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


Numerical Study of Quantum Transport
in Carbon Nanotube Based Transistors


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


Mahdi Pourfath


Spengergasse 4-8/2/1
A-1050 Wien, Österreich

Matr. Nr. 0326624
geboren am 18. Juli 1978 in Tehran


Wien, im Juli 2007

• Kurzfassung
• Abstract
• Acknowledgment
• Contents
• List of Abbreviations and Acronyms
• List of Symbols
  • Notation
  • Physical Quantities
  • Constants
  
  
• 1. Introduction
• 2. Fundamentals of Carbon Nanotubes
  • 2.1 Historical Overview
  • 2.2 Theoretical Background
  • 2.3 Nanotube Growth Methods
    • 2.3.1 Arc-Discharge and Laser Ablation
    • 2.3.2 Chemical Vapor Deposition
  • 2.4 Electronic Structure
    • 2.4.1 Electronic Band Structure of Graphene
    • 2.4.2 Electronic Band Structure of SW-CNTs
  • 2.5 Phonon Properties
    • 2.5.1 Phonon Dispersion Relations of Graphene
    • 2.5.2 Phonon Dispersion Relations of SW-CNTs
  • 2.6 Electron-Phonon Interaction
    • 2.6.1 Electron-Phonon Matrix Elements
  • 2.7 Transport Properties
    • 2.7.1 Ballistic Transport
    • 2.7.2 Diffusive Transport
  • 2.8 CNTs in Electronics
    • 2.8.1 Fabrication and Performance of CNT-FETs
    • 2.8.2 SCHOTTKY Barrier Model of CNT-FET Operation
    • 2.8.3 Environmental Influences on the Performance of CNT-FETs
    • 2.8.4 Prototype CNT-FET Circuits
  
  
• 3. Quantum Transport Models
  • 3.1 Equilibrium Zero Temperature GREEN's Function
    • 3.1.1 Definition of the GREEN's Function
  • 3.2 Equilibrium Finite Temperature GREEN's Function
    • 3.2.1 Equilibrium Ensemble Average
    • 3.2.2 MATSUBARA GREEN's Function
  • 3.3 Non-Equilibrium GREEN's Functions
    • 3.3.1 Non-Equilibrium Ensemble Average
    • 3.3.2 Contour-Ordered GREEN's Function
    • 3.3.3 KELDYSH Contour
  • 3.4 Perturbation Expansion of the GREEN's Function
    • 3.4.1 WICK Theorem
    • 3.4.2 First-Order Perturbation Expansion
  • 3.5 DYSON Equation
  • 3.6 Approximation of the Self-Energy
    • 3.6.1 Elecron-Electron Interaction
    • 3.6.2 Electron-Phonon Interaction
  • 3.7 Analytical Continuation
    • 3.7.1 Real Time Formalism
    • 3.7.2 LANGRETH Theorem
  • 3.8 Quantum Kinetic Equations
    • 3.8.1 The KADANOFF-BAYM Formulation
    • 3.8.2 KELDYSH Formulation
    • 3.8.3 Steady-State Kinetic Equations
  • 3.9 Relation to Observables
    • 3.9.1 Electron and Hole Density
    • 3.9.2 Spectral Function and Local Density of States
    • 3.9.3 Current Density
    • 3.9.4 Current Conservation
  • 3.10 Comparison of Transport Models
    • 3.10.1 Non-Equilibrium GREEN's Function
    • 3.10.2 Master Equation for the Density Matrix
    • 3.10.3 The WIGNER Distribution Function
  
  
• 4. Implementation
  • 4.1 Electrostatic Potential and the POISSON Equation
    • 4.1.1 Discretization of the POISSON Equation
    • 4.1.2 Boundary Conditions
    • 4.1.3 Interface Conditions
  • 4.2 Basis Functions and Matrix Representation
  • 4.3 Tight-Binding HAMILTONian
  • 4.4 Mode-Space Transformation
  • 4.5 Contact Self-Energies
    • 4.5.1 Semi-Infinite CNT Contacts
    • 4.5.2 SCHOTTKY Type Metal-CNT Contacts
  • 4.6 Scattering Self-Energies
    • 4.6.1 Scattering with Optical Phonons
    • 4.6.2 Scattering with Acoustic Phonons
  • 4.7 Evaluation of Observables
    • 4.7.1 Carrier Density
    • 4.7.2 Current
    • 4.7.3 Discussion
  • 4.8 Selection of the Energy Grid
    • 4.8.1 Confined States
    • 4.8.2 Non-adaptive Energy Grid
    • 4.8.3 Adaptive Energy Grid
  • 4.9 Self-Consistent Simulations
    • 4.9.1 Self-Consistent Iteration Scheme
    • 4.9.2 Convergence of the Self-Consistant Simulations
  
  
• 5. Applications
  • 5.1 Double-Gate Design
    • 5.1.1 Ambipolar Conduction
    • 5.1.2 Double-Gate CNT-FET
  • 5.2 Asymmetric Single-Gate Design
    • 5.2.1 Gate-Source Spacer Length
    • 5.2.2 Gate-Drain Spacer Length
  • 5.3 Device Optimization
    • 5.3.1 Gate-Delay Time of CNT-FETs
    • 5.3.2 Optimized Spacer Length
  • 5.4 Tunneling CNT-FETs
    • 5.4.1 Symmetric and Asymmetric Doping
  • 5.5 The Effect of Electron-Phonon Interaction
    • 5.5.1 Electron-Phonon Coupling Strength
    • 5.5.2 Phonon Energy
    • 5.5.3 Diffusive Limit
    • 5.5.4 Discussion
  
  
• 6. Summary and Conclusions
• A. First and Second Quantization
• B. Time Evolution Pictures
  • B.1 SCHRÖDINGER Picture
  • B.2 Interaction Picture
  • B.3 HEISENBERG Picture
  • B.4 The Evolution Operator $\hat{S}$
  • B.5 Imaginary Time Operators
  
  
• C. Review of Thermodynamics and Statistical Mechanics
  • C.1 FERMI-DIRAC Statistics
  • C.2 Bose-EINSTEIN Statistics
  
  
• D. Non-Interacting GREEN's Functions
  • D.1 Non-Interacting FERMIons
  • D.2 Non-Interacting Bosons
  
  
• E. FEYNMAN Diagrams
• F. Variational Derivation of Self-Energies
  • F.1 Electron-Electron Interaction
    • F.1.1 Screened Interaction, Polarization, and Vertex Function
  • F.2 Electron-Phonon Interaction
    • F.2.1 The Phonon GREEN's Function
    • F.2.2 The Phonon Self-Energy
  • F.3 Approximation of the Self-Energy
  
  
• G. Treatment of Contacts
  • G.1 Matrix Truncation
  • G.2 Contact Self-Energies
  • G.3 Surface GREEN's Function
  
  
• H. Recursive GREEN's Function Method
  • H.1 Recursive Algorithm to Calculate $G^\mathrm{r}$
  • H.2 Recursive Algorithm to Calculate $G^<$
  
  
• Bibliography
• Own Publications
  • Publications in Journals
  • Publications in Conference Proceedings
  • Chapter in Books
  • Patents
  
  
• Curriculum Vitae
• About this document ...

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors