Computational Electromagnetics --Dragica Vasileska and Stephen M. Goodnick:Computational Electromagnetics --Dragica Vasileska and Stephen M. Goodnick
Copyright ©2006 by Morgan & Claypool
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in
any form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotations
in printed reviews, without the prior permission of the publisher.
Computational Electronics
Dragica Vasileska and Stephen M. Goodnick
www.morganclaypool.com
1598290568 paperback Vasileska/Goodnick
1598290576 ebook Vasileska /Goodnick
DOI 10.2200/S00026ED1V01Y200605CEM006
A Publication in the Morgan & Claypool Publishers series
SYNTHESIS LECTURES ON COMPUTATIONAL ELECTROMAGNETICS
Lecture #6
Editor: Constantine A. Balanis, Arizona State University
Series ISSN Synthesis Lectures on Computational Electromagnetics
Print 1932-1252 Electronic 1932-1716
First Edition
10 9 8 7 6 5 4 3 2 1
Printed in the United States of America
ABSTRACT
Computational Electronics is devoted to state of the art numerical techniques and physical models
used in the simulation of semiconductor devices from a semi-classical perspective. Computational
Electronics, as a part of the general Technology Computer Aided Design (TCAD) field,
has become increasingly important as the cost of semiconductor manufacturing has grown
exponentially, with a concurrent need to reduce the time from design to manufacture. The
motivation for this volume is the need within the modeling and simulation community for a
comprehensive text which spans basic drift-diffusion modeling, through energy balance and
hydrodynamic models, and finally particle based simulation. One unique feature of this book
is a specific focus on numerical examples, particularly the use of commercially available software
in the TCAD community. The concept for this book originated from a first year graduate
course on Computational Electronics, taught nowfor several years, in the Electrical Engineering
Department at Arizona State University. Numerous exercises and projects were derived from
this course and have been included. The prerequisite knowledge is a fundamental understanding
of basic semiconductor physics, the physical models for various device technologies such as pn
diodes, bipolar junction transistors, and field effect transistors.
KEYWORDS
semiconductor device simulation, semiconductor transport, computational science and engineering,
integrated circuit technology, technology computer aided design
Contents
1. Introduction to Computational Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Semiconductor Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
2.1 Semiconductor Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Simplified Band Structure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Carrier Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Effective Mass in Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Semiclassical Transport Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
2.5.1 Approximations Made for the Distribution Function . . . . . . . . . . . . . . . . 17
2.6 Boltzmann Transport Equation (BTE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Scattering Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8 Relaxation-Time Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
2.9 Solving the BTE in the Relaxation-Time Approximation. . . . . . . . . . . . . . . . . . . .25
3. TheDrift–Diffusion Equations and TheirNumerical Solution . . . . . . . . . . . . . . . . . .33
3.1 Drift–Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Physical Limitations on Numerical Drift–Diffusion Schemes . . . . . . . .35
3.1.2 Steady-State Solution of Bipolar Semiconductor Equations . . . . . . . . . 37
3.1.3 Normalization and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.4 Gummel’s Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
3.1.5 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.6 Generation and Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44
3.1.7 Time-Dependent Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
3.1.8 Scharfetter–Gummel Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48
3.1.9 Extension of the Validity of the Drift–Diffusion Model . . . . . . . . . . . . . 50
4. Hydrodynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Extensions of the Drift-Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Stratton’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Balance Equations Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 Displaced Maxwellian Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
vi CONTENTS
4.3.2 Momentum and Energy Relaxation Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.3 Simplifications that Lead to the Drift-Diffusion Model . . . . . . . . . . . . . 70
4.4 Numerical Solution Schemes for the Hydrodynamic Equations . . . . . . . . . . . . . . 71
4.4.1 Von Neumann Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.2 Lax Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
4.4.3 Other Varieties of Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.4 Second-Order Accuracy in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.5 Fluid Dynamics with Shocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
5. Use of Commercially Available Device Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 The Need for Semiconductor Device Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.1 Importance of Semiconductor Device Simulators . . . . . . . . . . . . . . . . . . . 84
5.1.2 Key Elements of Physical Device Simulation . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.3 Historical Development of the Physical
Device Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Introduction to the Silvaco ATLAS Simulation Tool . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.1 The ATLAS Syntax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88
5.2.2 Choice of the Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.3 Solutions Obtained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.4 Advanced Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
5.2.5 Run-Time Output, Log Files, Solution Files,
and the Extract Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Examples of Silvaco ATLAS Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98
5.3.1 pn-Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.2 MOSFET Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3.3 Simulation of BJT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.4 Simulation of SOI Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3.5 Gate Tunneling Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3.6 Simulation of a MESFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6. Particle-Based Device Simulation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139
6.1 Free-Flight Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.2 Final State After Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3 Ensemble Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.4 Multicarrier Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.4.1 Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.4.2 Carrier–Carrier Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150
CONTENTS vii
6.4.3 Band to Band Impact Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152
6.4.4 Full-Band Particle-Based Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153
6.5 Device Simulation Using Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.5.1 Monte Carlo Device Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.5.2 Direct Treatment of Interparticle Interaction . . . . . . . . . . . . . . . . . . . . . 166
Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199
:11bb
[hide]
[/hide]
看看，谢谢分享！学习一下！非常感谢！
谢谢分享。。。。
:31bb :31bb :31bb
感谢管理员无私奉献:27bb :27bb
好书，感谢楼主分享:29bb :27bb
：cacakiki12de ：cacakiki12de ：cacakiki3de ：cacakiki3de ：cacakiki7de
感謝樓主
感覺這一本是計算半導體相關的書籍
也是很火熱的領域
謝謝囉
非常感谢
:27bb :27bb
看看，谢谢分享！学习一下！非常感谢！
看了目录不错！！下来看看！:11bb :11bb
这么好的书，正是喜爱的，奖励，虽然管理员太有钱了，还是要奖励的
多谢搂主，好人一生平安！！！！
好东西，我爱微网！爱麦克斯韦，爱楼主！2
还比较新的书，谢谢楼主:30bb
，仅限学习交流使用，请勿用于商业用途，如果
好书，值得学习的好书！
好书啊好书!管理员同志是个好同志!:30bb
:27bb :27bb :11bb
应该是本好书
:11bb :11bb :11bb :11bb
thanks....:11bb :11bb :11bb :11bb
:11bb :27bb :29bb :30bb :31bb :11bb :27bb :27bb :27bb
好shu，谢谢分享！
thanks
真是个不错的资料!呵呵!谢谢分享!:11bb
:30bb :30bb :30bb
GOOOOOOOOOOOOOOOOOOOOD
顶顶顶顶顶顶顶顶顶顶顶顶的顶顶顶顶顶顶顶顶的的
:29bb :29bb :29bb :29bb
see see kan ~~~~~~~~~~~
看看，谢谢分享！学习一下！非常感谢！
谢谢楼主的分享，
:43bb：15de：13de：9de
good，have a look
up!:13bb:14bb
：17de什么东东
下载学习下
谢谢楼主 感谢分享
ok, very good !
ok, very good !
好书，值得学习的好书！不知下得了不？
看看。谢谢分享。
Thank you for your sharing
哈哈，多谢多谢啊
好书啊好书!管理员同志是个好同志!
看看，谢谢分享！学习一下！非常感谢！
内容需要回复才可以浏览
管理员同志是个好同志!
谢谢呆呆，很好的书，自己总是不辞辛苦的上传，谢谢。
多谢多谢啊
{:6_944:}{:6_923:}
是否有可参考的内容？
是新书？瞧一睢！
好东西，多谢！！！
谢谢分享好书啊
谢谢楼主, 很需要这本书.
感谢分享啊，哈哈哈哈哈
是什么时候发表的？
好书啊好书!管理员同志是个好同志!
感谢楼主分享
看看，谢谢分享！学习一下！非常感谢！
好书，值得学习的好书！
:15bb分享
thank you .,,.
好书，感谢楼主分享
谢谢分享{:7_1235:}
好书，感谢楼主分享
好!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
thank youthank youthank you
Computational Electromagnetics --Dragica Vasileska and Stephen M. Goodnick
路过，看看！
kan yi xia，好东西
好书啊好书!管理员同志是个好同志
really good book for us, push.{:7_1235:}
meisha,灌水看帖子
excellent work, thanks.{:7_1238:}
多谢分享
哈哈东西不错
看一下，很有用的书
thanks a lot
感谢楼主分享
非常感谢
非常感谢
谢谢楼主 学习了
Computational Electromagnetics，谢谢楼主，看看
下来看看！！谢谢楼主！！
看看1gaghah
thanks
Computational Electromagnetics --Dragica Vasileska and Stephen M. Goodnick
我回复乐乐乐乐了
好书，谢谢楼主分享
学习啊学习啊
啥也不说了，楼主就是给力！
多谢分享
支持楼主，想要这本书看看
好書呀
感謝樓主大方分享
where is it
学习一下吧！！！！
Computational Electromagnetics --Dragica Vasileska and Stephen M. Goodnick: Computational Electromagnetics.rar
