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Dissertation R. Kosik
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Contents
I. Challenges for Modeling and Simulation 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. On the Road Through 2010: Ultimate Nanoscale CMOS Simulation Capability
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
2.2.1.1 Sets of Moments
2.2.1.2 Physical Quantities
2.2.2 Hierarchy of Moment Equations
2.2.2.1 Even Moments
2.2.2.2 Odd Moments
2.2.2.3 Non-Parabolicity Corrections
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. On the Road Beyond 2010: Simulation of Emerging Research Devices
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
6.1.1.1 Schrödinger Equation
6.1.1.2 Von Neumann Equation
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
7.1.2.1 Schrödinger: Transparent BCs
7.1.2.2 Riccati and Prüfer: Dirichlet BCs
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
7.2.6.1 Schrödinger Equation with an Imaginary Potential
7.2.6.2 Relaxation-Time-Like Models
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
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Acknowledgment
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Dissertation R. Kosik
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I. Challenges on the
R. Kosik: Numerical Challenges on the Road to NanoTCAD
Previous: Acknowledgment
Up: Dissertation R. Kosik
Next: I. Challenges on the
Previous: Acknowledgment
Up: Dissertation R. Kosik
Next: I. Challenges on the