晶体生长的高过饱和理论：1+1维非线性一步流动模型
Towards a high-supersaturation theory of crystal growth: Nonlinear one-step flow model in 1+1 dimensions
摘要：
从一个动力学的多原子主方程开始，限制固体-固体（KRSOS）模型与外部材料沉积，我们调查的非线性方面的通道，在1+1维的晶体表面的介观描述。后一种描述集中于原子线缺陷（即台阶）的运动，其通过KRSOS微观状态上的适当统计平均来定义。近热力学平衡和足够低的过饱和度，我们表明，这个中尺度图片是合理的忠实于伯顿-卡布雷拉-弗兰克（BCF）阶跃流模型。更具体地说，我们调用的最大值原理结合渐近误差估计，推导出的BCF模型的元素：（一）吸附的吸附原子的密度扩散方程;（二）一个步骤的速度法;和（iii）吸附原子的质量通量在步骤的线性关系。在这种情况下，我们还提供了一个标准，吸附原子通量保持线性过饱和，这表明一系列的非平衡条件下，BCF模型仍然有效。最后，我们利用动力学蒙特卡罗模拟来数值研究驱动系统达到更高过饱和度的效应-例如，从上面将材料沉积到表面上。利用这些结果，我们描述了经验修正的BCF模型，相当于一个非线性关系的吸附原子通量在台阶边缘。
英文摘要：
Starting with a many-atom master equation of a kinetic, restricted solid-on-solid (KRSOS) model with external material deposition, we investigate nonlinear aspects of the passage to a mesoscale description for a crystal surface in 1+1 dimensions. This latter description focuses on the motion of an atomic line defect (i.e. a step), which is defined via appropriate statistical average over KRSOS microstates. Near thermodynamic equilibrium and for low enough supersaturation, we show that this mesoscale picture is reasonably faithful to the Burton-Cabrera-Frank (BCF) step-flow model. More specifically, we invoke a maximum principle in conjunction with asymptotic error estimates to derive the elements of the BCF model: (i) a diffusion equation for the density of adsorbed adatoms; (ii) a step velocity law; and (iii) a linear relation for the mass flux of adatoms at the step. In this vein, we also provide a criterion by which the adatom flux remains linear in supersaturation, suggesting a range of non-equilibrium conditions under which the BCF model remains valid. Lastly, we make use of kinetic Monte Carlo simulations to numerically study effects that drive the system to higher supersaturations -- e.g. deposition of material onto the surface from above. Using these results, we describe empirical corrections to the BCF model that amount to a nonlinear relation for the adatom flux at the step edge.
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