1.1 Semiclassical Carrier Transport
1.1.1 The Drift-Diffusion Model
1.1.2 The Hydrodynamic Model
1.1.3 The Energy Transport Model
1.1.4 The Six-Moments Model
1.1.5 The Monte Carlo Method
1.2 Requirements of Modern TCAD
1.2.1 Accuracy
1.2.2 Charge Conservation
1.2.3 Self-Consistency
1.2.4 Resolution of Complicated Domains
1.2.5 Computational Feasibility
1.2.6 Extendibility
1.3 Outline
2 The SHE Equations
2.1 Historical Overview
2.2 Derivation of the SHE Equations
2.3
-Transform and MEDS
2.4 Self-Consistency with Poisson’s Equation
2.4.1 Gummel Iteration
2.4.2 Newton’s Method
2.5 Compatibility with Modern TCAD
3 Physical Modeling
3.1 Spherically Symmetric Energy Band Models
3.2 Incorporation of Full-Band Effects
3.3 Linear Scattering Operators
3.3.1 Acoustic Phonon Scattering
3.3.2 Optical Phonon Scattering
3.3.3 Ionized Impurity Scattering
3.4 Nonlinear Scattering Operators
3.4.1 Pauli Exclusion Principle
3.4.2 Carrier-Carrier Scattering
4 Structural Properties
4.1 Sparse Coupling for Spherical Energy Bands
4.2 Coupling for Nonspherical Energy Bands
4.3 Stabilization Schemes
4.4 Boundary Conditions
4.5 Solution of the Linear System
4.6 Results
5 SHE on Unstructured Grids
5.1 The Box Integration Scheme
5.2 Construction of Boxes
5.3 Box Integration for SHE
5.3.1 Discretization of the Even-Order Equations
5.3.2 Discretization of the Odd-Order Equations
5.4 Results
6 Adaptive Variable-Order SHE
6.1 Variable-Order SHE
6.2 Adaptive Control of the SHE Order
6.2.1 Rate of Decay of Expansion Coefficients
6.2.2 Target Quantity Driven Adaptive Control
6.2.3 Residual-Based Adaptive Control
6.3 Results
7 Parallelization
7.1 Energy Couplings Revisited
7.2 Symmetrization of the System Matrix
7.3 A Parallel Preconditioning Scheme
7.4 Results
8 Numerical Results
8.1 MOSFET
8.2 FinFET
9 Outlook and Conclusion
9.1 Possible Further Improvements of the SHE Method
9.1.1 Bipolar SHE
9.1.2 Energy Grid with Hanging Nodes
9.1.3 Fast Self-Consistency with Poisson’s Equation
9.1.4 More Flexible Discretization on Unstructured Grids
9.1.5 Preconditioner for the Compressed Matrix Scheme
9.2 Conclusion
A Mathematical Tools
A.1 The Kronecker Product
A.2 Wigner 3jm Symbols
Bibliography