ANALYSIS OF FREQUENCY-DEPENDENT BEHAVIORS IN FERROELECTRIC MATERIALS

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1 Proceedngs of the 6th Internton Conference on Mechncs nd Mters n Desgn, Edtors: J.F. Sv Gomes & S.A. Megud, P.Degd/Azores, Juy 205 PAPER REF: 5332 ANALYSIS OF FREQUENCY-DEPENDENT BEHAVIORS IN FERROELECTRIC MATERIALS Nng Lu, Yu Su,2(*) Deprtment of Mechncs, Schoo of Aerospce Engneerng, Beng Insttute of Technoogy, S. 5 Zhong-Gun- Cun Ave., Beng, P. R. Chn 2 Mechncs nd Aerospce Desgn Lbortory, Unversty of Toronto, 5 Kng s Coege Rd, Toronto, Ontro M5S 3G8, Cnd (*) Ems: dmyusu@bt.edu.cn, yusu@me.utoronto.c ABSTRACT In ths work the eectromechnc behvors of ferroeectrc mters re numercy studed wth dfferent eectrc odng frequences. The mter under nvestgton s brum ttnte (BTO 3 ), whch shows notcebe frequency-dependent response upon unxy pped terntng eectrc fed. Seprte modes re empoyed to nvestgte the dependence of pped frequency on the hysteress behvor, nmey the snge cryst mode nd the poycryst mode. The frequency-dependent ferroeectrc propertes, e.g., coercve eectrc fed, remnnt porzton nd deectrc permttvty, re numercy exmned bsed on these three modes, nd the mechnsm of the odng frequency dependence hs been dscussed through nyss on the grn-boundry effect. Keywords: ferroeectrc, frequency, grn boundry, phse fed. INTRODUCTION Ferroeectrc mters beong to n mportnt css of dvnced mters due to ther outstndng deectrc propertes nd ther couped eectromechnc behvors. Due to the spontneous porzton n ferroeectrc crysts, the domn vrnts re metstbe nd my experence domn reorentton upon rge pped eectrc fed or mechnc stress. The exstence of such metstbe sttes dds nonnerty to the couped behvors of ferroeectrcs. Among the fctors ffectng such mut-physcy couped nonner probem, the frequency dependence s qute mportnt nd worth of beng understood snce most of the ferroeectrc components n re ppcton re workng n n envronment wth terntng eectrc odng condtons. For nstnce, ferroeectrc thn fm n MEMS cn be subect to dfferent odng frequences, nd ferroeectrc mcro-ctutor my experence vrous odng rte of the pped eectrc fed. The functon of such components s very key to be ffected by the odng frequency snce some of ther mportnt mter coeffcents w chnge wth the frequency. These coeffcents ncude the deectrc coeffcent, the remnent porzton, the coercve fed nd the pezoeectrc coeffcent. Therefore, better understndng n how these mter propertes vry wth the eectrc odng frequency cn sgnfcnty beneft to the desgn of mcro- or nno-sce ferroeectrc sensors nd ctutors. The tme-reted behvors of ferroeectrc mters hve been observed n qute few experment studes (Lente, 2003, Ers, 2008 nd Jn, 202). However, the theoretc studes on the tme-dependent domn swtchng n ferroeectrc mters re not very wdey seen. Especy, the dets of the mcrostructur dynmcs ssocted to such process need -667-

2 Symposum_7 Mechnc Behvours of Advnced Mters nd Structures t A Sces to be further expored. Athough mny mcromechncs-bsed theoretc modes cn successfuy predct the homogenzed ferroeectrc propertes of snge crysts (L 999 nd Su 2005) nd cermcs (Su 2006 nd Su 2009), the tme dependence of the domn swtchng s genery not consdered n such pproch. Some of the few theoretc works reted to tme-dependent ferroeectrc mcrostructur evouton were done by Km nd hs ssoctes (Seeecke 2005 nd Km 2009) wth proposed evouton w of the phse frcton n ferroeectrc crystte. In ddton to these pproches the tempor evouton of the mcrostructures n ferroeectrc crysts cn be successfuy studed wth phse-fed computton. In phse-fed studes, the tme-dependent Gnzburg Lndu knetc equton s empoyed to compute the evouton of the order prmeter (the porzton vector). Ths pproch hs been successfuy utzed n the studes of ferroeectrc nno-dots (Su 2009b, Su 200, Wng 20 nd Su 203), nno-thn fms (Su 20) nd nno-grned poycrysts (Lu 203 nd Lu 204). In recent work by Su et. (Su 205), the frequency dependence of the nnocrystne ferroeectrc poycrsts ws numercy studed usng the phse-fed pproch. The effect of the grn boundry ffected zone (GBAZ) ner the grn-grn nterfce ws nvestgted. In the present work, we focus on the nyss of the frequency-dependent ferroeectrc property of both snge cryst nd poycrystte. The eectrc odng frequency-dependent hysteress behvors of brum ttnte (BTO 3 ) nd ther corretons to the mcrostructur dynmcs re numercy exmned by phse-fed smuton. Bsed on two dfferent mter modes, nmey the snge-cryst mode nd poycryst mode, the mcrostructur mechnsm for frequency dependence cn be reveed. In ddton, the effect of the grngrn nterfce on the frequency-dependent behvor cn so be studed by the comprson between the phse-fed resut wth GBAZ ssumpton nd the resut wth zero grnboundry thckness. PHASE FIELD APPROACH The tot free energy of the ferroeectrc system, F, s gven by the voume ntegr of the energy densty,, s F = ψ ( P, P,, ε, D ) dv, () V where P s the porzton vector, P, ts grdent, the strn tensor, nd D the eectrc dspcement vector. In phse fed, P s tken s the order prmeter of the system. As the system evoves towrd equbrum, ts tot free energy contnues to decrese. Its vrton wth respect to tme, t, cn be evuted from ts vrton dervtve, s df dt δf δf ψ ψ dv < 0 ; where = ( ). (2) δp t δp x = V From the ntegrnd t cn be dentfed tht the term, δ F / δp, s the thermodynmc conugte to P / t, nd thus ts negtve so serves s the thermodynmc drvng force for the evouton of the porzton vector, P. Its form s dentc to the eft-hnd sde of the Euer- Lgrnge equton n cssc mechncs. In the smpest cse one my tke the evouton of P to be proporton to ts drvng force, s, -668-

3 Proceedngs of the 6th Internton Conference on Mechncs nd Mters n Desgn, Edtors: J.F. Sv Gomes & S.A. Megud, P.Degd/Azores, Juy 205 ( x, t) t = L δf ; δp ( x, t) ψ ψ or β = ( ), (3) t x, where L re the components of the postve-defnte knetc coeffcents, nd β the components of the nverse mobty tensor. Eq. (3) s the we-known tme-dependent Gnzburg-Lndu (TDGL) knetc equton for the evouton of the order prmeter, P. It provdes the fundment mens of descrbng the knetcs of the porzton vectors s functon of the thermodynmc drvng force. From (2) nd (3), t cn be seen tht the tot free energy of the system w contnue to decrese s the domn structure evoves from non-equbrum towrd the equbrum stte, wth the end pont mrked by δ F / δp 0. A gener form of the Lndu-Gnzburg-Devonshre energy densty functon for BTO 3 hs been wrtten s ψ = g 2 kmnrs kmn k P, P k, k + { ( T ) PP + 2 PP P P P P P P } + { b ε Pk P Pm Pn } + ( D 2κ m n r 0 s k 4 P )( D k ε P P + k P P P P 2 P ). k c + H k 6 ε ε k kmn + f PP P P P kmn k k m ε ε P In ths functon, the frst term represents the grdent energy of the dffuse domn ws; t s so referred to s the exchnge energy. A greter vue of k penzes more grdent energy, nd therefore requres thcker domn w re for smoother spt reorentton of porzton. The four terms n the frst brce consst of the Lndu-type energy tht re used to crete the non-convex energy ndscpe of the free energy wth mnm octed t the spontneous porzton sttes. The four terms n the second pr re used to ft the spontneous strn ong wth the deectrc, estc, nd pezoeectrc propertes bout the spontneous stte. Note tht the estc, deectrc, nd pezoeectrc propertes re nonner, therefore the tensor components re ft to the tngent mter propertes t the spontneousy porzed stte. The fn term represents the energy stored wthn the free spce occuped by the mter, where κ 0 s the permttvty of free spce. Among the coeffcents, ony whch represent the deectrc stffness re temperture-dependent. In the ccutons, we used the qus-sttc form of the Mxwe equton, the mechnc equbrum equton, nd the TDGL equton together to construct wek form of the vrtu work formu, nd numercy sove them by the fnte eement method. COMPUTATION OF THE FREQUENCY-DEPENDENT BEHAVIOR The mcrostructur evouton of brum ttnte cn be computed through fnte eement procedure usng the forementoned phse-fed method. Unx terntng eectrc fed s pped to the brum ttnte wth specfed frequency. A tme-dependent snusod eectrc fed cn be cheved through the settng of eectrc potents n the boundry condtons, wth E= E sn(2π f t), (5) mx where E mx s the mptude of the pped fed nd f the odng frequency. The mptude s set to be suffcenty hgh to ensure tht sturton of the porzton swtchng coud be reched. The mechnc boundry condton s set to be trcton-free n most cses. In m P n P n (4) -669-

4 Symposum_7 Mechnc Behvours of Advnced Mters nd Structures t A Sces cses the oc stress fed stemmng from the pezoeectrc effect s fuy consdered n our ccutons. Both brum ttnte snge cryst nd poycryst re consdered for the frequencydependence nyss. For the cse of snge cryst, two-dmenson (2D) rectngur geometry wth perodc boundry condtons s seected to represent the re of the snge crystne mter. A unform cryst ttce orentton n dherence to the gob coordntes s ssumed for the snge-cryst mode. For the cse of cermcs, 2-D poycryst mode s constructed, s shown n Fgs. 2() nd 2(b), to represent the grn confgurton of the mter. A unt ce of the poycryst wth perodc geometry s dopted. The grn structure s determned wth respect to scnned topogrphy mge of dense nnocrystne BTO 3 cermc, nd the grns n the mode re mde n comprbe sze to ech other. Fg. 2() ustrtes the unt ce wth four typc grns,, 2, 3 nd 4, whch re tken to possess the oc cryst ttce orenttons of 0 o, 30 o, 45 o, nd 60 o, respectvey, wth respect to the gob fxed coordntes. The mode shown n Fg. 2() ssumes zero thckness of the grn-grn nterfce. As comprson, the GBAZ mode (Su 205) s ustrted n Fg. 2(b). Such core-she mode hs been prevousy dopted n mcromechncs settng to study the grn-sze dependence nd strn-rte senstvty of nnocrystne poycrysts [ Guo 20xx] nd the frcture n mnted mets wth nnogrned nterfce yer (). Unke the "ded yer" mode, the GB ffected zone st possesses some swtchbe spontneous porzton, but ke the ded yer mode ts deectrc constnt s so ower thn tht of the grn nteror. In our ccutons we hve set ts deectrc permttvty nd spontneous porzton to be 40% nd 35%, respectvey, of the grn nteror. These vues were set through dustment of ts Lndu coeffcents n Eq. (4). In ght of the reported 7-0 tomc spcng for the GBAZ or -3 nm for the ded yer, we took ts thckness to be 2nm, regrdess of the grn sze. GB Grn Intero () (b) Fg. - The 2D poycryst mode of brum ttnte: () the mode wth ssumed zero thckness of the grngrn nterfce; nd (b) the mode wth the ssumpton of GBAZ A the computtons re under the ssumpton of pne probem. The perodcty boundry condtons of the eectrc potent nd the porzton vector re mposed to the boundres of the representtve unt. On the ower nd upper horzont boundres, the perodcty condtons re gven by φ ( x,0) = φ( x, h) + E h nd P ( x,0) = P ( x, h) ( =, 2) for the eectrc potent nd porzton, respectvey. The constnt h s the heght of the representtve unt ce. On the eft nd rght vertc boundres, the perodcty condtons re gven by by φ ( 0, y) = φ( w, y) nd P ( 0, y) = P ( w, y) ( =, 2), where the constnt w s the wdth of the representtve unt ce. The equbrum of surfce chrge nd surfce -670-

5 Proceedngs of the 6th Internton Conference on Mechncs nd Mters n Desgn, Edtors: J.F. Sv Gomes & S.A. Megud, P.Degd/Azores, Juy 205 grdent fux t the boundres were ensured v the method of mutpont constrnts (MPCs) durng the fnte eement procedure. A the mter constnts used n ths phse-fed smuton re sted n the work of Su 205. A the obtned resuts re ment to be red ccordng to the current boundry-condton setup s descrbed n ths secton. RESULTS AND DISCUSSIONS Phse fed computton hs been conducted on snge-cryst nd poycryst modes respectvey. For convenence, we refer the snge-cryst mode to Mode-I, refer the poycryst mode wth zero thckness of grn-grn nterfce to Mode-II, nd fny refer the poycryst mode wth GBAZ ssumpton to Mode-III. The obtned hysteress oops nd butterfy oops of Mode-I re presented n Fgs. 3() nd 3(b), respectvey. Two odng condtons were mpemented n the numerc nyss. One odng condton s the qus-sttc odng condton, whch qures suffcenty mny tme steps to ow the evouton to rech stbe stte n ech odng ncrement. In the other odng condton, the ncrement of the unx eectrc fed s pped wth rte of 40 Hz. A remrkbe effect of the odng frequency s observed from the over ook of the resuts of the two odng condtons. The numerc resuts bsed on both odng condtons show very shrp swtchng of porzton ner the normzed coercve fed. Such feture grees wth most experment observton of snge cryst. In comprson, the resuts of 40 Hz odng show smoother domn swtchng process thn the qus-sttc odng cse. A remrkbe dfference n the coercve fed nd ctuton strn s observed between these two odng condtons. There s no notcebe dfference n the remnent porzton, the dfferent deectrc coeffcent nd the dfferent pezoeectrc coeffcent though. To demonstrte the tme-dependent behvour of the porzton swtchng process, the tot eectrc fed nd eectrc dspcement re potted n Fg. 3 s functon of tme. () (b) Fg. 2 - The computton resuts bsed on Mode-I: () the hysteress oop; nd (b) the butterfy oop E/E 0 D/D 0 Fg. 3 - The normzed tot eectrc fed nd tot eectrc dspcement s functon of the tme for the sngecryst mode wth the odng frequency of 40 Hz. -67-

6 Symposum_7 Mechnc Behvours of Advnced Mters nd Structures t A Sces In Fg. 4, the hysteress oop of the poycryst bsed on Mode Mode-II II s presented wth the odng frequency equency from 25 to 60 Hz. It s evdent tht the coercve fed w genery ncrese wth the eevted odng frequency whe the propertes t the remnent stte do not show notcebe chnge. The ncrese of the coercve fed wth the odng frequency frequ s potted n Fg. 5. As the computton ws run wth fxed mxmum mgntude of the eectrc fed, the normzed coercve fed shown n Fg. 5 hs n upper mt of 0.5, whch s the mxmum mgntude of the pped eectrc fed. Fg. 4 - The hysteress oops bsed on Mode Mode-II wth Fg. 5 - The ncrese of the normzed coercve fed odng frequency from 25to 60 Hz. s functon of the eectrc fed frequency (Mode-II) Fny, the computed hysteress oops nd butterfy oops re presented n Fgs. 6() nd 6(b), respectvey. The odng frequency rnges from 25 Hz to 000 Hz. It s observed tht strong frequency dependence exst n the ferroeectrc behvour of the brum ttnte poycryst. The coercve fed ncreses wth the odng frequency. The remnent porzton, the dfferent deectrc coeffcent nd the dfferent pezoeec pezoeectrc trc coeffcent vry wth the odng frequency. Especy, whe the odng frequency s bove 800 Hz, the hysteress oop nd butterfy oop become eptc nd kdney-shped.. Ths observton s n greement wth the ery theoretc nyss ooff the frequency dependence n mgnetc hysteress (Ro990), where sttstc phenomenoogc pproch ws used to descrbe the evouton ev of the order prmeter (the the mgnetzton vector vector). The knetc equton used tht work s Lngevn equton nstedd of the Gnzburg-Lndu equton. b Fg. 6 - The computton resuts bsed on Mode Mode-III: () the hysteress oop;; nd (b) the butterfy oop -672-

7 Proceedngs of the 6th Internton Conference on Mechncs nd Mters n Desgn, Edtors: J.F. Sv Gomes & S.A. Megud, P.Degd/Azores, Juy 205 CONCLUSIONS In ths work, three phse-fed modes were set up to nvestgte the frequency-dependent behvor n ferroeectrc mters. These three modes re bsed on dfferent mcrostructur chrcterstcs, nmey the snge-cryst mode (Mode-I), the poycryst mode wth zero thckness of grn-grn nterfce (Mode-II), nd the poycryst mode wth the ssumpton of GBAZ (Mode-III). In the study of the frequency dependence of ferroeectrc hysteress, we tke the vew tht t s resut of drect competton between the speed of porzton evouton nd the speed of extern odng. The phse-fed resuts bsed on these three modes showed tht the coercve fed genery ncreses wth frequency, but the propertes t the remnent stte showed very mted chnge wth the frequency, especy for the Mode-I nd Mode-II. The ccuted ow-frequency phenomen re wdey supported by experments, but the hgh-frequency ones re not commony found n terture except for the ery theoretc study n mgnetc hysteress. It s cery observed, from the resuts of Mode-I to III, tht the frequency dependence becomes more remrkbe s the eve of the compexty of the crystne mcrostructure ncreses. Therefore, by substnttng the resuts of three types of mter mcrostructure, we ssocted the frequency-dependent chrcterstcs of the ferroeectrc mters to the underyng domn dynmcs. ACKNOWLEDGMENTS The uthors grtefuy cknowedge the support from Nton Ntur Scence Foundton of Chn (Grnt No nd 22202) nd BIT Internton Reserch Coborton Proect (Grnt No. GZ204050), REFERENCES []-Ers JA, Lente MH. Domn reorentton n ferroeectrc-ferroestc cermcs. Ferroeectrcs, 2008, 363, p [2]-Guo X, Leung AYT, Chen AY, Run HH, Lu J. Investgton of non-oc crckng n yered stness stee wth nnostructured nterfce. Scrpt Mter, 200, 63, p [3]-Guo X, Weng GJ, Soh AK. Ductty enhncement of yered stness stee wth nnogrned nterfce yers. Comp Mter Sc, (3), p [4]-Guo X. Zhng WJ, Zhu LL, Lu J. Mesh dependence of trnsverse crckng n mnted mets wth nnogrned nterfce yers. Eng Frct Mech, 203, 05, p [5]-Jn LW, Su Y. The experment nvestgtons on frequency dependence of ferroeectrc propertes n soft PZT-5 cermcs. In: ASME 202 Internton Mechnc Engneerng Congress nd Exposton. Houston, 202, p [6]-Km, SJ. Predctons of tense creep behvor of PZT wfer by normy dstrbuted free energy mode. Mech Mter, 2009, 4, p [7]-Lente MH, Pcnn A, Rno JP, Ers JA. 90 degrees domn w rexton nd frequency dependence of the coercve fed n the ferroeectrc swtchng process. J App Phys, 2003, 95, p

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