(66e) Existence and Properties of Multiple Steady States to Horizontal Ribbon Growth

Global energy demands are increasing significantly due to population explosion and industrialization. Renewable energy like wind and solar offer an alternative to the limited availability of non-renewable energy sources, which are increasing in cost and have deleterious environmental impact. Solar energy is a promising source of energy due to its abundance, and has been advocated by reducing productions costs and rising global demand. Despite this the silicon wafer costs are still considered high and accounts for 40% of the photo-voltaic supply chain [1]. This is in part because current ingot based wafer production techniques like Czhokralski and Brigdman processes---which contribute more than 80% of the silicon substrate for solar cells---are batch processes with low production rates (~2mm/min) and are highly wasteful [2]. These processes produce considerable amount of material loss while sawing (up to 55%), which can be significantly reduced if the wafers could be directly solidified to its correct thickness [3].

Silicon, like water, has a lower density in its solid form than its liquid form. This property of silicon can be used to solidify single-crystal silicon on top of its melt to form a thin sheet of ribbon. The ribbon can then be continuously extracted horizontally by cooling the upper surface of the melt. This process, descriptively called horizontal ribbon growth (HRG), has the potential for significant cost savings over current wafer production methods. Because the wafers must only be cut to length, there is a substantial decrease in material loss. Additionally, this process has the potential to run continuously and at much higher production rate of 12cm/min [4], which can lower costs. Since the silicon ribbon is resting on top of its melt, the HRG process reduces thermal and mechanical stresses, and limits contamination from the crucible compared with other ribbon growth techniques such as the edge-defined film growth (EFG) and ribbon growth on substrate (RGS) [5].

The HRG furnace is a distributed parameter system(DPS), with limited theoretical understanding of the process. The process itself has not been commercialized yet due to the difficulty of manufacturing a stable mono-crystalline ribbon at the desired thickness of 0.2mm, and at an economic production rate. Experimental investigations on the horizontal ribbon growth process of mono-crystalline silicon have reported the formation of a {111} facet at the leading growth edge of the silicon wafer [6]. This low-index facet was found to limit the rate at which the wafer was being pulled and required severe under-cooling in order to maintain its growth [7]. The formation of this facet also makes it difficult to obtain appropriate thickness control in order to meet industry standards. Operations at commercial pull speeds requires intense heat removal which makes the process susceptible to dentritic growth and hence put an upper limit to the pulling velocity. The existence of wavy un-smooth surface at the bottom of the wafer has also been attributed to the formation of this facet [4].

We formulate a novel cellular automata model to numerically simulate the directional solidification of silicon wafer as observed in experiments. We aim to provide a systems-level understanding at the meso-scale, to simulate the distributed parameter system using tools from process systems engineering, and circumventing the molecular complexities involved in the faceted growth of silicon crystal. The main feature of this algorithm is its efficacy to converge to the {111} facet of the leading edge independent of chosen grid size. On one hand, by choosing a coarse grid, the algorithm allows for shorter simulation times to scan through the range of optimal operational conditions. On the other, choosing a finer grid provides high accuracy results to study the formation and stability of the leading edge for a given operation point. Since the DPS exhibits multiple timescales for solidification and thermal diffusion, the algorithm tracks the phase change at the interface using a different time-step, maintaining speed. The algorithm incorporates an analytic expression for solidification at the interface along with an alternating direction implicit (ADI) scheme for 2D heat diffusion to maintain numerical stability. Using the temperature information from the ADI scheme, the analytic expressions determine the phase change explicitly and provide a uniform bound on the change in temperature at each grid point. This helps in avoiding oscillations due to overshooting and resolves discretization errors due to sharp temperature gradients that are inherently present in the HRG process. This has made it possible to converge to the faceted solution for different operating conditions while maintaining numerical speed and stability.

The main contribution of this work is its agreement with experimental results using our novel cellular automata model. Depending on the initial condition, the leading edge converges to the {111} facet which is in agreement with the experimental observations. This faceted solution to the leading edge was found to be almost half-stable i.e. although the solution was stable to positive deviation to the orientation of the interface at the triple point, a modest negative deviation in the orientation would diverge the faceted solution to a second (stable) roughened solution. The roughened solution is similar in nature as predicted by Zoutendyk [8] and displays an inverse relationship for pull speed versus thickness. The roughened solution also does not produce severe under-cooled zones and offers a desirable shape of the interface with a triangular growth tip, allowing for the required thickness control. Hence the existence of a roughened solution would be more favorable in practice.

References:

[1] Ranjan, S., S. Balaji, Rocco A. Panella, and B. Erik Ydstie. "Silicon solar cell production." Computers & Chemical Engineering 35, no. 8 (2011): 1439-1453.

[2]Oliveros, German A., Ruochen Liu, Seetharaman Sridhar, and B. Erik Ydstie. "Silicon wafers for solar cells by horizontal ribbon growth." Industrial & Engineering Chemistry Research 52, no. 9 (2013): 3239-3246.

[3]Ke, Jiaying, Aditya S. Khair, and B. Erik Ydstie. "The effects of impurity on the stability of Horizontal Ribbon Growth." Journal of Crystal Growth 480 (2017): 34-42.

[4]Kellerman, Peter, Brian Kernan, Brian T. Helenbrook, Dawei Sun, Frank Sinclair, and Frederick Carlson. "Floating Silicon Method single crystal ribbon–observations and proposed limit cycle theory." Journal of Crystal Growth 451 (2016): 174-180.

[5]Schönecker, Andreas, L. J. Geerligs, and Armin Müller. "Casting technologies for solar silicon wafers: block casting and ribbon-growth-on-substrate." In Solid State Phenomena, vol. 95, pp. 149-158. Trans Tech Publications, 2004.

[6]Kellerman, Peter. Floating Silicon Method. No. DOE-FSM-00595. Applied Materials-Varian Semiconductor Equipment, 2013.

[7]Helenbrook, Brian T., Peter Kellerman, Frederick Carlson, Nandish Desai, and Dawei Sun. "Experimental and numerical investigation of the horizontal ribbon growth process." Journal of Crystal Growth 453 (2016): 163-172.

[8]Zoutendyk, John A. "Analysis of forced convection heat flow effects in horizontal ribbon growth from the melt." Journal of Crystal Growth 50, no. 1 (1980): 83-93.