5. Growth mechanism. 5.1 Introduction. Thermodynamically unstable ambient phase stable crystal phase

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1 5. Itrodutio 5. Growth mhim Thrmodmill utbl mbit h tbl rtl h Diffiult of ultio du to th didtg of urf rg Ar of ulu lrgr th th ritil o b flututio K iu: growth loit driig for iorortio rt t th itrf btw olid d mbi 65

2 5. Hlf rtl it Numbr of hmil bod 6 rt ighbor tom Adortio t kik 3 Hollow 5 Kik it r dir: hlf rtl it Imortt for rtl growth 5.3 Mimum growth rt 5.3. Vor growth Numbr of iidt rtil r uit tim duit r; If llimigig rtil r iorortd to th rtl; i i out t ir out At quilibrium, q out q i i N im N im mkt mkt mkt

3 At urturtio Numbr of rtil i rtl; N dn dt i out mkt 5.3 omlt odtio oditio Growth rt; Hrtz Kud q. mimum growth rt R m bf dn dt mkt 5.4 Proortiol to driig for dgr of urturtio / f ; Ar of o rtil o rtl f b; Thik of o molulr lr R m U mkt kt 5.4' 67

4 5.4 Diffuio /dortio ro d odtio ftor i im omlt odtio i rthr il i R im rt of migrtig tom l th ubtrt bfor big iorortd to rtl mkt mkt R m m ridtil tim ; τ Dortio rt E / kt ; odtio ftor ddt o dgr of urturtio, rtl f, rough t. Boltzm ftor E ibrtio frqu of dorbd molul dortio rg E / / kt/ E Sublimtio rg of i 5.3 ; tom dit urf ottil 676 l/g =.5 V, E ~.6 V, ~ 3 /, T = 58 K 5, ~ 8 68

5 5.4. dit of dorbd molul; d im im / 5.4 dt t / t / A B B / B t / A t im im / B A mkt im t / A/ B 5.6 mkt t / / t / urf diffuio lgth; rdom wlk diffuio qutio E Dortio Ltti ott Ed / / l l l + At t =, l =, t it l ftr t l / l / tim to right tim to lft l / l / l / l / l t

6 Wl, ; robbilit o it l ftr t dirt robbilit ditributio dirt ribl / / l l l + Dimio W l,, W l, [{ W l, W l, } { W l, W l, }] W l otiuou ribl t ', l; { W l, W l, } W, t W l, W, t ' W, t '... t W, t W l, W, t W, t W, t W, t 5.8 ' 5. t W, t t o t bfor W l, W, t 5.7 { W l, W l, W l, } otiuou robbilit ditributio 5.8 W, t W, t D, D ' ' Ltti ott O t W, t; robbilit o it t tim t 5.9 W, t... rdom wlk ld to diffuio qutio Diffuio qutio ' D; Diffuio offiit 7

7 for t it l ftr t l l l + l / tim to right origi t, l, l W l, l l 3 Dimio W l, l, ; W l, dl W l, dl W, t W, t D t 3 Dimio l l / z t ' W l, 5.6 W, t C Gu ditri., Dt Dt 5.8 l { 5.5 W, t, z l /! l}!{ D 6 ' Eiti rltio l}! tim to lft /6 /6 /6 /6 /6 /6 73

8 Diffuio lgth; l l l+ E Ed D 5.9 ; m ridtil tim o t; ' / ' / kt E d 5.3 / kt / E D ' D E d / kt E / kt / E E d / kt 5.3 E Ed / kt 5.3 grll, E E d Ag f ltti E 3 for, 4 for rg of hmil bod / kt 5 t K for, for 74

9 5.5 Kol mhim 3 D diffuio of rtil i mbi Surf diffuio of dorbd rtil Moig of kik it Ad of t Comltio of iomlt lr rt dtrmiig oditio Formtio of iomlt lr o flt urf Shmti iw of ioltd igl hight t growig through urf diffuio. i th flu of tom from th bulk or h towrd th rtl urf, i th flu of dtom diffuig to th t d i th rg ig btw kik of ig. i th m dit ord b th dtom durig thir lif tim o th urf rood b Kol lbortd b Burto, Cbrr, Frk BCF thor; Bi ro of or growth t dig ro; 5.5 r of w iomlt lr Frk mhim

6 τ;m ridtil tim fr from t it t tit ; ; 5.34 5.

10 5.5. Ltrl growth ro [A] d of igl t D modl rur ;urf otrtio ;urf otrtio t t ;urf otrtio fr from t, / mkt 5.6 τ;m ridtil tim fr from t it t tit ; ; diffui flow dit log th urf dd D D, D grd D 5.35 d grd, +d 76

11 qutio of otiuit from riil of mttr ortio t t di di D, di D 5.37 t Dgr of urturtio / Udr it of rtil flow from mbi or h D, D / D 5.38 Dgr of urturtio o th urf D D ot. flow dit from mbi uiform, iddt d Iid from mbi iddt / 5.33 Dortio to mbi / ddt ddt / 77

12 5.4 Stti qutio di di t udr th oditio tht t dig loit i muh l th rtil loit 3 D m for imliit D boudr oditio [ / / 3 / m / d / d ] [ / 5.33 ] 5.48,, 5.34 : dgr of urturtio di D : dgr of urturtio o th urf 78

13 Numbr of rtil rhig th t r uit lgth d uit tim r; Vloit of t / rl dit of ltti oit D d D D D D d ; itrl of t D / E / kt / ;urf dit of dorbd rtil udr quilibrium W / kt W / kt 5.5, [B] multi t itrl λ W ; rg to mo tom from bulk to dorbd tt W W E W; ublimtio rg 5.49 d d 5.46 λ A B / / 5.5 / / 79

14 5.5 /; ; d d Boudr oditio 5.57 th D B A D B A D d d D, 5.56, oh / oh /, /, , / ' / /, 5.33, / / B A A A B B A d d B A B A / / 8

15 multi t itrl λ th / / 5.57' th 3 For mll λ, omtitio btw dt t our to tur imigig rtil. Ol λ// of rtil i ilbl.

16 [C] urd t r D ultio ; t urf rg r lgth, f ; r of o rtil, ; ritil rdiu Fr rg hg of D domi r G r 5.58 f irumf dg r dr f G f f; 4.5 3D f 5.59 urf rg 5.6 A urd t h didtg omrd with tright t du to urf rg G bulk fr rg urf rg r kt log f / kt log f / kt log 8

17 ; quilibrium rur for urd t rdiuρ ; quilibrium rur for tright t rdiu 5.6,, kt f kt f kt f kt f Dr i dgr of urturtio 5.6 ltrl growth ffti dgr of urturtio 83 o; dlrt o; lrt o tright

18 5.5. Growth rt rmtr to hrtriz th growth mod ; loit of t forwrdig tim to omlt o lr formtio tio 5.5. J; frqu of grtio of ritil ulu r uit r, uit tim tio 4.4, 4.5 J;mll,;lrg o ulu grtd, thu omltio of o lr igl ultio J;lrg,;mll r of multi domi o th rtl urf multi ultio J;mll,;lrg J;lrg,;mll 84

19 witig tim wig tim [A] igl ultio t w / JS t S / S; urf r ; t loit J [m ] S S J;mll,;lrg t w t R / t JS w ftr log witig tim, rid formtio of o lr triggrd b ulu grtio growth rt; 5.65 ddt o th urf r [B] multi ultio t w t o ml of r ddt growth rt growth of ulu + ol omltio of o lr witig tim of formtio of dditi ulu o o ulu; t w ' t w J 3 J /3 ' t dt, t 5.66 w R t ' w 3 /3 /3 J / iddt of urf r S 85

20 R t ' w 3 /3 /3 J / W / kt Δ 3 6 J 4r * mkt 8 kt kt 3kT 4.5 growth our for Δμ>Δμ tio 4.4 J R R m Δ Δ, urturtio, urturtio Δ 86

21 5.6. rw dilotio d irl growth D ultio; r low Crtl growth; ot o lt 5.6. Frk mhim SS // lidig dirtio rw dilotio Qui rft rtl: rft t th rgio roud SS Rt limitig ftor ; D ultio, ; diffuio or ul of rtil 5.6. growth rt Ad with ST kt tright i imoibl, bu th outr rgio hould grow ftr S S loit ditributio o to mk lrgr th 5.6

22 olr oordit R r, From 5.6, 5.7, / r r' rdiu of urtur 5.7 r r' rr' whr dr d r r', r'' d d r dr dt 5.7 dr d r' d dt r r rr' ro / / r r' r r' r r' rr'' rr' 3/ r r' r r' For mll r, r d r ould b gltd with rt to thir driti / 5.74 r' 5.74 r', / r Arhimd irl r' λ ; width of irl r r ott width Et olutio f 9f kt 5.78 width i irl roortiol to 88

8 5.83 5.84 Surturtio mll; σ, lrg σ 5.8 5.8 R R m R irl growth R ul

23 growth rt R / t' t' / R / 5.79 from 5.5 d 5.57 W / ktth from 5.79, 5.8, 5.8 kt f f / kt W / kt R W / kt th th R W / kt R W / kt Surturtio mll; σ, lrg σ R R m R irl growth R ul ultio + ltrl growth Aiotro hgol ltti Lrd mtril GS / H Si Sitm Ui. K. Uo 89

24 Imortt rmtr Growth rt Kol ro irl growth Summr of tio 5 m ridtil tim / E / kt/ 5.3 dit of dorbd molul t 5.6 mkt urf diffuio lgth E Ed /kt 5.3 Vloit of t W / kt 5.5 multi t itrl λ th 5.57' urd t rdiu ρ 5.6 R t ' w 3 /3 /3 J / J; ultio rt R W / kt th 5.8 9

Chapter 3 Higher Order Linear ODEs

Chapter 3 Higher Order Linear ODEs ht High Od i ODEs. Hoogous i ODEs A li qutio: is lld ohoogous. is lld hoogous. Tho. Sus d ostt ultils of solutios of o so o itvl I gi solutios of o I. Dfiitio. futios lld lil iddt o so itvl I if th qutio