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


Physical Modeling of Electron
Transport in Strained Silicon and Silicon-Germanium


ausgeführt zum Zwecke der Erlangung des akademischen Grades
eines Doktors der technischen Wissenschaften


unter der Betreuung von
Ao.Univ.Prof. Dipl.-Ing. Dr.techn. Hans Kosina
und
O.Univ.Prof. Dipl.-Ing. Dr.techn. Siegfried Selberherr

eingereicht an der Technischen Universität Wien,
Fakultät für Elektrotechnik und Informationstechnik, von


Sergey Smirnov


Margaretenstrasse 30/21
A-1040 Wien, Österreich

Matr. Nr. 0127158
geboren am 6. März 1976 in Skhodnya, Russland


Wien, im Dezember 2003

• Kurzfassung
• Abstract
• Acknowledgment
• Contents
• List of Symbols
  • Physical Quantities
  • Constants
  
  
• List of Figures
• List of Tables
• 1. Introduction
• 2. The Semiclassical Transport Model
  • 2.1 Equations of Motion
    • 2.1.1 Real Space Equation of Motion
      • 2.1.1.1 Equation of Motion
      • 2.1.1.2 Velocity-Band Structure Relationship
    • 2.1.2 Time Evolution of the Quasi-Momentum
      • 2.1.2.1 Equation of Motion
      • 2.1.2.2 Force Expression
    • 2.1.3 General Properties of Semiclassical Dynamics
      • 2.1.3.1 Phase Space Domain Evolution.
      • 2.1.3.2 Constant External Electric Field
  • 2.2 Distribution Function
    • 2.2.1 Equilibrium Distribution
      • 2.2.1.1 Statistical Distribution
      • 2.2.1.2 Distribution of Fermions
      • 2.2.1.3 Distribution of Bosons
    • 2.2.2 Non-Equilibrium Distribution
      • 2.2.2.1 Non-Equilibrium State
      • 2.2.2.2 Interpretation Within the Relaxation Time Approximation
  • 2.3 Boltzmann's Transport Equation
    • 2.3.1 Liouville's Theorem
    • 2.3.2 Collision Integral
      • 2.3.2.1 Scattering Probability
      • 2.3.2.2 Pauli Exclusion Principle
      • 2.3.2.3 Out-Scattering and In-Scattering Terms
    • 2.3.3 Principle of Detailed Balance
  • 2.4 Band Structure
    • 2.4.1 Electron in a Periodic Potential
      • 2.4.1.1 Bloch's Theorem
      • 2.4.1.2 Energy Bands
      • 2.4.1.3 Band Structure Periodicity
    • 2.4.2 Analytical Band Structure
      • 2.4.2.1 Effective Mass Tensor
      • 2.4.2.2 Nonparabolic Band Structures
      • 2.4.2.3 Analytical Band Structures of Si and Ge
      • 2.4.2.4 Herring-Vogt Transformation
  • 2.5 Scattering Mechanisms
    • 2.5.1 Perturbation Theory
      • 2.5.1.1 First Order Perturbation
      • 2.5.1.2 Fermi's Golden Rule
    • 2.5.2 Scattering on Phonons
      • 2.5.2.1 Phonon Concept
      • 2.5.2.2 Intravalley Scattering by Acoustic Phonons
      • 2.5.2.3 Intravalley Scattering by Optical Phonons
      • 2.5.2.4 Intervalley Phonon Scattering
    • 2.5.3 Plasmon Scattering
      • 2.5.3.1 Plasmon Concept
      • 2.5.3.2 Scattering Rate
    • 2.5.4 Alloy Scattering
    • 2.5.5 Ionized Impurity Scattering
      • 2.5.5.1 Brooks-Herring Model
      • 2.5.5.2 Two-Ion Scattering
      • 2.5.5.3 Screening Theory
      • 2.5.5.4 Second Born Correction
      • 2.5.5.5 Ridley's Model
  
  
• 3. Strain Effects in $\textrm {Si}_{1-x}\textrm {Ge}_{x}$ Grown on a $\textrm {Si}_{1-y}\textrm {Ge}_{y}$ Substrate
  • 3.1 SiGe Strained Layers
    • 3.1.1 Critical Thickness and Dislocations
    • 3.1.2 Applications of Strain
    • 3.1.3 Strained Layers in Semiconductor Devices
      • 3.1.3.1 Modulation-Doped FET
      • 3.1.3.2 Strained MOSFET
  • 3.2 Linear Deformation-Potential Theory
    • 3.2.1 General Description of the Conduction Band Splitting in Strained SiGe
    • 3.2.2 Strain Tensor
    • 3.2.3 Stress Tensor
    • 3.2.4 Energy Shift
      • 3.2.4.1 Shift of Conduction Band Minima
      • 3.2.4.2 Shift of the Mean Energy
  • 3.3 Substrate Orientation and Strain Tensor
    • 3.3.1 Strain Tensor in the Interface Coordinate System
    • 3.3.2 Coordinate System Transformation
      • 3.3.2.1 Euler's Angles
      • 3.3.2.2 Transformation Operator
      • 3.3.2.3 Tensor Transformations
    • 3.3.3 Strain Tensor Elements in the Principle Coordinate System
  • 3.4 Band Structure of Strained SiGe layers
    • 3.4.1 Hydrostatic Strain
    • 3.4.2 Uniaxial Strain
      • 3.4.2.1 Splitting of the X Valleys
      • 3.4.2.2 Splitting of the L Valleys
    • 3.4.3 Effective Masses in Strained SiGe
  • 3.5 Scattering Mechanisms in Strained SiGe
    • 3.5.1 Electron-Phonon Scattering
    • 3.5.2 Ionized Impurity Scattering
    • 3.5.3 Plasmon Scattering
  
  
• 4. Monte Carlo Methods for the Solution of the Boltzmann Equation
  • 4.1 Perturbation Approach to the Boltzmann Equation
    Including the Pauli Principle
    • 4.1.1 The Zeroth Order Equation
    • 4.1.2 The First Order Equation
  • 4.2 Integral Form of the First Order Equation
    • 4.2.1 New Differential and Total Scattering Rates
    • 4.2.2 Integral Form for the Distribution Function Perturbation
    • 4.2.3 Free Term and Initial Distribution
    • 4.2.4 The Resolvent Series
    • 4.2.5 The Second Iteration of the Forward Resolvent Series
  • 4.3 Zero Field Monte Carlo Algorithm Including the
    Pauli Principle
    • 4.3.1 Low Field Carrier Mobility and Monte Carlo Techniques
    • 4.3.2 Specific of the Equilibrium Distribution
    • 4.3.3 Total Scattering Rate
    • 4.3.4 Expression for the Initial Distribution
    • 4.3.5 Monte Carlo Algorithm for the Mobility Tensor
  • 4.4 The High Field Small Signal Monte Carlo Algorithm
    • 4.4.1 Solution of the Zeroth Order Equation
      • 4.4.1.1 Initial Distributions of the Two Ensembles
      • 4.4.1.2 Normalization of the Stationary Distribution Function
      • 4.4.1.3 Integral Form of the Nonlinear Boltzmann Equation
    • 4.4.2 Solution of the First Order Equation
    • 4.4.3 Monte Carlo Algorithm for the Impulse Response
  
  
• 5. Modeling of Strained $\textrm {Si}_{1-x}\textrm {Ge}_{x}$ on $\textrm {Si}_{1-y}\textrm {Ge}_{y}$ Substrates
  • 5.1 Low Field Electron Mobility in Undoped Layers
    • 5.1.1 Si layers on $\textrm {Si}_{1-y}\textrm {Ge}_{y}$ substrates
      • 5.1.1.1 Ge Composition Dependence of Perpendicular and In-plane Components
      • 5.1.1.2 Substrate Orientation Dependence
    • 5.1.2 $\textrm {Si}_{1-x}\textrm {Ge}_{x}$ layers on $\textrm {Si}_{1-y}\textrm {Ge}_{y}$ substrates
  • 5.2 Low Field Electron Mobility in Doped Layers
  • 5.3 Small Signal Response
  
  
• 6. Summary
• A. Second Quantization
  • A.1 Many-Body Operators
  • A.2 Creation and Annihilation Operators
  
  
• B. Random Phase Approximation and Plasmons
• Bibliography
• Own Publications
• Curriculum Vitae
• About this document ...

S. Smirnov: