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In this paper, we perform the quantitative phase-field simulations based on the surface morphology and growth regime of the hexagonal GaN spiral structure. We investigate the highly anisotropic energy, the deposition rate and the kinetic attachment and detachment effects. A regularized equation including the modified gradient coefficient is employed to study the anisotropic effect. Results show that the highly anisotropic energy modulates the equilibrium state by changing the local curvature of the tip step and thus leading to the changed spiral spacing. Under the weak anisotropy, the spiral spacing and morphology keep stable with the increase of the anisotropic strength. In the case of facet anisotropy, however, the larger anisotropic strength facilitates the spiral growth due to the local interfacial instability caused by increasing the supersaturation for the tip step. As to the effect of deposition, the deposition rate imposes the reaction on the curvature of interface due to the variations of supersaturation and step velocity. The larger rate of deposition enables the shorter spacing for both anisotropic and isotropic spirals. We carry out a convergence study of spiral spacing with respect to the step width to estimate the precision of the phase-field simulation. Results show that the larger deposition rate and the higher anisotropy give rise to the lower convergence of the spiral model. Moreover, we find that the kinetic attachment affects the instinct regime of spiral growth by changing the step spacing and the scaling exponents of spiral spacing versus deposition rate. The anisotropic spiral exhibits the more significant hexagonal structure and the lower value of step velocity by reducing the value of kinetic coefficient. The scaling exponent decreases with anisotropy increasing, but it increases with kinetic effect strengthening. The highly anisotropic energy contributes to weakening the sensitivity of the spiral spacing to the kinetic effect.
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Keywords:
- Computer modeling and simulation /
- Surface patterning /
- Film growth and deposition /
- Kinetics
[1] Smereka P 2000 Physica D 138 282
[2] Sorge J B, van Popta A C, Sit J C, Brett M J 2006 Opt. Express 14 10550
[3] Hodgkinson I, Wu Q 2001 Adv. Mater. 13 889
[4] Liu Y, Li L 2011 Nanotechnology 22 3990
[5] Burton W K, Cabrera N, Frank F C 1951 Philos. Trans. R. Soc. London, Ser. A 243 299
[6] Bennema P 1984 J. Cryst. Growth 69 182
[7] Lin E Y, Zhang Y X, Liao Y J, Mo Y J, Jiang S 2014 J. Comput. Mater. Sci. 90 148
[8] Dam B, Rector J H, Huijbregtse J M, Griessen R 1998 Physica C 305 1
[9] Vezian S, Natali F, Semond F, Massies J 2004 Phys. Rev. B 69 125329
[10] Dong X L, Xing H, Sha S, Chen C L, Niu L W, Wang J Y, Jin K X 2015 Sci. China Technol. Sci. 58 753
[11] Kim S H, Dandekar P, Lovette M A, Doherty M F 2014 Cryst. Growth Des. 14 2460
[12] Cuppena H M, van Veenendaala E, van Suchtelena J, van Enckevorta W J P, Vlieg E 2000 J. Cryst. Growth 219 165
[13] Swendsen R H, Kortman P J, Landau D P, Muller-Krumbhaar H 1976 J. Cryst. Growth 35 73
[14] Ratsch C, Smilauer P, Vvedensky D D 1995 Sur. Sci. Lett. 329 L599
[15] Caflisch R E, Gyure M F, Merriman B, Ratsch C 1999 Phys. Rev. E 59 6879
[16] Liu F, Metiu H 1997 Phys. Rev. E 19 2601
[17] Pierre-Louis O 2003 Phys. Rev. E 68 021604
[18] Otto F, Penzler P, Ratz A, Rump T, Voigt A 2004 Nonlinearity 17 477
[19] Rtz A, Voigt A 2004 Appl. Anal. 83 1015
[20] Rtz A, Voigt A 2004 J. Cryst. Growth 266 278
[21] Beckermann C, Diepers H J, Steinbach I, Karma A, Tong X 1999 J. Comput. Phys. 154 468
[22] Karma A, Rappel W J 1996 Phys. Rev. E 53 3017
[23] Karma A, Rappel W J 1998 Phys. Rev. E 57 4323
[24] Echebarria B, Folch R, Karma A, Plapp M 2004 Phys. Rev. E 73 061604
[25] Ramirez J C, Beckermann C, Karma A, Diepers H J 2004 Phys. Rev. E 69 051607
[26] Folch R, Plapp M 2003 Phys. Rev. E 68 010602
[27] Wang Z J, Wang J C, Yang G C 2010 Chin. Phys. B 19 017305
[28] Xing H, Wang J Y, Chen C L, Jin K X, Du L F 2014 Chin. Phys. B 23 038104
[29] Xing H, Dong X L, Chen C L, Wang J Y, Du L F, Jin K X 2015 Int. J. Heat. Mass. Tran. 90 911
[30] Duan P P, Xing H, Chen Z, Hao G H, Wang B H, Jin K X 2015 Acta Phys. Sin. 64 60201 (in Chinese) [段培培, 邢辉, 陈志, 郝冠华, 王碧涵, 金克新 2015 64 60201]
[31] Karma A, Plapp M 1998 Phys. Rev. Lett. 81 4444
[32] Yu Y M, Liu B G, Voigt A 2009 Phys. Rev. B 79 235317
[33] Redinger A, Ricken O, Kuhn P, Rtz A, Voigt A, Krug J, Michely T 2008 Phys. Rev. Lett. 100 035506
[34] Kobayashi R 1993 Physica D 63 410
[35] McFadden G B, Wheeler A A, Braun R J, Coriell S R, Sekerka R F 1993 Phys. Rev. E 48 2016
[36] Fierro F, Goglione R, Paolini M 1998 Math. Mod. Meth. Appl. Sci. 8 573
[37] Eggleston J, McFadden G B, Voorhees P W 2001 Physica D 150 91
[38] Neugebauer J 2001 Phys. Stat. Sol. 227 93
[39] Cabrera N, Coleman R V 1963 The Art and Science of Growing Crystals (New York: John Wiley) p3
[40] van der Eerden J P 1981 J. Cryst. Growth 53 305
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[1] Smereka P 2000 Physica D 138 282
[2] Sorge J B, van Popta A C, Sit J C, Brett M J 2006 Opt. Express 14 10550
[3] Hodgkinson I, Wu Q 2001 Adv. Mater. 13 889
[4] Liu Y, Li L 2011 Nanotechnology 22 3990
[5] Burton W K, Cabrera N, Frank F C 1951 Philos. Trans. R. Soc. London, Ser. A 243 299
[6] Bennema P 1984 J. Cryst. Growth 69 182
[7] Lin E Y, Zhang Y X, Liao Y J, Mo Y J, Jiang S 2014 J. Comput. Mater. Sci. 90 148
[8] Dam B, Rector J H, Huijbregtse J M, Griessen R 1998 Physica C 305 1
[9] Vezian S, Natali F, Semond F, Massies J 2004 Phys. Rev. B 69 125329
[10] Dong X L, Xing H, Sha S, Chen C L, Niu L W, Wang J Y, Jin K X 2015 Sci. China Technol. Sci. 58 753
[11] Kim S H, Dandekar P, Lovette M A, Doherty M F 2014 Cryst. Growth Des. 14 2460
[12] Cuppena H M, van Veenendaala E, van Suchtelena J, van Enckevorta W J P, Vlieg E 2000 J. Cryst. Growth 219 165
[13] Swendsen R H, Kortman P J, Landau D P, Muller-Krumbhaar H 1976 J. Cryst. Growth 35 73
[14] Ratsch C, Smilauer P, Vvedensky D D 1995 Sur. Sci. Lett. 329 L599
[15] Caflisch R E, Gyure M F, Merriman B, Ratsch C 1999 Phys. Rev. E 59 6879
[16] Liu F, Metiu H 1997 Phys. Rev. E 19 2601
[17] Pierre-Louis O 2003 Phys. Rev. E 68 021604
[18] Otto F, Penzler P, Ratz A, Rump T, Voigt A 2004 Nonlinearity 17 477
[19] Rtz A, Voigt A 2004 Appl. Anal. 83 1015
[20] Rtz A, Voigt A 2004 J. Cryst. Growth 266 278
[21] Beckermann C, Diepers H J, Steinbach I, Karma A, Tong X 1999 J. Comput. Phys. 154 468
[22] Karma A, Rappel W J 1996 Phys. Rev. E 53 3017
[23] Karma A, Rappel W J 1998 Phys. Rev. E 57 4323
[24] Echebarria B, Folch R, Karma A, Plapp M 2004 Phys. Rev. E 73 061604
[25] Ramirez J C, Beckermann C, Karma A, Diepers H J 2004 Phys. Rev. E 69 051607
[26] Folch R, Plapp M 2003 Phys. Rev. E 68 010602
[27] Wang Z J, Wang J C, Yang G C 2010 Chin. Phys. B 19 017305
[28] Xing H, Wang J Y, Chen C L, Jin K X, Du L F 2014 Chin. Phys. B 23 038104
[29] Xing H, Dong X L, Chen C L, Wang J Y, Du L F, Jin K X 2015 Int. J. Heat. Mass. Tran. 90 911
[30] Duan P P, Xing H, Chen Z, Hao G H, Wang B H, Jin K X 2015 Acta Phys. Sin. 64 60201 (in Chinese) [段培培, 邢辉, 陈志, 郝冠华, 王碧涵, 金克新 2015 64 60201]
[31] Karma A, Plapp M 1998 Phys. Rev. Lett. 81 4444
[32] Yu Y M, Liu B G, Voigt A 2009 Phys. Rev. B 79 235317
[33] Redinger A, Ricken O, Kuhn P, Rtz A, Voigt A, Krug J, Michely T 2008 Phys. Rev. Lett. 100 035506
[34] Kobayashi R 1993 Physica D 63 410
[35] McFadden G B, Wheeler A A, Braun R J, Coriell S R, Sekerka R F 1993 Phys. Rev. E 48 2016
[36] Fierro F, Goglione R, Paolini M 1998 Math. Mod. Meth. Appl. Sci. 8 573
[37] Eggleston J, McFadden G B, Voorhees P W 2001 Physica D 150 91
[38] Neugebauer J 2001 Phys. Stat. Sol. 227 93
[39] Cabrera N, Coleman R V 1963 The Art and Science of Growing Crystals (New York: John Wiley) p3
[40] van der Eerden J P 1981 J. Cryst. Growth 53 305
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