# 臺灣博碩士論文加值系統

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 本篇論文針對半導體儀器作數值模擬，運用 inexact Newton''s method 對 drift-diffusion model 求解。考慮原型的 drift-diffusion model 包含：電子電壓，電子濃度，電洞濃度等三個未知變數。數值實驗使用 drift-diffusion model 模擬一個一維的二極體幾何模型。我們討論兩個不同的 non-dimensionalization approach 對 Newton''s method 的影響並分析 GMRES method 使用不同的 preconditioner 在 Newton''s method 的結果。實驗結果顯示使用不同的 non-dimensionalization approach 將影響 Newton''s method 的收斂情形。在實驗中我們使用 US non-dimensional approach (Uniform Scaling non-dimensional approach) 有效的提供 Newton''s method 一個良好的環境。根據實驗結果發現增加 block Jacobi preconditioner 中 block 的數量幾乎不影響 Newton''s method 的迭代次數，更甚者即便是增加網格點的數目 Newton''s method 的迭代次數依然不受影響。
 The aim of this thesis to employ an inexact Newton''s method to solve discrete drift-diffusion model in semiconductor device simulations, where the drift-diffusion model in the primitive form consists of the electrostatic potential , the electron concentrations and the hole concentrations. Consider a 1D diode simulations modeled by drift-diffusion as a test case. We discuss the effect on Newton''s method by two non-dimensionalization approaches and the application of GMRES method without/ with diagonal and block Jacobi. It is true that the non-dimensional approach will affect the converge of Newton''s method. In our case, we choose US non-dimensional approach (Uniform Scaling non-dimensional approach) and it will make a great environment for Newton''s method. From numerical experiment, we find that increasing number of blocks for a block Jacobi preconditioner almost doesn''t affect the number of Newton''s iterations and decreasing grid size for a block Jacobi preconditioner also doesn''t affect the Newton''s iterations neither.
 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The drift-diffusion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 The drift-diffusion model in semiconductor devices . . . . . . . . . . . . 22.2 Two Non-dimensionalization approaches . . . . . . . . . . . . . . . . . . 33 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1 Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 The inexact Newton method with backtracking for semiconductor algorithm 84 Numerical result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1 The drift-diffusion model for the n
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