單調迭代法半導體元件方程式數值解

Abstract

本論文主要是研究運用單調迭代法(Monotone Iterative Method)解二維半導體元件方程式。求解過程，首先藉由分離步驟(decoupled approach)將非線性泊松(Poisson)方程式、電子流以及電洞流連續方程式進行分離，再對每一分離後的偏微分方程式進行有限差分逼近轉換為非線性代數方程組，最後運用單調迭代法解每一非線性代數方程組。吾人所提出的方法其主要的觀點是源自於半導體方程式中非線性物理現象的特性，運用有限差分型單調迭代法求解每一分離後的方程式。 由於所提出的方法完全掌握半導體方程式中具有的強非線性特質，因此，本研究所提出的迭代法，它具有以下幾個重要優點:(1)所提出的迭代法其在分析半導體方程式計算上的收斂行為時，是全域性(global)的收斂性質；也就是說，由此法進行解行為分析與計算時，其計算的收斂性與初始猜測條件(initial guess)無關。(2)與一般的牛頓迭代法比較，此迭代法很容易實現；相對於牛頓迭代法，此迭代法具有速度快與計算時問短之特性。(3)再者，此法本身就具有可平行計算的基本架構，其在科學計算上將更易於平行化。 利用本研究所提出的迭代法，在各種不同偏壓情況下，吾人已成功地發展出N型通道金屬氧化物半導體元件二維模式問題的分析與計算，且模擬出數個典型的結果例如元件的電位分佈及直流特性等。相較於牛頓法，本研究所提出的求解方法其計算速度約為牛頓迭代法的30倍。

In this thesis, a monotone iterative method for solving two-dimensional semiconductor device equations is presented. We solve the semiconductor device equations by using decoupled approach to decouple three nonlinear PDE's, i.e., nonlinear Poisson equation, electron current continuity and hole current continuity equations. Then, finite difference approximation is applied to discretize the decoupled PDE's from which a system of nonlinear algebraic equations is obtained. Finally, monotone iterative method is applied to solve each system of nonlinear algebraic equations. The main concept of this method is that it takes some special nonlinear property of each equation, and analyzes each decoupled equation by using finite difference approximation and monotone iterative method. Based on the fact of a high nonlinear property of each semiconductor equation, the proposed monotone iterative method has several advantages. First, it is global convergence. In other words, it converges monotonically for arbitrary initial guess. Secondly, by comparing with a standard Newton's method, this method is easy for implementation, relatively faster with much less computation time, and its algorithm is inherently parallel in scientific computation. A two-dimensional N-MOSFET semiconductor device model problem under various bias conditions has been successfully implemented and several typical numerical results of this model problem, such as potential distribution and dc current characteristics of a device, are demonstrated. By comparing with a standard Newton's method, a speed-up factor of 30X can be achieved.