onlinear Dielectric Response of Perioic Composite aterials A.G. KOLPAKOV 3, Bl.95, 9 th ovember str., ovosibirsk, 639 Russia the corresponing author e-mail: agk@neic.nsk.su, algk@ngs.ru A. K.TAGATSEV Ceramics Laboratory, Swiss Feeral nstitute of Technology, EPFL, 5 Lausanne, Switzerlan L. BERLAD Department of athematics, Penn State University, University Park, PA, 68, USA A. KAAREK Eucli TechLabs LLC, Solon, OH 39 USA Abstract. The paper aresses the rigorous treatment of the tunability effect in highcontrast two imensional perioical composite a ferroelectric matri with the high value of ielectric constant fille with ielectric inclusions. The theoretical analysis showe that the tunability of the composite was weakly affecte by the nonlinear ferroelectric ilution with the ielectric inclusions of the relatively low value of the ielectric constant. The results of our simulations corroborates with eperimental ata on Ba Sr - TiO 3 BST ferroelectric with the g base aitives. The result of the numerical analyst gives no support to the ecouple approimation in the effective meium approach often use for the escription of the ielectric non-linearity of composites. Keywors: nonlinear composite, electrostatic problem, overall properties, local electric fiel, homogenization, tunability..ntrouction The potential usefulness of ferroelectric materials in high frequency tunable evise has been recognize on early stages of their investigation. n general, these materials can appreciably change their ielectric permittivity ε uner the application of a c electric fiel.
n other wors, these materials can ehibit high relative tunability efine as [] ε ε E n r ε where where εe is the ielectric permittivity of the material corresponing to the fiel E applie to the material. t is this property that can be use for fabrication of electronic evise with parameters that can be tune by the application of a controlling c voltage. However, ue to various reasons relate to both evice electronics an materials technology, it is only in the last few years that intensive evelopment efforts are being mae in this irection. One of the irections of the evelopment in the fiel is the use of ferroelectric/ielectric composites. Typical eample of such material is Ba,SrTiO 3 ceramics fabricate with aition of go, which for volume concentrations beyon -% precipitates in a reaily istinguishe seconary phase. Figure show a SE image ocumenting this kin of structure []. The ielectric constants of these two phases are very ifferent typically - for Ba,SrTiO 3 an about for go, so that the ilution of ferroelectric with the ielectric can obviously lea to an appreciable reuction of its ielectric constant of the material. The tunability values of the two phases are also iffer significantly, for ferroelectric one can reaily have n.. 3 whereas go oes not ehibit any measurable tuning at realistic values of the electric fiel. Thus, one might epect some reuction of tunability of material with increasing concentration of ielectric content. This reuction might be epecte to be much stronger than that of the ielectric constant, if one base corresponing estimates on the Lanau theory of pure ferroelectrics. At the same time the eperiment shows that, in fact, the aition of the ielectric up to appreciable concentrations oes not affect tunability of material at least as far as the effect of mechanical miing of the two phases is concerne [,3,]. This unepecte phenomenon has been aresse theoretically see for a review []. t has been shown that inee in a moel, where a small concentration of spherical ielectric inclusion is embee into ferroelectric an tuning is still small, no ecrease of tunability with ilution is epecte. At the same time, a slight increase of tunability is preicte [] nr, eff nr +.q r
whereas the same moel gives an appreciable reuction of the permittivity ε ε.5q. eff Here ε eff an r eff n, stan for the permittivity an relative tunability of the composite containing a volume fraction q of the ielectric whose permittivity is much smaller than that of the matri. Though, the qualitative agreement between the simple theory an eperiment is encouraging, one shoul recognize the essential limitations of the theoretical result. First, the theory aresses only the limit of small concentrations, so that the comparison of the theoretical preictions with the eperimental ata typically up to q. 3 is not fully justifie. Secon, the mathematical analysis use in the aforementione theoretical treatment is not fully justifie from the point of view of the rigorous mathematical theory of composites [5, 6]. Concerning the case non-linear response at noticeable ielectric concentrations, there eists a problem with the application of any analytical metho so that for further progress the authors have to introuce aitional assumptions into the theory. n a rather popular ecouple approimation in the effective meium approach [7,8] it was postulate that for the spatial istribution of the electric fiel in the composite E i there is a relation spatial average. < E >< E > i i between its momentums, where > <... stans for the n this paper, we present the results of rigorous mathematical treatment analytical an numerical of the problem of the ielectric non-linearity of the ferroelectric ielectric composite. We present the result obtaine for the case of D perioic composite, which correspons to the material where a system of ielectric ros is embee into a ferroelectric matri an epose to electric fiel that is normal to the irection of the aes of the ros. Though the consiere moel is quit far from the real eperimental situation in ferroelectric/ielectric composites, which are uner common eperimental investigations, we believe that our analysis oes contribute to the problem as presenting the first rigorous mathematical treatment of this kin of the problem. 3
.Tunability of composite materials The relative tunability of an electrically tunable material has been introuce by Eq.. We will aress the case of small tuning where the function εe can be approimate as see e.g.[] For, formula takes the form ε E + µ E ε. µ n r E. 3 ε Composite material is an inhomogeneous material Fig., i.e. material whose properties epen on the spatial variable, y. Then, in ε an µ are functions that epen on the spatial variable. Then, for composite, takes the form ε, φ ε, + µ φ. Here φ enotes the electrostatic potential E φ. The local electric isplacement is relate with φ by the formula D ε, φ φ an satisfies the equation Substituting in 5, we obtain equation ivd. 5 iv [ ε, φ φ]. 6 Solving 6 with a corresponing bounary conition, we can fin the potentialφ. n general, it is a very comple problem ue to inhomogeneity in. A composite forme of many small components looks like a homogeneous material with its own material properties calle in literature overall, microscopically, effective an homogeneous characteristics. Several authors aresse computations of the overall properties for the composite escribe see e.g. [7, 9]. Here we aress the case of perioical composite illustrate in Fig. an the case of weak nonlinearity, i.e where the fiel inuce relative change of the
ielectric constant is small n <<. n this case, the overall ielectric constant of the composite can be presente in the form [5] r where an A ε eff E A + BE, 7 ε, 8 B µ 9 here enote the volume of the elementary cell of the composite an is solution of the so-calle cellular problem see Appeni for etails iv[ ε, ] in,, / /, ote that the average value of,/ /,, y for y / an /. over the elementary cell of the composite is equal to unity. Really, taking into account the secon equality from, perioicity of in the variable y an irection of the normal vector n on bounary of the cell n / /., we have Since A an B are inepenent of the scaling transformation of the potential, the above equation can be written in imensionless form. Relations analogous to 7 an 8 has been earlier of obtaine for the case of arbitrary composite by Strou with coworkers see e.g., Ref. [9] however, the metho use in this work cannot be consiere as fully mathematically rigorous. Our erivation of this relation for D perioical composite base on the homogenization metho [5] guaranties valiity of the formulae 7-9 for small fiels. The overall tunability of composite escribe by equation 7 is thus B. A n r, eff E 5
3.Analysis of overall properties of composite Observing the formulas for the overall characteristics of composite see formulas 8, 9 an, we fin that the main problem in computation of these properties is computation of function. Let us iscuss this problem for materials uner consieration. Our analysis is carrie out for a two-phase composite, which consists of nonlinear matri with linear inclusions Fig.3. n accorance with the ata presente in ntrouction above, we can assume that the ielectric constant of the inclusions is rather small as compare to the ielectric constant of the matri. Then, we can approimately accept that ε in inclusions, ε ε in the matri. The material of inclusions is a linear one an the matri material is a non-linear accoringly. Then µ in inclusions, µ µ in the matri. For this case, we obtain from 8,, an 3 the following formula for the parameters A, B an n r, eff : where A ε, B µ n, eff, JE, 3 µ r ε, ote that tunability of pure ferroelectric is µ E ε, J., then, ue to the equation 3, J is equal to the ratio of tunability of composite to tunability of pure ferroelectric. nr, eff J. nr The formula below enables us to evaluate the impact of ilution on the n, factor. r eff.estimates for the homogenize tunability of a high-contrast composite For a high-contrast composite, the value of tunability in accorance with the formulas - can be epresse through the values of the integral functionals 6
7 an for the problem solution. As it was mentione in ntrouction above, a popular metho so-calle ecouple approimation was use for evaluation of the ielectric non-linearity of composites. t was postulate that for the spatial istribution of the electric file in the composite E i, there is a relation > >< < E E i i. t terms of our variable, this means that. Below we present our estimates for the quantities an respectively. n the net section it will be emonstrate that oes not approimate for the local fiel etermine out of the electrostatic problem solution... Low-sie estimate. Let us apply Caushy-Bounaykovski inequality to the integral in the following way / / / /. Then. Diviing by, we obtain. Thus Taking into account that < q, where q is volume fraction of the ielectric inclusion, we conclue that the relation, which makes the basic assumption of the ecouple approimation can near take place in the consiere system. However, the result
8 obtaine can eclue that. We will return to this problem later when treating the problem numerically... Upper-sie estimate. Let us apply Caushy-Bounaykovski inequality to the integral in the following way / / 8 / / 8. Then 8. Diviing the epression above by, one can obtain 8. Thus 8 or 8 q. Another upper-sie estimate can be obtaine as follows. We apply Caushy- Bounaykovski inequality to the integral in the following way / / 6 3. Then 6. Diviing again by, we obtain
9 8. The last multiplayer in the right-han sie of this equality is equal A. Then, we have A 6. 5. Computer simulations We consier a perioic cell with circular inclusion symmetric with respect to the coorinate ais. To etermine the function it woul be sufficient to solve so calle cellular problem for the perioicity cell of the composite. The cellular problem is equivalent to the problem of minimization of the integral functional ε on the set of function satisfying following bounary conitions,,, an, y n for y an y. as well. Using symmetry approach, one can solve the corresponing problem for ¼ of the perioicity cell. The erivatives an y were approimate using finite-ifferences approimation h j i i i,, + an h j i i i,, +, where h is the size of iscretization. As a result, we obtain a quaratic function of finite variables. inimizing this function by iterative metho, we obtain approimate solution of the problem above. We compute the integrals 8 an 9 numerically then. A computer coe for the -D cellular problem solution has been evelope. The computation results are presente in Tables an. Table shows the values of the integrals as functions of volume fraction q of ielectric inclusion for high-contrast composite ε iel. The quantities
, have been compute as well. The quantity an 8 8. A is the ratio of the ielectric constant ε of the composite to the ielectric constant of pure ferroelectric matri. The quantity J n r, eff is the ratio of tunability of composite to one of pure ferroelectric matri. n r For rather low contrast composite we compute the quantity A in accorance with the formula 8 A ε, Table shows that an vary approimately twice when the volume ratio varies from.9 to.35. From Table it is seen that. The ratio tunability value if it s multiplie by which is equal to the µ varies from.98 to.9 when the volume ratio ε varies from.9 to.35. t means that tunability is still a stable quantity in contrast to other quantities inicate in Table. Our numerical simulation also enables evaluation of the valiity of the basic assumption of ecouple approimation in the effective meium approach [7, 8], which postulates that < E i >< Ei >. t terms of our variable this means that it is clear form Table, in the case of the contrast composite rather than. As. Thus, our calculations give no support to the ecouple approimation in the effective meium approach. 6. Conclusions We have carrie out the theoretical analysis of weak ielectric non-linearity in highcontrast nonlinear composites of D perioic structure. The analysis was one using the rigor homogenization metho earlier evelope for mathematical treatment of composites. For the moel consiere, this approach provies the full mathematical justification of the
earlier obtaine relations for the ielectric non-linearity of the weakly nonlinear ferroelectric composites [,9]. Our numerical treatment of D perioic composites of this kin has shown that the tren establishe for the case of low concentrations namely, the impact of the ilution on the permittivity is much stronger than on tunability, can hol for appreciable ielectric concentrations. This corroborates with the recently obtaine eperimental finings on the Ba,SrTiO 3 -go system [,3,]. Using the result of our numerical simulations we have also probe the principle assumption of the ecouple approimation in the effective meium approach to fin no support for this assumption for the system consiere in this paper. 7. Acknowlegments This work was supporte by DoE SBR contract DE-FG-ER8396. The work of L. Berlyan was also supporte by SF grant DS-637, an the work of A.G. Kolpakov was in part supporte by SF grant DS-637. References. A.K. Tagantsev, V.O.Sherman, K.F.Astafiev, J.Venkatesh an.setter, Journal of electroceramics., 5-66 3.. E. enasheva, A. Deyk private communication. 3. L. C. Sengupta an S. Sengupta, at. Res. nnovat., 78 999. E.A.enasheva, A.D.Kanareykin,.F.Karpenko an S.F.Kramarenko, Journal of Electroceramics,3, 35-38. 5. A. Bensoussan, J-L, Lions an G.Papanicolaou, Asymptotic analysis for perioic structures orth - Hollan Publ.Comp. Amsteram, 978 6. A.L.Kalamkarov an A.G.Kolpakov, Analysis, esign an optimization of composite structures. John Wiley&Sons, Chichester, ew ork, 997. 7. L. Gao an Z. Li, Journal of physics: Conense matter, 5, 397-9 3. 8.. Sahimi, Heterogeneous aterials Springer, ew ork, 3 9. D. Strou an P.. Hui, Phys. Rev. B. 37, 5, 879-87 988...E.Tamm, Theory of electricity oscow. auka, 8th eition, 976 [in Russian].
. W.R. Smythe, Static an ynamic electricity cgrow Hill, ework, n eition, 95.. V.L.Berichevski, Variational principles of mechanics of solis auka, oscow, 983. 3. L.D.Lanau an E..Lifshitz, Electrostatics of continuum mei. Pergamon, Ofor, 98.
APPEDX Homogenization proceure for electrostatic problem arbitrary volume fraction of inclusions Let ielectric constant of nonlinear material has the form A ϕ a + λb ϕ, where λ is a small parameter. t means that we consier a nonlinear material with small nonlinearity too. The small nonlinearity can be use as moel of significantly nonlinear material but uner conition that the electric fiel magnitue is rather low as well. Let a particles of another material to be introuce into this material. Then A A, ϕ a + λb, ϕ. A Potential of the electric fiel in the composite satisfies the following ifferential equation iv [ A, ϕ ϕ]. A We consier -D composite of perioic structure for which computation of homogenize characteristics can be reuce to analysis of a problem for one perioicity cell only. We assume that the perioicity cell is a rectangle one with the center in the origin of coorinate aes an with the length of the sies equal to L an respectively. We assume that it contains one inclusion, which is symmetric with respect to the origin Fig.A. These assumptions are one in orer to simplify the mathematical evaluations to be presente below. We assume that the horizontal pair of sies are subjecte to voltage ± EL these values correspon to the average fiel magnitue E an on another pair of sies potential is perioic see. Fig.A. Due to the summery an equivalent-ness of all perioicity cells, we obtain the following conitions on the bounary of one perioicity cell ϕ, ± L / ± EL /, A3 ϕ ± /, y. A The total flu of the electric isplacement D A ϕ ϕ through the specimen is the sum of flues through all its sies. The flu through the upper sie Γ { L / < < L /, y / } of the perioicity cell, corresponing to ifference of 3
potentials U in A3 is equal to ϕ F A, ϕ. A5 Γ ultiplying A by ϕ an integrating by parts, we obtain with regar to A3 an A A, ϕ ϕ + A, ϕ ϕ. A6 Flu through the lateral sies of the cell is equal to zero see A then ± L / Γ A, ϕ ϕ. A7 Taking into account A7 an bounary conition A3 we obtain from A6 that ϕ A, ϕ ϕ + A, ϕ [ ± EL / ]. A8 y± L / Flues through the upper an low sies of the cell are equal to ϕ ϕ A, ϕ A n, ϕ. A9 n y L / y L / From A8 an A9 we obtain ϕ A, ϕ ϕ EL A, ϕ. A n y ± / y / From A8, A flu correspons to the ifference of potentials U we obtain D total Γ ϕ A, ϕ A, ϕ ϕ. A EL The specific flu D flu per unit length of the horizontal bounary y L / of the perioicity cell is D EL A, ϕ ϕ. A The formula A correspons to the homogenization theory, see [5] if we enote L the measure the square in the case uner consieration of the perioicity cell. Consiering small nonlinearity, we will carry out homogenization analysis in combination with the metho of classical small parameter. Following [6], we fin solution A-A in
the form of ϕ ϕ + λϕ... A3 + Substituting A3 in A-A, we have then iv a + λb, ϕ + λϕ +... ϕ + λϕ +...], A [ ϕ + +..., ± / ±U, A5 λϕ ϕ + λϕ +... ± /, y. A6 From A-A6 we obtain the equations of the th orer terms corresponing to iv [ a ϕ ], A7 ϕ, ± / ± /, A8 E ϕ ± /, y, A9 an equations of the st orer terms corresponing to λ iv a ϕ + b, ϕ ϕ ], A [ ϕ, ± /, A ϕ ± /, y Substituting A3 in A, we obtain. A F A, ϕ + λϕ +... ϕ + λϕ +.... E Saving terms which orer is not higher than the first orer of µ, we obtain λ F A, ϕ ϕ + E A3 λ + E a ϕ ϕ + b, ϕ ϕ ]. [ ϕ n A3 we omit the terms of the highest orer. Carry out same transformations. ultiplying A7 by ϕ an integrating, we obtain iv a ϕ ] ϕ. [ 5
ntegrating by parts, we have ϕ a ϕ ϕ + a ϕ. A Consier the first bounary integral in A5. For the upper an lower sies of the cell, we have y± / ϕ a ϕ, y. A5 The last equality takes place ue to ϕ, ±L/ in accorance with A. For lateral sies of the cell ± / ϕ a, y ϕ, y. ϕ The equality takes place ue to ± /, y in accorance with A9. Then the bounary integral in A is equal to zero an from A it follows that a ϕ ϕ. A6 ote that we cannot obtain an analog of equality A6 for arbitrary function. We obtain A6 using the fact that sies on which ϕ an. Due to A6 we can rewrite A3 as ϕ λ F A, ϕ ϕ + EL b, ϕ ϕ EL are zero cover all bounary of the cell. A7 The specific flu of electric fiel through the perioicity cell has the form F F + µ F, where F A, ϕ ϕ, A8 EL µ F b, ϕ ϕ EL. A9 The homogenize ielectric constant D of composite is introuce as ratio of total flu 6
F of the electric fiel to the corresponing initiating this flu ifference of potentials E. n the case uner consieration F D D + µ D, E where D ϕ a ϕ a, A3 E L L U λ ϕ. D b, ϕ ϕ λ b, ϕ E L L U A3 We introuce, y as solution of linear cellular problem A7-A9 with E /. Then ϕ E, A3 an we can write formulas A3, A3 in the following form where D a L, A33 D λ b, U, A3 L Write on the cellular problem. t has the following form iv[ a ], A35, ± L / ± L /, ± /, y. This problem correspons to the ifference of potentials equal to. Let us iscuss the formulas obtaine above. Formula A33 is known from paper [5]. t introuces the homogenize ielectric constant of linear composite. Let us consier composite «nonlinear matri linear inclusion». n this case b, E in matri, 7
Then formula A3 takes the form b, E in inclusion. D λ b U, A36 L t means that we integrate over the matri only. n this case a an b U o not epen on the spatial variables, they take values corresponing to the matri. Assume that b,. Uner this assumption the tunability n r, eff is compute in accorance with the formula n r, eff b U D U λ λ, D a The tunability of the material of the matri is b U n r λ. a A37 Diluting matri which has high ielectric constant with particles which have low ielectric constant, we believe that the overall ielectric constant D as well as ecrease. When D an D ecrease against a an b U, the ratio A37 can increase an ecrease against We can write U b U both. a b U D U b U b U D a a a a [ b U b U ] ad. A38 t is seen from A38 that tunability is etermine by istribution of local electric fiel solution of the cellular problem A35. The quaratic function A. n applications function A is often taken in the form of quaratic function. n this case b U µ U. 8
n this case A3 becomes D λ U µ. A36 t was consiere the case when the perioicity cell. Every square can be transforme into the cell. As a result of this transformation we obtain that for arbitrary square all integrals over an must be ivie by the square of the cell. This conclusion is in accorance with general homogenization theory [5]. 9
FGURES Fig.. SE images of the Ba,SrTiO 3 go samples at magnifications of 5. Two phases of the composite material are clearly seen: the ark phase is go an the light one is the main phase of BST with -5 µm crystallites [].
y/ -/ / cell y-/ Fig.. A moel of composite material left an its perioicity cell inclusion, matri Fig.3. A structural element of composite matri, - inclusion
yl/ / L Fig.A. Perioicity cell of composite
Table umerical computations for high-contrast composite. ntegrals as functions of volume fraction q of ielectric the inclusions. q 8 J.9.9369.8697.8786.3.98335.9.8578.67797.7755.35388.9575.35.7333.839.536.758886.9 3