Analytical and numerical studies of the meniscus equation in the case of crystals grown in zero gravity conditions by the Dewetted Bridgman technique

Size: px

Start display at page:

Download "Analytical and numerical studies of the meniscus equation in the case of crystals grown in zero gravity conditions by the Dewetted Bridgman technique"

Elwin Lyons
6 years ago
Views:

Anlytil nd numril studis of th mnisus qution in th s of rystls grown in zro grvity onditions by th Dwttd Bridgmn thniqu S Epur Abstrt On th physil point of viw, th dwtting phnomnon is govrnd by th

1 Anlytil nd numril studis of th mnisus qution in th s of rystls grown in zro grvity onditions by th Dwttd Bridgmn thniqu S Epur Abstrt On th physil point of viw, th dwtting phnomnon is govrnd by th Young-Lpl qution of pillry surf in quilibrium, whih is nonlinr prtil diffrntil qution of sond or Strting from this qution, n nlytil xprssion of th mnisus surf in zro grvity ondition ws stblishd, lding to importnt informtion bout th mnisus shp, usful for furthr stbility nlysis of th growth pross Th nlytil rsults wr vlidtd by th numril studis Thrfor, th Young-Lpl qution hs bn solvd numrilly, in th xi-symmtri s, using th dptiv 4 th or Rung-Kutt mthod for InSb rystls Kywords Dwttd Bridgmn rystl growth thniqu, Growth from th mlt, Nonlinr prtil diffrntil qution, Young-Lpl qution C I INTRODUCTION RYSTALS, usd s snsors, s lsr rdition sour dttors or solr lls, r ssntil omponnts of mny high thnology pprtuss produd in th optoltroni industry Th qulity of this kind of pprtus dpnds, on th qulity of th ggrgt rystls, whih n b obtind by diffrnt growth mthods Bfor its utiliztion in nginring, th rystls r onstrind to som supplmntry mhnil prosss (utting, polishing) for bringing thm to th dsird form Ths prosss r gnrting dfts nd mtril losss, so th finl produt hs low qulity nd it is mor xpnsiv For this rson thos growth mthods r prfrrd whih llow obtining th rystl dirtly in th finl dsird form (without dditionl mhining) nd with miniml dfts Thniqus of rystl ltrl surf shping without ontt with th ontinr wlls r prfrrd: Dwttd Bridgmn (DW), Edg-dfind Mnusript rivd Otobr 31, 009 This work ws supportd by th Romnin Ntionl Univrsity Rsrh Counil (Grnt PN II 131/ ) nd th Europn Sp Agny (Mp-CdT progrm), th Cntr Ntionl d Etuds Sptils (CNES) S Epur is with th Computr Sin Dprtmnt, Wst Univrsity of Timisor, Blv V Prvn 4, Timisor, 3003, ROMANIA nd Grnobl INP, SIMP-EPM, Phlm BP 75, F-3840 Sint Mrtin d Hrs, FRANCE (orrsponding uthor phon: ; fx: ; -mil: simon_pur7@ yhooom) film-fd growth (EFG), Czohrlski, Floting-zon; th bsn of ontt btwn th rystllizing substn nd ruibl wlls llows on to improv rystl struturs nd to s th mhnil strss lvl Clssil Bridgmn mthod involvs hting polyrystllin mtril bov its mlting point in ruibl nd slowly ooling it from on nd whr sd rystl is lotd (Fig 1 ()) Singl rystl mtril is progrssivly formd long th lngth of th ruibl Th disdvntg of this thniqu is tht th rystl ontts th ruibl wll, whih gnrlly rsults in inrsing th mhnil strsss, impurity lvl, nd dft dnsity in th grown rystls Th disdvntg n, howvr, b ovrom by th dwtting solidifition thniqu Fig 1 Shmti Bridgmn (), dwttd Bridgmn (b) rystl growth systms nd photogrph of n ingot showing tthd nd dthd rgions () Phnomnon of dwtting is hrtrizd by th lssil Bridgmn thniqu, but th rystl is grown without ontt with th ruibl wlls thnks to th stbility of smll liquid mnisus (Fig 1 (b)) rting gp btwn th rystl nd th ruibl wll [1] Issu 1, Volum 4,

2 This phnomnon ws first obtind spontnously, in sp xprimnts during th Bridgmn solidifition of InSb prformd on th Skylb-NASA mission-1974 [-3] Numrous othrs Bridgmn rystl growth xprimnts in sp showd th sm bhviour [4] In dwtting Bridgmn thniqu thr r two problms of intrst [5]: - Wht is th rystl-ruibl gp thiknss, thrfor th rystl rdius, r r? - Wht is th shp of th mnisus? This shp is rltd to th stbility of th pross Th min purpos of th prsnt ppr is to prform nlytil nd numril studis for th mnisus surf in zro grvity onditions, strting from Young-Lpl s qution [6]-[7] of pillry surf in quilibrium nd to stblish th proprtis of th funtion whih dsribs th mnisus surf, lding to importnt informtion for th stbility nlysis of th growth pross [8]-[10] z'' z' + + (5) 3 1 γ b 1+ ( z' ) r 1+ ( z' ) whr /b is du to th urvtur t th top whih dpnds on th ontt ngl θ nd on th mpoul rdius, r Un mirogrvity ondition it n b writtn s: 1 osθ [11] b r II MENISCUS SURFACE S EQUATION Th qution of pillry surf in quilibrium in th bsn of xtrior prssur is givn by th funtion: z z( x,y) (1) whih vrifis th Young-Lpl qution with prtil ivtivs: z z z z z z z y x x y x y x y () 3 ρlg H z z z γ b x y whr Δ P P Ph rprsnts th prssur diffrn btwn th old nd hot sids of th smpl, θ - th ontt ngl, H - th totl lngth of th mlt nd solid, ρ - th dnsity of th liquid, g - th grvittionl lrtion, γ - th surf tnsion of th mlt, r - th mpoul rdius nd th trm is du to th urvtur t th top [11] b Whn rfrring to systm of oordints s in Fig, du to th rdil symmtry, nd imposing z indpndnt of th polr ngl: x r os β y r sin β, r > 0, β [ 0, ] (3) z z( r) th mnisus qution is obtind by th rottion round Oz xis of th urv k whih stisfis th qution: z'' z' ρlg( H z) + + (4) 3 1 γ b 1+ ( z' ) r 1+ ( z' ) In zro grvity onditions, th Young-Lpl qution boms: l Fig Dwtting onfigurtion in mirogrvity onditions Th solution of (5) should stisfy th wtting bouny ondition: z( r) l+ h,z' ( r) tn θ, θ, (6) nd th hivmnt of th growth ngl: z( r) l,z' ( r) tn α, α 0; (7) whr l rprsnts th solid-liquid intrf oordint nd h is th mnisus hight Eqution (5) n b writtn s r z'' + z' 1 + ( z' ) osθ 3 r r γ 1+ ( z' ) whih is quivlnt to r z' osθ r 1 ( z' ) + r γ i, by intgrtion: r z' r osθ z' r γ Squring th bov rltion givs: Issu 1, Volum 4,

3 ( z' ) osθ r r + 1 r γ osθ r r r + 1 r γ from whr is obtind osθ r + 1 r z' ( r γ ) ± osθ r r + 1 r γ As th funtion z( r ) is stritly inrsing on [ r;r ] (8), whr r rprsnts th rystl rdius, in (8) th positiv sign should b hosn Th onstnt 1 is dtrmind from th bouny ondition z' ( r) tn θ, lding to: osθ r r + r z' ( r γ γ ) osθ r r r + r γ γ Th nlytil xprssion of th mnisus n b obtind intgrting rltion (9) As th intgrl n b xprssd using lmntry funtions only in som prtiulr ss, furthr two diffrnt ss will b trtd sprtly: Δ P 0 nd 0 Cs I: Δ P 0 On th physil point of viw, this mns tht thr is onntion btwn th old nd hot sids of th smpl, so tht th prssurs P nd P r qul h In this s (9) boms r osθ z' ( r ) (10) r r osθ Intgrting (10) givs: 1 z( r) r r os os θ θ + Using th ondition z( r ) l+ h, th nlytil xprssion of th mnisus surf in zro grvity, whn Δ P 0 is obtind: 1 z( r) ( r r os θ r sinθ) + l+ h, (11) osθ whr r [ 0,r ] Thus, by th rottion of th urv k round th vrtil oordint Oz, th mnisus qution is obtind: (9) [ 0 ] [ 0 ] x r os β, β, y r sin β, r,r 1 z ( r r os θ r sinθ) + l + h osθ whr θ, Furthr, som proprtis of th funtion z( r ) will b prsntd Proposition 1: Funtion z( r) whih dsribs th mnisus surf hs th following proprtis: (i) z( r ) is stritly inrsing for r [ 0,r ] (ii) z( r) is onvx for r [,r ] 0 ; Proof: (i) Driving th rltion (11) givs: 1 ros θ z ( r) osθ r r os θ rosθ > 0, ( ) r [ 0,r ] r r os θ (ii) In or to show th onvxity of th funtion z( r ) th sign of th sond ivtiv should b studid for [ 0,r ] : rosθ r osθ z'' ( r) 3 r r os θ ( r r os θ) > 0, with θ, Thus, funtion z( r) is onvx From th bov proprtis it rsults tht in zro grvity ondition nd null prssur diffrn th mnisus is lwys globlly onvx nd this is in grmnt with th numril rsults obtind in th s of InSb (Fig 3) Fig 3 Mnisus shp z( r) for r m InSb, θ +α , Issu 1, Volum 4, 010 5

4 Thiknss of th rystl-ruibl gp Dwtting ours whn th growth ngl α 0, (th ngl btwn th tngnt to th mnisus surf nd th vrtil) is hivd t lst t on point on th mnisus surf, i whn th qution: φ ( r) α (1) hs t lst on solution in th rng ( 0,r ); whr φ is th ngl btwn th pln z 0 nd th tngnt pln to th mnisus t point P( r,β ) For this ngl th qulity tnφ z' ( r) holds nd hn informtion onrning th hivmnt of th growth ngl is givn by th qution: rosθ tnφ r r os θ Rwriting th bov rltion s: sinφ rosθ 1 sin φ r r os θ it oms r osθ sinφ (13) r whih is quivlnt to r osθ φ rsin, for r [ 0,r ] (14) r Rltion (14) givs ondition of dwtting whih dpnds on th growth ngl α nd ontt ngl θ Th positivity of th ivtiv dφ osθ > 0, θ, r r os θ givs tht th funtion φ ( r) is stritly inrsing for r [ 0,r ] Tking into ount this monotony nd th bouny ondition z' ( r ) to φ tn θ whih is quivlnt θ, th growth ngl is hivd if θ to α, lding to r φ ( r) ss from α < θ, i, θ + α > In th opposit s whn θ + α <, th growth ngl n not b hivd du to th monotony of φ ( r) Un th hypothsis tht th growth ngl ritrion is stisfid, i, θ + α >, th Eqs (1) nd (13) giv: ( r ) osθ sin α (15) r whr rprsnts th rystl-ruibl gp thiknss nd r r th rystl rdius From (15) rsults th gp thiknss formul [11]: osθ + osα r (16) osθ vlid un zro grvity ondition, Δ P 0, nd θ + α > Thorm: For givn mpoul rdius r nd Δ P 0, if θ, nd α 0, stisfy th inqulity θ + α >, thn th mnisus hight in zro grvity is onstnt nd is givn by th following rltion: r h ( sinθ sin α) (17) osθ Proof: Rltion (17) is obtind imposing to rltion (11) th z r l, whih ondition of th growth ngl hivmnt givs: 1 h r r os θ r sinθ osθ nd by rpling r r, whr is givn by (16) Cs II: 0 Th physil mning of 0 is tht th gss btwn th old nd hot sids of th smpl do not ommunit, so tht prssur diffrn xists In or to obtin th mnisus qution, rltion (9) should b intgrtd, but if 0 th intgrl n not b xprssd using lmntry funtions Thn, for obtining informtion onrning th mnisus shp, hivmnt of th growth ngl, nd gp thiknss, qulittiv studis should b prformd tnφ z' r in rltion (9), givs: Introduing r osθ r r sinφ + (18) r γ γ whih is quivlnt to r osθ r r φ rsin + (19) r γ γ r,r for ny [ ] 0 In wy similr to prvious lultions, th sign of th ivtiv d φ will giv informtion bout th shp of th mnisus, nd bout th ondition whih should b imposd on th sum of th ontt nd growth ngls suh tht hivmnt of th growth ngl is fsibl Thus, iving th rltion (19) givs: Issu 1, Volum 4,

5 dφ 1 osθ r 1 1 r r+ r γ γ r osθ r 1 r r γ γ r Th sign of this ivtiv dpnds on th sign of th xprssion dpnding on r nd Δ P : osθ r E( r, Δ P) r (1) r γ γ nd thn, th following ss should b onsid: (i) If ( ;0], thn E( r, Δ P) > 0 nd hn dφ > 0 Morovr, d z 1 dφ > 0, i, th mnisus is os φ globlly onvx, nd th growth ngl n b hivd only if θ + α > (0) (ii) If γ osθ 0;, thn th mnisus hngs its r urvtur (onv-onvx) t th point r I r r γ osθ r, i, E( r, P) 0 I Δ whih is quivlnt to dφ d z ( ri ) ( r ) 0 nd th growth ngl n I b hivd on or twi, dpnding on its vlu (iii) If γ osθ ; + thn E( r, Δ P) < 0 nd r hn dφ < 0 In this s th mnisus is globlly onv, i, d z < 0, nd th growth ngl n b hivd only if θ + α < Un th hypothsis tht Δ P, θ nd α r hosn suh tht th growth ngl n b hivd, th growth ngl ritrion (1) is stisfid somwhr long th mnisus From (18): osθ sin α ( r ) r () r 1 ( r ) + γ γ r th following gp thiknss formuls [11] r obtind: γ osα + γ osθ +Δ Pr + δ 1 r Pr γ osθ Δ + γ osα + γ osθ +r δ r Pr γ osθ Δ + whr δ γ os α +Δ P r + r γδ P osθ (3) (4) Th first gp formul (3) is vlid whn th growth ngl is hivd on th onvx prt of th mnisus, nd th sond formul (4) is vlid whn th hivmnt of th growth ngl ours on th onv prt of th mnisus [1] III NUMERICAL RESULTS Th rsults obtind solving numrilly th Young- Lpl qution by Rung-Kutt mthod for InSb rystls grown in zro grvity by th dwttd Bridgmn thniqu (mtril prmtrs usd in numril omputtions: 3 γ 04 N m, ρ kg m, r m H 008 m), onfirm th bhviors obtind through th qulittiv study: (i) If ( ;0], thn th mnisus is globlly onvx nd th growth ngl n b hivd on In th s of hivmnt of th growth ngl th gp thiknss is givn by 1 xprssd in (3) Th numril rsults rvl this bhviour for Δ P 50 ( ; 0] nd θ + α >, s it n b sn in Fig 4 showing tht th mnisus is globlly onvx nd tht th growth ngl is hivd Th omputd gp thiknss r - r m is qul to th on givn by formul (3), i, m orrsponding to prssur diffrn Δ P 50 P nd θ + α for InSb Th pl whr th growth ngl (5 ) is hivd is shown by th blk dot Fig 4 Mnisus shp z(r) () nd mnisus ngl ( r) Issu 1, Volum 4,

6 (ii) If γ osθ 0;, thn th mnisus is onvonvx (hs n inflxion point) Whn th growth r ngl is hivd on th onv prt, th gp thiknss is givn by xprssd by (4), nd on th onvx prt, th gp thiknss is givn by 1 xprssd by (3) Th numril rsults onfirm ths bhviours Th mnisi r onv-onvx nd th growth ngl n b hivd on or twi: () for θ + α < nd γ osθ Δ P 8 0; 0; 8 6 r th growth ngl is not hivd (s Fig 5), but for P 0 ( 0; 8 6) growth ngl is hivd on, s it n b sn on Fig 6; (b) for θ + α > nd γ osθ Δ P 30 0; 0; r ngl is hivd twi (s Fig 7) Δ th th growth orrsponding to prssur diffrn Δ P 0 P nd Fig 6 Mnisus shp z(r) () nd mnisus ngl ( r) θ + α for InSb Th pl whr th growth ngl (5 ) is hivd is shown by th blk dot orrsponding to prssur diffrn Δ P 8 P nd Fig 5 Mnisus shp z(r) () nd mnisus ngl ( r) θ + α for InSb Th pl whr th growth ngl (5 ) is hivd is shown by th blk dot Issu 1, Volum 4,

7 Fig 7 Mnisus shp z(r) () nd mnisus ngl ( r) orrsponding to prssur diffrn Δ P 30 P nd θ + α for InSb Th pls whr th growth ngl INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES (5 ) is hivd r shown by th blk dots Th bov figurs show tht th mnisi r onvonvx, nd thr r situtions whr th growth ngl is not hivd, or th growth ngl is hivd on, or th growth ngl is hivd twi If th growth ngl is hivd on th onv prt of th mnisus, thn th omputd gp thiknss in Fig 6 r -r m is qul to m givn by formul (4) nd in Fig 7, r -r m is qul to m If th growth ngl is hivd on th onvx prt of th mnisus, thn th omputd gp thiknss r -r m is qul to m givn by formul (3), s n b obsrvd in th Fig 7 γ osθ (iii) If ; +, thn th mnisus is r onv nd th growth ngl n b hivd on In th s of hivmnt of th growth ngl th gp thiknss is givn by xprssd in (4) Th numril rsults show tht th mnisus is onv, nd tht for θ + α <, γ osθ Δ P 40 ; + [ 8 6; + ) th growth r ngl is hivd (Fig 8) Th omputd gp thiknss r -r m is qul to m givn by formul (4) IV DISCUSSION Un zro grvity onditions th prssur insid th liquid is imposd by th hot fr surf of th liquid nd dpnds only on th ruibl rdius r nd on th ontt nglθ (Fig ) Thn th urvtur of th mnisus t th solid-liquid intrf is totlly fixd Th xprimntl obsrvtions un mirogrvity onditions hv shown, tht th rystl-ruibl gp is rmrkbly stbl [5] whih is in grmnt with th bov nlysis: in mirogrvity, th mnisus is onvx (i, th sond ivtiv of th funtion whih dsribs th volution of th mnisus hight is positiv) s its urvtur is imposd by th mlt fr surf t th hot sid Only in s of lrg prssur diffrn Δ P, th shp of th mnisus t th liquid- solid -gs tripl lin n b onv orrsponding to prssur diffrn Δ P 40 P nd θ + α for InSb Th pls whr th growth ngl (5 ) is hivd r shown by th blk dots Fig 8 Mnisus shp z(r) () nd mnisus ngl ( r) ACKNOWLEDGMENT Author is grtful to th Europn Sp Agny (Mp-CdT progrm), th Cntr Ntionl d Etuds Sptils (CNES) nd th Romnin Ntionl Univrsity Rsrh Counil (Grnt PN II 131/ ) for th finnil support of this projt REFERENCES [1] T Duffr, I Prt-Hrtr, P Dussrr, Cruibl d-wtting during Bridgmn growth of smiondutors in mirogrvity, Journl of Crystl Growth, Vol 100, 1990, pp [] A F Witt, H C Gtos, Pro Sp Prossing Symp MSFC, Albm, NASA M74-5, Vol 1, 1974, pp [3] H C Gtos, A F Witt t l, Skylb Sin Exprimnts Pro Symp 1974, Si Thnol Sris, Vol 38, 1975 p7 [4] L L Rgl, W R Wilox, Dthd solidifition in mirogrvity-a rviw, Mirogrvity Si Thnol, Vol X1/4,1998, pp [5] T Duffr, L Syll, Vrtil Bridgmn nd dwtting, In Crystl growth prosss bsd on pillrity, Wily-Blkwll, in prss, 009 [6] P S Lpl, Trité d méniqu élst ; supplémnts u Livr X, Eouvrs Complèts Vol 4, Guthir-Villrs, Pris, 1806 [7] T Young, An ssy on th ohsion of fluids, Philos Trns Roy So London, Vol 95, 1805, pp65-87 [8] L Brsu, Nonlinr bouny vlu problm of th mnisus for th pillrity problms in rystl growth prosss, in Prodings of 11 th WSEAS Intrntionl Confrn on Mthmtil nd Computtionl Issu 1, Volum 4,

Mthods in Sin nd Enginring (Bltimor, USA), pp 197-0 (009) [9] L Brsu, Nonlinr bouny vlu problm of th mnisus for th dwttd Bridgmn rystl growth prosss, Intrntionl Journl of Mthmtil Modls nd Mthods in

8 Mthods in Sin nd Enginring (Bltimor, USA), pp (009) [9] L Brsu, Nonlinr bouny vlu problm of th mnisus for th dwttd Bridgmn rystl growth prosss, Intrntionl Journl of Mthmtil Modls nd Mthods in Applid Sins, to b publishd (009) [10] S Epur, T Duffr, L Brsu, On th pillry stbility of th rystlruibl gp during dwttd Bridgmn pross, Journl of Crystl Growth, to b publishd (009) doi:101016/jjrysgro [11] T Duffr, P Boiton, P Dussrr, J Abdi, Cruibl d-wtting during Bridgmn growth in mirogrvity, II Smooth ruibls, Journl of Crystl Growth, Vol 179, 1997, pp [1] L Brsu, S Epur, T Duffr, Mthmtil nd numril nlysis of pillrity problms nd prosss, in Crystl growth prosss bsd on pillrity, Wily-Blkwll, 010, h8 Simon Epur ws born in Romni in 1980 Sh is o-tutoril PhD studnt (Wst Univrsity of Timisor, Romni- Grnobl Institut of Thnology, Frn, Thsis: Stbility of th rystls grown by Dwttd Bridgmn mthod- dfns in 010) Th rsrh intrsts inlud: prtil diffrntil qutions; ordinry diffrntil qutions; nonlinr modls in pplid sins; modling of th rystl grown from th mlt: EFG nd Dwttd Bridgmn thniqus Hr rsrh hs ld to 6 pprs publishd in rfrd journls, 1 hptr in book (Wily & Sons) nd 3 pprs publishd in prodings Issu 1, Volum 4,

HIGHER ORDER DIFFERENTIAL EQUATIONS

HIGHER ORDER DIFFERENTIAL EQUATIONS Prof Enriqu Mtus Nivs PhD in Mthmtis Edution IGER ORDER DIFFERENTIAL EQUATIONS omognous linr qutions with onstnt offiints of ordr two highr Appl rdution mthod to dtrmin solution of th nonhomognous qution