低溫電漿平行化流體程式的發展及其應用

Abstract

本論文研究目的是發展與驗證一個平行化一維/一維軸對稱/二維/二維軸對稱低溫非熱平衡電漿流體模型程式，數值方法主要是使用Fully-Implicit有限差分法以及混和解析解與數值解的Jacobian矩陣，論文將詳細介紹模擬的實作方式。此程式可以廣泛的應用在各 種不同的氣壓以及不同的功率輸入頻率(從百萬赫茲射頻到數千赫茲的交流電壓)。模擬的結果將會與實驗結果驗證比較，並且討論其中的電漿物理與化學。

本研究主要可以分成三部份。第一部份，建立並且驗證一個平行化一維/一維軸對稱/二維/二維軸對稱低溫非熱平衡電漿流體模型程式。程式中使用的流體模型是由波茲曼方程式出發，推導出包含所有粒子的連續方程式、所有帶電粒子則使用Drift-Diffusion近似的動量方程式以及電子的能量方程式。Poisson方程式則用來解析空間中的電位分佈。所有的待定變數都經過無因次化，必且使用完全耦合的Newton-Krylov-Schwarz (NKS) 演算法將方程式離散。其中，Overlapping additive Schwarz 方法被使用來作為preconditioner，而Bi-CGStab 及GMRES 方法被使用來解析線性方程式矩陣。一系列使用氦氣與氮氣以及不同輸入頻率的一維電漿模擬結果與本研究室的實驗結果相互驗證。二維的GEC氦氣電漿模擬結果也與文獻中的實驗數據與數值解析結果互相驗證。平行程式的效率測試則是使用國立中央大學的V’ger cluster system (Xeon 3GHz dual-core dual-CPU)作為測試平台。測試結果顯示，在使用144個處理器的狀況下，平行效率還可以達到超線性。而最好的平行計算組合是使用LU分解法preconditioner，搭配GMRES 方法解析線性矩陣。

論文的第二部份：主要是利用第一部份所發展的平行化流體模型程式來研究氦氣DBD在輸入一個變形正弦波下的研究。研究中我們選用了兩組不同氦氣電漿反應方程式，模擬並驗證比較不同的實驗結果。結果顯示使用較為複雜的氦氣電漿反方程式可以如實地得到與實驗接近的計算結果。根據模擬的結果發現電漿在驅動的過程中經歷了複雜的模式轉換：從long secondary Townsend like 到dark current like, 接著short primary Townsend like 以及類short secondary Townsend like。

論文的第三部份：主要是模擬研究通入混合著氫氣與矽烷的電漿輔助化學氣相沉積電漿源。為了減少計算耗費的時間，模擬中是用了多尺度時間方法分別處理電子、離子與中性粒子的時間進行。模擬所需要的背景氣體密度分佈以及溫度分佈則是引用一個有限體積法Navier-Stokes 解析所計算的結果作為電漿模擬的初始條件。氫氣與矽烷的電漿總共使用了15種不同的粒子並引用28個電漿反應方程式。結果顯示SiH3是最主要的帶矽自由基，結果與文獻相符合。在假設表面反應site的比率為0.015的狀況下，我們可以成功的計算出符合實驗結果的沈積速率與均勻度。

除了總結論文的研究結果之外，同時在論文最後章節亦條列出建議未來應進行研究的研究部分。

Development of parallelized 1D/1D-axisymmetric and 2D/2D-axisymmetric fluid modeling codes using fully implicit finite-difference method with hybrid analytical-numerical Jacobian evaluation for low-temperature, non-equilibrium plasma simulation has been reported in this thesis. Implementation and validations against earlier simulations and experimental data are described in detail. Applications with wide range of pressures and frequencies (radio frequency in mega Hertz and alternating current in kilo Hertz) are demonstrated, compared with experimental data wherever possible, and related plasma physics and chemistry are discussed therein. Research in this thesis is divided into three major phases, which are briefly described in the following in turn.

In the first phase, parallelized 1D/1D-axisymmetric and 2D/2D-axisymmetric fluid modeling codes based on distributed memory framework were developed and verified. Fluid modeling equations, resulting from the velocity moments of Boltzmann equation, include continuity equations for all species, momentum equations with drift-diffusion approximation for all charged species, diffusion approximation for neutral species, energy density equation for electrons and Poisson’s equation for electrostatic potential distribution. All model equations were nondimensionalized and discretized using fully coupled Newton-Krylov-Schwarz (NKS) algorithm, in which the overlapping additive Schwarz method and Bi-CGStab/GMRES scheme were used as the preconditioner and linear matrix equation solver, respectively. An atmospheric 1D helium dielectric barrier discharge (DBD) (driven by 13.56 MHz and 20 KHz power sources) and an atmospheric nitrogen 1D DBD (driven by 60 KHz power source) are simulated and validated by excellent agreement of discharged currents with experimental results obtained in our laboratory. In addition, a 2D-axisymmetric RF driven GEC chamber with helium discharge (capacitively coupled plasma) was simulated and the results agree reasonably well with previously reported experimental data. Then, parallel performance of the fluid modeling code was investigated using the same GEC case, which was simulated on a PC cluster system (V’ger cluster system with Xeon 3GHz dual-core dual-CPU, the Center for Computational Geophysics, National Central University, Taiwan.) up to 144 processors. The parallel performance showed superlinear speedup up to 144 processors with the GMRES as the matrix solver combining with LU as the preconditioner.

In the second phase, a one-dimensional helium DBD driven by 20 KHz distorted sinusoidal voltages was investigated in detail using the developed fluid modeling code. Effect of selecting plasma chemistry on simulations of helium DBD was investigated by comparing simulations with experiments. Results show that the simulations, which include more helium related reaction channels, could faithfully reproduce the measured discharged temporal current quantitatively. Based on the simulated discharge properties, we have found that there is complicated mode transition of discharges from the long secondary Townsend like to the dark current like, then to short primary Townsend like and to short secondary Townsend like. Related plasma physics and chemistry are described in detail.

In the third phase, a chamber-scale gas discharge of plasma enhanced chemical vapor deposition (PECVD) with silane/hydrogen as the precursors, which was used for depositing hydrogenated amorphous silicon thin film, was simulated. A multiscale temporal marching scheme for electron, ion and neutral species is designed to reduce the computational cost. Neutral flow and thermal field obtained by a finite-volume Navier-Stokes equation solver was used as the background gas. The plasma chemistry includes 15 species and 28 reaction channels. Results show that SiH3 is the most dominating radical species with silicon that is directly related to silicon film deposition, which agree with previous experiments and simulations. The deposition rate and uniformity evaluated from the simulation results agree with experiment data if the fraction of reaction sites is assumed to be 0.015.

Recommendations of future research are also outlined at the end of this thesis.