D I S S E R T A T I O N

Numerical Challenges on the

Road to NanoTCAD

ausgeführt zum Zwecke der Erlangung des
akademischen Grades
eines Doktors der technischen Wissenschaften
eingereicht an der Technischen Universität Wien
Fakultät für Elektrotechnik und Informationstechnik
von

Robert Kosik

Wallensteinstraße 55/28
Wien XX

Matrikelnummer 8825185
geboren am 3. Oktober 1969 in Eisenstadt

Wien, im August 2004

• Kurzfassung
  • Momenten-Methoden
  • Quantenmechanischer Transport
• Abstract
  • Moment Methods
  • Quantum Transport
• Acknowledgment
  
• Contents
• I. Challenges on the Road to NanoTCAD
  • 1. From Micro- to Nanoelectronics
    • 1.1 The Road to NanoTCAD
    • 1.2 Extensions of Drift Diffusion Models
    • 1.3 Entering the Quantum Regime
    • 1.4 Outline of the Work
      • 1.4.1 Higher Order Moment Methods
      • 1.4.2 Quantum TCAD
  
  
• II. Nanoscale CMOS Simulation Capability
  • Introduction to Part II
  • 2. A Six Moments Model for the Boltzmann Equation
    • 2.1 The Boltzmann Poisson System
      • 2.1.1 The Boltzmann Equation for Parabolic Energy Bands
      • 2.1.2 Poisson Equation
      • 2.1.3 Dimensional Reduction
      • 2.1.4 Distribution Function Models
    • 2.2 The Method of Moments
      • 2.2.1 Macroscopic Observables
      • 2.2.2 Hierarchy of Moment Equations
      • 2.2.3 Closure Problem
      • 2.2.4 Boundary Conditions
    • 2.3 Closure Relations for the Scattering Operator
      • 2.3.1 Even Moments: Relaxation Times
      • 2.3.2 Odd Moments: Mobilities
      • 2.3.3 Analytical Mobility and Relaxation Times
      • 2.3.4 Consistency with Bulk Monte Carlo Results
      • 2.3.5 Hierarchy of Equations
  • 3. Highest Order Moment Closure
    • 3.1 Cumulant Closure
    • 3.2 Maximum Entropy Closure
      • 3.2.1 Distribution Families
      • 3.2.2 Critique and Modifications
      • 3.2.3 Diffusion Closure
    • 3.3 Higher Order Statistics
    • 3.4 Gaussian Invariant Closure
  • 4. Tuning of the Nonlinear Solver
    • 4.1 Discretization
      • 4.1.1 Variants of the Scharfetter Gummel Discretization
      • 4.1.2 Double Grid Discretization
      • 4.1.3 Discussion
    • 4.2 Consistent Highest Order Moment Closure
    • 4.3 Choice of Residual Function
    • 4.4 Linesearch
    • 4.5 Initial Values and Stepping Methods
    • 4.6 Variations of the Newton Method
    • 4.7 Comparison with Monte Carlo
  
  
• III. Quantum Simulation of Emerging Research Devices
  • Introduction to Part III
  • 5. A Test Case for NanoTCAD: Resonant Tunneling Structures
    • 5.1 Physics of Resonant Tunneling
    • 5.2 Resonant Tunneling Diode Structure
    • 5.3 Coherent Tunneling
    • 5.4 Influence of Scattering Events
  • 6. Mathematical Formulations of Quantum Mechanics
    • 6.1 Quantum Mechanics in Configuration Space
      • 6.1.1 Wave Mechanics
      • 6.1.2 Hydrodynamical Formulation
      • 6.1.3 The Riccati and Prüfer Equation
    • 6.2 Quantum Mechanics in Phase Space
      • 6.2.1 Phase Space Distribution Functions
      • 6.2.2 Definition of Wigner Function
      • 6.2.3 Operator-Theoretic Structure
      • 6.2.4 Probabilistic Structure
      • 6.2.5 A Caveat on Marginal Distributions
      • 6.2.6 A Feasible Phase Space Distribution
      • 6.2.7 Quantum Propagators and Trajectories
  • 7. Schrödinger and von Neumann Equation
    • 7.1 Single Particle Transport
      • 7.1.1 Open Quantum Systems
      • 7.1.2 Absorbing Boundary Conditions
      • 7.1.3 Comparison: Schrödinger, Riccati & Prüfer Equation
    • 7.2 The Open Von Neumann-Poisson Equation
      • 7.2.1 Separation of Neumann Equation
      • 7.2.2 Open System
      • 7.2.3 Self Consistency
      • 7.2.4 A Brute Force Approach: Parallelization with MPI
      • 7.2.5 Existence But Non-Uniqueness
      • 7.2.6 Scattering in the QTBM
  • 8. Finite Difference Wigner Function Method
    • 8.1 Wigner Equation
    • 8.2 Discrete Wigner Transform
    • 8.3 Conservation of Mass
      • 8.3.1 Continuity Equation
      • 8.3.2 Meshing Constraints
    • 8.4 Discretization
      • 8.4.1 Free Term
      • 8.4.2 Drift Term
      • 8.4.3 Relaxation Time Scattering
      • 8.4.4 Boundary Conditions
    • 8.5 Remarks on Implementation
  • 9. Stochastic Methods and Monte Carlo
    • 9.1 Quantum Electron-Phonon Scattering
    • 9.2 Boltzmann-Type Scattering
    • 9.3 A Scattering Interpretation of the Potential Operator
    • 9.4 The Negative Sign Problem
  • 10. Evaluation of Quantum Simulation Approaches
    • 10.1 Comparison of Simulation Results
      • 10.1.1 I-V curves
      • 10.1.2 Current Density
      • 10.1.3 Carrier Density
    • 10.2 Discussion of Discrepancy
      • 10.2.1 Coherence Length
      • 10.2.2 Mesh Size
      • 10.2.3 Boundary Conditions
      • 10.2.4 Effective Mass Model
      • 10.2.5 Discretization
  
  
• IV. Conclusion
  • 11. Resume
    • 11.1 The Highest Order Moment Closure Problem
    • 11.2 Merits of Quantum Simulation Approaches
    • 11.3 Closing Words
  
  
• List of Figures
• Bibliography
• List of Publications
  • Publications in Scientific Journals
  • Oral and Poster Presentations with Proceedings
  • Oral and Poster Presentations without Proceedings
• Curriculum Vitae