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

38 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy cntimtr) h th m dimnion th t-up, th lctric problm cn b trtd qui-ttionry problm nd th coutic problm wv trnmiion problm; Bd on th thory of coutic plt wv guid [5], th coutic wv producd by th lctric fild in nr-fild r vry complictd nd thr r diffrnt mod. According to th orthogonlity of coutic wv mod, ll th wv mod cn b pndd um of pln wv with diffrnt ptil trnmiion dirction. Bcu longitudinl trnducr i ud in th ytm to chng coutic longitudinl wv into lctric ignl, only th longitudinl wv trnmittd in dirction will b chngd into lctric ignl by th trnducr. A th lock-in tchniqu i ud to mplify th chngd lctric ignl with th m frquncy th lctric fild ourc, w nd only to conidr th coutic pln wv with th m frquncy tht of th ourc. For th coutic pln wv trnmittd only in dirction with th frquncy of th ourc fild, w cn u th trnmiion lin mod to clcult only th longitudinl pln wv in th nrfild pproimtly. A dtild clcultion of lctric nd coutic coupling of pln wv in BTiO i hown in Appndi A; Bcu th fild i concntrtd in th middl of th mpl nd trnducr, th boundry ffct in th trnctionl pln i nglctd for wv trnmiion. It mn th pln wv pproimtion for both lctric nd coutic fild cn b ud in th mpl nd in th trnducr; Th ignl i o wk tht th mll ignl pproimtion in coutic cn b ud; Th coordint r chon to coincid with th crytl in th mpl nd trnducr; Th contct btwn th mpl nd th lctrod of th trnducr nd btwn th trnducr nd th coppr lctrod, nd both lctrod of th trnducr r o thin tht thir ffct r nglctd; Th bckid of th coppr lctrod i couticlly mtchd o wll tht thr i no rflctd coutic wv bck to lctrod; Th impdnc of th lock-in mplifir i o lrg tht th trnducr cn b umd to work undr opn circuit condition pproimtly. Undr th umption bov, if n lctric fild ourc E r E r in nr-fild i givn, th hrmonic coutic longitudinl pln wv ( v, i govrnd by th qution (Appndi A): ) producd in th nr-fild by thi ourc 8

39 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy ρ t v Eq.-4 v c p + t c p E t Eq.-5 Th uffi t th right top cornr in th prion ( v, ) mn th r in nr-fild of th mpl. Th tbl hrmonic olution of th longitudinl pln wv in nr-fild cn b olvd by th on-dimnionl trnmiion lin mod (A, Appndi): ( ) / + jωt + Eq.-6 v ( + jωt ) / Z Eq ( ) jk + + () jk, in which + E jkζ () ( Z ) dζ ( d) Eq.-8 c t p ( ) jk ( d) jk ( d ) +, in which d E jkζ () (Z ) dζ ( d) Eq.-9 c t p Hr Z ρ c p i th chrctritic ritnc of wv ( v, + ) in nr-fild. ( ) nd ( ) r wv mplitud trm rulting from th ourc lctric fild in nr-fild. + () nd ( d) r contnt which r dtrmind by boundry condition hown in th t-up of Fig.-, if th ourc i givn. Th wv outid th nr-fild but till in th mpl r govrnd only by th wv qution without ourc (Eq.-) nd cn b writtn ( v, ): ( ) / jωt + + Eq.- v ( + jωt ) / Z Eq.- ( d) + jk ( d ) + Eq.- ( d) jk ( d ) Eq.- Hr + ( d) nd ( d) r contnt which cn b dtrmind by boundry condition of 9

40 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy th ytm nd th ourc, ω th ngulr frquncy, nd k coutic longitudinl wv numbr in th dirction in th mpl outid th nr-fild. Th uffi t th right top cornr indict th r outid th nr-fild but till in th mpl. It i clr tht th wv numbr inid nd outid th nr-fild i th m, ll th longitudinl wv trnmit inid nd outid th nr-fild with th m mtril contnt. Tht mn k k k. Th coutic boundry condition of th modl hown in Fig.- r gnrlly th m: if ; Eq.-4 if d; Eq.-5 v v if d; Eq.-6 if D v v if D Eq.-7 Eq.-8 ( D) R + ( D ) jk D if D + D Eq.-9 hr R i th coutic rflction contnt of th coppr lctrod to th longitudinl wv. Th bckid of coppr lctrod i wll mtchd couticlly. From th t-up hown in Fig.-, th coutic wv in nr-fild r dcribd by Eq.-6 to Eq.-9, wv in th mpl but outid th nr-fild r by Eq.- to Eq.-, nd wv in th trnducr by Eq.-4 to Eq.-7. If th ourc fild in nr-fild i givn, + ( d) nd () r two trm of ourc intgrtion in nr-fild nd r lo givn. Thr r i contnt + (), ( d), + ( d), ( d), + D ), nd D ). Thr r lo i indpndnt linr boundry qution from Eq.- to Eq.-9. Thr i only on ingl olution for ll th contnt. A dtild olution of ll th contnt cn b found in Appndi A. Th coutic longitudinl wv mplitud in th trnducr undr th opn circuit nd wk coupling condition cn b writtn : + ( D) ( d E 4 jz jkd in( kζ) dζ c p t jkd jkd + )( + R ) + αα β ( ( Eq.-4 in which α + jkd jkd α, R + R jk D jk D Z, nd β. Z

41 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy Th prion i vry complictd but th phyicl mning i vry clr. Th trm d E t in( t th right id of Eq.-4 i th ourc coupling in nr-fild. Th rt t kζ) dζ th right id of Eq.-4 i th ytm trnmiion function which i dtrmind by th boundry condition of th ytm. Th coutic wv mplitud + D ) ( i dpndnt on both th coupling ourc in nr-fild nd th trnmiion function of th ytm. From Eq.- 4, th output ignl of th trnducr undr th opn circuit nd wk coupling condition cn b ily found from Eq.-: V output jk c PZT + ( D ) ε PZT ( + R jk D ) (- jk D ) PZT P Th finl output ignl of th trnducr i: V output ( d )( c p E t PZT in( kζ) dζ ) jk D β ε PZT ( )( ω jkd )( + + αα β jkd ) Eq.-4 Th firt trm t th right id of Eq.-4 i only rltd to th mtril proprti in nrfild; th cond trm i th trm dtrmind by ourc lctric fild in nr-fild; th third trm i th prmtr of PZT trnducr; Th rt i dpndnt on th dtcting ytm hown in Fig.-. Eq.-4 mn tht if th ourc E r E r in nr-fild i givn, th output ignl of th trnducr t th opn circuit condition for t-up hown in Fig.- cn b obtind. For givn ytm, if th ourc fild E i kpt contnt during th cnning, th chng of Voutput i only rltd to mtril proprti ( ) in nr-fild. A it i hown in Eq.-4, th c p piolctric contnt d (i.. in Eq.-4) i dpndnt on polrition. Thrfor, if th coupld ourc lctric fild in nr-fild cn b kpt contnt during th cnning nd thr i diffrnc of polrition in nr-fild of vry cnning point, th finl coutic img of primnt t-up Fig.- i n img of th chng of frrolctric polrition in nr-fild. Thi i th contrt mchnim of nr-fild t-up hown in Fig.- to img frrolctric domin in BTiO mtril ytm. It cn b n ltr tht thi i lo th thorticl bckground for th complmntry tudy on frrolctric domin of BTiO mtril by SEAM nd SNAM tchniqu, th contrt mchnim of both tchniqu i th m. If, howvr, th coupld ourc lctric fild ditribution i cttrd by dfct in nr-fild,

42 . Elctric nd coutic coupling in Scnning Nr-fild Acoutic Microcopy th output ignl of th trnducr will b nturlly dpndnt on th cttring ourc nd th vrg piolctric ffct in nr-fild. Thi i lo th contrt origin of dfct in nrfild which cn b imgd by SEAM nd SNAM. Furthrmor, for th dvlopmnt of primntl t-up of both nr-fild tchniqu bd on th thory bov, it i lo ncry to timt th vlu of output ignl from th trnducr. From th Eq.-4, th vlu of piolctric contnt of piolctric BTiO nd PTZ trnducr hv lmot th m vlu. Th tiffnd piolctric contnt in nrfild i bout for BTiO nd dilctric contnt of PZT ε PZT i bout -. Th nrfild dpth i uully in th rng of µm. Th thickn of th mpl nd trnducr i lo in millimtr rng. If it i umd tht th typicl lctric fild in th nr-fild i mllr thn kv/m, which i mll nough not to chng th domin tructur in nr-fild, th output ignl of th trnducr i in th rng of µv. Thi ignl i nturlly in th bckground of whit noi. W hv to u th lock-in mplifir to dtct uch mll ignl ubmrgd in th whit noi bckground. How th two kind of nr-fild microcopy tchniqu coupl th ourc lctric fild in nr-fild, how th dtild primnt t-up for both ytm r tblihd bd on th thory bov, nd how lrg i th nr-fild dpth of both ytm will b dicud in prlll wy in th following chptr 4 nd 5.

43 4. Scnning Elctron Acoutic Microcopy 4 Scnning Elctron Acoutic Microcopy 4. Phyicl bckground, ignl gnrtion, nd contrt mchnim Th phyicl proc by injcting lctron in Scnning lctron microcopy (SEM) r dicud thoroughly by om work [55~6, 7~7]. Th Scnning lctron coutic microcopy (SEAM) w dvlopd from th commrcil SEM to nly th coutic proprti of mtril [55, 56]. According to diffrnt ppliction, th mchnim cn b gnrlly clifid into thrml coutic coupling, piolctric coupling, nd c crrir coupling [6]. For th thrml coutic coupling, om uthor introducd dtild dicuion of coupling qution nd output ignl of th trnducr in SEAM [58, 75] which will not b dicud in thi work. For th piolctric coupling mchnim, lthough om work h bn don to nly th coupling mchnim [6], dtild plntion of thi kind of mchnim for th imging of frrolctric domin in SEAM h not bn introducd until rcntly [6]. In tht work, th uthor hv dicud th piolctric coupling ffct in th frrolctric BTiO mtril by SEAM, but th coutic olution wr umd th m inid nd outid th nr-fild. It w n umption which did not tify th coutic wv qution Eq.- in th r outid th nr-fild but till in th mpl. For th futur nlyi of frrolctric mtril by nr-fild coutic microcopy tchniqu, it i obligtory to tudy thi mchnim furthr. A thi kind of mchnim concrn th coupling of mchnic nd lctric phnomn in nr-fild, dtild nlyi of phyicl proc of lctron injction in nr-fild by primry lctron (PE) i lo ncry. If only th inulting mtril r concrnd, om uthor introducd om modl to dcrib fild ditribution in th inulting mtril bombrdd by PE [7~74]. Mot of th modl hv umd tht th lctric fild in lctron injcting r i uniform in th injction dirction. Thi i nturlly gint Gu thorm. A mor ccurt modl to clcult lctric fild formd by injction lctron w introducd [74]. Although thi modl i impoibl to plin ll th ctul phnomn of bombrdmnt on inulting mtril by PE, it i th mot ccptd modl in thi pct tody. Thrfor thi modl i chon th b for th nly of th contrt mchnim of SEAM on frrolctric BTiO mtril. According to th modl, th PE injct into mpl nd form trppd chrg ditribution. For typicl t-up of SEAM ytm, th chrg ditribution in n inultor by bombrdmnt of primry lctron i hown in Fig. 4-. Th chrg ccumulting r h

44 4. Scnning Elctron Acoutic Microcopy two prt. Th firt i th poitiv chrgd r ρ +, which i cud by th miion of condry lctron (SE) with quntum yild δ. Th dpth + d of thi r i l thn vrl nm for mtl nd from to 5 nm for inultor. Th rt of PE injct through thi r nd furthr into th mpl until thy com to rt. A frction of lctron will b bck cttrd (with bckcttring cofficint chrgd lyr with dnity of ρ nd dpth η E ) nd th rt of lctron form ngtiv d. Th dpth of pntrtion d i i dpndnt on cclrting voltg nd th mtril proprti [57, 6]. + d + d which Dimtr of primry bm Elctron bm Mtl film D : Thickn of th Smpl Injcting dpth d + d - d (,, ) + - Y Elctric fild D : Thickn of th trnducr ε r PZT Acoutic wv v, ( ) Good coutic mtching Z To lock-in mplifir Fig.4-: Proc of lctron bombrdmnt on frrolctric mtril If th PE kp on injcting th r, th chrg will b ccumultd. Th conductivity of th mpl will dichrg thi chrg ditribution. At th m tim, th lctric fild which i formd by th ccumultd chrg will forc th lctron to migrt up to th urfc. If th mpl urfc i groundd by thin mtl lyr, th injcting lctron nd th lctron dichrgd nd migrtd will hv n quilibrium tt ftr tim τ. In thi tt, tbl lctric fild will b producd in th injctd r in th mpl. Unfortuntly, thi tim contnt τ i vry difficult to vlut, bcu it dpnd on both mtril proprti nd primnt rrngmnt nd currntly cn only b dcidd by primnt. 4

45 4. Scnning Elctron Acoutic Microcopy If th intnity of PE i modultd with n ngulr frquncy ω, th lctric fild ditribution cn lo b tblihd ftr dly tim fctor longr thn τ. In thi c, th lctric fild formd h lo n ngulr frquncy of ω. To tudy th problm furthr, om pproimtion hv to b md. A it cn b n ltr, vn undr uch pproimtion, th olution of piolctric mchnim in SEAM h vry complictd form. Bid th fundmntl pproimtion to nly nr-fild microcopy ytm in Chptr, om pcil pproimtion for SEAM mut b umd hr: Th dimnion of nr-fild of SEAM i o mll comprd to mpl nd wvlngth, tht th niotropic proprti for th clcultion of lctric fild in nr-fild cn b nglctd; Th miion of SE tk plc minly t th urfc of th mtllic lyr, o tht thr i only ngtivly chrgd r in th mpl with dnity of ρ ; Th pntrtion chrg will b umd tht it i chrg cylindr (dimtr nd lngth d ) in th mpl with dfocu condition, o tht th lctric fild componnt in rdil dirction i nglctd nd th lctric fild componnt lctric fild in th middl of th nr-fild; E i uniform nd qul th Th illumintion tim of PE t vry cn point i grtr thn th dichrg dly tim τ of th mtril, o tht tbl hrmonic fild ditribution i tblihd. From Fig. 4-, if th mpl h thin lyr of mtl on th urfc, th tbl lctric fild ditribution cn b clcultd by th u of n img chrg ditribution. With th pproimtion bov, th lctric fild long th i i [74]: E A jωt ( d) ( <<d, + y ) E ( >d nd <, + y ) Eq.4- whr ρ i th dnity of tbl chrg ditribution in nr-fild nd ε i th dilctric contnt in dirction of BTiO mtril. Th, nd y componnt of lctric fild r nglctd thy r mll comprd to componnt. Th lctric fild ditribution of Eq.4- i th ourc fild E in nr-fild of SEAM. Th lctric fild ourc in nr-fild with hrmonic tim vrition i: E A jωt ( d) τ η π ε ) I d ( E jωt Th coutic ourc in nr-fild for BTiO : ( d) Eq.4-5

46 4. Scnning Elctron Acoutic Microcopy c p t E A( d) j t ω in which A jω c p ( η π ε E ) I d τ Eq.4- Th output ignl of PZT trnducr undr th opn circuit condition i [A, ppndi]: V output j ( ρε ( ηe ) I )( π d τ PZT ). ε PZT in( kd) kd ω ( )( + jk D jkd β )( + α α β jkd ) Eq.4-4 In Eq.4-4, α + jkd jkd α, R + R jk D jk D, nd β Z. Th trm in th firt brckt of Eq.4-4 i dirctly rltd to th mtril proprti in nrfild; Th trm in th cond brckt i bout th chrg ditribution in nr-fild; Th trm in th third brckt i rltd to th proprti of th PZT trnducr; Th rmining four trm r rltd to dtcting t-up nd hv vry complictd frquncy dpndnc. A it i plind in chptr, if th trinic contribution of th domin wll to th piolctric ffct in frrolctric mtril i nglctd, th contnt of (or d ) i dirctly rltd to frrolctric polrition [ Eq.-4]. If thr r c- nd - domin on th two r on th urfc of BTiO mtril, th r of c-domin h polrition prlll to th dirction nd h non-ro vlu of in nr-fild. Whr th r of - domin, which h polrition prpndiculr to th dirction, will hv ro vlu of. Thi i th contrt mchnim of SEAM on th frrolctric BTiO ingl crytl. Eq.4-4 h vry complictd frquncy dpndnc, which i provd by SEAM primnt [4, 8]. Thi dpndnc rult from th frquncy trm in Eq.4-4 nd cn only b nlyd by computr imultion. Th computr imultion i compltd by th u of th fr Linu oftwr Octv Vr... nd th rult i hown in Fig.4-. A th typicl frquncy rng of mot lock-in mplifir i from vrl H to kh, th mplitud nd ph rpon of th output ignl of th trnducr in thi frquncy rng h dpndnc of frquncy hown in Fig.4-. Erly primntl work [8] rvld th m tndncy of th coutic output ignl in SEAM in thi frquncy rng. Z 6

47 4. Scnning Elctron Acoutic Microcopy µv 8 rd kh Fig.4-: Amplitud of th output ignl of th trnducr in SEAM kh Fig.4-b: Ph of th output ignl of th trnducr in SEAM If thr i domin contrt, th output ignl i dirctly proportionl to primry bm currnt. It i provd by primnt tht th domin contrt of SEAM i bttr with lrg bm currnt [8]. Howvr, th dpndnc of domin contrt with rpct to injcting dpth i rltivly difficult. Up to now, mot of th modl to dcrib SEAM contrt hv only mntiond th thrml mchnim who contrt i origintd from th priodicl thrml diffuion by th injction of modultd PE. For th contrt origintd from frrolctric domin, it i hown by th modl dicud bov, th nr-fild r i ctully th whol injcting dpth of primry lctron. Dtild dicuion bout thi injcting dpth h bn prntd [57, 7] nd th injction dpth will b ud dirctly hr. 4. Eprimnt t-up Th primntl t-up of SEAM i hown in Fig.4-. It cn b dvlopd from commrcil SEM. Th SEM ud in thi work i Modl S5 nd CS from Cmbridg Microcop Ltd. Th oftwr from Tripl-O Elctron Sourc Microcopy Compny Ltd. i ud to control Choppr th imultnou imging of condry lctron img (SEI) nd lctron coutic Function Gnrtor img (EAI). Th function gnrtor HP88 i ud th ignl ourc nd th lock-in mplifir Itho5 i ud to mplify th lctron coutic ignl. Th ytm cquir both SEI nd EAI imultnouly, o tht th Fig.4-: Stup of SEAM occionl noi will b minimid. Th bm currnt i murd by norml Frdy cup. 7

48 4. Scnning Elctron Acoutic Microcopy Th piolctric trnducr i P5 piolctric crmic from Compny Vlvo [76]. Som of th typicl rult r prntd [4, 6, 7, 8]. 4. Dicuion of th ignl nd noi To vlut n imging ytm, it i lwy to nly om importnt prmtr, uch tim rolution nd bndwidth, ltrl nd dpth rolution, nitivity nd ignl-noi rtio, tbility, nd th tt tim. A dicud bov, Th illumintion tim of PE t vry cn point i grtr thn th dichrg dly tim τ of th mtril, o tht tbl hrmonic fild ditribution i tblihd. Thi dichrg dly tim τ i dpndnt on both th injcting bm nd th mpl tudid. A typicl vlu for inulting mtril undr th bombrdmnt of th PE with currnt dnity of µa/mm i. m [74]. For typicl img of 4*4 rolution cnnd by computr controlld SEM, th cnning bm cn ty for m t vry pil. Tht mn thi dly tim h no ignificnt ffct on th typicl SEAM imging proc. A prntd by othr work [55~6], th ltrl nd dpth rolution of SEAM r minly dpndnt on th intrction volum of PE pntrtion, which i dfind th volum of nr-fild for SEAM in thi work. Whr th nr-fild volum h th dimnion of th thrml diffuion volum in c of thrml mchnim, th nr-fild volum by piolctric mchnim h dpth rolution of th whol PE pntrtion dpth. Typicl vlu of pntrtion dpth of SEAM img with kv primry lctron i bout ~ µm [55] which i dpndnt on th mtril tudid. Th ltrl rolution i lo dpndnt on th dimnion of th nr-fild. For th imging of frrolctric mtril, if thr i urfc polrition in nr-fild, thi polrition will nturlly ffct th ltrl ditribution of lctron of injcting PE. A rult, th dimnion of th nr-fild i chngd. Bcu th piolctric vibrtion mod chng nd ltrl lctric fild ditribution in nr-fild r vry complictd nd littl documntd in th litrtur up till tody, it cn only b timtd primntlly. Unfortuntly, from th primnt rult thi ffct i littl documntd too. Th ignl to noi rtio cn lo b timtd. If th noi from function gnrtor, SEM, nd th muring circuit nd lock-in mplifir r noi of coutic dtcting ytm σ G, σ SEM, nd σ m, it i ncry to dd th σ A which i compod of thrml noi in nr-fild, th coutic trnmiion noi, nd th thrml noi of trnducr. Th totl noi σ totl i: 8

49 4. Scnning Elctron Acoutic Microcopy σ totl σ G + σ SEM + σ m + σ A Eq.4-6 Th σ SEM, σ G, σ m r known for crtin SEM ytm, crtin function gnrtor, nd givn lock-in mplifir. But σ A i vry difficult to vlut nd could only b dcidd primntlly up to now. A typicl vlu of ignl to noi rtio t frquncy of kh ftr th lock-in mplifir i ~5 for SEAM. Thr r two fctor which ffct th tbility of SEAM by piolctric imging mchnim. On i th tbility of fild ditribution in nr-fild. Th othr i th tbility of SEM. Uully th tbility of SEM i givn for crtin ytm. According to th primnt, th tbility of fild ditribution in nr-fild i vry nitiv to th bm currnt, th thrml ffct by th injcting PE h conidrbl ffct on th tbility. Undr norml condition, if th bm currnt i l thn ~5 na, thr i no problm of tbility cud by thrml ffct. 9

50 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) 5 Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) 5. Phyicl bckground, ignl gnrtion, nd contrt mchnim To nly th frrolctric proprti by SPM, much work h bn don nd diffrnt primnt SNAM t-up bd on SPM hv bn dvlopd [9~7]. Only rcntly, modl for th tudy of frrolctric thin film by piolctric rpon mod of SNAM i prntd [, 77]. Evn in tht modl, th lctric fild olution could not tify th lctric boundry condition of th prntd modl. Nithr could th introducd thorticl mthod b ud ytmtic thorticl tchniqu vn only to nly th qui-ttionry lctric fild for diffrnt mpl uch bulk mtril or thin film. For ytmtic nlyi of th lctric nd coutic coupling fild in frrolctric mpl tudid by SNAM ytm, no othr work h bn documntd bcu of th complity. Th firt difficulty i tht th lctric nd mchnicl fild in th nr-fild in th mpl i vry difficult to clcult in SNAM ytm [77, 78]. Th cond difficulty rult from urfc of frrolctric mtril which r lo unknown to u. Although primnt rult r prntd from both piolctric rpon mod [9,] nd th ytm dvlopd in thi work [4, 7], n plntion of contrt of SNAM cn only b qulittiv, th lctric nd mchnicl coupling in nr-fild i till undr tudy. Although thr hv bn lot of numricl olution of th lctric fild ditribution undr th tip, it would b difficult to u th numricl olution to tudy th contrt mchnim of SNAM ytm. Thrfor, impl nlyticl olution of lctric nd mchnicl coupling will b importnt for both th nlyi of contrt mchnim nd th timtion of proprti of frrolctric mtril tudid by SNAM ytm. A brifly dicud in th introduction, nw t-up of SNAM bd on SPM for th chrctrition of BTiO mtril ytm i dvlopd in thi work. To chrctri th ytm dvlopd, ytmtic thorticl modl bd on clicl lctromgntic nd coutic thori for th t-up i prntd in thi chptr. Dtild thorticl clcultion cn b found in Appndi A4. Th SNAM ytm to b introducd in th following i dvlopd from commrcil SPM ytm. Th SPM work in contct mod nd n ltrnting voltg i pplid btwn th tip nd th bckid of th mpl. A trnducr i ud to dtctd coutic vibrtion producd by th convr piolctric ffct of th mpl [4~7]. A brifly dicud in Chptr, th u of th trnducr provid olid b for thorticl nd primntl comprion 4

51 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) of th contrt of SNAM ytm dvlopd in thi work to othr tblihd nr-fild coutic mthod, uch SEAM. Som work [9,, 5, ~] introducd th o-clld piolctric rpon mod of SNAM which u th lr ignl to dtct th coutic vibrtion in nr-fild. A comprion of rult of th dvlopd ytm with th rult of th piolctric rpon mod i lo prntd ltr. To nly th dvlopd t-up ytmticlly, om importnt prmtr of th ytm hould b introducd t firt. By th contct mod of SPM, th ditnc btwn th tip nd th mpl urfc i mllr thn nnomtr. Th tip ud in th SNAM h phricl form with curvtur rdiu of nm t th pinncl. Th c voltg btwn th tip nd th bck lctrod i uully fid t crtin vlu t which th frrolctric domin would not b ffctd. Th trnducr nd mpl holdr r ctly th m tho ud in SEAM, o tht th m coutic boundry condition for both nr-fild tchniqu cn b nurd for th complmntry tudy dicud in Chptr 8. Th mpl tudid i uully niotropic nd th SNAM h complictd boundry condition bcu of th conductiv tip. It i impoibl to obtin impl nlytic olution without crtin pproimtion. Uully, th nr-fild undr th tip in SNAM i uully mll, nd, for th nlyi of qui-ttionry lctric fild problm, th lctric niotropic proprti of th mpl cn b nglctd. Only for th nlyi of coutic problm, th niotropic proprti mut b tkn into ccount, th dimnion of th coutic trnmiion i much lrgr thn th dimnion of nr-fild. D D ε r PZT (,, h ) Scnning Prob (,, ) q (,, -h) Vcuum or ir Fig.5-: Simplifid modl for th t-up Th form of th tip prnt nothr problm for th nlytic olution of th lctric fild in SNAM. Undr th microcopic viw, th form of th tip i diffrnt from on nothr. Thorticlly, th urfc of th conductiv tip cn b qutd by n infinit ri of point chrg, but it would b too complictd to min phyicl mning in SNAM ytm. A dicud in Chptr 4, th mpl nd th trnducr in nr-fild microcopy ytm hv uully 4

52 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) ltrl dimnion of - cntimtr. Th thickn of th bulk crmic or crytl mpl i normlly millimtr nd typicl thin film mpl nm. Th cnning tip h only dimtr of nm t th pinncl. Bcu of thi diffrnc of th gomtricl dimnion btwn th mpl nd th tip, th tip cn b normlly implifid point chrg which i t th curvtur cntr of th pinncl of th tip [, 77, 79]. Th mpl nd trnducr, dicud in chptr for typicl nr-fild coutic ytm, r implifid plt with n unlimitd ltrl dimnion. Th implifid modl to chrctri th lctric fild ditribution of th t-up dvlopd in thi work i hown in Fig.5-. Th pproimtion for SNAM ytm dvlopd, in th c of SEAM, r gnrlid : For th clcultion of qui-ttionry lctric fild in nr-fild, th niotropic of lctric proprty of th mpl i nglctd; Only by th trtmnt of coutic trnmiion problm, th niotropic proprti of mtril will b tkn into ccount; Th tip will b quivlnt point chrg t th curvtur cntr of th pinncl of th tip; At th pinncl point of th tip (,,h ), th potntil will b givn th voltg of th tip, bcu it i th nrt point to th mpl nd h th trongt ffct on th fild ditribution [77, 79]. According to th modl nd undr th pproimtion bov, th lctric nd mchnic coupling in th ytm cn b nlyd. Th nlytic olution of th qui-ttionry lctric fild in th SNAM ytm dvlopd hould b olution of th Poion qution with point ourc undr th givn boundry condition hown in Fig.5-. Th mot ffctiv wy to olv thi qui-ttionry fild i th olution of th Grn function of th ytm. 5.. Grn function of th modl Th lctric fild ditribution cn b nlyd by th u of Grn function. Th fild ditribution of point chrg q t (,,-h) in cylindricl coordint ytm cn b obtind by prting vribl [8~8]: V q 4πε J β + h ( βr dβ ) Eq.5- Th Grn function for th modl hown in Fig.5- cn b writtn : In th rgion - <<; q β + h β V [ J ( βr) dβ + A( β) J ( βr) dβ 4πε ] Eq.5-4

53 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) In th rgion <<D ; q β β V [ B( β) J ( βr) dβ + C( β) J ( βr) dβ 4πε At D, th mtl intrfc i rthd, o th voltg t th intrfc: ] Eq.5- V Eq.5-4 A th intrfc btwn th mpl nd PZT trnducr i rthd, it i only ncry to chrctri th lctric fild ditribution in two rgion bov. By olving th boundry condition of th ytm, th contnt cn b dcidd [Appndi A4.]. Th Grn function V in th rgion of - << i dividd into thr trm: Th firt trm: q 4πε J β + h ( βr dβ ) It cn b trtd point chrg t (,, -h) Th cond trm: q 4πε (- η ) βh + η βd J ( β β r) dβ Eq.5-4 q 4πε β( h) (- η ) J ( βr) dβ + q 4πε η J ( βr) β[ (D + h)] dβ + q 4πε ( η β( 4 D + h ) +...) β β J ( r) dβ Eq.5-5 Hr η ε ε r r. + Th trm t th right id of Eq.5-5 cn b trtd img chrg Q m - η q, Q m η q,... t th poition of (h, D +h, 4D +h,...) t i. Th third trm: q 4πε (-) β( D + h ) + η βd J ( β β r) dβ q 4πε β[ (D + h)] (-) J ( βr) dβ + q 4πε η J ( βr) β[ (4D + h)] dβ 4

54 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) q + 4πε [ η β( 6 D + h ) +...] β β J ( r) dβ Eq.5-6 In th m wy, trm t th right id of Eq.5-6 cn b trtd th img chrg Q n - q, Q n η q,... t th poition of (D +h, 4D +h,...) t i. Th phyicl mning of th clcultion bov i tht th Grn function for th r bov th mpl in SNAM cn b quivlnt th um of potntil which r producd by ri of point img chrg. Th finl img chrg ditribution for Grn function V i hown in Fig.5-. Q i - η q ( ε r ) + [,, -(D +h)] Q i ( ε + q r ) (,, -h) (,, ) (,, D ) Q j - ( ε r + ) q [,, D +h] Q j ( ε + r ) η q [,, (D +h)+d ] Fig.5-: Equivlnt point chrg for th Grn function in th hlf pc < Fig.5-: Equivlnt point chrg for th Grn function in th pc (D >>). Th Grn function V in th rgion of D >> i obtind in th m wy nd dividd into two trm (Appndi A4.): 44

55 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) Th firt trm of V: q 4πε β B( β) J ( βr dβ ) q 4πε ( ε r + ) J ( βr) β( + h) dβ q 4πε ) ( + η ε r J ( βr) β[ + (D + h)] dβ q 4πε ) ( ε r + [ βh ( ) β -...] J ( βr) dβ D η β Eq.5-7 Th trm t th right id of Eq.5-7 cn b quivlnt img chrg Q i Q i ( ε + r ) Th cond trm of V: η q,... t th poition of (-h, -(D +h),...) t i. ( ε + q, r ) q 4πε k C( k) J dk ( kr) q (-) 4πε ( ε r + ) J ( kr) k [ ( h+ D )] dk + q 4πε η ( ε r ) + J ( kr) k [ ( h+ 4D ) dk + q 4πε [ k ( h+ D ) ( ) k +...] J ( kr) dk kd η Eq.5-8 Th trm t th right id of Eq.5-8 cn b quivlnt img chrg Q j ( ε Q j ( ε + r ) q, ) + r η q,... t th poition of (D +h, 4D +h,...) t i. Th img chrg ditribution of Grn function V i hown in Fig Modling of thick mpl Th nr-fild r of SNAM i jut bnth th mpl urfc. If th mpl i vry thick comprd to th ditnc h (D >>h), th chrg t (,,-h) nd th img chrg t (,,h) 45

56 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) with chrg Q m η q hv th grtt contribution to th Grn function of V in th vicinity of th tip contct r nd th contribution of othr img chrg cn b furthr nglctd ( Fig.5-). In th m wy nd undr th m pproimtion bov, th fild ditribution in nr-fild in th mpl cn lo b clcultd by th img chrg t (,, -h) with chrg of Q i q ε + ( Fig.5-). Th phyicl mning of th pproimtion i r tht th mpl i trtd n infinit hlf pc with dilctric contnt ε r pproimtly. Undr th pproimtion bov nd from th Grn function, if th rctngulr Crtin coordint ytm i ud for th k of furthr nly of coutic problm, th lctric potntil undr th pproimtion bov cn b writtn : V 4πε ( q + y + ( + h) Q m + ), for < Eq y + ( h) in which Q m - η q; Q V i 4πε + y + ( + h) Eq.5-, for > Eq.5- in which Q i q ε +. Eq.5- r Th ytm cpcitnc cn b clcultd by th Grn function bov. To mk th bt pproimtion for th tip, th point (,,h ) t th pinncl point of th tip will b givn th voltg of V. Th quivlnt chrg, mntiond bov, i t th curvtur cntr t th pinncl point with curvtur rdiu of nm. Th quivlnt chrg q : q C V q 4πε ( h h ) η ( h + h ) From Eq.5-7, th ourc fild ditribution in th mpl cn b writtn : E r V r V r y ( + y V r + ) V Eq.5- Th nd y nd componnt of E 4πε ( + y + ( + h) ) E r hv th form: Q i Eq

57 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) E y y 4πε ( + y + ( + h) ) Qi Eq.5-5 E ( + h) 4πε ( + y + ( + h) ) Qi Eq.5-6 To undrtnd th nr-fild ffct of th SNAM ytm, n timtion of lctric fild undr th tip i ncry. Th nd y componnt r ro in th middl of th nr-fild r nd mll in th vicinity comprd to componnt, o tht thy cn b nglctd for furthr nlyi. Th BTiO crytl h typicl vlu of ε r. In th contct mod of SPM, th ditnc h in th modl i uully mllr thn 5 nm. Th curvtur rdiu t th pinncl point i bout 5 nm. From Fig.5-4, h nm. With th hlp of fr mthmticl imultion progrm Octv., th lctric fild in th mpl of E for typicl BTiO ingl crytl mtril i imultd. Th imultion how clrly tht th fild i concntrtd itlf on vrl h undr th urfc. At th dpth of th mtril bout 9h, th fild i dcrd to % of th vlu t th urfc. Undr th modl bov, th fild i nturlly proportionl to th voltg if th cpcitnc i clcultd Eq.5- which i gnrlly ccptd [, 77, 79]. kvolt/m kvolt/m /h Fig.5-4: Th chng of E with dpth (/h) t th tip voltg of Volt Volt Fig.5-4b: Th chng of E with th tip voltg t mpl urfc ( ) Th lctric fild in dirction for SNAM i: E ( + h) 4πε ( + y + ( + h) ) Qi Eq.5-7 If th nr-fild pproimtion lik tht for SEAM i lo md hr: Th nr-fild dimnion in ltrl dirction i dfind th r whr th vlu of th 47

58 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) E t th boundry i dcrd to / of th vlu in th middl; Within th ltrl dimnion of th nr-fild dfind bov, th lctric fild E i homognou in ltrl dirction nd qul th fild in th middl of th nr-fild; Thn th lctric fild E E 4πε ( + h) E in th whol nr-fild r cn b pproimtly writtn : Qi Eq.5-8 If th ourc voltg i tim hrmoniou function with hight frquncy of vrl hundrd kh, it will produc qui-ttionry lctric fild in th dirction. Th mplitud of thi qui-ttionry fild i lo prd by Eq.5-8 with only tim vribl j t ω. By th u of trnmiion lin mod nd by olving th ytm coutic qution hown in Chptr, th mplitud of coutic wv in trnducr in SNAM cn b obtind in th m wy : + ( D) [ + R jk D 4 jz c ][ + jkd p jk D E t D ] + β[ R in( kζ) dζ jk D ][ jk D ] Eq.5-9 Th output ignl of trnducr i from Eq.-: V output jk c PZT ( R + jk D ) ε PZT (- jk D ) + ( D ) PZT P 48 Eq.5- If th ytm boundry i givn nd th ourc fild i kpt contnt during cnning, th 5 4 µv 4 6 D/h Fig.5-5: Th chng of output ignl of th trnducr with mpl thickn D output ignl i only proportionl to mtril contnt D D nd c E in( kζ) dζ p. Hr th trm E in( kζ) dζ mut b dicud in dtil to th phyicl mning of th nr-fild ffct in SNAM. At givn point, th mtril proprti t th point r givn, th output ignl of th trnducr in SNAM i only proportionl to E D in( kζ) dζ. Bcu thi intgrtion h no impl form, w cn only clcult th output ignl of th trnducr

59 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) through th Octv mthmticl progrm. Th output ignl of th trnducr t crtin frquncy (fkh) with th chng of thickn of th mpl i hown in Fig.5-5. It i clr tht, ftr D h, th ignl chng vry lowly nd rch lmot th finl vlu. Tht mn tht th contrt of SNAM com from th urfc lyr in th rng from th urfc to vrl hundrd nnomtr dp. Th ron i tht mot of th ourc lctric fild in th modl i concntrtd in thi r bcu of th form of th tip, hown in Fig.5-4. Aftr th dicuion of nr-fild r in th thick mpl, th contrt of frrolctric domin cn b plind in th m wy in th c of SEAM. If th trinic contribution of th domin wll to th piolctric ffct in frrolctric mtril i nglctd, th contnt of (or d ) i dirctly rltd to frrolctric polrition [ Eq.-4]. If thr r two r of c- nd -domin on th urfc of BTiO mtril, th r of c-domin h polrition in th dirction nd h non-ro vlu of in nrfild. Whn th cnning tip i on thi r, th output ignl of th trnducr h crtin vlu. Whr th r of -domin, which h polrition prpndiculr to th dirction, will hv ro vlu of. Whn th tip i on thi -domin r, th output ignl of th trnducr, will hv ro vlu (or mor ccurtly, vry mll vlu). A th mplitud ignl i ud th imging ignl, th c-domin will ppr bright nd -domin will ppr blck in coutic img of SNAM ytm. Thi i th contrt mchnim of SNAM on frrolctric BTiO ingl crytl. Hr w mut notic tht th nr-fild dimnion i much mllr thn tht of SEAM. 5.. Modling of thin mpl or film If th mpl i not o thick comprd to th ditnc h (D h), or th mpl i frrolctric thin film which uully h thickn of vrl hundrd to vrl thound nnomtr, th condition D >> h i not tifid in thi c. According to th Grn function of Eq.5-4 to Eq.5-8, th lctric fild bov th mpl nd in th mpl r dpndnt on ri of img chrg. Th finl fild ditribution i th um of fild contribution from vry ingl img point chrg. How mny img chrg hould b ud to dcid th Grn function of th ytm, i nturlly dpndnt on th thickn of th mpl nd th ccurcy of th clcultion. Th typicl thin film tudid in thi work h thickn of µm nd th ditnc of h i nm. For thi typicl thin film ytm with th firt ordr of pproimtion, only th fild contribution of th img point chrg t th ditnc to th urfc in th rng of (D +h) will b conidrd. Th contribution of othr img chrg t th ditnc outid thi rng i umd o mll tht th contribution cn 49

60 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) b nglctd. Undr th pproimtion bov nd ccording to th Grn function dicud bov, th Grn function V i contructd by q t poition of (,, -h), Q m η q t (,, h), Q m η q nd Q n q t (,,D + h). Th Grn function V (. Fig. A) cn b contructd by chrg of Q i q ε + t poition of (,, h), Q i ηq ε + t poition r r [,, (D + h)], nd Q j q t [,,(D + h)]. ε + r If th rctngulr Crtin coordint ytm i ud for th k of implicity of furthr nly of coutic problm, th Grn function for thin film undr th condition bov cn b writtn : q V 4πε ( + y + ( + h) + + y Q m + ( h) + + y Q m (D + + [ h)] + + y Q n (D + + [ h)] ), for < Eq.5- V 4πε ( Q i + y + ( + h) + + y Q i + (D + + [ h)] + + y Q j (D + + [ h)] ), for < < D Eq.5- Th ytm cpcitnc cn b clcultd in th m wy. With th ourc V, th quivlnt chrg q i: q C q V 4πε V ( h h ) η ( h + h ) ( η )[D + h + h ] Th fild ditribution in th mpl cn b obtind : E r V r r + V V r + ) ( y Eq.5- E 4πε ( + y + ( + h) ) Q i + Q i ( + y + [ + (D + h)] ) 5

61 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) + Q j ( + y + [ (D + h)] ) E y y 4πε ( + y + ( + h) ) y + Q j ( + y + [ (D + h)] ) E ( + h) 4πε ( + y + ( + h) ) (D + h) + Q j ( + y + [ (D + h)] ) 5 Eq.5-4 y Q i + Q i ( + y + [ + (D + h)] ) i Eq.5-5 Q + Q i ( + y + (D + h) + [ + (D If th nr-fild pproimtion lik tht of SEAM i lo md hr: + h)] ) Eq.5-6 Th nr-fild dimnion in ltrl dirction i dfind th r whr th vlu of th E t th boundry i dcrd to / of th vlu in th middl; Within th ltrl dimnion of th nr-fild dfind bov, th lctric fild E i homognou in ltrl dirction nd qul th fild in th middl of th nr-fild; Thn th lctric fild in th whol nr-fild cn b pproimtly writtn : E E in which Q 4πε ( ( + h) i q ε +, Q i ε r [ + (D + h)] Q i + r + η q, nd j [ (D + h)] Q ) Q i j Q ε q. + r Eq.5-7 Th output ignl of th trnducr cn lo b obtind by Eq.5-9 nd Eq.5-. It mut b notd tht, if th thickn of th film i D µm, th lctric fild in dirction in nr-fild h littl chng ccording to Eq.5-7 th lt two trm in Eq.5-7 r much mllr thn th firt on. Tht mn tht th two img chrg Q i nd Q j hv o mll n ffct to th lctric fild ditribution in th nr-fild tht th ffct cn b nglctd wll. Only if th thickn of th film i comprbl to th prmtr h, th fild ditribution hould b timtd ccording to diffrnt ordr of pproimtion to th Grn function Eq.5-4 nd Eq.5-8, nd finlly th quivlnt img chrg cn b dcidd. 5. Eprimnt t-up

62 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) Th dvlopd SNAM ytm i hown in Fig.5-6 if th witch i turnd to poition (). Th SPM ytm i topomtri Eplorr. Th tip ud i th lctric conductiv tip of th Compny Nnonor [8]. Th trnducr nd mpl holdr r th m tho ud in SEAM dicud in Chptr 4. Fig.5-6: St-up of SNAM with trnducr () nd of piolctric rpon mod (b) Fig.5-7: St-up of SNAM with forc modultion By th u of trnducr, it i lo poibl to compr th dirct nd convr piolctric ffct in nr-fild. Th t-up to chck th dirct piolctric ffct i hown in Fig.5-7. Th tip i modultd nd th contct forc btwn th tip nd th urfc of th mpl i modultd wll. Thi modultd forc will ct point tr ourc on th mpl urfc nd produc diffrnt coutic wv in th mpl mong which thr r coutic longitudinl wv trnmittd in dirction. According to th dicuion of th lctric nd coutic coupling prntd in Chptr, bcu th mpl of BTiO i piolctric, th coutic longitudinl pln wv will produc coupld qui-ttionry lctric fild in th mpl. Thi coupld fild cn b imgd by muring th voltg btwn th conductiv tip nd th bottom id of th mpl, if th tip nd th mpl cn b wll hildd from othr diturbnc ignl, uch th diturbnc ignl from th PZT ctutor in SPM. It mn tht it i thorticlly poibl to img th dirct piolctric ffct in thi wy. Th t-up of Fig.5-6 t th witch poition () cn mur th convr piolctric ffct. By comprion of th t-up of Fig.5-7 with th t-up of Fig.5-6 t th witch poition (), it i 5

63 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) thorticlly poibl to compr dirct nd convr piolctric ffct in nr-fild. On of th rult i hown in Fig.7-5. Th o-clld piorpon mod of SPM [9,,5,~] i bd on uch principl: An c voltg i lo pplid btwn th conductiv tip nd th bckid of th mpl to produc th vibrtion in nr-fild of frrolctric mpl. A th tip i in contct with th mpl urfc, th nr-fild vibrtion trnmit lo to th cntilvr nd th lr bm i modultd by th vibrtion. Th photodiod in SPM dmodult th lr ignl nd form th o-clld T-B-ignl, which i th diffrnc of ignl obtind from top nd bottom diod ction of qudrnt dtctor. It crri th informtion of vibrtion in nr-fild. By dtcting th T-B ignl, th informtion in nr-fild cn b dtctd indirctly, hown in Fig.5-6 with witch to (b). Th tip ud i commrcil lctricl conducting tip. Th c voltg V ud mut b o mll tht th frrolctric domin will not b ffctd. In ordr to compr contrt btwn th t-up dvlopd in thi work nd th piolctric rpon mod, th m mpl t th m nvironmnt i lo imgd by both tchniqu nd th rult r hown in Fig.7-8. Both of th two mthod r provd to b bl to img frrolctric domin in BTiO mtril [7]. Thorticlly, for th tudy of frrolctric domin, both mthod cn b ud principlly. For th contrt comprion mong diffrnt nr-fild coutic microcopy tchniqu, it i don in thi work, th ytm dvlopd in thi work i dvntgou, th u of th trnducr provid uniqu b of th m coutic boundry condition o tht th m coutic trtmnt cn b ud. Bcu th piolctric rpon mod of SPM u lr ignl to dtct th coutic vibrtion on th cntilvr of th tip nd th cntilvr h vry complictd vibrtion trnmiion function [84], it would b vry difficult to compr th contrt of thi tchniqu with othr tblihd nr-fild tchniqu quntittivly. Howvr, th piolctric rpon mod would b vry dvntgou to tudy th thrml dynmic bhvior of frrolctric domin, th thrml drift of coutic proprti of th trnducr cn b voidd. In thi work, for th dynmic tudy of frrolctric domin in BTiO, both mthod r ud to img th frrolctric domin t givn r t firt, nd thn th cond mthod i ud t th m r to tudy th m frrolctric domin dynmiclly. Th principl of dynmic tudy i hown in Fig.5-8. For thrml dynmic tudy, thrml lmnt i ud to ri th tmprtur of th mpl nd tmprtur nor i ud to chck th tmprtur of th mpl urfc. At diffrnt tmprtur point, th frrolctric domin r imgd. For lctric dynmic tudy, bi dc voltg V i ddd to th ourc c 5

64 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) Fig.5-8: St-up of SNAM for dynmic tudy voltg V. Thi bi dc voltg will dd bi ttic lctric fild in nr-fild bid th qui-ttionry fild. By chnging th bi dc voltg, th chng of domin t diffrnt bi fild cn b imgd dynmiclly [7]. 5. Dicuion of th ignl nd noi in SNAM dvlopd A th SNAM i minly ud n imging tchniqu, ptil rolution ply n importnt rol. A dicud bov, both th ltrl rolution nd dpth pntrtion of SNAM r dcidd by th prmtr h which i th ditnc of th quivlnt chrg of th tip to th urfc. From th pproimtion in thi work, thi ditnc i bout nm. According to th umption of nr-fild for SNAM bov, th ltrl dimnion of th dfind nr-fild cn b clcultd thorticlly ccording to th fild ditribution Eq At th urfc, th dimnion of nr-fild in ltrl dirction h rdiu of h 4.h nm. Tht mn th ltrl rolution of th SNAM dvlopd cn rch 86 nm, which i vrifid by th primnt rult vry wll [6, 7]. Th dpth of imging would b th intgrtion of th lctric fild in dirction in nr-fild for thick mpl nd, ccording to Eq.5- nd Fig.5-5, th dpth of imging by SNAM on thick bulk BTiO mpl i bout nm. Th imging tim of th ytm i minly dpndnt on th cnning pd of SPM. In ordr to gt good topogrphy, th cnning pd i uully low, mllr thn pil/cond. For n img of 44 pil, it tk bout ~ minut to tk pictur. Th ignl to noi rtio cn lo b timtd. Lik in th c of SEAM, If th noi from function gnrtor, SPM, nd th muring circuit nd lock-in mplifir r σ m, it i ncry to dd th ignl to noi rtio of coutic dtcting ytm σ G, σ SPM, nd σ A which i compod of th coutic trnmiion noi nd th thrml noi of trnducr. Th totl noi σ totl i: 54

65 5. Scnning Nr-fild Acoutic Microcopy bd on SPM (SNAM) σ totl σ G + σ SPM + σ m + σ A Eq.5-8 Th σ G, σ SEM, σ m r givn for crtin SEM ytm. Principlly, bcu thr i no thrml noi in th nr-fild of σ A comprd to SEAM, th ignl to noi rtio of SNAM would b much bttr thn tht SEAM. But σ A i lo vry difficult to vlut nd cn only b dcidd primntlly. According to our primnt, typicl vlu of ignl to noi rtio of th coutic ignl imgd ftr th lock-in mplifir cn b murd by th lin murmnt of th oftwr of SPM. At kh, typicl vlu i in th rng of 7~ [6]. It cn b pctd tht th contrt of SNAM coutic img would b bttr thn tht of SEAM undr th m primnt condition. 55

66 6. Eprimnt procdur 6 Eprimnt procdur 6. Dcription of Spcimn A it i plind in chptr, frrolctric BTiO mtril h th typicl - nd c- domin tructur. BTiO crmic mpl AC w fbrictd by th Intitut für Anorgnich Chmi, Mrtin-Luthr-Univrität Hll-Wittnbrg. On of mpl of ingl crytl for SEAM tudy w providd by Lb. of function crmic, Shnghi Intitut of crmic. Th ingl crytl mpl for SNAM tudy w providd by Dprtmnt of Applid Phyic, Univrity of Elctronic Scinc nd Tchnology of Chin. Th film mpl for th SEAM tudy w providd by Intitut für Schicht- und Ionntchnik (ISI), Forchungntrum Jülich, Grmny. Th othr BTiO film w providd by M-Plnck-Intitut für Mikrotrukturphyik, Grmny. All th mpl hd crud urfc nd no trtmnt w rquird for both SEAM nd SNAM tudi. Th crmic mpl AC w intrd with c % mol of TiO nd hd n vrg grin i from to 5 micromtr [86]. 6. Spcimn prprtion nd trtmnt A both tchniqu, SEAM nd SNAM, r nondtructiv tchniqu, th urfc of mpl nd no hndling nd trtmnt. In ordr to tudy th m r of mpl with two kind of microcopy tchniqu, it i ncry tht th urfc topogrphy nd th othr proprti of th mpl b kpt unchngd during th tudy. Furthrmor, to mk th comprion tudy of two microcop, th m mpl mut b tudid in on of th microcop, thn b movd to th othr microcop. A th r tudid i uully mllr thn micromtr nd th mpl hv lmot no pculir urfc chrctr, it rult in vry difficult tk to find th m r ftr th chng of microcop. For SEAM, in ordr to void th chrg ffct, thin gold lyr of bout to 5 nnomtr i uully cotd on th urfc of th mpl. To olv th problm bov, th SEAM primnt i rrngd t firt. Thr r two dvntg to do o. Firtly, th SEAM i rltiv tbl nd tblihd tchniqu. Although it h rolution of micromtr, it lrg cn rng, which cn rch vrl hundrd micromtr, provid uful wy to img th frrolctric domin brodly. Scondly, by th u of th trong intrction btwn th mpl nd lctron bm, it i poibl to mk om urfc chrctr, uch orinttion lin outid th tudid r, o tht th m r cn b loctd prcily for th ltr SNAM primnt. Thr i but on problm with thi procdur. For SEAM, to void th urfc chrg ffct 56

67 6. Eprimnt procdur nd thrml nrgy ccumultion t vry cn point, it i uully bttr to mk thin gold lyr on th urfc of th mpl. For th ltr SNAM primnt, th urfc mut b kpt cln poibl. To olv thi problm, th mpl urfc i pod to w vpor for vrl cond bfor it i cotd with gold for SEAM tudy. Th ccurt thickn of w lyr i difficult to timt but SPM tudy on th mpl urfc howd thr i no obviou topogrphy chng bfor nd ftr th w lyr. In thi wy, it i ffctiv to mk th urfc fr from gold lyr ftr SEAM tudy. 6. Schm of murmnt 6.. Choic of frquncy A thorticlly hown in Chptr, th coutic nr-fild tchniqu hv uully trong frquncy dpndnc. It i lwy importnt for both tchniqu to lct right frquncy to gt th contrt which rultd from th intrction in nr-fild. Th uul primnt procdur i to djut othr primnt prmtr to th bt tt t firt. With th lin cnning of th control oftwr for SEAM nd SNAM, th output ignl cn b nlyd lin by lin. By chnging working frquncy of th ignl gnrtor, on cn gt th bt contrt t crtin frquncy nd th img cn finlly b rcordd. Thi i th mot ffctiv wy to find th right coutic contrt from nr-fild. 6.. Amplitud nd ph imging In mot modrn lock-in mplifir, it i poibl to obtin th mplitud nd ph ignl. Both mplitud nd ph ignl, hown in Chptr, r thorticlly dpndnt on th nr-fild intrction nd dtcting ytm. If right frquncy i chon, th output ignl of th trnducr i olly dpndnt on th intrction of th mtril with th timultion ourc in nr-fild. 6.. Vrition of prmtr If th ytm i optimid, th dpndnc of th output ignl cn b rcordd with diffrnt prmtr, uch th frquncy dpndnc, chng of th ignl with th ourc, nd o on [8]. Whr th chng of prmtr i rltiv y in SNAM bd on SPM, th chng of prmtr in SEAM i rltivly difficult, th SEM i rltiv complictd nd th chng of on prmtr will omtim ffct othr prmtr lightly. Thi chng will rult in contrt dtriortion nd hould thrfor b voidd during primnt Dynmic imging For dynmic primnt, it i lo ncry to crry out n primnt until ll th primntl prmtr r tbl. It i pcilly difficult for thrml primnt, bcu 57

68 6. Eprimnt procdur frrolctric mpl uully hv poor thrml conductivity. Whr th thrml ffct from th nvironmnt cn b nglctd in th vcuum chmbr of SEAM, th thrml ffct from th ir nvironmnt of SNAM i n importnt fctor for thrml dynmic primnt. On ffctiv wy to olv thi problm i to wit o long tim until primntl prmtr uch th tmprtur of th mpl nd th nvironmnt nr th mpl r tbilid. A thrml ioltion of SPM ytm in vcuum chmbr i lo n ffctiv wy to kp th rult from trnl thrml diturbnc. 58

69 7. Rult 7 Rult Thi chptr how om typicl rult by SEAM nd SNAM nd om plntion rpctivly. Th rult of complmntry tudy of frrolctric domin by both tchniqu wll th dicuion will b prntd in Chptr Singl crytl A ingl crytl of BTiO (from Shnghi Intitut of Crmic, P. R. of Chin) i imgd by th SEAM [8]. Anothr ingl crytl of BTiO (from Univrity of Elctric Scinc nd Tchnology of Chin) imgd by SNAM tchniqu nd hown in Fig.7-. Th frrolctric domin contrt, which i totlly diffrnt from topogrphy, it only in coutic img. Fig. 7-: SEI of on ingl BTiO crytl Fig.7-b: Th in itu EAI of th crytl 78 nm Surfc dfct nm Surfc dfct Domin 5 µm Fig. 7-: Topogrphy of on ingl BTiO crytl by SPM Fig.7-b: Th in itu coutic img by SNAM 59

70 7. Rult 7. Crmic Frrolctric BTiO crmic r lo imgd by both kind of tchniqu. On kind of frrolctric domin tructur of BTiO crmic (from Mrtin-Luthr-Uivrität Hll- Wittnbrg, Grmny) r imgd by SEAM nd rult hown in Fig.7- [5, 6, 7]. Th m mpl i lo imgd by SNAM nd rult hown in Fig.7-4 [6, 7]. From Fig.7-, th urfc dfct r viibl in both SE img nd EA img. Domin tructur r imgd only in coutic img. From Fig.7-4, thr r only domin tructur of th crmic in coutic img by SNAM wll. By comprion of both coutic img on th m crmic, it cn b n tht th domin tructur of th crmic r imgd principlly th m by both tchniqu. Th contrt hv th typicl trip tructur of lmintd frrolctric domin of BTiO. But thr i diffrnc btwn th coutic Domin µm Surfc dfct Surfc dfct Fig. 7-: SEI of on BTiO crmic Fig.7-b: Th in itu EAI of th crmic 455 nm Grin boundry nm Grin boundry.5 µm Domin Fig. 7-4: Topogrphy of th m BTiO crmic by SPM Fig.7-4b: Th in itu coutic img of th crmic by SNAM 6

71 7. Rult contrt of both tchniqu. At firt, th coutic contrt from both tchniqu hv diffrnt width. Th SEAM how th bright trip of bout µm wid. Th SNAM how th bright trip with width of vrl hundrd nnomtr. Furthr, th contrt of blck trip in SEAM coutic img i vry nrrow nd th boundri btwn th bright nd drk contrt i not clr. In coutic img obtind by SNAM, thr r clr priod btwn th good priodicl rryd bright or drk tructur t th r tudid. Th boundri btwn th bright nd drk trip r lo clr. Furthrmor, dicud in Eprimntl t-up of SNAM in Chptr 5, if th lctric fild ditribution in nr-fild i cttrd by dfct, th min contrt of SNAM will b domintd by th dfct. A hown in Fig.7-5, th dfct t uburfc of on kind of crmic (from Shnghi Intitut of Crmic) cn b hown by th chck of dirct nd convr piolctric rpon by SNAM [4, 5]. It mut b notd tht two kind of crmic r imgd with th m SNAM tchniqu but th coutic img hv totlly diffrnt contrt (Fig.7-4b nd Fig.7-5). Th m phnomn r lo obrvd from othr crmic. Why i thr uch grt diffrnc btwn th coutic img with th m SNAM tchniqu? It i till qution which rquir furthr tudy. According to th modl prntd in Chptr, th domin contrt cn b plind only undr two condition: On i tht th trnl piolctric ffct rultd from domin wll movmnt r nglctd ( Eq.-8 nd Eq.-9); Th othr i tht th ourc lctric fild i concntrtd in nrfild nd th mtril in nr-fild i homognou ( chptr ). Only undr uch condition cn th 9 domin tructur b imgd clrly by SNAM nd plind by th modl of nr-fild coutic. If thr r dfct in th mpl, uch ir bubbl, tructur frctur, nd mimtch of crytl lttic, which it frquntly in crmic, th dfct will cttr th fild ditribution of ourc lctric fild wll chng lctric nd piolctric proprti in nr-fild in th mpl. A rult, th contrt of coutic img will b ffctd by th dfct nd will b th um of ll th ffct, uch fild cttring, inhomognity of piolctric ffct, nd inhomognity of ltic nd lctric proprti. A clr domin contrt by SNAM could not b poibl in thi c, hown in Fig.7-5. It i clifid roughly inhomognity [4]. Som imilr rult r lo rportd by SEAM tchniqu [89]. If, howvr, crmic i highly homognou nd h vry good frrolctric proprti, th frrolctric domin cn b imgd by SNAM clrly, hown in Fig.5-4b. Bcu thr r fw litrtur for th dicuion of ll th dfct of crmic in micro- or nnomtr rng, complt plntion for th dfct imgd could not b rchd yt. 6

72 7. Rult 5.5 µv 4.5 µv Dfct µv Dfct µv µm µm Fig.7-5: Th coutic img by n c voltg btwn th tip nd th bottom of th mpl. Fig.7-5b: Th voltg img by n modultd forc btwn th tip nd th mpl urfc. 7. Film Frrolctric film (Forchungntrum Julich, Grmny) r lo tudid by SEAM. Th thickn of th film w bout µm nd th ubtrt mtril w MgO which i not frrolctric. Th ubtrt hd thickn of millimtr. Th cclrting voltg of th primry lctron ud w kv. Th typicl rult r hown in Fig.7-6 [5]. Th dfct it in both SEI nd EAI, but thr i om coutic contrt which i not th m th condry lctron contrt. A th kv for PE w ud nd th thrml diffuion dpth i bout ~ µm in th mpl, th coutic contrt could contin both th informtion of frrolctric polrition nd th thrml diffuion. A littl work on frrolctric thin film i documntd in prnt litrtur, thr i no concrt vidnc bout th contrt mchnim up to now. By comprion of th BTiO film thickn nd th thrml diffuion dpth, it could b poibl tht th thrml mchnim could ply min rol for thi coutic contrt. Thrfor th contrt would b imply rfrrd to th inhomognity of tr in thi work. A clr plntion of th contrt mchnim nd to b tudid furthr. 6

73 7. Rult Inhomognity of tr 5 µm Dfct Dfct Fig.7-6: SEI of on BTiO film Fig.7-6b:Th in itu EAI of th film t f7 kh 9 nm nm Frrolctric domin Grin boundry Grin boundry 65 nm Fig.7-7: Topogrphy of on BTiO film by SPM Fig.7-7b:Th in itu coutic img of th film by SNAM t f89 kh Anothr frrolctric BBi 4 Ti 4 O 5 thin film (M-Plnk-Intitut für Mikrotruckturphyik, Hll, Grmny) i imgd by th SNAM. Th film i µm thick nd th ubtrt mtril i th conductiv LNO []. Th typicl rult r hown in Fig.7-7. Thr i coutic contrt on on grin who boundry i hown by n rrow. Th m contrt i lo prntd by othr work with piolctric rpon mod of SNAM nd th contrt i clifid domin on on grin of th film []. 7.4 Th img mod comprion of SNAM To compr th img contrt of th t-up dvlopd in thi work with othr SPM bd tchniqu, th m tructur of th m mpl i tudid by both th dvlopd SNAM tchniqu nd th piorpon mod of SPM. A dcribd in Chptr 5, lthough thi 6

74 7. Rult mod of SPM h complictd trnmiion function of th cntilvr in SPM [84] nd i not idl to compr th contrt btwn diffrnt coutic nr-fild tchniqu, thi primntl t-up provid bttr thrml dynmic img poibility thn th SNAM tup dvlopd in thi work, th lr dtction ytm, if th primnt tim i not o long, will not b ffctd by th thrml conduction in th mpl nd thrml lmnt. To vlut both tchniqu, comprion of contrt on th m tructur with th m primnt nvironmnt btwn th dvlopd SNAM tchniqu nd th piorpon mod i tudid nd only coutic img r hown in Fig.7-8. Th tudid r i bout th m tht hown in Fig.7-4. Th grin boundry nd frrolctric domin tructur r imgd in Fig.7-8 with th m contrt t crtin frquncy (f. kh). Grin boundry Domin tructur Grin boundry Domin tructur.5 µm.5 µm Fig.7-8: coutic img by SNAM with trnducr of th BTiO crmic Fig.7-8b: coutic img by piorpon mod t th idnticl r Fig Dynmic tudy of frrolctric domin tructur To tudy th tmporl tbility of domin of BTiO, on BTiO crmic mpl (from Mrtin-Luthr-Uivrität Hll-Wittnbrg, Grmny) w imgd by SNAM dvlopd in thi work in 998 nd th coutic img i hown in Fig.7-9 [6]. Aftr two yr, th m tructur t th m r wr imgd onc mor nd th coutic img i hown in Fig.7-9b [7]. Th grin boundry in both img cn b n clrly. Th tringl in both img r ud ign to idntify th m tructur. It i hown clrly tht th domin tructur hv lmot no chng for two yr. It impli lo tht th frrolctric domin of th crmic tudid r vry tbl. 64

75 7. Rult Grin boundry Grin boundry 5 µm 5 µm Fig.7-9: frrolctric domin tructur imgd by SNAM in 998 Fig.7-9b: Th m tructur imgd by SNAM in To tudy th thrml proprty of frrolctric domin of th BTiO crmic, th m mpl i thrmlly htd ovr C, th Curi tmprtur of BTiO, nd thn th mpl i coold down to room tmprtur. Th frrolctric domin tructur t th idnticl r, hown in Fig.7-9 nd Fig.7-9b, w imgd by th dvlopd SNAM ytm onc mor ftr th thrml trtmnt. Th coutic img i hown in Fig.7- [7]. Grin boundry Structur chngd 65 By comprion of th Fig.7- nd Fig.7-9, it i clr tht om tructur t crtin plc rmind th m, hown in th tringl. But om chngd, hown by rrow. It cn b plind tht th thrml trtmnt chngd th intrnl nrgy of th crmic. Whn th mpl w htd ovr th Curi tmprtur, th 5 µm ymmtry of th mpl w chngd to cubic nd th intrnl nrgy w chngd. Fig.7-: Th tructur t th m r ftr Whn th mpl w coold down from th thrml trtmnt Curi tmprtur to room tmprtur, th frrolctric polrition would chng th polrition in uch wy tht th intrnl nrgy hd minimum undr th itd thrml, ltic, nd lctric boundry condition. Th boundry condition undr which th mpl w coold down wr diffrnt from tho whn th domin tructur in Fig.7-9 hd

76 7. Rult bn tblihd, o tht om domin tructur mut chng thir ditribution to nur th intrnl nrgy hd minimum vlu. Thrfor, om r hd nw tructur hown by rrow nd om r hd no chng of tructur hown by th tringulr in Fig.7-. To tudy frrolctric domin tructur nr th Curi tmprtur, th m tructur r imgd by dynmic t-up of SNAM dvlopd from piorpon mod by th u of thrml lmnt nd tmprtur control unit. Th primnt t-up i hown in Fig.5-8. Th coutic img t diffrnt tmprtur r hown in Fig.7-. Th tudid r i th m tht tudid bov. All th img in Fig.7- hv th m dimnion of µm t th m r nd th grin boundry, which i not chngd during th thrml dynmic imging, cn b n clrly from Fig.7-b to Fig.7-f. Thrfor, th µ-br to how th dimnion of th img nd th rrow to how th grin boundry r only notd in th firt coutic img Fig.7-. Whn th mpl w htd grdully from room tmprtur to Curi tmprtur of BTiO crmic ( C), th domin tructur dipprd grdully, hown from Fig.7- to Fig.7-d. At th Curi tmprtur, frrolctric domin dipprd totlly, hown in Fig.7-d. Whn th m mpl w coold down grdully from Curi tmprtur to room tmprtur, frrolctric domin pprd grdully, hown in Fig.7- nd Fig.7-f. It mut b notd tht thr i contrt diffrnc btwn th Fig.7- nd Fig.7-f t th m room tmprtur. Th frrolctric domin chngd thir polrition ditribution if th tmprtur of th mpl i chngd. Th m phnomnon i lo hown in Fig.7-9b nd Fig.7-. By th u of bi voltg btwn th tip nd th bckid of th mpl, th lctric dynmic proprty of frrolctric domin of th m tructur r imgd by SNAM dvlopd. Th rult r hown in Fig.7-. Th tudid r i th m tht tudid bov. All th img in Fig.7- hv th m dimnion of µm nd th grin boundry w imgd clrly in ll th img. Thrfor, th µ-br to how th dimnion of th img nd th rrow to how th grin boundry r only notd in Fig.7- wll. In Fig.7-, domin tructur wr imgd undr V bi voltg by SNAM dvlopd. Undr 4V bi voltg, th m r w imgd by th m t-up nd th coutic img i hown in Fig.7-b. Thn th bi voltg w witchd off nd th m r w imgd immditly onc mor, hown in Fig.7-c. It i clr tht mot of tructur dipprd whn th r w cnnd by dc bi voltg of 4V btwn th tip nd th bckid of th mpl. Aftr th dc bi voltg w rmovd, om frrolctric tructur pprd onc mor but with chngd tructur. 66

77 7. Rult Bcu th frrolctric tructur t th top urfc of frrolctric BTiO mtril, which r blivd to b diffrnt from th bulk tructur [~4], r littl documntd in litrtur, th phnomn could only b plind by th u of th thorticl modl dvlopd. At firt, it i ncry to not tht crmic do not chng it polrition totlly in on dirction vn whn th crmic i undr n lctric fild which i much grtr thn it corc fild. Som domin will chng thir polrition undr th corc fild nd om rmin lwy in th m dirction, th polrition dirction i dcidd from th minimum of th intrnl nrgy undr th clmpd condition mong diffrnt grin ( Chptr ). Whn th 4 V dc bi btwn th tip nd th bckid of th mpl w pplid, ccording to th clcultion of th fild intnity in Chptr 5, th fild intnity h vlu vrid from kv/m t th urfc to 5 kv/m t dpth of 6h6 nm. Bcu th fild intnity within thi dpth i grtr thn th frrolctric corc fild intnity of th BTiO mtril [from 5 kv/m to kv/m], if th clmpd condition llowd, om domin would chng polrition to th dirction of th lctric fild within thi dpth. A rult, th priodicl rry of - nd c-domin in thi r would b chngd. Th contrt, which com from th intgrtion btwn th polrition nd th concntrtd fild, would b chngd. If th clmpd condition of on r w o trong tht th polrition of th domin t tht r could not b chngd vn undr th m corc fild, th contrt of th SNAM t tht r would not chng, o tht th intgrtion w th m th intgrtion without th dc bi fild. Th contrt in thi r would not b chngd. Nturlly, thi lctric fild intnity chngd th intrnl nrgy of th r, whr th domin wr chngd. Whn th tip movd wy from th r, th frrolctric domin tructur t th rcnt cnnd r would chng in uch wy tht th intrnl nrgy t th r would hv minimum undr th nw boundry condition. Th nw boundry condition, uch th ltic nd lctric fild from th nighboring r which w jut bing cnnd, wr nturlly not th m th boundry condition whn th domin tructur in Fig.7- hd bn tblihd. Thrfor, th chng of polrition of th crmic would tk plc oon th tip movd wy from th r tudid. Finlly, thi chng of frrolctric domin tructur w imgd clrly in Fig.7-c whn th dc bi voltg w witchd off. 67

78 7. Rult Grin boundry T5 C Domin tructur T5 C Domin tructur 5 µm Fig.7-: Acoutic img of SNAM t room tmprtur (5 C) T8 C Domin tructur Fig.7-b: Acoutic img of SNAM whn tmprtur wnt up to 5 C T C All th tructur dipprd Fig.7-c: Acoutic img of SNAM whn tmprtur wnt up to 8 C T C Domin tructur Fig.7-d: Acoutic img of SNAM t tmprtur ovr Curi point T5 C Domin tructur Fig.7-: Acoutic img of SNAM whn tmprtur wnt down to C Fig.7-f: Acoutic img of SNAM whn tmprtur wnt bck to 5 C 68

79 7. Rult V bi 4V bi Domin tructur Structur dipprd 5 µm Grin boundry Structur unchngd Fig.7-: Th coutic img of on tructur imgd undr V bi voltg Fig.7-b: Th coutic img of th m r imgd undr 4 V bi V bi Structur chngd Structur unchngd Fig.7-: Th coutic img ftr th bi voltg w witchd off 69

80 8. Dicuion of both nr-fild coutic tchniqu 8 Dicuion of both nr-fild coutic tchniqu 8. Qulity of imging comprd to othr tchniqu Th SNAM nd SEAM imging tchniqu hv mny dvntg ovr othr tchniqu to img frrolctric domin. Th mot citing chrctritic of both tchniqu r th nondtructiv imging of frrolctric domin with no mpl prprtion. Th y dvlopmnt from commrcil SPM or SEM ytm i nothr dvntg. SNAM h furthr dvntg ovr SEAM in th rpct, uch no vcuum ytm, bttr rolution down to nnomtr, nd flibility of dynmic imging of frrolctric domin of mtril. On th othr hnd, SEAM h uch pculiriti lrg imging dpth nd brod imging r which r lo not o y to obtin only by th u of SNAM. Thrfor, both tchniqu will b pplid complmntrily to tudy frrolctric mtril in th futur. 8.. Spd of primnt Th imging pd of SEAM i limitd by th nr-fild tblihing tim in frrolctric mtril. A it i dicud in Chptr 4, th tim rquird to tblihd th lctric fild in nr-fild in SEAM dpnd on th chrg ccumultion in nr-fild nd th proprti of mtril. typicl vlu for miconductor nd crmic r in th rng of -4 cond. To nur good coutic img of SEAM, th tim t vry pil i uully chon m. For on typicl img with 44 pil, it would tk bout ~ minut to tk on img. SNAM tchniqu h to cn th tip (or mpl) by piolctric cnning unit. Unlik SEAM, it nd lmot no dly tim to tblih th lctric fild in nr-fild. Th imging tim of th ytm i minly dpndnt on th cnning pd of SPM. In ordr to nur good topogrphy, th cnning pd i uully low, mllr thn pil/cond. For n img of 44 pil, it tk bout ~ minut to tk pictur. It i clr tht both tchniqu hv lmot th m img pd. 8.. How quntittiv Gnrlly pking, th SEAM nd SNAM r both cnning imging tchniqu which rcord th rltiv vrition mong cnning pil in givn cnnd r. If phyicl proprti of on cn pil or on r cn b dtrmind quntittivly, th rt of cn pil cn hrftr b imgd quntittivly. SEAM tchniqu h vry complictd intrction on th urfc of frrolctric mtril, whn th primry lctron bombrd th urfc. Although much ffort h bn pid to nly intrction of primry lctron bombrding on mtl mtril, miconductor, nd 7

81 8. Dicuion of both nr-fild coutic tchniqu om dilctric mtril, both primntl nd thorticl work on frrolctric mtril r littl documntd in prnt litrtur. A quntittiv nlyi rquir th timtion of prmtr of primntl nvironmnt nd thir rror. If om bic prmtr of SEAM on frrolctric mtril, uch th ccurt lctric fild intnity in nr-fild, th ccurt pntrtion dpth of PE, th clculbl rltionhip of diffrnt imging mchnim nd o on, r not quntittivly dtrmind, th quntittiv nlyi of frrolctric polrition by SEAM would b impoibl. Unfortuntly, th importnt fctor hv bn prly documntd in th prnt litrtur. Thrfor, th SEAM tchniqu rmin in principl qulittiv imging tchniqu up till tody. Whthr th SNAM or th rltd tchniqu r quntittiv tool, i till rcnt tudy which i hvily dbtd tody [~, 85]. To nly th problm, it would b bttr to rviw om bic pproimtion which r oftn ud in th prnt litrtur to chrctri frrolctric mtril by SNAM ytm. Firtly, dicud in Chptr, th piolctric contnt of frrolctric crmic dpnd on pontnou polrition only undr th condition tht trnl piolctric ffct i nglctd nd polrition will not b ffctd during th imging; Scondly, to dtrmin th lctric fild in th mpl, ll th litrtur um tht th mpl i iotropic; Thirdly, only th min mod of th vibrtion in nr-fild i clcultd nd th coutic mod convrion nd nonlinr ffct in nrfild r nglctd; Finlly, by dtcting th nr-fild vibrtion through lr bm nd photodiod in piolctric rpon mod, thr it vry complictd vibrtion trnmiion function of th cntilvr [84, 85] from which n ccurt timtion of rror would b difficult. On th othr hnd, by dtcting th nr-fild vibrtion through trnducr, it i don in thi work, ll th thorticl clcultion mut b undr th pproimtion of coutic pln wv. Poibl rror incurrd undr o mny pproimtion by th modling of SNAM would b difficult to timt t prnt wll. It would b ncry to y tht undr ll th pproimtion bov, th thorticl modl dvlopd for both SEAM nd SNAM in thi work cn b ud to clibrt th coutic contrt or domin tructur quntittivly. 8. Comprion of SEAM nd dvlopd SNAM ytm 8.. Img comprion nd nly To compr th img of two kind of microcopy tchniqu, it i ncry to compr th proprti t idnticl r. A it i concrnd with th nondtructiv tudy of th frrolctric domin, th domin tructur hould not b dmgd during th chng of 7

82 8. Dicuion of both nr-fild coutic tchniqu microcop. Th mthod i Point lrdy dcribd in Chptr 6. By Point th u of opticl microcopy, on r of mpl AC, from th Univrity of Hll, Grmny, i imgd tndrd to compr topogrphy, hown in Fig.8-. Th m r i imgd by 5 µm SEAM, hown in Fig.8-. From SEAM img of Fig.8-, Fig.8-: Opticl img of on r of mpl AC th contrt of domin tructur r imgd only in coutic img. Th contrt in coutic img h typicl 9 tripd tructur with th width from 5 to µm. By th comprion to th opticl img Fig.8- in which th grin i r uully from µm to 5 µm, th contrt from frrolctric domin in coutic img of SEAM h nturlly no rltionhip with grin i. Aftr th crmic w tudid by SEAM, two typicl cor grin wr chon to b tudid complmntrily by SNAM. On grin, which h dimtr of µm, i mrkd point in opticl img Fig.8- nd in SE img of Fig.8-c nd Fig.8-. Th coutic contrt of th grin by SEAM r mrkd whit inrt with dhd lin in SEAM coutic img Fig.8-d nd Fig.8-f. Th othr grin, which h dimtr 6 µm, i mrkd point in opticl img Fig.8- nd in SE img of Fig.8-c. Th coutic contrt of th grin by SEAM i mrkd whit inrt with dhd lin in SEAM coutic img Fig.8-d. Domin tructur r hown clrly by th coutic img of Fig.8- nd Fig.8-4. Th contrt from domin tructur hv width from on hundrd to vrl hundrd nnomtr. It h th m ordr of domin i which i clcultd thorticlly by othr work [6~]. Furthrmor, it i hown in Fig.8-f tht om domin hv corrpondnc of contrt, hown t th lowr prt in Fig.8-f. Thi cn b plind tht th mpl in thi r h th α tructur of domin tructur of cor grin in BTiO [6]. Th m tructur cn lo b n clrly t th uppr id in Fig.8-4f. Som domin tructur uch tho imgd in th uppr prt of Fig.8-f, hv no corrpondnc of contrt. Thi cn b plind th γ domin tructur of frrolctric domin dicud in Chptr. It mut b notd, If th coutic img by SEAM in Fig.8- nd th coutic img by SNAM in Fig.8- nd Fig.8-4 on th both grin r comprd, it cn b n clrly tht th coutic contrt on th m grin by both tchniqu r totlly diffrnt. 7

83 8. Dicuion of both nr-fild coutic tchniqu Surfc dfct Surfc dfct µm Domin Fig.8-: SE img of th m mpl Fig.8- Fig.8-b: SEAM in itu coutic img of Fig.8-5 µm Surfc dfct Surfc dfct Point Point Domin Fig.8-c: SE img of th inrt with blck lin in Fig.8- Surfc dfct Fig.8-d: SEAM in itu coutic img of Fig.8-c Surfc dfct Point Domin Fig.8-: SE img of th inrt with blck lin in Fig.8-c Fig.8-f: SEAM in itu coutic img of Fig.8-7

84 8. Dicuion of both nr-fild coutic tchniqu Domin Grin boundry Grin boundry Fig.8-: Topogrphy of point in Fig.8- by SPM 56 nm Fig.8-b: SNAM in itu coutic img of Fig.8- Domin nm Grin boundry Grin boundry Fig.8-c: Topogrphy of th inrt in Fig.8- by SPM 455 nm Fig.8-d: SNAM in itu coutic img of Fig.8-c Domin nm Grin boundry Grin boundry Fig.8-: Topogrphy of th inrt in Fig.8- c by SPM Fig.8-f: SNAM in itu coutic img of Fig.8-74

85 8. Dicuion of both nr-fild coutic tchniqu Point Fig.8-4: Topogrphy of th r nr th point in Fig.8- by SPM Fig.8-4b: SNAM in itu coutic img of Fig.8-4 Grin boundry Grin boundry Domin Fig.8-4c: Topogrphy of th point in Fig.8-4 by SPM Fig.8-4d: SNAM in itu coutic img of Fig.8-4c Domin Fig.8-4: Topogrphy of th inrt in Fig.8-4c by SPM Fig.8-4f: SNAM in itu coutic img of Fig

86 8. Dicuion of both nr-fild coutic tchniqu 8.. Eplntion of th rult Th contrt diffrnc of coutic img by SEAM nd SNAM t th idnticl r (point nd point ) cn b plind th dpth diffrnc of nr-fild of both tchniqu. A it i plind in Chptr 4 nd Chptr 5 thorticlly, th contrt of both nr-fild coutic imging tchniqu (SEAM or SNAM) com from intrction btwn th vrg piolctric ffct in nr-fild nd th lctric fild formd by injcting lctron (SEAM) or th concntrtd fild of th cnning tip (SNAM). A it i not wll undrtood up to now how th domin wll movmnt contribut to th piolctric, ltic nd dilctric proprti of frrolctric crmic triniclly [8, ~] nd how mny kind of domin tructur of BTiO crmic it totlly [9, ], n ccurt thory of vrg piolctric ffct in nr-fild r for both tchniqu on BTiO mtril would b impoibl. A rult, clr plntion of contrt of both nr-fild coutic mthod m lo difficult t prnt. Th following would b n plntion which b olly on th prnt thory nd primntl fct: Th typicl domin tructur in crmic hv th typicl width of vrl hundrd nnomtr for cor grin, prdictd by prnt thori nd primntl mthod [9, ]; Th lctric fild in th nr-fild i o wk tht it produc only vibrtion through intrinic convr piolctric ffct nd mll movmnt of domin wll but no chng of th whol domin tructur []; By SEAM, th nr-fild dimnion i lmot - µm wid nd - µm dp t kv cclrting voltg for PE, nd th intrction r i not homognou [57]. For th typicl domin tructur which r clr to u [9], uch hown in Chptr, th domin width i uully vrl hundrd nnomtr. Hr, w would tk cor grin of BTiO crmic n mpl. Th typicl domin tructur hv bn tudid [9, ]. Th domin hv typiclly four tructur, hown tructur -4 in Fig.8-5. A hown in Chptr 4, th coutic mplitud contrt of SEAM of vry cn point com from th vrg intrction btwn th lctric fild formd by injcting lctron nd th vrg piolctric ffct in th whol intrction r. For cn point N nd ny othr cn point in r of tructur, thr i no diffrnc of th vrg intgrtion in nr-fild. Tht mn tht thr i no contrt diffrnc by SEAM in th r of tructur. With th m rgumnt, thr i no contrt diffrnc for ny cn point M in th r of tructur. Similrly, ny cn point in th r of tructur or 4 h no contrt diffrnc with ny othr point in th m tructur rpctivly. 76

87 8. Dicuion of both nr-fild coutic tchniqu Domin width Scn point M Scn point N d: Dpth of nr-fild of SEAM Y Structur Structur Structur Structur 4 Fig:8-5: Th contrt mchnim of SEAM on th crmic mpl Furthrmor, ccording to Eq.-4 in Chptr, th piolctric ffct of frrolctric crmic i dpndnt on th polrition nd trinic ffct. If th lctric fild in nr-fild r in SEAM i mll nough, th trinic ffct (minly of domin wll movmnt) cn b nglctd []. Thu th piolctric ffct in nr-fild r of SEAM i olly dpndnt on th pontnou polrition. W cn drw qulittivly th pontnou polrition for tructur -4 which contribut to coutic longitudinl wv in dirction in nr-fild of SEAM ytm Fig.8-6. P P P P d p Z d p Z d p Z d p Z Fig.8-6: Structur Fig.8-6b: Structur Fig.8-6c: Structur Fig.8-6d: Structur 4 Bcu th output ignl of trnducr i th intgrtion btwn th convr piolctric ffct nd th ourc lctric fild in nr-fild r ccording to Eq.-8, ftr th linr mplifiction of th lock-in mplifir, th output ignl of th lock-in mplifir i proportionl 77

88 8. Dicuion of both nr-fild coutic tchniqu to thi intgrtion. A only mplitud ignl of th lock-in mplifir i ud th imging ignl of SEAM, th gry lvr of th coutic img of SEAM i nturlly proportionl to mplitud of thi intgrtion wll. A hown in Fig.8-6 nd Fig.8-6d, th vrg polrition in tructur nd tructur 4 h th m vlu but only chng it ign. If th ourc lctric fild i kpt th m during cnning, th mplitud of th intgrtion btwn th convr piolctric ffct nd th ourc lctric fild in th two r hv th m bolut vlu but only diffrnt ign. Tht mn th two r hv th m gry lvr in coutic img of SEAM. According to th m roning, thr would b no contrt diffrnc btwn tructur nd tructur ithr. At th b of roning bov, thr r only two tructur, tructur nd tructur hown in Fig.8-6 nd Fig.8-6b bov, in which th cnning pil, uch th pil M in tructur nd pil N in tructur hown in Fig.8-5, hv poibly diffrnt contrt in coutic img of SEAM. In othr word, undr th prnt knowldg of domin tructur in frrolctric BTiO crmic, th coutic img of SEAM would img only th diffrnc btwn diffrnt lmintd r. Th imgd domin wll could b th wll btwn diffrnt 9 domin lmintd tructur of BTiO crmic. If th m tructur r tudid by SNAM tchniqu, th contrt cn b nlyd in th m wy. Bcu of th form of th tip, mot of th ourc lctric fild i concntrtd jut bnth th contct point undr th tip, it i dicud nd clcultd in Chptr 5 in thi work, o tht th nr-fild r of SNAM i mll vrl hundrd nnomtr ( Eq.5-8 nd Fig.5-5). Undr th m wk fild pproimtion dicud bov nd ccording to Eq.5-, th output ignl of th trnducr in SNAM i proportionl to th intgrtion btwn th convr piolctric ffct nd th ourc lctric fild in nr-fild r of vry cn point of SNAM. Aftr th linr mplifiction of th lock-in mplifir, th output ignl of th lock-in mplifir i proportionl to thi intgrtion wll. A only th mplitud ignl of th lock-in mplifir i ud th imging ignl of SNAM, th gry lvr of th coutic img of SNAM i nturlly proportionl to th mplitud of thi intgrtion. Two cn point in r of tructur hown in Fig.8-7 r tkn n mpl. Th cn point N h polrition prlll to th ourc lctric fild. Th convr piolctric ffct in nr-fild in th dirction of th fild i non-ro. Th intgrtion btwn th convr piolctric ffct nd th fild i non-ro too. Th contrt in coutic img of SNAM t thi point h non-ro gry lvr. On th othr hnd, th cn point M h th polrition prpndiculr to th ourc lctric 78

89 8. Dicuion of both nr-fild coutic tchniqu fild in nr-fild nd th convr piolctric ffct in nr-fild in th dirction of th fild would b ro. Th intgrtion btwn th convr piolctric ffct nd th fild would b ro wll. Thi mn tht th contrt t thi point in coutic img of SNAM h ro gry lvr. A th point N nd M r rbitrrily chon in th r of tructur, if th ourc lctric fild i kpt th m during cnning, ny cn point which h polrition in th dirction of th ourc lctric fild in tructur h th m non-ro gry lvr in coutic img of SNAM; Whr ny point which h polrition prpndiculr to th dirction of th ourc lctric fild h th ro gry lvr. Domin width Scn point M Scn point N Dpth of nr-fild of SNAM Y Structur Structur Structur Structur 4 Fig.8-7: Th contrt mchnim of SNAM ytm dvlopd In othr word, th contrt of coutic img of SNAM com from diffrnc of frrolctric polrition dirction of 9 domin tructur. It cn b n ily tht th roning i lo vlid in r of tructur,, or 4. By comprion of Fig.8-5 nd Fig.8-7, it i clr tht th 9 dgr domin will how diffrnt width nd totlly diffrnt contrt by SEAM nd SNAM tchniqu. 79

90 9. Concluion 9 Concluion Invtigtion of frrolctric domin tructur r ntil for both fundmntl nd pplid prpctiv inc thy dtrmin, to lrg tnd, th mcrocopic rpon of frrolctric mtril. In microcopic rng, thy domint th bhvior of lmot ll frrolctric dvic in th microlctronic indutry. Th rult prntd in thi work on frrolctric BTiO mtril how th fcinting world of thi kind of mtril. Th dimnion of th imgd frrolctric domin tructur by SNAM provid primntl vidnc for th thory of th tructur in cor grind BTiO crmic, which r only provd by dtructiv mthod uch th chmicl tching tchniqu bfor now. Tmporl bhvior of frrolctric domin tructur t n idnticl r of BTiO crmic h bn imgd t ubmicromtr rng for th firt tim. Th domin tructur of th crmic rmin unchngd for two yr. It impli tht frrolctric polrition of thi kind of crmic i vry tbl nd it would b n idl mdium for ppliction uch mmory dvic. Thrml dynmicl proprti of frrolctric domin tructur t n idnticl r of BTiO crmic r imgd by SNAM t ubmicromtr rng. Whn th tmprtur of th crmic i ovr Curi tmprtur, th domin tructur dippr totlly. Th frrolctric domin chng thir polrition ditribution if th tmprtur of th mpl chng. Elctricl dynmicl tudy of frrolctric domin t n idnticl r of BTiO crmic h lo bn tudid. With bi voltg, th chng of th m tructur r imgd clrly. Bd on th dvlopd thorticl modl, th fild intnity undr th tip nd th phyicl proc of polrition chng on th mpl urfc r nlyd. Th dynmic invtigtion ly th foundtion for futur dynmic nlyi of frrolctric domin of othr mtril by SNAM. Th ytm dvlopd for dynmic rrch of frrolctric domin tructur in thi work cn b ily modifid from commrcil SPM ytm. For th furthr tudy of othr frrolctric mtril nd dvic, thi ytm i y to u, with ubmicromtr or nnomtr rolution, nd high thrml nd lctric tbility if impl control unit r ud. It prnt n idl tool for th non-dtructiv dynmic tting nd modling of both frrolctric mtril in mtril nginring nd frrolctric dvic in microlctronic indutry. Th plntion of th contrt from coutic non-dtructiv imging ytm rmin lwy chllng for rrchr in thi r. Th SNAM ytm dvlopd by th u of n coutic trnducr provid b for both thorticl nd primntl complmntry 8

91 9. Concluion nly of th coutic contrt mong diffrnt kind of coutic nr-fild ytm. A n mpl, th complmntry nlyi of SEAM nd SNAM rvl th coutic contrt mchnim of both nr-fild tchniqu. Th diffrnt contrt which r rvld by complmntry nlyi t idnticl r cn b wll-plind uing th currnt thory for BTiO crmic. Th primntl mthod dvlopd for th complmntry nlyi cn b furthr pplid to nly othr proprti of mtril nd dvic mong diffrnt microcopy ytm. Th thorticl modling of SNAM nlyi, i.. th Grn function to dtrmin th quittionry fild in th ytm nd th trnmiion lin mod of coutic longitudinl wv within th mpl, i pplid uccfully for th firt tim to chrctri lctricl nd mchnicl fild ditribution of BTiO mtril tudid by SNAM. Bd on th thorticl olution prntd in thi work, othr frrolctric mtril cn b nlyd imilrly. Undr th m thorticl pproimtion tho of SNAM ytm, th contrt of SEAM ytm on frrolctric mtril h bn chrctrid by th thorticl tchniqu wll. Although only coutic pln wv mod r nlyd, th impl olution rvl om bic bhvior of SEAM ytm on frrolctric mtril. Th modlling tchniqu would b lo pplicbl to othr bm injcting coutic imging ytm. 8

92 . Futur pct Futur pct Th chrctrition of frrolctric mtril t ubmicromtr or vn in nnomtr rng, both in ltrl dirction nd in dpth profiling, i of vitl importnc for th dvlopmnt of futur frrolctric mtril nd intgrtd dvic. Unfortuntly, th prnt knowldg of frrolctric domin t thi rolution rng i not nough to prdict th o-clld domin influnc on th mtril. Whr common mthod hv diffrnt limittion for thi purpo, th non-dtructiv mthod bd on SPM provid nw brkthrough for thi gol. Although much ffort on thi tchniqu h bn givn nd mny rult r prntd for diffrnt frrolctric mtril by diffrnt tchniqu bd on SPM, from our point of viw, clr rltionhip btwn th domin ffct nd proprti of frrolctric mtril rquir furthr invtigtion. Th quntittiv dcription of nrgy chng of frrolctric domin during witching proc i not wll undrtood. A dicud in mny litrtur [~], thi nrgy chng i proc including thrml, lctricl, nd ltic nrgy. Although thi work h hown om chng of domin tructur by trnl thrml nd lctricl influnc, th ccurt rltionhip mong th thrml, ltic, nd lctricl nrgy chng hould b furthr invtigtd. To undrtnd thi proc ccurtly poibl, diffrnt non-dtructiv mthod t thi rolution rng, uch thrml nd coutic nr-fild tchniqu, hould b ud complmntrily. Th thrml bhvior of frrolctric domin of BTiO i nothr problm which nd furthr tudy wll. Although currnt litrtur prnt mny thorticl nd primntl rult on thi ffct [~5], quntittiv tudy of thrml bhvior of domin tructur of BTiO t ubmicromtr or nnomtr rolution i till not rchd. Evn though om thrml dynmic invtigtion h bn prntd in thi work, quntittiv rltionhip btwn frrolctric domin tructur nd th trnl thrml nrgy chng h not bn obtind yt. To chiv thi gol, complmntry nlyi on frrolctric BTiO by thrml nd coutic nr-fild tchniqu would b idl. Th mtl-dopd BTiO i widly ud in indutry for diffrnt purpo [86]. Although om work h bn don to rvl th rltionhip mong grin i, lctricl, ltic, nd piolctric proprti, frrolctric domin tructur from dopd BTiO mtril r littl tudid. Th thorticl rltionhip btwn grin i nd frrolctric domin tructur for dopd BTiO i not chivd ithr. Th prntd SEAM nd SNAM tchniqu r th bt non-dtructiv tool for primntl invtigtion nd bi for th modling nd mnufcturing of thi kind of mtril. 8

93 . Futur pct Frrolctric thin film r ud intnivly in th micro-lctronic indutry nd om proprti of thin film r lo intnivly tudid [4,, ~5]. But, from our point of viw, frrolctric domin tructur nd thir formtion proc undr trnl influnc nd to b tudid furthr. Problm uch how th domin chng thir tructur during witching proc, how much i th witching tim of diffrnt tructur, wht r th min ourc of th ftigu of th frrolctric polrition nd o on rquir furthr tudy. To nwr ll th qution, thorough undrtnding nd modling of frrolctric thin film, both primntlly nd thorticlly, would b ncry. Although thi work h prntd ytmtic thorticl mthod to tudy th lctricl nd couticl fild ditribution from diffrnt mpl tructur in SNAM ytm, thr i till lot of work to do to chrctri ll th fild componnt quntittivly. Whr th lctric fild ditribution in th ytm cn b dtrmind by th Grn function, th coutic fild olution r only bd on pln wv mod. For quntittiv chrctrition of frrolctric mtril by SNAM, it i ncry to nly th coutic fild mor ccurtly. Th coutic fild mod convrion in nr-fild nd th non-linr ffct of frrolctric mtril hould b furthr tudid. Rcnt litrtur [87] point out tht modrn chip indutri fc vr problm for modling of dvic nd filur nlyi multilyrd ytm r ud intnivly in th nnomtr rgim. Th chrctrition of lctricl, thrml, nd couticl fild ditribution in th ytm i of vitl importnc to dvlop nw dvic. A th tructur of th dvic hv nnomtr dimnion, om bic condition for prnt phnomnologicl thori cnnot b fulfilld ny mor. Nw bic thorticl bckground including combintion of quntum-mchnic nd quntum-lctronic for th chrctrition of th dvic mut b thoroughly dvlopd. Howvr, th ytmtic phnomnologicl mthod r foundtion for thi futur dvncmnt. 8

94 r r t t. Appndi Appndi A: Solution of on-dimnionl piolctric tiffnd pln wv in BTiO nd PZT Th piolctric trin nd tr rltionhip r: E ~ + I I Di, J J d i, J I c I + J E, J J d I, j ε ~ i, j I, j E j E j E j Eq.A- Eq.A- Eq.A- Di i, J + J ε i, j E j Eq.A-4 To chrctri th typicl nr-fild coutic ytm of prnt work on BTiO, it i ncry to dicu t firt th coupling btwn lctric nd coutic fild gnrlly in both PZT trnducr nd BTiO mpl in on dimnion. Th polrid PZT crmic cn b nlyd by crytl lttic of 6mm nd th crytl lttic of ingl BTiO with only c- monodomin i 4mm [5]. Th piolctric tnor nd dilctric tnor for both mtril ytm hv th m form but diffrnt vlu of componnt. Thi llow u to trt th coupling in trnducr wll in nr-fild thorticlly togthr. Only if quntittiv vlu of ch mtril r concrnd, ctul vlu of th tnor componnt will b ud rpctivly. If dirction i chon th trnmiion dirction, th diffrntil nd r qul to y ro if th ltrl dimnion i unlimitd. Only mchnic fild componnt. i non-ro for vry lctric nd From Mwll qution Eq.-, if thr r no conducting nd ourc currnt ( J r nd c J r ) in givn mtril, lctromgntic fild nd wv which trnmit only in dirction will b dcribd [5]: E r y y E E E y µ 84 r H t,

95 . Appndi 85 or E E y µ t H t H t H y Eq.A-5 H r y H H H y y t D r, or H H y t D t D t D y Eq.A-6 Th qution bov cn b writtn : E t H y µ, H t D y Eq.A-7 E t H y µ, H t D y Eq.A-8 t D t H Eq.A-9 Eq.A-7 nd Eq.A-8 dcrib two lctric mgntic wv nd Eq.A-9 two qui-ttionry fild. If th body torqu i nglctd undr th mll ignl pproimtion, th bic coutic qution in mtril r: F u r r t ρ t Eq.A- t u r Eq.A-

96 . Appndi 86 in which t nd r r trin nd tr in th mtril rpctivly, nd ρ, u r, nd F r r dnity of th mtril, diplcmnt fild, nd body forc rpctivly. Th ign of nd mn th following: t y y y t Eq.A- t u r y y y y u u u Eq.A- Hr th Voigt nottion for tr nd trin tnor i ud nd th tr nd trin tnor r writtn : t ),,,,, ( t ),,,,, ( If th vlocity of prticl in olid i v r y y v v v r r r + +, it cn b writtn : v r t u r Eq.A-4 If thr i no body forc, th coutic qution Eq.A- nd Eq.A- cn b writtn by th u of prticl vlocity : t v ρ nd v t Eq.A-5

97 . Appndi 4 v y v ρ y nd t 4 t Eq.A-6 5 v ρ nd t v 5 t Eq.A-7 Eq.A-5 dcrib on longitudinl coutic pln wv; Eq.A-6 nd Eq.A-7 dcrib two hr coutic wv. According to piolctric coupling qution Eq.A- to Eq.A-4 nd piolctric tnor of BTiO, th lctric nd coutic coupling cn b writtn : D ε E + Eq.A-8 D ε y D ε yy E y E Eq.A-9 Eq.A- c 4 c E E y Eq.A- Eq.A- 5 c E Eq.A- From lctromgntic wv qution Eq.A-7 to Eq.A-9, coutic qution Eq.A-5 to Eq.A-7, nd lctric nd coutic coupling Eq.A-8 to Eq.A-, th gnrl coupling of lctric nd mchnic fild componnt in BTiO cn b nlyd. Th coutic longitudinl wv dcribd by Eq.A-5 will b tudid with dtil in th following. According to Eq.A-9, th lctric diplcmnt D i contnt. For hrmonic tudy of fild, D cn b trtd ro. From Eq.A-8, th qui-ttic lctric fild [5] in dirction i: E ε Form Eq.A-4 nd Eq.A-, th tr in dirction i: Eq.A-4 ( c + ε ) Th contnt c p c + ε Eq.A-5 i th o-clld piolctric tiffnd tiffn. If Eq.A-5 i combind with Eq.A-5, th o-clld piolctric tiffnd wv qution for th tiffnd 87

98 . Appndi longitudinl wv ( v, ) r: t v ρ Eq.A-6 v c p Eq.A-7 t Eq.A-6 nd Eq.A-7 dcrib tiffnd coutic wv in BTiO ingl crytl without ourc. It cn b rwrittn th wv qution: v ρ c p v t Eq.A-8 Th phyicl mning of th qution bov cn b plind by piolctric thory th tiffnd ffct [5]. It m tht thi wv bhv if th ltic tiffn c wr incrd to c p c +. Th lctric nd coutic coupling for thi wv i th coupling of ε th qui-ttic lctric fild nd th coutic longitudinl wv ( v, ). Th coupling of longitudinl wv in PZT trnducr i th m, if only th mtril contnt in ll th prion bov hould b chngd to mtril contnt of PZT crmic. If thr i hrmonic ourc lctric fild in dirction dirction in th BTiO crytl tudid cn b writtn : totl E r E r, th totl lctric fild in E E + E Eq.A-9 E i th coupld tiffnd lctric fild in th crytl. Bcu only th hrmonic tbl tt will b tudid nd th coutic fild ( v, ) mut tify th m tiffnd condition, th coupld tiffnd lctric fild in th crytl i th m th fild without ourc. Form Eq.A-8 nd Eq.A-, th tr in th crytl with ourc: ( c + ε - ) E Eq.A- in which th contnt c p c + ε i th piolctric tiffnd tiffn. Bcu of th ourc lctric fild cn b obtind in th m wy: E, th wv qution for coutic longitudinl wv 88

99 . Appndi t v ρ Eq.A- v c p + t c p E t Eq.A- Eq.A- nd Eq.A- cn lo b writtn th wv qution with ourc: v ρ c p v t c p E t Eq.A- Th m proc cn b pplid to th othr two coupld wv Eq.A-6 nd Eq.A-7 in th crytl tudid. Th coupling qution r: E µ ε E t µ 5 v t Eq.A-4 E y µ ε yy E t y µ 5 v y t ( ε yy ε for BTiO ) Eq.A-5 At th firt ight, th coupling mchnim m lmot th m th coupling dicud bov. But dtild dicuion [5] how tht th kind of coupling r much mor compl thn th tiffnd lctric nd mchnic coupling. A dtild dicuion i byond th cop of th prnt work nd om dicuion cn b found in othr work [5, 54]. A: Trnmiion lin mod of coutic wv in nr-fild In nr-fild of piolctric mtril, th lctromgntic nd coutic coupling phnomn r vn mor complictd thn th coupling in th trnducr, bcu th coutic wv r r hr i th piolctric coutic wv with ourc lctric fild E E. Undr th umption dcribd in Chptr, th coutic trnmiion cn b comprd to th trnmiion of voltg wv in trnmiion lin with ourc [5]. If thr r pln longitudinl hrmonic coutic wv ( v, ) producd in th nr-fild in th crytl tudid, th qution ccording to Eq.A- nd Eq.A- r writtn th following: ρ t v Eq.A-6 v c p + t c p E t Eq.A-7 89

100 . Appndi Hr, th uffi t th top right cornr indict th nr-fild r in th mpl. A comprion, th qution of trnmiion lin with ourc V nd I will b dicud: V (, t) I (, t) L t + + RI (, t) Eq.A-8 V I (, t) C V (, t) t + GV (, t) I + Eq.A-9 Hr ( V (, t), I (, t) ) i olution of voltg or currnt wv of on-dimnionl trnmiion lin nd h th form: V (, t) V m I (, t) I m j( ωt k ) j( ωt k) Eq.A-4 Eq.A-4 whr V m nd I m r mplitud of th voltg nd currnt wv which r dcidd by th ourc nd boundry condition. In th c of trnmiion lin without lo, th qution cn b writtn : V (, t) I(, t) I (, t) L t C V (, t) t + V Eq.A-4 + I Eq.A-4 If th on-dimnionl trnmiion lin with ourc limitd in th r < <, gnrl olution of th qution Eq.A-4 nd Eq.A-4 bov i: V (, t) ( + () + () )/ Eq.A-44 I (, t) ( + () () )/Z Eq.A-45 + jk jkζ + jk () ( ( V + Z I ) dζ + ( Z ) ) ( > ) jk jkζ jk ( ) () ( V Z I ) dζ + ( Z ) ( < ) Eq.A-46 Eq.A-47 L whr R T i th chrctritic ritnc of voltg wv of trnmiion lin. C By comprion of Eq.A-4 nd Eq.A-4 with Eq.A-6 nd Eq.A-7, if th following comprion i md: V (, t) : th voltg wv i comprd to ngtiv tr in dirction of th 9

101 . Appndi longitudinl wv; I (, t) v : th currnt wv i comprd to vlocity of th prticl; L ρ : th inductnc i comprd to dnity of th mpl; C / c p / ( c / ε ) ltic tiffn; I c p E t + : th cpcitnc i comprd to rciprocl of th tiffnd : th currnt ourc i comprd to th tim diffrntil of ourc lctric fild; Eq.A-6 nd Eq.A-7 will hv th m olution tht of Eq.A-4 nd Eq.A-4. For th nr-fild ytm hown in Fig.5- with n lctric fild ourc in nr-fild, th tbl hrmonic olution of th longitudinl wv in nr-fild cn b writtn from bov comprion : ( ) / + jωt + Eq.A-48 v ( + jωt ) / Z Eq.A ( ) jk + + () jk, in which + E jkζ () ( Z ) dζ ( d) Eq.A-5 c t p ( ) jk ( d) jk ( d ) +, in which d E jkζ () (Z ) dζ ( d) Eq.A-5 c t p Hr + () nd () r trm rultd from ourc in nr-fild. Z ρ c p i th coutic chrctritic ritnc of wv ( v, ) in nr-fild nd + () nd ( d) r contnt which cn b dtrmind by boundry condition in Fig.5-. Th coutic wv outid th ourc r cn b comprd to th trnmiion lin mod without ourc. Th coutic wv outid th nr-fild but till in th mpl i govrnd only by th wv qution without ourc Eq.A-6 nd Eq.A-7, nd th olution cn b writtn ( v, ): ( ) / jωt + + Eq.A-5 v ( + jωt ) / Z Eq.A-5 9

102 . Appndi ( d) + jk ( d ) + Eq.A-54 ( d) jk ( d ) Whr + ( d) nd ( d) Eq.A-55 r contnt which cn b dcidd by th boundry condition of th ytm, ω th ngulr frquncy, nd k coutic longitudinl wv numbr in dirction in th mpl outid th nr-fild. It i clr tht th wv numbr nd coutic chrctritic ritnc inid nd outid th nr-fild in th mpl r th m, tht i k k nd Z Z. Th uffi t th top right cornr indict th r outid th nrfild but till in th mpl. Th finl olution of th coutic wv cn b obtind by mtching of boundry condition in diffrnt r. Th coutic boundry condition for th typicl ytm hown in Fig.- r gnrlly th m. if ; Eq.A-56 if d; Eq.A-57 v v if d; Eq.A-58 if D v v if D Eq.A-59 Eq.A-6 ( D) R + ( D ) jk D if D + D Eq.A-6 hr R i th coutic rflct contnt of th longitudinl wv t th intrfc btwn PZT trnducr nd th coppr lctrod. Th bckid of coppr lctrod i wll mtchd couticlly o tht thr i no mor rflction ftr th coppr lctrod. Although th SNAM bd on SPM tchniqu h contnt forc of th tip t, for th dicuion of hrmonic olution of coutic wv, it ply no rol for th hrmonic olution. At thi b, gnrlly dicuion of olution of diffrnt cnning nr-fild coutic microcopy ytm i poibl. At boundry, thr i only on boundry condition Eq.A-56 for + + () + () + () + + () + () + () + ( d) ( d) jkd jkd, Eq.A-6 9

103 . Appndi At th boundry d, thr r boundry condition Eq.A-57 nd Eq.A-58 for both ( v, nd ( v, ): ) + ( d) ( d) jkd + + () jkd jkd + ( d) ( d) + + ( d) + ( d) + ( d) jkd + + () jkd + ( d) ( d) + + ( d) Eq.A-6 Z {[ + jkd + jkd jkd ( d) + () ] - [ ( d) + ( d) ]} Z { + ( d) - ( d) } + ( d) jkd + + () jkd - ( d) ( d) + - ( d) Eq.A-64 At boundry D, th boundry condition Eq.A-59 nd Eq.A-6 r + ( D) { + R jk D } ( d) jk ( D d ) + + ( d) jk ( D d ) Eq.A-65 Z Z + ( D) { - R jk D } ( d) jk ( D d ) + - ( d) jk ( D d ) Eq.A-66 + If th ourc fild in nr-fild i givn, ( d) nd () r two ourc intgrtion trm in nr-fild nd r givn from Eq.A-5 nd Eq.A-5 rpctivly. Thr r i contnt + (), ( d), + ( d), ( d), + D ), nd D ). Thr r i indpndnt linr qution from Eq.A-6 to Eq.A-66. Thr i only on ingl olution ( ( + for ll th contnt. All th contnt cn b writtn th function of ( d) nd (). Lt α + th following: From Eq.A-59: jkd, jkd R α + R jk D jk D, Th finl prion of + D ) cn b clcultd + E jkζ () ( Z ) dζ ( d), c t + w cn gt th prion of ( d) : + d E jkζ ( d) ( Z ) dζ c t p p ( 9

104 . Appndi From Eq.A-6, d E jkζ () ( Z ) dζ ( d), c t w cn gt th prion of () : d E jkζ () ( Z ) dζ. c t p p + If th ourc fild i givn, th trm ( d) nd () r obtind. From Eq.A-7, w cn gt th trm + () : + () [ () + ( d) jkd ] If w dd nd ubtrct Eq.A-7 with Eq.A-7, w cn gt th trm ( d) + nd ( d) : + + ( d) ( d) jkd + + () jkd ( d) ( d) If w put th trm + () into th trm + ( d) bov, w gt th following prion: + ( d) + ( d) + [ ( d) - () ] jkd jkd Th trm t th right id of th qution bov i th ourc intgrtion in nr-fild. If w dd nd ubtrct Eq.A-74 with Eq.A-75, w cn gt th following prion: + ( D) {[ + jk R D ]+ β [- R jk D ]} ( d) + jk ( D d ) + ( D) {[ + jk R D ]-β [- R jk D ]} ( d) jk ( D d ) W cn limint th ( d) + nd ( d) trm in th qution bov nd gt th rltionhip btwn th trm + D ) ( nd th ourc trm: + [ + ( d) ()] ( D) jkd jkd ( + )( + R ) + α α β + jkd Hr α jkd α jkd, R + R jk D Eq.A-67 cn b implifid th following: jkd, nd β Z Z Z Z. jkd Eq.A-67 94

105 . Appndi [ ( d) ()] 4 + jkd Z c p E d jkζ jkζ jkd dζ t 4j Z c p E d jkζ jkζ jkd dζ t j 4j Z c p E d jkd in( kζ) dζ t 4 jz + ( D) jkd jkd ( + )( + R ) c d p E t + α α in( kζ) dζ Th output ignl of th trnducr undr th opn circuit condition i: β V output ( d )( c p E t PZT jk in( kζ) dζ ) D ε PZT ( )( ω jkd + jkd β )( + α α β ) Eq.A-68 A: Solution of SEAM A. Th coutic boundry condition of SEAM cn b writtn : if ; Eq.A-69 if d; Eq.A-7 v v if d; Eq.A-7 if D v v if D Eq.A-7 Eq.A-7 ( D) R + ( D ) jk D if D + D Eq.A-74 A. Th ourc lctric fild in th nr-fild of SEAM nd output ignl of trnducr Th ourc lctric fild in th nr-fild of SEAM cn b obtind by uing th img chrg nd olving th Poion qution t th givn lctric boundry condition in th mpl. Bd on th umption in Chptr 4., th lctric fild undr vry cnning point cn b writtn [74]: E A jωt ( d) ( <<d, + y ) E ( >d nd <, + y ) Eq.A-75 ( η hr A π ε E ) I d τ 95

106 . Appndi in which η E i th bckcttring contnt, I th bm currnt, τ th chrg ccumultion tim, i th rdiu of th primry lctron bm. c p E t A( d) j t ω, nd A jω c p A jω c p ( η π ε E ) I d τ Th ourc trm in nr-fild of SEAM i: d E kζ) dζ c p t in( A [in( kd) kd] k Eq.A-76 Th output ignl of PZT trnducr undr opn circuit condition i: PZT jk V output A [in( kd) kd] k D ε PZT ( )( ω jkd + β )( + α α β jkd ) j c p ( η π ε E ) I d τ PZT jk [in( kd) kd] k D ε PZT ( jkd + β )( + α α β jkd ) j ρε E ( η π ) I d τ PZT in( kd) kd ( ) ε PZT ω + jk D ( jkd )( β + α α jkd β ) Eq.A-77 + jkd in which ρ i th dnity of BTiO mtril, α jkd β Z. Z α jkd, R + R jk D nd A4: Solution of SNAM dvlopd from SPM A4. Th coutic boundry condition of th ytm Th boundry condition for SNAM dvlopd in thi work r lmot th m tho of SEAM ytm. Th only diffrnc i t pln. Th ctul coutic boundry condition r: F if ; Eq.A-78 tip if d; Eq.A-79 v v if d; Eq.A-8 96

107 . Appndi if D v v if D Eq.A-8 Eq.A-8 ( D) R + ( D ) jk D if D + D Eq.A-8 hr R h th m mning tht dicud bfor. Th contct forc F tip i kpt contnt by th fdbck control unit of SPM nd h typicl vlu from vrl nn to vrl mn. It i ctully contnt forc which ctd on th boundry of th nr-fild. For th hrmonic olution of nr-fild coutic, it ply no rol in th olution, th tim diffrntil i ro. A4. Grn function of th modl Th lctric fild ditribution of th ytm cn b obtind by olving th Poion qution with point chrg undr th givn lctric fild boundry condition of th implifid modl hown in Fig.5-. If th potntil in th rgion - << i V nd in th rgion <<D i V, th lctric fild boundry condition cn b writtn : At : VV V V 97 Eq.A-84 Eq.A-85 At D : V Eq.A-86 Th fild ditribution of point chrg t (,,-h) in cylindricl coordint ytm cn b obtind [8~8]: q V 4πε J β + h ( βr dβ ) Eq.A-87 Hr th J ( β ) i th Bl function of ro ordr. β i th ptil priodicl contnt in r dirction. Th potntil V i lo clld th Grn function of point chrg in cylindricl coordint ytm. Th Grn function for th implifid SNAM ytm cn b writtn : In th rgion - <<; q β + h β V [ J ( βr) dβ + A( β) J ( βr) dβ 4πε In th rgion <<D ; ] Eq.A-88

108 . Appndi q β β V [ B( β) J ( βr) dβ + C( β) J ( βr) dβ 4πε ] Eq.A-89 In Eq.A-88 nd Eq.A-89, A (β), B (β), nd C (β) r contnt which r only dpndnt on th ytm gomtry nd vribl β. Th contnt cn b olvd by th thr indpndnt qution from lctric boundry condition Eq.A-84, Eq.A-85, nd Eq.A-86. Th contnt r olvd in th following wy. According to Eq.A-84 nd Eq.A-85: At : + A (β) B (β) + C (β) Eq.A-9 βh - βh + A (β) [- B (β) + C (β) ] r According to Eq.A-86, t D : ε Eq.A-9 B(β) βd D + C(β) β Eq.A-9 If w lt η ( ε ( ε r r ) + ), th contnt (β) Eq.A-9, Eq.A-9, nd Eq.A-9. Th contnt r: A(β) η βh + η βd A, B (β), nd C (β) cn b obtind by olving β( D + h) + η βd Eq.A-9 B(β) ( ε βh r + ) + η βd Eq.A-94 C (β) D β ( ε βh r + ) + η βd 98 Eq.A-95 Th Grn function for th ytm hown in Fig.5- r obtind ccording to Eq.A-88 nd Eq.A-89 if ll th contnt r known. To th phyicl mning of th Grn function mor clrly, w cn dicu th Grn function in th following wy. Th Grn function in rgion - <<, ccording to Eq.A-88: q V 4πε J β + h q ( βr dβ + 4πε ) β A( β) J ( βr dβ ) Th firt trm of V i th potntil of point chrg q t (,, -h). Th cond trm of V i furthr compod of two trm:

109 . Appndi Th firt prt of q 4πε β A( β) J ( βr dβ ) i: q 4πε (- η ) βh + η βd J ( β β r) dβ According Tylor ri + η Th trm + η βd βd [ η m m m m..., ± cn b pndd into Tylor ri [88]: βd ( D η β ) D β Eq.A-96 (η )...] Eq.A-97 Thn, th firt prt of q 4πε β A( β) J ( βr dβ ) i: q 4πε (- η ) βh + η βd J ( β β r) dβ q 4πε q β( h) (- η ) J ( βr) dβ + 4πε η J ( βr) β[ (D + h)] dβ + q 4πε ( η β( 4 D + h ) +...) β β J ( r) dβ Eq.A-98 Th trm t th right id of Eq.A-98 cn b trtd img chrg Q m - η q, Q m η q,... t th poition of (h, D +h, 4D +h,...) t i. Th cond prt of q 4πε β A( β) J ( βr dβ ) i: q 4πε (-) β( D + h ) + η βd J ( β β r) dβ According to Eq.A-97, it cn b writtn th following: q 4πε β[ (D + h)] (-) J ( βr) dβ + q 4πε η J ( βr) β[ (4D + h)] dβ q + 4πε [ η β( 6 D + h ) +...] β β J ( r) dβ Eq.A-99 99

110 . Appndi In th m wy, trm t th right id of Eq.A-99 cn b trtd th img chrg Q n -q, Q n η q..., t th poition of (D +h, 4D +h,...) t i. Th ditribution of img chrg for th Grn function of V i hown in Fig.5-. Th Grn function V in th rgion <<D ccording to Eq.A-89 i: q V 4πε β q B( β) J ( βr) dβ + 4πε β C( β) J ( βr dβ ) From Eq.A-97, th firt trm of V cn b writtn : q 4πε β q B( β) J ( βr) dβ 4πε ( ε r + ) J ( βr) β( + h) dβ q 4πε η ( ε r ) + J ( βr) β[ + (D + h)] dβ q 4πε ( ε r + ) [ βh ( ) β -...] J ( βr) dβ D η β Eq.A- Th img chrg r Q i...) t i. ( ε + r ) q, Q i - ( ε + r ) From Eq.A-97, th cond trm of V cn b writtn : q 4πε β q C( β) J ( βr) dβ (-) 4πε ( ε r + ) η q,... t th poition of (-h, -(h+d ), J β[ ( h+ D )] ( βr) dβ + q 4πε η ( ε r ) + J β[ ( h+ 4D ) ( βr) dβ + q 4πε [ η β( h+ D ) D ( β ) +...] β β J ( r) d β Eq.A- Th img chrg r Q j ( ε + r ) q, Q j ( ε + r ) η q,... t th poition of (D +h, 4D +h,...) t i. Th ditribution of img chrg for th Grn function of V i hown in Fig.5-. A5: Mtril contnt of PZT nd BTiO Th ltic tnor of BTiO ingl crytl with monodomin:

111 . Appndi t m /N, m /N, m /N, m /N, m /N, m /N c c c t c c c c c c c c 44 c 55 c 66 c 7.5 N/m, c 7.9 N/m, c 5. N/m c 6.5 N/m, c 44 c N/m, c 66. N/m If only th ltic or piolctric proprti of polrid BTiO crmic r tudid, th polrid BTiO crmic cn b dcribd mtril with 6mm crytl lttic nd th ltic tnor i: c c c t c c c c c c c c 44 c 55 c 66 c 5. N/m, c 6.6 N/m, c 6.6 N/m, c 4.6 N/m, c N/m Th ltic tiffn tnor of PZT crmic h th m form tht of BTiO crmic. Th vlu of tnor componnt of PZT crmic r: c.6 N/m, c 7.95 N/m, c 8.4 N/m, c.7 N/m,

112 . Appndi c 44. N/m Th piolctric tnor of BTiO for both ingl crytl nd crmic d t d d d d 5 d 5 d C/N, d C/N, d C/N t C/m,.7 C/ m, 5. C/ m Th piolctric tnor of PZT crmic h th m form. Th componnt vlu: PZT d C/N, PTT -6.5 C/m, PZT d PTT 9 C/ m, 59 - C/N, PTT C/ m Th dilctric tnor of BTiO mtril i: ε t ε ε ε, ε 9 ε, ε 68 ε PZT d C/N hr th t th top right cornr dnot th dilctric tnor undr contnt tr. ε t ε ε ε, Hr th t th top right cornr dnot th dilctric tnor undr contnt trin. Th dilctric tnor of PZT crmic h th m form tht of BTiO crmic but diffrnt vlu of tnor componnt. Th corrpondnt vlu r: ε, PZT, ε ε 4 PZT, ε ε, PZT, ε 7 PZT, ε ε 47

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116 . Rfrnc Schwi) 7... Liu, R. Hidrhoff, H.-P. Abicht, nd L. J. Blk: Chrctrition of frrolctric domin by th u of cnning nr-fild coutic microcopi, Anlyticl Scinc, ccptd for publiction 8... Liu, B. Y. Zhng, Q. R. Yin, nd L. J. Blk: Scnning lctron coutic microcopy for th vlution of domin tructur in BTiO ingl crytl nd crmic, Journl of Mtril Scinc, (998) J. F. Ny: Phyicl of crytl, Oford Univrity Pr. 4. A. F. Dvonhir, Thory of Brium Titnt - Prt, Phil. Mg. 4 (949) V. A. Zhirnov: Sov. Phy. JETP 5 (959), E. V. Chnki: Sov. Phy.-Solid St. 4 (97), J. Bouillot nd R. Mchl: Proc. Europ. Mtg on Frrolctricity, Srbrückn, Wi. Vrlg, Stuttgrt 969, J. C. Burfoot nd G. W. Tylor: Polr dilctric nd thir ppliction, 979 Mcmilln Pr Ltd. 45. V. Holy U. Pitch, nd T. Bumbch: High rolution -ry cttring from thin film nd multilyr, Springr Vrlg, B. M. Prk, S. J. Chung, H. S. Kim, W. Si, nd M. Dudly, Synchrotron whit-bm - ry topogrphy of frrolctric domin in BTiO ingl crytl, Philoophicl Mgin A, 75 (), 997, C. L. Ji, K. Urbn, M. Mrtin, S. Hoffmnn, nd R. Wr, Th tructur nd formtion of nnotwin in BTiO thin film, Philoophicl Mgin A, 77(4) (998), I. T. Kim, J. W. Jng, H. J. Youn, C. H. Kim, nd K. S. Hong, 8 frrolctric domin in polycrytllin BTiO thin film, Appl. Phy. Ltt. 7() (998), R. C. Millr: Opticl Hrmonic Gnrtion in Singl Crytl BTiO[ub], Phy. Rv. 4 (964), A-A8 5. M. Flörhimr, R. Pchott, U. Kubitchck, Ch. Brillrt, D. Hofmnn, L. Hur, G. Schribr, C. Vrbk, W. Sohlr, nd H. Fuck: Scond-hrmonic imging of frrolctric domin in LiNbO with micron rolution in ltrl nd il dirction, Appl. Phy. B 67 (998)

117 . Rfrnc 5. I. I. Smolyninov nd C.C. Dvi: Nr-fild opticl tudy of photorfrctiv urfc wv in BTiO, Optic Lttr, 4 No.9, (999), S. N. Zhu nd W. Co, Imging of frrolctric domin in LiTiO by nvironmnt cnning lctron microcopy, Frrolctric, (999), B. A. Auld: Acoutic fild nd wv in olid, John Wily & Son, Inc S. W. Mk nd B. A. Auld: Opticl nd coutic dvic ppliction of frroltic crytl, Advnc in lctronic nd lctron phyic, 7 (998) E. Brndi nd A. Roncwig, Thrml-wv microcopy with lctron bm, Applid Phyic Lttr, 7, (98), G. S. Crgill, Ntur, 86, (98), L. J. Blk: Advnc in Elctronic nd Elctronic Phyic, 7, (988) J. Cntrll nd M. Qin, Piolctric dtction of ignl in cnning lctron coutic microcopy, Mtril Scinc nd Enginring, A (989) L. J. Blk: Scnning Acoutic Microcopy, Anlyi of Microlctronic Mtril nd Dvic, Ed. by M. Grrbur nd H. W. Wrnr, John Wily & Son Ltd. (99) B. Y. Zhng nd Q. R. Yin: Piolctric coupling ffct of frrolctric in cnning lctron coutic imging, Progr in nturl cinc 6(upplmnt), 996, S-~S-6 6. S. W. Fu, S. Y. Zhng, Z.S. Wu, L. B. Luo, Z. Zhng, nd H. I Chn, Photocoutic invtigtion of chmicl rction by piolctric cylindricl tub trnducr, Progr in nturl cinc 6(upplmnt), 996, S-54~S F. G. Stkiwic, J. C. Murphy nd L. C. Amodt: in Tchicl digt of th Fourth Intrntionl Topicl Mting on Photocoutic, Thrml, nd Rltd Scinc (Montrl, 985) 6. D. N. Ro, H. R. Turnr, K. Lgg: in Tchicl digt of th Fourth Intrntionl Topicl Mting on Photocoutic, Thrml, nd Rltd Scinc (Montrl, 985) 64. P. Günthr, U. Ch. Fichr, nd K. Drnfld: Scnning nr-fild coutic microcopy, Applid Phyic, B48, (989), U. Rb nd W. Arnold: Acoutic Microcopy by tomic forc microcopy, Applid Phyic Lttr, 64(), (994),

118 . Rfrnc 66. M. Mywld, G. Brockt, nd L. J. Blk: Scnning Prob Acoutic Microcopy, Int. of Phy. Conf. Sr. No. 46 (995) U. Grunwld, K. Brtk, T. Antrck, Appliction of th ndl nor for microtructur murmnt nd tomic forc microcopy, Thin Solid Film, 64, (995), A. Michl, F. Minn, T. Murdfild, W. Göhd, U. C. Fichr, E. Bckmnn, nd H. Fuch: MH qurt lngth tnion rontor prob for cnning nr-fild coutic microcopy, Thin Solid Film, 64, (995), T. Hjdl, E. Chill, nd H. J. Fröhlich: Scnning coutic forc microcop invtigtion of urfc coutic wv, Surfc nd Intrfc Anlyi, 5, (997), R. M. Crmr, V. Biltki, P. Lpidi, L. J. Blk: Suburfc nly of dfct in intgrtd dvic by cnning prob coutic microcopy, Microlctronic Rlibility 9 (999) F. Ohuchi, D. E. Clrk, L. L. Hnch: Effct of crytllition on th Augr lctron ignl dcy in n LiO SiO gl nd gl-crmic, J. Am. Crm. Soc., 6 (979), F. Ohuchi nd P. H. Hollowy. Gnrl modl of odium dorption nd diffuion during lctron bombrdmnt of gl. Journl of Vcuum Scinc nd Tchnology, (98), L. Rimr: Scnning Elctron Microcopy, Spring-Vrlg, J. Cu: Som conidrtion on th lctric fild inducd in inultor by lctron bombrdmnt, Journl of Applid Phyic, 59(5), 986, S. G. Qin, S. Y. Zhng, J. C. Chng, nd Z. Q. Wng, Thr dimnionl thory of cnning lctron-coutic imging, 6(upplmnt), 996, S-7~S- 76. Vlvo: Piooid (PE), Photo Copi Gmbh, Hmburg, (984) 77. C. Durkn nd M. E. Wllnd: Invtigtion into locl frrolctric proprti by tomic forc microcopy, Ultrmicrocopy, 8, (), C. H. Ahn, T. Tybll, L. Antogn, K. Chr, R. H. Hmmond, M. R. Bly, Ø. Fichr, nd J.-M. Tricon: Locl, nonvoltil lctronic writing of pitil Pb(Zr.5Ti.48)O/SrRuO htrotructur, Scinc 76, (997), - 8

119 . Rfrnc 79. S. Blidi, P. Girrd, nd G. Lvqu: Elctrottic forc cting on th tip in tomic forc microcopy: Modulition nd comprion with nlytic prion, Journl of pplid Phyic, 8(), 997, - 8. J. A. Strtton: Elctromgntic thory, McGrw-Hill Book compny, Inc W. R. Smyth: Sttic nd dynmic lctricity, McGrw-Hill Book compny, Inc V. W. Hnn: Numricl olution of ntnn in lyrd mdi, John Wily & Son Inc Nnonor Dr. Olf Woltr GmbH, Wtlr-Blnknfld, Grmny 84. S. Hirkorn, U. Rb, nd W. Arnold: Thorticl dcription of th trnfr of vibrtion from mpl to th cntilvr of n tomic forc microcop, Nnotchnology, 8 (997), U. Rb, S. A. Mlio, E. Ktr, V. Schrr, S. Hirkorn, W. Arnold: Quntittiv dtrmintion of contct tiffn uing tomic forc coutic microcopy, Ultronic 8 (), H. T. Lnghmmr, T. Müllr, A. Polity, K.-H. Flgnr, nd H.-P. Abicht: On th crytl nd dfct tructur of Mngn-dopd brium titnt crmic, Mtril Lttr, 6, 996, Ntionl tchnology rodmp for miconductor, Autin: SEMATECH Inc I. N. Brontin nd K. A. Smndjjw: Tchnbuch dr Mthmtik, Q. R. Yin, J. Lio, B. Y. Zhng, Y. Yng, nd F. M.Jing, Elctron coutic imging of frrolctric domin nd mchnim nlyi on BTiO crmic, Frrolctric (999), -8 9

120 Publiction nd prnttion Publiction nd prnttion Prt of thi work hv bn publihd in intrntionl journl or prntd t th intrntionl confrnc, cintific minr, nd invitd cintific tlk. Th following i lit of th publiction nd prnttion of th uthor in rcnt yr.. B. Y. Zhng,.. Liu, Q. R. Yin, nd L. J. Blk,: Scnning nr fild coutic microcopi for th vlution of polycrytllin mtril, Acouticl Imging,, (997) Liu, L. J. Blk, H.-P. Abicht, A. Ecku, nd Q. R. Yin: Scnning coutic microcop for th invtigtion of frrolctric proprti of frrolctric mtril, Solid Stt Phnomn, 6-64, (998), Liu, B. Y. Zhng, Q. R. Yin, nd L. J. Blk: Scnning lctron coutic microcopy for th vlution of domin tructur in BTiO ingl crytl nd crmic, Journl of Mtril Scinc, (998) Liu, R. Hidrhoff, H.-P. Abicht, nd L. J. Blk: Chrctrition of frrolctric domin by th u of cnning nr-fild coutic microcopi, Anlyticl Scinc, Vol.7, Spcil Iu No Liu, R. Hidrhoff, H.-P. Abicht, nd L. J. Blk: "Dynmic chrctrition of frrolctric domin of BTiO by th u of cnning nr-fild coutic microcopy", ccptd for prnttion t "Intrntionl Confrnc on Mtril for Advncd Tchnologi" (ICMAT,.-6.7., Singpor), to b publihd in Mtril Chmitry nd Phyic Prnttion 6. L. J. Blk, G. Fig, nd.. Liu: Imging of Thrml nd Mchnicl Structur by Scnning Forc Microcopy, Invitd Tlk t "SPMM 97, Intrntionl Confrnc on Scnning Prob Microcopy Microytm" ( , Wroclw, Polnd) 7... Liu nd L. J. Blk: Scnning coutic microcop for th invtigtion of frrolctric mtril, prntd t " th Confrnc on Dvlopmnt nd Tchnologicl Appliction of Scnning Prob Mthod (SM)", from Sp. 4 to 7, Liu: Chrctrition of frrolctric domin by th u of cnning nr-fild coutic microcopy, Prntd on Jn. 7,, in Fchbrich Elktrotchnik und Informtiontchnik, BUGH Wupprtl

121 Publiction nd prnttion 9... Liu: Chrctrition of frrolctric domin by th u of cnning nr-fild coutic microcopy, Prntd on Fb. 4,, in Fchbrich Phyik, Mrtin-Luthr- Univrität Hll-Wittnbrg... Liu: Chrctrition of frrolctric domin by th u of cnning nr-fild coutic microcopy, Prntd on Fb. 5,, in Bundntlt für Mtrilforchung und Prüfung, Brlin... Liu: Chrctrition of frrolctric domin by th u of cnning nr-fild coutic microcopy, Prntd on Jul. 5,, in Dprtmnt of Elctricl Enginring nd Informtion, Ntionl Univrity of Singpor... Liu: Chrctrition of frrolctric domin by th u of cnning nr-fild coutic microcopy, Prntd on Jul.,, in Shnghi Intitut of Crmic, Chin Acdmy of Scinc