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Formation of step bunching on 4H-SiC (0001) surfaces based on kinetic Monte Carlo method

Li Yuan Shi Ai-Hong Chen Guo-Yu Gu Bing-Dong

Formation of step bunching on 4H-SiC (0001) surfaces based on kinetic Monte Carlo method

Li Yuan, Shi Ai-Hong, Chen Guo-Yu, Gu Bing-Dong
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  • Wide-band gap SiC is a promising semiconductor material for microelectronic applications due to its superior electronic properties, high thermal conductivity, chemical and radiation stability, and extremely high break-down voltage. Over the past several years, tremendous advances have been made in SiC crystal growth technology. Nevertheless, SiC will not reach its anticipated potential until a variety of problems are solved, one of the problem is step bunching during step flow growth of SiC, because it could lead to uneven distribution of impurity and less smooth surfaces. In this paper, step bunching morphologies on vicinal 4H-SiC (0001) surfaces with the miscut toward $\left[ {1\bar 100} \right]$ or $\left[ {11\bar 20} \right]$ directions are studied with a three-dimensional kinetic Monte Carlo model, and then compared with the analytic model based on the theory of Burton-Cabera-Frank. In the kinetic Monte Carlo model, based on the crystal lattice of 4H-SiC, a lattice mesh is established to fix the positions of atoms and bond partners. The events considered in the model are adsorption and diffusion of adatoms on the terraces, attachment, detachment and interlayer transport of adatoms at the step edges. The effects of Ehrlich-Schwoebel barriers at downward step edges and inverse Schwoebel barrier at upwards step edges are also considered. In addition, to obtain more elaborate information about the behavior of atoms in the crystal surface, silicon and carbon atoms are treated as the minimal diffusing species. Finally, the periodic boundary conditions are applied to the lateral direction while the " helicoidal boundary conditions” are used in the direction of crystal growth. The simulation results show that four bilayer-height steps are formed on the vicinal 4H-SiC (0001) surfaces with the miscut toward $\left[ {1\bar 100} \right]$ direction, while along the $\left[ {11\bar 20} \right]$ direction, only bunches with two-bilayer-height are formed. Moreover, zigzag shaped edges are observed for 4H-SiC (0001) vicinal surfaces with the miscut toward $\left[ {11\bar 20} \right]$ direction. The formation of these step bunching morphologies on vicinal surfaces with different miscut directions are related to the extra energy and step barrier. The different extra energy for each bilayer plane results in step bunches with two-bilayer-height on the vicinal 4H-SiC (0001) surface. And the step barriers finally lead to the formation of step bunches with four-bilayer-height. Finally, the formation mechanism of the stepped morphology is also analyzed by a one-dimensional Burton-Cabera-Frank analytic model. In the model, the parameters are corresponding to those used in the kinetic Monte Carlo model, and then solved numerically. The evolution characteristic of step bunching calculated by the Burton-Cabera-Frank model is consistent with the results obtained by the kinetic Monte Carlo simulation.
      Corresponding author: Li Yuan, li-yuan-email@qq.com
    [1]

    Kimoto T 2016 Prog. Cryst. Growth Charact. Mater. 62 329

    [2]

    Tsunenobu K 2015 Jpn. J. Appl. Phys. 54 040103

    [3]

    唐超, 吉璐, 孟利军, 孙立忠, 张凯旺, 钟建新 2009 物理学报 58 7815

    Tang C, Ji L, Meng L J, Sun L Z, Zhang K W, Zhong J X, 2009 Acta Phys. Sin. 58 7815

    [4]

    冯倩, 郝跃, 张晓菊, 刘玉龙 2004 物理学报 53 626

    Feng Q, Hao Y, Zhang X J, Liu Y L 2004 Acta Phys. Sin. 53 626

    [5]

    杨慧慧, 高峰, 戴明金, 胡平安 2017 物理学报 66 216804

    Yang H H, Gao F, Dai M J, Hu P A 2017 Acta Phys. Sin. 66 216804

    [6]

    La V F, Severino A, Anzalone R, Bongiorno C, Litrico G, Mauceri M, Schoeler M, Schuh P, Wellmann P 2018 Mater. Sci. Semicond. Process. 78 57

    [7]

    Müller S G, Sanchez E K, Hansen D M, Drachev R D, Chung G, Thomas B, Zhang J, Loboda M J, Dudley M, Wang H, Wu F, Byrappa S, Raghothamachar B, Choi G 2012 J. Cryst. Growth 352 39

    [8]

    Tomoki Y, Ohtomo K, Sato S, Ohtani N, Katsuno M, Fujimoto T, Sato S, Tsuge H, Yano T 2015 J. Cryst. Growth 431 24

    [9]

    Schwoebel R L 1969 J. Appl. Phys. 40 614

    [10]

    Kimoto T, Itoh A, Matsunami H, Okano T 1997 J. Appl. Phys. 81 3494

    [11]

    Kimoto T, Itoh A, Matsunami H 1995 Appl. Phys. Lett. 66 3645

    [12]

    Ohtani N, Katsuno M, Aigo T, Fujimoto T, Tsuge H, Yashiro H, Kanaya M 2000 J. Cryst. Growth 210 613

    [13]

    Heuell P, Kulakov M A, Bullemer B 1995 Surf. Sci. 331-333 965

    [14]

    Borovikov V, Zangwill A 2009 Phys. Rev. B 79 245413

    [15]

    Krzyżewski F 2014 J. Cryst. Growth 401 511

    [16]

    Xie M H, Leung S Y, Tong S Y 2002 Surf. Sci. 515 L459

    [17]

    Krzyżewski F, Załuska–Kotur M A 2014 J. Appl. Phys. 115 213517

    [18]

    Li Y, Chen X J, Su J 2016 Appl. Surf. Sci. 371 242

    [19]

    Battaile C C 2008 Comput. Methods Appl. Mech. Engrg. 197 3386

    [20]

    Chien F R, Nutt S R, Yoo W S, Kimoto T, Matsunami H 1994 J. Mater. Res. 9 940

    [21]

    Heine V, Cheng C, Needs R J 1991 J. Am. Ceram. Soc. 74 2630

    [22]

    Li Y, Chen X, Su J 2017 J. Cryst. Growth 468 28

    [23]

    Camarda M, La Magna A, La Via F 2007 J. Comput. Phys. 227 1075

    [24]

    Ohtani N, Katsuno M, Takahashi J, Yashiro H, Kanaya M 1999 Phys. Rev. B 59 4592

    [25]

    Sato M 2018 Phys. Rev. E 97 062801

    [26]

    Ranguelov B, Müller P, Metois J J, Stoyanov S 2017 J. Cryst. Growth 457 184

    [27]

    Markov I V 2003 Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth and Epitaxy (London: World Scientific)

    [28]

    Vasiliauskas R, Marinova M, Hens P, Wellmann P, Syväjärvi, Yakimova R 2012 Cryst. Growth Des. 12 197

    [29]

    Mochizuki K 2008 Appl. Phys. Lett. 93 222108

  • 图 1  4H-SiC晶体结构示意图

    Figure 1.  Schematic crystal structure of 4H-SiC.

    图 2  (a)邻位关系计算模型; (b)中间层俯视图

    Figure 2.  (a) The calculation model of neighbors; (b) top view of medial layer.

    图 3  4H-SiC (0001)面偏向$\left[ {1\bar 100} \right]$的台阶形貌演化 (a)初始台阶; (b) AB (蓝)与 BC (浅蓝)聚并台阶的形成; (c) 四层台阶聚并

    Figure 3.  The evolution of stepped morphology on 4H-SiC (0001) surface with miscut toward $\left[ {1\bar 100} \right]$ direction: (a) Initial stage; (b) formation of two-bilayer-height AB (blue) steps and CB (light blue); (c) bunching of four bilayers.

    图 4  4H-SiC (0001)面偏向$\left[ {11\bar 20} \right]$的台阶形貌演化 (a)初始台阶; (b) AB (蓝)与 BC (浅蓝)聚并台阶的形成; (c) 两层台阶形成的聚并扭折台阶

    Figure 4.  The evolution of stepped morphology on 4H-SiC (0001) surface with miscut toward $\left[ {11\bar 20} \right]$ direction: (a) Initial stage; (b) formation of two-bilayer-height CB (light blue) and AB (blue) steps; (c) formation of two-bilayer-height steps with zigzag shapes.

    图 5  不同类型台阶键的配置方式 (a)SN型台阶; (b)SD型台阶; (c)SM型台阶

    Figure 5.  The difference in bond configurations are shown schematically: (a) Type of SN step edge; (b) type of SD step edge; (c) type of SM step edge.

    图 6  台阶流动生长中SiC晶体邻晶面示意图 (a)台阶表面事件与能量势垒; (b)台阶侧面

    Figure 6.  Schematic top and side view of a vicinal surface during step-flow growth: (a) The events occurring on the surface and energy barriers; (b) side view of a vicinal surface.

    图 7  4H-SiC (0001)面偏向$\left[ {1\bar 100} \right]$方向台阶轨迹

    Figure 7.  Step trajectories of the vicinal surface with miscut angles towards $\left[ {1\bar 100} \right]$ direction.

    图 8  4H-SiC (0001)面偏向$\left[ {11\bar 20} \right]$方向台阶轨迹

    Figure 8.  Step trajectories of the vicinal surface with miscut angles towards $\left[ {11\bar 20} \right]$ direction.

  • [1]

    Kimoto T 2016 Prog. Cryst. Growth Charact. Mater. 62 329

    [2]

    Tsunenobu K 2015 Jpn. J. Appl. Phys. 54 040103

    [3]

    唐超, 吉璐, 孟利军, 孙立忠, 张凯旺, 钟建新 2009 物理学报 58 7815

    Tang C, Ji L, Meng L J, Sun L Z, Zhang K W, Zhong J X, 2009 Acta Phys. Sin. 58 7815

    [4]

    冯倩, 郝跃, 张晓菊, 刘玉龙 2004 物理学报 53 626

    Feng Q, Hao Y, Zhang X J, Liu Y L 2004 Acta Phys. Sin. 53 626

    [5]

    杨慧慧, 高峰, 戴明金, 胡平安 2017 物理学报 66 216804

    Yang H H, Gao F, Dai M J, Hu P A 2017 Acta Phys. Sin. 66 216804

    [6]

    La V F, Severino A, Anzalone R, Bongiorno C, Litrico G, Mauceri M, Schoeler M, Schuh P, Wellmann P 2018 Mater. Sci. Semicond. Process. 78 57

    [7]

    Müller S G, Sanchez E K, Hansen D M, Drachev R D, Chung G, Thomas B, Zhang J, Loboda M J, Dudley M, Wang H, Wu F, Byrappa S, Raghothamachar B, Choi G 2012 J. Cryst. Growth 352 39

    [8]

    Tomoki Y, Ohtomo K, Sato S, Ohtani N, Katsuno M, Fujimoto T, Sato S, Tsuge H, Yano T 2015 J. Cryst. Growth 431 24

    [9]

    Schwoebel R L 1969 J. Appl. Phys. 40 614

    [10]

    Kimoto T, Itoh A, Matsunami H, Okano T 1997 J. Appl. Phys. 81 3494

    [11]

    Kimoto T, Itoh A, Matsunami H 1995 Appl. Phys. Lett. 66 3645

    [12]

    Ohtani N, Katsuno M, Aigo T, Fujimoto T, Tsuge H, Yashiro H, Kanaya M 2000 J. Cryst. Growth 210 613

    [13]

    Heuell P, Kulakov M A, Bullemer B 1995 Surf. Sci. 331-333 965

    [14]

    Borovikov V, Zangwill A 2009 Phys. Rev. B 79 245413

    [15]

    Krzyżewski F 2014 J. Cryst. Growth 401 511

    [16]

    Xie M H, Leung S Y, Tong S Y 2002 Surf. Sci. 515 L459

    [17]

    Krzyżewski F, Załuska–Kotur M A 2014 J. Appl. Phys. 115 213517

    [18]

    Li Y, Chen X J, Su J 2016 Appl. Surf. Sci. 371 242

    [19]

    Battaile C C 2008 Comput. Methods Appl. Mech. Engrg. 197 3386

    [20]

    Chien F R, Nutt S R, Yoo W S, Kimoto T, Matsunami H 1994 J. Mater. Res. 9 940

    [21]

    Heine V, Cheng C, Needs R J 1991 J. Am. Ceram. Soc. 74 2630

    [22]

    Li Y, Chen X, Su J 2017 J. Cryst. Growth 468 28

    [23]

    Camarda M, La Magna A, La Via F 2007 J. Comput. Phys. 227 1075

    [24]

    Ohtani N, Katsuno M, Takahashi J, Yashiro H, Kanaya M 1999 Phys. Rev. B 59 4592

    [25]

    Sato M 2018 Phys. Rev. E 97 062801

    [26]

    Ranguelov B, Müller P, Metois J J, Stoyanov S 2017 J. Cryst. Growth 457 184

    [27]

    Markov I V 2003 Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth and Epitaxy (London: World Scientific)

    [28]

    Vasiliauskas R, Marinova M, Hens P, Wellmann P, Syväjärvi, Yakimova R 2012 Cryst. Growth Des. 12 197

    [29]

    Mochizuki K 2008 Appl. Phys. Lett. 93 222108

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  • Received Date:  21 November 2018
  • Accepted Date:  23 January 2019
  • Available Online:  23 March 2019
  • Published Online:  01 April 2019

Formation of step bunching on 4H-SiC (0001) surfaces based on kinetic Monte Carlo method

    Corresponding author: Li Yuan, li-yuan-email@qq.com
  • 1. School of Transportation, Qinghai Nationalities University, Xining 810007, China
  • 2. School of Chemistry and Chemical Engineering, Qinghai Nationalities University, Xining 810007, China

Abstract: Wide-band gap SiC is a promising semiconductor material for microelectronic applications due to its superior electronic properties, high thermal conductivity, chemical and radiation stability, and extremely high break-down voltage. Over the past several years, tremendous advances have been made in SiC crystal growth technology. Nevertheless, SiC will not reach its anticipated potential until a variety of problems are solved, one of the problem is step bunching during step flow growth of SiC, because it could lead to uneven distribution of impurity and less smooth surfaces. In this paper, step bunching morphologies on vicinal 4H-SiC (0001) surfaces with the miscut toward $\left[ {1\bar 100} \right]$ or $\left[ {11\bar 20} \right]$ directions are studied with a three-dimensional kinetic Monte Carlo model, and then compared with the analytic model based on the theory of Burton-Cabera-Frank. In the kinetic Monte Carlo model, based on the crystal lattice of 4H-SiC, a lattice mesh is established to fix the positions of atoms and bond partners. The events considered in the model are adsorption and diffusion of adatoms on the terraces, attachment, detachment and interlayer transport of adatoms at the step edges. The effects of Ehrlich-Schwoebel barriers at downward step edges and inverse Schwoebel barrier at upwards step edges are also considered. In addition, to obtain more elaborate information about the behavior of atoms in the crystal surface, silicon and carbon atoms are treated as the minimal diffusing species. Finally, the periodic boundary conditions are applied to the lateral direction while the " helicoidal boundary conditions” are used in the direction of crystal growth. The simulation results show that four bilayer-height steps are formed on the vicinal 4H-SiC (0001) surfaces with the miscut toward $\left[ {1\bar 100} \right]$ direction, while along the $\left[ {11\bar 20} \right]$ direction, only bunches with two-bilayer-height are formed. Moreover, zigzag shaped edges are observed for 4H-SiC (0001) vicinal surfaces with the miscut toward $\left[ {11\bar 20} \right]$ direction. The formation of these step bunching morphologies on vicinal surfaces with different miscut directions are related to the extra energy and step barrier. The different extra energy for each bilayer plane results in step bunches with two-bilayer-height on the vicinal 4H-SiC (0001) surface. And the step barriers finally lead to the formation of step bunches with four-bilayer-height. Finally, the formation mechanism of the stepped morphology is also analyzed by a one-dimensional Burton-Cabera-Frank analytic model. In the model, the parameters are corresponding to those used in the kinetic Monte Carlo model, and then solved numerically. The evolution characteristic of step bunching calculated by the Burton-Cabera-Frank model is consistent with the results obtained by the kinetic Monte Carlo simulation.

    • SiC是一种新型宽带隙半导体材料, 具有大禁带宽度、高饱和电子漂移速率、高临界击穿场强和高热导率等优异的物理性质, 同时具有极高的化学稳定性. 这些特殊的物理化学性质使得SiC材料在高温、高频、高功率、抗辐射、抗腐蚀电子器件和光电集成器件等方面具有广阔的应用前景, 被认为是应用潜力巨大的“极端电子学”半导体器件材料[1,2]. 此外, SiC晶体可作为衬底材料, 被广泛地应用在生长石墨烯、氮化镓等材料[3-5]. 尽管在过去的几十年里, SiC晶体生长技术取得了巨大的进步, 然而, 大规模商业化生产SiC晶体仍然受到诸多因素的限制. 其中, 晶体表面台阶聚并是外延生长SiC晶体过程中最为普遍的现象之一, 它可以引起各种各样的缺陷, 如: 位错、孪晶、堆叠层错[6] 等. 另外, 它还与微管缺陷的形成与消亡过程有紧密的关系, 而微管缺陷被认为是电子器件的“杀手”[7]. 因此, 研究SiC晶体表面台阶聚并对提高SiC晶体质量与微电子器件发展等方面均有重要意义.

      目前, 外延生长SiC晶体过程中表面形貌的稳定性以及演化过程已经得到了广泛的研究. Tomoki等[8]报道了用实验方法研究4H-SiC晶片碳面的表面形貌特征, 研究发现晶面中心处台阶比较凌乱, 而边缘处由于台阶密度较大形成了聚并台阶. Schwoebel[9]提出了一个基于Burton-Cabera-Frank (BCF)理论的分析模型, 该模型在一定程度上揭示了台阶聚并形成机制. 但是很难解释SiC晶体表面形成的特殊台阶, 因为在实验中观察到SiC晶体表面形成了一个或半个晶胞高度的聚并台阶[10-12]. 另一方面, 蒙特卡罗模拟(kinetic Monte Carlo, KMC)作为一种有效的方法已被成功应用于研究和分析SiC晶体表面形貌的演化过程. Heuell等[13]基于随机行走(random walk)算法建立了一个一维模型来研究在SiC外延生长过程中4H和15R-SiC(0001)面形貌演化过程, 并观察到4个和5个Si-C双原子层台阶聚并现象. Borovikov和Zangwill[14]采用KMC模型模拟6H-SiC(0001)邻晶面上台阶聚并演化过程, 模拟发现该邻晶面在偏向$\left[ {1\bar 100} \right]$方向容易形成一个晶胞高度的聚并台阶, 而在$\left[ {11\bar 20} \right]$方向上形成了半个晶胞高度的台阶形貌. 但是, 该模型中并没有考虑Ehrlich-Schwoebel (ES)和Inverse Schwoebel Barrier (ISB)能量势垒的影响. ES能量势垒是指当台阶从其上方捕获原子时受到的能量势垒, 而ISB能量势垒是台阶从其下方捕获原子时所受到的能量势垒. 研究表明, 这些能量势垒对晶体形貌演化过程至关重要[15,16]. 此外, 他们并没有把Si原子和C原子当作最小的扩散粒子进行处理, 所以无法得到更为详细的SiC晶体表面形貌演化过程. 最近, Krzyżewski和Załuska-Kotur[17]应用KMC模拟方法研究了4H-SiC (0001)面的台阶形貌特征以及在不同沉积和扩散速率下晶体表面的稳定性, 研究表明在较高能量势垒和较低生长温度条件下, 台阶聚并更容易发生. Li等[18]详细研究了6H-SiC(0001)邻晶面上台阶形貌发展过程, 并阐述了6H-SiC晶体台阶聚并形成机理. 然而, 用KMC方法模拟4H-SiC(0001)面聚并台阶形貌演化未见报道.

      本文利用KMC模型, 研究了4H-SiC(0001)面聚并台阶形貌演化过程. 该模型充分考虑了ES和ISB能量势垒影响, 并将硅原子和碳原子分开考虑, 实现了对微观原子动力学过程信息更加精确的捕捉和计算效率的提高. 其次, 建立了一种基于BCF方程的理论分析模型, 并利用该模型进一步揭示了台阶聚并形貌演化机理.

    2.   计算方法与模型
    • 在该模型中, 首先基于SiC晶体结构, 建立了一个计算SiC晶体生长过程的晶格网格, 用来确定Si原子和C原子晶格坐标以及连接它们之间的化学键. 图1给出了4H-SiC晶体多型结构. 对于4H-SiC晶体, 它需要4个Si-C双原子层以ABCB的方式排列组成单元晶胞, 沿[0001]方向进行重复堆叠, 并形成了两种不同类型的台阶, AB和CB, 它们之间的夹角是60°, 如图1所示.

      Figure 1.  Schematic crystal structure of 4H-SiC.

    • 该模型计算邻位原子数目达到16个. 其中, 位于上层的三个Si原子被一个C原子连接属于同一个四面体, 中间层的六个Si原子由三个C原子连接, 最下层有三个Si原子, 它们属于不同的四面体, 如图2所示.

      Figure 2.  (a) The calculation model of neighbors; (b) top view of medial layer.

      图2中红色小球表示所求邻位关系的中心原子, 红色和黑色的线所连接的是两种不同类型的邻位原子, 分别是最邻近C原子(nearest neighbor, NN)与次邻近Si原子(next nearest neighbor, NNN). 它们形成了两种不同类型的共价键: 一种是最邻近原子间的价键即NN bond, 它是由不同类型原子形成, 即Si原子与C原子化合的键; 另一种是次邻近原子间的价键即NNN bond, 它是由两种相同原子形成的价键, 即Si—Si键或C—C键.

    • 针对SiC晶体外延生长过程的KMC模拟可以描述为跟踪或捕捉不同事件在晶体表面上的演化过程. 其中, 扩散过程是吸附原子最主要的迁移方式, 通过扩散吸附原子迁移到表面能量相对稳定的位置, 晶体生长过程才能继续进行. 在给定温度下, 吸附原子从初始位置扩散到目的位置速率由Arrhenius方程给出[19]:

      其中, k为玻尔兹曼常数, ${\nu _0}$为原子振动频率1 × 1013 Hz. $\Delta {E_i}$为吸附原子与其邻近原子之间相互作用能, 它与其最邻近和次邻近原子数目以及原子类型有关, 计算公式如下:

      式中, 指数i值设为1或2, 分别表示Si或C原子; 而NijNir分别为可能被占据的最邻近与次邻近原子个数; JiJii表示原子与原子之间的键能, 对于SiC晶体J1(J2) = 0.75 eV, J11 = 0.35 eV以及J22 = 0.65 eV, 分别对应于Si—C (C—Si), Si—Si以及C—C化合的键[14,17].

      ${E_{{\rm{H}}i}}$表示某个SiC多型生长新的一层需要的额外能量[20,21],

      此外, 当原子向台阶边沿扩散时, 将会受ES或ISB能量势垒. 所以, 原子向台阶边沿扩散的速率可表示为

      式中, j的值为1和2, 即${E_{{\rm{B}}1}}$${E_{{\rm{B}}2}}$, 分别表示ES和ISB能量势垒.

    • 对于每个沉积事件, 以相同概率随机地选择某个空位格点, 然后原子通过随机行走的方式到达晶体表面, 同时将其变为吸附原子. 在模拟过程中基质温度设为1500 K, 沉积事件速率设置为F = 0.1 ML·s–1, 相当于SiC晶体生长速率为0.5 ${\text{μm}}$·h–1. Li等[22]通过研究SiC晶体成核对台阶形貌的影响表明在该情况下晶体表面几乎不会出现晶核, 其对台阶形貌的影响有限. 这是因为在该条件下, 吸附原子扩散速率较大, 有充分的时间扩散到台阶边缘, 使其成为台阶上的原子, 而没有机会成为晶体核. 所以在该模型中为了有效地利用计算机资源, 提高计算效率, 忽略了原子成核的影响. 对于边界条件, 在非晶体生长方向应用了循环边界条件(periodic boundary condition), 而在晶体生长的方向上则应用了垂直边界条件(helicoidal boundary conditions)[23].

    3.   KMC模拟结果与讨论
    • 图3给出了偏向$\left[ {1\bar 100} \right]$方向4H-SiC(0001)邻晶面表面形貌演化过程, 该面由72 × 30个Si原子和C原子组成, 形成了24个均匀分布台阶, 如图3(a)所示. 图中, 不同深度蓝色用于区分SiC晶体不同Si-C双原子层, 蓝色表示4H-SiC晶体的AB台阶, 而淡蓝色表示CB台阶. 由图可知, 4H-SiC(0001)邻晶面在偏向$\left[ {1\bar 100} \right]$方向上, 首先形成了半个晶胞高度的聚并台阶AB和CB, 如图3(b)所示. 其次, 随着台阶的发展, 速度较快的AB台阶赶上了CB台阶. 而后, AB台阶会覆盖CB台阶, 并且其推进速度受到CB台阶抑制. 最终, 它们形成了一个晶胞高度的聚并台阶ABCB, 一起向前推进, 如图3(c)所示.

      Figure 3.  The evolution of stepped morphology on 4H-SiC (0001) surface with miscut toward $\left[ {1\bar 100} \right]$ direction: (a) Initial stage; (b) formation of two-bilayer-height AB (blue) steps and CB (light blue); (c) bunching of four bilayers.

      图4给出了沿$\left[ {11\bar 20} \right]$方向斜切的4H-SiC (0001)面的台阶形貌特征. 该面由120 × 20个Si原子和C原子组成, 形成了20个均匀分布台阶, 如图4(a)所示. 由图4(b)可知, 偏向$\left[ {11\bar 20} \right]$方向的邻晶面形成了半个晶胞高度的聚并台阶AB和CB, 且其台阶扭折不平. 随着生长过程的进行, 这些台阶并没有出现一个晶胞高度的聚并台阶ABCB, 而是保持半个晶胞高度的台阶形貌不变, 如图4(c)所示.

      Figure 4.  The evolution of stepped morphology on 4H-SiC (0001) surface with miscut toward $\left[ {11\bar 20} \right]$ direction: (a) Initial stage; (b) formation of two-bilayer-height CB (light blue) and AB (blue) steps; (c) formation of two-bilayer-height steps with zigzag shapes.

      图3图4所示, 在偏向 $\left[ {1\bar 100} \right]$方向的4H-SiC (0001)邻晶面上, 最终形成了一个晶胞高度的聚并台阶形貌, 而在偏离$\left[ {11\bar 20} \right]$方向邻晶面上, 仅形成了半个晶胞高度的台阶. 该模拟结果与实验结果相符合[11,24].

      偏向不同方向的SiC(0001)邻晶面聚并台阶形貌形成与${E_{{\rm{H}}i}}$${E_{{\rm{B}}i}}$能量势垒有关. 首先, 对于偏向$\left[ {1\bar 100} \right]$$\left[ {11\bar 20} \right]$方向的4H-SiC(0001)邻晶面, 用方程(1)计算的扩散速度是不同的, 因为其生长新的一层所需要的额外能量势垒${E_{{\rm{H}}i}}$不一样. 由于台阶H2与H4的额外能${E_{{\rm{H}}i}}$大于台阶H1和H3的额外能, 所以台阶H1和H3的迁移速率较大. 因此在台阶推进过程中, H1台阶能够赶上H2台阶, H3台阶能赶上H4, 从而形成了半个晶胞高度的聚并台阶形貌AB与CB, 如图3(b)图4(b)所示.

      其次, 对于SiC(0001)面, 有三种不同类型的台阶: SN型, 垂直于$\left[ {1\bar 100} \right]$, $\left[ {01\bar 10} \right]$$\left[ {\bar 1010} \right]$方向, 仅有一个悬空键; SD型, 垂直于 $\left[ {\bar 1100} \right]$, $\left[ {0\bar 110} \right]$$\left[ {10\bar 10} \right]$方向, 有两个悬空键; SM型台阶垂直于$\left[ {11\bar 20} \right]$等方向, 它是由50% SN型和50% SD型台阶混合而成.

      图5给出了SN, SD和SM三种台阶上键的配置方式. 从图可以看出SN有单个的悬空键, 而SD是双悬空键, 所以SD台阶吸附原子能力要强于SN台阶. 另外, SN和SD型台阶是平直的, 所以在没有晶核干扰下, 它们不会形成扭折不平的台阶, 如图5(a)图5(b)所示. 而SM型台阶由SN和SD型混合而成, 具有一定数量的扭折(kink), 它可以进一步发展为较大的扭折, 最终这些扭折会形成曲折不平的台阶, 如图5(c)所示.

      Figure 5.  The difference in bond configurations are shown schematically: (a) Type of SN step edge; (b) type of SD step edge; (c) type of SM step edge.

      对于偏向$\left[ {1\bar 100} \right]$方向4H-SiC(0001)邻晶面CB层全为SN型台阶, 而AB层属于捕捉原子能力更强的SD型台阶, 所以AB台阶推进速度更快, 从而导致了4个Si-C双原子层高度的台阶聚并, 即形成了ABCB层聚并的大台阶, 如图3(c)所示. 而对于偏向$\left[ {11\bar 20} \right]$方向的邻晶面, AB和CB台阶都属于SM型台阶, 其对原子捕捉能力相同, 所以台阶具有相同的推进速率, 最终保持了2个Si-C双原子层高度台阶不变, 如图4(c)所示. 此外, 由于SM型台阶是由SN和SD型混合而成, 其台阶内具有一定数量的扭折, 它可以进一步发展为较大的扭折. 因此在偏向$\left[ {11\bar 20} \right]$方向的邻晶面有较大扭折的台阶形成, 如图4(b)图4(c)所示.

    4.   BCF理论模型
    • 在SiC晶体生长过程中, 当碳原子与硅原子射入到基底表面成为吸附原子. 经过一段时间的扩散后, 一些原子脱离生长表面返回到气相中, 一些原子则通过克服ES和ISB能量势垒到达台阶生长为晶体. 因此, 发生在晶体表面的事件包括原子的吸附、扩散以及原子在台阶上的附着、分离和传输等事件, 如图6(a)所示.

      Figure 6.  Schematic top and side view of a vicinal surface during step-flow growth: (a) The events occurring on the surface and energy barriers; (b) side view of a vicinal surface.

      根据BCF理论, 该过程可以用稳态扩散方程描述[18,25,26]:

      式中, Ds是表面扩散系数, n(x)为台阶上吸附原子浓度, F是沉积通量, $\tau$是吸附原子平均蒸发寿命. 该方程物理意义比较明确, 台阶上吸附原子浓度减少是由于吸附原子向台阶扩散生长成为晶体(方程左侧第一项)以及原子蒸发(方程左侧第二项). 来流原子沉积通量F使得吸附原子浓度增加, 该方程的通解为

      式中, ${\lambda _{\rm{s}}} = \sqrt {\tau {D_{\rm{s}}}} $为吸附原子表面扩散特征长度, 积分常数A和B由边界条件决定.

      4H-SiC单元晶胞由4个Si-C双原子层组成, 并且每层所具不同能量势垒. 因此, 需要给4个台阶设置边界条件, 每个台阶需要两个边界条件, 一个施加于台阶上面, 另一个施加于台阶下面, 如图6(b)所示.

      式中, 指数因子i, j分别代表第i个台阶、第j层的4H-SiC晶体(1 ≤ iN, 1 ≤ j ≤ 4); nse是台阶上吸附原子平衡浓度; $k_{i, j}^ + $$k_{i + 1, j + 1}^ + $分别表示上升台阶与下降台阶动力因子[18,27], 它与台阶上能量势垒有关.

      将边界条件(8)与(9)式代入方程(7), 可以求得台阶推进的速率:

      其中

      式中, ${n_0}$$n_{\rm{s}}^{\rm{e}}$分别表示台阶上吸附原子的密度与平衡浓度, V+V分别表示上升与下降台阶向前推进速率, ${l_{i, j}} = {x_{i + 1, j + 1}} - {x_{i, j}}$表示第i个台阶第j层的宽度.

      为了研究聚并台阶形貌的演化过程, 本文计算了台阶轨迹随生长时间的变化规律. 根据方程(10)可得到每一个台阶推进速率Vi, j, 那么台阶轨迹变化表示为Li, j = xi, j + Vi, jt, 式中, xi, j为当前台阶位置, t为模拟时间. 在整个过程中, 共计算了120个台阶, 即i = 120; 对于4H-SiC每隔4个台阶进行循环, 即j = 1, 2, 3, 4. 当台阶与台阶间距离达到零时, 用它们的平均速度, 作为新台阶移动速度. 计算过程中各个参数设置如下[28,29]: 温度T = 1500 K, 吸附原子扩散系数${D_{\rm{s}}} = 0.33 \times $10–2cm2·s–1, Si面表面扩散特征长度为$ {\lambda _{\rm{s}}} = 0.3 \times $${10^{ - 4}}$ cm, 台阶初始宽度为${l_0} = 1.0 \times {10^{ - 5}}$ cm. 台阶上吸附原子平衡浓度$n_{\rm{s}}^{\rm{e}}{{ = }}0.21 \times {10^8}$ cm–2, $ F\tau = $$1.0 \times {10^9}$ cm–2, 吸附原子的密度${n_0} \!=\! 1.06 \!\times\! {10^{15}}$ cm–2.

      图7给出了偏向$\left[ {1\bar 100} \right]$方向4H-SiC(0001)邻晶面上台阶演化过程, 图中红色和绿色线条分别表示考虑能量势垒和没有考虑能量势垒情况下台阶轨迹的变化. 由图可知, 在没有能量势垒时, 台阶轨迹是一条直线, 且台阶宽度保持不变, 如图7绿色虚线所示. 考虑能量势垒时, 台阶轨迹变化比较复杂. 当模拟时间在0—100 s时, 有些线条相互靠近, 而有些线条则相互远离, 说明台阶正以不同的推进速率生长, 其彼此之间靠近或者远离. 随着晶体生长过程的进行, 当模拟时间超过100 s时, 这些台阶逐渐集聚重合, 最终每四条实线集聚在一起, 说明形成了4个Si-C双原子层高度的聚并台阶形貌特征, 如图7红色线条所示. 该演化过程与KMC模拟结果相一致, 如图3所示.

      Figure 7.  Step trajectories of the vicinal surface with miscut angles towards $\left[ {1\bar 100} \right]$ direction.

      图8给出了台阶轨迹在$\left[ {11\bar 20} \right]$方向发展过程. 如图所示, 有些线条在彼此接近, 有些线条在彼此远离, 说明在该过程当中, 有些台阶逐渐开始堆叠在一起, 向多个高度的聚并台阶方向发展. 最终, 每两条实线集聚在一起, 说明形成了半个晶胞高度的聚并台阶形貌, 该演化过程与KMC模拟结果相符合, 如图4所示.

      Figure 8.  Step trajectories of the vicinal surface with miscut angles towards $\left[ {11\bar 20} \right]$ direction.

      SiC台阶形貌演化过程与相邻层与层间相互作用的能量势垒${E_{{\rm{H}}i}}$以及ES(ISB)能量势垒${E_{{\rm{B}}i}}$有关. 较高的EBi能量势垒降低了吸附原子向台阶扩散的能力, 相应的台阶推进速率也会减少. 同样, 较低的能量势垒, 提高了台阶推进速率. 对于4H-SiC多型, 相邻层与层间相互作用的能量势垒${E_{{\rm{H}}i}}$以及ES(ISB)能量势垒EBi是不同的, 因此台阶向前推进速率不同, 使得邻晶面上台阶的宽度发生变化. 一些台阶宽度会变窄, 而另外一些台阶宽度会变宽. 较宽的台阶意味着能给台阶边缘提供更多吸附原子, 从而使得台阶推进速率更快. 相反, 较窄的台阶给台阶边缘提供吸附原子很少, 所以台阶推进速率较慢. 因此, 随着生长过程的进行, 较宽的台阶会变得越来越宽, 较窄的台阶会变的越来越窄, 最终导致了台阶聚并形貌的出现.

    5.   结 论
    • 针对SiC外延生长中晶体表面动力学过程, 发展了一种三维动力学蒙特卡罗模型. 研究了4H-SiC(0001)邻晶面台阶聚并形貌演化过程, 并用BCF理论阐明了其形成机制. 结果表明: 在偏向$\left[ {1\bar 100} \right]$方向的邻晶面上, 首先形成了半个晶胞高度的聚并台阶, 随着生长过程进行, 发展较快的台阶赶上了发展较慢的台阶, 最终形成了一个晶胞高度的聚并台阶形貌; 而沿着$\left[ {11\bar 20} \right]$方向的邻晶面上只有半个晶胞高度的聚并台阶形貌形成. 4H-SiC (0001)邻晶面上形成AB与BC聚并台阶形貌的主要原因是台阶每生长新的一层所需的能量不一样, 导致台阶扩散速度不同. 而由AB与BC聚并台阶发展为ABCB的聚并台阶形貌, 主要是由于台阶边上ES与ISB能量势垒不一样导致的. 最后, 利用BCF理论计算了偏向$\left[ {1\bar 100} \right]$$\left[ {11\bar 20} \right]$方向的4H-SiC (0001)面台阶生长轨迹, 得到了聚并台阶形貌演化过程. 该分析结果与KMC的模拟结果以及实验结果相符合.

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