(612c) A Cellular Automata Approach for Simulation of Crystal Growth

Ke, J., Purdue University
Ydstie, B. E., Carnegie Mellon University
Khair, A. S., Carnegie Mellon University
Cellular Automata (CA) provides a computational platform for combining micro-scale fluctuation theory with macro-scale field models to predict discontinuities, sharp transitions and moving boundaries in solidification of impure systems. The CA system can consider solute and heat conservation equations subject to complex boundary conditions at the solid-liquid interface. Constitutional and curvature undercooling on the equilibrium interface tare easy to incorporate. These variables are needed to predict the boundary between stable crystal growth and the unstable dendritic growth that lead to defect formation in many applications. The stochastic procedures in place for nucleation and dendritic growth can also be applied without making strong assumptions about the underlying rate processes. The price one has to pay can be significant computational cost due to the detailed modeling of the micro-structure.

The aim of the current paper is to consider the application of CA to the Horizontal Ribbon Growth (HRG) process. HRG was proposed by Shockley for producing single crystalline silicon sheets [1]. It is a continuous process and it has potential to overcome the limitations of traditional batch-scale wafer processes based on Czochralski crystallization and wire sawing. In HRG, a thin silicon solid sheet is produced and extracted continuously from a molten silicon pool, minimizing in this way material losses. In order to achieve fast production speed, the pulling rate has to be determined in conjunction with proper cooling set up and the presence of impurities may introduce defects and other instabilities that must be controlled.

Many technical challenges have been encountered in the effort to stable and fast ribbon production in the HRG process. Experiments show that sudden dendritic growth occurs at the crystal front when the pulling speed exceeds a threshold which depends on a number of parameters in the system including the accumulation of impurities [2]. Experimental results have shown cases of non-smooth and unstable solid-liquid interface and sharp wedge has occurred [3].

In the current paper we develop the CA approach for thin film crystal growth and show how the models can be used to derive stability/instability boundaries and control strategies to suppress these. We show results that demonstrate computational tractability and we present parametric tests to show how different cooling conditions, impurity concentrations, and other process related parameters relate to process stability. A method to control the process is proposed and it shown how model results relate to experimental results reported in the open literature.


  1. William, S. (1962). U.S. Patent No. 3,031,275. Washington, DC: U.S. Patent and Trademark Office.
  2. Kudo, B. (1980). Improvements in the horizontal ribbon growth technique for single crystal silicon. Journal of Crystal Growth50(1), 247-259.
  3. Kellerman, P., Kernan, B., Helenbrook, B. T., Sun, D., Sinclair, F., & Carlson, F. (2016). Floating Silicon Method single crystal ribbon–observations and proposed limit cycle theory. Journal of Crystal Growth, 451, 174-180.