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六方小面相螺旋在各向异性、表面吸附、界面动力学作用下生长的相场

董祥雷 邢辉 陈长乐 沙莎 王建元 金克新

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六方小面相螺旋在各向异性、表面吸附、界面动力学作用下生长的相场

董祥雷, 邢辉, 陈长乐, 沙莎, 王建元, 金克新

Phase-Field Modeling of Facet Hexagonal Spirals with Anisotropy, Deposition, and Kinetic Effects

Dong Xiang-Lei, Xing Hui, Chen Chang-Le, Sha Sha, Wang Jian-Yuan, Jin Ke-Xin
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  • 利用定量相场模型研究了强各向异性、表面吸附率以及界面动力学作用条件下六方GaN螺旋结构的表面形貌与生长机理. 通过引入小面相各向异性的相场修正方程, 研究了不同各向异性的稳态螺旋形貌, 发现各向异性通过改变台阶尖端的曲率作用影响螺旋生长. 弱各向异性下稳态螺距及界面动力学特征相对稳定, 各向异性较强时尖端的过饱和度随着各向异性的增强而增大, 并使得界面平衡态向着有利于螺旋台阶推进的方向移动. 研究了表面吸附率对小面相螺旋生长的作用机理, 发现吸附率的增加导致了稳态螺旋间距的降低, 通过分析螺旋间距随台阶宽度的变化趋势, 发现增强的表面吸附和各向异性强度降低了螺旋间距的收敛性, 并且具体分析了收敛性误差; 通过探讨界面动力学作用条件下螺旋形貌特征以及螺旋间距变化趋势, 发现界面动力学系数通过改变稳态螺旋间距与特征指数因子调控螺旋生长的动力学机理, 与各向同性相比小面相螺旋生长表现出较低的界面动力学系数依赖性.
    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.
      通信作者: 陈长乐, chenchl@nwpu.edu.cn
    • 基金项目: 国家自然科学基金 (批准号: 61471301, 51172183, 51402240, 51471134)、陕西省自然科学基金 (批准号: 2015JQ5125)、西北工业大学博士论文创新基金(批准号: CX201325)、中央高校基本科研业务费 (批准号: 3102015ZY078)和高等学校博士学科点专项科研基金(批准号: 20126102110045)资助的课题.
      Corresponding author: Chen Chang-Le, chenchl@nwpu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61471301, 51172183, 51402240, 51471134), the National Natural Science Foundation of Shaanxi Province, China (Grant No. 2015JQ5125), the Doctorate Foundation of Northwestern Polytechnical University, China (Grant No. CX201325), the Fundamental Research Funds for the Central Universities (Grant No. 3102015ZY078), and the Specialized Research Fund for the Doctoral Program of Higher Education, China (Grant No. 20126102110045).
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    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

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    Kim S H, Dandekar P, Lovette M A, Doherty M F 2014 Cryst. Growth Des. 14 2460

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    Swendsen R H, Kortman P J, Landau D P, Muller-Krumbhaar H 1976 J. Cryst. Growth 35 73

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    Ratsch C, Smilauer P, Vvedensky D D 1995 Sur. Sci. Lett. 329 L599

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    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

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    Rtz A, Voigt A 2004 J. Cryst. Growth 266 278

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    Beckermann C, Diepers H J, Steinbach I, Karma A, Tong X 1999 J. Comput. Phys. 154 468

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    Karma A, Rappel W J 1996 Phys. Rev. E 53 3017

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    Karma A, Rappel W J 1998 Phys. Rev. E 57 4323

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    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 物理学报 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

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    Cabrera N, Coleman R V 1963 The Art and Science of Growing Crystals (New York: John Wiley) p3

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    van der Eerden J P 1981 J. Cryst. Growth 53 305

  • [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 物理学报 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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出版历程
  • 收稿日期:  2015-09-07
  • 修回日期:  2015-12-03
  • 刊出日期:  2016-01-20

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