Limitations to High-speed Crystal Growth Based on Conservation Laws

The main application motivating this thesis is the design of a high-speed crystal growth process, called the Horizontal Ribbon Growth (HRG), that can reduce the costs of manufacturing silicon wafers by 50%. Silicon wafers serve as the primary photovoltaic material for solar cells, therefore, innovations in HRG can make solar energy more affordable for everybody. 

The main challenge with HRG is that stable operating conditions for the process are not very well known. In addition, the process is not economically viable due to low production speeds. To add to this, current models of crystal growth cannot predict the limitations in production speed. Without a predictive model, it is not possible to diagnose the limitations of the HRG process. 

The main goal of this thesis is to develop models that can predict limitations to the HRG process. As we do so, we build on a wider set of mathematical tools, which can be used to model other kinds of solidification processes, like crystal growth and droplet freezing, as well. 

To find the stable operating conditions, we develop a parametric free energy formulation and use Weierstrass’ variational theory to analyze stable ribbon growth configurations under static conditions. The parametric formulation allows us to find multivalued meniscus shapes which are currently not known in the crystal growth field. The stability of the meniscus shapes is analyzed using second order variation to the free energy. The systems exhibits saddle node bifurcations and shows no solution for the meniscus in the horizontal ribbon configuration. The range of stable operating conditions is plotted as a function of pull angle and melt height. We also perform a simple proof of concept “kitchen” experiment to illustrate the instability of the HRG configuration. 

A novel numerical algorithm based on energy conservation is developed to model the heat transfer and phase transition near non-smooth interfaces. The algorithm uses a conservative discretization scheme to simulate non-smooth interface motion, for e.g., in the case of crystal growth. Simulation of the HRG process demonstrates the phenomena of pull speed limitation observed in experiments. A series of simulation studies are performed to quantify the effects of active cooling on the ribbons’ growth rate and thickness. A linear scaling relationship between the limiting pull speed and the total heat removed is derived empirically for a family of Gaussian cooling profiles. These scaling relationships show that the intensity and spread of a cooling profile are directly tied to the growth rate limit and the ribbon’s thickness, respectively.

Conservation laws are used to find constraints on the angles at the solid-liquid-gas triple junction. Energy and mass conservation imply a 90^◦ angle for the solid and liquid phase. The problem of pull speed limitation is directly attributed to the perpendicular shape of the solid-liquid interface. The perceived advantage of the HRG process with vertical heat transfer is found not true. The experimental observations of a 55^◦ facet angle are reconciled with the theoretical 90^◦ angle using a multiple-scale theory. A cellular automata simulation algorithm is outlined to explain this point of view. Results from the simulations exhibit a 55^◦ solid angle at the triple junction, in line with the multiple-scale theory. The formation of a facet angle at the triple junction is shown to have a negative effect on the pull speed limitation. Results also include the first simulation evidence for the formation of a dual facet at the triple junction.