arX

iv:0

903.

0764

v1 [

cond

-mat

.mtr

l-sc

i] 4

Mar

200

9

Origin of Ferroelastic Domains in Free-Standing Single Crystal Ferroelectric Films

I. A. Luk’yanchuk,1, 2 A. Schilling, J. M. Gregg,3 and G. Catalan, J. F. Scott4

1Laboratory of Condensed Matter Physics, University of Picardie Jules Verne, Amiens, 80039, France2L. D. Landau Institute for Theoretical Physics, Moscow, Russia3Centre for Nanostructured Media, School of Maths and Physics,

Queen’s University of Belfast, University Road, Belfast BT7 1NN4Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ

(Dated: July 29, 2021)

The origin of the unusual 90°ferroelectric / ferroelastic domains, consistently observed in recentstudies on meso and nanoscale free-standing single crystals of BaTiO3 [Schilling et al., PhysicalReview B, 74, 024115 (2006); Schilling et al., Nano Letters, 7, 3787 (2007)], has been considered.A model has been developed which postulates that the domains form as a response to elastic stressinduced by a surface layer which does not undergo the paraelectric-ferroelectric, cubic-tetragonalphase transition. This model was found to accurately account for the changes in domain periodicityas a function of size that had been observed experimentally. The physical origin of the surfacelayer might readily be associated with patterning damage, seen in experiment; however, when allevidence of physical damage is removed from the BaTiO3 surfaces by thermal annealing, the domainconfiguration remains practically unchanged. This suggests a more intrinsic origin, such as theincreased importance of surface tension at small dimensions. The effect of surface tension is alsoshown to be proportional to the difference in hardness between the surface and the interior of theferroelectric. The present model for surface tension induced twinning should also be relevant forfinely grained or core-shell structured ceramics.

PACS numbers: 77.80.Bh, 77.55.+f, 77.80.Dj

Introduction

In 1935 Landau and Lifshitz predicted the appearanceof periodic, thermodynamically stable, domains, with op-positely oriented magnetic moments, in ferromagneticcrystals. The existence of these domains minimized theenergy of the depolarizing field, caused by the abruptdiscontinuity of the spontaneous magnetization at thesample surface [1]. Additional consideration of the as-sociated domain wall energies allowed them to predicta square root dependence of the domain period on thesample thickness. This square root relationship is oftenreferred to as the Kittel law because of its independentformulation in 1946 by Kittel [2]. The law is also validfor 180°domains in ferroelectric crystals [3, 4], where theunfavourable depolarizing electric field is provided by theabrupt polar discontinuity at the surface. The Landau-Lifshitz-Kittel theory was later extended by Roytburd [5]to describe the behaviour of ferroelastic domains formedas a result of substrate clamping effects in thin film sys-tems.

In previous studies we have shown that the Kittel lawworks perfectly for a wide class of ferroic materials (fer-romagnetic, ferroelectric or ferroelastic) over six ordersof magnitude in film thickness [6, 7], and that it can beintuitively expressed in terms of the domain wall thick-ness [8, 9, 10]. Moreover, we have demonstrated that theKittel approach can be extended to three dimensionalstructures [11, 12, 13], to ferroelectric superlattices [9]and to multiferroic materials [14, 15]. Much of the exper-imental work has been in association with observations of

periodic 90°ferroelectric-elastic domains that have beenconsistently observed in free-standing single crystal thinfilms (see Fig.1a) and nanowires of BaTiO3. In all our ex-periments, the size of the 90°domains as a function of sizeis indeed found to be well described by a Kittel-Roytburdformalism.

At first glance, however, the very existence of such do-mains in our free-standing samples is quite surprising.Ferroelastic domains normally appear in response to anexternal stress (such as that imposed by clamping to arigid substrate, for example) which forces the sample to-wards shape preservation in the clamped directions. Thedomain configuration is such that the macroscopic shapedifference between the paraphase and the ferrophase isminimized, while the domain size responds to an equi-librium between domain energy and domain wall energy[5]. However, our BaTiO3 lamella are free standing sin-gle crystals and therefore free from epitaxially-inducedstress, so they have no Roytburd-like interface-inducedelastic driving force. The question, then, is the following:if there is no external stress, what causes the appearanceof the ferroelastic domains?

In the present work we explain the appearance ofthe self-organized domain patterns in free-standing nano-samples of BaTiO3 by assuming that the driving stressis provided by an encapsulating surface layer. In muchof our experimental work, this encapsulation layer couldeasily be associated with the surface damage caused byfocused ion beam milling. However, we have also ob-served here that 90°domains persist, with only slightlyaltered periodicities, even when surface damage has been

2

(a)

200nm

<100>pseudocubic

<100>pseudocubic

(b)

y x

z

“X”“Y”

ASL

AFL

d

FL

SL

SL

FIG. 1: (Color Online)(a) Scanning Transmission Elec-tron Microscopy Image of periodic domain structure infree-standing single crystal lamela of BaTiO3. (b) Sketchof our model system, comprising a ferroelectric layer(FL) sandwitched between two surface encapsulation lay-ers (SL) which do not undergo the ferroelastic transforma-tion. The stress imposed by these untransformed ”dead”layers onto the ferroelectric layer induces the appearance of90°ferroelectric/ferroelastic domains

repaired by thermal annealing; this suggests that thestrain effects may in fact arise intrinsically from surfacetension, being therefore unavoidable even in nominally”perfect” free-standing ferroelectric nanostructures.

Model

The development of the model focuses on the geome-try of the single crystal lamellae first reported in ref.[7],an example of which is shown in Figure 1(a). Figure1 (b) shows a schematic cross-section of our model rep-resentation for such BaTiO3 lamella, with the surfaceencapsulation layers (SL’s) that are potentially respon-sible for the formation of 90°domains present and theirobeyance to Kittel behaviour. The model comprises atetragonal ferroelectric/ferroelastic layer (FL) of thick-ness AFL, exemplified by BaTiO3, sandwiched betweentwo cubic paraelectric surface layers (SL) of thicknessASL. The surface layers provide the stress for the cre-ation of the ferroelectric-ferroelastic domains in the FL.The assumption of cubic surfaces is supported by the ob-

servation that barium titanate nanoparticles have a core-shell structure with tetragonal interior and cubic surfaces[16, 17]. However, the cubic symmetry is not a key fea-ture of the model: for the SL to impose stress on the inte-rior, it is just sufficient that it does not undergo the phasetransition at TC . A non-ferroelectric capsule/matrix wasalso assumed in recent phase-field simulations of domainpatterns in ferroelectric nanostructures[18].In the present model, the SL thickness is an adjustable

parameter to be determined from experimental measure-ments. The cartesian z-axis is oriented perpendicular tothe plane of the lamella, and the x and y axes coincidewith the crystal axes of BaTiO3 as shown in Fig. 1a. Theequilibrium lattice parameters of the FL are assumed tobe those of the bulk ferroelectric tetragonal BaTiO3 crys-tal and, in order to minimize depolarization fields, thepolarization will tend to lie within the XY plane, point-ing parallel to either the X or Y directions. In terms ofnotation, [c0, a0, a0] if P is parallel to x or [a0, c0, a0] ifP is parallel to y. The formation of the SL with differentequilibrium lattice constants [b0, b0, b0] results in elasticstress, provided by lattice matching at the FL and SLinterface.Consider to start with the simplest possibility when

the matching stress is uniform, i.e., there are no stressgradients. Because SL is much thinner than FL, the lat-ter remains undeformed, keeping the equilibrium BaTiO3

lattice parameters ([c0, a0, a0] or [a0, c0, a0]). Deforma-tion concerns only the SL which, because of the matchingconditions, should conserve the same XY-plane parame-ters as the FL but can relax in the Z direction (s′zz = 0).The corresponding deformation energy is caused by thein-plane misfit strains of SL and contains the tension andshear parts [19, 20, 21, 22] :

WSL = 2GASL

1− ν

(

s′2a + 2νs′as′

c + s′2c)

= ASLG1 + ν

1 − ν(δab + δcb)

2 +ASLGδ2ca (1)

where the diagonal components of the plain strain tensorof SL (s′a, s

′

c, 0) = (s′xx, s′

yy, s′

zz) are expressed via themismatch parameters:

s′a = δab =a0 − b0

b0, s′c = δcb =

c0 − b0b0

,

s′a − s′c = δca =c0 − a0

b0, (2)

G is the shear modulus, ν is the Poisson ratio and thefactor 2 corresponds to two (top and bottom) SLs on bothsides of FL. We assume that G ≈ 40− 55 · 109Nm−2 [23]and ν ≈ 0.28 [24] are approximately the same for SL andFL.Such system can be unstable towards formation

of the experimentally observed periodic structure of

3

90°ferroelastic domainsX/Y/X/Y shown in Fig. 1. Usu-ally, such domains have 45°domain walls and ”head-to-tail” polarization contacts to avoid the formation of de-polarization charge. The reason of the instability is thatdomain formation can reduce the elastic energy (1) of theSL by allowing the misfit stress to gradually relax insidethe FL, within an interfacial region whose thickness isof the order of the domain width d. The domains willappear when the elastic energy imposed onto the SL bythe ferroelectric is bigger than the energy required for theformation of domain walls. We emphasize that, althoughthe net effect of ferroelastic twinning is to minimize themacroscopic deformation of the SL, this twinning neces-sarily requires a strain gradient near the interface. Thisis because, while afar away from the interface the latticeparameters of the FL are those of the bulk ferroelectric,at the interface they must become closer to cubic in or-der to match the SL. Accordingly, an inhomogeneouslystrained region must appear for which flexoelectric effectsmay be important ([25, 26, 27, 28]). The flexoelectriccontribution, however, has been left out of this model forthe sake of simplicity.

The situation described here is in fact quite similar tothe appearance of periodic 90°domains in epitaxial ferro-electric films strained by thick undeformable substrates[19, 20, 22] in which the film/substrate mismatch strainrelaxes in the near-surface layer of the film. The only dif-ference is that the contact elastic SL in our case is thinand its deformation should be considered self-consistentlywith that of FL. In a way, the model discussed here canbe seen as a generalization of the substrate/film mod-els for the case of a substrate that has finite thicknesscompared with the film.

The mechanism of nonuniform strain relaxation in thedomain-populated FL gives rise to two new energy con-tributions: the near-surface deformation energy of theferroelectric layer, WFL, and the energy of the domainwallsWDW . The energy balance betweenWSL, WFL andWDW optimizes the domain period 2d and the matchingplane lattice constants changing periodically at SL/FLinterface from (c, a) at X-domain to (a, c) at Y-domain.

Before proceeding to the derivation of the total energyof the system

W = 2WFL + 2WSL +WDW (3)

(the factors 2 correspond to two-side SL and two-siderelaxation near-surface layers in FL), we assume that theoptimal domain width d is thinner than FL but thickerthan SL :

ASL < d < AFL. (4)

Observed in experiment and discussed in detail later,such hierarchy simplifies the calculation of the differentcontributions to (3).

We express the deformation energy WFL of the ferro-electric layer in terms of periodically changing strains ofdomains (sa, sc)/(sc, sa) at the surface of the FL,

sc =c− c0c0

, sa =a− a0a0

, (5)

These are taken as variational parameters in the gen-eral expression for the elastic energy of the 2-dimensionalcrystal, periodically strained as (sa, sc)/(sc, sa) and withrelative domain population equal to φ (See also Eq.(30)in [22]):

WFL =GAFL

1− ν

(

s2a + 2νsasc + s2c)

+GAFL(sa − sc)2

[

−2φ+2d

AFL

fa

(

2d

AFL

, φ

)]

. (6)

Adapting Eq.(6) to the case of equally-populated (φ =1/2) and thin (d < AFL) domains, the universal dimen-sionless function can be simplified as: fa ≈ AFL/4d +7ζ(3)/8π3 [22], and therefore:

WFL =1

2AFLG

1 + ν

1 − ν(sa + sc)

2 +1

2κ−1dG(sa − sc)

2,

κ =2π3

7ζ(3)≈ 7.4 (7)

Physically, the first term in (7) corresponds to the av-erage interface-induced strain that propagates throughthe whole thickness AFL of the FL. The second termis produced by the superposition of alternative strainsof an infinite series of domains, compensating inside theFL and relaxing in the near-surface gradient layer of athickness of order d. Technically this sum is expressedvia the zeta-function ζ(3), like the electrostatic energy ofalternative depolarization charge compensation in Kittelformula for ferroelectric domains [1, 2].Before minimizing (7) we can ensure first the vanishing

of the largest first (volume) term, selecting sa + sc = 0;this is justified because the domain pattern can compen-sate for shear strains (sa−sc) but not for volume changes.The number of variational parameters then reduces toone: s ≡ sa = −sc and the second (surface) term takesthe form:

WFL = 2κ−1dGs2. (8)

Consider now the deformation energy WSL of the SLsubjected to 2d−periodic domain-induced surface strains(s′a, s

′

c)/(s′

c, s′

a), taking into account that s′a and s′c areexpressed via variational parameters sa and sc (5) as:

s′c =c− b0b0

≈ sc + δcb, s′a =a− b0b0

≈ sa + δab (9)

If SL is thinner than the domain structure period:(ASL ≪ d) the periodic surface strain does not man-age to relax across the SL. Then, SL can be presented

4

as a piecewise (s′a, s′

c)/(s′

c, s′

a) strained film, having uni-form deformation for each section. Summing the givenby the first part of Eq. (1) elastic energies from all thepieces (that are equal because of the s′a ↔ s′c symmetry)and taking into account the discussed above constrainsa = −sc = s we present the elastic energy of SL as asuperposition of tension and shear contributions:

WSL =1

2ASLG

1 + ν

1 − ν(δab + δcb)

2 +1

2ASLG(2s− δac)

2.

(10)Note that the tension contribution:

WT =1

2ASLG

1 + ν

1− ν(δab + δcb)

2 (11)

does not depend on strain variational parameter s andcoincides with the tension energy of the uniformly de-formed SL in (1).Combining (8) and (10) with the energy of domain

walls

WDW = σAFL

d(12)

(σ is the surface energy density of the domain walls), thetotal energy (3) is:

W = WT + σAFL

d0

[

d

d0r2 + κ

ASL

d0(r − 1)2 +

d0d

]

, (13)

where r = 2s/δca and

d0 =√

κDAFL = 2.7√

DAFL. (14)

with the length-scale parameter

D =σ

Gδ2ca≈ 0.5nm, (15)

that can be interpreted as ”domain wall half-thickness”.This value is ca. one-two lattice constants and agreeswith previously reported theoretical and experimentalvalues [29, 30, 31, 32].Minimization of (13) over r and d gives the expression

for optimal domain width

d =d0

1− d0

κASL

(16)

When d0 ≪ κASL, one can neglect the responsiblefor the upward curvature denominator in (16) and ob-tain the Kittel-type dependence (14) for d(AFL). Thisexplains why our experimental results could be reason-ably well fitted assuming a simple square root depen-dence of domain periodicity on film thickness [7]. Notehowever that we are still in the thin surface layer limitASL < d0. The opposite (irrelevant for our system) limitd0 < ASL will lead to the Roytburd-Pompe-Pertsev situ-ation of thick substrate with domain pattern also obeying

BaTiO3

lamella

c

Gabeam

BaTiO3

lamella

BaTiO3 bulksingle crystal

a b

d

0.5 µm

FIG. 2: Different stages of a FIB process to fabricate aBaTiO3 lamella for cross-sectional view: (a) schematic draw-ing of a lamella milled by FIB; prior to milling, a protectiverim of either Au or Pt (pale blue strip in the top edge of thelamella in the sketch) was deposited in order to help mini-mize Ga damage during milling, (b) FIB image of a lamellatilted at 45 for better viewing, (c) drawing of the process tocross-section a lamella, and (d) cross-sectional TEM image ofa lamella milled by FIB.

the Kittel law but with a different numerical constant.When d0 → κASL, the domain width diverges, implyinga transition to a mono-domain state. Domains thereforeexist only when:

d0 < κASL (17)

or, taking into account (14) and (16) when:

AFL . 7.4A2

SL

D(18)

where D is estimated as (15). This represents a rathernarrow constraint, providing assumption (4) is satisfied.Substitution of the optimal parameters r and d into

(13) gives the energy of the domain state

W = WT + 2d0

κASL

[

1−1

2

d0κASL

]

ASLδ2acG (19)

which is smaller than (1) (since 2x(1 − x/2) ≤ 1 withx = d0/κASL) in the limits of applicability of the the-ory. This confirms the instability of the ferroelectric free-standing lamella with surface tension towards domainformation.

Comparison with experiment

An attempt was made to use the model developedabove to describe the variation in domain periodicity ob-served for single crystal BaTiO3 lamellae quantitatively.

5

50 nm

Pt

BaTiO3Crystalline BaTiO3

Amorphous BaTiO32 nm

a b

FIG. 3: (a) HRTEM image of a cross-sectional BaTiO3

lamella after FIB milling showing a 20nm thick damaged layerat the Pt/BaTiO3 interface (the Pt epilayer was deposited topreserve the original surface structure associated with FIBprocessing of the lamellae) (b) Zoom in on the damage layer,showing its amorphous structure.

To do this it was noted that the previously published do-main period data [7] had all been taken from lamellae forwhich there had been no attempt to repair surface dam-age caused by focused ion beam (FIB) processing. It wasexpected that physical ’encapsulation layers’ of amor-phous BaTiO3 should exist on the top and bottom lamel-lar surfaces. To establish the thickness of the physicallydamaged layers, cross-sectional transmission electron mi-croscopy was used, with the lamellar cross-sections pre-pared by FIB according to the schematic shown in figure2. As can be seen in figure 3, a surface layer of amorphousmaterial, approximately 20nm in thickness, was indeedobserved. In conjunction with energy dispersive x-raydata, this layer was categorized as a gallium-impregnatedbarium titanate glass.Substitution of this glassy layer thickness as that of

an encapsulation layer (ASL) and using a domain wallenergy density of σ =3Ö10−3Jm−1 [29] produced a re-markably good quantitative description of the observeddomain periodicity data, as can be seen in figure 4. Thisstrong agreement was obtained without any free fittingparameters, and one might naturally conclude that all ofthe encapsulation suffered by the BaTiO3 lamellae wasindeed due to the constraint from the ion beam damagedlayers.However, we have spent some considerable effort to

develop processing methodologies to repair the surfacedamage caused by focused ion beam processing. Ther-mal annealing in air at 700 � has been seen to both re-crystallise the damage and expel the implanted gallium(forming thin gallium oxide platelets), recovering pris-tine single crystal BaTiO3 [33, 34, 35](see figure 5). Ifthe thermal annealing is performed in oxygen, then func-tional measurements even suggest that the permittivityof the surface region is the same as that seen in bulk[33, 35].The domain structure seen in a lamella which had been

thermally annealed and is expected to have no surfacedamage is shown in figure 5-c. The domain appearance

FIG. 4: (Colour online) Solid data in black colour: experi-mentally measured domain width as a function of thicknessfor free-standing single crystal films of ferroelectric BaTiO3

with a dead layer thickness of ∼ 20 nm. The red colour pointrepresents the domain width after annealing at 700 �for 1 hin air. For a lamella thickness of 150 nm, the domain widthchanges from 76.6 ±6.5 nm (before annealing) to 88.4 ±3.5nm (after annealing). Dotted line, calculation using eq. (16)assuming ASL = 20nm and a domain wall energy density ofσ ≈ 3 · 10−3

Jm−1 [29]. The agreement is good without free

fitting parameters

is almost identical to the unannealed sample in Figure 1,with the domain walls still of 110-type, indicating that90°domain sets have again formed. While gallium oxideplatelets have precipitated on the annealed surface (thewhite blotches in figure 5-c), these platelets certainly donot form a continuous layer, and it seems unlikely thatthey can provide the homogeneous stress needed to in-duce the 90°domain sets observed. Further, when theperiodicity of the domains is analysed and compared tothat obtained in unannealed lamellae, only a slight in-crease in domain period is observed; the increase in do-main period is in fact consistent with an effective increasein the thickness of the lamella by 40nm (that associatedwith the recrystallisation of the amorphous glassy regionsdescribed above). The essential physics at play thereforeappears to be unchanged even when the extrinsic damagelayer is removed.

Overall, then, it appears that the quantitative agree-ment between the model and our data may be fortuitous,and that the stress responsible for the domains is not nec-essarily coming from the glassy barium titanate. Instead,a more fundamental source of stress must be at play forall of the FIBed single crystal barium titanate lamellaeinvestigated to date. This suggests the existence of anintrinsic surface relaxation layer, probably due to surfacetension.

The existence of surface layers in BaTiO3 has beenknown for quite some time [16, 17, 36, 37, 38, 39, 40, 41,

6

(c)

FIG. 5: (a) HRTEM image of a cross-sectional BaTiO3

lamella after FIB milling and annealing at 700C in air. (b) Aninverse FFT image of the Au/BaTiO3 interface showing thecomplete reconstructed structure by post-annealing. Latticefringes are evident up to the boundary with an Au protec-tive layer, with no evidence of amorphization (c) Plan-viewTEM image of the annealed lamella, showing the expelled Gaas clusters on the surface and the persistence of the regulardomain pattern.

42, 43], yet there is surprisingly little agreement abouttheir properties. Experimentally, their thickness seemsto be on the region of 5-10 nm or more [16, 17, 36, 37, 38], whereas first principles calculations give a much smallervalue, about 1nm [39, 40, 41, 42] . In some works the SLis found to be tetragonal at all temperatures, even aboveTc [36, 37], where in others it is cubic even below Tc[16, 17]. In fact, the structure of the SL can be rathercomplex and depends on processing conditions [43]. Nev-ertheless, what is important from the point of view of ourmodel is not so much the exact symmetry of the surface, but the fact that it does not undergo the same ferro-electric/ferroelastic phase transition as the inside of thefilm. Thus, when the film becomes ferroelectric, it auto-matically becomes stressed by the untransformed surfacelayer. The present model requires only the induced stressis isotropic or orthotropic, and this is the case not onlywith cubic SL’s, but also with either amorphous SL or

with tetragonal SL provided that the tetragonal axis isout-of-plane.On the other hand, the elastic energy stored by the sur-

face layer is proportional to its thickness. The outstand-ing question, then, is whether an intrinsic and very thinSL due to surface tension can lead to the same domainsize as would be expected from the thicker encapsulationlayers seen in our unnanealed samples.Indeed, Eq.16 states that the domain size is essentially

d0, independent of the SL thickness; however, this isequation is only valid when d0 ≪ κASL, which is nottrue if ASL ≈ 1nm. On the other hand, our model hasimplicitly assumed that the stiffness (the shear modulusG) is the same for FL and SL, and this is unlikely whenthe SL is an intrinsic surface-tension layer; for these, thebonds are known to be shorter [42] and the SL shouldbe expected to be harder than the FL. It is relativelystraight-forward to incorporate the different shear mod-ulus of the surface layer (GSL) and the ferroelectric layer(GFL) onto the model, by simply substituting G for GFL

and GSL in eqs. (8) and (10) respectively. Minimizationof the total energy then leads to the new generalized ex-pression for domain size:

d =d0

1− d0

κASL

GFL

GSL

(20)

This expression is almost identical to (16) except forthe appearance of the factor GFL/GSL in the denomi-nator. This factor can compensate for a reduced thick-ness of the surface layer insofar as its hardness is greaterthan that of the ferroelectric layer. Presently we haveno quantitative estimates for the hardness of the intrin-sic epilayer, and we very much encourage the theoreticalcommunity to perform first principles calculations of thevalue of GSL. What we can say is that, if the thickness ofthe SL is only 1-2nm, as suggested by the ab-initio calcu-lations [39, 40, 41, 42], its shear modulus would need tobe roughly 10 times bigger than that of the FL to havethe same effect on domain size as our 20nm extrinsic en-capsulation layer. If, however, the true thickness of theintrinsic SL is ca. 10nm, as suggested by experimentalmeasurements [16, 17, 36, 37, 38] then the SL need notbe more any more rigid than the FL.

Conclusions

In sum, we have shown that, even in the absence ofrigid substrates or any other source of external stress,ferroelastic twinning can appear due to the self-stress im-posed by surface layers. The importance of such layersobviously increases as the size of the system decreasessuch that this effect becomes particularly important atthe nanoscale. Furhermore, there need not be an extrin-sic surface layer for the ferroelastic domains to appear;

7

surface tension, which is intrinsic and therefore unavoid-able, can also provide the necessary stress for domain for-mation. Finally, we note that epilayers can be expectedto be important not only for isolated nano-ferroelectrics,but also in macroscopic devices such as ceramic capaci-tors made with nano-powders or core-shell grains.This work was supported by the EC project FP6-

STREP-MULTICERAL and by the French-UK collab-oration program “Alliance”.

[1] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153(1935).

[2] C. Kittel, Phys. Rev. 70, 965 (1946).[3] T. Mitsui and J. Furuichi, Phys. Rev. 90, 193 (1953)[4] S. K. Streiffer, J. A. Eastman, D. D. Fong, Carol Thomp-

son, A. Munkholm, M.V. Ramana Murty, O. Auciello, G.R. Bai, and G. B. Stephenson, Phys. Rev. Lett. 89, 67601(2002).

[5] A. L. Roytburd, Phys. Status Solidi A 37, 329 (1976).[6] J. F. Scott, J. Phys. Cond. Mat. 18, R361 (2006).[7] A. Schilling, T. B. Adams, R.M. Bowman, J. M. Gregg,

G. Catalan, and J. F. Scott, Phys. Rev. B 74, 024115(2006).

[8] G. Catalan, J. F. Scott, A. Schilling, and J. M. Gregg, J.Phys. Cond. Mat. 19, 022201 (2007).

[9] V. A. Stephanovich, I. A. Luk’yanchuk, and M. G.Karkut, Phys. Rev. Lett., 94 047601 (2005).

[10] F. De Guerville, I. Luk’yanchuk, L. Lahoche and M. El-Marssi, Mater. Sci. Eng. B 120, 16 (2005).

[11] A. Schilling, R. M. Bowman, J. M. Gregg , G. Catalan,J. F. Scott, Appl. Phys. Lett. 89, 212902 (2006).

[12] G. Catalan, A. Schilling, J. F. Scott and J. M. Gregg, J.Phys. Cond. Mat. 19, 132201 (2007).

[13] A. Schilling, R. M. Bowman, G. Catalan, J. F. Scott, andJ. M. Gregg Nano Lett. 7, 3787 (2007).

[14] G. Catalan, H. Bea, S. Fusil, M. Bibes, P. Paruch,A. Barthelemy, and J. F. Scott, Phys. Rev. Lett. 100,027602 (2008).

[15] M. Daraktchiev, G. Catalan, J. F. Scott, Ferroelectrics375, 122-131 (2008).

[16] T. Takeuchi, K. Ado, T. Asai, H. Kageyama, Y. Saito, C.Masquelier, O. Nakamura, J. Am. Ceram. Soc. 77, 1665(1994).

[17] M. Tanaka and Y. Makino, Ferroelec. Lett. 24, 13 (1998).

[18] J. Slutsker, A. Artemev, and Alexander Roytburd, Phys.Rev. Lett. 100, 087602 (2008).

[19] W. Pompe, X. Gong, Z. Suo, and J. S. Speck, Appl. Phys.74, 6012 (1993)

[20] J. S. Speck and W. Pompe, J. Appl. Phys. 76, 466 (1994).[21] J. S. Speck, A. C. Datkin, A. Siefert, A. E. Romanov and

W. Pompe, J. Appl. Phys. 78, 1696 (1995).[22] N. A. Pertsev and A. G. Zembilgotov, J. Appl. Phys. 78,

6170 (1995).[23] B. L. Cheng, M. Gabbay, and Fantozzi G, J. Mat. Sci.

31, 4141 (1996).[24] G. Bradfield, Nuovo Cim. 7, 182 (1950).[25] W. Ma, L.E. Cross, Flexoelectricity of BaTiO3, Appl.

Phys. Lett. 88, 232902 (2006).[26] G. Catalan, L. J. Sinnamon, J. M. Gregg, J. Phys.: Con-

dens. Matter 16, 2253 (2004).[27] G. Catalan, B. Noheda, J. McAneney, L. J. Sinnamon,

and J. M. Gregg, Phys. Rev. B 72, 020102 (R) (2005).[28] M. S. Majdoub, R. Maranganti, P. Sharma,

arXiv:0812.2083 (2008).[29] A. Zhirnov, Sov. Phys. JETP 35, 822 (1959).[30] B. Meyer and D. Vanderbilt, Phys. Rev. B 65, 104111

(2002).[31] N. Floquet and C. Valot, Ferroelectrics 234, 107 (1999).[32] D. Shilo, G. Ravichandran and K. Bhattacharya, Nat.

Mater. 3, 453 (2004).[33] M. M. Saad, R. M. Bowman, and J. M. Gregg, Appl.

Phys. Lett. 84, 1159 (2004).[34] A. Schilling, T. Adams, R. M. Bowman, J. M. Gregg,

Nanotechnology 18, 035301(2007)[35] L. W. Chang, M. McMillen, F. D. Morrison, J. F. Scott,

and J. M. Gregg Appl. Phys. Lett. 93, 132904 (2008).[36] M. Anliker, H. R. Brugger, and W. Kanzig, Helv. Phys.

Acta 27, 99 (1954).[37] W. Kanzig, Phys. Rev. 98, 549 (1955).[38] Feng Tsai and J. M. Cowley, Applied Physics Letters 65,

1906 (1994).[39] J. Padilla and D. Vanderbilt, Phys. Rev. B 56, 1625

(1997).[40] B. Meyer, J. Padilla, and D. Vanderbilt, Faraday Discuss.

114, 395 (1999).[41] C. Bungaro and K. M. Rabe, Phys. Rev. B 71, 035420

(2005).[42] R. I. Eglitis and D. Vanderbilt, Phys. Rev. B 76, 155439

(2007).[43] A. M. Kolpak, D. Li, R. Shao, A. M. Rappe, and D. A.

Bonnell, Phys.Rev.Lett. 101, 036102 (2008).