本研究使用粒子網格方法模擬在低壓、高氣體密度與低電離率，兩極板在瞬間給定一穩定電壓差條件下之氬氣電漿粒子的非穩定三度空間動態行為。電漿之用途極廣，可用之於材料加工製造、薄膜製造、核融合、光源等；電漿性質更是物理、化學、光電、航空太空、工程科技熱門之研究範疇。本研究考慮粒子碰撞，包括電子與中性粒子間彈性、非彈性及電離碰撞，離子與中性原子間彈性及電荷交換碰撞，及電子與離子間的庫倫碰撞。本模式忽略磁場效應，不考慮二次電子散射，中性粒子為均勻分布，速度滿足麥斯威爾分布，及不考慮電子與離子的複合碰撞。計算結果將顯示粒子彈性、非彈性碰撞對於電漿行為及工件表面電荷不平衡鞘層的影響。理論與實驗完滿之比較將可確定並深入了解電漿經過鞘層到工件表面的非穩態質量、動量、能量傳遞現象。

This study uses the PIC (Particle-in-cell) method to simulate unsteady three-dimensional dynamics of particles in argon plasma under low pressure, high density, and weak ionization between two planar electrodes subject to a sudden biased voltage. Plasma has been widely used in materials processing, film manufacturing, nuclear fusion, lamps, etc. Properties of plasmas are also becoming important area for research. This work includes 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, and Coulomb collisions between electrons and ions. The model ignores magnetic field, secondary electron emission, recombination between ions and electrons, and assumes uniform distribution of the neutrals having velocity of Maxwellian distribution. The computed results show the effects of elastic and inelastic collisions on the characteristics of plasma and sheath (space charge region) in front of the workpiece surface. Unsteady mass, momentum and energy transport from the bulk plasma through sheath to the workpiece is confirmatively and exploratorily studied after successful comparison between PIC prediction and experimental data has been made.

中文提要 I
英文提要 II
致謝 III
目錄 IV
圖表目錄 VII
符號說明 XI

第一章 緒論 1
1.1 研究背景與目的 1
1.2 粒子網格(PIC)方法簡介 2
1.3 本論文研究內容簡介 7
1.4 本論文結構安排 8
第二章 理論分析 9
2.1 基本問題 9
2.1.1 模擬粒子 9
2.1.2 模擬粒子的電場和受力計算 9
2.1.3 粒子邊界條件 10
2.1.4 無因次化 12
2.2 物理模型與基本假設 13
2.3 粒子運動的處理 14
2.3.1 粒子的初始設定 14
2.3.2 粒子運動方程 15
2.4 粒子碰撞行為的處理 16
2.4.1碰撞理論 16
2.4.2 粒子碰撞後的分析 18
2.4.3 庫倫碰撞 22
2.4.4 碰撞截面積 25
2.5 模擬流程 27
第三章 數值方法求解 28
3.1 泊松方程式 28
3.1.1 邊界條件的設定 28
3.1.2 計算電荷密度 29
3.1.3 泊松方程式的求解 31
3.2 電場的求解 33
第四章 研究結果與討論 35
4.1 複合碰撞 35
4.2 收斂比對 36
4.3 鞘層厚度比對 36
4.4 電離碰撞 38
第五章 結論 40
參考文獻 41

[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] D. Bohm, The Characteristics of Electrical Discharges in Magnetic Fields, A. Guthrie and R.K. Wakerling, Eds. New York and London: McGraw-Hill, 1949.
[4] R.N. Franklin, “The plasma-sheath boundary region,” J. Phys. D: Appl. Phys., vol. 36, pp. 309-320, 2003.
[5] R.N. Franklin, “Where is the ‘sheath edge’？,” J. Phys. D: Appl. Phys., vol. 37, pp. 1342-1345, 2004.
[6] K.-U. Riemann, "Kinetic theory of the plasma sheath transition in a weakly ionized plasma," Physics of Fluids, vol.24, pp.2163-2172, 1981.
[7] K.-U. Riemann, "The plasma sheath transition of a weakly ionized plasma with reflecting walls," Proc. XVIIth Int. Conf. on Phenomena in Ionixed Gases, Budapest, pp. 519-521, 1985.
[8] K.-U. Riemann, "The Bohm criterion and the field singularity at the sheath edge," Physics of Fluids B, vol.1, pp.961-963, 1989.
[9] K.-U. Riemann, "The Bohm criterion and sheath formation," J. Physics D: Applied Physics, vol.24, pp.493-518, 1991.
[10] K.-U. Riemann, "The Bohm criterion and boundary conditions for a multicomponent system," IEEE Transactions on Plasma Science, vol.23, pp.709-716, 1995.
[11] O. Buneman, “Dissipation of currents in ionized media,” Phys. Rev., vol. 115, pp. 503-517, 1959.
[12] J.M. Dawson, “One-dimensional plasma model,” Phys. Fluids, vol. 5, pp. 445-459, 1962.
[13] S. Kondo and K. Nanbu, “PIC/MC analysis of three-dimensional DC magnetron discharge,” Rep. Inst. Fluid Sci., vol. 12, pp. 111-142, 2000.
[14] J.P. Verboncoeur, “Particle simulation of plasmas: review and advances,” Plasma Phys. Control. Fusion, vol. 47, pp. 231-260, 2005.
[15] R.J. Procassini, C.K. Birdsall, and E.C. Morse, “A fully, self-consistent particle simulation model of the collisionless plasma-sheath region,” Phys. Fluids B, vol. 2, pp. 3191-3205, 1990.
[16] G.A. Emmert, R.M. Wieland, A.T. Mense, and J.N. Davidson, “Electric sheath and presheath in a collisionless, finite ion temperature plasma,” Phys. Fluids, vol. 23, pp. 803-812, 1980.
[17] R.C. Bissell and P.C. Johnson, “The solution of the plasma equation in plane parallel geometry with a Maxwellian source,” Phys. Fluids, vol. 30, pp. 779-786, 1987.
[18] J.T. Scheuer and G.A. Emmert, “Sheath and presheath in a collisionless plasma with a Maxwellian source,” Phys. Fluids, vol. 31, pp. 3645-3648, 1988.
[19] B. Briehl and H.M. Urbassek, “Note on boundary conditions in plasma sheath simulations using the particle-in-cell algorithm,” IEEE Trans. Plasma Sci., vol. 29, pp. 809-814, 2001.
[20] M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing. New York: Wiley, 1994.
[21] R.T. Farouki, M. Dalvie, and L.F. Pavarino, “Boundary-condition refinement of the Child-Langmuir law for collisionless DC plasma sheaths,” J. Appl. Phys, vol. 68, pp. 6106-6116, 1990.
[22] K. Nanbu, “Probability theory of electron-molecule, ion-molecule, molecule- molecule, and coulomb collisions for particle modeling of materials processing plasmas and gases,” IEEE Trans. Plasma Sci., vol. 28, pp. 971-990, 2000.
[23] R.W. Boswell and I.J. Morey, “Self-consistent simulation of a parallel-plate RF discharge,” App. Phys. Lett., vol. 52, pp. 21-23, 1988.
[24] V. Vahedi and M. Surendra, “A Monte Carlo collision model for the particle- in-cell method: applications to argon and oxygen discharges,” Comput. Phys. Commun., vol. 87, pp. 179-198, 1995.
[25] 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.
[26] 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.
[27] M. Surendra, D.B. Graves, and I.J. Morey, “Electron heating in low-pressure RF glow discharges,” Appl. phys. Lett., vol. 56, pp. 1022-1024, 1990.
[28] C.K. Birdsall, “Particle-in-cell charged-particle simulations, plus Monte Carlo collisions with Neutral Atoms, PIC-MCC,” IEEE. Plasma Sci., vol. 19, pp. 65-85, 1991.
[29] V. Vahedi, C.K. Birdsall, M.A. Lieberman, G. DiPeso, and T.D. Rognlien, “Capacitive RF discharges modeled by particle-in-cell Monte Carlo simulation. I: analysis of numerical techniques,” Plasma Source Sci. Technol., vol. 2, pp. 261-272, 1993.
[30] T. Takizuka and H. Abe, “A binary collision model for plasma simulation with a particle code,” J. Comput. Phys., vol. 25, pp. 205-219, 1977.
[31] K. Nanbu, “Theory of cumulative small-angle collisions in plasmas,” Phys. Rev. E, vol. 55, pp. 4642-4652, 1997.
[32] E. Kawamura and C.K. Birdsall, “Effect of Coulomb scattering on low-pressure high-density electronegative discharges,” Phys. Rev. E, vol. 71, 026403, 2005.
[33] J.D. Blahovec, Jr., L.A. Bowers, J.W. Luginsland, G.E. Sasser, and J.J. Watrous, “3D ICEPIC simulations of the relativistic klystron oscillator,” IEEE Trans. Plasma Sci., vol. 28, pp. 821-829, 2000.
[34] P.C. Liewer and V.K. Decyk, “A general concurrent algorithm for plasma particle-in-cell codes,” J. Comput. Phys., vol. 85, pp. 302-322, 1989.
[35] B. Di Martino, S. Briguglio, G. Vlad, and P. Sguazzero, “Parallel PIC plasma simulation through particle decomposition techniques,” Parallel Comput., vol. 27, pp. 295-314, 2001.
[36] C.K. Birdsall and A.B. Langdon, Plasma Physics via Computer Simulation. New York: McGraw-Hill, 1985.
[37] R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles. New York: McGraw-Hill, 1981.
[38] C.K. Birdsall and J.M. Dawson, “Plasma physics,” Computers and their Role in the Physical Sciences, S. Fernbach and A. Taub, Eds. New York: Academic, 1970, pp. 247-310.
[39] J. Denavit and W.L. Kruer, “How to get started in particle simulation,” Comments Plasma Phys. Contr. Fusion, vol. 6, pp. 35-44, 1980.
[40] J.M. Dawson, “Particle simulation of plasmas,” Rev. Mod. Phys., vol. 55, pp. 403-447, 1983.
[41] J.P. Verboncoeur, A.B. Langdon, and N.T. Gladd, “An object-oriented electromagnetic PIC code,” Comput. Phys. Commun., vol. 87, pp. 199-211, 1995.
[42] 錢振型 主編，固體電子學中的等離子體技術，電子工業出版社，1987。
[43] 邵福球 編著，等離子體粒子模擬，科學出版社，2002。
[44] A. Anders, “Fundamentals of pulsed plasmas for materials processing,” Surface & Coatings Technology, vol. 183, pp. 301-311, 2004.
[45] K.U. Riemann, “The Bohm Criterion and Boundary Conditions for a Multicomponent System,” IEEE Trans. Plasma Sci., vol. 23, pp. 709-716, 1995.
[46] E. Zawaideh, F. Najmabadi, and R.W. Conn, “Generalized fluid equations for parallel transport in collisional to weakly collisional plasmas,” Phys. Fluids, vol. 29, pp. 463-474, 1986.
[47] A. Bogerts, R. Gijbels, and J. Vlcek, “Collisional-radiative model for an argon glow discharge,” J. Appl. Phys., vol. 84, pp. 121-136, 1998.