(359g) Modeling Interface Dynamics in Crystal Growth Processes: Applications to Silicon Wafer Manufacturing

Many industrial crystal growth processes require the careful management of the moving solid-liquid interface to avoid quality imperfections in the final product. In these processes it is necessary to control the formation of dendrites to guarantee a micro- and macro structurally good crystal. Dendrites are a consequence of a morphologically unstable interface; understanding how these instabilities form is the first step to addressing how they should be controlled. Our goal is to link the theory of stability of interfaces with the tools of systems engineering and apply them to a novel crystal growth process. Specifically, we incorporate the morphological stability analysis into the development and mathematical modeling of the Horizontal Ribbon Growth (HRG) process to manufacture silicon wafers. In this process a thin film of silicon is produced continuously from its melt making use of the fact that silicon floats on its melt just like ice floats on water. Our specific goal in this work is to analyze the effects of an applied perturbation to the growing solid-liquid interface of the HRG process.

The seminal investigation in the area of the stability of solid-liquid interfaces was carried out in the early ‘60s by Mullins and Sekerka [1]. They considered a binary alloy and analyzed the effect of applying a sinusoidal perturbation of infinitesimal amplitude to an interface moving at a constant velocity under purely diffusive transport. Under the steady state and infinite domain assumptions, they found an analytical expression for the rate of growth –or decay- of the perturbation as a function of the temperature gradients at each side of the interface, the concentration gradients in the melt, and the surface tension. A hill or valley of the sinusoidal wave ahead or behind the interface finds itself in a thermally unfavorable situation. If a tip of solid protruding into the liquid is surrounded by a domain that is at a temperature above the melting point, heat will flow from the liquid to the solid tip to equilibrate the thermal profile; this results in a stabilizing force. Surface tension offers resistance to deformation by minimizing the surface area of the system; hence the low free energy planar morphology is preferential over the high free energy curved interface. Concentration gradients, on the other hand, are a destabilizing force. The effect of melting point depression due to the presence of impurities causes an undercooling close to the front. Thus, when a sufficiently undercooled tip protrudes into the melt, it might not be immediately surrounded by a domain of a higher temperature since the undercooled melt lowers the temperature around the tip and counteracts the heat flow coming from the liquid. If the undercooled tip overcomes this “heat barrier”, the interface breaks down resulting in the formation of cellular and dendritic morphologies.

In previous work we applied the Mullins-Sekerka stability criterion directly to an unperturbed silicon film under the assumption that the film is growing at a relatively low velocity [2]. In this work we integrate these two analyses and apply them to the growth of a silicon film in a bounded domain under radiative cooling. Aspects of the formulation that are reformulated are the use of proper boundary conditions and the incorporation of the initial transient effects of the growing film.  The mathematical formulation of the new system yields a set of linear and nonlinear partial differential equations. The unperturbed and perturbed modes are independent of each other in both subdomains (solid and liquid) due to the linearity of the diffusion equations. The only coupling occurs at the solid-liquid interface, where the perturbation is applied. The system is modeled using standard numerical methods techniques.

[1] Mullins, W.W., Sekerka, R.F. Stability of a Planar Interface during solidification of a dilute binary alloy, Journal of Applied Physics, 35,444-451, 1964.

[2] Oliveros G.A., Liu R., Sridhar S., Ydstie B.E., Model development and control strategies for multicomponent alloy solidification processes and its application to solar cell manufacturing, Presented at the 2011 AIChE Annual Meeting, MN, USA.