臺灣博碩士論文加值系統

本論文研究目的是發展、驗證與應用一個平行化二維軸對稱的電漿流體模型程式。本研究所使用的電漿流體模型程式包含了流體模型方程式、計算感應電磁場的Maxwell’s equations、計算靜電場的Ambipolar 擴散模型、計算蝕刻二氧化矽表面鍵結情形的表面平衡方程式。我們是使用半隱式(semi-implicit)有限差分法離散流體模型所考慮的方程式。這些離散後的矩陣方程式則是使用Krylov subspace (KSP)和矩陣前處理器(preconditioner)數值方法來求解。在計算的過程中，我們利用區域切割的平行計算方法來加快計算時間，處理器在計算期間所需要的資訊交換是透過MPI的函式庫來完成。接著，我們利用此一電漿流體模型來探討不同型式的CF4低溫電感式電漿源在蝕刻SiO2基版時所發生之電漿物理與化學特性。研究結果顯示CF3+是主要的離子氣體，而負離子F-的濃度相較於正離子氣體稍微低一點，但是和電子相比，F-的濃度卻相當的接近電子濃度，代表所模擬的電漿源為負離子電漿源。在中性氣體的模擬結果顯示活性自由基F原子氣體僅次於CF4進氣氣體，為最多的活性氣體; 另外，SiF4、COF2 和O2為蝕刻SiO2時所產生最主要的蝕刻氣體，和實驗所量測之蝕刻產物氣體成分相當吻合。最後，模擬顯示螺旋式線圈和進氣口均圍繞在傳統腔體側邊的電漿源的蝕刻效率最好，因為不但有高蝕刻速率並且有良好的蝕刻均勻度。本研究主要可以分成四個部份。第一部分為二維軸對稱電漿流體模型方程式之描述，第二部分為我們所使用的離散方法、數值方法和平行化的方法之簡介，第三部分為平行程式之驗證與平行效能測試，第四部分主要是利用所驗證的平行化流體模型程式來研究各種不同CF4電感式電漿源。以下將一一簡單介紹每一部份。
在第一部分，我們將詳細介紹由波茲曼方程式推導出之電漿流體模型、Maxwell’s equations和表面動力模型。此電漿流體模型包含所有粒子的連續方程式、帶電粒子的漂移-擴散(drift-diffusion) 近似的動量方程式以及電子的能量方程式。流體模式中所需要的靜電場是依據Ambipolar擴散模型所得到。電子能量方程式所需要的電子吸收功率，則是計算Maxwell’s equations得到軸向的感應電場並帶入Ohmic heating得到電漿吸收功率。程式裡面用來計算二氧化矽覆蓋率和表面化學特性的表面動力學也會在此部份中做一詳盡的介紹。
在第二部分，我們將介紹程式中計算流體的數值方法和平行化的計算。此部份一開始會詳細介紹離散化後之方程式。在每一時間步進中，我們使用Krylov subspace method (KSP)和矩陣前處理(precondition)之組合求解離散化後之矩陣方程。為了加速計算的速率，我們使用區域切割計算方法來平行化這些計算之數值方法。
在第三部分，我們將驗證與測試所發展之平行化電漿流體程式的正確性和平行計算效能。程式之驗證是和Fukumoto等人[26][74]之前的研究結果作一比較，在同一模擬條件下所模擬的結果和他們所作之實驗和模擬數據非常一致。我們將所發展的平行化電漿流體模型在台灣國家高速電腦中心IBM1350的機器上作一效能測試，發現疊代法使用GMRES和矩陣前處理器使用Block Jacobi在網格數為122 ?e 179之電漿模擬能線性加速計算效能至26顆處理器。
在第四部分，我們利用所發展和所驗證的平行化電漿流體模型程式探討不同型式的CF4電感式電漿源內之物理機制和化學特性。此部份所探討之電感式電漿源包括標準的GECRC電漿源、平板式線圈設計在傳統腔體上方的電漿源、螺旋式線圈設計在傳統腔體側邊的電漿源以及圓頂式腔體的電漿源。本部份的電漿流體化學模型總共考慮96條氣相化學反應式和27條二氧化矽表面反應。首先，我們將詳盡地討論在GECRC電感式電漿源和圓頂腔體式電感式電漿源內之電漿氣體的空間分布、化學特性、蝕刻產物和化學反應速率。結果顯示不論在GECRC或是圓頂式腔體內部主要的離子氣體為CF3+，原因為解離CF4產生CF3+的解離能在所有的解離反應為最低。而負離子F-的濃度相較於正離子氣體稍微低一點，但是由於其電子附著反應能很低，使得F-的濃度相當的接近電子濃度。此外，在中性氣體的結果發現氟原子氣體F僅次於CF4背景氣體為最多的活性氣體，因為氟原子在游離反應中被大量的生成。此外，蝕刻過程所產生的蝕刻氣體大部份為SiF4、COF2 和O2，其結果和很多實驗所量測結果一致。同時，我們應用此模型來探討各種電感式電漿源的蝕刻速率和蝕刻均勻度並幫助改進電感式電漿源的腔體設計。模擬結果顯示把螺旋式線圈和環狀進氣口兩者圍繞在傳統腔體測邊的電漿源之蝕刻效率最好，因為不但有高蝕刻速率並且有良好的蝕刻均勻度，其原因為電漿生成區域接近外圍基版，不會過度集中於腔體中間區域，使的蝕刻所需的活性離子氣體如氟原子氣體均勻地傳輸到基版。
此外，我們會將本論文研究的主要發現以及未來應進行之研究部分條列說明於論文之最後。

This thesis reports development and validation of a parallel 2-D axisymmetric plasma fluid model which includes the fluid modeling equations for plasma transport, the Maxwell’s equations for induced electric field, ambipolar diffusion for electrostatic electric field, and surface kinetic model for modeling SiO2 etching. This model is discretized by using the semi-implicit finite difference method with preconditioned Krylov subspace (KSP) method for discretized modeling equations. The fluid modeling code is parallelized using domain decomposition method through the use of MPI protocol. We employ this plasma fluid model to study plasma physics and chemistry of tetrafluoromethane (CF4) gas discharge considering the etching process of a SiO2 substrate in different inductively coupled plasma (ICP) sources. The results show that CF3+ is the dominant charged species, and F- concentration is comparable to that of electron. Reactive F atom is the most dominant radical in CF4 discharge. In addition, the major etching products from the the substrate are SiF4, COF2 and O2 in all ICP reactors considered in this thesis. Finally, the results show that it is possible to design an ICP reactor with relatively high and uniform etching rate with both gas inlet and coil arranged along the cylindrical wall. Researches in this thesis are divided into four major parts. The first part is the description of the fluid moldeing equations. The second part is the numerical schemes and algorithms for solving the fluid modeling equations and corresponding parallel computing method. The third part is the description of validation and parallel performance of the parallel fluid modeling code. The fourth part is the description of applications of the developed fluid modeling code for study the CF4 discharge in various inductively coupled plasma sources.
In the first part, the fluid modeling equations, derived from the velocity moments of Boltzmann equation, the Maxwell’s equations and the surface kinetic model are introduced in detail. The fluid modeling equations include the continuity equations for charged and neutral species, the electron energy density equation, the momentum equations by the drift-diffusion approximation for the charged species, the ambipolar diffusion approximation for the electrostatic field. The power absorption that is needed in electron energy equation is solved through the Maxwell’s equation. The surface kinetic model (or site balance equations) for CF4 discharge etching SiO4 is also introduced in this part.
In the second part, the numerical schemes and algorithms for solving fluid modeling equations and the corresponding parallel computing method are introduced. The discretized equations are presented in detail. A combined method of preconditioning and Krylov subspace method (KSP) are proposed to solve the large sparse algebraic linear system formed at each time step. Parallel computing of the fluid modeling code using domain decomposition is also reported.
In the third part, the validation and parallel performance of the developed parallel 2D axisymmetric plasma fluid model are reported. Simulations are compared reasonably well with the previous simulation and experimental results by Fukumoto et al. [26][74]. Parallel performance study shows that the fluid code is scalable up to 26 processors on the IBM-1350 at National Center for High-Performance Computing in Taiwan using the combination of GMRES and Block Jacobi with sub-preconditioner ILU with a problem size of 122×179.
In the fourth part, we employ our developed and validated parallel plasma fluid model to study plasma physics and plasma chemistry of CF4 discharge in different geometries of ICP reactors: a typical ICP reactor, a GECRC, and a dome-shaped ICP reactor. The plasma chemistry includes 96 gas-phase reaction channels and 27 surface reaction channels. The spatial distributions of various plasma properties, etching characteristics, and production rate are described in detail in the both GECRC and dome-shaped ICP reactor. The numerical results indicate that CF3+ is the dominant charged species because the threshold energy of electron-impact dissociative ionization reacting with the feedstock that produces CF3+ is the lowest among all dissociative ionization. In addition, F- concentration is comparable to that of electron in the CF4 discharge because dissociative-attachment energy of CF4 is the lowest. Furthermore, the reactive F atom is found to be the most dominant radical in CF4 ICP discharge. The major etching products from the substrate are SiF4, COF2 and O2, which is consistent with the previous experimental observations. Meanwhile, this plasma fluid model is employed to study the etching characteristics in different geometries of ICPs reactor for predicting etching rate and corresponding uniformity on the substrate surface. The results show that the typical cylindrical ICP reactor with both gas inlet and coils arranged along the cylindrical wall gives relatively high and uniform etching rate because the dissociative ionization tends to peak off-axis to affect plasma density profile most likely off-axis so that the discharge region is flat and wide near the substrate. It proofs that this simulation tool could help to optimize the designs of large reactors with very low cost.
Finally, major findings and recommendations for future study are outlined at the end of the thesis.

TABLE OF CONTENTS
中文摘要 I
ABSTRACT IV
ACKNOWLEDGMENTS VIII
TABLE OF CONTENTS IX
LIST OF TABLES XIV
LIST OF FIGURES XV
NOMENCLATURE XXIII
CHAPTER 1 INTRODUCTION 1
1.1 Background and Motivation 1
1.2 Overview of Low-Pressure Non-Equilibrium Plasma Sources 3
1.2.1 Classification of Low-Pressure Gas Discharges in Semiconductor Fabrication 4
1.2.1.1 Direct Current Plasma (DCP) 4
1.2.1.2 Capacitively Coupled Plasma (CCP) 5
1.2.1.3 Electron Cyclotron Resonance Plasma (ECR) 6
1.2.1.4 Inductively Coupled Plasma (ICP) 7
1.2.2 Some Fundamentals of Plasma Physics 8
1.2.2.1 Debye Length 9
1.2.2.2 Plasma Sheath 10
1.2.2.3 Bohm Velocity 10
1.2.2.4 Plasma Oscillations (plasma frequency) 11
1.2.3 Classification of Plasma Chemistry 12
1.2.3.1 Electron-Impact Reactions 12
1.2.3.2 Ion Related Reactions 13
1.2.3.3 Neutral Related Reactions 14
1.3 Overview of Inductively Coupled Plasma Source (ICPs) 14
1.3.1 Dielectric Constant and Conductivity 17
1.3.2 Wave and Skin Depth in ICPs 17
1.3.3 Ohmic Heating in an ICPs 18
1.4 Literature Survey for Modeling ICP 19
1.4.1 Numerical Modeling of ICPs 20
1.4.2 Electropositive ICP 21
1.4.3 Electronegative ICP 23
1.4.4 Tetrafluoromethane CF4 Discharge in ICPs 25
1.5 Specific Objectives and Organization of this Thesis 27
CHAPTER 2 NUMERICAL METHODS 29
2.1 Introduction 29
2.2 Fluid Model Equations 30
2.2.1 Transport Equations for Charged Species 34
2.2.2 Transport Equations for Neutral Species 35
2.2.3 Electron Energy Density Equation 36
2.3 Maxwell’s Equations for Electron Power Absorption Calculation 37
2.3.1 Vector Wave Equation 37
2.3.1.1 Plasma Region 38
2.3.1.2 Quartz Region 39
2.3.1.3 Coil Region 39
2.3.1.4 Air Region 39
2.3.1.5 Maxwell’s Equations in Alternative Form 40
2.3.2 Power Absorption 41
2.4 Heat eqution 41
2.5 Surface Kinetic Model 42
2.6 Boundary Conditions 44
2.7 Gas-Phase Reactions of CF4 ICP Source 46
2.8 Numerical Discretizations and Algorithms 47
2.8.1 Discretizations of Conservation Equations 47
2.8.1.1 Discretizations of Continuity Equations 48
2.8.1.2 Discretizations of Electron Energy Density Equation 49
2.8.1.3 Discretizations of Maxwell’s Equation 49
2.8.1.4 Discretizations of Heat Equation 51
2.8.2 Numerical Algorithms for Solving Discretized Equations 52
2.8.3 Numerical Procedures 53
2.8.4 Implementation of Parallel Computation 54
CHAPTER 3 VALIDATION AND PARALLEL PERFORMANCE 55
3.1 Validation 55
3.2 Parallel Performance 56
CHAPTER 4 RESULTS AND DISSCUSION 58
4.1 CF4 Discharge in Gaseous Electronics Conference Reference Cell (GECRC) 58
4.1.1 Spatial Profiles 59
4.1.1.1 Induced Electric Field, Power Absorption and Electron Temperature 59
4.1.1.2 Production Rate 60
4.1.1.3 Electron and Negative Charge Ions 62
4.1.1.4 Positive Ions 63
4.1.1.5 Neutral and Radical Species 64
4.1.2 Chemical Ingredients 67
4.1.3 Surface Coverage 68
4.1.4 Etching Rate and Fluxes to Substrate 68
4.2 CF4 Discharge in a Dome-Shaped ICP Reactor 70
4.2.1 Spatial Profiles 71
4.2.1.1 Induced Electric Field, Power Absorption and Electron Temperature 71
4.2.1.2 Production Rate 72
4.2.1.3 Electron and Negative Charge Ions 74
4.2.1.4 Positive Ions 75
4.2.1.5 Neutral and Radical Species 76
4.2.2 Chemical Ingredients 78
4.2.3 Surface Coverage 79
4.2.4 Etching Rate and Fluxes to Substrate 79
4.3 Comparison of Etching Rates in Different Type of Inductively Coupled Plasma Reactors 81
4.3.1 Reactor Geometries 81
4.3.2 Testing Conditions 82
4.3.3 Spatial Profiles 82
4.3.4 Fluxes, Coverage and Etching Rate 85
CHAPTER 5 CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORK 88
5.1 Summary 88
5.2 Recommendations for Future Work 90
REFERENCES 91
TABLES 97
FIGURES 111
LIST OF PUBLICATIONS 166
APPENDIX A 168
APPENDIX B 170

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