Crystal growth in condensed phase and phase diagrams

Transcription

1 , (011) DOI: /jp/ Ownd by th authors, publishd by EDP Sins, 011 Crystal growth in ondnsd phas and phas diagrams C. Goutaudir Univrsité d Lyon, Univrsité Claud Brnard Lyon 1, Laboratoir Multimatériaux t Intrfas UMR CNRS 5615, 696 Villurbann dx, Fran hristll.goutaudir@univ-lyon1.fr Abstrat. h rystal growth thniqus usd in industry usually tak pla in mlt and mor than 80% of th rystals produd hav a ongrunt mlting omposition. Howvr, th omplxity of singl-rystal matris is onstantly inrasing (4 onstitunts or vn mor) and in ths multi-onstitunt systms, vry fw ar fully stoihiomtri ompounds with a ongrunt mlting omposition. Whatvr th thniqu usd, to mastr th growth pross, a rquird first stp is to know th stability domain of th solid and uid phass onrnd in a tmpratur-prssur-omposition oordinat systm. h hoi of growth thniqu to b implmntd will dpnd primarily on th thrmodynami paramtrs and th nulation onditions whih will rsult in rystal gnration. 1 Introdution h trm krystallos appars in antiquity and oms from th Grk word ryo whih dsignats i. A rystal is a solid matrial whih rprsnts on th on hand an xtrnal symmtry dlinatd by a st of rystallin surfas, on th othr hand an intrnal symmtry rsulting from th thr-dimnsional layout of ntitis aording to a rgular modl alld th latti systm. A monorystallin solid is idally mad of a singl rystal, for whih th spatial arrangmnt is a rgular and priodi rptition of a larg numbr of atoms, moluls or ions. h singl rystal may b onsidrd as th prft rystal, and its rlvan lis in its grat strutural homognity, i.. th nvironmnt of ah similar ntity is xatly th sam. It is ssntial to th study and utilisation of th solid-stat physial proprtis whih dpnd on th strutur (mhanial, ltromagnti, aousti proprtis, t.). Singl rystals ar at th root of a numbr of high-thnology appliations in suh filds as ltronis, optis, radiation dttion, t. h industrial prodution of bulk singl rystals startd in th 1990s and has dvlopd ontinuously to rah about tn thousand tons today [1]. A fw xampls of rystals as wll as th distribution of industrial prodution among th various appliation filds ar prsntd in abl 1. h miroltronis and smiondutors industris onstitut th major part and this is onstantly inrasing. h fundamntal aspts of rystal growing wr ddud from th first rystallisation xprimnts at th bginning of th 18 th ntury, thn improvd with th dvlopmnt of thrmodynamis (lat 19 th ntury) and finally with th thoris of nulation/growth and th rol of transport proprtis []. h hoi of rystal growth thniqu to b implmntd will rsult from a ompromis btwn: - on th on hand, th main four growth paramtrs, i.. growth tmpratur, suprsaturation, tmpratur gradint and fluid phas dynamis; - on th othr hand, th rystal qualitis imposd by th thnologial appliation and th onomi fator: stoihiomtry, impurity rat, homognity of omposition, inlusions, strutural prftion, rapidity and profitability. h purpos of this papr is to addrss rystal growth through a thrmodynami approah, and mor spifially th hoi of mthodology to b implmntd and th ontrol of rystal omposition from th knowldg of quilibria btwn phass. his is an Opn Ass artil distributd undr th trms of th Crativ Commons Attribution-Nonommrial Lins 3.0, whih prmits unrstritd us, distribution, and rprodution in any nonommrial mdium, providd th original work is proprly itd. Artil publishd by EDP Sins and availabl at or

2 abl 1. A fw industrial singl rystal xampls by appliation fild. Appliation fild Exampls of rystals Distribution of industrial prodution [1] Smiondutors Si 60% Sintillation rystals Optial rystals Aousti optis rystals Lasr rystals Jwllry and wath industry Bi 4 G 3 O 1 Lu SiO 5 Alkali halids Quartz LiNbO 3 Y 3 Al 5 O 1 KH PO 4 Saphir Colord spinl 1% 10% 10% 5% 3% h most omplx ass appar whn a fourth phas is obsrvd, i.. two solid phass that orrspond to two allotropi varitis of th pur onstitunt. hr ar two typs. Eithr th strutural transition ours during a hang in tmpratur (Figur 1a): thr is only on solid-uid quilibrium urv and it will b impossibl to rystallis th low tmpratur varity from th uid phas. Or th strutural transition is linkd to a hang in prssur (Figur 1b): thr is a solid-uid quilibrium urv for ah rystal varity and diffrnt prssurs should b usd to rystallis on or th othr; this is th as, for instan, of arbon (graphit or diamond). P uid hrmodynamis Solid I Solid II A rystal is formd through ions, atoms or moluls going from a disordrd stat to an ordrd stat. h disordrd mothr phas may b a gas, a uid or vn a solid. It will initially b maintaind in thrmodynami quilibrium but this quilibrium must b disturbd to rsult in th phas-transition pross. (a) Vapor.1 Phas rlation In ordr to mak massiv singl rystals, th rystal growth mthods most widly usd industrially usually all for a uid-solid transition. h vapour phas thniqus ar ssntially kpt for thin layr growth. In any as, to mastr th growth pross, a rquird first stp is to know th stability domain of th solid and uid (or gas) phass onrnd in a tmpraturomposition or vn tmpratur-prssur-omposition diagram. P Solid I Solid II uid Vapor.1.1 h as of unary systms h xistn domain of solid-uid-gas phass of a unary systm is onvntionally rprsntd in a P- diagram and th simplst ass inlud 3 phass. h rlationships btwn ah of th phass ar dsribd by Clapyron s rlation whih, in th as of solid-uid quilibrium, is xprssd by: H fus dp dln (1) V fus whr H fus is th nthalpy variation during th transition phas, onsidrd hr as indpndnt of tmpratur, and V fus is th molar volum variation. It is thn asy to dfin th tmpratur domain whih is favourabl to rystal growth. (b) Fig. 1. Existn domain of four phass in a unary systm. (a) nantiotropi as (b) monotropi as.1. h as of binary systms Whn th rystal to b mad onsists of two indpndnt onstitunts, it is asy to rfr to tmpraturomposition isobari rprsntations in ordr to guid th hoi of rystal growth thniqu. hrfor th solid-uid quilibrium urvs ar rprsntd for ah onstitunt i by th rlation: ai, sol H i, fus dln d () a R i, 0000-p.

3 whr a i is th ativity of onstitunt i in th binary mixtur, and H i, fus is its mlting nthalpy undr th prssur onsidrd. h simplst as of solid-uid quilibria utilisation will b for ompounds with a ongrunt mlting omposition, i.. ompounds for whih th mltd uid omposition is idntial to th omposition of th solid from whih it oms (Figur a). h prinipl of rystalgrowing in mlt will onsist in slowly rystallising th matrial from its uid phas, at th quilibrium mlting tmpratur. Most of th massiv stoihiomtri rystals industrially fabriatd hav a ongrunt mlting omposition. Howvr, th mlting quilibrium may not b prossabl somtims. his is th as of: - ompounds with a mor or lss xtndd solid solution domain (Figur b). Controlling th solid omposition during ooling is triky and will rsult in high inhomognitis in th singl rystal, - ompounds whih brak down by a pritti ration (Figur ). h omposition of th mltd uid may b vry diffrnt from that xptd for th solid phas, - ompounds prsnting svral strutur varitis (figur d). Cooling a singl rystal mad in its high tmpratur form will rsult in its dstrution upon going through th transition isothrm on th way to th low tmpratur phas. fus L+A L+A x B y L+A x B y A+A x B y B+Ax B y A A x B y B (a) fus L + L A 1-x B x + A 1-x B x A 1-x B x A+A 1-x B x B+A 1-x B x A B (b) In ths spifi ass, th thniqus implmntd will onsist in prossing a solid-uid quilibrium for whih th uid phas has a diffrnt omposition from that of th dsird solid: this will b alld solution or flux growth. A+L.1.3 h as of highr ordr systms h thnologial advans mad in th last fw dads imply that th omplxity of singl-rystal matris is onstantly inrasing (4 onstitunts or vn mor), so that th homognity of th matrial s hmial omposition onditions its pross through th strutur/proprty rlationship. In ths multi-onstitunt systms, vry fw ar fully stoihiomtri ompounds with a ongrunt mlting omposition. h gnral as looks mor lik an aumulation of all abnormalitis dsribd in th as of binary ompounds (solid solution, domposition, rystal varitis). Only th thniqus starting from an initial omposition flux diffrnt from that of th rystal will lad to rystal growth in th dsird phas. his typ of mthod invitably rquirs knowldg of th solid-uid quilibria in th trnary or vn highr-ordr systms. As an xampl, w an it th KY(WO 4 ) ompound for whih th singl rystals ar studid for thir ivilian lasr appliations. Study of th xistn domain of KY(WO 4 ) in th trnary systm K O-Y O 3 -WO 3 dmonstratd that: A A x B y +L A x B y +A A x B y fus () S +L S S 1 +L (d) A x B y +B B+L Fig.. Diffrnt ass of solid-uid quilibrium in binary systm. (a) ongrunt mlting of stoihiomtri oumpound (b) ongrunt mlting of solid solution () pritti domposition (d) strutural transition B 0000-p.3

4 - this ompound is not stoihiomtri and its atual omposition is diffrnt from its thortial omposition [3]: ongrunt mlting is obsrvd at = 1085 C for stoihiomtry K O-1.173Y O 3-4WO 3. - it has a nonstoihiomtri domain whih xtnds to approximatly mol% in Y O 3, - thr is strutural transition at tmpratur = 1034 C los to mlting, btwn a monolini low tmpratur form () and a quadrati high tmpratur form (). Only th low tmpratur phas shows th dsird opti proprtis [4]. Figur 3 shows th polythrmal projtion of th rystallisation domains for th two polymorphous varitis and of th KY(WO 4 ) ompound [5]. Givn th wid rystallization domain of phas, svral uid phass may b usd to fabriat singl rystals and th hoi of thir omposition may b adaptd aording to th rystal growth prossing onditions and th atual omposition of a singl rystal [6].. Equilibrium disruption Equilibrium disruption is prformd by hanging th prssur, onntration or tmpratur in ordr to rat a suprsaturation of th mothr phas, i.. to kp this mothr phas byond th quilibrium point without a transition ourring. For instan, in th as of rystallisation from a mlt, th mothr phas is maintaind uid blow its mlting quilibrium tmpratur: this is th suprooling phnomnon. Suprsaturation is dfind by th hmial potntial diffrn of th onstitunt i btwn th mothr phas whih is maintaind in mtastabl quilibrium and th nw ordrd phas (3) i i, mothr phas i, nw phas his suprsaturation may b asily xprssd during a hang in isothrmal onntration whn working from a solution or during a hang in isobari tmpratur whn onsidring th simpl modl of idal solution for th mothr phas, and whn th ondnsd phass ar inomprssibl...1 Suprsaturation by onntration hang In th as of rystallin growth from solution, th suprsaturation will orrspond to th inras in mothr solution onntration C ompard to th solubility quilibrium onntration C at tmpratur. h hmial potntial of th suprsaturatd mothr solution is thn xprssd as a funtion of th onntration ratio. solution k ln (4) whr is th hmial potntial at quilibrium and k is Boltzmann s onstant. If w onsidr that th hmial potntial of rystal rystal dos not hang ompard to th stabl quilibrium rystal (5) Rlation (3) thn boms solution rystal k ln (6).. Suprsaturation by tmpratur hang Whn rystal growth is prformd from th mltd ompound, usually undr atmosphri prssur, suprsaturation is obtaind at a tmpratur lowr than th rystallisation (or fusion) quilibrium tmpratur. h mothr phas and ordrd phas hmial potntials ar thn rsptivly xprssd aording to th tmpratur gap pr rlations (7) and (8) in whih S and S rystal rprsnt th molar ntropy of ah phas: S ( ) (7) S ( ) (8) rystal rystal Considring th onstant ntropy on a narrow tmpratur domain, suprsaturation is thn givn by rlation (9) in whih th ntropy hang is rgardd as mlting ntropy. h tmpratur gap, always positiv, is alld suprooling. S ( ) (9) 3 Nulation rystal fusion Disturban of thrmodynami quilibrium is not nough to xplain rystal ration. For instan, a uid phas may b kpt for a long tim at a tmpratur lowr than fusion ; this phas is mtastabl and it will only tak a shok to obsrv spontanous rystallisation orrsponding to a rturn to quilibrium. Forming a nw phas in th mtastabl mothr phas will thn rquir a nulation stag followd by a growth. Suprsaturation will loally lad to th rationalisation of ntitis and th formation of sds or lustrs whih form th rystallin latti. It is only from ths sds, i.. ntitis of suffiint siz grouping togthr, that rystals will b abl to grow aording to variabl growth kintis. Nulation has two origins: a) thrmal agitation produs random loal atom (or molul) arrangmnts in th mothr phas this is homognous nulation; b) forign partils suspndd in th mdium or a surfa (substrat) will atalys a nw phas nulation this is htrognous nulation p.4

5 h nrgy variation that gos with th ration of a nulus rsults from two fators [7-9]: - a dras of fr nrgy du to th transition of th disordrd mothr phas into an ordrd phas ; - an nrgy inras assoiatd with th ration of th intrfa btwn th two phass. For a sphrial nulus of radius r formd by homognous nulation, th nrgy balan rlativ to th volum unit is xprssd by rlation (10) whr is th surfa fr nrgy of th nulus/uid intrfa. G V 4 3 r 4r 3 (10) Whatvr th nulus gomtry, th volution of this nrgy aording to nulus siz will go through a maximum that will orrspond to th ritial siz from whih th sd will b stabl and abl to grow. As an xampl, abl ompars th nrgy balan to rat a sphrial sd with a radius of urvatur r in th as of homognous and htrognous nulation. h variation may b rgardd as th diffrn btwn th fr nthalpis of th phass masurd on marosopi sampls, i.. G fusion (or G rystallisation ) in th as of a solid-uid transition. 4 Conlusion h thrmodynami approah to rystallin growth provids an ssntial thortial basis for any singlrystal fabriation pross. his is all th mor important as dsird matris bom mor and mor omplx and th homognity of th matrial s hmial omposition onditions its pross through th strutur/proprty rlationship. Knowldg of th rlations btwn phass is an ssntial prrquisit to hoosing th rystal growth thniqu. h most widly usd industrial thniqus for massiv singl rystals inlud: - rapid growth thniqus (a fw mm.h -1 ) suh as Vrnuil, Czohralski, Bridgman, whih ar haratrisd by a rystallisation dirtd from a mlt and ar totally adaptd to ongrunt mlting ompounds for whih th mltd uid has th sam omposition as th rystallisd solid; - slow growth thniqus (a fw mm.day -1 ) whih ar solution or flux growths; thy usually oprat at a lowr tmpratur and ar usd for non-ongrunt mlting rystals (uid and solid do not hav th sam omposition) or thos with a strutural transition. hrmodynami onsidrations also involv th dpndn of th suprsaturation phnomna with nulation nrgy and sd siz that will rsult in th irrvrsibl growth of marosopi ntitis. abl. Comparison of homognous nulation / htrognous nulation for th ritial radiuss and nrgy barrirs. r* Homognous nulation G rist. 16 G* 3 ; 3 G rist : surfa fr nrgy of th intrfa sin r* Htrognous nulation G rist 3 16 G* 3 G rist (1 os ) ( os ) 4 : surfa fr nrgy of th intrfa : wtting angl Rfrns 1. H.J. Shl, J. Crystal Growth, 11, 1-1 (000). D.J.. Hurl, Handbook of Crystal Growth, vol. 1, (Elsvir, Amstrdam) 3. E. Gallui, C. Goutaudir, G. Boulon, M.. Cohn- Adad, Eur. J. of Solid Stat Inorg. Chm., 35, (1998) 4. S.V. Borisov, R.F. Kltsova, Sov. Phys. Cryst., 13(3), 40 (1968) 5. C. Goutaudir, E. Gallui, G. Boulon, M.. Cohn- Adad, J. Phas Equilibria, (3), (001) 6. E. Gallui, C. Goutaudir, G. Boulon, M.. Cohn- Adad, B.F. Mntzn, J. Cryst. Growth, 09, (000) 7. D. urnbull, Solid Stat Physis, vol.3 (Aadmi Prss, Nw York, 1956) 8. C. Hrring, Strutur and proprtis of solid surfas ( R. Gomr, C.S. Smith Eds, 1969) 9. J. Burk, Cinétiqu d hangmnts d phas dans ls métaux (Masson & Ci, Paris, p.5

6 Fig. 3. Crystallization filds of th - and -KY(WO 4 ) phass in th trnary sub-systm K WO 4 -Y WO 6 -WO p.6