This IMA Volume in Mathematics and its Applications SEMICONDUCTORS, PART II is based on the proceedings of the IMA summer program "Semiconductors." Our goal was to foster interaction in this interdisciplinary field which involves electrical engineers, computer scientists, semiconductor physicists and mathematicians, from both university and industry. In particular, the program was meant to encourage the participation of numerical and mathematical analysts with backgrounds in ordinary and partial differential equations, to help get them involved in the mathematical as pects of semiconductor models and circuits. We are grateful to W.M. Coughran, Jr., Julian Cole, Peter Lloyd, and Jacob White for helping Farouk Odeh organize this activity and trust that the proceedings will provide a fitting memorial to Farouk. We also take this opportunity to thank those agencies whose financial support made the program possible: the Air Force Office of Scientific Research, the Army Research Office, the National Science Foundation, and the Office of Naval Research. A vner Friedman Willard Miller, J r. Preface to Part II Semiconductor and integrated-circuit modeling are an important part of the high technology "chip" industry, whose high-performance, low-cost microprocessors and high-density memory designs form the basis for supercomputers, engineering work stations, laptop computers, and other modern information appliances. There are a variety of differential equation problems that must be solved to facilitate such mod eling.

Circuit simulation has been a topic of great interest to the integrated circuit design community for many years. It is a difficult, and interesting, problem be cause circuit simulators are very heavily used, consuming thousands of computer hours every year, and therefore the algorithms must be very efficient. In addi tion, circuit simulators are heavily relied upon, with millions of dollars being gambled on their accuracy, and therefore the algorithms must be very robust. At the University of California, Berkeley, a great deal of research has been devoted to the study of both the numerical properties and the efficient imple mentation of circuit simulation algorithms. Research efforts have led to several programs, starting with CANCER in the 1960's and the enormously successful SPICE program in the early 1970's, to MOTIS-C, SPLICE, and RELAX in the late 1970's, and finally to SPLICE2 and RELAX2 in the 1980's. Our primary goal in writing this book was to present some of the results of our current research on the application of relaxation algorithms to circuit simu lation. As we began, we realized that a large body of mathematical and exper imental results had been amassed over the past twenty years by graduate students, professors, and industry researchers working on circuit simulation. It became a secondary goal to try to find an organization of this mass of material that was mathematically rigorous, had practical relevance, and still retained the natural intuitive simplicity of the circuit simulation subject."

Semiconductor and integrated-circuit modeling are an important part of the high-technology "chip" industry, whose high-performance, low-cost microprocessors and high-density memory designs form the basis for supercomputers, engineering workstations, laptop computers, and other modern information appliances. There are a variety of differential equation problems that must be solved to facilitate such modeling. This two-volume set covers three topic areas: process modeling and circuit simulation in Volume I and device modeling in Volume II. Process modeling provides the geometry and impurity doping characteristics that are prerequisites for device modeling; device modeling, in turn, provides static current and transient charge characteristics needed to specify the so-called compact models employed by circuit simulators. The goal of these books is to bring together scientists and mathematicians to discuss open problems, algorithms to solve such, and to form bridges between the diverse disciplines involved.

The motivation for starting the work described in this book was the interest that Hewlett-Packard's microwave circuit designers had in simulation techniques that could tackle the problem of finding steady state solutions for nonlinear circuits, particularly circuits containing distributed elements such as transmission lines. Examining the problem of computing steady-state solutions in this context has led to a collection of novel numerical algorithms which we have gathered, along with some background material, into this book. Although we wished to appeal to as broad an audience as possible, to treat the subject in depth required maintaining a narrow focus. Our compromise was to assume that the reader is familiar with basic numerical methods, such as might be found in [dahlquist74] or [vlach83], but not assume any specialized knowledge of methods for steady-state problems. Although we focus on algorithms for computing steady-state solutions of analog and microwave circuits, the methods herein are general in nature and may find use in other disciplines. A number of new algorithms are presented, the contributions primarily centering around new approaches to harmonic balance and mixed frequency-time methods. These methods are described, along with appropriate background material, in what we hope is a reasonably satisfying blend of theory, practice, and results. The theory is given so that the algorithms can be fully understood and their correctness established.

The motivation for starting the work described in this book was the interest that Hewlett-Packard's microwave circuit designers had in simulation techniques that could tackle the problem of finding steady state solutions for nonlinear circuits, particularly circuits containing distributed elements such as transmission lines. Examining the problem of computing steady-state solutions in this context has led to a collection of novel numerical algorithms which we have gathered, along with some background material, into this book. Although we wished to appeal to as broad an audience as possible, to treat the subject in depth required maintaining a narrow focus. Our compromise was to assume that the reader is familiar with basic numerical methods, such as might be found in [dahlquist74] or [vlach83], but not assume any specialized knowledge of methods for steady-state problems. Although we focus on algorithms for computing steady-state solutions of analog and microwave circuits, the methods herein are general in nature and may find use in other disciplines. A number of new algorithms are presented, the contributions primarily centering around new approaches to harmonic balance and mixed frequency-time methods. These methods are described, along with appropriate background material, in what we hope is a reasonably satisfying blend of theory, practice, and results. The theory is given so that the algorithms can be fully understood and their correctness established.

Circuit simulation has been a topic of great interest to the integrated circuit design community for many years. It is a difficult, and interesting, problem be cause circuit simulators are very heavily used, consuming thousands of computer hours every year, and therefore the algorithms must be very efficient. In addi tion, circuit simulators are heavily relied upon, with millions of dollars being gambled on their accuracy, and therefore the algorithms must be very robust. At the University of California, Berkeley, a great deal of research has been devoted to the study of both the numerical properties and the efficient imple mentation of circuit simulation algorithms. Research efforts have led to several programs, starting with CANCER in the 1960's and the enormously successful SPICE program in the early 1970's, to MOTIS-C, SPLICE, and RELAX in the late 1970's, and finally to SPLICE2 and RELAX2 in the 1980's. Our primary goal in writing this book was to present some of the results of our current research on the application of relaxation algorithms to circuit simu lation. As we began, we realized that a large body of mathematical and exper imental results had been amassed over the past twenty years by graduate students, professors, and industry researchers working on circuit simulation. It became a secondary goal to try to find an organization of this mass of material that was mathematically rigorous, had practical relevance, and still retained the natural intuitive simplicity of the circuit simulation subject."