使用平衡方程式的半導體元件模擬

Abstract

本研究從波茲曼方程式推導出平衡方程式，用來模擬半導體元件。吾人採用蒙地卡羅模擬方法所求出的弛懈率近似值，而此近似值考慮到非拋物線能帶結構。這種流體力學模擬方法可用來描述次微米的拋射物二極體。我們是利用牛頓法來線性化這些耦合在一起的平衡方程式和帕森方程式，且用左上右下三角矩陣(LU)分解法求解線性系統。在平衡狀態下和施予偏壓的情況下，描述電位、載子濃度、速度和能量（溫度）的暫態分佈。並於數個不同的偏壓下，觀察這些變量在不同的偏壓下在穩態的不同情況。這個流體力學的模型可以正確的預測載子在次微米元件中受熱而增加溫度和速度變化的情形，且速度的異常超射的情況並沒有在我們的模擬結果中出現。

The balance equation method derived from Boltzmann transport equation is presented. The relaxation rate approximation is accounted to nonparabolic band structure by using Monte Carlo method. The hydrodynamic simulation of submicron ballistic diode is described. The coupled system of the balance equations and Poisson’s equation are linearized by Newton’s iteration and solved by LU decomposition method. The transient distributions of the variables such as electrostatic potential, carrier concentration, velocity, and energy under equilibrium and during voltage applied are illustrated. The distributions of all variables under steady state at various applied voltages are also shown. The hydrodynamic model can accurately predict the carrier heating phenomena in sub-micron device. However, the spurious velocity overshoot has not been observed in this work. English Abstract ii Acknowledges iii Contents iv Figure Captions v Chapter 1 Introduction 1 Chapter 2 Fundamental Equations and Physical Models 4 2.1 Boltzmann Transport Equation 4 2.2 Derivation of Balance Equations 5 2.2.1 Carrier Balance Equation 5 2.2.2 Momentum Balance Equation 6 2.2.3 Energy Balance Equation 7 2.3 Collision Terms 9 2.4 Balance Equations for Silicon 11 2.5 Poisson’s Equation 11 2.6 Boundary Conditions 12 Chapter 3 Numerical Method and Solution 14 3.1 Normalization 14 3.2 Discretization 15 3.3 Linearization 18 3.4 Solution Procedure 21 Chapter 4 Results and Discussion 24 4.1 Transient Response at Thermal Equilibrium 24 4.2 Transient Response with Applied Voltage 32 4.3 Steady State Distribution under Various Applied Voltages 38 4.4 The Current Density 45 4.5 The Effect of Mesh Size 49 Chapter 5 Conclusion and Future Work 53 References 54