Band Structure Calculations

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Table of contents

Introduction 
I Overview of Graphene Physics and Mesoscopic Transport
1 Physical Properties of Graphene 
1.1 Crystal Structure of Graphene
1.2 Graphene Band Structure
1.2.1 Basic Principles
1.2.2 Band Structure Calculations
1.2.3 Low-Energy Excitations
1.3 Properties of Dirac Fermions
1.3.1 Probabilty Current Density
1.3.2 Zitterbewegung, Chirality and Klein Tunneling
2 Transport in Mesoscopic Systems 
2.1 Mesoscopic Scale
2.1.1 Classical Discription of Transport
2.1.2 Quantum Phase Coherent Systems
2.1.3 Effects of the Coherence on Transport Properties
2.2 Scattering Approach
2.2.1 Framework, Hypotheses, Formulation
2.2.2 Scattering Approach at Work
2.3 Quantum Hall Effect
2.3.1 Landau Quantization
2.3.2 Integer Quantum Hall Effect
3 Mesoscopic Transport in Graphene 
3.1 Conducance and Shot Noise in Graphene
3.2 Quantum Hall Effect in Graphene
3.3 Experimental Results
II Experimental Setup and Device
4 Measurement System Principle 
4.1 Experimental Requirements and Techniques
4.1.1 Typical Scales
4.1.2 Noise Measurement Techniques
4.2 Technical Realisation
4.2.1 Device Design
4.2.2 Measurement System
4.2.3 Cryogenic Amplification System
5 Device Fabrication 
5.1 Methods
5.1.1 Obtaining Graphene
5.1.2 Making Graphene Visible
5.1.3 Raman Spectroscopy of Graphene
5.1.4 Graphene Oxygen Plasma Etch
5.2 Processes
5.2.1 Wafers Preparation
5.2.2 Graphite Deposition
5.2.3 Graphene Flakes Detection
5.2.4 Microcircuit Deposition
5.2.5 Nano-Constriction in Graphene
5.2.6 Device Test
5.2.7 Difficulties and Solutions
III Experimental Results 
6 Measurement System Calibration 
6.1 Low Frequency Calibration
6.1.1 Lines Tuning
6.1.2 Two-point Measurement
6.1.3 Calibration
6.2 High Frequency Calibration
6.2.1 Lines Tuning
6.2.2 Current Measurement
7 Conductance Measurements at Zero Magnetic Field 
7.1 Conductance at Zero Bias
7.1.1 Ballistic Regime Hypothesis
7.1.2 Diffusive Regime Hypothesis
7.1.3 Discussion
7.1.4 Model for the “altered” Curves
7.2 Conductance Spectroscopy
8 Conductance in the Magnetic Field 
8.1 Measurement Principle
8.2 Results and Discussion
9 Shot Noise in Graphene 
9.1 Measurement Principle
9.2 Shot Noise in Presence of Joule Heating
9.2.1 Cooling by Electron Diffusion
9.2.2 Data Fit
9.2.3 Cooling by Phonon Emission
9.3 Noise Power Fluctuations
9.4 Results and Discussion
Conclusion
Appendices
A Device Fabrication 
A.1 Recipes
A.2 Introduction to Raman scattering
A.3 Common Nano-fabrication Techniques
A.3.1 Microlithography Principle
A.3.2 Optical and E-Beam lithography
A.3.3 Thin Films Deposition
B Measurement System 
B.1 Cryogenic Inset
B.2 Data Acquisition Module
B.3 RLC-Filter Pass-Band Calculation
Bibliography