Designing Piezoelectric Interdigitated Microactuators using COMSOL

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1 Excrpt from th Procdings of th COMSOL Confrnc 2008 Boston Dsigning Pizolctric Intrdigitatd Microactuators using COMSOL Olivr J. Myrs *1, M. Anjanappa 2 and C. Fridhoff 3 1 Mississippi Stat Univrsity, 2 Univrsity of Maryland Baltimor County, 3 Northrop Grumman Corporation, Elctronics Systms Sctor, Baltimor, MD *Corrsponding author: 219 Carpntr Hall, Box 9552, Mississippi Stat, MS 39762, myrs@m.msstat.du Abstract: This papr prsnts a mthodology towards dsigning, analyzing and optimizing pizolctric intrdigitatd microactuators using COMSOL Multiphysics. Th modls usd in this study wr basd on a circularly intrdigitatd dsign that taks advantag of primarily th d 33 lctromchanical pizolctric constant cofficint. Bcaus of th symmtric natur of th dvics, a small numbr of 2-D axisymmtric modls wr dvlopd to charactriz th bhavior of th diaphragms. Th variation in th dsign paramtrs and thir ffct on dflction was capturd using ths modls. Th modls also showd that svral of th dsign paramtrs wr naturally coupld. Discrt modls wr thn usd to captur th variations in ky dsign paramtrs during fabrication. Th numrical modls corrlat wll to th maximum dflction of th xprimntal dvics. Kywords: Microactuators, pizolctric, modling. 1. Introduction Micromchanical dvics that mploy activ pizolctric matrials, typically in thin-film form, show promis for a varity of applications, particularly actuation [1,2]. As th dvics bcom incrasingly divrs and sophisticatd, th nd ariss for incrasingly accurat and fficint modling of thir bhavior for dsign purposs. Th lctrically activ rgion of th pizolctric matrial prforms ithr as a snsor (strain input, lctrical output) or as an actuator (lctric-fild input, displacmnt output) or both. Th pizolctric layr is usually a dpositd film with activ-ara dimnsions that ar 100X or mor than th thicknss. Othr lctrically activ and passiv layrs ar prsnt and ovrlap ach othr to form gomtrically and functionally complx layrd structurs. Sinc th dvics ar on th micro-scal, th activ part of th structur maks its bhavior mor snsitiv to lctrical aspcts of its nvironmnt; hnc good lctrical modling is rquird [3]. Th task of dsigning and optimizing microactuators or MEMS dvics brings uniqu challngs of analyzing intrdpndnt physical phnomna, dvic snsitivity and small scal gomtris. MEMS simulation rquirs multidisciplinary softwar to captur th multiphysical natur of MEMS dvics [4]. This papr prsnts th dsign initiation of intrdigitatd pizolctric thin-film micoactuators. Paramtric analysis is initially usd to gathr a larg battry of solutions from a fw modls. Exprimnts wr thn conductd on discrt actuators with varying cntr disk diamtrs and lctrod pattrns. Finit lmnt modls of th particular actuator diaphragms wr cratd to corroborat th xprimntal obsrvations. 2. Thortical Background 2.1 Elasticity Equations Th dflction modl of th intrdigitatd mmbran bgins with th quation for lastic dflction: w q 64D 2 2 ( a r ) = (1) whr q is th prssur, D is th flxural rigidity, a is th radius and r is th radial position. Taking this quation a stp furthr, th shar componnt is introducd to account for intrfacial ffct of th layrs h 2 2 ( a r ) + ( a r ) q w = 64D 1 ν whr h is th thicknss and ν is th Poisson s ratio of th plat. Accounting for th multipl layrs, quivalnt D, ν, wr drivd as: (2)

2 E h = (3) 3 D = K2 12 ν ( 1 ) K 3 ν = ν a (4) K2 wt t E E = (5) 3 b a b a EI K1 12 a a + b b ( t E t E ) 2.2Pizolctric Constitutiv Equations Pizolctricity is th intraction btwn lctrical filds and mchanics. To obtain a rasonabl modl of this intraction, linar lasticity quations ar coupld with lctrostatic charg quations by mans of lctric constants. Th strss-charg form of th quations is: E S s d T t = T D d ε E (6) whr S is th strain, s E is th complianc matrix at a constant lctric fild, d t is transpos of th pizolctric coupling matrix rlating strain to lctric fild, ε T is th prmittivity at constant strss, T is th strss, E is th lctric fild and D is th lctric displacmnt. Th strss-charg form of ths quations is mor usful for finit lmnt analysis bcaus th strss-charg form matchs th PDEs for Navir s quations for mchanical strss and Gauss law of lctric charg. Th quations of convrsion ar as follows: c [ ] E = s E 1 (7) E = d[ s ] 1 (8) S T E 1 ε = ε d[ s ] dt (9) whr c E is th stiffnss matrix, is th pizolctric coupling matrix rlating strss to lctric fild and S is th prmittivity matrix at constant strain. Th strss-charg form is thn: E T c S t = S D ε E (10) 3. Initial Modl Dsign Paramtrs Initial modls wr cratd with th matrials listd in Tabl 1. Th masurd rsidual strsss inducd from fabrication wr also incorporatd into th modls as shown in figur 1 [5]. Symmtric Boundary Tabl. Matrial Proprtis [6,7,8,9] Matrial E (GPa) ν ρ (kg/m3) R=0 Gold ZrO SiO Al2O GND +140 V GND +140 V GND PZT = 125 MPa Figur 1. Rsidual strss boundary conditions +140 V Th modl variations includ matrial thicknss with an mphasis on pizolctric thicknss, lctrod width, lctrod sparation and cntr disk diamtr. Simulations wr conductd to dtrmin a viabl pizolctric matrial thicknss for ths diaphragms. Figur 2 shows th sharp dclin in dflction with rspct to matrial thicknss as th pizolctric matrial is incrasd from 2 to 3μm ZrO2 = 250 MPa SiO2 = -147 MPa PZT Thicknss μm) ( Figur 2. Dflction vs. pizolctric matrial thicknss Modls wr also cratd to simulat th various clamping conditions that wr anticipatd during th fabrication and rlas procsss of ths diaphragms. Ths boundary conditions includd a clampd boundary at th outr circumfrnc of th diaphragm, a clampd boundary with an offst lctrod pattrn, a partially clampd lctrod and a fully clampd lctrod. Th rsults of figur 3 show that ach of th spcific boundary conditions gav a slight progrssiv dcras in dflction.

3 4. Exprimntal Validation Dflction ( m) Figur 3. Conditions PZT Thicknss (μm) Flush Elctrods Offst Elctrods Partially Clampd Elctrods Clampd Elctrods Dflction vs Clamping Boundary Elctrod sparation modls and varid cntr disk diamtr modls showd a natural coupling with th numbr of lctrods and furthr highlightd a dflction that was dpndnt on th numbr of lctrods. As th pitch incrasd, th dflction incrasd, howvr th dflction furthr incrasd for diaphragms with an vn numbr of lctrods. Evn though th incrasing cntr diamtr had a ngativ ffct on th dflction, th sam varianc btwn an vn and odd numbr of lctrods was prdictd. Th ZYGO NwViw 100 Whit Light Profilomtr was usd to obsrv th static dflction of th micro-actuators with xprimntal stup as shown in Figur 10. Data was collctd by varying th applid DC voltag from 0 to 180V. Th polarity of th applid voltag was st at positiv on th intrior lctrods. Th sampls wr pold at approximatly thr tims th Curi voltag (3Vc), which is 100V for 15 minuts [10]. Simulations of th abov intrdigitatd dsigns wr run varying th voltag btwn 0 and 180V and incorporatd th rsidual strss data as applid to th prvious simulation modls. Th polaritis will also b st in th sam mannr as th xprimnts. Elctrod Spacing (Evn Numbr Elctrods) Dflction (Microns) Elctrod Spacing (Odd Numbr Elctrods) Figur 4. Dflction vs. lctrod sparation Dflction (microns) Cntr Disk Diamtr (Evn Numbr Elctrods) Cntr Disk Diamtr (Odd Numbr Elctrods) Figur 5. Dflction vs. cntr disk diamtr Odd Elctrods Evn Elctrods Odd Numbr Elctrods Evn Numbr Elctrods Figur 6. Static Dflction Masurmnt Stup Th xprimnts wr conductd on a st of mmbrans that wr mad of Al 2 O 3, SiO 2, ZrO 2 and PZT. Ths mmbrans wr 650μm in diamtr with cntr disk diamtrs of 90, 150, and 210μm. Th lctrods and cntr disk covrd 80 prcnt of th mmbran with a 5μm width and pitch. Th numbr of lctrods for ach mmbran rangd from 15 to 21 with rspct to th cntr disk. Data was collctd only at 100, 140 and 180V to gt a sns of th ffct of th numbr of lctrods and cntr disk. Th lctrods wr polarizd so th positiv potntial was on th lctrod just insid th outrmost lctrod which corrlatd with th xprimntal procdur usd. Th cntr disk was not activatd. 5. Discussion

4 Ths mmbrans xprincd a grat amount of dflction bcaus th 0.25μm thick protctiv layr of Al 2 O 3 was sputtrd which allowd for good manufacturing tolrancing, good adhsion to itslf and to th SiO 2 layr Micron Mmbran with 90 Micron Cntr Disk μm Mmbran with 90μm Cntr Disk Th xprimntal dflction valus of th 90μm Cntr Disk mmbran wr 3.93, 6.44 and 7.93μm for 100, 140, and 180 Volts rspctivly. Whn corrctd simulations wr conductd, th rsidual strsss in th pizolctric matrial only appard to b rlvant at 100 V and th rsidual strsss of th ZrO 2 and SiO 2 wr rlvant at 140 and 180V. This only partially corrsponds to th data Pnn Stat gathrd. Th rsidual strsss wr masurd only aftr th mmbrans hav bn rlasd. Th silicon layr also crosss th strss thrshold from bing comprssiv to tnsil at a voltag bfor 100V, th zirconium layr gratly incrass in tnsion and th pizolctric matrial dos not appar to b as tnsil as initially dtrmind. Tabl 1: Rsidual Strss Valus Applid During Numrical Analysis of 650μm Diamtr Actuator with 90 μm Cntr Disk. Voltag ZrO 2 SiO 2 PZT 100 V 270 MPa 206 MPa 60 MPa 140 V 350 MPa 245 MPa 0 MPa 180 V 400 MPa 305 MPa 0 MPa Th xprimntal dflction as shown in Figur 7 shows a flattnd cntr and an inconsistnt curv along th radius of th mmbran whr th numrical simulations show a curvd cntr dflction and small stair stp pattrn along th radius of th mmbran whr th lctrods ar positiond Radial Position ( μm) 100 V Simulation 140 V Simulation 180 V Simulation 100 V Exp. Data 140 V Exp. Data 180 V Exp. Data Figur 7. Corrctd Numrical vrsus Exprimntal Dflction Comparison of 650 Micron Diamtr Actuator with 90 Micron Cntr Disk Figur 8. Corrlation of Numrical Dflction, Elctric Fild, and Von Miss Strsss in 650 Micron Diamtr Mmbran with 90 Micron Cntr Disk at 180 Volts μm Mmbran with 150μm Cntr Disk Th xprimntal dflction valus of th 150 μm Cntr Disk mmbran wr 2.12, 5.86 and 7.17 μm for 100, 140, and 180 Volts rspctivly. Whn corrctd simulations wr conductd, th rsidual strsss of th pizolctric matrial, again, only appard to b rlvant at 100 V and th rsidual strsss of th ZrO 2 and SiO 2 wr rlvant at all voltags. Th silicon layr also crosss th strss thrshold from bing comprssiv to tnsil at a voltag bfor 100V and th pizolctric matrial dos appar to b as tnsil as initially dtrmind.

5 Tabl 2: Rsidual Strss Valus Applid During Numrical Analysis of 650μm Dia. with 90 μm Cntr Disk Actuator Voltag ZrO 2 SiO 2 PZT 100 V 270 MPa 205 MPa 130 MPa 140 V 360 MPa 255 MPa 0 MPa 180 V 410 MPa 315 MPa 0 MPa Th xprimntal dflction as shown in Figur 9 shows a flattnd cntr and an inconsistnt curv along th radius of th mmbran whr th numrical simulations show a curvd cntr dflction and small stair stp pattrn along th radius of th mmbran whr th lctrods ar positiond Microm Mmbran w/ 150 Micron Cntr Disk and 1.40 μm for 100, 140, and 180 Volts rspctivly as shown in Figur 11. Whn corrctd simulations wr conductd, th rsidual strsss of th pizolctric matrial only appard to b rlvant at 140 and 180V and th rsidual strsss of th ZrO 2 and SiO 2 wr rlvant 100 and 140V. This dos not corrspond to th data Pnn Stat gathrd. Th rsidual strsss of all th matrials wr significantly lowr than initially dtrmind. Th larg cntr disk and lack of lctrods causd th actuator to bhav as a sandwich pizolctric unimorph that utilizd th d 31 lctromchanical coupling cofficint rathr than th dsird d 33 coupling cofficint. Tabl 3: Rsidual Strss Valus Applid During Numrical Analysis of 650μm Dia. with 90 μm Cntr Disk Actuator Voltag ZrO 2 SiO 2 PZT 100 V 29 MPa 29 MPa 0 MPa 140 V 30 MPa 35 MPa 30 MPa 180 V 400 MPa 305 MPa 30 MPa Radial Position ( μm) 100 V Simulation 140 V Simulation 180 V Simulation 100 V Exp. Data 140 V Exp. Data 180 V Exp. Data Figur 9. Corrctd Numrical vrsus Exprimntal Dflction Comparison of 650 Micron Diamtr Actuator with 150 Micron Cntr Disk Micron Mmbran w/ 210 Micron Cntr Disk Radial Position ( μm) 100 Simulation Simulation 100 V Exp. Data 140 V Exp. 180 V Simulation 140 V 180 V Data V Exp. Data Figur 11. Corrctd Numrical vrsus Exprimntal Dflction Comparison of 650 Micron Diamtr Actuator with 210 Micron Cntr Disk Figur 10. Corrlation of Numrical Dflction, Elctric Fild, and VonMiss Strsss in 650 Micron Diamtr Mmbran with 90 Micron Cntr Disk at 180 Volts μm Mmbran with 210μm Cntr Disk Th xprimntal dflction valus of th 210 μm Cntr Disk mmbran wr 1.44, 1.45

6 Figur 12. Corrlation of Numrical Dflction, Elctric Fild, and Von Miss Strsss in 650 Micron Diamtr Mmbran with 90 Micron Cntr Disk at 180 Volts 7. Conclusions A dsign and analysis mthodology using 2- D axis-symmtric multi-physical modls is dvlopd for circularly intrdigitatd pizolctric micro-actuators. Ky physical dsign paramtrs wr varid using th mbddd dsigning modul and th matrials and corrsponding proprtis wr also varid to gain an undrstanding of which matrials ar bst suitd for th application. Exprimnts wr conductd on a discrt st of microactuators that naturally coupld th cntr disk diamtr and numbr of lctrods. Good dflction corrlation was obtaind btwn th numrical and limitd xprimntal data, howvr mor work nds to b don numrically to bttr rplicat th shap functions of ths particular dvics. Th numrical modls can srv as an activ dsign tool to optimiz intrdigitatd pizolctric microactuator mmbrans for various applications. Microlctromchanical Systms, Volum, 3, pp , Sptmbr S. Xu and T. Koko, Finit Elmnt Analysis and dsign of activly controlld pizolctric smart structurs, Finit Elmnts in Analysis and Dsign, Volum 40, pp , W. Moussa, MEMS dsign optimization with FEA, ALGOR Cntr for Mchanical Dsign Tchnology Whit Paprs, Hong, E, R.L. Smith, S.V. Krishnaswamy, C.B. Fridhoff and S. Trolir-McKinstry, Strss Dvlopmnt in PZT/ZrO2/SiO2 Stacks For MEMS Pizolctric Unimorph Diaphragms Using Intrdigitatd Transducr (IDT) Elctrods, CRC Matrials Scinc and Enginring Handbook. 7. IEEE Transactions on Elctron Dvics, vol. 70, p. 421, May IEEE Transactions on Elctron Dvics, vol. 25,p. 1249, Octobr Thin Solid Films Y. K. Hong, H.-K. Park, S. Q. L, K. S. Moon, R. R. Vanga, and M. Lvy, Dsign and prformanc of a slf-snsing, slf-actuating pizolctric monomorph with intrdigitatd lctrods, Procding of th SPIE Intrnational Confrnc on Opto-mchanical Actuators, Snsors and Control, Volum, 5602, pp , Octobr Acknowldgmnts Th authors would lik to thank Northrop Grumman Corporation for support. 8. Rfrncs 1. B. Chn, B. Chsman, A. Safari, S. Danforth, and T. Chou, Thortical and numrical prdictions of th lctromchanical bhavior of spiral-shapd lad zirconat titanat (pzt) actuators, IEEE Transactions on Ultrasonics, Frrolctrics and Frquncy Control, Volum 49, pp , Marchs D. Bri and J. Blchschmidt, Dsign and static modling of a smicircular polymric pizolctric microactuator, Journal of

7 1. B. Chn, B. Chsman, A. Safari, S. Danforth, and T. Chou, Thortical and numrical prdictions of th lctromchanical bhavior of spiral-shapd lad zirconat titanat (pzt) actuators, IEEE Transactions on Ultrasonics, Frrolctrics and Frquncy Control, Volum 49, pp , Marchs D. Bri and J. Blchschmidt, Dsign and static modling of a smicircular polymric pizolctric microactuator, Journal of Microlctromchanical Systms, Volum, 3, pp , Sptmbr S. Xu and T. Koko, Finit Elmnt Analysis and dsign of activly controlld pizolctric smart structurs, Finit Elmnts in Analysis and Dsign, Volum 40, pp , W. Moussa, MEMS dsign optimization with FEA, ALGOR Cntr for Mchanical Dsign Tchnology Whit Paprs, Hong, E, R.L. Smith, S.V. Krishnaswamy, C.B. Fridhoff and S. Trolir-McKinstry, Strss Dvlopmnt in PZT/ZrO2/SiO2 Stacks For MEMS Pizolctric Unimorph Diaphragms Using Intrdigitatd Transducr (IDT) Elctrods, CRC Matrials Scinc and Enginring Handbook. 7. IEEE Transactions on Elctron Dvics, vol. 70, p. 421, May IEEE Transactions on lctron dvics, vol. 25,p. 1249, Octobr Thin Solid Films Y. K. Hong, H.-K. Park, S. Q. L, K. S. Moon, R. R. Vanga, and M. Lvy, Dsign and prformanc of a slfsnsing, slf-actuating pizolctric monomorph with intrdigitatd lctrods, Procding of th SPIE Intrnational Confrnc on Opto-mchanical Actuators, Snsors and Control, Volum, 5602, pp , Octobr 2004.