Semiconductor Equations

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 Schrodinger 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.