Crystal Structure. Fig. 11: The fourteen Bravais lattices

Transcription

1 rytal tructur It i uful t bgin any dicuin f crytal tructur with an lmntary dfinitin f m crytallgraphic trm. Firt f all, crytal ar mad up f idntical, rpating arrangmnt f atm calld unit cll. By dfinitin, a unit cll i th mallt vlum f a crytallin lid that hibit th ymmtry prprti f th whl crytal. Jut vn baic crytal ytm ar knwn. Th ar, cubic, hagnal, ttragnal, rthrhmbic, trignal, mnclinic, and triclinic. Th may b furthr claifid int furtn Bravai lattic typ: Fig. : Th furtn Bravai lattic Of th typ, fr micnductr th fac cntrd cubic (F ytm i f mt imprtanc. Frtunatly, cubic lattic ar al th ait Bravai lattic t viualiz and undrtand.

2 It i clar frm th lmntary tructur f th Bravai lattic that ach unit cll ha vral lattic pint. In trm f th actual phyical tructur f a lid matrial, ach lattic pint i aciatd with a bai grup. Thu, th lattic bai grup fr a particular crytal i a dfinit grup f atm aciatd with ach lattic pint. In th implt ca (which, fr ampl, ccur in th ca f m lmntal mtal th bai grup cnit f jut a ingl atm, in which ca n atm ccupi ach lattic pint. Of cur, th lattic bai grup mut b idntical fr all lattic pint. Obviuly, fr cmpund matrial th bai grup mut cnit f mr than n atm inc it cannt b dfind a n kind f atm at n lattic pint and anthr kind f atm at a diffrnt lattic pint. Thu, n find that in gnral, th bai grup f a lattic cnit f a dfinit rpating grup f atm. In m ca, th bai grup may b idntifid with an actual cvalnt mlcul that maintain it idntity vn whn th lattic brak up during mlting r ublimatin, (.g., a in th ca f ic, watr, and watr vapr. Hwvr, in many ca, th bai grup al brak up with th lattic itlf during chang f pha (.g., a in th ca f mtallic r inic lid. Finally, bai grup and unit cll huld nt b cnfud. Bth ar rpating grup f atm, hwvr, th bai grup d nt hibit all f th ymmtry prprti f th whl crytal. It i fund that vn fr m lmntal matrial, th bai grup cnit f mr than n atm. Thi i prcily th ca fr lmntal ilicn fr which th lattic bai grup cnit f tw atm. T undrtand why thi i, n brv that th Bravai lattic fr ilicn i aily idntifid frm pwdr -ray diffractin pattrn a F. Hwvr, n al rcall frm prviu cnidratin f lctrnic tructur, that ilicn ha ttrahdral crdinatin du t th ttrahdral gmtry f th p 3 hybrid rbital. larly, if a ingl ilicn atm i inrtd at ach pint f an F lattic, th rulting atmic crdinatin i nt ttrahdral. Hwvr, if a tw-atm bai grup i inrtd in th F lattic, ttrahdral crdinatin can b ralizd. Th rult i th diamnd cubic tructur, which can b thught f a tw intrpntrating F lattic fft n quartr f th unit cll dimnin in ach dirctin: Fig. 2: Diamnd cubic crytal tructur

3 Obviuly, th archtyp f thi tructur i lmntal carbn, viz., diamnd. It i wll knwn that th diamnd cubic tructur ha a high dgr f ymmtry including vral mirrr plan and tw, thr, and furfld rtatin a. Thi i a dirct cnqunc f th high dgr f ymmtry aciatd with ttrahdral crdinatin. In gnral, th grup f ymmtry prprti rv a a uniqu pcificatin f any crytal tructur. In paing, n huld nt that th cmpund micnductr, gallium arnid, GaA, ha ntially th am tructur a lmntal ilicn r grmanium. Hwvr, fr GaA, th lattic bai grup mut cnit f tw diffrnt typ f atm, viz., n Ga and n A, intad f tw idntical atm, i.., ithr i r G. Thi al crrpnd t a cubic crytal tructur, imilar t th diamnd cubic tructur, calld zincblnd (aftr th naturally ccurring minral frm f zinc ulfid, Zn. Anthr imprtant trm n ftn ncuntr in cnnctin with micnductr matrial i crytal rintatin. T what i mant by crytal rintatin, n rturn t th baic unit cll which in th mt gnral ca, i.., a triclinic ytm, mut b dcribd by i indpndnt lattic paramtr (thr lngth and thr angl. Hwvr, again bcau f it high ymmtry, a cubic ytm rquir nly a ingl lattic paramtr, a, namly, th unit cll lngth. (larly, fr a cubic ytm, th unit cll lngth i th am in all thr dirctin and all th angl ar 90. Thu, n naturally dfin a crdinat ytm fr th lid with th lattic paramtr a th baic unit f lngth. Thu, in thi dirct pac, any pint within th crytal having all intgr crdinat crrpnd t a crnr lattic pint, i.., vrt, f a unit cll. rytal rintatin i gnrally pcifid by Millr indic, (h,k,l. Th act dfinitin f Millr indic a crdinat in rciprcal pac i quit tchnical in natur and i bynd th cp f th prnt cur; hwvr a impl dfinitin may uffic in trm f th thr dirct pac crdinat ai intrcpt. inc thr pint uniquly dtrmin a gmtric plan, th thr ai intrcpt dfin a crytallgraphic plan. Prhap, th implt ampl i prvidd by a plan that intrct ach ai at a ditanc f n lattic paramtr frm th rigin, i.., th plan ha unit intrcpt n ach crdinat ai. Thi i primitivly idntifid a th [] crytallgraphic plan. In cntrat, if n cnidr a plan that i paralll t th y and z a, but ha unit intrcpt n th ai, hw huld thi plan b dignatd? larly, frm a trictly gmtric pint f viw, th crrpnding y and z ai intrcpt fr uch a nnintrcting plan hav rcdd t infinity. It i fund that a cnitnt dignatin rult if th Millr indic, h, k, and l, ar idntifid a rciprcal f th, y, and z ai intrcpt intad f th intrcpt thmlv. Thu, th rciprcal f frmally crrpnd t zr and th plan in qutin i cnquntly dignatd a [00]. Uing thi chm, a plan that i paralll t th z ai, but ha unit intrcpt n th and y a i dignatd a [0]. larly, th ca f [] rmain unchangd inc unity i lf-invr. Indd, it turn ut that th dfinitin f Millr indic a rciprcal f crdinat ai intrcpt quit naturally fllw frm a gnral dcriptin f crytal diffractin phnmna in trm f thr dimninal Furir tranfrm. A furthr imprtant brvatin t b mad at thi pint i that inc th crdinat rigin in dirct pac i alway tranlatabl by an intgral numbr f lattic paramtr, th dignatin [hkl] rally dfin a family f plan rathr than a ingl uniqu plan. (Th famili f plan tak n particular imprtanc fr dcriptin f crytal diffractin phnmna. Furthrmr, althugh crytallgraphic plan can b arbitrarily pcifid, thy ar

4 gnrally nly uful whn thy crrpnd t a plan f atm having a rgular pattrn, i.., a tw dimninal lattic. Digring brifly, it i imprtant t nt that within th gnral cntt f micnductr prcing, th dignatin f crytallgraphic plan (and crytal rintatin i gnric in natur. Of cur, in a trict n Millr indic, [00], [00], and [00] mut dnt diffrnt famili f crytallgraphic plan; hwvr, all f th plan crrpnd t a fac f th unit cll. Accrdingly, again, a a cnqunc f th high dgr f ymmtry charactritic f th diamnd cubic tructur, all f th plan ar lctrically and tructurally quivalnt. Thu, any crytallgraphic plan crrpnding t a fac f th unit cll i gnrically calld a [00] plan. ikwi, any crytallgraphic plan that cut a unit cll fac diagnally and i al paralll t an dg f th unit cll i calld a [0] plan. imilarly, any crytallgraphic plan that intrct thr nn-adjacnt vrtic f th unit cll i calld a [] plan. Th gnric plan ar illutratd blw: O O O [00] [0] [] Fig. 3: Rprntativ crytallgraphic plan larly, in ach unit cll, thr mut b thr diffrnt, i.., nn-paralll, [00] plan, i diffrnt [0] plan, and fur diffrnt [] plan. In practic, phyical crytal ar gnrally dignatd by th rintatin f thir urfac. In th ca f micnductr crytal, i.., ilicn ubtrat, thi i th crytallgraphic plan paralll t th urfac ud fr dvic fabricatin. Thu, a ilicn crytal, i.., a ubtrat, which ha a urfac paralll t th unit cll fac, i dignatd [00]. imilarly, if th urfac can b thught f a intrcting thr ppit crnr f th unit cll, thn th crytal i dignatd []. Th tw rintatin ar ntially th nly n vr ud fr dvic fabricatin with [00] bing much, much mr cmmn inc it i ud cluivly fr fabricatin f MO dvic. A mall amunt f [] matrial i till ud fr fabricatin f biplar dvic fr which vry hallw dping i dirabl. In principl, [0] ubtrat culd b manufacturd; hwvr uch matrial i f n ral practical u and, a uch, i vry rar. larly, th atmic arrangmnt fr ach f th urfac, i.., [00], [], and [0], i diffrnt. Dirctly rlatd t thi i th dnity f dangling bnd that ar lft bhind if n brak r clav a crytal paralll t a particular crytallgraphic plan. Naturally, thi dnity i dirctly rlatd t th urfac nrgy f a crytal f a particular rintatin. Thrfr, n can primitivly timat rlativ urfac nrgy by imply

5 cunting th numbr f bnd brkn pr unit cll if n clav th crytal alng a givn plan and thn dividing by th urfac ara f that plan within th unit cll. (In rality, thi impl pictur i gnrally cmplicatd by th ccurrnc f urfac rcntructin, which lwr urfac nrgy by vrlapping nighbring dangling bnd. On find that in th diamnd cubic tructur, fr a [00] plan, fur bnd pr unit cll mut b brkn in rdr t clav th crytal. Obviuly, th unit cll ara aciatd with thi clavag jut qual th quar f th lattic paramtr, a 2. imilarly, fr clavag paralll t a [] plan, nly thr bnd pr unit cll mut b brkn. Obviuly, a [] plan frm an quilatral triangl within th unit cll with ach id having a lngth f a 2. Th hight f th quilatral triangl i a 3 2, thu, th crrpnding ara i 2 2 a a 3 2. Finally, fr [0] clavag, again nly thr bnd mut b brkn. Th crrpnding ara f a [0] plan within th unit cll i, f cur, a 2 2. On, thrfr, can dtrmin primitiv dangling bnd dniti, n 00, n, and n 0 a fllw: n 4 n n a a a 2 a a larly, th dangling bnd dnity fr a [0] plan i much lwr than ithr fr a [00] r a [] plan. Thrfr, it fllw that th crytal binding nrgy acr a [0] urfac i particularly lw. Hnc, a diamnd cubic crytal huld b much mr aily brkn paralll t a [0] urfac than paralll t ithr a [00] r [] urfac. It turn ut that [0] urfac d indd crrpnd t th natural clavag plan f a ilicn crytal. Thi can b dmntratd rathr dramatically by braking ilicn ubtrat f diffrnt rintatin frm th cntr uing a diamnd-tippd crib. On find that [00] ubtrat naturally parat int quartr and [] ubtrat int ith.

6 rytal Grwth nidring mr practical mattr, intgratd circuit ar fabricatd n ingl crytal ilicn ubtrat which ar mirrr-lik, plihd circular dik calld wafr (4 /00 mm dia. 525 m thick, 6 /50 mm dia. 675 m thick, 8 /200 mm dia., and 2 /300 mm dia., tc.. At prnt, th largt wafr cmmnly ud in intgratd circuit manufacturing ar 300 mm althugh thr i till ubtantial uag f mallr iz, viz., 200 mm. In additin, ubtrat f 450 mm (8 ar undr dvlpmnt by wafr manufacturr and ar currntly bing ud by a fw manufacturr. afr ar cut a lic frm larg ingl crytal ingt f ilicn calld bul. Of cur, thi ilicn ultimatly cm frm quartzit and, which i a naturally ccurring minral frm f ilicn diid, io 2. Typically, th raw id i rducd t mtallurgical grad ilicn in an lctric furnac uing carbn a th rducing agnt: io 2 2 i 2 O Thi matrial till cntain ubtantial impurity and t btain lctrnic grad ilicn, mtallurgical grad ilicn i ractd with hydrgn chlrid r chlrin: i 3Hl ihl H i 2 l il Thi prduc vlatil chlrid, i.., trichlrilan (ihl 3 r ttrachlrilan (il 4, which ar thn carfully ditilld and rducd again t ilicn by pyrlyi in pur hydrgn (r m thr high quality rducing agnt. Th rulting lctrnic grad plycrytallin matrial i quit pur and ha a littl a 0.05 ppb (50 ppt f ridual brn a th mt cmmn impurity. Thr ar tw majr crytal grwth tchniqu. Th ar th zchralki r Z prc and th flat zn r FZ prc. In th Z prc, lctrnic grad plycrytallin ilicn i placd in a quartz crucibl urrundd by graphit hat hilding and thn hatd t th mltn tat in an inrt atmphr by lctrical hating lmnt. A d rd having prpr rintatin i dippd int th mlt and thn cntrllably withdrawn. Naturally, th d i much mallr in diamtr than th dird crytal and, thu, th initial tag f Z grwth rquir lidificatin utward frm th d t tablih th dird diamtr. Onc diamtr i tablihd, th bul i lwly and cntrllably withdrawn uch that mltn ilicn lidifi with th dird rintatin and a larg crytal i built up r pulld. During th grwth prc, bth th bul and th crucibl ar rtatd t nhanc unifrmity. Of cur, all f thi rquir vry prci maurmnt and cntrl f tmpratur and hat flu, which gnrally can b achivd fr larg diamtr crytal nly by phiticatd cmputr fdback and cntrl. Typically, a Z apparatu can hld many kilgram f mltn ilicn. (Thi, f cur, dpnd n th iz f th wafr bing prducd and ha gratly incrad vr th lat frty yar. Th rulting bul i ubtantially fr f crytal dfct; hwvr, it d cntain ygn cntaminatin ariing frm th quartz crucibl and carbn cntaminatin frm lwhr (graphit hild, ucptr, tc.. In cntrat, in th FZ mthd, a lid

7 lctrnic grad plycrytallin ilicn rd i rcrytallizd and thn rfind uing a ndl y furnac. Thu, th FZ mthd i ntially a claical zn rfining tchniqu. It prduc th hight purity ilicn availabl; hwvr FZ ilicn i gnrally quit brittl. Indd, ygn and carbn cntaminatin in Z ilicn bth nhanc mchanical trngth and allw fr th applicatin f intrnal gttring mthd. Fr th ran, th vat majrity f ilicn wafr ud in intgratd circuit fabricatin ar manufacturd uing th Z prc. Th zchralki Prc Frm th pint f viw f intgratd circuit manufacturing, it i dirabl fr th tarting ilicn wafr nt t b intrinic, but rathr t b unifrmly dpd with m hallw lvl impurity, i.., B, P, A, b, tc. Thu, a wll a crytal rintatin, backgrund dping (hnc, majrity carrir typ and ritivity i gnrally pcifid whn wafr ar purchad. Thrfr, it i uual fr th mlt, hnc, th grwn ingt t b intntinally cntaminatd with a knwn quantity f hallw lvl dpant impurity. Thrfr, cntrl f dpant cncntratin and ditributin during a Z grwth prc i f fundamntal imprtanc. A an initial dcriptin f zchralki grwth, it i uual t aum rapid tirring cnditin, which impli that any c impurity that might it in th immdiat vicinity f th grwth intrfac i quickly diprd int th mlt. nquntly, uch cnditin imply that th mlt i thrughly mid and, accrdingly, cncntratin f any impurity i unifrm thrughut th mlt, i.., right up t th frzing intrfac. Phyically, thi crrpnd t vry lw grwth f th crytal uch that th rat that impurity i diprd thrughut th mlt i larg in cmparin t th rat that impurity i incrpratd int th frzing crytal. Furthrmr, n can afly aum that impurity diffuin within th lid crytal itlf i unimprtant inc th diffuin cfficint f impurity in th liquid i many rdr f magnitud largr than th crrpnding lid diffuin cfficint. Th Z grwth prc can b rprntd pictrially a fllw: =0 d l Fig. 4: chmatic diagram f th zchralki (Z prc

8 Th arrw dnt th pulling dirctin. Th dimnin,, i th lngth f crytal pulld frm th mlt, hnc, th ma f crytal lidifid,, i jut A, uch that A i th cr ctinal ara f th ingt (aumd t b unifrm and i th dnity f ilicn. Naturally, l i th vlum cncntratin f lut, i.., dpant impurity, atm in th mlt and i th vlum cncntratin f lut atm in th lid crytal. larly, at any givn pint during crytal grwth and undr th aumptin f rapid tirring, l i unifrm thrughut th mlt; hwvr, i a functin f th pitin alng th bul, i.., a functin f. Accrdingly, if i dfind a th ttal numbr f lut atm within th mlt, thn th diffrntial numbr f lut atm lt frm th mlt du t frzing a diffrntial lngth f crytal i: d Ad d Hr, th ngativ ign i frmally includd t indicat that atm ar lt frm th mlt t th grwing crytal. larly, fr a diffrntial chang in th lngth f th bul, i.., diffrntial pull ditanc, d, a crrpnding diffrntial ma f lid, d, i addd t th crytal. Obviuly, th diffrntial quantiti ar rlatd imply a fllw: d Ad At an arbitrary pint in th crytal grwth fr which a crytal f lngth,, and ma,, ha bn lidifid, th ma f th rmaining mlt i jut th diffrnc,, uch that i th initial ma f th mlt bfr any crytal ha bn lidifid, i.., pulld. Thu, it fllw that th cncntratin f lut in th mlt i givn by: l On frmally lv thi prin fr and cmbin it with th diffrntial prin t btain: d d d l l ( Thi i a diffrntial prin that rlat th cncntratin f impurity atm in th mlt t th ma f th crytal at any tag during th grwth prc. At thi pint, n might naivly aum that th cncntratin f impurity in th mlt and in th lid ar actly qual; hwvr, thi i nt th ca. At a dfinit tmpratur uch ituatin ar dcribd by a thrmdynamic ditributin quilibrium charactrizd by a cntant cfficint,, which i frmally dfind in th prnt ca a th cncntratin rati, / l. In phyical trm, n can rgard thi quilibrium a cnqunc f th fact that impurity atm d nt fit int th crytallin ilicn lattic

9 a wll a ilicn atm. Thrfr, n pct that impurity atm will b incrpratd int th frzing lid at a lwr intrinic rat than ar ilicn atm thmlv and, thu, impurity atm will tnd t b rjctd back int th mlt. Accrdingly, can b pctd t b gnrally l than n. A i tablihd by primntal brvatin and hwn in th fllwing tabl, thi i gnrally fund t b th ca: Dpant B 0.72 P 0.32 A 0.27 b Ga Al In Tabl : Ditributin fficint fr hallw lvl impuriti in ilicn Thrfr, th prcding prin tak th frm: d d Thi prin i radily intgratd dirctly a fllw: d 0 d A with thr paramtr, i dfind a th initial numbr f lut atm in th mlt, which, f cur, i dtrmind by th impl frmula: Hr, i jut th initial impurity cncntratin in th mlt. Fr cnvninc, th intgratin variabl i frmally rplacd with a nw variabl, w, dfind a, hnc: d dw w larly, th indicatd intgratin i lmntary, thu:

10 ln (ln( ln ln Mrvr, it fllw immdiatly frm th lmntary prprti f lgarithm that: ln ln Obviuly, n invrt th lgarithm n ach id t btain: Thi i mr cnvnintly prd in trm f initial mlt wight and impurity cncntratin, which ar gnrally knwn bfr crytal grwth tart, hnc: Furthrmr, ha bn prviuly rlatd t mlt cncntratin, l : l ( Thrfr, it immdiatly fllw that: l Of cur, it i cncntratin f dpant in th lid crytal that i rally f intrt, but thi i trivially btaind frm th ditributin quilibrium: Thu, th impurity cncntratin in th lid and th mlt i dtrmind at all tag f crytal grwth undr rapid tirring cnditin. Thi prin may b rcat a a functin f ingt lngth a fllw: A

11 Dpant ncntratin Rati Phyically, thi quatin dcrib variatin f impurity cncntratin alng th lngth f a Z grwn ingt undr rapid tirring cnditin. larly, at th d nd, i.., 0, f th bul, impurity cncntratin jut crrpnd t th impl ditributin quilibrium. Hwvr, a th grwth prc prcd, impurity i rjctd frm th grwing crytal, and th cncntratin f th mlt incra. Thi cau th cncntratin f impurity t incra in th bul a a functin f ditanc frm th d nd. Thi i hwn in th fllwing figur fr vral valu f (th individual curv ar labld by th crrpnding valu f : ngth Fractin Fig. 5: Dping prfil f a Z crytal auming rapid tirring By dfinitin, th d nd f th bul crrpnd t a lngth fractin f zr. In cntrat, th butt nd i ppit f th d nd (i.., a lngth fractin narly unity and cntain th lat matrial lidifid. Of cur, dpant cncntratin rati a indicatd in th figur i jut /. larly, th clr that i t unity, th mr unifrm th dping prfil btaind. A might b pctd, fr ralitic valu f miing, grwth, and tranprt rat, th rjctin rat f impurity at th grwing crytal intrfac cd th rat at which impuriti can b tranprtd back int th bulk f th mlt. Hnc, th rapid tirring cnditin i nt atifid and brak dwn. nquntly, in th vicinity f th grwth intrfac impurity cncntratin build up abv th valu brvd in th bulk mlt. Accrdingly, thi cau th crytal t b dpd mr havily than wuld b pctd

12 undr rapid tirring cnditin. Thi bhavir can b undrtd by auming that a tagnant bundary layr f thickn,, it btwn th lid/mlt intrfac and th bulk f th mlt. Thrfr, if D rprnt th impurity diffuin cfficint within th mlt and if R i th intantanu pull rat, i.., d/dt, thn auming narly tadytat cnditin, th impurity cncntratin in th tagnant rgin nar th grwth intrfac may b dcribd by th prin: 2 d D 2 d d R 0 d Hr, i lcal lut cncntratin within th mlt and i ditanc frm th grwth intrfac. larly, mut cnvrg t th unifrm cncntratin, l, if bcm larg. Phyically, th firt trm in th prcding prin accunt fr diffuin f impurity atm away frm th grwth intrfac back int th mlt, and i a dirct cnqunc f Fick cnd aw. ikwi, th cnd trm accunt fr rmval f impurity frm th mlt du t frzing. Mathmatically, thi trm ha th apparanc f a nt drift du t m trnal ptntial, but intad phyically dcrib nt mtin f matrial du t pulling th crytal. Indd, thr mut indd b a nt mtin f impurity atm with rpct t th grwth intrfac imply du t lidificatin, i.., in principl, all impurity atm vntually mut pa thrugh th grwth intrfac. Thi amunt t an vrall rlativ mtin f impurity atm that can b rgardd quivalntly a bing du t mvmnt f th grwth intrfac with rpct t a tatinary mlt, r t mvmnt f th mlt with rpct t a tatinary grwth intrfac, r a m cmbinatin f th tw. (Trm dcribing cnvctiv flw f th mlt ar abnt inc th bundary layr i aumd t b tagnant. Naturally, undr cnditin f narly tady tat, diffuin and frzing trm mut ffctivly balanc ach thr. Fr implicity, i idntifid with th drivativ, d/d, thu, th prcding quatin bcm: d R d D larly, thi prin can b trivially intgratd; hwvr, intad f a dfinit intgral it i cnvnint t cntruct an indfinit frm a fllw: ( 0 R D Hr, 0 and ar dfind, rpctivly, a lut cncntratin gradint at th grwth intrfac ( 0 and far away frm th grwth intrfac (. Naturally, numrical valu f 0 and crrpnd t impitin f uitabl bundary cnditin. Accrdingly, a artd prviuly, far away frm th grwth intrfac cncntratin f dpant impurity can b pctd t b unifrm, which impli that vanih. Thu, th prcding frmula can b furthr implifid:

13 R D 0 T dtrmin 0 diffuin and rjctin flu f impurity atm at th grwth intrfac ar aumd t b qual. Of cur, thi i cnitnt with th riginal diffrntial quatin dcribing th ffct f diffuin and frzing in th tagnant bundary layr, and impli that th lut cncntratin prfil acr th grwth intrfac and bundary layr rmain ntially in a tady tat (r at lat nly vry lwly varying n. Thu, fr a unit ara f grwth intrfac, n can writ: D R( l 0 larly, th trm n th lft jut cm frm Fick Firt aw, which rlat diffuin flu linarly t cncntratin gradint. Obviuly, th numbr f impurity atm pr unit vlum rjctd back int th mlt mut jut b th diffrnc f impurity cncntratin in th mlt and in th lid actly at th grwth intrfac. (By dfinitin, l i th impurity cncntratin actly at th grwth intrfac. Thrfr, th trm n th right i th rjctin flu and i jut th prduct f th intrfacial cncntratin diffrnc and th intantanu pull rat, R; hnc, it fllw that: d R ( l d D R D It i a impl mattr t intgrat thi prin acr th ntir bundary layr t btain th prin: l l R ( l D R d p ( l 0 D D R p Hr, th lut cncntratin at th bundary layr dg, i..,, ha bn aumd t b th unifrm bulk cncntratin, l. Of cur, thi cnditin i nt trictly ralizd unl ; hwvr, it i ranabl t aum that i ufficintly larg that th cncntratin fr i nly ngligibly diffrnt frm l. Thi prin i aily rarrangd int a mr cnvnint frm: l l R D At thi pint, n rdfin th rati, / l, a an ffctiv grgatin cfficint,, inc an nrichd bundary layr li btwn th bulk and th grwth intrfac. (Of cur, th rati, / l, mut qual th actual grgatin cfficint,, inc l i dfind a th cncntratin at th actual grwth intrfac. Thrfr, th prcding prin bcm:

14 Obviuly, n lv fr a fllw: R D R D Upn frmally taking th rciprcal th dird prin i btaind, thu: ( R D Thi rult i applid t an actual Z grwth prc by jut ubtituting int prin btaind prviuly fr rapid tirring cnditin. Th paramtr, R/D, i calld nrmalizd grwth paramtr and in practic i dtrmind mpirically fr a givn crytal grwing apparatu. larly, if th nrmalizd grwth paramtr i mad ufficintly larg (fr ampl, by incraing th pull rat, thn tnd tward unity, and thi will rult in a mr unifrm ditributin f dpant alng th lngth f th bul. nvrly, if th pull rat and, thrfr, th nrmalizd grwth paramtr bcm mall, thn tnd tward. Obviuly, thi jut crrpnd t a rturn t rapid tirring cnditin. Bth D and R can b adjutd by a judiciu chic f prc cnditin,.g., tmpratur and pull rat. hat abut? It turn ut that i a functin f th rtatin rat f th bul fr which an mpirical rlatinhip ha bn dtrmind primntally:.8d Hr, i th vicity f th mlt and i rtatin rat. Th pull rat, R, i clly rlatd t th actual grwth rat f th crytal, hwvr, th intantanu grwth rat may diffr frm R bcau f thrmal fluctuatin, uprcling, tc. Indd, if th pull rat i rlativly mall, th intantanu grwth rat may bcm ngativ (thi i calld r-mlting. Thi can advrly affct bth dfct tructur and dping ditributin n a micrcpic cal. In particular, if r-mlting i nt trngly upprd by th u f a ufficintly larg pull rat, th crytal may hibit dfct wirl pattrn and dpant triatin. Mrvr, cnvctiv tranprt within th mlt may al rditribut impuriti nn-unifrmly. Thi i pcially ignificant in th ca f ygn which i dilvd frm th quartz crucibl at th mlt priphry. Indd, bcau f th kind f variatin it i difficult t prduc larg diamtr, lightly dpd Z wafr, i.., with a ritivity cding 00 cm. Rcntly, immrin f th crucibl during crytal grwth in a trng magntic fild (th magntic zchralki r MZ prc ha bn 3 6 2

15 fund t allw cntrl f cnvctiv tranprt, which imprv unifrmity f larg diamtr crytal (>300 mm. A wafr iz incra thi may b pctd t bcm an indutry tandard. Additinally, if th pull rat i lw, th lidifid crytal may b hld fr quit a lng tim abv 950. Thi may allw ufficint tim fr thrmally gnratd micrdfct t frm. Th frmatin f uch micrdfct i ffctivly qunchd if th pull rat cd a rat f rughly 2 mm/min. larly, th nrmalizd grwth paramtr and, hnc, th ffctiv grgatin cfficint can b mdifid by changing rtatin and pull rat during crytal grwth. Indd, it i cmmn cmmrcial practic t prgram grwth paramtr a t btain a unifrm impurity cncntratin vr a larg fractin f an ingt. Accrdingly, crytal grwth prcd in ditinct pha: Firt, during an initial grwth pha, a artd prviuly, crytal diamtr i built up t th dird dimnin. Nt, prgrammd pull and rtatin rat ar applid. Thi rult in a crytal f cntant impurity cncntratin vr a larg fractin f it lngth. Of cur, at m pint th mlt i ubtantially hautd and it bcm impibl t utain a unifrm cmpitin. Thi dfin a third grwth pha during which th crytal i rapidly pulld ut f th mlt. Zn Rfining A ntd at th utt, in Z grwth thr can b cnidrabl carbn and ygn cntaminatin that cm frm th quartz and graphit cmpnnt f th prc quipmnt. In gnral, thi cntaminatin cau n prblm and prhap may b bnficial inc, a brvd prviuly, ultrapur ilicn i actually mchanically mr fragil than rdinary Z ilicn and, al, a will b mad vidnt ubquntly, ygn cntaminatin may b ud t gd ffct t t up an intrnal gttring chm. Hwvr, thr ar m ca fr which ultrahigh purity i dird, viz., -0 cm. Thi matrial i mt cnvnintly fabricatd by zn rfining. Again, t ritrat, within th indutry ubtrat fabricatd thi way ar calld flat zn (FZ wafr. T undrtand zn rfining, upp that jut a in Z grwth, m impurity dilvd in mltn ilicn i in quilibrium with lid crytal. Onc again, th ditributin (r grgatin cfficint,, i dfind: l Of cur, i th impurity cncntratin within th lid and l i th impurity cncntratin in th liquid. Mrvr, again, jut a fr zchralki grwth it i pibl t maintain th ytm in a tady tat, but nt in rigru thrmal quilibrium, uch that an ffctiv ditributin cfficint,, i applicabl intad f ablut ditributin cfficint,. In practic, th baic tchniqu f zn rfining i t pa a lid pic f matrial,.g., an ingt f ilicn, thrugh a circular hating lmnt, viz., ndl y. Thi crat a mltn zn that lwly mv frm n nd f th ingt t th thr. (Of cur, migratin f th mltn zn alng th ingt can b accmplihd ithr by mving th ingt thrugh a fid hating lmnt r by mving th hating lmnt hlding th ingt in a fid pitin. In any ca, a th mltn zn migrat and a a

16 cnqunc f th ditributin quilibrium, impuriti ar cllctd in th mltn matrial and wpt prfrntially t n nd f th ingt. nquntly, th rat that th mltn zn pa thrugh th ingt i analgu t th pull rat in th zchralki prc. Thu, th ffctiv ditributin cfficint huld b ntially dtrmind by th zn migratin rat. ithin thi cntt, n pct that huld b ignificantly largr than (r vn, prhap, apprach unity if th mltn zn i mvd vry rapidly thrugh th ingt and, in cntrat, huld apprach r ntially qual if th mltn zn i mvd vry lwly. Fr clarity, it i intructiv t cnidr a ingl pa zn rfining prc, which may b rprntd pictrially a fllw: =0 d Fig. 6: chmatic diagram f th Flat Zn (FZ prc Hr, i th lngth f th mltn zn, i rdfind a th impurity cncntratin in th rfind ctin f th ingt, and i th impurity cncntratin in th unrfind ctin. Th variabl,, rprnt linar ditanc alng th ingt (with th zn rfining prc initiatd cnvntinally at 0. If i th numbr f impurity atm in th mltn zn and A i th cr ctinal ara f th ingt, thn th diffrntial chang in a th mltn zn pa thrugh th ingt i givn by: d Ad Ad Thi quatin i jut a frmal prin f th diffrnc in th rat that impurity atm ar addd t th mltn zn du t mlting f unrfind matrial t th rat that thy ar lt t th mlt at th frzing intrfac. larly, th impurity cncntratin in th liquid, l, i jut /A. Hnc, if n aum that th ditributin f impurity at th frzing intrfac i gvrnd by th ffctiv ditributin cfficint,, thn it fllw that: d d A (Obviuly, th trm, A, i prprtinal t th rat that impurity atm ntr th liquid at th mlting intrfac and, likwi, / i prprtinal t th rat that impurity atm ar rmvd frm th mlt at th frzing intrfac. If n aum that th initial impurity cncntratin,, i unifrm thrughut th unrfind ctin f th ingt, thn,

17 thi diffrntial quatin i aily intgratd by man f an pnntial intgrating factr. Accrdingly, if n dfin a nw lut paramtr, Q, a p( /, thn n btain: Thrfr, it fllw trivially that: dq d A A Q( Q(0 Rcating thi prin in trm f yild th rult: A ( (0 Th bundary cnditin fr (0 mut jut b A inc mlting an unrfind prtin f th ingt f vlum A frm th initial mltn zn. Upn ubtitutin, n btain: A ( ( If n u th dfinitin f l and th ditributin quilibrium, it fllw that: ( ( Thi prin dcrib th cncntratin f initially unifrmly ditributd impurity aftr ingl pa zn rfining. Furthr purificatin can b achivd by additinal zn rfining. Th quatin dcribing th prc i jut th am a bfr, hwvr, th initial impurity cncntratin i n lngr unifrm; hnc, th riginal diffrntial quatin mut b mdifid a fllw: d d ( A Hr, ( i an arbitrary (i.., nn-unifrm initial impurity ditributin. Again, thi prin i rcat in trm f Q: d d dq Q ( A d Q

18 Thrfr, n btain: A d dq ( Naturally, thi diffrntial quatin i aily intgratd frmally t giv: d A Q Q 0 ( (0 ( Fllwing th rult fr cntant, n idntifi (0 Q a (0 A and rcat thi prin in trm f : A d A (0 ( 0 It immdiatly fllw frm th ditributin quilibrium at th frzing intrfac that th lut cncntratin in th rfind ingt i givn by th prin: d (0 ( ( 0 Of cur, t dtrmin ( numrically, prir knwldg f ( i rquird. Naturally, ( may b th rult f a prviu zn rfining pa r may b a grwnin ditributin du t a particular t f prc paramtr fr a Z grwth prc. ithin thi cntt, if n ubtitut th cncntratin prfil btaind prviuly fr ingl pa zn rfining f an initially unifrm ingt int th prcding prin, n btain: d 2 0 ( ( Hr, ( i vidntly th cncntratin prfil f th ingt fllwing tw zn rfining pa. larly, th intgral ar all f lmntary frm and can b cntructd plicitly t giv th rult: ( ( 2 Of cur, rtain th uual dfinitin a initial unifrm impurity cncntratin in th ingt. Naturally, n can cmpar thi prin fr tw pa t th ingl pa rult

19 Dpant ncntratin Rati and al t th initial unifrm impurity cncntratin. larly, at th tarting, i.., d, nd f th ingt, th impurity cncntratin wa, f cur, jut prir t zn rfining. Aftr a ingl pa, thi i rducd t. Aftr tw pa thi i furthr rducd t 2. larly, inc i mallr than n, thi hw that th d nd f th ingt i prgrivly purifid. ncntratin prfil fr n and tw zn rfining pa with variu ffctiv ditributin cfficint ar hwn in th fllwing figur: Zn ngth Fig. 7: Dping prfil f an FZ crytal fllwing n (narrw lin and tw (wid lin zn rfining pa Of cur, th dpant cncntratin rati i, again, jut /. Th hrizntal ai i th numbr f zn lngth rfind. (Hnc, by dfinitin, th d nd f th ingt crrpnd t zr zn lngth. On brv that in cntrat t th zchralki prc fr which an ffctiv ditributin cfficint nar unity i dird, in zn rfining bttr rult ar btaind th mallr th valu f. (Of cur, can nvr bcm mallr than itlf. larly, thi impli that a lw rat f migratin f th mltn zn thrugh th ingt i dirabl. Phyically, zn rfining litrally wp impurity atm tward n nd f th ingt, i.., th nd tward which th mltn zn mv. inc, n impurity i phyically rmvd frm th ingt thi nd actually bcm mr impur. Hwvr, zn rfining prc paramtr can b t up in uch a way that th impur nd i a ranably mall part f th ttal vlum f th ingt. larly, if zn rfining i rpatd many tim, th bulk f an ingt can b rfind t ultrahigh purity and th impur nd can imply b

20 rmvd rulting in a larg amunt f highly rfind matrial. Of particular imprtanc i th prductin f wafr with lw ygn cntnt inc th cannt b prducd uing th Z prc. Of cur, zn rfining d nt lav impurity unifrmly ditributd. If thi i a rquirmnt, thn nc th impur nd ha bn rmvd, th ingt can b hat tratd t rditribut impurity mr unifrmly. T ritrat, th main advantag f th flat-zn prc i th vry lw impurity cncntratin in th ilicn crytal. In particular th ygn and carbn cncntratin ar much lwr a cmpard t th Z prc, inc th mlt d nt cm int cntact with a quartz crucibl, and n ht graphit cntainr i ud. A a practical mattr, FZ ingt ar prducd frm an initial plycrytallin ilicn ingt, which i dd with a crytal f dird rintatin at th tart f th prc. Altrnativly, a mncrytallin ingt can b furthr purifid by zn rfining. Evn, th FZ prc i mr pniv than thn Z prc, and, at prnt, crytal diamtr i limitd t 200 mm.