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2001 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Quasi-hydrodynamic Semiconductor Equations

verfasst von: Ansgar Jüngel

Verlag: Birkhäuser Basel

Buchreihe : Progress in Nonlinear Differential Equations and Their Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

In this book a hierarchy of macroscopic models for semiconductor devices is presented. Three classes of models are studied in detail: isentropic drift-diffusion equations, energy-transport models, and quantum hydrodynamic equations. The derivation of each of the models is shown, including physical discussions. Furthermore, the corresponding mathematical problems are analyzed, using modern techniques for nonlinear partial differential equations. The equations are discretized employing mixed finite-element methods. Also, numerical simulations for modern semiconductor devices are performed, showing the particular features of the models.
Modern analytical techniques have been used and further developed, such as positive solution methods, local energy methods for free-boundary problems and entropy methods.
The book is aimed at applied mathematicians and physicists interested in mathematics, as well as graduate and postdoc students and researchers in these fields.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
In this chapter a hierarchy of kinetic and quasi-hydrodynamic semiconductor equations is introduced and described. The connections between the various (semi-) classical and quantum models are explained (Section 1.1). Three quasi-hydrodynamic equations are considered in more detail, and an outline of the contents of this work is presented (Section 1.2)
Ansgar Jüngel
Chapter 2. Basic Semiconductor Physics
Abstract
This chapter presents a short summary of the physics and properties of semiconductors. Only those subjects relevant to the presented mathematical and numerical analysis in the following chapters are included here. We explain basic notions of the structure of homogeneous semiconductors (Section 2.1) and describe the inhomogeneous semiconductor devices pn-junction diode, bipolar transistor, MOSFET, ballistic diode, and resonant tunneling diode (Section 2.2). We refer to the standard textbooks, e.g. [342, 284, 33, 61, 144] for a detailled consideration of solid-state physics and to [36, 69, 129, 139] for new developments in semiconductor physics.
Ansgar Jüngel
Chapter 3. The Isentropic Drift-diffusion Model
Abstract
In this chapter the isentropic drift-diffusion equations are studied. First, the model is derived from current relations involving Fermi-Dirac statistics (Section 3.1). The existence and uniqueness of solutions to the time-dependent problem are proved in Sections 3.2 and 3.3. Since the model contains parabolic equations of degenerate type, solutions may exist for which the carrier densities vanish locally. These solutions are calledvacuum solutions. In Section 3.4 we study the space and time localization of the corresponding vacuum sets. The isentropic drift-diffusion equations are discretized in one and two space dimensions by using an exponentially fitted mixed finite element method. Numerical examples modeling junction diodes and transistors are presented (Section 3.5). Section 3.6 is devoted to the study of the static current-voltage characteristics of diodes.
Ansgar Jüngel
Chapter 4. The Energy-transport Model
Abstract
This chapter is concerned with the analysis and numerical approximation of the energy-transport model. Some ideas of the derivation of this model are given in Section 4.1. The energy-transport model is a special case of systems arising in nonequilibrium thermodynamics. In Section 4.2 we define the entropy function of this general system and transform the equations to a symmetrized problem via the dual entropy variables. The existence of transient solutions is shown (Section 4.3) and the long-time behavior of the solutions to the thermal equilibrium state is studied (Section 4.4). Section 4.5 is devoted to the proof of some regularity properties and of a uniqueness result. The existence and uniqueness of steady-state solutions is investigated in Sections 4.6 and 4.7. Finally, the equations are discretized by using mixed finite elements and the numerical solution for a ballistic diode is presented (Section 4.8).
Ansgar Jüngel
Chapter 5. The Quantum Hydrodynamic Model
Abstract
In this chapter we are concerned with the quantum hydrodynamic model.
Ansgar Jüngel
Backmatter
Metadaten
Titel
Quasi-hydrodynamic Semiconductor Equations
verfasst von
Ansgar Jüngel
Copyright-Jahr
2001
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-8334-4
Print ISBN
978-3-0348-9521-7
DOI
https://doi.org/10.1007/978-3-0348-8334-4