Czochralski法是生長高品質矽單晶的重要技術。其中矽晶片的電阻率和差排密度與晶棒中雜質有很重要關係，特別是氧和碳的濃度。因此雜質控制便成為Czochralski法製造單晶矽中的重要問題，然而矽晶棒中的碳濃度流動與設定參數之間的關係在現存文獻中尚未確立，在這項研究中，軸對稱數值模型用於研究Cz過程中碳的質量傳遞現象，考慮晶體和坩堝的不同轉速和方向，研究不同流動模式下的質量傳遞現象。
我們研究晶體和坩堝旋轉的哪種組合具有較低的碳濃度，並分別進行晶體坩堝同向與反向旋轉，找到造成碳濃度降低的物理機制。我們發現晶體旋轉10 rpm和反向坩堝旋轉3 rpm具有最低的碳濃度。而碳傳輸的行為與熔融矽中的對流和晶體下方的速度有關，介面的溫度、熔湯氧濃度也會影響碳雜質溶入熔湯的多寡。其中有三個重要的渦流，首先是Taylor-Proudman cell在晶體 - 熔體界面下，它是碳濃度重要的來源。其次，浮力渦流將碳帶入熔體，熔湯的自由表面的範圍成為一個重要因素。這是因為碳在浮力渦流中也會再透過自由液面蒸發，如果被其他渦流抑制，碳便不會在一開始時蒸發而直接流入下個渦流。第三，二次渦流位於Taylor-Proudman cell和浮力渦流之間，適當大小的二次渦流能使碳滯流於此渦流，使碳有更多機會流回到浮力渦流讓碳蒸發。

Czochralski (Cz) method is widely used for the production of high quality silicon single crystal. Under high temperature condition of growth process, the undesirable impurities, such as oxygen and carbon, enter the silicon melt and their content strongly affects the resistivity and the dislocation density of the silicon wafer. A precise control of these impurities at a low concentration and uniform distribution, therefore, has played an important role for improving the quality of silicon crystals, especially large-sized crystals. To our best knowledge, there are few publications showing the effects of the operation parameters of CZ growth process on carbon concentration. In this study, a 2D axisymmetric numerical model is used to study the heat and carbon transport during the growth of a 6 inch-diameter silicon ingot. Different rotation speed and direction of the seed and crucible are considered to investigate their effects on the variation of heat, flow, and carbon characteristics.
The numerical simulations show that the carbon concentration gets lowest when the counter rotation rates of seed and crucible are 10 rpm and -3rpm, respectively. The behavior of carbon movement is related to the melt convection and the velocity under crystal-melt interface. While the temperature on free melt surface and oxygen in the melt will affect the quantity of carbon. The flow structure is included three main vortices: Taylor-Proudman cell (1), under the crystal-melt interface, buoyancy driven cell near the crucible wall (3), and the secondary cell (2) between (1) and (3). It was found that the carbon atoms are carried by cell (3) into the silicon melt. The carbon atoms are got out of the melt from the free melt surface. The larger effective evaporation area may reduce the carbon content in the melt due to the larger evaporation rate of carbon. Moreover, the secondary vortex (2) also affects the carbon transportation. Appropriate cell (2) may keep the carbon atoms stay longer in the melt and buoyancy cell (3) is easier to bring them to the free melt surface.

[1] J. Czochralski, “Ein neues Verfahren zur Messung der Kristallisation geschwindigheit der Metalle,” Zeitschrift fur Physikalische Chemie, Vol.92, pp. 219-221, 1918.
[2] P. E. Tomaszewski: Jan Czochralski Restored, Oficyna Wydawnicza ATUT Wroclaw 2013.
[3] Reinhard Uecker, “The Historical Development of the Czochralski Method,” Journal of Crystal Growth, Vol. 401, pp. 7-24, 2014.
[4] H. Walther, Rev. Sci. Instrum. Vol. 8, pp. 406, 1937.
[5] 林明獻，矽晶圓半導體材料技術，全華圖書，台北，民國九十六年
[6] FEMAG直拉法單晶矽生長數值模擬方案，中仿仿真智領創新
http://solution.cntech.com/femag/201412/femag_simulation.html, accessed on April 23, 2019.
[7] O. Anttila, “Czochralski growth of silicon crystals”, Silfex Incorporated-A division of Lam Research Corporation, Eaton, OH, USA
[8] W.C. Dash, “Growth of silicon crystals free from dislocations”, J. Appl. Phys. Vol.30, pp.459, 1959.
[9] Fumio Shimura, Single-Crystal Silicon: Growth and Properties, Springer International Publishing, Japan, 2017.
[10] H. P Utech and M. C. Flemings, “Elimination of Solute Banding in Indium Antimonide Crystals by Growth in a Magnetic Field”, Journal of Applied Physics, Vol. 37, pp. 2021, 1966.
[11] H. A. Chedzey and D. T. J. Hurle, “Avoidance of Growth-striae in Semiconductor and Metal Crystals grown by Zone-melting Techniques”, Nature, Vol. 210, pp. 933, 1966.
[12] U. P. Utech and M.C. Flemings, Journal of Applied Physics, Vol. 37, pp. 2021, 1966.
[13] H. A. Chedzey and D. T. J Hurle, Nature, Vol. 210, pp. 933, 1966.
[14] K. Hoshikawa, X. Huang,“Oxygen transport during Czochralski silicon crystal growth,”Materials Science and Engineering B, Vol. 72, pp. 73-79, 2000.
[15] Y. S. Lee, C.H. Chun,“Effects of a cusp magnetic field on the oscillatory convection coupled with crucible rotation in Czochralski crystal growth,”Journal of Crystal Growth, Vol.197, pp. 307-316, 1999.
[16] K. Seigo, M. Yoshiaki, K. Masaru, I. Takashi, “ Thermally induced microdefects in Czochralski-grown silicon: nucleation and growth behavior,” Jpn. J. Appl. Phys. Vol. 21, pp. 1–12, 1982.
[17] Y. Nagai, S. Nakagawa, K. Kashima, “ Crystal growth of MCZ silicon with ultralow carbon concentration,” Journal of Crystal Growth, Vol.401, pp.737–739, 2014.
[18] Wen Lin (AT and T Bell labs), “The incorporation of oxygen into silicon crystals, ”
Semiconductors and Semi-Metals, Academic Press, New York, Vol. 42 pp.9-52, 1994.
[19] D.E. Bornside, R.A. Brown, “The Effects of Gas-Phase Convection on Carbon Contamination of Czochralski-Grown Silicon,” J. Electrochem. Soc. Vol.142, pp.2790, 1995.
[20] B. Gao, K. Kakimoto, “Global simulation of coupled carbon and oxygen transport in a Czochralski furnace for silicon crystal growth,” Journal of Crystal Growth, Vol. 312, pp. 2972-2976, 2010.
[21] Y. Nagai, S. Nakagawa, K. Kashima, “Crystal Growth of MCZ Silicon with Ultralow Carbon Concentration,” Journal of Crystal Growth, Vol. 401, pp. 737-739, 2014.
[22] Xin Liu, Bing Gao, Koichi Kakimoto, “Numerical investigation of carbon contamination during the melting process of Czochralski silicon crystal growth,” Journal of Crystal Growth, Vol. 417, pp. 56-64, 2014.
[23] Xin Liu, Satoshi Nakano, K. Kakimoto,“Effect of the packing structure of silicon chunks on the melting process and carbon reduction in Czochralski silicon crystal growth,” Journal of Crystal Growth, Vol. 468, pp. 595-600, 2017.
[24] Xin Liu, Bing Gao, Satoshi Nakano, K. Kakimoto, “Reduction of carbon contamination during the melting process of Czochralski silicon crystal growth,” Journal of Crystal Growth, Vol. 474, pp. 3-7, 2017.
[25] Xin Liu, Xue-Feng Han, Satoshi Nakano, K. Kakimoto, “Effect of controlled crucible movement on melting process and carbon contamination in Czochralski silicon crystal growth,” Journal of Crystal Growth, Vol. 483, pp. 241-244, 2018.
[26] Thi Hoai Thu Nguyen, Jyh Chen Chen, Chieh Hu, Chun Hung Chen, “Numerical simulation of heat and mass transfer during Czochralski silicon crystal growth under the application of crystal-crucible counter- and iso-rotations,” Journal of Crystal Growth, Vol. 507, pp. 50-57, 2019.
[27] 鄧應揚，「多晶矽太陽能電池晶碇固化生長之熱流場研究」，國立中央大學，博士班資格考計畫書，民國 97 年。
[28] T. Nozaki, Y. Yatsurugi, and N. Akiyama, “Concentration and Behavior of Carbon in Semiconductor Silicon,” J. Electrochem. Soc., Vol. 117, pp. 1566–1568, 1970.
[29] A.D. Smirnov, V.V. Kalaev, “Development of oxygen transport model in Czochralski growth of silicon crystals,” Journal of Crystal Growth, Vol. 310, pp.2970- 2976, 2008.
[30] Y. Y. Teng, J. C. Chen, C. W. Lu, C. Y. Chen, “Numerical and experimental study for improving the concavity of the crystalline front in multicrystalline silicon ingots during the directional solidification process,” solidification process, submit to Solar Energy Material & Solar Cells.
[31] O. R. Asadi Noghabi and M. M'Hamdi, “Sensitivity analyses of furnace material properties in the Czochralski crystal growth method for silicon”, Measurement Science and Technology, Vol. 24, 2013.