結晶成長とパターン形成 - S研へようこそ ...

1 1 Makio UwahaDept. of Physics, Nagoya 2008 8 9 Crystal Growth and Pattern Formation ( ) ( ) ( ) Ellen WilliamsK. Thuermer, J. E. Reutt-Robey, E. D. Williams, M. Uwaha, A. Emundts and H. P. Bonzel, Phys. Rev. Lett. 87(2001) 186102 710 C NaCl 5-10 m 620 C NaClSEM images J, C. Heyraud and J. J. Metois, J. Cryst. Growth 84(1987) 503 4He (0001) (1-100) ? (1-101) Sebastien Balibar2005 London prize 2 ( ) ( ) 2)2(2)1(2)2(2)1(2)2(2)1(254253)12(222222 )11(222)10()01(aaaaaabbbbbb +=+=+== 2 wulff iiiiiiiiiiiiiiiiihAAhhAhAhAVhAVhnr 2)(3131 = +==== r A h nihiri Ai wulff + =+ =+ =iiiisiiiiiisiiisAhvAAhvGAvVG )21(21 hi wulff ()()siiisiisiiiiiisiiisiiivhvnvnrAnnrvAn vGAn22 2 0 2) 2(0 = = = = + = += rrrrrrr 3 Ai r - = nihi 3 wulff )

2 Siivh2 = nihi Burkov, J. Physique 46(1985) 317 Landau (1950) J2=0 2 J2>0 2 J2<0C. Rottman, M. Wortis, Phys. Rev. B 29(1984) 328 Michael Wortis ( ) 710 C NaCl 5-10 m 620 C NaClJ. C. Heyraud and J. J. Metois: J. Cryst. Growth 84 (1987) 503 SEM images SEM images by Heyraud, Metois ( 1000 C 5 m) ( 300 C 6 m)Jean-Jacque Metois4 ( ) ( ) =KV2 2 (2 ) h dsndAGsurfacecrystal + =) ( ()dsshhRhdsG ++ = 1 = //1 RshdsRG ++ =221 1 ( ) RhGeff ~ = = 22~ += Herring equilibrium shape 0)(2211= ++ + = RReff (roughening transition) kBT/ = H.

3 Gilmer5 (roughening transition) ( ) ( ) dxdyzEdxdyyxzH20221),(1 + += 22212220yxdkdkzkLEH += ==>=kk ()()arTkdkdkzLzzByxlncos124)()(222 = = rk0rk (normal growth) (adhesive growth) =KV normal, adhesive growth ( ) Peq(T) jin=jout TmkPTmkTknTkenmpdpdpdpPTjjjVBBBgBTmkpgzx yzinoutinB 22)2()(),()(2/32/02== = = =sgthermBeqnnmvTmkTPPV12)( ( ) TkeaeeaeeaVaeejjjBTkTkTkTkTkTkTkoutinBdB BdBdBdBdBd = = = = + + ////)(/2/)(/)1()(/)( 6 lateral growth, layer growth (lateral growth) layer growth 2 D nucleation growth 2 2 RRRG2)(22+ = spiral growth cRl219=222)(19119 = = =aKRaKKlaVscss =sstepKV By H.

4 Bethge 2 TkconstVB2exp =sKV22)(191 =aKVs 7 ]10[]11[ V(n) R =) () (nKnV + =2211~~) () (RRvnKnVs R Cf. =) () (nKnV v0 v0sin << L= 1/ N t ~ L / v0sin ~ L / v0 tNtNLtDD11322 L ~ 1/N ~ 1/ After Chernov Modern Crystallography III 3D Smereka, Li, Russo, Srolovitz, Acta Materialia53(2005) 1191 8 By Furukawa William W.

5 MullinsRobert Sekerka - Martin Glicksman PWA-1480 Phase-Field 3 Ryo Kobayashi Yokoyama, Kuroda: Phys. Rev. A 41 (1990) 20382 MS (DLA:diffusion-limited aggregation)lD: dc: 9 DLA (diffusion-limited aggregation)N=100N=1000N=10000 (self-similar) 10 101/D D N DfBy C. H. Lam ( ) 1 mm Miyashita, Y. Saito and M. Uwaha, J. Phys. Soc. Jpn. 66(1997) 929S. Miyashita, Y. Saito and M. Uwaha, J. Cryst. Growth 283(2005) 533 AgNo3 c= M. Uwaha and Y. Saito, Phys. Rev.

6 A 40(1989) 4716 System size:1024 900ng= (DLA) Df= Uwaha, Saito: J. Phys. Soc. Jpn. 57(1988) 3285, Phys. Rev. A 40(1989) 4716 N r Df D: n r Df-d10 h(t)c =f(tc2 )fDd =1 h: c: d: Df: DLA) heightMiyashita, Saito, Uwaha: J. Phys. Soc. Jpn. 66 (1997) 929 Miyashita, Saito, Uwaha: J. Cryst. Growth, 283(2005) 533 : : : : : : : : : heighth(t)c =f(tc2 )1mm ( ) cSilver aggregateInterference fringe MC Uwaha, Saito: J. Phys. Soc. Jpn. 57(1988) 3285, Phys. Rev. A 40(1989) 4716 2 Thomas Michely11 Uwaha, Saito: J. Cryst. Growth 99(1990) 175, ( ) (1/ ) ( t=1/(4Ns)) (Ps) (Pm)]/)[(11,TkEExpPBsms m += c c / - c / - jj = - D c fccDtcs+ = 212 ()()()eqseqseqsnsnsolidifycycKcycKccKyjy jj + = = =++ + )()()()(TkFcTRkccTkFceccBeqBeqeqBeqTkFeq eqBint0000/0~1 += = ()eqsolidifyccKjv = = - (ES)

7 0<y<l ()eqccKv =() + ++=ccceeeycsxlxlyxysss//)(/1)()2/tanh()/ 2(110ssseqxlxDKccv+ = vicinal face inphase wandering bunching bending zxy (ES ) (step wandering) MC Uwaha, Saito: Surf. Sci. 283 (1993) 36613 MC tiqxqqeytVtxy ++=0),(42qqq cff 4422xyxyty = 42qqq cff 4422xyxyty = 14 (Kardar-Parisi-Zhang term) -Sivashinsky(KS) 201),( += xyVttxy2442221 + = xyxyxyty + 20211xyVBenna, Misbah, Valance: Phys.

8 Rev. B 47, 7408 (1993)vyvnKS KS SOS(solid-on-solid) Non-SOS (10) (11) Saito, Uwaha: Phys. Rev. B 49(1994) 10677KS 244222211 + = xyxyxyxyaty Saito, Uwaha: J. Phys. Soc. Jpn. 65(1996) 3576A= ES ( : drift) 2 Si(111) at 950 Cwith downward DCDegawa et alHomma, Finnie, Uwaha: Surf. Sci. 492 (2001) 125 Hibino, Homma, Uwaha, Ogino, Surf. Sci. 527(2003) L222Si(111) at 860 Cduring growthSi(111) during growthMaroutian, Douillard, Ernst,Phys. Rev. Lett83 (1999) 4353Cu(1,1,17) during growth ES xxyjsx )( 22~xyxxjsx 15 ( ) + ++ + = 2/3222221111xyxyxxyxyxyxty ()relaxxgrowthxxjjxjxtxy+ = = )( Pierre-Louis, Misbah, Saito, Krug, Politi: Phys.

9 Rev. Lett. 80(1998) 4221 t 1/2 MC 2 Si(001) (coarsening)t = 6/1t (Paulin, Gillet, Pierre-Louis, Misbah,Phys. Rev. Lett. 86 (2001)5538) 2 Si(001) (coarsening)Sato, Uwaha, Saito, Hirose: Phys. Rev. B 67(2003) 125408 MC Saito, Uwaha: Phys. Rev. B 49 (1994) 10677 vicinal face inphase wandering bunching bending zxy16 (ES ) =ssnR2 1 ( )tinklnneyvtnly +++=)})(({)(tyFttynn= 432kikkk + Sato, Uwaha: Phys.

10 Rev. B 51(1995) 11172 1 ( (y)=1/|yn-yn-1|) Benney ( = -dz/dx)yyyyt = 443322)})(({)(tyFttynn= 244332221 + = yzyzyzyztz Sato, Uwaha: Europhys. Lett. 32(1995) 639 Benney Benney 0 KS KdV xxxt = 4422xxxxt = 443322xxt = 33T. Kawahara: Phys. Rev. Lett. 51(1983) 380 Benny 1 1 Benny GaAs(111)B Ren, Shen, Nishinaga: J. Cryst. Growth 166(1996) 21717 ES cccDtcs = v2Si(111) dsvccLccDj2+ + ++ vdSi(111) Homma, Aizawa: Phys.