本研究使用粒子網格PIC(Particle-in-cell)方法模擬在兩電極板間，低壓、高氣體密度和低電離率，瞬間給定一偏壓電位條件下之氬氣電漿粒子的非穩態三度空間的輸送變數。近年來電漿其用途已經被廣泛而且有效地運用在材料加工製造、薄膜製造、核融合發電、光源等，研究電漿的輸送現象因此非常重要。本模型忽略磁場效應，不考慮二次電子散射和離子與電子間的複合碰撞，並且假設中性粒子為均勻分布，速度滿足麥斯威爾分布。本研究考慮電子與中性粒子間的彈性與非彈性碰撞(激發碰撞和電離碰撞)，離子與中性粒子間的彈性和電荷交換碰撞。本模型之計算結果將顯示，極板在瞬間給定負直流偏壓電位條件下，離子通過鞘層到達工作表面的非等向性壓力、剪應力及熱傳現象。

This study uses the PIC (Particle-in-cell) method to simulate unsteady three-dimensional transport variables in argon plasma under low pressure and weak ionization between two planar electrodes suddenly biased by a negative voltage. Plasma has been widely used in etching, ion implantation, light source, and encountered in nuclear fusion, etc. Studying transport processes of plasmas therefore is important. This work ignores magnetic field, secondary electron emission, recombination between ions and electrons, and assumes a uniform distribution of the neutrals having velocity of a Maxwellian distribution. Accounting for elastic collisions between electrons and neutrals, ions and neutrals, and inelastic collisions resulting in ionization from impacting neutrals by electrons, and charge exchange between ions and neutrals, the computed results in this work quantitatively show non-isotropic pressures, shear stresses and heat conduction of the ions across the sheath to the surfaces suddenly biased by a dc negative voltage.

謝誌 Ⅰ
目錄 Ⅱ
圖目錄 IV

符號說明 Ⅴ
中文摘要 VII

英文摘要 VIII

第一章　緒論 1
　　　1.1　研究背景與目的 1
　　　1.2　粒子網格(PIC)方法 3
　　　1.3　本論文研究內容簡介 5
1.4 本文架構 5
第二章　理論分析 6
2.1　模擬流程 6
　　　2.2　無因次化 7
　　　2.3　系統模型與假設 8
2.4 粒子初始設定與運動方程式 9
2.5 外加電場 10
2.5.1 泊松方程式與邊界條件的設定 10
2.5.2 計算電荷密度 12
2.5.3 求解泊松方程式 13
2.5.4 電場的求解 17
2.6 蒙地卡羅方法處理粒子間的碰撞 18
2.6.1 碰撞理論 18
2.6.2 粒子碰撞後的分析 19
2.6.3 各種碰撞類型的碰撞截面積 24
第三章　研究結果與討論 25
3.1　鞘層厚度 25
　　　3.2　壓力、剪應力與熱傳 28
　　　3.3　離子能量分布 32
第四章　結論 37
參考文獻 38

[1] F.F.Chen,Introduction to Plasma Physics and Controlled Fusion.New York and London:Plenum Press,second edition, 1983.
[2] L.Tonks and I.Langmuir,“A general theory of the plasma of an Arc,”Phys. Rev., Vol. 34, pp. 876-922,1929.
[3] Bohm, D., The Characteristics of Electrical Discharges in Magnetic Fields, A. Guthrie and R.K. Wakerling, Eds. New York and London: McGraw-Hill, 1949.
[4] Anders, A., 2004, “Fundamentals of Pulsed Plasmas for Materials Processing,” Surface and Coating Technology, Vol. 183, pp.301-311.
[5] Lieberman, M.A. and Lichtenberg, A.J., 1994, Principles of Plasma Discharges and Materials Processing. Wiley, New York.
[6] Widner, M., Alexeff, I., Jones, W. D., and Lonngren, K. E., 1970,“Ion Acoustic Wave Excitation and Ion Sheath Evolution,” Physics of Fluids, Vol.13, pp.2532-2540.
[7] Lieberman, M. A., 1989, “Model of Plasma Immersion Ion Implantation,” J. Applied Physics, Vol.66, pp.2926-2929.
[8] Vahedi, V., Lieberman, M. A., Alves, M. V., Verboncoeur, J. P., and Birdsall, C. K., 1991, “A One-Dimensional Collisional Model for Plasma-Immersion Ion Implantation,“ J. Applied Physics, Vol.69, pp.2008-2014.
[9] Stewart, R. A., and Lieberman, M. A., 1991, “Mode of Plasma Immersion Ion Implantation for Voltage Pulses with Finite Rise and Fall Times,“ J. Applied Physics, Vol.70, pp.3481-3487.
[10] Riemann, K.-U. and Daube, Th., 1999, “Analytical Model of the Relaxation of a Collisionless Ion Matrix Sheath,” J. Applied Physics, Vol.86, pp.1202-1207.
[11] Daube, Th., Meyer, P., Riemann, K.-U., and Schmitz, H., 2002, “Relaxation Phenomena in Pulsed Discharges,” J. Applied Physics, Vol.91, pp.1787-1796.
[12] Riemann, K.-U., 1991,"The Bohm criterion and sheath formation," J. Physics D: Applied Physics, Vol.24, pp.493-518.
[13] Hsu, K. C., and Pfender, E., 1983,"Analysis of the Cathode Region of a Free-Burning High Intensity Argon Arc," J. Applied Physics, Vol.54, pp.3818-3824.
[14] Montierth, L. M., Neuman, W. A., and Morse, R. L., 1992, “Fluid Model Treatment of Surface Plasma Structures,” Physics of Fluids B, Vol.4, pp.784-795.
[15] Scheuer, J. T. and Emmert, G. A., 1990,"A Fluid Treatment of the Plasma Presheath for Collisionless and Collisional Plasmas" Physics of Fluids B, Vol.2, pp.445-451.
[16] Stangeby, P. C., 2000, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing, Bristol, UK
[17] Bissell, R. C., Johnson, P. C., and Stangeby, P. C., 1989,"A Review of Models for Collisionless One-Dimensional Plasma Flow to a Boundary," Physics of Fluids B, Vol.1, pp.1133-1140.
[18] Hong, M. P., and Emmert, G. A., 1995, “Two-Dimensional Fluid Simulation of Expanding Plasma Sheaths,” J. Applied Physics, Vol.78, pp.6967-6973.
[19] Sheridan, T. E. and Alport, M. J., 1994, “Two-dimensional Model of Ion Dynamics during Plasma Source Ion Implantation,” Applied Physics Lett., Vol.64, pp.1783-1785.
[20] Kim, H. C., Iza, F., Yang, S. S., Radmilovi -Radjenovi , M., and Lee, J. K., 2005, “Particale and Fluid Simulations of Low-Temperature Plasma Discharges: Benchmarks and Kinetic Effects,” J. Physics D: Applied Physics, Vol.38, pp.R283-R301.
[21] Wei, P. S., and Yeh, F. B., 2000, "Fluid-like Transport Variables in a Kinetic, Collisionless Plasma near a Surface with Ion and Electron Reflection," IEEE Transactions on Plasma Science , Vol. 28, pp. 1233-1243.
[22] Sutton, G. W., and Sherman, A., 1968, Engineering Magnetohydrodynamics, McGraw-Hill, New York.
[23] Procassini, R.J., Birdsall, C.K. and Morse, E.C., 1990, “A fully, self-consistent particle simulation model of the collisionless plasma-sheath region,” Phys. Fluids B, Vol. 2, pp. 3191-3205.
[24] Briehl, B. and Urbassek, H.M., 2001,“Note on boundary conditions in plasma sheath simulations using the particle-in-cell algorithm,” IEEE Trans. Plasma Sci., Vol. 29, pp. 809-814.
[25] Kostov, K. G., Barroso, J. J., and Ueda, M., 2004, “Two Dimensional Computer Simulation of Plasma Immersion Ion Implantation,” Brazilian J. Physics, Vol.34, pp.1689-1695.
[26] Briehl, B. and Urbassek, H. M., 2003, “Simulation of Sheath Dynamics and Current Nonuniformity in Plasma-Immersion Ion Implantation of a Patterned Surface,” J. Applied Physics, Vol.93, pp. 4420-4431.
[27] Lee, H. J., and Choi, Y. W., 2004, “Particle-in-Cell Simulation of the Collisional Sheath in an Argon DC Discharge,” J. Korean Physical Society, Vol.44, pp.1044-1048.
[28] Birdsall, C.K. and Langdon, A.B., 1985, Plasma Physics via Computer Simulation. McGraw-Hill, New York.
[29] Hockney, R.W., and Eastwood, J.W., 1988, Computer Simulation Using Particles, Institute of Physics Publishing, Bristol.
[30] Birdsall, C. K., 1991,“Particle-in-Cell Charged-Particle Simulations, Plus Monte Carlo Collisions with Neutral Atoms, PIC-MCC,” IEEE Transactions on Plasma Science, Vol. 19, pp. 65-85.
[31] Vahedi, V., and Surendra, M., 1995, “A Monte Carlo Collision Model for the Particle- in-Cell Method: Applications to Argon and Oxygen Discharges,” Computer Physics Communications, Vol. 87, pp. 179-198.
[32] Nanbu, K., 2000, “Probability Theory of Electron-Molecule, Ion-Molecule, Molecule- Molecule, and Coulomb Collisions for Particle Modeling of Materials Processing Plasmas and Gases,” IEEE Trans. Plasma Science, Vol. 28, pp. 971-990.
[33] Kondo, S., and Nanbu, K., 2000, “PIC/MC Analysis of Three-Dimensional DC Magnetron Discharge,” Rep. Inst. Fluid Sci., Vol. 12, pp. 111-142.
[34] Verboncoeur, J. P., 2005, “Particle Simulation of Plasmas: Review and Advances,” Plasma Phys. Control. Fusion, Vol.47, pp. A231-A260.
[35] S.L.Lin and J.N. Bardsley, “Monte Carlo simulation of ion motion in drift tubes,” J.chem.phys.,vol.66,pp. 435-445,1977.
[36] J.P. Boeuf and E.Marode, “A Monte Carlo analysis of an electron swarm in a non-uniform field: the cathode region of a glow discharge in helium,”J.phys. D: Appl. Phys., vol. 15,pp. 2169-2187, 1982
[37] D. A. Anderson, J.C. Tannehill and R. H. Pletcher: “Computational Fluid Mechanics and Heat Transfer”, McGraw-Hill, New York,1984, pp. 128-130.
[38] Theodoros Panagopoulos and Demetre J. Economou,1998, “Plasma sheath model and ion energy distribution for all radio frequencies,” vol.85.pp.3435.
[39] E-Kawamura and V. Vahedi and M. A. Lieberman and C. K. Birdsall, 1999, “Plasma Sources Sci. Technol,”vol.8, R45-R64.
[40] Benoit-Cattin P, Bernard L. C. and Bordenave-Montesquieu A 1967 Dispersion en energie des ions produits par une source a excitation de haute frequence Entropie 18 29.