Contents

1 Introduction
 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 H  -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