柴氏長晶法是目前最常用來生長單晶矽的方法，柴氏長晶法適合生長大尺寸且高品質的矽晶體並有良好的經濟效應。
本研究的目的以有限元素法為基礎的COMSOL Multiphysics軟體模擬不同長晶階段的爐體溫度分佈與氬氣的速度分佈，以獲得爐體內之熱流場分佈情況與相關資訊。
輻射隔絕器可以減少熱損失並引導氬氣流動減少上爐壁氧化矽沉積，有了優化的熱區，在液固界面處的溫度梯度會增加，晶體可以以更快的速度生長。所以輻射隔絕器和拉速是影響溫度與晶體生長的主要參數。通過晶體與熔湯溫度分佈的分析，不同的熱傳條件與熔湯流動會導致凹型液固界面、凸型液固界面與W型液固界面的形成。
結果發現，矽熔湯內部的溫度場會受到流場浮力影響而扭曲變形，隨著固化分率的增加，熔湯內部溫度及速度會趨於緩和，且因晶體生長長度增加，增加了晶棒表面的散熱速度，使得液固界面會更凸向熔湯。坩堝旋轉會造成矽熔湯產生三個渦旋，使得熔湯更為穩定。在未改變熱場與拉速之前，液固界面為W型。改變熱場後液固界面為凸型，改變熱場與拉速後，且得到較平坦的凹型液固界面。

Nowadays, the Czochralski (CZ) method is the most commonly used scheme for the single crystal growth of silicon, which is good for growing a large and high-quality crystal. The study is to simulate the furnace temperature and argon velocity distributions at different stages of silicon single-crystal growth, utilizing the Czochralski method. The numerical simulation is made by employing the finite element software COMSOL Multiphysics. Heat shields can prevent the heat loss in the upper portion of the furnace and guide the argon flow to make the SiO2 deposition decrease on the upper furnace wall. With the optimized hot zone, the increase of temperature gradient near the crystal/melt interface has the crystal grow faster. Accordingly, the heat shields and pulling rate are the primary factors affecting the heat transfer and crystal growth. Through analyses of the temperature distributions in the crystal and melt, it can be found that different heat transfer conditions and melt flow patterns lead to the formation of convex, concave or W shape interfaces. The results show that the temperature field affected by buoyant force causes the distortion of isothermals in melt. As the solid fraction is raised, the temperature and velocity fields of melt become eased and the increase of ingot surface enhances the heat dissipation, which makes the interface shape more convex. The crucible rotation creates three vortex in the melt, which results in the stable melt. Before adding the heat shield and changing pulling rate, the crystal/melt interface is W-shaped. After adding the heat shield, the interface becomes concave. With the increase of the pulling rate, the shape of the interface becomes flatter than before.

[1]林明獻, "太陽電池技術入門," 全華圖書,臺北, 中華民國九十七年.
[2]J. Z. Czochralski, phys.Chem., p. 219, 92(1989).
[3]G. K. Teal and J. B. Little, "Growth of Germanium Single Crystals Containing p-n Junctions," Phys. Rev., 647(1950).
[4]H. J. Scheel and T.Fukuda, "The Development of Crystal Growth Technology," Crystal Growth Technology, p. 647, 3-14(2003).
[5] 林明獻, "矽晶圓半導體材料技術," 全華圖書,臺北, 中華民國九十六年.
[6]W.C.Dash, J. of Applied physics, 459(1959).
[7]V. V. Voronkov and J., Crystal Growth, p. 625, 59(1982).
[8]D. T. J. HURLE, "Analytical represention of the shape of the meniscus in Czochralski growth" Crystal Growth Technology, 13-17(1983).
[9]J. J. Derby and R. A. Brown, "Thermal-capillary analysis of Czochralski and liquid encapsulated crystal growth I. Simulation," Journal of Crystal Growth, 605-624(1986)
[10]J. J. Derby and R. A. Brown, "Thermal-capillary analysis of Czochralski and liquid encapsulated crystal growth II. Simulation," Journal of Crystal Growth, 227-240(1986)
[11]F. Dupret and Y. Ryckmans, "Numerical calculation of the global heat transfer in a Czochralski furnace," Journal of Crystal Growth, 84-91(1986).
[12]Y. Ryckmans, P. Nicodeme, and F. Dupret, "Numerical Simulation of crystal growth : Infuluence of melt convection on global heat transfer and interface shape," Journal of Crystal Growth, 702-706(1990).
[13]K.-W. Yi, H.-T. Chung, H.-W. Lee, and J.-K. Yoon, "The effects of pulling rates on the shape of crystal/melt interface in Si single crystal growth by the Czochralski method," Journal of Crystal Growth, 451-460(1993).
[14]N. V. d. Bogaert and F.Dupret, "Dynamic global simulation of the Czochralski process I. Principles of the method," Crystal Growth Technology, 65-76(1997).
[15]N. V. den and F.Dupret, "Dynamic global simulation of the Czochralski process II. Analysis of the growth of germanium crystal," Crystal Growth Technology, 77-93(1997).
[16]A. Lipchin and R. A. Brown, "Hybrid finite-volume/finite-element simulation of heat transfer and melt turbulence in Czochralski crystal growth of silicon," Journal of Crystal Growth, 192-203(2000).
[17]K. Takano and e. al, "Global simulation of the CZ silicon crystal growth upto 400 mm in diameter," Crystal Growth Technology, 26-30(2001)
[18]黃禮翼, "高效率太陽能矽單晶提拉之熱場設計與分析," 國立臺灣大學,臺北, 中華民國九十三年.
[19]C. J. Jing, "Global analysis of heat transfer in CZ crystal growth of oxide taking into account three-dimensional unsteady melt convection:Effect of meniscus shape," Crystal Growth Technology, 204-213(2008).
[20]C. Jianweia, "Simulation aided hot zone design for faster growth of CZ silicon mono crystals," RARE METALS, vol. 30, p. 155, 3-14(2003).
[21]O. A. Noghabi, "Analysis of W-shape melt/crysta linterface formation in Czochralski silicon crystal growth," Crystal Growth Technology, 77-82(2013).
[22]J. A. Dantzig, "Modelling liquid-solid phase changes with melt convection," International Journal for Numerical Methods in Engineering, vol. 28, pp. 1769-1785, 1989.
[23]陳柏瑋, "數值模擬晶棒成長之熱流分析," 國立交通大學,新竹, 中華民國九十九年.
[24]陳俊宏, "泡生法生長氧化鋁單晶之數值模擬分析," 國立中央大學,桃園, 中華民國一零一年.
[25]L. B.E. and S. D.B., "The Numerical Computation of Turbulent Flow," Computer Method in Applied Mechanics and Engineering vol. 3, pp. 269-289, 1974.
[26]M. Kirpo, "Global simulation of the Czochralski silicon crystal growth in ANSYS FLUENT," Journal of Crystal Growth, vol. 371, pp. 60-69, 2013.
[27]O. A. Noghabi, M. Jomâa, and M. M'Hamdi, "Analysis of W-shape melt/crystal interface formation in Czochralski silicon crystal growth," Journal of Crystal Growth, vol. 362, pp. 77-82, 2013.
[28]A. Muiznieks, A. Krauze, and B. Nacke, "Convective phenomena in large melts including magnetic fields," Journal of Crystal Growth, vol. 303, pp. 211-220, 2007.
[29]O. Asadi Noghabi, M. M'Hamdi, and M. Jomâa, "Effect of crystal and crucible rotations on the interface shape of Czochralski grown silicon single crystals," Journal of Crystal Growth, vol. 318, pp. 173-177, 2011.
[30]Y.-Y. Teng, J.-C. Chen, C.-C. Huang, C.-W. Lu, W.-T. Wun, and C.-Y. Chen, "Numerical investigation of the effect of heat shield shape on the oxygen impurity distribution at the crystal–melt interface during the process of Czochralski silicon crystal growth," Journal of Crystal Growth, vol. 352, pp. 167-172, 2012.
[31]J. Cao, Y. Gao, Y. Chen, G. Zhang, and M. Qiu, "Simulation aided hot zone design for faster growth of CZ silicon mono crystals," Rare Metals, vol. 30, pp. 155-159, 2011.
[32]W. Su, R. Zuo, K. Mazaev, and V. Kalaev, "Optimization of crystal growth by changes of flow guide, radiation shield and sidewall insulation in Cz Si furnace," Journal of Crystal Growth, vol. 312, pp. 495-501, 2010.
[33]O. V. Smirnova, N. V. Durnev, K. E. Shandrakova, E. L. Mizitov, and V. D. Soklakov, "Optimization of furnace design and growth parameters for Si Cz growth, using numerical simulation," Journal of Crystal Growth, vol. 310, pp. 2185-2191, 2008.
[34]J. Virbulis, T. Wetzel, E. Tomzig, and W. von Ammon, "Silicon melt convection in large size Czochralski crucibles," Materials Science in Semiconductor Processing, vol. 5, pp. 353-359, 2002.