Flexoelectric effect in finite samples
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Flexoelectric effect in finite samples
Alexander K. Tagantsev
Ceramics Laboratory, Swiss Federal Institute ofTechnology (EPFL), CH-1015 Lausanne, Switzerland
Alexander S. Yurkov (Dated: October 3, 2018)
Abstract
Static flexoelectric effect in a finite sample of a solid is addressed in terms of phenomenologicaltheory for the case of a thin plate subjected to bending. It has been shown that despite an explicitasymmetry inherent to the bulk constitutive electromechanical equations which take into accountthe flexoelectric coupling, the electromechanical response for a finite sample is ”symmetric”. ”Sym-metric” means that if a sensor and an actuator are made of a flexoelectric element, performanceof such devices can be characterized by the same effective piezoelectric coefficient. This behavioris consistent with the thermodynamic arguments offered earlier, being in conflict with the cur-rent point of view on the matter in literature. This result was obtained using standard mechanicalboundary conditions valid for the case where the polarization vanishes at the surface. It was shownthat, for the case where there is the polarization is nonzero at the surface, the aforementioned sym-metry of electromechanical response may be violated if standard mechanical boundary conditionsare used, leading to a conflict with the thermodynamic arguments. It was argued that this conflictmay be resolved when using modified mechanical boundary conditions. It was also shown that thecontribution of surface piezoelectricity to the flexoelectric response of a finite sample is expectedto be comparable to that of the static bulk contribution (including the material with high values ofthe dielectric constant) and to scale as the bulk value of the dielectric constant (similar to the bulkcontribution). This finding implies that if the experimentally measured flexoelectric coefficientscales as the dielectric constant of the material, this does not imply that the measured flexoelectricresponse is controlled by the static bulk contribution to the flexoelectric effect.
PACS numbers: 77.22.-d, 77.65.-j, 77.90.+k . INTRODUCTION The flexoelectric effect consists of a linear response of the dielectric polarization to astrain gradient. This is a high-order electromechanical which is expected, in general, to berather weak. However, some of its features make this effect to be of interest from both funda-mental and applied points of view. This has stimulated recent intensive experimental andtheoretical activity in the field.On the fundamental side, it is of interest that this effect cannot be considered just as anon-local generalization of the piezoelectric effect. In contrast to the later, the flexoelectriceffect (response) is controlled by 4 mechanisms of different physical nature, the contributionsof which can be comparable .On the practical side, of first importance is that this effect, in contrast to the piezoelectriceffect, is allowed in centro-symmetric material. It is believed that it is the flexoelectric effectthat is responsible for the generation of an electric field in acoustic shock waves propagatingin centro-symmetric solids . It was recently shown that this effect plays also an essentialrole in electromechanical properties of materials with a moderate level of electronic andionic conductivity . However, the most applied interest is focused on the ”piezoelectricmetamaterial” – composites made of non-piezoelectric components, which exhibit effectivepiezoelectric response generated due to the flexoelectric effect. The work in this direction wasinitiated by pioneering experimental studies by Professor Cross with coworkers and waslater also supported by theory . Presently, (Ba,Sr)TiO -based composites have been shownto yield effective piezoelectric coefficients comparable to those of commercial piezoelectricceramics . It was argued, based on the constitutive equation for the flexoelectric response,that a mechanical sensor made of such metamaterials should exhibit a very unusual property.Specifically, in contrast to piezoelectric based devices, it will not behave as an actuator .There are several reasons to question such statement. First, already in the 60’s of the pastcentury, the group of Professor Bursian reported experimental data on BaTiO crystals andgave arguments based on equilibrium thermodynamics , which contradict this statement.Second, the existence of a linear sensor-not-actuator may come into conflict with the generalprinciples of thermodynamics.The goal of this paper is to address theoretically this conflict situation to demonstratethat despite an explicit asymmetry of the constitutive equations for the bulk flexoelectric2ffect, when this effect is characterized in a realistic finite sample the apparent asymmetryof the electromechanical response will vanish. In particular, this implies that the aforemen-tioned piezoelectric metamaterial should exhibit the identical piezoelectric constants whencharacterized in ”direct” and ”converse” regimes. We will demonstrate this for two leadingcontributions to the static flexoelectric response: the contribution of static bulk ferroelec-tricity and that of surface piezoelectricity. II. STATIC BULK FLEXOELECTRICITY
In the present paper, being interested in the static or quasi-static situation, we are fac-ing three contributions to the flexoelectric response, which are associated with (i) staticbulk ferroelectricity, (ii) surface piezoelectricity, and (iii) surface flexoelectricity . In thissection we will discuss the problem of the relation between the direct and converse effectsfor contribution (i), reserving the next section for contribution (ii). In view of vanishingpractical importance of the surface flexoelectric effect (it is expected not to be enhanced onhigh-dielectric-constant materials which are the only ones suitable for applications) we willnot be treating it in this paper.The static bulk flexoelectric effect is customarily described by the following free energy(density) expansion (see e.g. ): F = χ − ij P i P j − f ijkl (cid:18) P k ∂u ij ∂x l − u ij ∂P k ∂x l (cid:19) + c ijkl u ij u kl (1)where P i and u ij are the polarization vector and strain tensor, respectively, and where theEinstein summation convention is adopted. We will consider this thermodynamic potentialas having the differential dF = E i dP i + σ ij du ij . Then calculating the electric field, E i , andstress tensor, σ ij , as variational derivatives of the free energy of the sample, given by theintegral of F over its volume, one arrives at the following constitutive equations: E k = χ − kj P j − f ijkl ∂u ij ∂x l (2)and σ ij = f ijkl ∂P k ∂x l + c ijkl u kl . (3)The first equation describes a linear polarization response to strain gradient (direct flexo-electric effect). The second one describes the converse flexoelectric effect, implying that to3et a ”mechanical yield”, spatial inhomogeneity of the polarization is needed. From this, onemight infer (as customarily done in relevant papers) that the application of a homogeneouselectric field to a sample will not lead to its deformation. Even being nearly evident, inreality, the last statement is not correct. h /2 -h /2 FIG. 1: Plate of the material exposed to bending and the reference frame used in calculations.
Let us show this for the flexural mode. Consider, a (001) plate of a cubic materialof thickness h in the reference frame specified in Fig.1, with the X and Y dimensionsbeing L and b respectively. To make the analysis transparent, we allow only a cylindricalbending of the plate about OX axis. To simplify the discussion further, we set, for themoment, c = c = 0. In such simplified model, the plate bending is associatedwith ∂u /∂x = 0, whereas u = u = 0 so that Eq.(2) suggests the appearance of P component of the polarization controlled by f component of the flexoelectric tensor. Toaddress the reversibility of this effect, one should check if the application of an electric fieldnormal to the plate will cause its bending. A straightforward way do this is to derive theequation of balance of the bending moment for the plate subjected to a homogeneouselectric field E normal to its suface by integrating Eq.(3) across a Y Z cross-section of thesample: b Z h/ − h/ σ zdz = bf Z h/ − h/ ∂P ∂z zdz + bc Z h/ − h/ u zdz. (4)At mechanical equilibrium, the lhs term must be equal to the minus the component of themechanical moment of the external forces applied to the lefthand (with respect to the cross-section of the integration) part of the plate, − M . Without the first rhs term, this equationdescribes that bending of the sample caused by this moment. To identify the role of thisterm, we first evaluate it using integration by parts: Z h/ − h/ ∂P ∂z zdz = − Z h/ − h/ P dz = − h h P i (5)4here h P i is the averaged polarization induced by the field E in the bulk of the plate. Indoing so we assume that the polarization changes continuously from its bulk value to zero onthe plate boundary. If, however, one explicitly considers nonzero polarization at the samplesurface (as was done previously in Ref. ), then one should revise the traditional boundaryconditions of the elasticity theory . We will return to this issue later in the paper. Since thespatial scale of the polarization variation at the interface is much smaller than the thicknessof the plate, with a good accuracy h P i ≈ P , where P – polarization in the bulk. Thus, theequation for the moment balance can be rewritten as − M /b + f hP = c Z h/ − h/ u zdz. (6)It is clear from this equation that the application of a homogeneous electric field to theplate is equivalent to that of an external bending moment. Thus, we conclude that a finite,mechanically free ( M = 0) sample, placed in a homogeneous electric field, will be bent.This conclusion is closely related to that drawn by Eliseev et al . These authors have shownthat a ferroelectric plate with the out-of-plane orientation of the spontaneous polarizationshould exhibit spontaneous bending due to the flexoelectric coupling. It was found that thiseffect is controlled by a factor R h/ − h/ ( ∂P /∂z ) zdz which was calculated using a numericalsolution for the polarization profiles P ( z ) in the sample. Here it is also worth mentioningthat the bending effect addressed, though being proportional to a component of the bulk flexoelectric coefficient and the bulk value of the induced polarization, is actually controlledby forces applied to the surface of the plate.It is instructive to illustrate quantitatively the ”symmetry” of the direct and converseflexoelectric effects in a finite sample for a situation which is readily mathematically track-able. We will consider the case of a (001) plate of a cubic material in symmetrical flexuralmode with the polarization P normal to the plate and homogenous in its bulk. In the caseof symmetric bending, the curvature of the plate in all crossections normal to it, G , is thesame. Enjoying the results of the theory of thin plates we can express the components ofthe strain tensor in terms of this curvature: u = u = zG ; u = − z c c G ; u = u = u = 0 . (7)Integrating the free energy density, Eq.(1), with strain coming from Eq.(7), over the platethickness, one finds the free energy density per unit area of the plate as a function of P and5 : Φ = χ − hP + D s G − hP G ( f − c c f ) . (8) D s = h c + c c − c c . (9)where D s is a coefficient controlling the flexural rigidity of the plate for this kind of bending.In derivation of Eq.(8), we have again used Eq.(5) (this gave a factor of 2 in the couplingterm from this equation). Similar expression for the free energy of flexoelectric plate in thecylindrical bending mode was offered by Bursian and Trunov , based on purely symmetryarguments. Minimizing Eq.(8) with respect to P and G one finds the equation for the directand converse effects for the plate in symmetric flexural mode: G = 2 hD s µ pl E. (10) D = P = 2 µ pl G. (11)where the electric displacement, D , and µ pl = χ c f − c f c (12)can be treated as an effective flexoelectric coefficient of the plate. The flexural response,given by Eq.(10), is compatible with the results obtained by Eliseev et al for the case ofspontaneous bending of thin plates with the blocking boundary condition for the polariza-tion.Obviously, elements of such plate will work as both actuators and sensors. If roundpieces of the plate with central loading and symmetric free-edge side support are used aselements of a piezoelectric metamaterial, electromechanical properties of the latter will becharacterized by a single effective piezoelectric coefficient d . Using the relation betweenthe cross-section curvature, G , and the maximal deflection, ξ max , for symmetric bending ofa circular plate: ξ max = GR . (13)(where R is the radius of the plate) and using Eq.(10) one readily finds d = µ pl R D s . (14)6 II. CONTRIBUTION OF SURFACE PIEZOELECTRICITY
As was recognized at the first thorough treatment of the flexoelectric response , thepolarization response to a strain gradient in a finite sample, generally speaking, may not befully controlled by the contribution of the bulk static flexoelectricity, even in materials withhigh values of the dielectric constant (high- K materials). The competing effect that is dueto surface piezoelectricity, was not, however, properly addressed theoretically. Thus, it isnot clear if it can in fact compete in high- K materials with the static bulk flexoelectricity.In high- K materials, the contribution of the static bulk flexoelectricity is enhanced, sinceit scales as the dielectric susceptibility (cf. Eq.(11) and Eq.(12)). At the same time, theeffect associated with surface piezoelectricity originates from the presence of the interfaceadjacent layers where the piezoelectricity is induced by the inversion-symmetry-breakingeffect of the interface. Since the sign of the effective piezoelectric coefficients of the layerson the opposite sides of a plate should be opposite (as controlled by the orientation of thesurface normal), bending of the plate should result in dipole moments in these layers, thesign of which are the same. The dipole moment in a layer is proportional to the strain in it,which, in turn, is proportional to the product of the strain gradient and the plate thickness.Having calculated the resulting change of the average polarization of the whole system, thiswill give rise to a net polarization proportional to the strain gradient. From this reasoningit is not obvious that such response will be enhanced once that the dielectric constant of thebulk of the material is high. However, such reasoning does not provide a proper vision ofthe whole effect. In what follows, we will show that such enhancement does take place andthe considered bulk and surface contributions to the flexoelectric response can be readilycomparable in high- K materials. We will also address the problem of the relation betweenthe direct and converse effect for the mechanism related to the surface piezoelectricity.To be specific we will address these problems for the case of symmetric bending of a thin(001) plate of a cubic material. We will model the effect by considering a system consistingof a plate of an ”ideally homogenous” material (i.e. its material parameters are the samethroughout the plate) with the bulk flexoelectric effect being neglected and two thin surfacepiezoelectric layers (Fig.2). The thickness of each layer, λ , is much smaller that that of theplate, h . The top layer is characterized by piezoelectric moduli h and h = h , whereasfor the bottom layer these moduli have the same the absolute value but are of opposite sign.7 λλ FIG. 2: Model for the contribution of surface piezoelectricity to the flexoelectric response of a non-piezoelectric material. The surface layers of thickness λ model the surface adjacent (atomicallythin) layers of the material where the piezoelectricity is induced by the symmetry breaking impactof the surface. We also ascribe to these layers an out-of-plane component of the dielectric constant, ε λ .Let us find the extra free energy associated with the top piezoelectric layer when the plateis symmetrically bent with a cross-sectional curvature, G , and when out-of-plain polarizationin the layer equals P λ . We assume no difference in the elastic properties of the piezoelectriclayers and the plate.We start with the free energy density in the layer defined as F λ = α P λ − P λ [ h u + h ( u + u )] + c u + u + u ) + c ( u u + u u + u u )(15)where c = c , c = c , and α is the inverse dielectric susceptibility of the layer if itwere fully mechanically clamped. Because the plate is thin we set in the layer u = u = hG/ . (16)As for u , we find it from the condition that the surface of the film is mechanically free, ∂F λ /∂u = 0: u = − c c hG + h c P λ . (17)Inserting Eqs.(16) and (17)) into Eq.(15) and multiplying the result with λ , one finds thefree energy of top piezoelectric layer. The energy of the bottom piezoelectric layer is thesame since this layer differs from the top layer by the sign of piezo-moduli and by that ofthe strain; in the expression for energy these signs cancel each other out. Thus, for thecontribution of the two piezoelectric layers to the free energy of the system, we find:Φ λ = 2 λ (cid:20) χ − λ P λ − ( h − c c h ) hGP λ (cid:21) + Φ (18)8here χ λ = ( α − h /c ) − is the true (under the mixed mechanical conditions) dielectricsusceptibility of the piezoelectric layers and Φ is their mechanical bending energy.To describe the direct flexoelectric response we use the equation of state for the polar-ization in the piezoelectric layer, ∂ Φ λ /∂P λ = E λ ( E λ is the electric filed in the layer), thecondition of continuity of the electric displacement, D , in the layer, and the short-circuitcondition. This leads to the following set of equations: P λ = χ λ E λ + ehG (19) D = ε f E f = ε E λ + P λ (20)2 λE λ + hE f = 0 (21)where E f , ε f , ε are the electric field in the bulk of the plate, its dielectric constant, andthe dielectric constant of the free space, respectively, and e = χ λ ( h − c c h ) . (22)Solving this set of equations we find the relation for the direct flexoelectric response: D = 2 e eλG. (23)where e e = e hε f λε f + hε λ (24)and ε λ = ε + χ λ .To describe the converse flexoelectric response we present the elastic energy of the systemat fixed P λ in the form Φ = D s G − hλeGE λ . (25)When writing this equation we have neglected the elastic energy of the surface layer, Φ ,compared to that of the plate and have simplified (19) down to P λ ≈ χ λ E λ . This ap-proximation means that we neglect feedback effect of G on P λ . This effect will yield somerenormalization of D s , but practically such a renormalization is negligible indeed. Minimiz-ing Φ with respect to G and applying the electrostatic relation used above we arrive at theset of equations G = 2 hλeD s E λ (26)9 f E f = ε λ E λ (27)2 λE λ + hE f = E ( h + 2 λ ) (28)where E is the applied (average) electric field. This set leads us to the equation for theconverse flexoelectric effect in the system: G = 2 hD s e eλE. (29)When writing this relation, only the leading terms to within a small parameter λ/h werekept.The following remarks are to be made concerning the results obtained. First, the relationsobtained for the contribution of the surface piezoelectricity into the flexoelectric response,Eqs. (23) and (29), are identical to those obtained for the case of static bulk flexoelectricity,Eqs. (11) and (10), to within the replacement µ pl ⇒ e eλ . Thus, all conclusions about thedirect-converse-effect symmetry drawn in the previous section for the static bulk flexoelec-tricity hold for the contribution associated with the surface piezoelectricity. Second, in ahigh-K material the latter contribution scales as its dielectric constant (similar to the caseof the static bulk flexoelectricity). Formally, this follows from the expression for e e , Eq. (24).Taking into account that the thickness of the surface piezoelectric layer is expected to be ofthe order of the lattice constant, for realistic values of the plate thickness, e e can be evaluatedas e e = e ε f ε λ . (30)Note that there is no reason to consider ε λ in a high-K material (typically it is a ”regular”of incipient ferroelectric in the paraelectric phase) to be high, since the special interplay ofthe atomic forces responsible for the high value of the bulk permittivity will be inevitablydestroyed in the surface layer. Such enhancement of a surface driven effect by a factor ofthe bulk permittivity looks surprising. However, the physical mechanisms behind this effectcan be identified.For the converse effect it is quite transparent. The bending of the system is controlledby the value of the field in the surface layer (cf. Eq. (26)). Due to its small thickness, thisfield is enhanced by a factor of ε f /ε λ , compared to the applied field.For the direct effect, the explanation is less straightforward. This time, the bendingcreates polarization in the surface layer. Because of the inhomogeneity of the system, the10hort-circuiting does not guarantee the absence of the electric field in it so that the polar-ization in the surface layer induces a depolarizing field both in itself and in the bulk of theplate. It occurs that if the surface layer is thin enough whereas ε f /ε λ is large, the polar-ization response is controlled by the depolarizing field in the bulk of the plate. This waythe polarization response of the system becomes sensitive to the bulk value of the dielectricconstant.Another important conclusion is that, taking into account the aforementioned effect ofenhancement, one expects both contributions to the flexoelectric response discussed to beof the same order of magnitude even in high-K materials. These contributions would becomparable if λ e e/ε f were about the typical value of the components of the flexoelectrictensor f ijkl , 1 −
10 V (see e.g. ). For the ”atomic values” of the entering parameters( λ = 0 . e = 1 C / m ) and ε λ /ε = 10, we evaluate λ e e/ε f ≃ IV. DISCUSSION AND CONCLUSIONS
The analysis presented clearly demonstrates that starting from continuous constitutiveelectromechanical equations one can derive the relations for the direct and converse flexo-electric effects in a finite sample, which exhibit the symmetry required by thermodynamics.On the practical level such symmetry implies that a piezoelectric meta-material based onthe flexoelectric effect will exhibit the same effective piezoelectric coefficient in the testingregimes for the direct and converse piezoelectric effect. This is in conflict to the belief ofthose dealing with such meta-materials .At the same time, it is also clear that the analysis, based on the constitutive electrome-chanical equations presented above, is valid only for the situation where the polarization atthe plate surface can be treated as continuously changing from its bulk value to zero. If itis not the case, formally following this analysis one readily finds that the aforementionedsymmetry is violated. For example, for the case of free boundary conditions for polarization( ∂P /∂z = 0 at the boundary), corresponding to nonzero polarization at the sample surface,the equation of mechanical equilibrium, Eq. (4), implies the absence of the converse effect.Such conclusion would be fully consistent with that by Eliseev et al who argued that themanifestation of the converse flexoelectric effect in a plate is strongly dependent on the11oundary conditions for the polarization. At the same time, there is no reason to expectthat the free boundary conditions for the polarization will suppress the direct flexoelectriceffect. Thus, if we followed the calculating scheme employed by Eliseev et al (and usedin Sect.II) we would find, for the free polarization boundary conditions, the absence of theconverse flexoelectric effect in the presence of the direct effect. This would make an apparentcontradiction between the results obtained from the continuous constitutive equations andthose obtained from thermodynamics.We suggest the following resolution to this contradiction. The point is that incorporatingthe flexoelectric coupling into the free energy density of a material leads to a modificationof the boundary conditions for the bulk constitutive equations. Eliseev et al have derivedmodified boundary conditions for the polarization, however these authors have postulatedthat the classical mechanical boundary conditions are not affected by such incorporation.However, as was recently shown by one of the authors , generally, the mechanical boundaryconditions should be modified as well. It has been shown that such boundary conditionsreduce down to the classical mechanical boundary conditions when the polarization vanishesat the surface. This justifies the calculations based on the classical mechanical boundaryconditions, which we have presented in Sect.II. For the general case, the problem of theconverse flexoelectric effects should be revisited with the correct mechanical boundary con-ditions which contain the surface value of the polarization. We expect that such treatmentwill yield the results consistent with the symmetry between direct and converse flexoelectriceffects in a finite sample dictated by thermodynamic arguments.Another important conclusion follows from the results obtained in Sect.III. There, it wasshown that the contribution of the surface piezoelectricity to the flexoelectric response isexpected to be comparable to that of the static bulk contribution (including the materialwith high values of the dielectric constant) and to scale with the bulk value of the dielectricconstant (similar to the bulk contribution). The latter statement actually implies identical(or at least similar) temperature dependences of these contributions. Note that in earlierpublications it was hypothesized that these depedences are expected to be different. Basedon this hypothesis, the fact that the experimentally measured flexoelectric coefficient scalesas the dielectric constant of the material was customarily taken as an indication that themeasured flexoelectric response is controlled by the static bulk contribution to the flexoelec-tric effect. The results from Sect.III essentially change the situation. Now one can state12hat the fact that the experimentally measured flexoelectric coefficient scales as the dielectricconstant of the material does not imply that the measured flexoelectric response is controlledby the static bulk contribution to the flexoelectric effect. V. ACKNOWLEDGEMENTS
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