A thermal-capillary dynamic model for Czochralski (Cz) growth of silicon single crystal is presented that accounts for convection in the melt, conduction in the crystal, and radiation from the melt free surface and crystal. The shapes of the crystal/melt interface, moving crystal, and free surface are governed by the balance of energy and stresses. Decrease of the melt volume as the crystal is grown is also considered. A control algorithm based on the adjustment of the pull rate and/or crucible wall temperature has been developed. The diameter of the crystal can remain constant or vary with time based on the strategies implemented to control the growth process.
Diameter-Controlled Czochralski Growth of Silicon Crystals
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Zhang, H., Zheng, L. L., Prasad, V., and Larson, D. J., Jr. (November 1, 1998). "Diameter-Controlled Czochralski Growth of Silicon Crystals." ASME. J. Heat Transfer. November 1998; 120(4): 874–882. https://doi.org/10.1115/1.2825906
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