Excrpt from th Procdings of th COMSOL Confrnc 200 Paris Finit Elmnt Modl of a Frrolctric A. Lópz, A. D Andrés and P. Ramos * GRIFO. Dpartamnto d Elctrónica, Univrsidad d Alcalá. Alcalá d Hnars. Madrid, España. *Pablo@dpca.uah.s Abstract: Frrolctric tchnology continus attracting significant intrst du to its wid rang of applications (capacitors, nonvolatil mmoris, acclromtrs... In this work w prsnt a finit lmnt modl of a simpl frrolctric structur. Pizolctric and frrolctric matrics ar includd. Th coupling btwn th pizolctric and frrolctric paramtrs will b introducd formally using a thrmodynamic approach. Using th Comsol finit lmnt nonlinar analysis, th frrolctric loop (polarization vrsus lctric fild and buttrfly loop (strain vrsus lctric fild in frrolctric matrials is obtaind and discussd. Th xprimntal rsults ar compard to th finit lmnt simulations Morovr, th two opposit dirctions of th polarization vctors ar stabl. This proprty, known as bistability, allows th us of frrolctric matrials as nonvolatil mmory lmnts []. In addition to its application in computr mmoris [2], [3], frrolctric matrials can b usd in th fabrication of many novl dvics that can tak advantag of th spcial proprtis of frrolctric matrials. A larg numbr of ths dvics hav a potntial commrcial dvlopmnt; figur 2 shows svral applications of thin film frrolctric matrials. Kywords: Frrolctric, pizolctric, finit lmnt, frrolctric loop, buttrfly loop.. Introduction Frrolctric matrials ar charactrizd by having prmannt lctric polarization vctors, which can b orintd in opposit dirctions by applying an xtrnal lctric fild, bing abl to switch btwn both stats (figur. D D r Figur 2: Svral applications of frrolctric matrials. Pizolctricity is on of th spcial proprtis prsnt in frrolctric matrials. It can b applid in th manufactur of micromachins, such as acclromtrs and in th manufacturing of svral typs of control dvics. Thr ar two kinds of pizolctricity: dirct and indirct. In dirct pizolctricity th frrolctric layr xpands or contracts whn a crtain voltag is applid across it. Figur. Hystrsis cycl of a frrolctric matrial. On th othr hand, whn th frrolctric layr is comprssd, a voltag appars btwn th opposit sids of th frrolctric layr. This typ of pizolctricity, known as indirct pizolctricity, is applid in th construction of prssur snsors.
Pyrolctricity is anothr particular proprty of frrolctric matrials. Whn a hating procss is applid to a frrolctric crystal, positiv and ngativ ions ar cratd in th opposit sids of th frrolctric layr. This ffct involvs th apparition of lctric voltag btwn both sids of th crystal that can b usd in infrard tmpratur snsors. Thr ar frrolctric matrials that also xhibit lctro-optic activity. This mans that th rfractiv indx of a frrolctric matrial changs whn a voltag is applid across it. This proprty is usd in color-filtring dvics, displays, imag rcording systms. In this work, w hav prformd a finit lmnt modl of frrolctric matrials with Comsol Multiphysics, linkd to th modl of a pizolctric matrial prsnt in th program. Th charactristic curvs of th matrial wr simulatd, vrifying th propr opration of th modl. 2. Frrolctric Modl Frrolctric matrials hav two stats of polarization (p and p - and can switch btwn thm whn an xtrnal lctric fild is applid. In Figur 3 it can b sn that thr is no tndncy towards on of th two polarization stats in th absnc of applid voltag. Figur 4 shows th volution whn an xtrnal lctric fild is applid. In this cas, applying a positiv voltag V, a rsulting positiv stat of polarization (p, is obtaind, bcaus som of th dipols switch thir orintation. Applying a ngativ voltag -V, a frrolctric stat in opposit polarization (p - would b stablishd. ESTADO STATE p- p STATE ESTADO Figur 4: Effct in th polarization stats whn a positiv voltag is applid. Whn an xtrnal lctric fild is applid to a frrolctric capacitor, th polarization stat of th capacitor changs, if th applid lctric fild dirction is th opposit to th prvious on. Th amount of ngativ polarization (p - which bcoms positiv tndncy (p whn applying a positiv voltag V, i, th incrasing of th p polarization stat, is givn by th following quation [4]: p = ( p f V ( whr p is th incrmnt of th positiv polarization stat, V is th incrmnt of th applid voltag and (f V is th probability of th occurrnc of a switching of polarization stat du to an incrmnt in th applid voltag V. In this quation, f is obtaind as: f = f V V ( V VC V 0 (2 whr Vc y ar th corcitiv and thrmal voltags rspctivly. Ths two voltags ar vry rlvant paramtrs in frrolctric capacitors. Th thrmal voltag is dfind by th nxt quation: V=0.0 Figur 3: Polarization stats in th absnc of applid of lctric fild. K. T = q (3
whr K is th Boltzmann constant (,38047.0-23 J/K, T is th tmpratur (K and q is th lctron charg (,60864.0-9 C. From th prvious xprssions w can obtain an xprssion for p as follows: ( ViVc p = ( p i. ( V Vc (4 whr V i is th initial voltag and p i is th initial positiv polarization. Finally, th lctric displacmnt (D, obtaind for positiv applid voltags has th valu: = Ps ( 2 p D (5 In a similar way, w can obtain for ngativ voltags: D = Ps ( 2 p (6 Figur 5: Gomtry usd in th modl. In th first cas, in th lctric fild, th valu of D /- is introducd as a paramtr of rmannt displacmnt. In th scond on, th xisting thortical rlationship btwn d 33 and th valu of D /-, obtaind from th thrmodynamic bhavior of frrolctric matrials, is introducd in th pizolctric matrix [5]. whr p = ( p i. ( VVc 3. FEM modl ( ViVc (7 Th mployd gomtry in th modl is a plan capacitor with cylindrical shap (figur 5. It is dfind as an lmnt of lctric currnts insid AC/DC physics and as a pizolctric matrial insid th Pizolctric Dvic physics. / =2 / (9 / = / =2 / (0 As xpctd rsult, whn introducing this non linarity, w should obtain a shap of s 33 vs E as shown in figur 6, known as buttrfly cycl of a pizolctric matrial [5]. Th lctrical boundary conditions hav bn dfind. A sinusoidal voltag is applid to th uppr lctrod. Th lowr lctrod rmains connctd to ground. Th primtr of th structur is lctrically isolatd. W hav mployd a triangular msh with tim dpndnt rsolution. In both physics, a non linal solvr is usd.
S 33 Figur 6. Buttrfly cycl of a pizolctric matrial [5]. 4. Simulation rsult and discussion. Th obtaind rsults wr simulatd for th uppr part of th structur, wr th sinusoidal voltag is applid. W valuatd th D 33 vs E 33 in AC/DC physics and S 33 vs E 33 in Pizolctric Dvic physics. First, an lctrical fild blow th corcitiv fild was applid, obtaining a linal rspons of a dilctric matrial (figur 7 and th typical rspons of a pizolctric matrial (figur 8 for a linal matrial dscribd in Comsol. Figur 8. Rspons of a pizolctric matrial for E<Ec. Onc th linar rspons was obtaind, w simulatd th rspons for an lctrical fild highr than Ec, obtaining th typical non linar rspons of a frrolctric matrial and th ffct of this nonlinarity in th pizolctric rspons (figurs 9 and 0. Comparing thos graphs with th xpctd from thory (graphs and 6, w can conclud that th modl prforms th bhavior of a frrolctric matrial from both points of viw, lctrical and mchanical. Figur 7. Rspons of a frrolctric matrial for E<Ec. Figur 9. Simulatd buttrfly cycl.
[4] Kyunam Lim, Kyuhyon Kim, Songchol Hong, Kywro L; "A smi-mpirical cad modl of frrolctric capacitor for circuit simulation ; Intgratd frrolctrics; vol. 7; 97-04 (997. [5] Pizolctric PZT Cramics Springr Sris in Matrials Scinc, 2008, lum 4, I, 89-30. G. Hlk and K. Lubitz. Figur 0. Simulatd frrolctric cycl. 5. Conclusions Th dvlopd modl allows to adquatly dscribing frrolctric matrials, not includd in th Comsol matrial libraris. Th prdictd lctrical and mchanical rsponss of th matrials ar confirmd, this shows than th pizolctric and frrolctric charactristics hav bn wll dscribd with this modl. This will allow dvloping modls for mor complx structurs with rlvant tchnological intrst, such as multifrroic matrials. 6. Rfrncs [] J.F. Scott, F.M. Ross, C.A. Paz d Araujo, M.C. Scott and M. Huffman; Structur and dvic charactristics of SrBi 2 Ta 2 O 9 basd nonvolatil random-accss mmoris ; Mrs Bulltin; 33-39 (julio 996. [2] O. Aucillo, J.F. Scott and Ramamoorthy Ramsh; Th physics of frrolctric mmoris ; Physics Today; 22-27 (julio 998. [3] Robrt E. Jons, Jr. and Sshu B. Dsu; Procss intgration for nonvolatil frrolctric mmory fabrication ; Mrs. Bulltin; 55-58 (junio 996.