(657b) High-Speed Growth of Crystalline Wafers for Solar Cells

Currently the global market for mono-crystalline wafers for solar cells is about $20B and growing at about 15% per year. The main application motivating this work is how to design of a production process, called the Horizontal Ribbon Growth (HRG) process, that can reduce the costs of manufacturing silicon wafers for solar cells by 50%, leading to savings exceeding $10B per year. Other applications of the theory proposed here include the manufacture of sapphire windows for laptops and other displays.

The main challenge with HRG is that stable operating conditions for the process are not 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 goal of this work is to develop models to predict limitations to the HRG process. As we do so, we apply mathematical tools, that can be used to model other kinds of solidification processes 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. The parametric formulation allows us to find multivalued meniscus shapes which were previously 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.

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 calculate interface motion when it is non-smooth. Simulation of the HRG process demonstrate 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 90◦conclusion of the theory with 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.

The analysis shows that to develop a practical and economically feasible approach to HRG of single crystals it is necessary to apply dynamic control. At the end of the paper we show how we can use unsteady state periodic control to stabilize the process at high pull speed. The main idea behind the use of oscillation is that it improvers improves the heat transfer around the triple point, increases the growth speed of the solid, and homogenizes the melt. This prevents the significant undercooling that cause problems in the current implementation of HRG and may help to reduce the effects of impurity segregation close to the solid-liquid interface.

From a computational standpoint, GPU acceleration can be used to carry out highfidelity simulations in the unsteady state. This can be especially important with cellular automata algorithms, which are computationally irreducible. Advancements in GPU simulations and control of cellular automata process can also contribute to related areas of crystal growth.