Flex o Electric

MR43CH16-Tagantsev ARI 24 May 2013 15:27

Flexoelectric Effect in SolidsPavlo Zubko,1 Gustau Catalan,2,3

and Alexander K. Tagantsev41Department of Condensed Matter Physics, University of Geneva, CH-1211 Geneva 4,Switzerland; email: [email protected] Catala` de Recerca i Estudis Avancats (ICREA), Barcelona 08010, Catalunya, Spain3Institut Catala` de Nanocie`ncia i Nanotecnologia (ICN2), CSIC-ICN, Campus UniversitatAutonoma de Barcelona, Bellaterra 08193, Spain; email: [email protected] Laboratory, Swiss Federal Institute of Technology (EPFL), Lausanne 1015,Switzerland; email: [email protected]

Annu. Rev. Mater. Res. 2013. 43:387421

The Annual Review of Materials Research is online atmatsci.annualreviews.org

This articles doi:10.1146/annurev-matsci-071312-121634

Copyright c 2013 by Annual Reviews.All rights reserved

Keywords

strain gradients, electromechanical coupling, piezoelectricity,ferroelectricity, domain walls

Abstract

Flexoelectricitythe coupling between polarization and strain gradientsisa universal effect allowed by symmetry in all materials. Following its discov-ery several decades ago, studies of exoelectricity in solids have been scarcedue to the seemingly small magnitude of this effect in bulk samples. The de-velopment of nanoscale technologies, however, has renewed the interest inexoelectricity, as the large strain gradients often present at the nanoscale canlead to strong exoelectric effects. Here we review the fundamentals of theexoelectric effect in solids, discuss its presence in many nanoscale systems,and look at potential applications of this electromechanical phenomenon.The review also emphasizes the many open questions and unresolved issuesin this developing eld.

387

Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


Piezoelectricity: thelinear response ofpolarization tomechanical strain andvice versa; possibleonly innoncentrosymmetricmaterials

1. INTRODUCTION

Flexoelectricity is a property of all insulators whereby they polarize when subject to an inhomoge-neous deformation. The exoelectric coupling is between polarization and strain gradient, ratherthan between polarization and homogeneous strain. This difference is crucial to understandingboth the advantages and the limitations of exoelectricity compared with its close relative,piezoelectricity.

Strain, like stress, does not by itself break centrosymmetry: If a material is centrosymmetricto start with, it will continue to be centrosymmetric under a homogeneous deformation. Thisis intuitively clear: When a plate of a centrosymmetric material is subjected to a homogeneousdeformation (Figure 1a), we cannot rationalize the appearance of polarization, because there is nopreferred sense of direction for the polarization vector. By contrast, a strain gradient does breakcentrosymmetry. Under a strain gradient (e.g., one resulting from bending, as in Figure 1b),the top and bottom surfaces of the plate are no longer equivalent and therefore dene a senseof direction for the induced polarization vector. Mathematically, the exoelectric effect is con-trolled by a fourth-rank tensor and is therefore allowed in materials of any symmetry, whereaspiezoelectricity is controlled by a third-rank tensor, which is allowed only in materials that arenoncentrosymmetric. In piezoelectrics, the polarization thus occurs due to the low symmetry ofthe material rather than due to the symmetry-breaking effect of the perturbation.

Flexoelectricity as a strain gradientdriven breaking of the local centrosymmetry can also bevisualized at the microscopic level. For a simple ionic lattice, such as that sketched in Figure 1b, avertical gradient of in-plane strain (induced, for instance, by bending) may cause the central cationto be squeezed up like a pea inside a peapod, breaking the local centrosymmetry and inducingpolarity. This analogy, based purely on steric considerations, is crude but close to the idea ofBursian & Zaikovskii (1) that the central Ti ion in the ferroelectric perovskite BaTiO3 must shiftup on bending or, vice versa, that bending must appear due to the presence of a Ti ion in theupper side of the unit cell, which causes it to expand while the more empty lower half contracts.

P 0P = 0

a b

Figure 1Cartoon illustrating the microscopic mechanism of exoelectricity. (a) Uniform strain (e.g., that due touniaxial compression) does not break inversion symmetry and hence cannot generate polarization in acentrosymmetric material. (b) A strain gradient (e.g., that due to bending), however, does break inversionsymmetry and can therefore lead to relative displacements of the centers of negative and positive charges andto the appearance of polarization in any material.

388 Zubko Catalan Tagantsev

Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


Electrostriction: thecontribution to thestrain proportional tothe square of thepolarization (or thesquare of an externalelectric eld); possiblein any material

An alternative picture is provided by Harris (2), who noticed that the strain gradient in a shockwave makes unequal the distances between the atomic planes, resulting in a local breaking ofcentrosymmetry.

The above are all simple ionic pictures, with rigid ions shifting and causing polarization. Theexoelectric effect is nevertheless a subtle physical phenomenon, and its intuitive vision is oftendeceptive. For example, the conclusion about the sign of the exoelectric response that one mightdraw from the cartoon shown in Figure 1b may readily be wrong (3, 4).

Of course, strain gradients do not affect only ionic positionsan asymmetric redistribution ofthe electron density will take place as well, contributing to the total polarization. The exoelec-tricity in graphene (5) is controlled by this mechanism. The ionic and electronic components ofexoelectricity are complementary, but in this article we keep an ionic picture for simplicity. Wealso exclude from this article the exoelectricity of liquid crystals (6) [which is actually the eld inwhich the term exoelectricity was rst coined and was later adopted by Indenbom et al. (7) forsolid dielectrics] and biological materials (810), as they originate from different physics and havevery different applications (11).

So why should we care about exoelectricity? Electromechanical properties play an essentialrole in the physics of solids and their practical application. Until recently, when referring toelectromechanical properties, one generally meant piezoelectricity and electrostriction, whereasexoelectricity was hardly mentioned on account of its relative weakness. However, it is becomingclear that this neglect is not justied, for several reasons. First, exoelectricity, in contrast to piezo-electricity, is a universal property allowed by symmetry in any structure and therefore broadens thechoice of materials that can be used for electromechanical sensors and actuators. Second, reduceddimensions imply larger gradients: A strain difference over a small distance gives a large straingradient. The small length scales involved in nanotechnology thus lead to the growing impact ofexoelectricity, which at the nanoscale may even be competitive with that of piezoelectricity. Inaddition, a number of experiments have reported giant exoelectric coupling constants exceedingtheoretical estimates by several orders of magnitude. Finally, the polar nature of the exoelec-tric effect means that strain gradients can effectively play the role of an equivalent electric eldand can be used, for example, to switch the spontaneous polarization of a ferroelectric material.By contrast, switching of polarization by homogeneous strain is allowed by symmetry only in alimited class of ferroelectrics (those that are piezoelectrics in the paraelectric phase). Therefore,exoelectricity is not just a substitute for piezoelectricity at the nanoscale; it also enables additionalelectromechanical functionalities not available otherwise.

Several reviews on exoelectricity have been published, and the reader is referred to Refer-ences 3, 4, and 1216. The subject is, however, evolving very rapidly, withmany new developmentshappening in the past few years, particularly in the area of nanoscale materials such as thin lms.Here we therefore focus more attention on recent developments, both in experiment and in the-ory, as well as on controversial or unresolved issues. We begin with a brief introduction to thephenomenological theory of bulk exoelectricity in Section 2, where many of the characteristicfeatures of exoelectricity are derived. The concepts discussed here are referred to throughout therest of the review. In Section 3 we focus on the importance of nite size effects that are inherentin all real exoelectric systems. In Section 4, we then turn our attention to the magnitude of theexoelectric response and discuss the progress made and challenges faced in the development ofa microscopic theory of exoelectricity. Section 5 is devoted to experimental studies of exoelec-tricity in both bulk and nanoscale materials, and Section 6 addresses possible applications basedon the exoelectric effect. Finally, we conclude this review with a brief summary of open questionsand proposed future research in Section 7.

www.annualreviews.org Flexoelectric Effect in Solids 389

Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


2. PHENOMENOLOGICAL THEORY OF THE BULK EFFECT

Theoretical work on exoelectricity dates back to the seminal papers by Mashkevich & Tolpygo(17, 18), who rst proposed the effect, and by Kogan (19), who formulated the rst phenomeno-logical theory. For a historical overview of the early developments in the theory of exoelectricity,the reader is referred to the comprehensive review by Tagantsev (4) as well as to the more recent,concise summary by Maranganti et al. (13). Here we briey summarize the basic phenomenologi-cal description of the direct and converse exoelectric effects, emphasizing the differences betweenexoelectricity and piezoelectricity, and the analogies between strain gradients and electric elds.

2.1. Static Response

In contrast to the piezoelectric response, the exoelectric effect in the static (e.g., in a bent plate)and dynamic (e.g., in a sound wave) situations generally requires separate treatments (4, 20). Letus start with the static case.

2.1.1. Constitutive equations. We introduce the static exoelectric effect as a linear responseof polarization Pi to a strain gradient ukl/x j in the absence of electric eld. It is governed by afourth-rank exoelectric tensor, ijkl ,

klij =(

Pi(ukl/x j )

)E=0

, 1.

whereE denotes electric eld.This effect can readily be incorporated into theLandau phenomeno-logical framework. Despite the need for tensors for the description of exoelectricity, an insightinto this phenomenon can be gained by considering a 1D model involving one component ofpolarization and one of strain, where the tensor sufxes can be omitted.

In such a model, the macroscopic description of the static bulk exoelectric response can beobtained by generalizing the thermodynamic potential used for the description of the piezoelectricresponse by introducing a linear coupling between the polarization and strain gradient and viceversa into the system. In the most general form, a thermodynamic potential density suitable forsuch a description reads

G = 12 P2 + c

2u2 Pu f1P u

x f2u P

x PE u, 2.

where is the relevant stress component. This expansion does not contain anharmonic terms(i.e., the electrostriction term is omitted). When nonlinear effects are of interest, these terms canreadily be incorporated into the framework, as in, for example, References 2123. In particular,electrostrictive coupling terms are always allowed by symmetry and are essential for the descriptionof perovskite ferroelectrics; for these materials, the linear piezoelectric term Pu in Equation 2has to be substituted by the electrostrictive coupling term quP2 (24).

If we set to zero the coefcients for the gradient-containing terms, the bulk equations of stateof the material can be found by a simple minimization of the potential density in Equation 2 withrespect to the polarization and strain. Such minimization leads to the standard linear electrome-chanical equations of a piezoelectric:

P = E + eu, 3. = eE + c Eu, 4.

where e = is the so-called strain-charge piezoelectric coefcient and c E = c 2 is the elasticconstant at xed electric eld. From this we see that the term Pu in Equation 2 controls the bulk


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


piezoelectric response. We drop this term in further discussion so as to see the electromechanicalconsequences of pure exoelectricity without piezoelectricity. However, this discussion can bereadily generalized to piezoelectrics by taking this term into account.

Thus, we address exoelectricity by using the thermodynamic potential density fromEquation 2 with = 0. Here it is convenient to present G as the sum of two contributions:

G = f1 + f22(uP )x

, 5.

= 12

P2 + c2u2 f

2

(P

ux

u Px

) PE u, 6.

where f f1 f2 is the exocoupling coefcient (a tensor in the general case).Now that the potential density contains gradient terms, to get the equation of state, one should

minimize the thermodynamic potential of the sample as a whole,GdV (integrating over the

volume of the sample); i.e, one should apply the Euler equations G/ A ddx (G/( A/x)) =0, where A stands for the variables of the problem (in this case P and u). Such minimization yieldsthe bulk constitutive electromechanical equations (7)

P = E + ux

, 7.

= c u +

Px

, 8.

f. 9.In Equation 7, the rst right-hand-side term describes the dielectric response with the clampeddielectric susceptibility , and the second right-hand-side termdescribes the exoelectric responsewith the exoelectric coefcient . In Equation 8, the rst right-hand-side term describes Hookslaw, with the elastic constant at xed polarization c, and the second right-hand-side term describesthe converse exoelectric response. The latter term has the physical meaning of a linear responseof stress (or strain) to a polarization gradient in a mechanically free sample and was studied byMindlin (25) and others within the mechanics-of-materials community, in which the theory of(converse) exoelectricity appears to have developed independently of that of the condensedmatterphysics community outlined here (for details, see Reference 13). Converse exoelectricity may beparticularly important in piezoresponse forcemicroscopy experiments inwhich the voltage appliedto the conducting tip of an atomic force microscope gives rise to large and highly nonuniformelectric elds that, through converse exoelectricity, may give rise to an apparent piezoelectricresponse, even in nonpiezoelectric materials (26).

As is clear fromEquation 7, the vector f ux (or fijkl

uijxl

in full tensor notation) has the same effecton polarization as the external electric eld E. It is sometimes termed the exoelectric eld andis a convenient concept when one is considering the strength of exoelectric poling effects. Forexample, when polarization switching or poling caused by strain gradients is discussed (2731), itis this quantity that should be compared with the coercive or built-in electric elds.

The last term in Equation 5 does not contribute to the bulk constitutive electromechanicalequations. This can be concluded directly from the fact that its contribution to the thermody-namic potential of the sample can be transformed to an integral over the surface of the sample: 12 ( f1 + f2)

uPdS. Thus, the thermodynamic potential density in Equation 6 provides a full

phenomenological description of the static bulk exoelectric effect. Consequently, both the di-rect and converse exoelectric effects in Equations 7 and 8 are described by the same coefcientf f1 f2.


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


The bulk exoelectric effect describes a strain gradientinduced polarization analogous to thestrain-induced polarization of the piezoelectric effect. The constitutive equations (Equations 79)also highlight an important feature of the bulk exoelectric effect: Equation 9 explicitly shows thatthe bulk exoelectric coefcient is proportional to the dielectric permittivity of the material (7,19, 32, 33), suggesting that the exoelectric response should be enhanced in materials with highdielectric constants (high-K materials) such as ferroelectrics.

2.1.2. Direct and converse flexoelectricity. It is instructive to rewrite the constitutive equationsby taking into account that the exoelectric effect is relatively weak so that P in Equation 8 can bereplaced with E. This gives a set of constitutive equations,

P = E + ux

, 10a.

= Ex

+ c u, 10b.suitable for comparison with those for the piezoelectric response, which for our one-component1D model are given by Equations 3 and 4. Although for both effects the direct and converseresponses are controlled by exactly the same coefcient, there exists a strong asymmetry betweenthe converse and direct exoelectric effects. Specically, for the direct effect, in the absence of anelectric eld, a strain gradient induces a homogeneous polarization. However, for the converseeffect, in a mechanically free sample, a homogeneous electric eld does not cause a strain gradientor any other linear mechanical response (of course, nonlinear effects such as electrostriction stillexist). This asymmetry is in strong contrast to the symmetric piezoelectric effect, in which straininduces polarization and electric eld induces stress, as is clear from Equations 3 and 4.

As pointed out in Section 1, however, exoelectricity is a subtle phenomenon, and the afore-mentioned asymmetry of the bulk exoelectric effect does not lead to an asymmetry of the linearelectromechanical response of a sample as a whole. Actually, a homogeneous electric eld doescause an inhomogeneous deformation for nite-size samples, as is discussed in Section 3.2.

It is worth making a note on terminology: Whereas in the liquid crystals community the termconverse exoelectricity generally refers to bending of the material induced by an electric eld, inthe solid-state community this term is generally reserved for the appearance of stress or strain dueto a polarization gradient, as discussed above [although Indenbom et al. (7) used the term inverseexoelectric effect to refer to polarization-induced bending in their article].

2.2. Dynamic Response

The dynamic bulk exoelectric response can be described by minimizing the action

(T +u )dVdt (the integral is taken over the volume of the sample and time). Here comes fromEquation 6, and the kinetic energy density is dened as (20)

T = 2U 2 + MUP, 11.

where and U stand for the density and the acoustic displacement, respectively (in the 1D model,u = U/x); the dot over P and U refers to the time derivative. Such minimization with respectto polarization and strain leads to the following equations of motion for these variables:

P = E + ux

MU, 12.

U = c ux

MP +

2Px2

. 13.


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


+

+

= s + d

+

Figure 2Illustration of the dynamic exoelectric effect. An acoustic wave passing through a medium generates atime-dependent strain gradient that generates a polar displacement of the ions. For an acoustic wave, theacceleration of the atoms is proportional to the strain gradient. Thus, on top of the static exoelectricresponse, s ux , the acceleration of ions of different masses gives rise to an additional polar displacement,d (m1 m2) 2Ut2 (m1 m2)

ux .

In Equation 13, the last two terms play an essential role in the lattice dynamics, controlling, forexample, the specic shape of the dispersion curve for acoustic phonons in perovskite ferroelectrics(34). However, for macroscopic situations, in which the typical spatial scales are much largerthan the lattice constant, these terms can be neglected. Then, eliminating U from the set ofEquations 12 and 13, we get the following equation for the polarization accompanying the straingradient in a moving medium (e.g., in the case of an acoustic wave):

P = E + ( + d )ux

, 14.

where

d = c M/ 15.is the coefcient describing the dynamic exoelectric response. The physical meaning of thedynamic exoelectric effect is clear from Equation 12: It is the polarization induced by the accel-eration of the medium. Melnikovsky (35) also recently discussed polarization due to acceleration.On the microscopic level, the dynamic exoelectric effect can be related to the mass difference ofthe ions constituting the material (2, 20).

The M coefcient can be calculated in terms of the dynamic lattice theory (20). Figure 2illustrates the static and dynamic contributions to the polarization in an acoustic wave. As is clearfrom Equation 15, the dynamic contribution scales as the bulk dielectric constant of the material,just like the static one. Of key importance is that these contributions are expected to be of the sameorder of magnitude, according to estimates (20). The dynamic contribution to the exoelectriceffect makes the exoelectric effect qualitatively different from the piezoelectric effect, for whichthe polarization and strain in a moving medium are linked by the same relation as in the staticcase, i.e., P = E + eu.

The above treatment gives a qualitatively correct picture of the exoelectric effect in a soundwave in which the amplitude of the polarization is controlled by the sum of the static and dynamiccontributions. Remarkably, both exoelectric coefcients are independent of the frequency ofthe wave. Unlike the exoelectric coefcients, however, the magnitude of the polarization doesdepend on frequency, as the strain gradients in two acoustic waves of the same amplitude butdifferent frequency will be different.

For the quasi-static case, however, our 1D treatment is oversimplied. Here a full tensorialtreatment is needed to evaluate the contribution of dynamic exoelectricity to the total polarizationresponse (P. V. Yudin & A. K. Tagantsev, submitted manuscript). It can then be shown that this


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


contribution can be neglected in the quasi-static regime, i.e., when the modulation frequency islower than that of the relevant acoustic resonance of the sample. The treatment of this effect isformally similar to that of the piezoelectric resonance in a nite sample.

The framework formulated above for the 1D model can readily be generalized to the real 3Dsituation. Now the coefcients f, , and d become fourth-rank tensors symmetric with respectto the permutation of the rst pair of sufxes. As these tensors are of even rank, they can benonzero in materials of any symmetry, including amorphous substances (36). The number of theirindependent components is controlled by the symmetry of the material (see, e.g., References 3739). For example, this number is two for isotropic materials and three for nonpiezoelectric cubicmaterials. Because these tensors do not exhibit permutation symmetry with respect to the secondpair of sufxes, the two-sufx Voigt notation cannot be consistently introduced. Nevertheless, thisarticle uses such two-sufx representation (e.g., 11 1111), as is customarily done in papers onexoelectricity.

3. FLEXOELECTRICITY IN FINITE SAMPLES

Real samples are always nite, and thus a complete phenomenological description must includethe effects of surfaces or interfaces. In this section we briey discuss two effects associated withthe nite size of a real sample: (a) the contribution of surfaces to bending-induced polarizationand (b) the polarization-induced bending that appears in thin samples. Although our discussionexplicitly focuses on themodel systemof a cylindrically bent capacitor, the results derived here haveimportant implications for all nite exoelectric systems, including thin lms and nanodevices,which are discussed in the later sections.

3.1. Contribution of Surface Piezoelectricity

In a nite sample there will always be surface contributions to any effect. Typically, these are small,as they are usually controlled by the surface/volume ratio, but in some cases they can competewith the bulk contribution of another, weaker effect. For example, in centrosymmetric materials,due to the symmetry-breaking impact of the surface, a thin surface layer of thickness generallybecomes piezoelectric and can mimic the bulk exoelectric response. Let us discuss this effectby considering the cylindrical bending of the thin parallel-plate capacitor sketched in Figure 3.The piezoelectric coefcients, e, of the surface layers on the opposite sides of the plate should beof opposite signs (as controlled by the opposite orientation of the surface normal); the same isvalid for the bending-induced strains in these layers. Therefore, the induced polarizations in theselayers are of the same sign.

The normal component of the electric displacement D = 0E+P across any dielectric interfacemust be preserved, i.e., Dz = P + 0E = Ph + 0Eh , where 0 is the vacuum permittivity.Therefore, the presence of a polarization P within the piezoelectric surface layers must give riseto internal electric elds in the sample, thus polarizing it. For a short-circuited capacitor, thepotential difference = 2E + hEh across the capacitor must vanish; here h is the thickness ofthe nonpiezoelectric bulk (see Figure 3). The electric displacement induced by the strain gradientcan then be calculated as (40)

D = e hh2h + h

u11x3

, 16.

where = 0 + is the dielectric constant of the surface layer and h is that of the bulk. Forthin-enough surface layers (/h /h), Equation 16 yields the effective exoelectric coefcient


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


z

Ez

E

E

Eh

z

hh

P

P

Ph

Figure 3Surface piezoelectricity. Upon bending, as shown on the left, the tensile/compressive strains in thetop/bottom surface layers give rise to a polarization P in the piezoelectric surface layers of thickness .Because the normal component of the electric displacement must remain constant, this surface polarizationgives rise to electric elds Eh and thus to a polarization Ph within the nonpiezoelectric bulk. The measuredaverage polarization of the whole structure therefore depends not only on the dielectric properties of thepiezoelectric surface layers but also on those of the bulk. The potential and eld E proles along thesample thickness are shown on the right.

associated with surface piezoelectricity:eff1133 = e

h

. 17.

Thus, the polarization of the system arising from surface piezoelectricity is sensitive to the bulkvalue of the dielectric constant and, for thin-enough surfaces layers, is independent of the surface/volume ratio.

Finally, to evaluate the size of this surface effect, let us get an estimate for the effective exocou-pling coefcient f eff eff/h e/. For a conservative lower-bound estimate, we considerthe surface layer to be atomically thin ( = a few angstroms). Then, using e 1 C m2 and/0 10, we nd f eff of a few volts. This value is approximately the typical value of the compo-nents of the exocoupling tensor fijkl110 V (see Section 4). Thus, surface piezoelectricity canreadily compete with bulk exoelectricity.

3.2. Polarization-Induced Bending

As discussed in Section 2, the electromechanical constitutive equations (Equations 10a and 10b)describing the bulk exoelectric effect suggest an asymmetry between the direct and converseexoelectric responses. Specically, in the absence of an electric eld, a strain gradient inducesa homogeneous polarization, whereas a homogeneous electric eld does not induce a straingradient. Meanwhile, a thermodynamic analysis by Bursian & Trunov (33) shows that this isnot the case and that a plate of centrosymmetric material should bend under the action of theeld due to exoelectric coupling, as experimentally observed in BaTiO3 platelets in both theparaelectric and ferroelectric phases (1).

The reason for this discrepancy is that the polarization-induced bending predicted by Bursian& Trunov (33) is a nonlocal effect that can be obtained only by considering the thermodynamicsof the nite-size sample as a whole. Thus, this effect is not captured by the local theory developedin Section 2.1.


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


The origin of the polarization-induced bending moment can be understood by consideringwhat happens at the sample surface. If the polarization at the surface of the plate changescontinuously from its bulk value P to zero outside the plate, then according to Equation 8, thepolarization gradients at the plate surfaces must create some stress via the converse exoelectriceffect, resulting in forces applied to the opposite surfaces of the plate (40). The mechanicalmoment of these forces causes the bending predicted by the Bursian-Trunov approach. TheBursian-Trunov theory, although written as a bulk theory, actually treats the total free energyof the plate, implicitly taking into account the surface effects (i.e., the aforementioned forcesapplied to the surfaces). A similar bending effect was considered by Eliseev et al. (41). It can thusbe shown that a bending-mode exoelectric sensor working as an actuator will be characterizedby the same effective piezoelectric constant (40).

For more general boundary conditions, the situation is more complex and requires the use ofmodied mechanical boundary conditions (40, 42). Modied boundary conditions for the polar-ization were derived by Indenbom et al. (7) and by Eliseev et al. (41), whereas modiedmechanicalboundary conditions were obtained by Yurkov (42). The paper by Indenbom et al. (7) also gavealternative modied mechanical boundary conditions, although their explicit derivation was notprovided. Failure to take into account these exoelectricity-induced modications of mechanicalboundary conditions can lead to explicit conicts with thermodynamics. This applies, for instance,to the results in Reference 41, in which the validity of classical mechanical boundary conditionswas postulated. In view of the importance of the modied boundary conditions, it would nowbe useful to reexamine the ideas of Cross and coworkers [who proposed that, in piezoelectriccomposites based on the exoelectric effect, a separation of the direct and converse piezoelectriceffects can be achieved (12, 118)] by explicitly taking the boundary conditions into account. Theexperimental observation of polarization-induced bending is discussed in Section 5.2.4.

4. MICROSCOPIC CALCULATIONS OF BULK FLEXOELECTRICITY

Above, we do not address the magnitude of the exoelectric effect. A rough, order-of-magnitude,theoretical estimate of the exocoupling coefcient was rst provided by Kogan (19). Subse-quently, other approaches led to essentially the same result (2, 45). The most straightforward wayto arrive at Kogans estimate is to consider a simple lattice of point charges q with interatomicspacing a. Let this lattice be distorted by an atomic-scale strain gradient of order 1/a and withan atomic-scale polarization of the order of (ea)/a3. Such a strong perturbation is expected tomodify the energy density in the material, which is of the order of q240a

1a3 , by a comparable

amount. Assigning this energy change to the exoelectric term f P ux yields Kogans estimate of

the exocoupling coefcient,

f q/(40a) 110V, 18.using the electronic charge for q and a of the order of a few angstroms.

Since Kogans (19) seminal work in 1964, there have been a number of attempts to properlyquantify the exoelectric response in solids. On the microscopic level, static bulk exoelectricitycan be viewed as strain gradientdriven charge redistribution in the material. Roughly speaking,one can distinguish two contributions to this redistribution: ionic and electronic. The rstattempts to quantify the ionic contribution were undertaken by Askar et al. (46), who useda shell model for the lattice dynamics of ionic crystals to calculate a full set of exocouplingcoefcients for a number of biatomic cubic crystals. In the late 1980s, Tagantsev (4, 20) thendeveloped a microscopic theory of ionic exoelectricity based on a rigid-ion model. The rst stepstoward a microscopic description of the electronic contribution to exoelectricity were recently


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


made by Resta (47) and Hong & Vanderbilt (48), who generalized Martins (49) treatment ofpiezoelectricity to the exoelectric effect.

To gain some microscopic understanding of the ionic contribution to exoelectricity, let usbegin by briey discussing the theory developed by Tagantsev (4, 20). When a crystalline latticeis deformed, the ith component of the displacement of the nth atom, wn,i , can be expressed as asum of two terms:

wn,i = Rn, j

x0j

U ix j

d y j + wintn,i , 19.

where x0j and Rn, j are the coordinates of an immobile reference point and that of the nth atombefore the deformation. Here, summation over the three Cartesian coordinates is assumed forall repeating sufxes. The rst right-hand-side term in Equation 19 represents the contributionof the unsymmetrized strain U i/x j taken in the elastic medium approximation, also known asexternal strain. In the case of homogeneous strain, the external strain has a simple meaning: Thedisplacement of each atom is proportional to the distance to it from a certain immobile point. Fora material in which all atoms are centers of inversion, the external strain is sufcient to describe allatomic displacements caused by a homogeneous deformation. Consider, for instance, the simplecentrosymmetric diatomic lattice sketched in Figure 4a with c = 2a . Under the homogeneousdeformation shown in Figure 4b, centrosymmetry is preserved (c = 2a), and the displacementsof both types of atoms follow the external strain eld, which is represented by the deformation ofthe square lattice in Figure 4a.

However, if the material is noncentrosymmetric or if the deformation is inhomogeneous,the two types of atoms are no longer constrained by symmetry and may undergo additionalinternal displacements within the deformed unit cell. These internal displacementsthe dif-ference between the real displacement of an atom and that calculated from the local externalstrainare known as internal strains (50) and are described by the second right-hand-side term inEquation 19. Figure 4c shows the same centrosymmetric atomic lattice stretched inhomoge-neously along x1. The longitudinal strain gradient breaks the centrosymmetry, and thus the dis-placements of the atoms no longer have to follow the external strain eld.

For a piezoelectric (noncentrosymmetric) material, the internal strains are responsible for thechange in polarization under homogeneous strain and are, to lowest order, proportional to themacroscopic strain uij = (U i/x j + Uj/xi )/2. For a centrosymmetric material, the internalstrain of the nth atom is, to lowest order, proportional to the elastic gradient:

wintn, j = wexn, j N ikln, juikxl

. 20.

If the body is considered to be made up of point charges Qn (this includes all charges, not onlyions) with the coordinates Rn,i, the response of the polarization to an inhomogeneous deformationcan be found by calculating the variation of the average dipole-moment density caused by thisperturbation:

Pi = V 1n

n

Qn(Rn,i + wn,i ) V 1

n

Qn Rn,i , 21.

where V and Vn are the sample volume before and after the deformation, respectively. Substitut-ing Equation 19 into 21, one nds the polarization response to be composed of two contributions:One is conditioned by internal strain, and the other by external strain. The rst describes piezo-electricity (in noncentrosymmetric materials). It also describes the bulk exoelectric effect with aexoelectric coefcient given by

ikjl = v1

n

QnN ikln, j , 22.


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


a

b

c

c

a

w int

x1

x3

c = uc

a = ua

Figure 4When uniformly strained along x1, the centrosymmetric lattice in panel a will deform as shown in panel b.All atomic displacements are restricted by symmetry and will follow the elastic medium approximation (bluelattice). For example, if the homogeneous strain is u = u11, then c = uc and a = ua . However, aninhomogeneous deformation lifts this symmetry restriction, and the atomic displacements will not, ingeneral, follow the elastic medium approximation (as shown for the black atoms in panel c). The differencewint between the actual displacement (a) and that expected from the elastic medium approximation (ua)gives rise to internal strain.

where the summation is taken over the ions in a unit cell of volume v. The interpretation ofthe second contribution is not straightforward. It is termination dependent, and according toTagantsev (4, 20), it can be interpreted as the surface exoelectric effect. Resta (47) recently ques-tioned the existence of this effect [and claimed that the dynamic contribution to the exoelectriceffect (see Section 2.2) does not exist either]. Its contribution, however, is not expected to be en-hanced in high-K materials and is therefore of minor importance for applications; we thus excludeit from further consideration.

The tensor N ikln, j above, which links the internal strains and strain gradients, can be calculatedfrom the dynamic matrix of the material (20), which in turn can be obtained from ab initio latticedynamics simulations. Using this tensor and the transverse Born ionic charges (obtained, e.g.,from Berry-phase calculations), one can obtain the tensor by using Equation 22. This approachwas implemented by Maranganti & Sharma (51) to calculate the exoelectric coefcients for anumber of materials.

An alternative way to obtain the tensor consists of direct calculations of the polarizationresponse in an inhomogeneously deformed crystalline lattice. In view of the periodic boundary


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


conditions typically required for rst-principles calculations, consideration of a periodic distri-bution of the strain gradient (as the source of a static wave of external strains) is a reasonableoption. Then once the transverse Born ionic charges are available, the amplitude of the polariza-tion wave can be found. Hong et al. (52) directly implemented this approach in their calculationsfor some ferroelectric ABO3 perovskites. These authors introduced the static strain wave via xingthe positions of the A-site atoms as a sinusoidal function of the distance. Both the direction of theatomic displacements and the modulation direction were parallel to a cubic crystallographic axisso that the simulated exoelectric response was related to the 11 component of the exoelectrictensor. One drawback of this approach is that such a simulation does not provide all the infor-mation needed to evaluate this component in view of the depolarizing-eld problem. Specically,such conditions of the simulation imply conservation of the longitudinal component of electricaldisplacement, and the effective coefcient D extracted from such a simulation is related to thetrue coefcient via D = /(1+ /0). Thus, in the case of high-K materials (like ferroelectricperovskites) with 0, the calculated D does not yield , whereas D/0 appears to be closeto the exocoupling coefcient f = / . To calculate the coefcient within this framework,additional simulations of are required.

Ponomareva et al. (53) circumvented the depolarizing-eld problem in nite-temperature sim-ulations of and f tensors. Here the effect was addressed by employing Monte Carlo simulationswith an ab initiocalculated effective Hamiltonian; the contribution of the depolarizing energywas deliberately eliminated. A drawback of this work is that, in the ab initio calculations of thef tensor (also used in the Monte Carlo simulations of ), only the interaction between the localdipole and strain inside one unit cell was taken into account, and such approximation can readilyentail some 50% inaccuracy.

A disadvantage of the three aforementioned methods is that the purely electronic contributionto bulk exoelectricity is missing. This is probably a minor drawback when it comes to high-Kmaterials, in which the ionic contribution is enhanced and dominates the total tensor, buta complete theory should also incorporate the electronic contributions. Recently, Resta (47)demonstrated how the classical work by Martin (49) can be extended to the description ofthe purely electronic contribution to bulk exoelectricity. Also inspired by Martins work,Hong & Vanderbilt (48) calculated the electronic contribution from rst principles for anumber of materials. A remarkable feature of this contribution is that, in contrast to theionic one, it appears even in the absence of internal strains. These calculations were alsoperformed under conditions of xed electrical displacement so that the evaluated electronicexoelectric coefcient is underestimated by a factor of (1 + el/0), where el is the electroniccontribution to the permittivity. First-principles calculations of the purely electronic contribu-tion to the total exoelectric response (including possible surface conditioned contributions)were also performed by Dumitrica et al. (54) and by Kalinin & Meunier (5) for carbonsystems.

5. EXPERIMENTAL STUDIES OF FLEXOELECTRICITY

5.1. Quantifying Flexoelectricity in Bulk

The static exoelectric response is most commonly measured by using some variation of the twomethods sketched in Figure 5. The rst method consists of dynamically bending the material ina cantilever-beam geometry to generate a transverse strain gradient, as shown in Figure 5a. Theexoelectric polarization can then be measured by recording the displacement current owing


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


F

F

a b

x3

u11

u11x3

P = ~

12u33x3

P = ~

11

A

PPP

Figure 5Methods most commonly used to quantify the exoelectric response. (a) The cantilever bending methodused to measure the effective transverse exoelectric coefcient 12. (b) The pyramid-compression methodused to obtain the effective longitudinal exoelectric coefcient 11. Metallic electrodes are shaded in gray.

between the metallic plates with a lock-in amplier. In this way the coefcient 12, where

P3 = 12 u11x3

,

can be calculated; here the exoelectric tensor is expressed by using the two-sufx notation. Thecantilever bending method has been used extensively to investigate exoelectricity (also referredto as bending piezoelectricity) in polymers (5557).Ma&Cross (5862) recently used this methodto systematically quantify the exoelectric response in a number of perovskite ceramics. Variationsof the method involving three- or four-point bending geometries have also been successfullyemployed (12, 63, 64). Importantly, the cross section of a bent beam is modied due to anticlasticbending (as sketched in Figure 5a), giving rise to nonzero u22 and u33 components and theircorresponding gradients, related to u11 through the Poisson ratio . Thus, 12 is an effective exo-electric coefcient involving a combination of exoelectric tensor components that depends onthe precise geometry of the system (33, 64, 65). In the case of an isotropic bent beam, for example,

12 = 11 + (1 )12.

A second method for measuring direct exoelectricity, employed by the Cross group (12),involves uniaxial compression of a truncated pyramidshaped sample, as illustrated in Figure 5b.The stress 33 = F/A, generated by the pair of forces F, is different at the top and bottom surfacesof the truncated pyramid due to their different areas A, setting up a longitudinal strain gradientand thus generating a exoelectric polarization P3 = 11 u33x3 . Again, 11 is an effective coefcient.If we assume that 33 is the only nonzero stress component (14), for an isotropic material thiscoefcient can be related to the exoelectric tensor components through

11 = 11 212.

In practice, the pyramid-compression approach is complicated by the fact that the strain gradientis strongly inhomogeneous (66) (concentrated mainly at the sample edges), making it difcult toextract reliable values of .


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


Conversely, application of a bias across such a structure gives rise to a nonuniform electric elddistribution and hence polarization gradients that, in turn, generate strain in the sample throughthe converse exoelectric effect, which can be measured by using interferometric techniques (12,67). Such measurements always include contributions from electrostriction, which usually domi-nate the signal. However, the eld dependence is different for electrostriction (which is quadraticin eld) and exoelectricity (which is linear in eld), and therefore the two effects can, in principle,be separated. Fu and coworkers (67) used this method to measure the converse exoelectric effectand thus to estimate the exoelectric coefcient 11 for (Ba,Sr)TiO3. The result was in excellentagreement with measurements of the direct exoelectric effect. Hana and colleagues (68, 69) useda similar method to study converse exoelectricity in Pb(Mg1/3Nb2/3)O3-PbTiO3 (PMN-PT).

Extracting the full exoelectric tensor is nevertheless challenging, even for the simplest cubicdielectrics with only three independent coefcients. Zubko et al. (64, 65) employed a dynamicalmechanical analyzer in the three-point bending conguration to generate exoelectric polariza-tion in nonpiezoelectric SrTiO3 single crystals of different crystallographic orientations.However,pure bending experiments yield only two independent equations for the three exoelectric tensorcomponents and thus must be combined with a different method to obtain all three tensor com-ponents (65). One such method involves accurate measurements of transverse acoustic phononfrequencies, which are renormalized by the exoelectric terms and can thus be used to quantifythe exoelectric coefcients (21, 87). This method, however, will generally give the sum of thestatic and dynamic responses.

The exoelectric coefcients for a number of perovskite ceramics are listed in Table 1.Particularly high exoelectric coefcients (10100 C m1) have been measured close to theferroelectric-to-paraelectric phase transitions of (Ba,Sr)TiO3, relaxor PMN-PT, and (Pb,Sr)TiO3ceramics, for which the dielectric constants reach values exceeding 10,00020,000. Measurementsof the exoelectric response as a function of temperature conrm the expected scaling of with , as illustrated in Figure 6 for several perovskite compounds in their paraelectric phases. Theexact proportionality between and predicted by Equation 9, however, does not always hold,as can be seen most clearly in Figure 6b. Due to the large differences in dielectric permittivitiesof different compounds, it is more instructive to compare the normalized (or exocoupling)

Table 1 Measured experimental coefficients for perovskite ceramics in the paraelectric phasea

Compound Coefficient Method Value (C m1) /0 f = / (V)BaTiO3 12 (Tc + 3.4 K) CB 50 (62) 10,000 560 (62)Ba0.67Sr0.33TiO3 11 PC (0.5 Hz) 150 (12) 20,000 850

11 CFE (400 Hz) 120 (67)12 CB (1 Hz) 100 (60) 16,000 700

Ba0.65Sr0.35TiO3 12 CB 8.5 (144) 4,100 234PbMg0.33Nb0.67O3 12 CB 34 (58) 13,000 2645PMN-PT 11 PC (410 Hz) 612 (69) 21,000 3265

PC (0 Hz) 2050 (69) 110270Pb0.3Sr0.7TiO3 11 PC (0.5 Hz) 20 (12) 13,500 170Pb(Zr,Ti)O3 12 4PB 0.5 (63) 2,200 25

CB (1 Hz) 1.4 (61) 72

aThe values in the last column were calculated from the measured exoelectric coefcient and the experimental dielectric susceptibilities. References in thefourth and six columns are in parentheses. All values are room temperature values unless stated otherwise. Measurement techniques: CB, cantileverbending; PC, pyramid compression; CFE, converse exoelectric effect; 4PB, four-point bending.


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


11 (

C m

1)

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

20

40

60

80

100

120

140

160

0

5,000

10,000

15,000

20,000

Relative permittivity

a (Ba,Sr)TiO3 ceramics

c SrTiO3 single crystals

b (Pb,Sr)TiO3 ceramics

Temperature (C)

Temperature (C)

20 30 40 50 60 70

60

50

40

30

20

10

I/z 0

(pA

m

1)

Capacitance (nF)

0

2

4

6

8

111

101

100

Strain gradient (m1)

|P3|

1

010

(C m

2)

0.00 0.02 0.04 0.06 0.08 0.10 0.12

250 200 150 100 50 0 50

11 (

C m

1)

Relative permittivity

5

10

15

20

2,000

4,000

6,000

8,000

10,000

12,000

14,000

Temperature (C)20 30 40 50 60 70

Pola

riza

tion

(C

m2

)

23C

28C30C35C40C50C60C70C

0.0008 0.00100.00040.00000.00

0.04

0.08

0.12

0.16

Strain gradient (m1)

Figure 6Temperature evolution of the effective longitudinal exoelectric coefcient and the dielectric permittivity of (a) (Ba,Sr)TiO3 and(b) (Pb,Sr)TiO3 ceramics above the Curie temperature. The inset in panel a shows the linear relationship between polarization andstrain gradient. Adapted from Reference 12 with kind permission from Springer Science+Business Media. (c) Temperature evolution ofthe ratio of exoelectric current I and bending displacement z0 (with I/z0 12) and capacitance of a (100)-oriented crystal of SrTiO3.The anomaly at 105 K is related to ferroelastic domain wall motion in SrTiO3. The inset shows the linear relationship betweenpolarization and strain gradients for three crystals of different orientation. Reprinted with permission from Reference 64. Copyright2007, the American Physical Society.

coefcients f = / measured in volts; these coefcients were predicted by Kogan (19) to be ofthe order of f q/40a = 110 V (Equation 18) for simple ionic solids.

For all the materials listed in Table 1, the absolute magnitude of the response greatly exceedsthis simple theoretical estimate. BaTiO3-based ceramics, in particular, show enormous exoelec-tric and exocoupling coefcients. Such large exoelectric coefcients are particularly puzzling,as theoretical considerations suggest that exocoupling coefcients in excess of 1015 V should


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


make the perovskite structure unstable to the formation of incommensurate phases; this instabilityis discussed in more detail in Section 5.2.5. For PMN-PT, the measured coefcients vary by anorder of magnitude, depending on the measurement method used (68, 69).

Experimental data for single crystals are shown inTable 2 together with theoretical coefcientscalculated by using density functional theory (DFT). Although direct comparison with ceramicsis difcult due to the scarcity of single-crystal data and the different measurement techniquesused in most cases, the exocoupling coefcients for single crystals are generally considerablylower. For SrTiO3the only monocrystalline material whose full static exoelectric tensor hasbeen quantiedthe magnitude of the exoelectric response is in good agreement with Kogansestimate (64, 65) (although there is still some uncertainty about the signs of some tensor com-ponents). Neutron and Brillouin scattering measurements on BaTiO3 and KTaO3 single crystalsgive values of the same order of magnitude (34, 7072). However, all the values given inTables 1and 2 should be treated as order-of-magnitude estimates only. Challenges to properly quantifyingexoelectricity arise from difculties in separating the bulk exoelectric response from the contri-butions of surface piezoelectricity and dynamic exoelectricity. It has been generally assumed thatin high-K dielectrics, the contributions from surface piezoelectricity should not be signicant, asthey have not been expected to scale with the bulk dielectric constant of the material. This, how-ever, appears not to be the case, as is discussed in Section 3.1, and thus all the experimental bulkstatic exoelectric coefcients may well be affected by this contribution. Meanwhile, neutron andBrillouin scattering measurements of phonon dispersion curves include the dynamic exoelectriceffect. Moreover, they cannot be used to determine the signs of the coefcients (they can be usedto determine only the relative signs of some of the coefcients) (P.V. Yudin & A.K. Tagantsev,submitted manuscript).

We deliberately exclude polymers and elastomers from Tables 1 and 2, as the nature of theeffect is very different in these materials and is beyond the scope of this review. Nevertheless,it is worth mentioning that the exibility of these materials makes them in principle attractivecandidates for future exoelectric devices and has motivated many studies on exoelectricity inpolymers. Recently, Baskaran et al. (66, 73, 74) reported extremely large exoelectric coefcientsfor polyvinylidene uoride (PVDF) thin lms, but there is still no consensus on the best methodsfor quantifying exoelectricity in these materials, leading to very large variations in the valuesreported for different polymers by different groups (5557, 75, 76). For PVDF alone, the reportedcoefcients range from 13 nC m1 (76) to 80 C m1 and higher (66, 73, 74).

Theoretically quantifying the exoelectric tensor has been equally challenging. Nevertheless,a number of different approaches have been developed to calculate the exoelectric coefcients,as reviewed in Section 4. The results of such calculations for a variety of materials are summarizedin Table 2.

For SrTiO3, the theoretical and experimental values are of the same order of magnitude,although there is still signicant disagreement (including in the signs) between values obtainedby using different theoretical methods. For other materials, notably BaTiO3, the disagreementbetween theory and experiment is much larger. In all cases, however, the calculations yield f-valuescomparable to or smaller than Kogans estimate (Equation 18) and thus cannot account for thelarge coefcients measured for ferroelectric ceramics.

In summary, despite signicant experimental and theoretical effort, reliable quantication ofthe exoelectric response remains challenging. Although there is some convergence on the orderof magnitude of the response for some monocrystalline oxides, there are still large, unexplaineddiscrepancies between experimental data on ceramics and experimental data on single crystalsand equally large differences between theoretical values obtained by using different techniques.The importance of building an accurate database of exoelectric tensor components for a wide


Ann

u. R

ev. M

ater

. Res

. 201

3.43

:387

-421

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rg A

cces

s pro

vide

d by

WIB

6283

- U

nive

rsita

etsb

iblio

thek

Dor

tmun

d on

05/

11/1

5. F

or p

erso

nal u

se o

nly.


Table

2Experim

entaland

theo

reticalfl

exoe

lectriccoeffic

ientsforsing

lecrystals

a

Sing

lecrystals

Experim

ent

The

ory

Com

poun

dCoe

fficien

tValue

(nCm

1)

/0

f=

/(V

)

Value

(nCm

1)

f(V)

BaT

iO3

|11

12

||

44|

Flex o Electric

Documents

Transcript of Flex o Electric