現代的半導體製造因為晶圓尺寸的微縮，導致頻繁的使用低氣壓電漿製程，從文獻回顧可以了解到高氣壓的電漿模擬所使用的物理假設並不適用於低氣壓的操作條件，使用計算形式的電子能量分布函數來考量更多的物理效應對低氣壓電漿的模擬是十分必要的，低氣壓下電子的運動過程除了受到局部電場影響之外還必須考量空間變化造成的影響，考慮非局部效應的動力模型會對關鍵的高能粒子分布產生明顯的影響，為了了解低氣壓電漿的物理特性，有許多研究相繼提出不同模型來完成低氣壓電漿的模擬。純流體模型透過假設電子能量分布函數呈現馬克斯威爾分布等固定形式的分布進行模擬，而本研究所採用的模型是透過求解二項近似簡化的波茲曼方程式結合流體模型的混合模型來完成低氣壓電漿的模擬，並建立了簡單的一維模型對其EEDF(Electron Energy Distribution Function)進行分析。
本研究使用了兩種不同的氣相反應式，除了考慮不同的氣相反應之後會對結果直接產生影響，藉由對不同模型的結果分析也可以了解高能電子的分布情形主導著活性粒子的密度差異。另外使用非局部動力模型分析不同氣壓的結果時，可以看到在低氣壓時非局部效應變的很具有影響力，非局部效應明顯的提升了高能電子的分布，在非彈性碰撞反應對高能電子分布敏感以及對模擬結果影響明顯的前提下，非局部動力模型的重要性便是顯而易見的，因此在低氣壓的條件下使用非局部模型是非常必要的，同時通過對EEDF 的分析能為粒子分布帶來更清晰、完整的物理解釋。

Due to the scaling of wafer size in modern semiconductor manufacturing, low pressure plasma processes have become common. From the literature review, the physical assumptions used in high pressure plasma simulations are not suitable for low pressure. Therefore, it is
essential to consider a computational form of the electron energy distribution function(EEDF) to account for more physical effects in low pressure plasma simulations. In addition, the motion of electrons at low pressure is influenced not only by local electric fields but also by spatial variations. The inclusion of nonlocal effects in the kinetic model significantly impacts the distribution of critical high energy particles.
To understand the physical characteristics of low pressure plasma, various research studies have proposed different models for simulating low pressure plasma. A pure fluid model assumes fixed forms of distribution, such as the Maxwellian distribution, for the electron
energy distribution function. The model used in this study combines a simplified Boltzmann equation solved using a binomial approximation with a fluid model, achieving a hybrid model for simulating low pressure plasma. A simple one dimensional model was established to
analyze the EEDF.
Two different gas phase reaction models were employed in this study. Analyzing the results from different models allows us to understand how the distribution of high energy electrons dominates the density differences of particles. When analyzing results at different pressures using a nonlocal kinetic model, it's evident that nonlocal effects are significant at low pressures, leading to a notable enhancement of the high energy electron distribution. Considering the sensitivity of inelastic collision reactions to the distribution of high energy electrons and their significant impact on simulation results, the importance of nonlocal kinetic models becomes evident. Therefore, the use of non local models is essential under low pressure conditions, and analyzing the EEDF provides a clearer and more comprehensive physical explanation for particle distribution.

摘要---i
Abstract---ii
目錄---iv
圖目錄---vi
表目錄---viii
第一章 前言---1
1.1 研究背景---1
1.2 電容式耦合電漿源之簡介---2
1.3 研究動機與目的---2
第二章 文獻回顧---4
2.1 氣壓對於電子能量分布函數的影響---4
2.2 不同的動力模型對結果之影響---8
2.3 電子能量分布函數與空間相關的非局部效應---9
2.4 文獻回顧結論---13
第三章 物理模型與研究方法---14
3.1 模擬軟體介紹---14
3.2 純流體模型---15
3.2.1 電子---15
3.2.2 電子通量邊界條件---16
3.3 混合模型(流體模型+動力模組)---16
3.3.1 波茲曼方程式---16
3.3.2 電子分布函數邊界條件---19
3.3.3 電子的傳輸特性---20
3.3.4 總能量邊界條件---21
3.4 離子與中性粒子---21
3.4.1 離子的守恆方程式---21
3.4.2 離子的傳輸特性---22
3.5 電磁場---23
3.6 幾何結構---23
3.7 反應資料庫---24
第四章 不同模型之結果與影響---26
4.1 馬克斯威爾電子能量分布函數(Maxwellian)---26
4.1.1 條件與參數設定---26
4.1.2 模擬結果與分析---27
4.2 非局部動力模型EEDF---33
4.2.1 條件與參數設定---33
4.2.2 不同反應式之結果與分析---34
4.2.3 不同操作氣壓之模擬與分析---40
4.3 不同模型之差異---41
第五章 總結---45
參考文獻---46
附錄 A 局部動力模型---49
A.1 局部模型EEDF---49
A.2 條件與參數設定---49
A.3 結果與分析---50
附錄 B 質點網格法 Particle in cell---55
附錄 C 總能量守恆的概念---56

[1] S. Wilczek, J. Schulze, R. P. Brinkmann, Z. Donko, J. Trieschmann, and T. Mussenbrock, "Electron dynamics in low pressure capacitively coupled radio frequency discharges," Journal of Applied Physics, vol. 127, no. 18, pp. 181101-1-181101-26, 2020
[2] V. I. Kolobov, "Advances in electron kinetics and theory of gas discharges," Physics of Plasmas, vol. 20, no. 10, pp. 101610-1-101610-14, 2013
[3] D. B. Graves, "Plasma Processing," IEEE Transactions on Plasma Science, vol. 22, no. 1, pp. 31-42, 1994
[4] V. A. Godyak, B. M. Alexandrovich, and V. I. Kolobov, "Measurement of the electron energy distribution in moving striations at low gas pressures," Physics of Plasmas, vol. 26, no. 3, pp. 033504-1-033504-5, 2019
[5] U. Kortshagen, C. Busch, and L. D. Tsendin, "On simplifying approaches to the solution of the Boltzmann equation in spatially inhomogeneous plasmas," Plasma Sources Sci. Technol., vol. 5, no. 1, pp. 1-17, 1996
[6] L. D. Tsendin, "Electron kinetics in non-uniform glow discharge plasmas," Plasma Sources Sci. Technol., vol. 4, no. 2, pp. 200-211, 1995
[7] C. X. Yuan, E. A. Bogdanov, S. I. Eliseev, and A. A. Kudryavtsev, "1D kinetic simulations of a short glow discharge in helium," Physics of Plasmas, vol. 24, no. 7, pp. 073507-1-073507-16, 2017
[8] M. A. Lieberman, Principles of Plasma Discharges and Materials Processing, 2005
[9] V. A. Godyak and R. B. Piejak, "Abnormally Low Electron Energy and Heating-Mode Transition in a Low-Pressure Argon rf Discharge at 13.56 MHz," Physical Review Letters, vol. 65, no. 8, pp. 996-999, 1990
[10] V. A. Godyak, R. B. Piejak, and B. M. Alexandrovich, "Measurements of electron energy distribution in low-pressure RF discharges," Plasma Sources Sci. Technol., vol. 1, no. 1, pp. 36-58, 1992,
[11] V. Godyak, "Hot plasma effects in gas discharge plasma," Physics of Plasmas, vol. 12, no. 5, pp. 055501-1-055501-11, 2005
[12] V. I. Kolobov and R. R. Arslanbekov, "Simulation of electron kinetics in gas discharges," IEEE Transactions on Plasma Science, vol. 34, no. 3, pp. 895-909, 2006
[13] V. Ivanov, O. Proshina, T. Rakhimova, A. Rakhimov, D. Herrebout, and A. Bogaerts, "Comparison of a one-dimensional particle-in-cell-Monte Carlo model and a one-dimensional fluid model for a CH4/H-2 capacitively coupled radio frequency discharge," Journal of Applied Physics, vol. 91, no. 10, pp. 6296-6302, 2002
[14] M. Tatanova, Y. B. Golubovskii, A. S. Smirnov, G. Seimer, R. Basner, and H. Kersten, "Electron stochastic heating in a capacitively coupled low-pressure argon rf-discharge," Plasma Sources Sci. Technol., vol. 18, no. 2, pp. 025026-1-025026-6, 2009
[15] A. Ranjan and P. L. G. Ventzek, "Simulations of hybrid direct current radiofrequency (dc/rf) capacitively coupled plasmas," Japanese Journal of Applied Physics, vol. 58, no. 3, pp. 036001-1-036001-8, 2019
[16] L. Xu et al., "Diagnostics of ballistic electrons in a dc/rf hybrid capacitively coupled discharge," Applied Physics Letters, vol. 93, no. 26, pp. 261502-1-261502-3, 2008
[17] J. Schulze, Z. Donko, T. Lafleur, S. Wilczek, and R. P. Brinkmann, "Spatio-temporal analysis of the electron power absorption in electropositive capacitive RF plasmas based on moments of the Boltzmann equation," Plasma Sources Sci. Technol., vol. 27, no. 5, pp. 055010-1-055010-15, 2018
[18] G. J. M. Hagelaar and L. C. Pitchford, "Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models," Plasma Sources Science & Technology, vol. 14, no. 4, pp. 722-733, 2005
[19] V. Kolobov and R. Arslanbekov, "Four dimensional Fokker-Planck solver for electron kinetics in collisional gas discharge plasmas," Computer Physics Communications, vol. 164, no. 1-3, pp. 195-201, 2004
[20] S. V. Berezhnoi, I. D. Kaganovich, L. D. Tsendin, and V. A. Schweigert, "Fast modeling of the low-pressure capacitively coupled radio-frequency discharge based on the nonlocal approach," Applied Physics Letters, vol. 69, no. 16, pp. 2341-2343, 1996
[21] H. M, "Bibliography of electron and photon cross sections with atoms and molecules," Published in the 20th Century —Argon Report NIFS-DATA-72, 2003.
[22] R. H. McFarland and J. D. Kinney, "Absolute Cross Sections of Lithium and Other Alkali Metal Atoms for Ionization by Electrons," Physical Review, vol. 137, no. 4A, pp. A1058-A1061, 1965
[23] S. Rauf, "On uniformity and non-local transport in low pressure capacitively coupled plasmas," Plasma Sources Sci. Technol., vol. 29, no. 9, pp. 095019-1-095019-13, 2020
[24] V. Vahedi and M. Surendra, "A Monte Carlo collision model for the particle-in-cell method: applications to argon and oxygen discharges," Computer Physics Communications, vol. 87, no. 1-2, pp. 179-198, 1995
[25] C. K. Birdsall, "Particle-in-Cell Charged-Particle Simulations, Plus Monte-Carlo Collisions With Neutral Atoms,PIC-MCC," IEEE Transactions on Plasma Science, vol. 19, no. 2, pp. 65-85, 1991
[26] J. V. Dicarlo and M. J. Kushner, "Solving the spatially dependent Boltzmann’s equation for the electron-velocity distribution using flux corrected transport," Journal of Applied Physics, vol. 66, no. 12, pp. 5763-5774, 1989
[27] V. I. Kolobov and V. A. Godyak, "Nonlocal Electron Kinetics in Collisional Gas Discharge Plasmas," IEEE Transactions on Plasma Science, vol. 23, no. 4, pp. 503-531, 1995
[28] U. Kortshagen, I. Pukropski, and L. D. Tsendin, "Experimental investigation and fast two-dimensional self-consistent kinetic modeling of a low-pressure inductively coupled rf discharge," Physical Review E, vol. 51, no. 6, pp. 6063-6078, 1995
[29] L. J. Overzet, "Microwave Diagnostic Results from the Gaseous Electronics Conference RF Reference Cell," Journal of Research of the National Institute of Standards and Technology, vol. 100, no. 4, pp. 401-414, 1995
[30] M. Tatanova, G. Thieme, R. Basner, M. Hannemann, Y. B. Golubovskii, and H. Kersten, "About the EDF formation in a capacitively coupled argon plasma," Plasma Sources Sci. Technol., vol. 15, no. 3, pp. 507-516, 2006
[31] V. Vahedi and M. Surendra, "A Monte Carlo collision model for the particle-in-cell method: applications to argon and oxygen discharges," Computer Physics Communications, vol. 87, no. 1-2, pp. 179-198, 1995