High Coupling Factor Piezoelectric Materials for Bending Actuators: Analytical and Finite Elements Modeling Results

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1 Excerpt from te Proceeding of te COSOL Conference 009 ilan Hig Coupling Factor Pieoelectric aterial for Bending Actuator: Analytical and Finite Element odeling Reult I.A. Ivan *1,. Rakotondrabe 1 and N. Caillet 1 1 FETO-ST Intitute / Univerity of France-Comté *Correponding autor alex.ivan@femto-t.fr ; addre: FETO-ST, Dpt. AS, Rue Alain Savary, 5000 Beançon, FRANCE Abtract: New giant pieoelectric factor material uc a PN-PT and PZN-PT were reearced during te lat decade and are actually becoming commercially available. A tey eem very attractive for actuator deign, we tudied teir potential in replacing PZT ceramic. In a firt comparative approac, we teted a erie of claic rectangular compoite bimorp tructure of different combination of ticknee. Te goal conited in evaluating te quantitative improvement in term of maximum diplacement range and blocking force value. Te reulted ratio i ituated between to 5. Given te very ig coupling factor of PN-PT and PZN-PT tat may reac 0.97 (te PZT range around 0.), we noticed large difference between te exiting analytical reult and te finite element imulation performed under COSOL ultipyic, wic finally, for ti type of material, proved to be muc more reliable. Keyword: pieoelectric actuator, PZT, PN- PT material 1. Introduction In te recent year tere a been an increaing interet in growing and caracteriing of xpb(g 1/ Nb / )O (1-x)PbTiO (PN PT) [1][][] and of xpb(zn 1/ Nb / )O (1-x)PbTiO (PZN PT) [][5] olid olution. Initially ued a ceramic, tey can be actually grown into ingle crytal by modified Bridgman or olidtate ingle crytal growt metod. Tee material exibit ultra-ig pieoelectric propertie for compoition near morpotropic pae boundary wic i located at x65% for PN-PT and x91.5% for PZN-PT. Compared to bet PZT [6] ceramic, tey are rougly to 5 time more pieoelectric, finding early actuating application [7]. Electro-mecanical coupling coefficient are very ig, uually ranging from 0.8 up to Reported denity i iger tan tat of PZT ceramic, of around kg/m. Electrotrictive propertie are alo important, being already ubject to few application uc a [8]. Dielectric contant are T ε alo iger tan tat of PZT: for intance can exceed a value of A drawback, tey are more mecanically fragile, teir compliance i iger, and bipolar capability i limited. Teir compliance coefficient are - time iger, uggeting tat tee ingle-crytal material are bet uited to low induced-tre, ig train and deflection application a in te cae of compoite bimorp actuator. An accurate modeling of tee material i recommended to etimate te performance increae related to te optimal deign. Replacing PZT ceramic wit te new generation of giant pieoelectric coefficient material will primarily allow an increaed diplacement range for te ame electric field or a reduction of te working voltage for te ame diplacement. Secondly, being monocrytalline, material like PN-PT and PZN-PT are more compatible wit microtecnology procee. We conidered a imple compoite bimorp (alo called unimorp ) deign for tee preliminary tet, a in figure 1.a. Te pieoelectric layer i conidered fixed, of 100µm tick. Te paive ubtrate i varying between 5 and 150 µm. An optimum tickne ratio exit for eac pair of material/ubtrate (figure -5). Firt, te material ad to be defined into te program and ten a erie of comparative analytical and COSOL imulation were performed. Legend: 1 - pieoelectric material layer, paive material (uually metal), tin film electrode, conductive adeive layer Figure 1. Cro ection of different pieoelectric bender tructure: a) compoite bimorp (unimorp), b) tree-layer bimorp, c) two-layer bimorp.

2 te above PN-9PT [0] poe alo important pieoelectric propertie: d m/v, data from [5].. Analytical imulation.1 Background formulae Figure. A compoite bimorp pieo-cantilever model wit Silicon ubtrate a) under external blocking force and b) free ended. Dimenion are 5.0x0.x0.1mm. Te clamped-free bending tructure i evaluated in free diplacement condition (figure.a) and in blocking force at null diplacement condition (figure.b).. Conidered pieoelectric material Te ubtrate wa conidered a [0] ilicon wafer. Several material type ave been teted: 1. PZT-5H from [] wic i one of te mot advanced type of oft PZT ceramic. Pieoelectric tranvere coefficient equal d m/v ; m /N ;. PN-0PT poled along [001] wic prove a larger tranvere pieoelectric coefficient but alo a larger compliance tan te above PZT: d m/v; m /N ; material contant were entered from ref. [1].. PN-9PT poled along [0] from []. Ti type of poling i quite particular for tee domain engineered ingle crytal. It enable a large negative tranvere pieoelectric coefficient in Y-direction ( d ) and a maller and poitive d coefficient. We ued te following contant: d m/v; 10-1 m /N.. PZN-7PT poled along [001]. Altoug le available on te market tan teir omologue PN-PT, due to growt and mecanical tability problem, we imulated from [] te following contant: d m/v; m /N. 5. PZN-7PT poled along [0] and analogue to Let u conider a blocked-free beam of lengt L and widt w. Paive material i denoted a and pieoelectric material i denoted a ; teir related tickne i and repectively. Lengt i along X-axi wile te Z-ai i vertical to te beam. For te analytical formulation of te problem we will reference to te et of contitutive equation developed by W.G. Smit and W-S. Coi [9] and by. S. Weinberg [10]. Te contituent equation et of Smit [9] applie for directly for compoite bimorp at termodynamic equilibrium. Starting from te well-known pieoelectric train-carge coupled equation, te energy denity i integrated and ummed for te pieoelectric and te paive element. Ten te canonical conjugate wit te external quantitie (voltage, momentum etc.) are determined by partial derivative, providing element for a contitutive matrix. Tip diplacement δ i decribed a a function of te following external quantitie: a mecanical moment, an external tranvere force F at te free end of te beam, a uniform load p and an electrical voltage V : 6L L L d BL δ A F p V Kw Kw K K were A ), K 6 and ( ( ( ) B. Reulted blocking force F d Bw B V ) F B i: ( ) L. Weinberg [10] preented a rater different approac, more related to tructural mecanic and dedicated for multilayer beam. Contitutive equation et wa extended to include axial

3 tenion/compreion. Neutral axi i determined, ten pieoelectric axial force and torque per unit voltage. Curvature related to different parameter i derived and contitutive coefficient are provided. If we particularie te Weinberg equation for te multilayer tructure to te given cae, diplacement δ i: C L C L C V L δ F V Blocking force F become: B V FB V L were: /, / 1 A i w pi i te i layer cro-ection area, 1 / A / A C A1 A I1 I 1 A1 1 1 p p p p i te beam curvature per unit torque, ( d A ( )) ( ) V i te torque per unit voltage acro electrode, p w I 1 i i Ai 1 i te area momentum of inertia, and finally 1 A i te neutral fiber poition (te origin of te Z- axi i conidered at te interface). Even toug te deriving metod of δ and F are different, imulation owed identical B reult. Te free diplacement varie wit te quare of te lengt L wile te blocking force depend linearly of te widt w and i inverely proportional wit te cubic lengt. µm, for PN-0PT [001] it i 19 µm and for PN-9PT [0] it ow 1µm. In te cae of te PZN-7PT [001] and [0] optimal ilicon tickne are 15 µm and 1 µm repectively. Smaller tiffne coefficient of PN-PT and PZT-PT tan PZT favour diplacement to blocking force (maller blocking force gain, figure ), wic, owever, i not a real drawback for microactuator application. aximum ratio of diplacement gain i.5 for PN-0PT [001] (actuator along X-direction) and 8. in te cae of PN-9PT [0] (actuator along Y- direction). Figure. Analytical comparative diagram of δ free tip diplacement. Pieoelectric layer i 0.1 mm tick, lengt i 5 mm, widt i 0. mm and applied voltage 80V. Silicon layer tickne i varied from 5 to 150 µm.. Analytical reult We conider fixed pieoelectric layer tickne 100µm.an electric field of 8 kv/cm in te polariation direction, Geometry i L5.0 mm by w 0. mm. Paive layer tickne i conidered a variable and it optimum value for te five given material from Section- i invetigated. A een from figure, optimal Silicon layer tickne for bet PZT-5H bending train i Figure. Analytical comparative diagram of F B free tip diplacement. Same condition and legend a in figure.

4 . COSOL Simulation Figure 5 and 6 depict imulation reult performed by COSOL analyi conidering te ame bimorp a in previou ection. Reult ow tat blocking force obtained wit FE i imilar to te analytical one. However, ignificant difference in term of maximum free tip diplacement i noticed, wic i lower epecially for PN-PT and PZN-PT. Te relative difference between FE and analytical reult reac -7%, -69%, -50% and -58% for PN-0PT[001], PN-9PT[0], PZN- 7PT[001] and PZN-7PT[0] repectively. For PZT-5H te difference i more acceptable, of around -%. Several argument may explain tee difference. Firt, Smit and Coi [1] a well a Weinberg [10] metod treat in-plane problem were electric field ditribution i like tat from a plane capacitor. Te effect of mecanical train and tre on electric field i not taken into account. Alo, analytical metod don t take into account te electromecanical coupling factor E T k d ε tat influence on electrical diplacement. For PZT-5H k equal 0.8 and for [001]-poled ingle crytal it larger: 0.9 and 0.58 for PN-0PT and PZN-7PT repectively. For material poled along [0], omologue k factor are 0.9 and 0.86 for PN-9PT and PZN-7PT. Tere i a correlation between tee coupling factor value and difference noticed and depicted te paragrap above. Obviouly, working expreion (1) and (7) mi out toe coupling coefficient tat over-etimate te electrical field and increae te actual mecanical tiffne. Finally, for te analytical metod, te neutral fiber poition i fixed. However, in fact, for compoite bimorp deign and epecially for ig coupling factor material, te neutral fiber poition i varying according to te applied voltage. Hence, te immediate conequence i te reduction of te free tip diplacement. Notice tat in our cae, te utilied widt (0. mm) i quite comparable to total tickne ( mm). Te analyi of uc a cae ould be done wit D rater tan actual D boundary condition. Ti i wy te difference between analytical and FE metod are primarily explained by inerent implification of analytical model. Wile FE metod treat te coupling problem more rigorouly, in D, te lateral deformation are not taken into account by analytical metod. Hence, tere exit a curvature in te YZ cro-ection of te cantilever, due to interfacial tre between pieoelectric of te ilicon layer, tat analytical metod do not take into account. Figure 5. Free tip diplacement w.r.t ubtrate tickne (FE analyi). Same condition and legend a in figure. Figure 6. Blocking force w.r.t paive Si ubtrate tickne (FE analyi). Same condition a in figure. Same legend a in figure.

5 5. Concluion In te preent work we evaluated te actuating performance of te monocrytalline giant pieoelectric coupling factor material known a PN-PT and PZN-PT. Te new material contant were defined under COSOL ultipyic. Alternate analytical imulation were performed under atlab. Te ubtrate wa conidered te Silicon, in an attempt for future PieoES device. Compared to te regular PZT ceramic, te expected gain in term of free diplacement i between 00% and 00% and in term of blocking force between 10% and 100%. Tere wa noticed a very large difference between te analytical reult and te finite element one. In fact, analytical metod fail to accurately model te electric field of tee ig coupling coefficient material. Alo, neiter interface tre nor lateral deformation are taken into account by te analytical formulae. Finite element reult ould be conidered a mot appropriate. 5. Reference [1] R. Zang, W. Jiang, B. Jiang and W. Cao, Elatic, dielectric and pieoelectric coefficient of domain engineered 0.70Pb(g1/Nb/)O- 0.0PbTiO ingle crytal, AIP Conf. Proc. vol. 66, pp , 00 [] H. Cao, V. H. Scmidt, R. Zang and W. Cao, "Elatic, pieoelectric, and dielectric propertie of 0.58Pb(g1/Nb/)O- 0.PbTiO ingle crytal", J. of Appl. Py., vol. 96 no. 1 pp , 00 [] F. Wang, L. Luo, D. Zou, X. Zao and H. Luo, Complete et of elatic, dielectric and pieoelectric contant of ortorombic 0.71Pb(g1/Nb/)O - 0.9PbTiO ingle crytal, Appl. Py. Lett. 90, 190, 007 crytal poled along [0], Applied Py. Lett. 89, 908, 006. [6] A.J. oulon and J.. Herbert, Electroceramic: aterial, Propertie, Application nd Edition, Jon Wiley and Son, 00 [7]. Sitti, D. Campolo, J. Yan, R.S. Fearing, T. Su, D. Taylor and T. Sand, Development of PZT and PZN-PT Baed Unimorp Actuator for icromecanical Flapping ecanim IEEE Int. Conf. Robotic and Automation, pp , Seoul Korea, 001 [8] A. Hall,. Allaverdi, E.K. Akdogan and A. Safari, "Development and Electromecanical Propertie of ultimaterial Pieoelectric and Electrotrictive PN-PT onomorp Actuator", Journal of Electroceramic, 15, 1 150, 005 [9] J. G. Smit and W-S. Coi., Te contituent equation of pieoelectric eterogeneou bimorp, IEEE Ultraonic Sympoium, pp , 1990 [10]. S. Weinberg, Working equation for pieoelectric actuator and enor, ASE/IEEE Journal of ES, vol8, no., pp71-77, 1999 [] ttp:// roduct-material/data-eet-directory/ 6. Acknowledgement Te work wa upported from te European Project FP7-PEOPLE-IEF icropads: New icro-robotic Sytem featuring Pieoelectric Adaptive icrostructure for Sening and Actuating, wit Aociated Embedded Control. [] R. Zang, B. Jiang, W. Cao, Complete et of material contant of 0.9Pb(Zn1/Nb/)O- 0.07PbTiO domain engineered ingle crytal, J. of at. Sci. Lett. 1, pp , 00. [5] R. Zang, B. Jiang, W. Cao, Complete et of elatic dielectric and pieoelectric coefficient of 0.9Pb(Zn1/Nb/)O-0.07PbTiO ingle