Quasi-hydrodynamic Semiconductor Equations

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.- 3.1.1 Semiconductor equations based on Fermi-Dirac statistics.- 3.1.2 The isentropic model-scaling.- 3.1.3 The convergence result.- 3.2 Existence of transient solutions.- 3.2.1 Assumptions and existence result.- 3.2.2 Proof of the existence result.- 3.3 Uniqueness of transient solutions.- 3.4 Localization of vacuum solutions.- 3.4.1 Main results.- 3.4.2 Proofs of the main results.- 3.4.3 Numerical examples.- 3.5 Numerical approximation.- 3.5.1 The mixed finite element discretization in one space dimension.- 3.5.2 Numerical examples in one space dimension.- 3.5.3 The mixed finite element discretization in two space dimensions.- 3.5.4 Numerical examples in two space dimensions.- 3.6 Current-voltage characteristics.- 3.6.1 Numerical current-voltage characteristics.- 3.6.2 High-injection current-voltage characteristics.- 4 The Energy-transport Model.- 4.1 Derivation of the model.- 4.1.1 General non-parabolic band diagrams.- 4.1.2 A drift-diffusion formulation for the current densities.- 4.1.3 A non-parabolic band approximation.- 4.1.4 Parabolic band approximation.- 4.2 Symmetrization and entropy function.- 4.3 Existence of transient solutions.- 4.3.1 Assumptions and main results.- 4.3.2 Semidiscretization.- 4.3.3 Proof of the existence result.- 4.4 Long-time behavior of the transient solution.- 4.5 Regularity and uniqueness.- 4.5.1 Regularity of transient solutions.- 4.5.2 Uniqueness of transient solutions.- 4.6 Existence of steady-state solutions.- 4.7 Uniqueness of steady-state solutions.- 4.8 Numerical approximation.- 4.8.1 The mixed finite element discretization in one space dimension.- 4.8.2 Numerical results.- 5 The Quantum Hydrodynamic Model.- 5.1 Derivation of the model.- 5.2 Existence and positivity.- 5.2.1 Existence of steady-state solutions.- 5.2.2 Positivity and non-positivity properties.- 5.3 Uniqueness of steady-state solutions.- 5.4 A non-existence result.- 5.5 The classical limit.- 5.5.1 The classical limit of the thermal equilibrium state.- 5.5.2 The classical limit in the `subsonic' steady state.- 5.5.3 Numerical examples.- 5.6 Current-voltage characteristics.- 5.6.1 Scaling of the equations.- 5.6.2 Analytical and numerical current-voltage characteristics.- 5.7 A positivity-preserving numerical scheme.- 5.7.1 Semidiscretization in time.- 5.7.2 Stability bounds and convergence results.- 5.7.3 Numerical examples.- References.