單調迭代法應用於一個半導體元件能量傳播模型的適應性有限元素解

Abstract

在本篇博士論文，一個有關於半導體元件能量傳播模型的新的自伴隨 公式被提出並詳細的討論。由於當我們利用適應性有限元素法去計算 半導體元件能量傳播模型時，必須要解決一組非線性聯立方程組。而 這一個公式能夠導致方程組具備對稱與單調的良好性質，因此Picard， Gauss-Seidel 和Jacobi 這三種單調迭代法便可應用於此一組非線性 聯立方程組的計算。這三種方法具有眾多的有利特徵：全域性的收斂 性，不需組合聯立方程組的矩陣，也不需組合聯立方程組的Jacobian 矩陣。然後，我們將詳細地描述一個將這些方法移植入計算機的演算 法來闡明本論文的主要架構：適應性、逐點計算與全域性收斂。最後， 我們以數個深次微米的二極體和場效應電晶體等半導體電子元件來展 示這個演算法的精確性與效率性。

A self-adjoint formulation of the energy transport model of semiconductor devices is proposed. This new formulation leads to symmetric and monotonic properties of the resulting system of nonlinear algebraic equations from an adaptive finite element approximation of the model. Picard, Gauss-Seidel, and Jacobi monotone iterative methods are then presented for solving the system. These are globally convergent methods that do not require the assembly of the global matrix system and full Jacobian matrices. An adaptive algorithm implementing these methods are described in detail to illustrate the main features of this thesis, namely, adaptation, node-by-node calculation, and global convergence. Numerical results of simulations on deep-submicron diode and MOSFET device structures are given to demonstrate the accuracy and efficiency of the algorithm.