the molecular orien-tation; we can see that almost all of the particles align along the samedirection. (D) An antiferroelectric phase observed for η= 1.6 and T = 0.6.Two kinds of ferroelectric planes (distinguished by color) are stacked alter-nately, leading to cancellation of macroscopic polarization. The snapshotsin C and D are obtained from the low-T part on cooling from the simula-tions displayed in Fig. 2 A and B, respectively. In C and D, color fluctuationreflects orientation fluctuation, which is stronger in the ferroelectric state(C) than in the antiferroelectric one (D) (Fig. 3). We note that time-averagedorientation reduces fluctuation and coincides with that expected for theassigned space groups in the main text. (E) Color legends of the dipoleorientation.

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first order (SI Appendix, B). When the short-range interaction isstrongly anisotropic (η > 1.7), there is only one phase transitionfrom the liquid to the antiferroelectric crystal, strongly indicatingthat the antiferroelectric phase is stabilized by the increase of theparticle anisotropy (Fig. 1A). The liquid–crystal phase boundaryhas a V shape, although the slope for the large-η side is quitesmall. This appearance of the minimum in the melting-pointcurve is a universal feature of systems with competing phaseordering (24, 48, 49).

Phase Transitions Induced by Temperature Change and External Field.Next we show how ferroelectric order and antiferroelectric orderrespond to the change in temperature (SI Appendix, D). Theresults of the paraelectric–ferroelectric transition at small η areshown in Fig. 2A. The dipolar order 〈P〉, the average of the mag-nitude of local dipolar order (see SI Appendix, E for the detaileddefinition), grows steeply around T = 0.75ε/kB upon cooling.We can see almost the same polarization behavior for both cool-

dipolar order <P>

nematic order <Q>

=1.0

T=0.6

-1

0

1

-0.2 0 0.2External electric field

0.8

-0.4

0.4

0

Polarization Strain(%)

External electric field

Polarization

1 / f

-1

0

1

-0.2 0 0.2

400800

400040000

<P>

<Q>

=1.6

Temperature

0

0.2

0.4

0.6

0.8

1

0.6 0.7 0.8

AF structural order parameter

Temperature

dielectricconstant

101

102

103

104

0.6 0.7 0.8Temperature

0

0.2

0.4

0.6

0.6 0.7 0.8 0

0.2

0.4

0.6

0.8

0.6 0.7 0.8Temperature

A

D

E F

C

B

=1.0

=1.0=1.6

=1.6

=1.6

T=0.65=1.6

Fig. 2. Emergence of ferroelectric and antiferroelectric order. (A and B)Changes in dipolar order 〈P〉 and nematic order 〈Q〉 for a temperature cycleat η= 1.0 and η= 1.6, respectively. (C) The change in the dielectric constantas a function of temperature on both cooling and heating for η= 1.0 and1.6. (D) Transitions of the antiferroelectric phase as seen using an antifer-roelectric structural order parameter. (E) Polarization (red curve) and strain(blue curve) response of the antiparallel ordered state (η= 1.6 and T = 0.6)to an external electric field. (F) Frequency dependence of the field responseof the antiferroelectric state (η= 1.6 and T = 0.65). See SI Appendix, D–Gfor the sample preparation protocol, the definition of the order parameters,and the calculation of the dielectric constant and the field response.

ing and heating paths, indicating the second-order nature ofthis phase transition. We note that the local nematic order 〈Q〉,the average of the magnitude of local nematic order (see SIAppendix, E for the detailed definition), also develops upon cool-ing, since the dipole moment is set along the shape anisotropy (orlong) axis of each particle.

On the other hand, the paraelectric–antiferroelectric transi-tion at large η is accompanied by a large temperature hysteresis,as shown in Fig. 2B. We can see that the dipolar order 〈P〉abruptly decreases and almost vanishes upon cooling (Fig. 2B,red curve), whereas the nematic order increases discontinuouslyto have a large value even when the polarization vanishes (Fig.2B, blue curve). This is a clear signature of the emergence of anti-ferroelectric long-range order below T ∼= 0.66. The macroscopicpolarization should be zero for both paraelectric and antifer-roelectric phases. However, since we define 〈P〉 as the averageof the magnitude of local polarization, it need not be zeroand reflects the degree of local polarization fluctuation. Thus,the abrupt change in 〈P〉 upon phase transition is correlatedwith the change in the dielectric constant (Fig. 2C). The phasetransition shows a large hysteresis on cycling the temperature,indicating the strong first-order nature of the antiferroelectricphase transition. We also show the temperature dependenceof the dielectric constant for η= 1.0 and 1.6 (SI Appendix, F).The paraelectric–ferroelectric transition at small η is accompa-nied by a much larger change in the dielectric constant than theparaelectric–antiferroelectric one at large η (Fig. 2C). This isprimarily because smaller anisotropy allows larger orientationalfluctuations of dipoles due to less steric hindrance. Furthermore,dipole fluctuations are larger in the ferroelectric phase than inthe antiferroelectric one even for the same η, which is discussedlater (Fig. 3). The transformation of the crystal structure can alsobe clearly seen in Fig. 2D, where we show the change of a spe-cial structural order parameter that has a nonzero value onlyin the antiferroelectric phase (SI Appendix, E). To further con-firm that this ordered state really has antiferroelectric order, wealso examine the dielectric response to an external electric field.As shown in Fig. 2E, we observe a double hysteresis response(red curve) to the applied electric field, a characteristic of theantiferroelectric phase. Since the dipole moment is set alongthe long axis of the spheroid particle in our model, the polar-ization response is automatically accompanied by a large strainresponse, as shown by the blue curve in Fig. 2E. We note that thisresponse is electrostrictive, and not piezoelectric, since the spacegroup of the antiferroelectric phase is centrosymmetric P2/m .

The large hysteresis loops in Fig. 2 B and E imply a large free-energy barrier for the phase transformation. This indicates thatthere should be a long incubation time for spontaneous nucle-ation of antiferroelectric domains in the ferroelectric phase (seeSI Appendix, H for the dynamical process of nucleation). Indeed,this is confirmed by the frequency dependence of the polarizationresponse (Fig. 2F): With increasing the frequency, the double-hysteretic response characteristic of the antiferroelectric phasedisappears. This frequency dependence indicates that there isnot enough time for the system to transform back to the anti-ferroelectric state under such a high-frequency modulation. Thiscan be confirmed by the fact that the relaxation time ('100t0) (SIAppendix, K and Fig. S8) is longer than the time period of fieldswitching at the highest frequency. This highlights the interest-ing possibility of controlling both the strain and the polarizationresponses to an external electric field by tuning the degree ofparticle anisotropy (η in our model). Indeed, the field requiredto induce a ferroelectric state from an antiferroelectric one cru-cially depends on η. The hysteresis loop becomes smaller for ηclose to the phase boundary, as shown in Fig. 1B.

The Origin of Slow Dynamics of the Ferro-to-Antiferro Transforma-tion. Now we consider why the response is so slow. According to

Takae and Tanaka PNAS | October 2, 2018 | vol. 115 | no. 40 | 9919

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Ferroelectric Antiferroelectric

0 1 2 3 4 0

2

4

6

8

10

0 1 2 3 4 0

2

4

6

8

10

wavenumberfre

quen

cywavenumber

frequ

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Fig. 3. Rotational dynamical structure factor SR(q,ω) of ferroelectric andantiferroelectric phases, showing the difference in vibrational dynamicsbetween ferroelectric and antiferroelectric states. See SI Appendix, J for thedefinition of SR(q,ω) and the calculation of vibrational density of states.

classical nucleation theory, this timescale is determined by thefree-energy barrier to overcome and the dynamics of elemen-tary excitation. The free-energy gain comes from the formationof the more stable phase, whereas the penalty comes from theformation of the domain interface. In our system, the nucleationshould be accompanied by cooperative displacement and reori-entation of the particles in the same hexagonal plane over thesize of a critical nucleus. This is because such cooperativity is nec-essary to avoid charging (represented by ∇·P, where P standsfor polarization field) at the interface, since it costs large elec-trostatic energy (SI Appendix, H). Note that flipping of a dipolein a ferroelectric state inevitably results in charging at both endsat the dipole, but not in perpendicular directions. This requirescooperativity in dipole rotation along the direction of the longaxis of spheroids. The probability to have such cooperative reori-entation may steeply decrease with domain size, simply becausethe number of particles involved increases. This extra physicalconstraint arising from the avoidance of charging, which is a uni-versal electrostatic factor governing the kinetics of the polariza-tion ordering, explains why the ferroelectric-to-antiferroelectrictransition exhibits a large hysteresis and requires a long incuba-tion time.

Furthermore, the dipolar interaction also affects anotherimportant factor governing the slow dynamics: It is the dis-tance between adjacent spheroids, which controls steric hin-drance to rotational motion. This structural factor is responsiblefor the asymmetry in the response between ferro-to-antiferroand antiferro-to-ferro transformations. To see this, we calcu-late the vibrational entropy difference in these phases, whichstems from the difference in the degree of steric hindrance toparticle rotation. As can be seen in Fig. 1A, spheroids havemore space to fluctuate for the ferroelectric arrangement thanfor the antiferroelectric one: The nested structure of the lat-ter tends to inhibit rotational fluctuations. Indeed the averageangular amplitude of rotational vibration of the dipoles relativeto their average orientations becomes 23◦ for the ferroelectricstate and 16◦ for the antiferroelectric state, although these fluc-tuations are isotropic for both states. The antiparallel dipolearrangement in the antiferroelectric phase causes both electro-static and steric constraints on the particle motion: The strongelectrostatic attraction between adjacent antiparallel spheroidscauses a strong constraint on the relative motion of spheroids.Furthermore, the distance between the nearest antiparallelneighbors becomes shorter by approximately 5% upon phasetransformation from the ferroelectric to the antiferroelectricstate by the attractive interaction (SI Appendix, I and Fig. S6),causing a strong geometrical constraint on rotational motions.These electrostatic and geometrical constraints result in suppres-

sion of the rotational motion in the antiferroelectric phase. As wecan see in Fig. 3, the ferroelectric phase has more low-energy (orsofter) excitation modes of rotational nature compared with theantiferroelectric phase. The presence of the distinct gap in theexcitation frequency for the antiferroelectric phase indicates notonly the absence of low-frequency coherent rotational modes butalso the presence of a resonance frequency, the latter of whichis reminiscent of the magnon dispersion relation in ferromag-netic and antiferromagnetic materials with uniaxial anisotropy(50). We note that in our system the uniaxial anisotropy stemsfrom the steric hindrance to the dipole moment. This implies thatthe resonance frequency can be controlled by varying the shapeanisotropy of the spheroids. We note that we confirm that thereis little difference in the translational vibrational modes and con-figurational entropy between the two phases (SI Appendix, J andFig. S7).

Here we note that the entropy difference between ferroelectricand antiferroelectric phases implies an inverse electro-caloriceffect (SI Appendix, K and Fig. S8), which has been observed insome antiferroelectric inorganic oxides (19, 51).

Mechanical Control of Antiferroelectric Phase. Finally we showinteresting electro-mechanical coupling observed under exter-nal stress, indicating the possibility of mechanical manipula-tion of the antiferroelectric phase. Larger particle anisotropyη means a stronger coupling between anisotropic mechani-cal stress and particle alignment. Thus, an externally appliedanisotropic stress may affect the phase behavior via electro-mechanical coupling. In Fig. 4, we show examples of such apolarization response to a uniaxial stress. Depending on thedirection of the applied stress and temperature, the initial

AF to FAF to P

stretch

T=0.65

compressT=0.65

T=0.67

compress

AF to AF

Fig. 4. Mechanical switching of antiferroelectric order. By applying auniaxial compression to an antiferroelectric state, we can induce an anti-ferroelectric to antiferroelectric (AF to AF) polarization reorientation. Byapplying a uniaxial stretching and compression to the same initial state inappropriate directions, on the other hand, we can induce antiferroelectricto ferroelectric (AF to F) and antiferroelectric to paraelectric (AF to P) phasetransitions, respectively, which are accompanied by large changes in thedielectric permittivity. See SI Appendix, L for the details of the calculationsand the kinetics of the phase changes.

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antiferroelectric state transforms to a paraelectric, a ferroelec-tric, and another antiferroelectric state with a different orien-tation (see SI Appendix, Figs. S9–S11 for the kinetics of thesephase transformations). These phase transformations lead toa large change in the dielectric permittivity, pointing to aninteresting possibility of selective colossal enhancement of thedielectric permittivity by applying a mechanical stress. Stress-induced phase transformation can generally be realized bychanging the (anisotropic) relative position between adjacentdipoles, as depicted in Fig. 1A. This suggests an interestingpossibility to induce ferroelectric–antiferroelectric phase trans-formations by applying an anisotropic stress, which has so farnot been reported for simple particle systems (e.g., organicsubstances).

SummaryWe propose a simple particle-based physical model of ferro-electric–antiferroelectric phase transitions accompanying crystal-structure change. Our model reveals that the transition betweenferroelectric and antiferroelectric crystals can be induced by anontrivial interplay between the two types of anisotropic inter-actions favoring different symmetries, the dipole–dipole inter-action and short-range steric interaction. The modification ofthe latter can enhance the energetic frustration between thesetwo interactions, which induces a structural phase transitionaccompanying the crystal symmetry transformation when thefrustration becomes strong enough. Such a competition betweendipolar and steric orderings may indeed explain the emer-gence of antiferroelectricity observed in an organic solid (11,30), where the distance between dipole chains is controlledby using different acids. Ferroelectricity (antiferroelectricity)emerges for an acid molecule with large (small) distance betweenparallel chains, consistent with our picture that the antipar-allel configuration should be more stable in the latter. Wealso elucidate physical factors controlling electro-mechanicaleffects associated with the antiferroelectric phase transition,which, for example, suggest that there is an optimal molecu-lar shape to realize large switching ability. Such informationmay be useful for material design of molecular ferroelectrics/antiferroelectrics.

Here it may be worth noting that our model can experimentallybe realized in colloidal systems: Recent experimental develop-ments allow a simultaneous addition of dipolar interactions ofelectric or magnetic origin and shape anisotropy to colloidal par-ticles (52–57), and thus tunable steric and dipolar anisotropyof colloids may make experimental realization of our resultspossible.

Finally, we mention a general implication of our work. Webelieve that the physical mechanism based on competing order-ings, i.e., a competition of dipolar ordering with another type ofordering, may generally be relevant for the emergence of antifer-roic order in a wide class of materials, including even magneticand multiferroic materials because of the common dipolar natureof the interactions. We hope that our study will initiate furtherinvestigations along this direction.

Materials and MethodsModel. Many computational models of spheroidal particles have beendeveloped to study structure and thermodynamic phase behaviors (42, 58–60) and jamming transitions (61) of ellipsoids. Here we use an interparticlepotential proposed in refs. 43 and 62, which can incorporate an orientation–strain coupling in a simple manner. The potential is modified from theLennard-Jones interaction to include the dependences on both position rand orientation n of particles as follows:

vLJij = 4ε

[(1 + Aij)

(σ

rij

)12

−(σ

rij

)6], [1]

Aij = η[(ni · rij)2+ (nj · rij)

2]/r2

ij . [2]

For small η, this potential well represents interacting spheroids and pro-duces a ferroelastic phase (62). See SI Appendix, A for further details. Wealso introduce a dipole moment to spheroid i in parallel to ni as µi =µ0ni .Then the dipolar interaction is described as (43)

vdij =µi ·µj/r3

ij − 3(µi · rij)(µj · rij)/r5ij . [3]

For η= 0, this potential corresponds to the well-known Stockmayerpotential, whose phase behavior has been well understood (41).

Equation of Motion and Numerical Units. The numerical integration of parti-cle positions and orientations is obtained by solving the equation of motion

for uniaxial molecules (43, 62, 63), mri =−∂U/∂ri and I(↔1 −nini) · ni =

−(↔1 −nini) · ∂U/∂ni under an appropriate ensemble (below). Here,

↔1 is the

unit tensor and the inertia moment I is assumed to be that of correspond-ing spheroids: I =σ2(1 + p)m/20 with p = (1 + 2η)1/6 being the aspect ratio(43, 62). All of the quantities in this article are presented in dimensionlessform, where the units of energy, length, and mass are ε, σ, and m, respec-tively. Then the units of dipole moment and electric field are

√εσ3 and√

ε/σ3, respectively. The electrostatic interaction is treated by the smoothparticle mesh Ewald method (64) to reduce numerical costs. Here the Ewaldparameter is 0.52/σ, the interaction cutoff in real space is 6.9σ, the meshsize for fast Fourier transformation (FFT) is 643, and we adopt fourth-order B-spline interpolation to reduce the root-mean-square error in theforce and the torque (65) less than 10−3ε/σ. We also adopt the “conduct-ing” boundary condition (63, 66) in numerical simulations, which ignoresthe depolarization field by assuming that the system is surrounded bymetallic walls.

Phase Diagram. To obtain the phase diagram (Fig. 1B), we calculate thechemical potential of each state, including the metastable condition, wherethe magnitudes of the dipole moment and the pressure are fixed tobe µ0 = 1.6

√εσ3 and P = 5ε/σ3. The crystal structure of each phase is

described in SI Appendix, C. Each structure is relaxed under the Nose–Hoover thermostat and the Parrinello–Rahman barostat (67) at very lowtemperature, and then we raise the temperature to examine phase equi-librium. To obtain crystalline states without defects, the numbers of par-ticles are N = 4,000 for the FCC-like lattice, N = 4,032 for the HCP-likelattice, and N = 4,320 for the BCC-like lattice. For a liquid state, we fixthe shape of the simulation box to be cubic by applying the Andersenbarostat and equilibrate the system with N = 4,000. The method to cal-culate the chemical potential is as follows (66). For a crystal, we applythe thermodynamic integration method, where the reference state is anEinstein crystal. For a liquid, on the other hand, we apply Widom’s par-ticle insertion method to calculate the chemical potential. We calculatethe chemical potentials for various (η, T) and obtain the phase equilibriumtemperatures at η marked in the phase diagram in Fig. 1B (red circlesfor the paraelectric–ferroelectric equilibrium, blue squares for the liquid–paraelectric and liquid–antiferroelectric equilibria, and black diamonds forthe antiferroelectric–ferroelectric and antiferroelectric–paraelectric equi-libria) and connect them by using the Clausius–Clapeyron equation (66)in the η− T plane. The generalized Clausius–Clapeyron equation forour model reads dT/dη= (λ1−λ2)/β2(h1− h2) on the phase bound-ary between phases 1 and 2, where λ= β(∂µ/∂η) = β〈∂U/∂η〉/N andh is the enthalpy per particle. By calculating the statistical average of∂U/∂η and the enthalpy, one can obtain the slope of the phase bound-aries. For example, the liquid–paraelectric phase boundary and liquid–antiferroelectric phase boundary meet at (η, T) = (1.7, 0.755), wheredT/dη=−0.20ε/kB for the former and dT/dη= 0.01ε/kB for the latter.Thus, the V shape of the phase diagram is also confirmed by the Clausius–Clapeyron equation.

ACKNOWLEDGMENTS. We thank Y. Takanishi and A. Ikeda for valuable dis-cussions. We are also grateful to T. Yanagishima for the careful readingof the manuscript. The numerical calculations were partially performed onCRAY XC40 at Yukawa Institute for Theoretical Physics (YITP) at Kyoto Uni-versity and on the SGI ICE XA/UV hybrid system at The Institute for SolidState Physics (ISSP) at the University of Tokyo. This study was supported byGrants-in-Aid for Specially Promoted Research (Grant JP25000002), Innova-tive Areas of Softcrystal (JP17H06375), and Scientific Research (JP18H03675)from the Japan Society for the Promotion of Science.

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1. Lines ME, Glass AM (1977) Principles and Applications of Ferroelectrics and RelatedMaterials (Oxford Univ Press, Oxford).

2. Horiuchi S, Tokura Y (2008) Organic ferroelectrics. Nat Mater 7:183–208.3. Furukawa T (1997) Structure and functional properties of ferroelectric polymers. Adv

Colloid Interface Sci 71–72:183–208.4. Takezoe H, Gorecka E, Cepic M (2010) Antiferroelectric liquid crystals: Interplay of

simplicity and complexity. Rev Mod Phys 82:897–937.5. Zhang W, Xiong RG (2012) Ferroelectric metal–organic frameworks. Chem Rev

112:1163–1195.6. Tayi AS, Kaeser A, Matsumoto M, Aida T, Stupp SI (2015) Supramolecular ferro-

electrics. Nat Chem 7:281–294.7. Landau LD, Lifshitz EM, Pitaevskii LP (1984) Electrodynamics of Continuous Media

(Butterworth-Heinenann, Amsterdam).8. Chaikin PM, Lubensky TC (1995) Principles of Condensed Matter Physics (Cambridge

Univ Press, Cambridge, UK).9. Onuki A (2002) Phase Transition Dynamics (Cambridge Univ Press, Cambridge, UK).

10. Blinc R (2011) Advanced Ferroelectricity (Oxford Univ Press, Oxford).11. Horiuchi S, Kumai R, Ishibashi S (2018) Strong polarization switching with

low-energy loss in hydrogen-bonded organic antiferroelectrics. Chem Sci 9:425–432.

12. Zehe CS, et al. (2017) Mesoscale polarization by geometric frustration in columnarsupramolecular crystals. Angew Chem Int Ed 56:4432–4437.

13. Scott JF (2007) Applications of modern ferroelectrics. Science 315:954–959.14. Sando D, Barthelemy A, Bibes M (2014) BiFeO3 epitaxial thin films and devices: Past,

present and future. J Phys Condens Matter 26:473201.15. Martin LW, Rappe AM (2016) Thin-film ferroelectric materials and their applications.

Nat Rev Mater 2:16087.16. Damodaran AR, et al. (2016) New modalities of strain-control of ferroelectric thin

films. J Phys Condens Matter 28:263001.17. Pesic M, Hoffmann M, Richter C, Mikolajick T, Schroeder U (2016) Nonvolatile random

access memory and energy storage based on antiferroelectric like hysteresis in ZrO2.Adv Funct Mater 26:7486–7494.

18. Uchino K (2016) Antiferroelectric shape memory ceramics. Actuators 5:11.19. Moya X, Kar-Narayan S, Mathur ND (2014) Caloric materials near ferroic phase

transitions. Nat Mater 13:439–45020. Liu Y, Scott JF, Dkhil B (2016) Direct and indirect measurements on electrocaloric

effect: Recent developments and perspectives. Appl Phys Rev 3:031102.21. Aida T, Meijer EW, Stupp SI (2012) Functional supramolecular polymers. Science

335:813–817.22. Han S, Zhou Y, Roy VAL (2013) Towards the development of flexible non-volatile

memories. Adv Mater 25:5425–5449.23. Kan D, Cheng CJ, Nagarajan V, Takeuchi I (2011) Composition and temperature-

induced structural evolution in La, Sm, and Dy substituted BiFeO3 epitaxial thin filmsat morphotropic phase boundaries. J Appl Phys 110:014106.

24. Tokura Y (2006) Critical features of colossal magnetoresistive manganites. Rep ProgPhys 69:797–851.

25. Reyes-Lillo SE, Rabe KM (2013) Antiferroelectricity and ferroelectricity in epitaxiallystrained PbZrO3 from first principles. Phys Rev B 88:180102.

26. Mani BK, Chang CM, Lisenkov S, Ponomareva I (2015) Critical thickness forantiferroelectricity in PbZrO3. Phys Rev Lett 115:097601.

27. Hochli UT, Knorr K, Loidl A (1990) Orientational glasses. Adv Phys 39:405–615.28. Li PF, et al. (2017) Unprecedented ferroelectric–antiferroelectric–paraelectric phase

transitions discovered in an organic–inorganic hybrid perovskite. J Am Chem Soc139:8752–8757.

29. Horiuchi S, et al. (2012) Above-room-temperature ferroelectricity and antiferroelec-tricity in benzimidazoles. Nat Commun 3:1308.

30. Horiuchi S, Kumai R, Tokura Y (2007) A supramolecular ferroelectric realized bycollective proton transfer. Angew Chem Int Ed 46:3497–3501.

31. King-Smith RD, Vanderbilt D (1994) First-principles investigation of ferroelectricity inperovskite compounds. Phys Rev B 49:5828–5844.

32. Ghosez P, Cockayne E, Waghmare UV, Rabe KM (1999) Lattice dynamics of BaTiO3,PbTiO3, and PbZrO3: A comparative first-principles study. Phys Rev B 60:836–843.

33. Rabe KM (2013) Antiferroelectricity in oxides: A reexamination. Functional MetalOxides, eds Ogale SB, Venkatesan TV, Blamire MG (Wiley-VCH Verlag GmbH & Co.KGaA, Weinheim, Germany), pp 221–244.

34. Ponomareva I, Lisenkov S (2012) Bridging the macroscopic and atomistic descriptionsof the electrocaloric effect. Phys Rev Lett 108:167604.

35. Tagantsev AK, et al. (2013) The origin of antiferroelectricity in PbZrO3. Nat Commun4:2229.

36. Hlinka J, et al. (2014) Multiple soft-mode vibrations of lead zirconate. Phys Rev Lett112:197601.

37. Ishii F, Nagaosa N, Tokura Y, Terakura K (2006) Covalent ferroelectricity in hydrogen-bonded organic molecular systems. Phys Rev B 73:212105.

38. Osipov MA, Fukuda A (2000) Molecular model for the anticlinic smectic-CA phase.Phys Rev E 62:3724–3735.

39. Weis JJ, Levesque D (2005) Simple dipolar fluids as generic models for soft matter.Adv Polym Sci 185:163–225.

40. Hynninen AP, Dijkstra M (2005) Phase diagram of dipolar hard and soft spheres:Manipulation of colloidal crystal structures by an external field. Phys Rev Lett94:138303.

41. Groh B, Dietrich S (2001) Crystal structures and freezing of dipolar fluids. Phys Rev E63:021203.

42. Groh B, Dietrich S (1997) Orientational order in dipolar fluids consisting of nonspheri-cal hard particles. Phys Rev E 55:2892–2901.

43. Takae K, Onuki A (2017) Ferroelectric glass of spheroidal dipoles with impurities: Polarnanoregions, response to applied electric field, and ergodicity breakdown. J PhysCondens Matter 29:165401.

44. Johnson LE, Benight SJ, Barnes R, Robinson BH (2015) Dielectric and phase behaviorof dipolar spheroids. J Phys Chem B 119:5240–5250.

45. Resta R, Vanderbilt D (2007) Theory of polarization: A modern approach. Physics ofFerroelectrics: A Modern Perspective, eds Rabe KM, Ahn CH, Triscone JM (Springer,Berlin), pp 31–68.

46. Luttinger JM, Tisza L (1946) Theory of dipole interaction in crystals. Phys Rev 70:954–964.

47. Concha A, Aguayo D, Mellado P (2018) Designing hysteresis with dipolar chains. PhysRev Lett 120:157202.

48. Tanaka H (2012) Bond orientational order in liquids: Towards a unified description ofwater-like anomalies, liquid-liquid transition, glass transition, and crystallization. EurPhys J E 35:113.

49. Russo J, Romano F, Tanaka H (2018) Glass forming ability in systems with competingorderings. Phys Rev X 8:021040.

50. Lifshitz EM, Pitaevskii LP (1980) Statistical Physics, Part 2 (Pergamon, Oxford).51. Geng W, et al. (2015) Giant negative electrocaloric effect in antiferroelectric La-doped

Pb(ZrTi)O3 thin films near room temperature. Adv Mater 27:3165–3169.52. Yethiraj A, van Blaaderen A (2003) A colloidal model system with an interaction

tunable from hard sphere to soft and dipolar. Nature 421:513–517.53. Assoud L, et al. (2009) Ultrafast quenching of binary colloidal suspensions in an

external magnetic field. Phys Rev Lett 102:238301.54. Sacanna S, Pine DJ, Yi GR (2013) Engineering shape: The novel geometries of colloidal

self-assembly. Soft Matter 9:8096–8106.55. Newman HD, Yethiraj A (2015) Clusters in sedimentation equilibrium for an

experimental hard-sphere-plus-dipolar Brownian colloidal system. Sci Rep 5:13572.56. Li B, Zhou D, Han Y (2016) Assembly and phase transitions of colloidal crystals. Nat

Rev Mater 1:15011.57. Gundermann T, Cremer P, Lowen H, Menzel AM, Odenbach S (2017) Statistical anal-

ysis of magnetically soft particles in magnetorheological elastomers. Smart MaterStruct 26:045012.

58. Frenkel D, Mulder B (1985) The hard ellipsoid-of-revolution fluid I. Monte Carlosimulations. Mol Phys 55:1171–1192.

59. Rex M, Wensink HH, Lowen H (2007) Dynamical density functional theory foranisotropic colloidal particles. Phys Rev E 76:021403.

60. Dijkstra M (2014) Entropy-driven phase transitions in colloids: From spheres toanisotropic particles. Adv Chem Phys 156:35–71.

61. Schreck CF, O’Hern CS (2010) Computational methods to study jammed systems.Experimental and Computational Techniques in Soft Condensed Matter Physics, edOlafsen J (Cambridge Univ Press, Cambridge, UK), pp 25–61.

62. Takae K, Onuki A (2014) Orientational glass in mixtures of elliptic and circular par-ticles: Structural heterogeneities, rotational dynamics, and rheology. Phys Rev E89:022308.

63. Allen MP, Tildesley DJ (1989) Computer Simulation of Liquids (Oxford Univ Press,Oxford).

64. Essmann U, et al. (1995) A smooth particle mesh Ewald method. J Chem Phys103:8577–8593.

65. Deserno M, Holm C (1998) How to mesh up Ewald sums. I. A theoretical and numericalcomparison of various particle mesh routines. J Chem Phys 109:7678–7693.

66. Frenkel D, Smit B (2001) Understanding Molecular Simulation: From Algorithms toApplications (Academic, San Diego).

67. Parrinello M, Rahman A (1981) Polymorphic transitions in single crystals: A newmolecular dynamics method. J Appl Phys 52:7182–7190.

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