Contents

• 1. Introduction
  • 1.1 TCAD Tools
    • 1.1.1 Numerical Computing
    • 1.1.2 Typical Computational Analysis Cycle
  • 1.2 Mesh Adaptation
    • 1.2.1 Related Mesh Generation Developments
    • 1.2.2 Related Mathematical Modeling Developments
    • 1.2.3 Mesh Adaptation Remark
  • 1.3 Outline of this Thesis
  
  
• 2. Mesh Refinement in General
  • 2.1 General Definitions
    • 2.1.1 $n$ -Simplex
    • 2.1.2 Standard $n$ -Simplex
    • 2.1.3 $m$ -Face
    • 2.1.4 Convex Hull
    • 2.1.5 Covering Up
    • 2.1.6 Triangulation
    • 2.1.7 Mesh
    • 2.1.8 Conformal Mesh
    • 2.1.9 Mesh Element
  • 2.2 Tetrahedral Partitioning Methods
    • 2.2.1 Vertex Insertion Methods
    • 2.2.2 Tetrahedral Bisection
    • 2.2.3 Recursive Tetrahedral Bisection
  • 2.3 Anisotropy
    • 2.3.1 Anisotropic Metric
    • 2.3.2 Anisotropic Tetrahedral Bisection
  
  
• 3. Data Driven Refinement Methods
  • 3.1 Some Words on Data
    • 3.1.1 Tensors
    • 3.1.2 Data $m$ -Face Relation
    • 3.1.3 Interpolation Functions on the Standard $3$ -Simplex
  • 3.2 Layer Refinement Method
    • 3.2.1 Laplace Equation
    • 3.2.2 Plate Capacitor
    • 3.2.3 Metric Function
    • 3.2.4 Propaedeutic Example
  • 3.3 Gradient Refinement Method
    • 3.3.1 Gradient Field
    • 3.3.2 Propaedeutic Example
  • 3.4 Hessian Refinement Method
    • 3.4.1 Least Squares Fitting of the Model Function
    • 3.4.2 Propaedeutic Example
  • 3.5 Two Examples
    • 3.5.1 Non-Planar Surface
    • 3.5.2 EEPROM Cell
  
  
• 4. Mesh Refinement for Diffusion
  • 4.1 Error Estimation
  • 4.2 Propaedeutic Example
  • 4.3 A Further Diffusion Example
  
  
• 5. Dynamic Mesh Adaptation for Simulation of Electromigration
  • 5.1 Electromigration in a Nutshell
  • 5.2 Interface Mesh Modeling
    • 5.2.1 Explicit Interface
    • 5.2.2 Implicit Interface
    • 5.2.3 Diffuse Interface Model
    • 5.2.4 Refined Diffuse Interface
    • 5.2.5 Hierarchical Mesh Refinement-Coarsement Scheme
  • 5.3 Example
  
  
• 6. Mesh Refinement for Full Band Monte Carlo Simulation
  • 6.1 Geometric Properties of the First Brillouin Zone
  • 6.2 Geometric Properties of the Irreducible Wedge
  • 6.3 Silicon Band Structure Models
    • 6.3.1 Analytical Approximations
    • 6.3.2 Full Band Approach
  • 6.4 k-Space Mesh Generation
    • 6.4.1 Cubic Grid Based Meshes
    • 6.4.2 Unstructured Tetrahedral Meshes
    • 6.4.3 Resulting Unstructured k-Space Tessellation
  • 6.5 Examples
    • 6.5.1 Density of States Example
    • 6.5.2 Average Kinetic Energy Example
    • 6.5.3 Electron Velocity Example
  
  
• 7. Summary and Outlook
  • 7.1 Summary
  • 7.2 Outlook
  
  
• A. Principal Component Analysis
  • A.1 Statistical Background
    • A.1.1 Mean
    • A.1.2 Standard Deviation
    • A.1.3 Covariance and Covariance Matrix
    • A.1.4 Eigendecomposition
  • A.2 PCA of a Tetrahedron
  • A.3 Glyph Visualization
  
  
• B. Mathematical Goodies
  • B.1 Euler's Rotation Theorem
  • B.2 Least Squares Approximation
  • B.3 Volume Integral
    • B.3.1 Gaussian Quadrature
    • B.3.2 Integral Transformation
    • B.3.3 Legendre Polynomials
    • B.3.4 Numerical Quadrature Library
  
  
• Bibliography
• Own Publications
• List of Figures
• List of Tables
• Curriculum Vitae