Microanalyses of Ferroelectric Properties of BaTiO 3

Transcription

1 Micronly of Frrolctric Proprti of BTiO io i Liu Domin Grin boundry

2 Micronly of Frrolctric Proprti of BTiO Im Fchbrich Elktrotchnik und Informtiontchnik dr Brgichn Univrität-Gmthochchul Wupprtl Zur Erlngn d kdmichn Grd in Doktor-Ingniur ingricht Dirttion von M.Sc. Eng. io i Liu u Chngdu, Sichun Volkrpublik Chin Rfrnt: Korrfrntn: Prof. Dr. rr. nt. Ludwig Jof Blk Prof. Dr.-Ing. Wolfgng Mthi und Prof. Qingrui Yin Tg dr mündlichn Prüfung: 7. 6.

3 Acknowldgmnt Acknowldgmnt I would lik to pr my grtitud to Profor Dr. rr. nt. Ludwig Jof Blk for th opportunity to crry out thi work in Lhrtuhl für Elktronik, Brgich Univrität Gmthochchul Wupprtl undr hi uprviion. Hi continul intrting in my work, hi hlpful dicuion nd ncourgmnt, nd hi trict cintific point of viw hv ubtntilly contributd to th ucc of thi work. Through hi kindly hlp, I hv obtind th chnc to cooprt nd dicu with mny othr cintit in th r of frrolctric mtril nd coutic nr-fild imging. I m vry grtful for ll th hlp which h h givn m during my ty in hi Lhrtuhl. I lik to thnk Prof. Dr.-Ing. Wolfgng Mthi nd Prof. Qingrui Yin for co-mining thi work nd much hlpful dvic. I m lo vry grtful to Prof. mritu Dr. Klu Drnfld, Fchbrich Phyik, Univrität Kontn, who h rd ll th work crfully nd gv m much vlubl dvic. Th fruitful cooprtion btwn th Lhrtuhl für Elktronik, BUGH Wupprtl nd Lb. of function crmic, Shnghi Intitut of Crmic hlp m grt dl to bgin prnt work nd I m vry grtful for ll th hlp which Prof. Yin h givn m. I would lik pcilly to thnk Prof. Dr.-Ing. Volkrt Hnn, Lhrtuhl für thortich Elktrotchnik, BUGH Wupprtl, for th thorticl upport of th dirttion. For th hlp from Dr. Rlf Hidrhoff, who h hlpd m to complt th whol work, I would pr my pcil thnk. Mny thnk lo to Mr. Mchthild Knippchild, who h continully hlp m in ll th rpct. I m lo vry grtful to my dr collgu from th Lhrtuhl für Elktronik for th ctiv cooprtion nd upport. Thi work i prtly idd from DFG (Dutch Forchunggminchft) projct Nnochrktriirung frrolktrichr Domänn mittl Nhfld-kutichr- Rtrmikrokopin. I m grtful for thi finncil id. Th ncourgmnt nd hlp from my fmily nd, pcilly, from my wif Zhng Yun, hv crtinly contributd grt dl to th ucc of thi work nd I would lik to pr my thnk wholhrtdly. I

4 Contnt Micronly of frrolctric proprti of BTiO Acknowldgmnt Contnt Abbrvition, vribl, ymbol nd contnt I II VI Introduction. Poibl u of BTiO high dnity mmory mtril. Prnt rrch of frrolctric domin. Aim of prnt work 4.4 Structur 4 Thortic dcription of lctric nd frrolctric proprti of BTiO 6. Dfinition of frrolctric domin 6. Stt qution nd thrmodynmic of mtril 6.. Th tt qution 6.. Linr tt qution nd Mwll rltion 8.. Non-linr tt nd pproimtion 9. Thorticl dcription of BTiO mtril 9.. Crytl ymmtry nd frrolctric ph of BTiO ingl crytl 9.. Gnrl domin tructur of BTiO ingl crytl in ttrgonl ph.. Gnrl domin tructur of BTiO crmic.4 Stndrd mthod to img frrolctric domin 6.4. Chmicl tching 6.4. Powdr mthod 6.4. Opticl polriing microcopy 6 II

5 Contnt.4.4 -ry diffrction nd topogrphy SEM TEM 7.5 Nw mthod nd work to img frrolctric domin 7.5. Opticl mthod bd on th cond-hrmonic gnrtion 7.5. Scnning lctron nvironmnt microcopy 8.5. SEAM Scnning nr-fild coutic microcopy bd on SPM tchniqu 8.6 Limittion 8 Elctric mchnic coupling in typicl Nr-fild Acoutic Microcopy 9. Gnrl qution of lctric nd mchnicl coupling in olid 9. Dirct imging of lctric nd mchnic coupling. Combintion of coupling mthod to nr-fild mthod 4 4 Scnning lctron coutic microcopy 4. Phyicl bckground, ignl gnrtion, nd contrt mchnim 4. Eprimnt t-up 7 4. Dicuion of ignl to noi 8 5 Scnning nr-fild Acoutic Microcopy bd on SPM 4 5. Phyicl bckground, ignl gnrtion, nd contrt mchnim Grn function of th modl Modling of thick mpl Modling of thin mpl or film Eprimnt t-up 5 5. Dicuion of ignl to noi in SNAM 54 III

6 Contnt 6 Eprimnt procdur Dcription of Spcimn Spcimn prprtion nd trtmnt Schm of murmnt Choic of frquncy Amplitud nd ph imging Vrition of prmtr Dynmic imging 58 7 Rult Singl crytl Crmic 6 7. Film Th img mod comprion by SNAM Dynmic tudy of frrolctric domin tructur 64 8 Dicuion of both nr-fild coutic tchniqu 7 8. Qulity of imging comprd to othr tchniqu Spd of primnt How quntittiv 7 8. Comprion of SEAM nd SNAM dvlopd Img comprion nd nly Eplntion by th prnt thory 76 9 Concluion 8 Futur pct 8 Appndi 84 IV

7 Contnt Rfrnc Publiction nd prnttion V

8 Abbrvition, vribl, ymbol nd contnt Abbrvition, vribl, ymbol nd contnt AE i+, i Hlmholt fr nrgy Amplitud of + nd - trnmittd longitudinl coutic wv t ith mdium (i,,, which mn nr-fild in th mpl, r in th mpl but outid th nr-fild, nd in th PZT trnducr) B r, B Mgntic flu dnity nd it i th componnt (i,, or, y, ) i PZT c P c t, d c I, J, c P Stiffnd tiffn contnt of PZT trnducr nd BTiO ingl crytl Eltic tiffn tnor, nd it componnt with implifid uffi (I to 6 nd J to 6) Dpth of nr-fild d t, d i, J Piolctric trin contnt tnor, nd it componnt with implifid uffi (i,, or, y, nd J to 6) d w, D D d w Th domin width of fin nd cor grin in BTiO crmic Smpl thickn Thickn of PZT trnducr D r, D, Di Elctric diplcmnt, it mgnitud, nd it i th componnt (i,, or, y, ) PZT Piolctric tr contnt of PZT in -dirction. t, i, J Piolctric tr contnt tnor, nd th tnor componnt with implifid uffi (i,, or, y, nd J to 6) E r, E, Ei Elctric fild, it mgnitud, nd it i th componnt (i,, or, y, ) E F r g Gb Gb Gb Sourc lctric fild in nr-fild Body forc Grin i in crmic Gibb nrgy Eltic Gibb nrgy Elctric Gibb nrgy G t, G i Body torqu nd it i th componnt,( i,, or, y, ) H r, H Mgntic fild nd it i th componnt,( i,, or, y, ) HE i Enthlpy VI

9 Abbrvition, vribl, ymbol nd contnt HE HE J r J r k c k Eltic nthlpy Elctric nthlpy Conducting currnt Sourc currnt Stiffnd coutic longitudinl wv numbr in BTiO mtril Stiffnd coutic longitudinl wv numbr in PZT trnducr P r, P, Pi Polrition vctor, it mgnitud, nd it i th componnt, (i,, or, y, ) Q Q R Ht Elctrotrictiv cofficint Rflction contnt t th boundry btwn th trnducr nd th bck lctrod t,, Eltic complinc tnor, nd it componnt with implifid uffi (I to 6 I J nd J to 6) S Entropy T Tmprtur u r (r, r t), U Diplcmnt fild of prticl nd it i th componnt (i,, or, y, ) i U Intrnl nrgy v r, v i Vlocity of prticl nd it i th componnt (i,, or, y, ) Voutput WE Wm Output ignl of th PZT trnducr in nr-fild coutic microcopy ytm Elctric work Eltic work t, ij, I Strin tnor, it componnt, nd componnt with implifid uffi (I to 6) t, ij, I Str tnor, it componnt, nd componnt with implifid uffi (I to 6) Z Chrctritic ritnc of tiffnd coutic longitudinl wv in BTiO Z β ingl crytl with monodomin Chrctritic ritnc of tiffnd coutic longitudinl wv in PZT trnducr Sptil priodicl contnt in dirction for Bl function. β Th rtio of coutic chrctritic ritnc in th mpl nd in PZT VII

10 Abbrvition, vribl, ymbol nd contnt ε t, Z trnducr ( ). Z εi, j Prmitivity tnor, nd it componnt (i,, or, y, ) ε ε r, ε ri Prmitivity in vcuum (8.8548* - A/Vm) Rltiv prmittivity nd rltiv prmittivity of i th lyr of multilyr dvic PZT ε Prmittivity of PZT trnducr in -dirction ηe Bckcttring cofficint µ Mgntic prmbility in vcuum ρ Dnity of mtril (Kg/m ) σ ω EAI SEM SEAM SEI SNAM SIAM SPM SPAM TEM Noi of n quipmnt Angulr frquncy of th ytm Elctron Acoutic Img Scnning Elctron Microcopy Scnning Elctron Acoutic Microcopy Scondry Elctron Img Scnning Nr-fild Acoutic Microcopy bd on SPM Scnning Ion Acoutic Microcopy Scnning Prob Microcopy Scnning Photo Acoutic Microcopy Trnmiion Elctron Microcopy VIII

11 . Introduction Introduction. Poibl u of BTiO high dnity mmory mtril A rly 95 whn th dmnd for high-cpcity computr mmori cm, frrolctric mtril wr intnivly tudid in th world, thy md to b prim cndidt of nw mtril for binry mmori [~]. Ltr in th priod of 965 to 975, trmndou ffort wr tkn to dvlop frrolctric-miconductor mmori by th u of thin film of miconductor dpoitd on bulk frrolctric ingl crytl or crmic mtril. Though th bic concpt w vlid, th intbility of miconductor thin film t tht tim did not prmit vibl mmory to b built. In rcnt yr, frrolctric mtril hv ttrctd much mor ttntion bcu of th combintion of thir uniqu proprti of pontnou polrition, i.. th o-clld frrolctric domin of th mtril, to CMOS tchniqu of microlctronic indutry [4, 5]. Thi combintion h ld to lrg vrity of nw dvic in computr tchnology nd trnducing dvic in lctromchnic, lctrochmic, lctrooptic, nd coutooptic fild. Among ll th frrolctric ud during th dvlopmnt of modrn mmory dvic, Brium titnt (BTiO ) mtril ytm i on of th mot intrting frrolctric mtril ytm up to now []. BTiO ingl crytl h frrolctric tructur which r fr implr thn tho of ny othr frrolctric known nd thu provid good b for th rrch nd undrtnding of whol frrolctric phnomn. It i chmiclly nd mchniclly tbl nd h Curi tmprtur t bout C. It hytri loop h rthr hrp cornr nd good rctngulr pprnc. Th vlu of it corciv fild, murd t room tmprtur, vri from minimum of 5 kv/m to mimum of kv/m. Th dilctric contnt in th dirction of polrition ( ε r 6) i much mllr thn tht prpndiculr to it ( ε r 9 ) nd thy hibit pronouncd nomli t th trnition from ttrgonl to orthorhombic nd from orthorhombic to rhombohdrl tt. BTiO polycrytllin mtril, th o-clld crmic, nd thir modifiction offr vn mor ppliction in vriou fild of nginring. It i y to produc hrd BTiO crmic body by tndrd intring proc nd it body form cn lo b ily modifid ccording to th indutry ppliction. Th polrition dirction of crmic cn b chon rquird. BTiO thin film, nd othr provkit-typ film uch Ld Zircont Titnt (PZT), Strontium Bimuth Titnt film (SBT) nd o on, r intnivly tudid rcntly du to th intgrtion of th kind of film to CMOS circuit to produc vriou novl dvic [4, 5]. In gnrl,

12 . Introduction BTiO nd othr provkit typ frrolctric mtril provid tody n intnivly ctiv rrch nd ppliction fild. Evn though it tchnicl nd commrcil importnc i ubtntil, mny brkthrough ppliction my till li hd of u.. Prnt rrch on frrolctric domin Although prnt ppliction of frrolctric mtril in th modrn microlctronic indutry provid grt propct nd lmot unlimitd opportuniti, thr i much work to b don [5~]. Undr mcrocopic viw, problm ocitd with ppliction of BTiO nd othr currnt frrolctric r tht thir proprti r oftn controlld by th contribution from th o-clld trinic ffct which r rponibl for polrition ftigu, ging, frquncy nd fild dpndnc of piolctric, ltic nd dilctric proprti. Th contribution r gnrlly dcribd domin-wll ffct [~]. Th thorticl trtmnt nd primntl tudy on ll th contribution prnt u big chllng inc long tim. Although ubtntil inight into th ntur of frrolctric hv bn chivd, ll th thorticl modl, mot of which r phnomnologicl in ntur, nd primnt rult provid only globl or mcrocopic viw of th frrolctric. Th prnt ppliction of frrolctric mtril combind with CMOS intgrtd circuit rquir tht uch problm b furthr tudid undr micromtr or vn in difficult c in nnomtr rng. To kp pc with thi nw tchnologicl trnd, non-dtructiv tchniqu to invtigt th frrolctric domin undr uch ptil rolution mut b dvlopd. Whr th rolution of convntionl opticl microcopy i limitd by th diffrction limit, numbr of non-dtructiv mthod hv bn dvlopd to tudy frrolctric domin [4~7]. Scnning lctron microcopy (SEM) [4] i non-dtructiv but h didvntg tht contrt nd rolution r dpndnt on tim. In combintion with coutic, nondtructiv tchniqu with rolution nd contrt indpndnt of tim, cnning lctron coutic microcopy (SEAM), i ud to viuli frrolctric domin [5, 6]. Unfortuntly, SEAM h only rolution down to vrl micromtr du to th intrction r formd by primry lctron injctd into th mpl. Although trnmiion lctron microcopy (TEM) [7] h rolution down to nnomtr rng, thi tchniqu nd difficult mpl prprtion. Whthr th prprtion proc would ffct frrolctric domin i not clr. With th invntion of cnning prob microcopy (SPM) [8], non-dtructiv mthod to img frrolctric domin with ubmicromtr or nnomtr ptil rolution hv mrgd rcntly [9~]. F. Surnbch, t. l. [9] imgd th frrolctric domin of GMO ( Gd (MoO4 ) ) mtril ytm by th u of SPM

13 . Introduction in non-contct topogrphy mod. R. Lüthi t. l. [] prntd frrolctric contrt of GASH ( ( NH ) Al( SO ) H O ) nd M.-K. B t. l. [] of TGS C 4 6 ( ( CH COOH ) H SO4 NH ) mtril by th u of contct topogrphy mod of SPM rpctivly. By pplying n c voltg nd muring of th firt hrmonic ignl from topogrphy fdbck, K. Frnk t. l. [] introducd imging nd modifiction of domin in PZT film. For BTiO mtril ytm, S.-I. Hmki t. l. [] prntd rult by th u of SPM contct topogrphy mod on frrolctric BTiO ingl crytl nd uggtd tht th topogrphy img of SPM i lo frrolctric domin img. L. M. Eng t. l. [4] lo howd rult of domin contrt of frrolctric BTiO ingl crytl by th u of th non-contct mod wll th friction mod of SPM. Thrftr, thr wr minly two kind of mod of SPM which wr ud frquntly to img frrolctric domin of BTiO mtril ytm in th litrtur. On i th o-clld topogrphy mod [] nd th othr piorpon mod [9, ]. To tudy th contrt mchnim of both mthod, A. Gruvrmn t. l. [5] plind th contrt of topogrphy mod of SPM on BTiO rvld th diffrnc of -c domin boundry nd piolctric rpon mod imgd th c-c domin boundry. Bd on th principl of topogrphy mod of SPM, om litrtur [6-9] prntd th comprion of domin contrt from topogrphy mod of SPM with th rult from opticl microcopy, SEM, nd urfc potntil microcopy bd on SPM rpctivly. A th topogrphy mod of SPM rquir n bolutly flt mpl urfc, thi mod i impoibl to pply to img frrolctric domin of mpl with rough urfc uch crmic or lctronic dvic non-dtructivly. A ytmtic nlyi of contrt mchnim of thi mod nd th contrt comprion to othr tchniqu r lo difficult. At th b of th piolctric rpon mod of SPM, U. Rb t. l. tudid frrolctric domin from PZT crmic []. L. Eng. t. l. prntd th domin imging on PZT nd BTiO crmic, writing nd witching on bulk BTiO ingl crytl []. C. Hrng tudid th domin imging nd witching on BBi 4 Ti 4 O 5 thin film []. In ordr to nly th frrolctric domin by SPM quntittivly, C. Durkn t. l. prntd thorticl modl for th clcultion of th lctric fild in th ytm of th piolctric rpon mod of SPM []. Although th work [] h md comprion of th rult by piolctric rpon mod of SPM on BTiO crmic with rult by chmicl tching mthod, th chmicl tching mthod i ctully dtructiv mthod nd n plntion of th contrt diffrnc btwn th two mthod would b difficult. Mnwhil, lthough th thorticl modl [] for th piolctric rpon mod of SPM on frrolctric film h bn

14 . Introduction introducd, from our point of viw, th thorticl trtmnt of th modl cn not b th right thorticl nlytic mthod. Nithr cn th modl b ud ytmticlly to nly lctric fild ditribution in diffrnt mpl, uch bulk mtril, thin film or multilyrd film. Elctric nd mchnic fild ditribution nd th nrgy chng btwn thm in th ytm of piolctric rpon mod of SPM rmin unolvd up to tody.. Aim of prnt work In thi work, nw t-up, Scnning nr-fild Acoutic Microcopy bd on SPM (SNAM), i dvlopd to img frrolctric domin of both ingl crytl nd crmic of BTiO mtril ytm [4, 5, 6]. Th rult of crmic r comprd to th rult t idnticl r of th m crmic by nothr tblihd non-dtructiv coutic tchniqu, Scnning Elctron Acoutic Microcopy (SEAM) [5, 6, 7]. Bd on th clic phnomnologicl thory, thorticl modl for both th SNAM t-up dvlopd nd th SEAM on frrolctric BTiO mtril i groundd. Th frrolctric domin r nlyd quntittivly ccording to th modl tblihd. Diffrnt mod bd on SPM r lo comprd. Th frrolctric domin of BTiO crmic r imgd tmporlly, thrmlly dynmiclly, nd lctriclly dynmiclly [7]. A hown t th nd of thi work, both SEAM nd SNAM tchniqu r complmntry tool for th futur rrch nd ppliction of frrolctric mtril nd dvic. Th thorticl nd primntl mthod prntd cn furthr b pplid to othr nr-fild coutic microcopy tchniqu nd othr frrolctric mtril..4 Structur Thi work i minly dividd into th following chptr: Chptr will mntion brifly min thori on lctric nd frrolctric proprti of BTiO crytl nd crmic. Th typicl mthod nd nw work to img frrolctric domin r lo dicud. From th dicuion, two nr-fild tchniqu, SEAM nd SNAM, r chon mpl for th furthr dicuion. Chptr i concntrtd to lctric nd mchnic coupling of nr-fild coutic microcopy nd BTiO ingl crytl with monodomin tructur i chon n mpl to implify th dicuion. Th contrt mchnim for both nr-fild coutic microcopy tchniqu i nlyd. Chptr 4 trt th phyicl bckground, ignl gnrtion, contrt mchnim, nd primnt t-up of SEAM on BTiO mtril ytm. Chptr 5 dicu th phyicl bckground, ignl gnrtion, contrt mchnim, nd primnt t-up of SNAM dvlopd in thi work on BTiO mtril ytm. Thn th comprion of th contrt of th dvlopd ytm to tht of nothr mod of SPM, th o- 4

15 . Introduction clld piolctric rpon mod, i tudid. Finlly th domin tructur of BTiO crmic r tudid dynmiclly. Chptr 6 dicu th primntl dtil in thi work nd Chptr 7 prnt om typicl primnt rult. Chptr 8 dicu th complmntry tudy by both nr-fild tchniqu on idnticl r nd prnt n plntion for th tudy t th b of th thorticl modl groundd. A mll ummry will b givn in Chptr 9 nd futur propct will b brifly mntiond in Chptr. All thorticl clcultion will b prntd in th Appndi in dtil. 5

16 . Thortic dcription of lctric nd frrolctric proprti of BTiO Thortic dcription of lctric nd frrolctric proprti of BTiO. Dfinition of frrolctric domin Frrolctric domin r rfrrd to volum of pontnou polrition in frrolctric who polrition cn b chngd by n lctric fild [, ]. Th boundri prting domin r rfrrd domin wll. If frrolctric crytl i brought to th frrolctric tt from prlctric ph by dcring tmprtur, th lctrottic intrction on th urfc nd th inhomognity of tr in th mtril ffct th intrnl nrgy nd thu rult in frrolctric domin in frrolctric ph. A for th polycrytl mtril, th domin quilibrium i [5~7] in ny c i dtrmind by th minimum in nrgy which i ncry to prrv th hp of th grin whn ping from th prlctric to frrolctric tt. A thi frrolctric ph chng i rltd to lctric, ltic, nd thrml nrgy chng, it i ncry to dicu nrgy function nd tt qution of mtril t firt.. Stt qution nd thrmodynmic of mtril.. Th tt qution According to th thrmodynmic, it i umd tht th thrml, ltic, nd dilctric bhvior of homognou dilctric i fully dcribd by i vribl: tmprtur T, ntropy S, Strin t, Str t, lctric fild E r, nd lctric diplcmnt D r. According to th firt lw of thrmodynmic, th chng in intrnl nrgy U (pr unit volum) whn n infinitiml quntity of ht dq i rcivd by unit volum of dilctric i givn by: du dq + dw Eq.- whr dw i th work don on thi m volum during th rulting qui-ttic trnformtion. Auming rvribility, th cond lw of thrmodynmic rlt dq to th bolut tmprtur nd ntropy in th form: dq TdS Eq.- 6

17 . Thortic dcription of lctric nd frrolctric proprti of BTiO If only th mchnic nd lctric work r rltd, th firt lw of thrmodynmic i: du dq + dw E + dw m Eq.- Whr dw E nd dw m r lctric nd ltic work, which cn b writtn : dw E dd i E i dw m d I I Eq.-4 Hr th Voigt nottion ( I mn th tnor uffi nd i from to 6; i mn vctor uffi nd i from to.) r ud to pr tnor nd vctor [9]. Thi mthod provid om brifn for th prion of tnor nd vctor. Som work ud th m implifid nottion for tnor but diffrnt nottion for vctor ( i mn vctor uffi nd i, y, or.) [5] nd thi nottion mthod i clor to th cutom of lctricl nginr. For th dicuion of th prnt work, both nottion mthod r ud intrchngbly, th nottion of i from both nottion mthod giv th m phyicl mning. Th Eintin nottion for th ummtion of vctor nd tnor i lo ud. To dcrib ytm, ight diffrnt thrmodynmic potntil r dfind: Hlmholt fr nrgy: AE U TS Enthlpy: HE U I I E D i i Eltic nthlpy: HE U I I HE U E D i Elctric nthlpy: i Gibb fr nrgy: Gb U TS I I E D i i Eltic Gibb nrgy: Gb U TS I I Elctric Gibb nrgy: Gb U TS E i D i Eq.-5 Th ltic Gibb nrgy Gb i frquntly ud in th litrtur on frrolctric ph trnition. By chooing T, I, D ) indpndnt vribl, th diffrntil form of ltic ( i Gibb nrgy cn b writtn : 7

18 t r t r r t r r t t t r t t t r r r r t t t t. Thortic dcription of lctric nd frrolctric proprti of BTiO dgb SdT d + E dd Eq.-6 I Th othr vribl cn b clcultd : I i i S Gb T, I, D Gb I, Ei T, D Gb D i Eq.-7 T,.. Linr tt qution of mtril nd Mwll rltion If only linr form will b clcultd, th othr vribl of tt cn b clcultd : ds d I S S dt T d I I D, I T + i D D, T i, T d I I dt + I D, i I D D, T i, T S + dd Eq.-8 + dd Eq.-9 E i E i de i dt T d I I D, i + i D D, T i, T E + dd Eq.- Th cofficint in th bov qution r clld complinc. In th m wy, by chooing diffrnt vribl, diffrnt cofficint cn b obtind. For th c of iothrml (or dibtic), th qution cn b gnrlly dcribd : d I c I D, T, J dj ~ T h I, j dd j Eq.- de i T h i, J d + J T k, i, j dd j Eq.- d I I d ~ + D, T, J J g T I, j dd j Eq.- de i T g i, J d + J T k, i, j dd j Eq.-4 d I I ~ d + E, T, J J d T I, j de j Eq.-5 dd i T d i, J d + ε J, T i, j de j Eq.-6 d I c I E, T, J dj ~ T I, j de j Eq.-7 8

19 t. Thortic dcription of lctric nd frrolctric proprti of BTiO dd i T i, J d + ε J, T i, j de j Eq.-8 Th lt four qution r clld piolctric tin nd tr qution. A th iothrml (or dibtic proc) will b uully dicud, th ign of contnt T in th prion bov i omittd in th following dicuion... Non-linr tt nd pproimtion Th linr tt qution bov r dcribd by th linr diffrntil qution. But om of th mot importnt chrctritic of frrolctric uch hytri loop, lctrotriction, polrition rvr nd o on r fundmntlly non-linr ffct nd hnc rquir n tnion of th thory to highr ordr. Gnrlly th nonlinr tt of mtril cn b dcribd by pnion of th tt qution to rbitrrily high ordr to dfin th non-linr complinc, but th prcticl difficulti of tnor mthmtic t high ordr mk it lmot impoibl. In ordr to how th phyicl mning mor clrly, om pproimtion hv to b ddd: Th prototyp tt i D i I, which mn th originl tt h nithr polrition nor tr; Th tt with polrition h it polrition long on of th crytllogrphic ; Non-polr tt i cntroymmtric. Undr th umption bov, th ltic fr nrgy cn b pndd Tylor ri [5, 4]: Gb Gb 4 ( T ) + αd + α D + αd Q D + L Eq.-9 Hr, for th k of implicity, th uffi of vctor nd tnor r omittd it will not chng th phyicl mning of th prion. In th following dicuion, th m implicity i lo ud until it i ncry to nly th componnt of vctor nd tnor.. Thorticl dcription of BTiO mtril.. Crytl ymmtry nd frrolctric ph of BTiO ingl crytl Two kind of tructur of BTiO r known. On blong to th hgonl ytm, nd th othr i provkit. Th firt on do not how frrolctricity nd hnc only th provkit tructur will b furthr dicud in thi work. Th provkit-typ tructur, which i th 9

20 . Thortic dcription of lctric nd frrolctric proprti of BTiO lrg fmily of compound with th gnrl formul ABO, h th cubic non-polr ph (cntroymmtricl) nd th ptil A lttic i hown in Fig.-. Th i th B lttic contnt. Th provkit BTiO blong to thi kind of mtril nd i O cubic nd nonpiolctric bov th Curi point (bout C). It i th ttrgonl lttic ytm from Curi point to C, Fig.-: Provkit-typ tructur ABO orthorhombic from C to 9 C, nd rhombohdrl blow 9 C. Th diffrnt frrolctric polrition dirction (hown rrow) t diffrnt ph obrvd by primnt r illutrtd in Fig.-. c c b r r r Fig.-: Cubic Fig.-b: Ttrgonl Fig.-c: Fig.-d: T > C C > T > C Orthorhombic Rhombohdrl C >T> 9 C -9 C >T Undr th umption tht ph chng i mll nd th crytl h only monodomin t frrolctric ph, phnomnologicl thory [4] w prntd to plin th polrition in BTiO mtril. Th principl i to minimi th ltic Gibb nrgy whn th mtril go through th ph chnging point. From th qution Eq.-9, by nglcting th trm highr thn E Gb D 6 D, th lctric fild cn b writtn : 5 α D + α D + α Eq.- D hr D i qul to P, if lctric fild do not it. During th ph chng, P r h thr componnt:

21 . Thortic dcription of lctric nd frrolctric proprti of BTiO D r P r P r + P y r y + P r Eq.- According to th Eq.-, th ( P, P y, P ) cn b olvd for four poibiliti: ) ( P, P y, P ), Gb b) P P y, P, Gbb Ttrgonl tt, >T> c) P, d) P P y P y P, P, Gbc Gbd Orthorhombic tt, >T>-9 Rhombohdrl tt, -9 >T Hr Gb, Gb b, Gb c, nd Gb d r th Gibb nrgy t th corrponding tt. Th four tt bfor nd ftr pontnou polrition dcrib th fct which hv bn obrvd in primnt, hown bov in Fig.-. To th m pproimtion, if nd E do not it during th ph chng, ccording to Eq.-7 nd Eq.-9, it mut b notd tht th pontnou trin: Gb P Q Eq.- Hr Q i th lctrotrictiv cofficint. Bcu th lctrotrictiv ffct it in ll mtril, it mn tht pontnou trin lwy ccompni pontnou polrition. Th piolctric voltg cofficint i: g Gb D Q P Eq.- Th piolctric trin contnt in th dirction of polrition cn b writtn [5]: d εqp Eq.-4 Tht mn tht piolctric ffct of frrolctric mtril t monodomin tt i proportionl to pontnou polrition through lctrotrictiv contnt... Th gnrl domin tructur of BTiO ingl crytl in ttrgonl ph In rlity, th monodomin tt of mtril t frrolctric ph, which i umd bov

22 . Thortic dcription of lctric nd frrolctric proprti of BTiO for th implicity of dicuion, do not it [~]. Frrolctric domin tructur t frrolctric ph r lwy formd by th lctrottic intrction on th urfc nd th inhomognity of tr in th mtril, whn BTiO ingl crytl go through th ph chng. At ttrgonl ph, thr r totlly four tructur of frrolctric domin of BTiO ingl crytl mtril. Th domin with pontnou polrition prpndiculr to Surfc Domin boundry Domin boundry Surfc P P c P P Fig.-: 9 - domin wll [] Fig.-b: 9 -c domin wll [] Surfc Surfc Domin boundry Domin boundry P P P c Pc Fig.-c: 8 - domin wll[] Fig.-d: 8 c-c domin wll th urfc of th mtril i clld c-domin nd tht with pontnou polrition on th urfc of th mtril -domin. Thr r totlly four poibiliti of domin boundri, tht i, if th urfc i umd th [] pln, th 9 - [], 9 -c [], 8 - [], nd 8 c-c domin boundri (domin wll). Th lt on h n rbitrry boundry form which i vrticl to th urfc. It hould lo b mphid hr tht th domin wll it poibly in ny quivlnt pln. On th urfc of th c-domin, thr r urfc crn chrg. Fig.- how th domin tructur on thin plt of BTiO ingl crytl. For th tructur bov, th domin wll width ftr th ph trnition cn b dcribd by th minimiing th totl fr nrgy [, 4~4]. Th Gibb fr nrgy undr th umption bov cn b writtn : 4 Gb Gb + WE + Wdip + Wm + ( α D + α D ) dv V 4 Eq.-5

23 . Thortic dcription of lctric nd frrolctric proprti of BTiO Hr W E, W dip, nd W m r lctric nrgy, nrgy by th dipolr intrction, nd th ltic 6 nrgy. Th high ordr trm of D i furthr nglctd for th implicity. According to th thory bov nd by minimiing th totl fr nrgy, om work [4, 4] hv nlyd th domin wll nd ld to n timtion of wll thickn nd wll nrgy of om mtril. Th typicl vlu of BTiO ingl crytl r hown in tbl. Thi prdiction i nturlly undr th umption tht ll th prmtr nd dimnion of domin wll cn b dcribd mcrocopiclly. Th rult of th wll width clcultd bov r nturlly not of mcrocopic dimnion. Thr i lo microcopic pproch which giv th m vlu of wll nrgy [4]. Th mthod nd vlu hr will b thrfor ccptd in thi work, othr work do. Mtril Wll thickn (nnomtr) Wll nrgy (mj/m ) BTiO 8 wll.5~ BTiO 9 wll 5~ ~4 Tbl : Th typicl proprti of domin wll of BTiO ingl crytl.. Th gnrl domin tructur of BTiO crmic Th formtion of domin in crmic i diffrnt from tht in ingl crytl bcu domin tructur of grin r formd undr clmpd condition, whr ingl crytl i fr. Th domin i t quilibrium tt in ny c i dtrmind by th minimum of th intrn nrgy which i ncry to prrv th hp of th grin whn ping from th prlctric to frrolctric tt. Although thr h bn lot of ffort to plin th rltionhip mong grin i, domin width, ltic, piolctric, dilctric proprti nd o on, gnrlly ccptd thory to dcrib nd plin ll th rltion nd lo to b tblihd [5~]. Furthrmor, how mny kind of tructur of frrolctric domin t th ttrgonl ph of BTiO crmic it gnrlly i till not clr. Th following i th mot widly ccptd dcription of frrolctric domin tructur of BTiO crmic t ttrgonl ph in rcnt litrtur [6~]. It i found tht thr r diffrnt tructur of domin in BTiO crmic but two kind of domin tructur in BTiO crmic r typiclly obrvd by chmicl tching mthod. On it minly in th grin with i mllr thn µm nd hv impl lmintd 9 domin tructur. Th othr kind compo minly of bndd 9 domin if grin i r

24 . Thortic dcription of lctric nd frrolctric proprti of BTiO lrgr thn µm [6, 7]. For th firt kind of domin in BTiO crmic, ccording to th m phnomnologicl thory bov, om work to clcult th rltionhip btwn th domin width nd th grin i i prntd. Th domin width ( d ) with rpct to th grin i g i clcultd [7]: w / d w (g) Eq.-6 Th typicl width of 9 domin i vrl hundrd nnomtr. Anothr kind of frrolctric domin it who domin tructur r typiclly 9 bndd domin nd hown in Fig.-4 [6, 8, 9, ]. Thi kind of tructur of BTiO crmic w obrvd motly if th grin i i lrgr thn µm. Th two 9 domin tructur of thi kind of domin r hown in Fig.-4. Th α tructur h on-to-on corrpondnc of conjugtd domin r whr th γ tructur without ny corrpondnc. Th domin width d w cn b clcultd []:.4 d w (µm).4g / ( + ) / g.4 Eq.-7 If cor grin of BTiO crmic h dimnion of 5 µm, th domin width i bout nm. It i clr tht th domin width who tructur r known tody r in th rng of vrl hundrd nnomtr in BTiO crmic. Grin i g Domin width dẃ Y Y Fig.-4: α tructur of 9 domin lmintd tructur Fig.-4b: γ tructur of 9 domin lmintd tructur 4

25 . Thortic dcription of lctric nd frrolctric proprti of BTiO Othr typ of domin of frrolctric BTiO crmic, uch tho with no rltionhip to grin i, bcu of thir compl ntur nd P r in th bulk mtril, r l tudid nd littl documntd. Th complictd rltionhip mong domin tructur, tmprtur, lctric nd frrolctric proprti, nd grin i r uully tudid primntlly. In rcnt litrtur, domin nd thir ffct on mcrocopic proprti of frrolctric crmic r tudid primntlly. Th dilctric, ltic, piolctric proprti of frrolctric crmic r dcribd by rcnt tudi th um of intrinic nd trinic proprti [~]. Th intrinic proprty of frrolctric mtril i dfind proprti of th mtril with monodomin nd th trinic proprty th contribution from othr prt of th mtril, uch domin wll movmnt. Th dilctric contnt ε, piolctric contnt d, nd ltic complinc, r thrfor writtn follow: ε t t t ε + ε d t t in t t d in + d t t in + Eq.-8 Th ubcript in nd dnot th intrinic nd trinic contribution. Although modl for frrolctric crmic to dcrib th rltionhip btwn th intrinic nd trinic contribution undr wk fild h only bn prntd rcntly, gnrlly ccptd thory vn by phnomnologicl mthod i till not wll dvlopd []. Th piolctric rltion of Ryligh modl for frrolctric crmic will b ccptd in thi work []: d εε rqp +. 7 Eq.-9 m whr d i th ctul piolctric contnt for frrolctric crmic; Q, P, nd m r th lctrotrictiv contnt, pontnou polrition, nd th mimum vlu of priodicl tr by trnl fild. Th firt trm i th o-clld intrinic piolctric ffct. Th dtild dicuion of Ryligh modl nd th dilctric nd ltic rltion will not b mntiond hr for th implicity of our dicuion of th nr-fild coutic imging in thi work. 5

26 . Thortic dcription of lctric nd frrolctric proprti of BTiO.4 Stndrd mthod to img frrolctric domin Th microcopic chrctrition of frrolctric domin i lwy chllng for cintit in thi r. Th clic work will b hortly mntiond hr..4. Chmicl tching Thi i th rlit wy to viuli frrolctric domin in frrolctric. Concntrtd HCl tch th - nd c-domin of BTiO t diffrnt rt, nd lo +c nd c nd of th domin, o tht diffrnt ttur or hd ppr whn th tchd pcimn i mind undr microcop []. It i dtructiv mthod nd how uch n tching proc influnc ubqunt domin i lo not documntd..4. Powdr mthod Thi mthod u colloidl upnion of chrgd prticl, which dpoit prfrntilly on ithr poitiv or ngtiv nd of domin. It i nondtructiv mthod, but th difficult choic of powdr nd th limitd rolution nd contrt mk it not fibl for indutry. Furthrmor it i lo vry difficult to u it on mtril with rough urfc bcu th urfc topogrphy will dd to th contrt of powdr contrt..4. Opticl polriing microcopy To obrv frrolctric domin, th uul mthod i th opticl microcopy with polrir nd n nlyr. Whn th polrir nd nlyr of th opticl microcop r crod t 9 orinttion, no light i trnmittd through microcop unl th pcimn inrtd produc ph chng btwn two diffrntly polrid light ry ping through it. If it i umd tht th opticl i i lo polr i, for BTiO, th c-domin will not chng th ph of th light nd ppr drk. Th -domin will chng th ph of th light nd ppr bright. It i lo hown tht on cn img +c nd c domin by tudying th trininducd biil mtril long ch domin wll [44]. Unfortuntly, th opticl mthod i limitd by th diffrction limit of th focud light bm nd th rolution i only bout hlf of th light wvlngth ud. It will b difficult to obtin th rolution in ubmicromtr or nnomtr rng. Morovr, lthough thi tchniqu i th mot common mthod to img frrolctric domin of ingl crytl, it i vry difficult to img frrolctric domin of crmic non-dtructivly, th urfc h to b polihd to gt ufficint contrt. To wht tnd th polihing proc chng th domin tructur i till not wll tudid. 6

27 . Thortic dcription of lctric nd frrolctric proprti of BTiO.4.4 -ry diffrction nd topogrphy Both mthod rquir th polihing of th urfc to incr th img contrt. Th -ry diffrction mthod i typicl mthod for th rrch of ingl crytl nd film. It i n indirct mthod nd by nlying th diffrction ngl, th urfc polrition cn b indirctly hown [44]. Th contrt mchnim of -ry topogrphy [, 45, 46] i tht th nomlou diprion of -ry cu diffrnc btwn th -ry intnity rflctd from th poitiv nd ngtiv nd of domin. By uing wvlngth clo to n borption dg of contitunt lmnt thi diffrnc cn b mimid. Th domin in BTiO hv bn uccfully obrvd by thi mthod. Although it i clic mthod to nly frrolctric domin, it rolution nd rquirmnt of urfc roughn limit it ppliction..4.5 SEM Mthod uing SEM to obrv domin r lo prntd [4]. Th principl i bd on chng of urfc lctric potntil from domin to domin which will b imgd in condry lctron img. It h only rolution of bout vrl µm. Th tbility of th contrt of thi mthod nd it comprion with othr tchniqu nd to b tudid..4.6 TEM Thi mthod i th mot powrful mthod to obrv frrolctric domin [5, 47, 48]. It prnt good rolution down to vrl nnomtr or lowr. But th mpl prprtion of thi mthod i difficult nd dtructiv, nd whthr th mpl prprtion would chng th domin on th mpl urfc rquir furthr invtigtion..5 Nw mthod nd work to img frrolctric domin Thr r lrg mount of litrtur which rportd nw tchniqu to img frrolctric domin. It i but impoibl to mntion ll th tchniqu hr. Only om typicl tchniqu will b brifly dicud in th following..5. Opticl mthod bd on th cond-hrmonic gnrtion Thi tchniqu cn b ud in principl for ny crytl which cn b mtchd in ph for cond-hrmonic gnrtion with light propgtion clo to th polr i [, 49, 5]. Som work hv lo combind thi tchniqu with nr-fild opticl mthod [5]. Although om rult on ingl crytl r prntd, thi mthod h no dvntg for th nly of 7

28 . Thortic dcription of lctric nd frrolctric proprti of BTiO BTiO crmic which r uully vry rough t th urfc nd opqu..5. Scnning Elctron Environmnt Microcopy Thi i rltivly nw mthod [5] nd th principl i lmot imilr tht of SEM to img frrolctric domin. But, bcu it u pcil nvironmnt in th vcuum chmbr in SEM, th img contrt cn b hld for vrl hour. For crytl mtril, om rult hv bn obtind but tho on crmic hv not bn prntd up to now. It rolution i lo in th rng of vrl micromtr..5. SEAM Thi mthod u vry thin gold film on th urfc of th mtril to void urfc chrging ffct nd th rult of both ingl crytl nd crmic r hown [6, 7, 7, 8]. Th dtild dcription of th mthod will b dicud ltr in Chptr Scnning nr-fild coutic microcopy bd on SPM tchniqu Th nm, Scnning nr-fild coutic microcopy (SNAM), w introducd in 989 [64] nd thr r lot of dvlopmnt with thi tchniqu up to tody. For th k of implicity of th prnt work, th nm SNAM would b ud to mn ll th ytm bd on SPM tchniqu to img coutic proprti. Th mthod r nw nd thr i till lot of dicuion on it. To chrctri th contrt of SNAM tchniqu on BTiO mtril, lthough diffrnt t-up nd diffrnt rult hv bn prntd dicud in Chptr, nw nondtructiv mthod with nnomtr rolution bd on th combintion of SPM nd coutic microcopy i introducd in thi work [4~7]. Thi mthod provid poibility to compr contrt of SEAM nd SNAM tchniqu both thorticlly nd primntlly. A dtild dcription of thi t-up of SNAM to img frrolctric domin tructur of BTiO will b prntd ltr in Chptr 5..6 Limittion: qui-ttic For ll th dicuion bov, on of th mot fundmntl umption i ud. Th dicuion nd imging of frrolctric domin r only dicud qui-tticlly. Th high frquncy proprti of frrolctric domin will not b dicud hr bcu th imging mchnim of SEAM nd SNAM will b minly concrnd in thi work. 8

29 r r t t. Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy. Gnrl qution of lctric nd coutic coupling in olid Elctricl nd coutic coupling qution r brifly dicud in Chptr. Thir ppliction combind with th clic lctromgntic nd coutic thori in ctul nrfild coutic microcopy ytm i dicud hr. From Eq.-5 to Eq.-8, th piolctric trin nd tr qution cn b obtind by intgrtion undr th umption of ro vlu of ll th fild componnt t quilibrium tt: E ~ + Eq.- I I, J J d I, j E j Di d i, J I c I + J E, J J ε ~ - i, j I, j E j E j Eq.- Eq.- Di i, J + J ε i, j E j Eq.-4 Eq.- nd Eq.- r piolctric trin qution nd Eq.- nd Eq.-4 tr qution. ~ Th contnt d, ( d I, j contnt of th trnpod mtri) i piolctric trin contnt nd i, J ( I, j i J ~ contnt of th trnpod mtri) piolctric tr contnt. Th Mwll qution to dcrib gnrl lctric nd mgntic phnomn r [5]: E r H r r B t E r + t J r c + J r Eq.-5 Eq.-6 whr E r nd H r r lctric nd mgntic fild; D r nd B r r lctric diplcmnt nd mgntic flu dnity; J r c nd J r r conducting nd ourc currnt. Acoutic vibrtion or wv in olid mtril r govrnd by th Nwton lw for dynmic motion undr th clic viw. Motion cn b clifid into two kind: trnltionl nd rottionl motion. For trnltionl motion, th Nwton lw cn b writtn : t r u r ρ F t Eq.-7 whr u r i th diplcmnt fild of prticl in mtril nd F r i n trnl body forc. 9

30 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy For th implicity of dicuion blow, th vlocity of prticl ( v r u r ) i ud nd Eq.-7 t cn b writtn : t v r ρ F t Eq.-8 For rottionl motion, th Nwton lw cn b writtn : + G Eq.-9 ji ij nd ij k ij r tr componnt nd G k i body torqu. Undr th mll ignl nd wk piolctric coupling pproimtion, th body torqu cn b nglctd vn if mot of th mtril of piolctric trnducr r frquntly frrolctric, o tht th tr tnor i lwy ymmtric tnor. Combining th lctromgntic nd coutic qution with th piolctric qution, on cn obtin th gnrl coupld qution for fild nd wv in piolctric mtril [5]: r r r v t r ρ ( c v) t : t E F + t Eq.-9 t r t E µ ε E r r r r t v J µ : J µ t µ c Eq.- t t t Th qution bov r thorticlly brif but phyicl mning r not y to dirctly. In ordr to tudy th contrt mchnim in nr-fild coutic ytm clrly, it i ir to nly th coupling by prting Eq.-9 nd Eq.- to diffrnt coupld group. It i lo ncry tht th ytm b implifid o tht nlyticl mthod cn b ud.. Dirct imging of th coupling by trnducr A gnrl t-up of nr-fild coutic microcopy i hown in Fig.-. In nr-fild r, if thr i crtin timultion which produc coutic vibrtion, th vibrtion will trnmit to trnducr which i in olid contct with th mpl nd will chng th coutic vibrtion into lctric ignl. To undrtnd th ytm ytmticlly, w mut tudy how th timultion produc coutic vibrtion, how th vibrtion trnmit to th trnducr, nd how th trnducr chng th coutic wv into lctric ignl in th typicl t-up. At firt, th lctric nd coutic ignl chng in th trnducr will b dicud. Th coutic wv in nr-fild nd thir trnmiion in th mpl, thy r mor complictd, will b dicud in th nt ction. Som typicl dt bout thi t-up r: th thickn of th mpl i uully ~5 mm nd th

31 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy Nr-fild dpth d Nr-fild width w y D: Thickn of th Smpl ε r D : thickn of th trnducr PZT D : thickn of th coppr lctrod Good coutic mtching Z To lock-in mplifir Fig. -: Typicl t-up of nr-fild coutic microcopy ltrl dimnion of th mpl vrl ~ cntimtr. Th trnducr h normlly thickn of ~ mm nd ltrl dimnion of ~ cntimtr. Th width of th nr-fild in ltrl dirction i dpndnt on th ytm nd th mtril proprti of th mpl. Typicl vlu of SEAM tchniqu on dilctric mtril i vrl micromtr [57, 7]. A for SNAM dicud ltr in Chptr 5, th nr-fild width i vrl tn of nnomtr. Bcu th ltrl dimnion of th mpl nd th trnducr i much lrgr thn thir thickn, th mpl nd PZT trnducr cn b pproimtly trtd thin plt with unlimitd dimnion in th ltrl dirction. Undr uch n pproimtion, th coupling in both kind of mtril cn b nlyd th lctric nd coutic coupling in on dimnion. Principlly, th coupling in on dimnion cn b olvd by th gnrl coupling qution Eq.-9 nd Eq.-, but it i clrr to th phyicl mning by prting th bic qution to find diffrnt kind of coupling. For typicl u of trnducr th coutic dtctor, lctric ignl, which r chngd from coutic ignl, r lwy in th rng of µv nd cn b trtd mll ignl. Th o-clld mll ignl nd wk piolctric coupling cn b ud. Undr uch pproimtion, dtild clcultion of lctric nd coutic coupling in on dimnion for BTiO crytl nd PZT i prntd in Appndi A. If th dirction i chon th trnmiion dirction, thr r thr kind of coupling in both mtril. Th firt i th coupling btwn qui-ttic lctric fild E [5] nd n coutic longitudinl pln wv ( v, ) which cn b dcribd by th coupling qution:

32 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy v ρ c p v t, nd E ε, Eq.- ( v, ) r th vlocity of prticl nd th tr in dirction; i th trin in dirction. Th qui-ttic lctric fild E i coupld to coutic wv ( v, ) through trin componnt. Th othr two typ of coupling r coupling btwn two lctromgntic pln wv E, H y ) nd ( E y, H ) nd two hr coutic pln wv ( v, 5 ) nd ( v y, 4 ): ( E µ ε E t µ 5 v t Eq.- E y µ ε yy E t y µ 5 v y t ( ε yy ε for BTiO ) Eq.- Th phyicl mning of Eq.- i tht, if qui-ttic lctric fild in crtin mtril i givn, th lctric nd coutic tiffnd coupling cn b dtctd by muring th coutic longitudinl pln wv producd by th lctric fild in tht mtril; Or, if n coutic longitudinl pln wv i givn, thr it crtinly qui-ttic lctric fild in dirction nd by muring thi lctric fild, th coupling cn b dtctd wll. Bcu longitudinl trnducr of th dvlopd ytm i ud to dtct th longitudinl wv producd in th ytm in dirction, only th firt coupling of Eq.- will b furthr dicud in thi work. In th trnducr, th hrmonic coutic longitudinl pln wv ( v, th u of norml mod [5]: ( ) / ) cn b writtn by + jωt + Eq.-4 v ( + jωt ) / Z Eq jk ( D ) ( D ) jk ( D ) ( D ) Eq.-6 Eq.-7 in which + D ) nd D ) r mplitud, Z th chrctritic ritnc, nd k th ( ( wv numbr of th coutic longitudinl wv in dirction in th trnducr. Th uffi t th right top cornr mn wv in th trnducr, nd + nd - indict th wv trnmiion dirction. At th intrfc btwn th PZT trnducr nd th coppr lctrod, if th coutic rflct

33 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy contnt t th boundry btwn th trnducr nd th bck lctrod i R, th wv form + nd R mut tify th condition: ( D ) jk D + jk D Eq.-8 ( D ) ( D) R + ( D ) jk D Eq.-9 Th lctric fild in th trnducr in dirction undr th opn circuit condition cn b prd [Appndi A]: E ε PZT PZT Eq.- Hr PZT c P, PZT ε, nd PZT r tiffnd tiffn, dilctric, nd piolctric contnt of PZT in -dirction; i th trin in -dirction of longitudinl wv nd h th form: v jω Z V p ( + + ) jk ( ) D Eq.- From Eq.-9 nd Eq.-, th qui-ttic lctric fild in th trnducr i: E ε PZT PZT PZT ( R jk ε PZT D + ) + jk ( D ) ( ) Z V D Eq.- p Th voltg btwn two lctrod of th trnducr undr th opn circuit condition i: D D V output D + E d jk c PZT + ( D ) ε PZT ( + R jk D ) (- jk D ) Eq.- PZT P In ll th qution bov, + D ) ( i contnt dtrmind by both th ytm boundry condition nd th timultion in nr-fild. If th coutic wv mplitud + D ) ( qul to on unit, which mn tht homognou coutic longitudinl pln wv with unit mplitud trnmit to th trnducr, th mplitud nd ph rpon of th trnducr output ignl to th frquncy of th typicl t-up hown in Fig.- cn b obtind. Th mplitud nd ph ignl rpon of th

34 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy output to frquncy r hown in Fig.- nd Fig. -b. µv Rd kh Fig. -: Amplitud rpon of th trnducr to unit longitudinl pln wv kh Fig. -b: Ph rpon of th trnducr to unit longitudinl pln wv Although Eq.- dcrib th quntittiv rltionhip btwn th coutic longitudinl wv nd th output ignl of th trnducr t opn circuit condition, in th ctul c of cnning nr-fild coutic microcopy ytm, th contnt + D ) i dpndnt not only on th boundry condition but lo on th coupling ourc in th nr-fild. Thrfor, it i ncry to nly th coupling mchnim in th nr-fild nd th whol trnmiion proprty of th ytm.. Piolctric coupling in nr-fild of th cnning nr-fild coutic microcopy To coupl coutic wv in mpl, diffrnt mthod r ud in non-dtructiv tting indutry [59]. Convntionl cnning coutic microcop (CSAM) u n coutic ln to focu nd injct th coutic wv into th mpl nd thrfor th rolution dpnd on th coutic wv lngth ccording to th Ryligh critrion. Although tchnologiclly ubl coutic wv in olid cn rch to vrl GH up to now, th rquird coupld liquid in CSAM ytm h uch high ttnution t thi frquncy rng tht th CSAM cn only u th frquncy blow thi rng nd th rolution cn only rch vrl micromtr t mot. Thr i nothr kind of tchniqu who rolution h no uch limittion, th o-clld nr-fild coutic mthod [55~6]. A typicl t-up i hown in Fig.-. Th principl of thi kind of microcopy i to produc coutic wv by vriou intrction ffct within tiny volum t th dirct vicinity of th mpl urfc. Thi volum i frquntly dfind coutic nr-fild nd it dimnion i much mllr thn th wv lngth of coutic wv ( 4

35 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy ud. Both ltrl nd dpth profiling of th mthod r gnrlly dpndnt on thi dimnion [57~6]. Th min intrction btwn th trnl timultion nd th mtril in nr-fild r photo, ion, lctron, nd, rcntly, prob coutic coupling intrction. Th coutic wv gnrtd in nr-fild by crtin intrction or vrl intrction will furthr trnmit through th mpl, b dtctd by trnducr, nd finlly b imgd by th ytm. Gnrlly, th min intrction by diffrnt mthod cn b gnrlid thrml coupling, gnrtion of intrnl lctric fild, chng of lttic contnt nd o on. Th min typ of microcop ytm which u th nr-fild intrction cn b gnrlid in th following. Scnning photo coutic microcopy (SPAM) u focud lr bm to produc coutic wv in nr-fild [6]. Th lr bm ud h wvlngth typiclly in th viibl rng nd powr from vrl mw to vrl hundrd mw. According to th mpl ttd, th contrt mchnim r minly thrml nd optocoutic coupling. Th formr cn b plind tht th choppd lr bm wrm th mpl in th nr-fild priodiclly, nd, bcu of thi priodicl thrml nrgy chng, th tiny pic of mtril in th nr-fild pnd nd contrct priodiclly. Th priodicl pnion nd contrction produc coutic wv which r rltd to thrml nd coutic proprti. In th pcil c of optocoutic mpl, th dirct optocoutic coupling i ud to produc coutic wv. A th dimnion of nr-fild r dpnd on th bm width of th injcting lr bm, which i formd by focuing ytm, th rolution of thi mthod i lo dpndnt on fctor uch th wvlngth of th lr bm nd th mtril proprti of th mpl. Ecpt for th optocoutic tructur, th limitd lr powr prnt min drwbck to th lctric nd coutic coupling in nr-fild. Scnning ion coutic microcopy (SIAM) h modultd microprob with ithr low [6] or high [6] ion nrgtic ( kv) bm implntr. Th coupling mchnim i thrml coutic coupling nd th gnrtion of c crrir which form n intrnl lctric fild. Thi fild cn produc coutic wv whn th mpl i piolctric. Although low nrgy ion bm cn b dirctly concntrtd on th illumintd urfc nd obtin good il rolution, it prnt grt rik of mpl dmg. Convrly, ft ion cn pntrt furthr into th pcimn, but th pntrting dpth i o lrg tht th il rolution bcom wor. To chrctri frrolctric domin with high rolution nd nondtructivly, thi kind of microcopy will not b convnint nough for th purpo. Scnning lctron coutic microcopy (SEAM) h bn dvlopd conidrbly inc it introduction in 98 [55, 56] nd diffrnt coupling mchnim r dicud thoroughly 5

36 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy [6, 55~6]. It i principlly nr-fild tchniqu which u tiny volum t th injcting point to gnrt coutic wv. Th coutic coupling mchnim r minly clifid thrml coutic coupling, piolctric coutic coupling, nd c crrir coutic coupling. Which contrt mchnim ply th mot importnt rol during th imging i nturlly dpndnt on th mtril imgd [6]. For mot mtl, thrml coupling i th dominnt ffct [55~59]. If miconductor mtril concrnd, th c crrir coupling ply min rol by SEAM [6]. Whn piolctric mtril r imgd, th piolctric coupling in th nr-fild h min ffct on ound gnrtion [6, 6]. During th lt twnty yr, diffrnt thori for th clcultion of diffuion dpth nd ltrl rolution of thi tchniqu hv bn dvlopd [57, 58] nd diffrnt rult on diffrnt mtril hv lo bn prntd. Howvr, lthough much ffort h lo bn givn to img frrolctric mtril by SEAM [6, 6], th contrt mchnim of SEAM on frrolctric mtril i till not wll undrtood bcu of th complity of frrolctric mtril nd SEAM tchniqu. It i thrfor ncry to nly th contrt mchnim of SEAM on frrolctric mtril furthr. Morovr, SEAM i non-dtructiv coutic mthod with rolution in micromtr rng, it prnt itlf prfct tool for complmntry nlyi nd n primntl bi for th dvlopmnt of nw kind of nr-fild microcopy with rolution of ubmicromtr or nnomtr rng. With th invntion of Scnning prob microcop (SPM) in 986 [8], diffrnt principl nd primntl t-up of Scnning nr-fild coutic microcopy (SNAM) bd on SPM hv bn dvlopd rcntly [9~7, 64~7]. Th principl of ll th SNAM b on coupling coutic wv in nr-fild nd dtcting thm by vriou mthod. According to th t-up dvlopd, th min work cn b roughly clifid into two cl. On cl u tunnl fork n coutic coupling ourc [64, 67, 68]. Th othr u common tip to coupl or dtct coutic wv [9, ~7]. According to th oprtion mod, SNAM cn thn b clifid SNAM with dirct coutic vibrtion coupling [64, 67, 68], dirct contct forc coupling [, 4~5, 65, 66, 69, 7], or piolctric coupling [9,,, ~, 6, 7]. A dtild dicuion on ll th work i byond th rng of thi work. Only th tchniqu which i dvlopd by th u of trnducr to dtct th coutic longitudinl wv producd in nr-fild for th nlyi of frrolctric mtril [6, 7] will b dicud in dtil in thi work. Th rolution of thi mod of SNAM i biclly dpndnt on th dimnion of th nr-fild which i formd jut bnth th contct point of cnning prob by n pplid c voltg. Bcu th cnning prob h vry hrp form nd dimtr down to nm t th vry tip, th ltrl dimnion of th nr-fild undr th 6

37 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy prob h lo ltrl rolution with lmot th m ordr. Thi provid n idl condition to tudy frrolctric domin nd mtril proprti t ubmicromtr or nnomtr rolution. Furthrmor, th u of n coutic trnducr t th bckid of th mpl for th dtction of coutic vibrtion producd in nr-fild provid olid b for th comprion of contrt mong diffrnt nr-fild coutic microcopy ytm both primntlly nd thorticlly. For th comprion of th contrt in thi work, two typicl nr-fild tchniqu which hv th hight rolution mong ll th nr-fild coutic microcopy tchniqu, th dvlopd SNAM t-up nd SEAM, will b chon nd dicud. For th nlyi of frrolctric proprti of BTiO mtril by both tchniqu, only th mchnim of th gnrtion of intrnl lctric fild in nr-fild will b dicud hr, both tchniqu rly on thi mchnim to gnrt coutic vibrtion in nr-fild. For SEAM, th lctric fild in nrfild i producd by th trppd chrg in th mpl. For SNAM ytm dvlopd, th lctric fild i concntrtd jut undr th tip bcu of th vry hrp form of th tip nd will b dicud in Chptr 5 in dtil. Nturlly, for th rlibility of complmntry nly, both tchniqu u th m primnt t-up, uch th m trnducr nd mpl holdr, to invtigt th m mpl t idnticl r. Although it i ncry to mk thin gold film (bout vrl nnomtr thick) on th mpl for SEAM tudy, thi film i o thin tht it will not chng th coutic boundry condition nd thrfor h no ffct on th hrmonic coutic wv olution. In th m wy, lthough th tip of SNAM i in contct with th mpl urfc, th contct forc btwn th tip nd th mpl urfc i kpt contnt during th cnning by th topogrphicl fdbck control unit of SPM. For th hrmonic coutic wv, thi contnt forc h no ffct on th hrmonic coutic wv olution ithr. Bd on th dicuion bov nd from th point viw of coutic trnmiion, w cn gnrli both ytm on frrolctric BTiO mtril on typicl t-up hown in Fig.- with n lctric fild timultion in nr-fild. In thi typicl nr-fild coutic ytm with n lctric fild timultion in nr-fild, om bic umption hv to b introducd in ordr to nly th ytm quntittivly: Th working frquncy i uully in th rng from vrl kh to vrl hundrd kh. Th wv lngth of lctromgntic wv i vrl kilomtr nd tht of coutic wv vrl cntimtr. Th t-up h typicl dimnion of vrl cntimtr. Bcu th wvlngth of lctromgntic wv (vrl km) in th ytm i much grtr thn th dimnion of th t-up nd th wvlngth of n coutic wv (vrl 7