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Mathematical and Numerical Modelling of Heterostructure Semiconductor Devices: From Theory to Programmingby E. A. B. Cole
Synopses & Reviews
A comprehensive account is given of the processes involved in the mathematical and numerical modelling of semiconductor devices. This account will follow three main strands: a presentation of the physical theory behind the mathematical equations involved in the modelling, a presentation of the main numerical methods involved in the solution of the equations, and a discussion of the practical aspects involved in applying these numerical methods to physical device shapes. Although most consideration is given to the modelling of MESFET and HEMT devices, all of the material is either immediately relevant, or can be easily relevant, or can be easily adapted, to the modelling of other devices.
Part I presents the main background physical theory. This theory consists of chapters on basic quantum mechanics, thermodynamics and statistical mechanics, and the main equations of device modelling. These notes are based on notes of lectures I have presented at final year undergraduate level.
Part II presents the numerical methods involved in the solution of these modelling equations. This Part includes discussions of the basic Newton method, relaxation methods, the upwinding method, the phaseplane method (used for rapid inclusion of new terms in the modelling equations), mulitgrids, and genetic algorithms with simulated annealing. These chapters are self-contained, and they assume no previous knowledge on behalf of the reader on these topics. Practical aspects of applying these various methods to actual devices are discussed and, where appropriate, short sections of computer code are presented for this purpose. This code is written in the C programming language, and is written in a simple and transparent way in order that the reader should find it straightforward to re-write it in his or her own favourite programming.
This book covers the basic ideas of quantum mechanics and statistical mechanics, and it clearly explains numerical techniques such as genetic computing. It includes case studies of those techniques being tailored to specific problems of device modeling.
The commercial development of novel semiconductor devices requires that their properties be examined as thoroughly and rapidly as possible. These properties are investigated by obtaining numerical solutions of the highly nonlinear coupled set of equations which govern their behaviour. In particular, the existence of interfaces between different material layers in heterostructures means that quantum solutions must be found in the quantum wells which are formed at these interfaces. This book presents some of the mathematical and numerical techniques associated with the investigation. It begins with introductions to quantum and statistical mechanics. Later chapters then cover finite differences; multigrids; upwinding techniques; simulated annealing; mesh generation; and the reading of computer code in C++; these chapters are self-contained, and do not rely on the reader having met these topics before. The author explains how the methods can be adapted to the specific needs of device modelling, the advantages and disadvantages of each method, the pitfalls to avoid, and practical hints and tips for successful implementation. Sections of computer code are included to illustrate the methods used. Written for anyone who is interested in learning about, or refreshing their knowledge of, some of the basic mathematical and numerical methods involved in device modelling, this book is suitable for advanced undergraduate and graduate students, lecturers and researchers working in the fields of electrical engineering and semiconductor device physics, and for students of other mathematical and physical disciplines starting out in device modelling.
Table of Contents
Part I Overview and physical equations.- 1 Overview of device modeling.- 2 Quantum mechanics.- 3 Equilibrium thermodynamics and statistical mechanics.- 4 Density of states and applications - 1.- 5 Density of states and applications - 2.- 6 The transport equations and the device equations.- Part II Mathematical and numerical methods.- 7 Basic approximation and numerical methods.- 8 Fermi and associated integrals.- 9 The upwinding method.- 10 Solution of equations: the Newton and reduced method.- 11 Solution of equations: the phaseplane method.- 12 Solution of equations: the multigrid method.- 13 Approximate and numerical solutions of the Schr¨odinger equation.- 14 Genetic algorithms and simulated annealing.- 15 Grid generation.- A The theory of contractive mapping.
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