Quasi-hydrodynamic Semiconductor Equations

This book presents a hierarchy of macroscopic models for semiconductor devices, studying three classes of models in detail: isentropic drift-diffusion equations, energy-transport models, and quantum hydrodynamic equations. The derivation of each, including physical discussions, is shown. Numerical simulations for modern semiconductor devices are performed, showing the particular features of each. The author develops modern analytical techniques, such as positive solution methods, local energy methods for free-boundary problems and entropy methods.

1 Introduction.- 1.1 A hierarchy of semiconductor models.- 1.2 Quasi-hydrodynamic semiconductor models.- 2 Basic Semiconductor Physics.- 2.1 Homogeneous semiconductors.- 2.2 Inhomogeneous semiconductors.- 3 The Isentropic Drift-diffusion Model.- 3.1 Derivation of the model.- 3.2 Existence of transient solutions.- 3.3 Uniqueness of transient solutions.- 3.4 Localization of vacuum solutions.- 3.5 Numerical approximation.- 3.6 Current-voltage characteristics.- 4 The Energy-transport Model.- 4.1 Derivation of the model.- 4.2 Symmetrization and entropy function.- 4.3 Existence of transient solutions.- 4.4 Long-time behavior of the transient solution.- 4.5 Regularity and uniqueness.- 4.6 Existence of steady-state solutions.- 4.7 Uniqueness of steady-state solutions.- 4.8 Numerical approximation.- 5 The Quantum Hydrodynamic Model.- 5.1 Derivation of the model.- 5.2 Existence and positivity.- 5.3 Uniqueness of steady-state solutions.- 5.4 A non-existence result.- 5.5 The classical limit.- 5.6 Current-voltage characteristics.- 5.7 A positivity-preserving numerical scheme.- References.