Piezoelectric anisotropy of a KNbO3 single crystal

Transcription

1 Piezelei anispy f a KNbO3 single ysal Linyun Liang, Y. L. Li, S. Y. Hu, Lng-Qing Chen, and Guang-Hng Lu Ciain: J. Appl. Phys. 18, (21); di: 1.163/ View nline: hp://dx.di.g/1.163/ View Table f Cnens: hp://jap.aip.g/esue/1/japiau/v18/i9 Published by he Ameian Insiue f Physis. Relaed Ailes Sess-nlled Pb(Z.52Ti.48)O3 hik films by hemal expansin mismah beween subsae and Pb(Z.52Ti.48)O3 film J. Appl. Phys. 11, (211) Elei field nlled magneizain ain in exhange biased anifemagnei/femagnei/piezelei mpsies Appl. Phys. Le. 99, (211) Enhaned piezelei espnse f BaTiO3 KNbO3 mpsies Appl. Phys. Le. 99, 2292 (211) A upled analysis f he piezespnse fe mispy signals Appl. Phys. Le. 99, (211) Celain beween dielei ppeies and hemial mpsiin f he umaline single ysals Appl. Phys. Le. 99, (211) Addiinal infmain n J. Appl. Phys. Junal Hmepage: hp://jap.aip.g/ Junal Infmain: hp://jap.aip.g/abu/abu_he_junal Tp dwnlads: hp://jap.aip.g/feaues/ms_dwnladed Infmain f Auhs: hp://jap.aip.g/auhs Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins

2 JOURNAL OF APPLIED PHYSICS 18, Piezelei anispy f a KNbO 3 single ysal Linyun Liang, 1,3 Y. L. Li, 2 S. Y. Hu, 2 Lng-Qing Chen, 3 and Guang-Hng Lu 1,a 1 Depamen f Physis, Beijing Univesiy f Aenauis & Asnauis, Beijing 1191, China 2 Paifi Nhwes Nainal Labay, Rihland, Washingn 99352, USA 3 Depamen f Maeials Siene & Engineeing, The Pennsylvania Sae Univesiy, Pennsylvania 1682, USA Reeived 27 July 21; aeped 2 Obe 21; published nline 11 Nvembe 21 Oienain dependene f he lngiudinal piezelei effiiens d 33 f a KNbO 3 single ysal has been invesigaed as a funin f empeaue by using he Landau Ginzbug Devnshie phenmenlgial hey. I is shwn ha he maximum f d 33 is n always alng he plaizain diein f he feelei. The enhanemen f d 33 alng a nnpla diein is aibued a feelei ansiin a whih a plaizain hanges is diein. In he eagnal, he maximum f d 33 a high empeaues is alng he eagnal pla diein and hen hanges is diein wad he pla diein f he hhmbi when he empeaue is lse he eagnal-hhmbi ansiin. The maximum f d 33 f he hhmbi depends n bh he high-empeaue and lw empeaue feelei ansiins. In he hmbhedal, he maximum f d 33 is elaively insensiive empeaue due he absene f any fuhe ansiins in he lw empeaue egime. These esuls an be genealized he ansiins indued by exenal elei field, pessue, and mpsiin vaiains. 21 Ameian Insiue f Physis. di:1.163/ I. INTRODUCTION Pb Z,Ti O 3 PZT eamis ae widely used in aua devies due hei exellen piezelei ppeies nea he mphpi bunday MPB. 1,2 Hweve, hee has been ineasing inees in lead-fee maeials due envinmenal nens. Candidae maeials f lead-fee piezelei eamis inlude BaTiO 3, Bi.5 Na.5 TiO 3, Bi.5 K.5 TiO 3, Na.5 K.5 NbO 3, and KNbO As a simple pevskie, BaTiO 3 has been inensively sudied bh in single ysal and eami fms f many yeas. Hweve, muh less is knwn abu KNbO 3, f whih he feeleiiy was fis bseved abu 6 yeas ag. The main easn may me fm he diffiulies f gwing KNbO 3 single ysals and hen pling a mulidmain suue in single-dmain a m empeaue. 7 Fuheme, he high ansiin empeaue beween eagnal and hhmbi feelei s indues fuhe mplexiy due addiinal dmain suue fmain upn ling. As a esul, i is diffiul measue he empeaue dependene f spnaneus plaizain and he single dmain dielei ppeies a highe empeaues. Reen disvey f exepinally lage piezelei espnses alng nnpla dieins in lead-based elaxfeelei slid sluins 8 has geneaed inees in simple pevskie feelei maeials suh as BaTiO 3 and KNbO 3. These lassi feelei maeials may display effes simila mplex slid sluins. The sudies f simple suue maeials an avid mplexiies assiaed wih messpi suues f elax-feelei slid sluins. F he eagnal BaTiO 3 single ysal, a highe piezelei espnse was bseved alng n-pla 111 a Eleni mail: lgh@buaa.edu.n. dieins, whee he supesip efes he ubi. 9 While in he hhmbi, he highes piezelei espnses wih d 33 ve 5 pc/n wee bseved when an elei field was applied alng 1 n-pla diein. 1 Hweve, his exellen piezelei pefmane f mnlini BaTiO 3 ysal ann be used a m empeaue sine he mnlini f BaTiO 3 is sable belw 5 C. Due a lage piezeleiiy and a high Cuie pin, 11,12 KNbO 3 eamis has been nsideed as ne f he andidae maeials f fuue lead-fee piezelei appliains. The elemehanial upling fa f he hikness-exensinal mde, k, in a KNbO 3 ysal eahes as high as.69 f he aed X-u abu he y-axis, whih is he highes amng all uen knwn piezelei maeials. 11 Meve, he pedied lngiudinal piezelei upling fa f he widh-exensinal mde, k 11,isas high as 82.4% f he 43.5 aed Z-u plae abu he y-axis. This value is mpaable ha f Pb Zn 1/3 Nb 2/3 O 3 PbTiO Wiesendange 14 measued d 15 and d 24 f KNbO 3 single-dmain ysals by applying a esnane-aniesnane mehd. Gune 15 measued d 31, d 32, and d 33 f KNbO 3 single-dmain ysals using he quasisai mehd. Zgnik e al. 16 alulaed he mplee se f piezelei d ij nsans. Wada e al. 12 invesigaed he piezelei ppeies f KNbO 3 ysals alng he pla 11 diein and 1 f engineeed dmain diein. Piezelei effiien d 31 alng nnpla 1 diein was 2.8 imes highe han ha alng he pla 11 diein. I is nied ha all expeimens wee pefmed a m empeaue and hee ae signifian disepanies amng hem. Fuheme, hee is a lak f mplee piez /21/18 9 /94111/9/$3. 18, Ameian Insiue f Physis Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins

3 Liang e al. J. Appl. Phys. 18, elei effiiens f all hee feelei s and hei dependene n ysal ienain, empeaue, elei field, and pessue. The enhanemen f piezelei effiiens was aibued he plaizain ain unde an exenal elei field, dmain wall nibuins in engineeed dmain suues, 17 and ansiins indued by empeaue exenal field. 18 The piezelei espnse an be analyzed using he Landau Ginzbug Devnshie LGD phenmenlgial appah. 19,2 F example, he ienain dependenes f lngiudinal piezelei effiien a vaius empeaues, sess, and mpsiin wee analyzed f BaTiO 3, PbTiO 3, and PZT maeials. 6,21 In his pape, we emply he LGD hemdynami phenmenlgial hey alulae and pedi he piezelei effiiens in he hee feelei s f KNbO 3 ysal unde vaius empeaues and hei ienain dependene f he lngiudinal piezelei effiien d 33. The enhanemen in piezelei effiiens indued by empeaue alng pla and n-pla dieins was disussed in em f plaizain ain and nain, and he flaening f he fee enegy landsape nea a ansiin. II. PHENOMENOLOGICAL THEORY In he famewk f he LGD-ype phenmenlgial hey, he fee enegy funin is expanded as a plynmial f he mpnens f plaizain P= P 1, P 2, P 3. An eighhde plynmial is emplyed desibe he fee enegy f KNbO 3 single ysal sine a six-de plynmial is n enugh desibe he hee ansiins in he feelei empeaue egime unless we assume he high-de expansin effiiens depends n he empeaue. Using he fee enegy f he paaelei as a efeene, he fee enegy f KNbO 3 single ysal an be expessed as f LGD P 1,P 2,P 3 = f hem + f = 1 P P P P 1 + P P P 1 2 P P 2 2 P P 1 2 P P P P P 4 1 P P P 4 2 P P P 4 3 P P P 2 1 P P P P P P 1 6 P P P 2 6 P P P 3 6 P P P 1 4 P P 2 4 P P 1 4 P P 1 4 P 2 2 P P 2 4 P 3 2 P P 3 4 P 1 2 P s s s Q 11 1 P P P 3 2 Q 12 1 P P P P P P 2 2 Q 44 4 P 2 P P 1 P P 1 P 2, 1 whee wih subsip index epesens enegy expansin effiien, i is sess mpnen, s ij is elasi mpliane a nsan plaizain, and Q ij is he elesiive upling effiien beween plaizain and sess. The dielei siffness effiien ij an be bained via a send paial deivaive f he fee enegy funin as ij = 2 f LGD / P i P j, i, j = 1,2,3. 2 The dielei susepibiliy effiiens ij an be deemined fm he eipal f he dielei siffness ij using he fllwing elain, ij = A ji /, i, j = 1,2,3, 3 whee A ji and ae he fa and deeminan f he ij maix. The piezelei effiien g ij epesening he upling beween plaizain and sess is deived fm he fllwing equain: g ij = 2 f LGD / P i j, i = 1,2,3, j = 1,2,3,4,5,6. 4 The piezelei effiiens d ij epesening he upling beween plaizain and sain an be alulaed by d ij = ik g kj, k = 1,2,3. 5 I is nvenien wie piezelei effiiens in he ubi ysallgaphi dinae sysem. We use a supesip indiae he ubi ysallgaphi dinae sysem. The piezelei effiiens f he hee feelei s in he ubi ysallgaphi dinae sysem have he fllwing fms: 22 F he eagnal wih P 1 = P 2 =, P 3 = P 3T, d 33T d 31T d 15T d 11T d 21T =2 33T Q 11 P 3T, = d 32T =2 33T Q 12 P 3T, = d 24T = 11 Q 44 P 3T, = d 12T = d 22T = d 13T = d 23T = d 14T = d 25T = d 16T =, = d 26T = d 34T = d 35T = d 36T =. 6 F he hhmbi wih P 1 =, P 2 = P 3 = P 3O, d 15O d 24O d 31O d 32O d 33O = d 16O = d 34O = d 21O = d 23O = d 22O = 11O = 33O =2 32O Q 44 P 3O, + 32O =2 Q 11 32O =2 Q 12 32O + 33O Q 44 P 3O, Q 12 P 3O, + Q 12 33O + Q 11 33O P 3O, P 3O, Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins

4 Liang e al. J. Appl. Phys. 18, d 11O = d 12O = d 13O = d 14O = d 25O = d 26O = d 35O = d 36O =. 7 F he hmbhedal wih P 1 = P 2 = P 3 = P 3R, d 11R d 12R d 15R d 14R = d 22R = d 13R = d 33R = d 21R =2 11R = d 23R = d 31R + 12R Q 11 + Q 12 P 3R, = d 16R = d 24R = d 26R + 12R Q 44 P 3R, = d 25R = d 36R = d 34R Q 11 P 3R = d 32R Q 12 P 3R, =2 11R Q 12 P 3R = d 35R = 11R =2 12R Q 44 P 3R. 8 The subsips P=T,O,R f he dielei susepibiliy ijp, plaizain P ip, and piezelei effiien d ijp epesen he eagnal, hhmbi, and hmbhedal s, espeively. The supesip index indiaes ha physial quaniies ae measued in he ubi ysallgaphi dinae sysem. In he feging, he Vig s nain f he piezelei effiien d ij i=1,2,3; j=1,2,3,4,5,6 is used. Bu is ens nain d ijk i, j,k=1,2,3 has be emplyed f dinae ansfmain. Thei elain beween he Vig s nain and ens nain is d im =d ijk m= j=k=1,2,3 and d im =2d ijk j k, m=9 j k, m=4,5,6. The piezelei effiiens, d p ijk, in he new dinae ha is assiaed wih ne f he hee feelei s an be bained by he ansfmain fm he ubi ysallgaphi dinae sysem d lmn as 23 d p ijk = il jm kn d lmn, 9 whee ij is he elemen f he ansfmain maix ha desibes he ain fm he iginal ubi dinae sysem he new dinae sysem naed by he supesip p wih p=,,, epesening he dinae sysem assiaed wih he eagnal, hhmbi, and hmbhedal s, espeively. If a fuhe ain is made wih espe he feelei dinae, he piezelei effiien d p ijk defined in he aed dinae an be alulaed by d p p ijk = a il a jm a kn d lmn, 1 whee,, and ae he Eule angles and a ij is he elemen f he Eule maix a ha desibes he ain defined by he Eule angles. A auin is neessay when mpaing daa fm diffeen sues sine Eule angles ae n uniquely defined in he lieaue. Hee, desibes he fis unelkwise ain aund he iginal x 3 axis, is he send unelkwise ain aund he new x 1 axis and is he hid unelkwise ain aund he new x 3 axis. The ansfm maix a fm he Eule angles is given by 24 a = s s s sin sin s sin + s sin s sin sin sin s s s sin sin sin + s s s sin s sin sin sin s s. 11 Wih Eqs. 1 and 11 ne an alulae he ienain dependene f he piezelei effiien. III. TEMPERATURE DEPENDENCE OF PIEZOELECTRIC PROPERTIES T alulae he piezelei ppeies f KNbO 3 ysals, we use he effiiens f an eighh-de fee enegy funin deemined peviusly. 25 The elesiive effiiens and elasi mpliane nsans ae Q 11 =.12 m 4 /C 2, Q 12 =.53 m 4 /C 2, and Q 44 =.52 m 4 /C 2 and s 11 = m 2 /N, s 12 = m 2 /N, and s 44 = m 2 /N, espeively. 15,26,27 The dielei pemiiviy p p ij is elaed he dielei susepibiliy ij wih p ij =1+ p ij p ij. The empeaue dependene f dielei susepibiliy effiiens p ij and piezelei effiiens d p ij f KNbO 3 ysals f all hee feelei s in he ubi ysallgaphi dinae sysem an be alulaed by using Eqs. 3 and 5. By aing dinae sysem fm he ubi ysallgaphi dinae sysem, ne an bain p ij and d p ij in he new dinae sysem as shwn in Figs. 1 and 2, espeively. The new dinae axes ae expessed by he iginal sysem ae 1 = 1, 1 = 1, and 1 = 1 f he eagnal, 1 = 1, 1 = 11, and 1 = 11 f he hhmbi, and 1 = 11, 1 = 2 11, and 1 = 111 f he hmbhedal, espeively. p ij Rhmbhedal 33 Ohhmbi = Teagnal Tempeaue T ( C) FIG. 1. Cl nline Calulaed empeaue dependene f he dielei susepibiliy effiiens p ij f KNbO 3 single ysals in all hee feelei s. Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins

5 Liang e al. J. Appl. Phys. 18, Rhmbhedal Ohhmbi Teagnal 4 d 24 d p (p/n) ij 2 d 15 d 15 d 15 d 33 d 33 d 33 d 32 d 31 d 32 d 31-2 d Tempeaue ( C) FIG. 2. Cl nline Calulaed empeaue dependene f piezelei effiiens d p ij f KNbO 3 single ysals in all hee feelei s. A. Teagnal In he eagnal, he KNbO 3 ysal has 4mm symmey. Sine he dinae sysem f he eagnal is hsen as he same as he ne in he ubi, d ij =d ij. Afe a ain f angle wih espe he 1, he lngiudinal piezelei effiien d 33 in he aed dinae an be expessed as d * 33 = s d 15 sin 2 + d 31 sin 2 + d 33 s FIG. 3. Cl nline The ienain dependene f piezelei effiiens d 33 f KNbO 3 in he eagnal f hee seleed diffeen empeaues, a T=23 C; b T=3 C; and T=35 C. Angle max a whih maximum d 33 us is indiaed f eah empeaue. Thee dinae axes espnd x 1 =d 33 sin s, x 2 =d 33 sin sin, and x 3 =d 33 s. The numeial values maked n he axes have uni pc/n. Only he uppe half f he dinae spae is shwn. The Eule angle =45 is espnding he dinaes assiaed wih he hhmbi. We pl an ienain dependene f alulaed d * 33 f he eagnal f hee seleed empeaues 23, 3, and 35 C as shwn in Fig. 3. I is shwn ha he sufae f d * 33 hanges upn ling fm he Cuie empeaue. The diein f he lages d * 33 is alng 1 diein a 35 C hen hanges max =31.1 a 3 C, and finally max =49.8 a 23 C. Analyzing he expessin f d * 33, ne an easily see ha d * 33 is deemined by hee paamees d 33, d 31, and d 15, whih an be alulaed by Eq. 6 Fig. 2. I shws ha d 33 and d 15 inease apidly as he empeaue appahes he eagnal ubi and eagnal hhmbi ansiin empeaues while d 31 nly hanges slighly in mpaisn wih d 33 and d 15. The inease in d 15 wih ineasing empeaue is simila he behavi f he dielei pemiiviy in he ubi, whih ineases when he ysal is led wad he feelei. 28 Fm Eq. 6 i is easily seen ha d and d , in whih dielei susepibiliy is pependiula and paallel he pla diein in he eagnal, espeively. The alulaed dielei nsans in he eagnal ae shwn in Fig. 1, 11 and 33 exhibis ppsie behavis in he whle eagnal empeaue ange. This leads a maximum d * 33max alng he pla diein 1 in he high-empeaue ange. As he empeaue deeases wad he hhmbi, he lages d * 33max develps alng a diein he han 1. A pl f d * 33 as a funin f espnding angle a seveal seleed empeaues lealy shws he end f maximum d * 33max wih empeaue in Fig. 4. The maximum d * 33max alng 1 diein an be expeed a he high-empeaue nea he eagnal-ubi ansiin pin. As he empeaue is lweed, d 15 ineases and d * 33 gadually develps a lal minimum alng 1 diein and hen a glbal maximum a an angle away fm 1. Ne ha he lages d * 33max sill lies alng 1 max = diein in he empeaue ange f eagnal. Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins

6 Liang e al. J. Appl. Phys. 18, T = 23 C T = 3 C T = 43 C 6 (d + d )/d d * 33 (pc/n) -1-2 max deg (deg.) FIG. 4. Cl nline The piezelei effiien d 33 KNbO 3 as a funin f angle a vaius empeaues. in he eagnal F example, a he empeaue T=43 C, lse he ubi, he maximum d * 33max = pc/n is alng he diein 1, while a he empeaue T=23 C, lse he hhmbi, he maximum d * 33max =118. pc/n is alng he diein defined by max =49.8. The maximum d * 33max deeases wih empeaue abve abu 314 C, and hen ineases belw i. I is ineesing pin u ha i is diffeen fm he feelei maeials like BaTiO 3 he maximum d * 33max lies alng he pla diein nea he eagnal-ubi ansiin and has he lages value upn ling f empeaue wih he diein alng n-pla diein and PbTiO 3 he maximum d * 33max always lies alng he pla diein in he feelei and he maximum lses he eagnal-ubi ansiin. 24 The angle max f he maximum d * 33max an be alulaed by he elain f s 2 max = d 31 +d 15 /3 d 31 +d 15 d 33, whih is als pled in Fig. 4. Damjanvi e al. 29 nluded ha max shuld saisfy a ndiin max f all eagnal pevskie maeials. Ou alulains shw max appahes 5.28 lse he hhmbi. A he empeaue belw 314 C, whee d 31 +d 15 /d , max deviaes fm Fig. 5. The value f q q=3/2 Q 12 /Q 11 defined by Damjanvi 29 is alulaed as 1.9. I is als easily shwn ha he maximum d 33max aes fm he pla diein ne q 1.9 belw 314 C. In he eagnal, he signifian inease in he dielei susepibiliy alng he 1 1 axes as he hhmbi is appahed upn ling, whih implies ha he eagnal KNbO 3 bemes dieleially sf alng ysallgaphi dieins pependiula he plaizain diein 1. This ase is pedied heeially in maeials exhibiing empeaue diven feelei-feelei ansiins. 21 d 33 is mainly deemined by d 15 lse he eagnalhhmbi ansiin. F KNbO 3, alhugh he plaizain ain effe is sng in his empeaue ange, i sill des n dminae he piezelei espnse. In he high-empeaue ange appahing he ubi, d 33 bemes dminan while d 15 is elaively small. The enhaned d 33 appahing he eagnal-ubi ansiin an als be pedied fm he flaening f he fee enegy pfile Fig. 6 a. B. Ohhmbi In he hhmbi, he empeaue dependene f piezelei effiien d 33 in KNbO 3 is me mplex han he eagnal. The hhmbi has symmey mm2 pssessing w disin diffeen shea effiiens d 15 and d 24. In de sudy he ienain depen- dene f d 33 due he ansiin, he Eule angles ae hsen as = and and vaying abiaily. = /2 and =asin 1/ 3 espnds he dinae assiaed wih he hmbhedal while = and = /4 espnds he dinae assiaed wih he eagnal. Wih suh a dinae ansfm, d 33 an be expessed by d 33, = s d 15 + d 31 sin 2 sin 2 + d 24 + d 32 sin 2 s 2 + d 33 s Sine he ansfm maix beween he ubi and hhmbi dinae sysems is 2/2 2/2 2/2 2/2 ij = 1, 14 whee he elain f piezelei nsans in he w diffeen dinae sysems an be expessed via Eqs. 7 and 9 as d 15 = 2d15O, d 24 = 2 d 31 2 d 33O = 2d31O, d 32 = 1 d 33 = Tempeaue ( C) 2 d 33O 2 d 33O d 32O, + d 32O + d 32O d 24O, + d 24O, [1] FIG. 5. Cl nline The angle max indiaing diein alng whih d 33 he lages as a funin f empeaue and d 31 +d 15 /d 33 ai. is Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins

7 Liang e al. J. Appl. Phys. 18, f LGD (Jm ).2.1 P s (C/m 2 ) T=4 C T=35 C T=25 C f LGD (Jm ).8.4. P s (C/m 2 ) T=22 C T=25 C T=-3 C f LGD (Jm ) T=-7 C T=-15 C T=-25 C. P s (C/m 2 ) (a) (b) () FIG. 6. Cl nline Calulaed LGD-fee enegy as a funin f plaizains P s in all hee feelei s a vaius seleed empeaues. d 11 = d 12 = d 13 = d 14 = d 16 = d 21 = d 22 = d 23 = d 25 = d 26 = d 34 = d 35 = d 36 =. 15 By using P 3 = 2P3O, = 33O + 23O d 15 = 11 Q 44 P 3,, ne an ge d 24 =2 22 Q 11 Q 12 P 3, d 31 =2 33 Q 12 P 3, = 11O, 22 = 33O 23O, and d 32 = Q 11 +2Q 12 Q 44 P 3, d 33 = Q 11 +2Q 12 + Q 44 P The empeaue dependene f piezelei effiiens in KNbO 3 ysals in he hhmbi is alulaed wih Eqs. 15 and 16 and is shwn in Fig. 2. The shea piezelei effiiens d 15 and d 24 exhibi a sng empeaue dependenes and als have an ppsie endeny, while he hee mpnens d 31, d 32, and d 33 ae elaively insensiive empeaue. Thus, he empeaue dependene f d 33 is dminaed by d 15 and d 24 as seen fm Eq. 13. The piezelei nsans a m empeaue ae given in Table I, whih shws ha he alulaed d 33, d 32, and d 31 agee well wih pevius esuls while d 15 and d 24 ae slighly veesimaed. The hee-dimensin d 33 sufaes a hee hsen empeaues 5, 25, and 22 C ae pled in Fig. 7. The TABLE I. Calulain bained m empeaue values f he piezelei effiiens, mpaed wih peviusly published esuls. Ppeies pc/n Wada a Zgnik b Gunhe Liang d This wk d 33 d 32 d 31 d 15 d 24 a Refeene 7. b Refeene 16. Refeene 15. d Refeene FIG. 7. Cl nline The ienain dependene f piezelei effiiens d 33 f KNbO 3 in he hhmbi f hee seleed diffeen empeaues, a T= 5 C; b T=25 C; and T=22 C. Angle max a whih maximum d 33 us is indiaed f eah empeaue. The hee dinae axes espnd x 1 =d 33 sin s, x 2 =d 33 sin sin, and x 3 =d 33 s. The numeial values maked n he axes have uni pc/n. Only he uppe half f he dinae spae is shwn. Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins

8 Liang e al. J. Appl. Phys. 18, d * 33 (pc/n) T =-5 C T =-3 C T =25 C T =1 C T =22 C (deg. ) (a) d * (pc/n) deg. ) (b) T =-3 C T = C T =15 C T =2 C T =22 C FIG. 8. Cl nline Piezelei effiiens d 33 in he hhmbi KNbO 3 as a funin f angle a vaius empeaues in diffeen planes, a =9 and b =. maximum d 33 hanges is diein upn ling fm he eagnal-hhmbi ansiin empeaue. As a esul, i leads a ain f maximum d 33 away fm he pla diein 1, whih is aibued he qualiaively ppsie dependenes f d 15 and d 24 n empeaue. Appahing he eagnal-hhmbi ansiin empeaue, d 24 dminaes ve d 33 Eq. 13, while d 15 bemes dminan n ling he empeaue nea he hhmbi-hmbhedal ansiin pin. Diffeen fm he eagnal, d 33 is dependen n w mpeiive shea piezelei effiiens in he hhmbi. In he hhmbi, he w shea piezelei effiiens d 15 and d 24 ae elaed he pemiiviies pependiula he diein f he spnaneus plaizain axis 1. d 15 is diely elaed he pesene f he hhmbi-hmbhedal ansiin. The signifian inease in he dielei susepibiliy 11 as he hmbhedal is appahed n ling implies ha he hhmbi KNbO 3 bemes dieleially sfened alng he ysallgaphi diein pependiula he plaizain diein 1. The LGD-fee enegy well as a funin f plaizain shwn in Fig. 6 b bemes shallwe wih ineasing empeaue, leading he inease in dielei susepibiliy, and hus he inease in is piezelei espnse. d 24 is elaed he eagnal-hhmbi ansiin and ineases wih ineasing empeaue. The inease in 22 lse he eagnal an be easnably explained by he ain f plaizain as he ansiin us. Theefe, d 33 in he hhmbi is influened by w adjaen feelei ansiins. The lage lngiudinal espnses alng n-pla dieins ae nsisen wih he esuls fm fis piniple alulains, whih inepeed he enhaned piezelei effiiens alng n-pla dieins by aing he plaizain indued by he sng exenal elei fields. 19,3 Is maximum as a funin f angle depends n he mpeiin f d 15 and d 24. As shwn in Fig. 2, d 33 is dminaed by d 15 lse he hhmbi-hmbhedal ansiin empeaue and by d 24 lse he eagnal-hhmbi ansiin empeaue. We analyze he values f d 33 n 1 and 1 assuming = and =9 and pl hem as a funin f angle unde vaius empeaues in Fig. 8. On he 1 plane wih =9, he diein f maximum d 33 hanges wih deeasing empeaue while i lies alng he same diein as n he 1 plane wih =. This an be easily seen fm Fig. 9, in whih he maximum d 33max and is espnding max ae shwn as a funin f empeaue. F =9, d 33max ineases wih deeasing empea- ue and max ineases apidly and eahes 5.6 wih deeasing empeaue and hen bemes independen f d * (pc/n) 33max max (deg.) d * (pc/n) 33max max (deg.) Tempeaue ( C) Tempeaue ( C) (a) (b) FIG. 9. Cl nline Maximum d 33 and is espnding angle max as a funin f empeaue f he hhmbi KNbO 3 in diffeen planes, a =9 and b =. Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins

9 Liang e al. J. Appl. Phys. 18, empeaue belw 1 C. Hweve, in he ase f =, d 33max deeases wih deease in empeae and max emains nsan. Ou alulains shw he sufae f d 33 is nealy symmeial n he 1 plane aund 1 C. The diein f he maximum d 33max is aed by 9 wih ineasing empeaue. The alulaed esuls shw ha d 33 has he lages value in he high-empeaue ange appahing he eagnal. This is diffeen fm BaTiO 3, in whih i has he lages d 33 in he lw empeaue ange lse he hhmbi-hmbhedal ansiin. This als implies ha he eagnal-hhmbi ansiin has snge effes n he piezelei espnse han he hhmbihmbhedal ansiin in KNbO 3. A m empeaue, he alulaed d 33max is nealy 1.5 pc/n wih = and max =5.6. This is quaniaively nsisen wih he Nakamua s measuemen esuls 11 whih shw he highes piezelei effiien 92 pc/n hugh evaluaing he sain vesus elei field uve f he aed 1 -u abu he 1 -axis f singledmain KNbO 3. Hweve, based n u alulain esuls, hee shuld exi anhe maximum d 33 wih =9 and max =5.6. d 11 = d 12 = d 13 = d 14 = d 23 = d 25 = d 26 = d 34 = d 35 = d 36 =. 19 All piezelei nsans an als be expessed as funins f he dielei nsans and plaizains by d 15 d * (pc/n) deg. = d 24 = Q 11 Q 12 + Q P 3, T =-5 C T = -9 C T = -15 C T = -23 C FIG. 1. Cl nline Piezelei effiiens d 33 in he hhmbi KNbO 3 as a funin f angle a vaius empeaues. C. Rhmbhedal F he lw empeaue hmbhedal, he ienain dependene f d 33 is given by d 33, = d 15 s sin 2 d 22 sin 3 + d 31 sin 2 s + d 33 s Amng he hee Eule angles, = is fixed f simpliiy. The es w angles and ae vaied invesigae he ansiin fm hmbhedal hhmbi. =aan 1/ 2 and = /2 gives he dinaes assiaed wih he hhmbi. The ansfm maix beween he ubi and hmbhedal dinae sysems is ij = 2/2 2/2 2/ 6 1/ 6 1/ 6 1/ 3 1/ 3 1/ Thus he elainship f he piezelei nsans beween w efeene sysems f he hmbhedal is given as d 15 d 16 d 31 d 33 = d 24 =2d 21 = d 32 = d 33R = 2d 33R = 2d 22 = d 33R +2d 32R 2d 32R + d 35R d 36R, 3 = 2 3 d 33R +2d 32R d 32R d 35R d 36R /2, 3 +2d 35R + d 36R, 3 + d 36R d 35R, d 16 d 31 =2d 21 2 = 2d 22 = 3 2Q 11 2Q 12 Q P 3, = d 32 = 1 3 2Q 11 +4Q 12 Q P 3, d 33 = 2 3 Q 11 +2Q 12 + Q P 3, 2 whee he elain f plaizains and dielei susepibiliies, P 3 = 3P3R, 11 = 22 = 11R 12R, and 33 = 11R +2 12R, beween he w dinae sysems ae used f alulains. The alulaed empeaue dependene f d 15, d 22, d 32, and d 33 ae given in Fig. 2. d 15 and d 22 negaive value inease wih ineasing empeaue while d 32 and d 33 hange slighly. Due he absene f ansiins in he lwe empeaue egime, he shea piezelei effiien enss ae elaively empeaue aund K. The alulaed hee-dimensinal sufaes f d 33 in he hmbhedal a hee seleed empeaues 7, 15, and 25 C ae nealy independen f empeaue and he maximum d 33max is edued wih deeasing empeaue. F example, a 7 C, d 33max =239.8 pc/n f max =61.8, while f 15 C, d 33max =158. pc/n f max =61.7 C. d 33max deeases in 11.9 pc/n f max =61.6 a 25 C. The piezelei effiien d 33 in he hmbhedal KNbO 3 is pled as a funin f unde vaius empeaues in Fig. 1. d 33 exhibis n-symmey wih espe he axis defined by =9. This an be easily seen fm he expessin f d 33 Eq. 17, whih inludes a em d 22 sin 3. The n-ze em d 22 sin 3 a =9 gives diffeen d 33 values in Fig. 1. Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins

10 Liang e al. J. Appl. Phys. 18, d * (pc/n) 33max The maximum d 33max and is espnding angle max as a funin f empeaue ae given in Fig. 11. d 33max and max deease wih deeasing empeaue. The angle max espnding he maximum d 33 is nealy independen f empeaue. Fm Eq. 2, d 15 is deemined by he dielei susepibiliy 11 22, i.e., by he plaizabiliy f a ysal pependiula he plaizain diein, while d 33 is nlled by he dielei susepibiliy 33 alng he pla diein. As disussed abve, he enhaned dielei susepibiliy is pependiula he pla diein and hus he enhaned piezelei espnse an be aibued he ain f plaizain lse he ansiin. This leads he inease in d 15 wih empeaue appahing he hhmbi. Theefe, d 33 is nly affeed by he ain f plaizain aused by he hhmbihmbhedal ansiin. The flaening f he fee enegy well Fig. 6 als implies ha he enhanemen f piezelei effiiens is alng a n-pla diein. The behavi f d 33 in he hmbhedal KNbO 3 is vey simila ha in BaTiO IV. CONCLUSIONS Tempeaue ( C) FIG. 11. Cl nline Maximum d 33max and is espnding angle max ae as a funin f empeaue f he hmbhedal KNbO 3. max (deg.) The piezelei effiien enss and he ienain dependene f lngiudinal piezelei effiien d 33 in KNbO 3 single ysals a diffeen empeaues ae analyzed using he LGD hemdynami hey. The dielei sfening alng he diein pependiula he pla diein is shwn be he main fa ha nibues he empeaue dependene f he diein f he maximum d 33. Shea piezelei effiiens inease due feelei ansiins leads a signifianly enhaned d 33 alng nnpla dieins. Simila behavi an be expeed wih espe ansiins aused by hemial mpsiin vaiain, exenal elei field and mehanial pessue. ACKNOWLEDGMENTS The wk is paially supped by NSF unde Gan N. ECCS and DMR B. Jaffe, R. S. Rh, and S. Mazull, J. Appl. Phys. 25, H. Jaffe, J. Am. Ceam. S. 41, Y. Hiuma, H. Nagaa, and T. Takenaka, Jpn. J. Appl. Phys., Pa 1 45, K. Yshii, Y. Hiuma, H. Nagaa, and T. Takenaka, Jpn. J. Appl. Phys., Pa 1 45, K. Masum, Y. Hiuma, H. Nagaa, and T. Takenaka, Jpn. J. Appl. Phys., Pa 1 45, E. Hllensein, M. Davis, D. Damjanvi, and N. See, Appl. Phys. Le. 87, S. Wada, K. Muaka, H. Kakem, T. Tsuumi, and H. Kumagai, Jpn. J. Appl. Phys., Pa 1 43, S. E. Pak and T. R. Shu, J. Appl. Phys. 82, S. Wada, S. Suzuki, T. Nma, T. Suzuki, M. Osada, M. Kakihana, S.-E. Pak, L. E. Css, and T. R. Shu, Jpn. J. Appl. Phys., Pa 1 38, S.-E. Pak, S. Wada, L. E. Css, and T. R. Shu, J. Appl. Phys. 86, K. Nakamua, T. Tkiwa, and Y. Kawamua, J. Appl. Phys. 91, S. Wada, A. Seike, and T. Tsuumi, Jpn. J. Appl. Phys., Pa 1 4, S. Saih, T. Kbayashi, K. Haada, S. Shimanuki, and Y. Yamashia, Jpn. J. Appl. Phys., Pa 1 37, E. Wiesendange, Feeleis 6, P. Güne, Jpn. J. Appl. Phys., Pa 1 16, M. Zgnik, R. Shlesse, I. Biaggi, E. Vi, J. Tshey, and P. Gune, J. Appl. Phys. 74, R. Ahluwalia, T. Lkman, A. Saxena, and W. Ca, Phys. Rev. B 72, L. Bellaihe, A. Gaıa, and D. Vandebil, Phys. Rev. B 64, M. Budimi, D. Damjanvi, and N. See, Phys. Rev. B 72, M. Budimi, D. Damjanvi, and N. See, Phys. Rev. B 73, M. Budimi, D. Damjanvi, and N. See, J. Appl. Phys. 94, M. J. Haun, E. Fuman, S. J. Jang, and L. E. Css, Feeleis 99, J. F. Nye, Physial Ppeies f Cysals Oxfd Siene Publiains, New Yk, D. Damjanvi, M. Budimi, M. Davis, and N. See, J. Mae. Si. 41, L. Liang, Y. L. Li, L.-Q. Chen, S. Y. Hu, and G.-H. Lu, Appl. Phys. Le. 94, L. E. Css and G. A. Rssei, J. Appl. Phys. 69, L. Liang, Y. L. Li, L.-Q. Chen, S. Y. Hu, and G.-H. Lu, J. Appl. Phys. 16, M. Budimi, D. Damjanvi, and N. See, Appl. Phys. Le. 85, D. Damjanvi, F. Bem, and N. See, Appl. Phys. Le. 8, H. Fu and R. E. Chen, Naue Lndn 43, Dwnladed 23 De Redisibuin subje AIP liense pyigh; see hp://jap.aip.g/abu/ighs_and_pemissins