Charge Transport in Low Dimensional Semiconductor Structures: The Maximum Entropy Approach

Preface
Contents
About the Authors
1 Band Structure and Boltzmann Equation
	1.1 Crystal Structure
	1.2 The Energy Band Structure
	1.3 The Si Band Structure and the Semi-classical Picture
	1.4 The Boltzmann Equation
	1.5 H-Theorem and the Null Space of the Collision Operator
	1.6 Quantum Confinement and Quasi 2DEG
	1.7 Derivation of the Transport Equation Along the Longitudinal Direction
2 Maximum Entropy Principle
	2.1 The Entropy
		2.1.1 Properties of the Shannon–Jaynes Entropy
		2.1.2 Shannon–Jaynes Entropy in the Continuous Case
	2.2 Maximum Entropy Inference of a Distribution: The Discrete Case
	2.3 Examples of Distribution Functions Deduced with MEP
		2.3.1 Maxwell–Boltzmann Distribution
		2.3.2 Fermi–Dirac and Bose–Einstein Distributions
	2.4 Maximum Entropy Inference of a Distribution: The Continuous Case
3 Application of MEP to Charge Transport in Semiconductors
	3.1 The Electron Transport Equation and the Maximum Entropy Principle
	3.2 Further Considerations
	3.3 Solvability of the MEP Problems in Semiconductors
		3.3.1 Statement of the Main Result
		3.3.2 A General Result by Csiszar
		3.3.3 The Weight Functions
		3.3.4 The Moment Cone
		3.3.5 The Entropy Functional
		3.3.6 The Lagrange Multipliers
		3.3.7 Proof of the Main Theorem
4 Application of MEP to Silicon
	4.1 Moment Equations and Closure Problem
	4.2 Lagrange Multipliers
	4.3 Constitutive Equations for the Fluxes
	4.4 Closure Relations for the Production Terms
		4.4.1 Acoustic Phonon Scattering
		4.4.2 Non Polar Optical Phonon Scattering
		4.4.3 Scattering with Impurities
	4.5 Parabolic Band Approximation
	4.6 Application to Bulk Silicon
	4.7 The Energy Transport Limit Model
	4.8 The Drift-Diffusion Limiting Model
		4.8.1 Parabolic Band Case
		4.8.2 Kane\'s Dispersion Relation
	4.9 Formulation of the Model in the Framework of Linear Irreversible Thermodynamics
	4.10 A Numerical Approach Based on Finite Elements
		4.10.1 Simulation of a n+-n-n+ Silicon Diode
		4.10.2 Simulation of a 2D Silicon MESFET
		4.10.3 Simulation of a 2D Silicon MOSFET
5 Some Formal Properties of the Hydrodynamical Model
	5.1 Hyperbolicity of the MEP Hydrodynamical Model
	5.2 Nonlinear Asymptotic Stability of the Equilibrium State
		5.2.1 Basic Equations and Formulation of the Problem
		5.2.2 Formulation of the Auxiliary Problems
		5.2.3 Asymptotic Stability of the Equilibrium State
		5.2.4 Explicit Expressions the Production Terms
		5.2.5 Estimates for J(0), J(2)
6 Quantum Corrections to the Semiclassical Models
	6.1 Wigner Equation
	6.2 Equilibrium Wigner Function
	6.3 The Collision Operator
	6.4 Quantum Corrections in the High Field Approximation
	6.5 The Quantum Moment Equations
	6.6 Entropy Balance Equation
	6.7 Energy-Transport and Drift-Diffusion Limiting Models
7 Mathematical Models for the Double-Gate MOSFET
	7.1 Semiclassical Model for DG-MOSFET
	7.2 The Moment System and Its Closure by the MEP
	7.3 Energy-Transport Model
	7.4 Boundary Conditions and Initial Data
		7.4.1 Boundary Conditions and Initial Data for the SP-Block
		7.4.2 Boundary Conditions and Initial Data for the ET Block
8 Numerical Method and Simulations
	8.1 Discretization of the Schrödinger–Poisson Equations
	8.2 Discretization of the Energy-Transport Equations
	8.3 Numerical Simulations
9 Application of MEP to Charge Transport in Graphene
	9.1 Kinetic Description
	9.2 Moment Equations
	9.3 Closure Relations
	9.4 Production Terms of Acoustic Phonon Scattering
	9.5 Production Terms of Optical Phonon Scattering
		9.5.1 Intraband Optical Phonon Scattering Production Term
		9.5.2 Interband Optical Phonon Scattering Production Term
	9.6 Production Terms of K-Phonon Scattering
		9.6.1 Intraband K-Phonon Scattering Production Term
		9.6.2 Interband K-Phonon Scattering Production Term
	9.7 Numerical Results with Constant Lattice Temperature
	9.8 Inclusion of the Crystal Heating
		9.8.1 Carrier Moment Equations
		9.8.2 The Phonon Moment System
	9.9 The Closure Problem
	9.10 Inversion of the Constraint Relations and Definition of the Temperature
		9.10.1 Phonons
		9.10.2 Temperature
	9.11 Closure Relations
		9.11.1 Electrons and Holes
		9.11.2 Phonons
	9.12 Hyperbolicity of the Model
	9.13 Numerical Results with Variable Lattice Temperature
	9.14 Conclusions
A 2DEG: Closure Relations for the Kane Case
B Closure Relations for the Parabolic Case
C Useful Computational Relations
D Crystal Vibrations and Phonons
E Simulation Codes
References
Index