Semiconductor Equations

ISBN-13: 9783709174524 / Angielski / Miękka / 2011 / 248 str.

Peter A. Markowich; Christian A. Ringhofer; Christian Schmeiser

Opis

In recent years the mathematical modeling of charge transport in semi conductors has become a thriving area in applied mathematics. The drift diffusion equations, which constitute the most popular model for the simula tion of the electrical behavior of semiconductor devices, are by now mathe matically quite well understood. As a consequence numerical methods have been developed, which allow for reasonably efficient computer simulations in many cases of practical relevance. Nowadays, research on the drift diffu sion model is of a highly specialized nature. It concentrates on the explora tion of possibly more efficient discretization methods (e.g. mixed finite elements, streamline diffusion), on the improvement of the performance of nonlinear iteration and linear equation solvers, and on three dimensional applications. The ongoing miniaturization of semiconductor devices has prompted a shift of the focus of the modeling research lately, since the drift diffusion model does not account well for charge transport in ultra integrated devices. Extensions of the drift diffusion model (so called hydrodynamic models) are under investigation for the modeling of hot electron effects in submicron MOS-transistors, and supercomputer technology has made it possible to employ kinetic models (semiclassical Boltzmann-Poisson and Wigner Poisson equations) for the simulation of certain highly integrated devices."

Spis treści

1 Kinetic Transport Models for Semiconductors.- 1.1 Introduction.- 1.2 The (Semi-)Classical Liouville Equation.- Particle Trajectories.- A Potential Barrier.- The Transport Equation.- Particle Ensembles.- The Initial Value Problem.- The Classical Hamiltonian.- The Semi-Classical Liouville Equation.- Magnetic Fields.- 1.3 The Boltzmann Equation.- The Vlasov Equation.- The Poisson Equation.- The Whole Space Vlasov Problem.- Bounded Position Domains.- The Semi-Classical Vlasov Equation.- Magetic Fields—The Maxwell Equations.- Collisions—The Boltzmann Equation.- The Semi-Classical Boltzmann Equation.- Conservation and Relaxation.- Low Density Approximation.- The Relaxation Time Approximation.- Polar Optical Scattering.- Particle-Particle Interaction.- 1.4 The Quantum Liouville Equation.- The Schrödinger Equation.- Tunneling.- Particle Ensembles and Density Matrices.- Wigner Functions.- The Quantum Transport Equation.- Pure and Mixed States.- The Classical Limit.- Nonnegativity of Wigner Functions.- An Energy-Band Version of the Quantum Liouville Equation.- 1.5 The Quantum Boltzmann Equation.- Subensemble Density Matrices.- The Quantum Vlasov Equation.- The Poisson Equation.- The Quantum Vlasov Equation on a Bounded Position Domain.- The Energy-Band Version of the Quantum Vlasov Equation.- Collisions.- 1.6 Applications and Extensions.- Multi-Valley Models.- Bipolar Model.- Tunneling Devices.- Problems.- References.- 2 From Kinetic to Fluid Dynamical Models.- 2.1 Introduction.- 2.2 Small Mean Free Path—The Hilbert Expansion.- 2.3 Moment Methods—The Hydrodynamic Model.- Derivation of the Drift Diffusion Model.- The Hydrodynamic Model.- 2.4 Heavy Doping Effects—Fermi-Dirac Distributions.- 2.5 High Field Effects—Mobility Models.- 2.6 Recombination-Generation Models.- Problems.- References.- 3 The Drift Diffusion Equations.- 3.1 Introduction.- 3.2 The Stationary Drift Diffusion Equations.- 3.3 Existence and Uniqueness for the Stationary Drift Diffusion Equations.- 3.4 Forward Biased P-N Junctions.- The Equilibrium Case.- The Non-Equilibrium Case.- Asymptotic Validity in the One-Dimensional Case.- Velocity Saturation Effects—Field Dependent Mobilities.- 3.5 Reverse Biased P-N Junctions.- Moderately Reverse Biased P-N Junctions.- P-N Junctions Under Extreme Reverse Bias Conditions.- The One-Dimensional Problem.- The Two-Dimensional Case.- 3.6 Stability and Conditioning for the Stationary Problem.- 3.7 The Transient Problem.- 3.8 The Linearization of the Transient Problem.- 3.9 Existence for the Nonlinear Problem.- Asymptotic Expansions for the Transient Drift Diffusion Equations.- 3.10 Asymptotic Expansions on the Diffusion Time Scale.- 3.11 Fast Time Scale Expansions.- The Case of a Bounded Initial Potential.- Fast Time Scale Solutions for General Initial Data.- Problems.- References.- 4 Devices.- 4.1 Introduction.- Static Voltage-Current Characteristics.- 4.2 P-N Diode.- The Depletion Region in Thermal Equilibrium.- Strongly Asymmetric Junctions.- The Voltage-Current Characteristic Close to Thermal Equilibrium.- High Injection—A Model Problem.- Large Reverse Bias.- Avalanche Breakdown.- Punch Through.- 4.3 Bipolar Transistor.- Current Gain Close to Thermal Equilibrium.- 4.4 PIN-Diode.- Thermal Equilibrium.- Behaviour Close to Thermal Equilibrium.- 4.5 Thyristor.- Characteristic Close to Thermal Equilibrium.- Forward Conduction.- Break Over Voltage.- 4.6 MIS Diode.- Accumulation.- Depletion—Weak Inversion.- Strong Inversion.- 4.7 MOSFET.- Derivation of a Simplified Model.- A Quasi One-Dimensional Model.- Computation of the One-Dimensional Electron Density.- Computation of the Current.- 4.8 Gunn Diode.- Bulk Negative Differential Conductivity.- Traveling Waves.- The Gunn Effect.- Problems.- References.- Physical Constants.- Properties of Si at Room Temperature.