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Table of contents
0.1 Introduction and overlook
0.2 Experimental motivation
0.3 This thesis
0.3.1 Part one
0.3.2 Part two
1 Introduction to scanning gate microscopy
Summary of chapter
1.1 Two dimensional electronic gases (2DEG)
1.1.1 2DEG in Heterojunctions
1.1.2 Fabrication of 2DEG
1.1.3 Density of states
1.1.4 advantages of 2DEG
1.2 Quantum Point Contact in 2DEG
1.2.1 Conductance quantization
1.2.2 0.7 anomaly
1.3 Models for quantum point contacts
1.3.1 Adiabatic constriction
1.3.2 Saddle-point constriction
1.3.3 Hyperbolic model
1.3.4 Wide-Narrow-Wide model
1.4 Scanning Gate Microscopy
1.4.1 SGM on Quantum Point Contacts
1.4.2 The charged Tip in SGM techniques
1.4.3 Modelization of the charged tip
1.4.4 Questions on experiment
2 Numerical tools for quantum transport
Summary of chapter 2
2.1 Quantum transport
2.1.1 Characteristic lengths
2.1.2 Quantum transport and scattering matrix
2.1.3 Quantum conductance
2.1.4 The Green’s function formalism
2.1.5 Application to quasi 1D wire
2.2 Dyson equation and recursive Green’s function
2.2.1 Dyson equation
2.2.2 Recursive Green’s function
2.3 Including the charged tip
3 Quantum transport and numerical simulation
Summary of chapter 3
3.1 Zero temperature conductance change
3.1.1 Comparing different QPC models
3.1.2 Conductance change as a function of the tip position
3.2 The charged tip effect
3.2.1 Dirac delta potential tip model
3.3 Short range impurity
4 Resonant level model and analytical solution for electron transport through nanoconstrictions
Summary of chapter 4
4.0.1 The conclusions of numerical simulations
4.0.2 Toy Model: 2D resonant level model
4.1 The 2D lead self energy in the absence of the charged tip
4.1.1 Presentation of the lattice model
4.1.2 Method of mirror images and self energy of a semi-infinite lead
4.1.3 Expansion of the self energy in the continuum limit
4.2 Self energy of a 2D semi-infinite lead in the presence of a charged tip
4.3 Decay law of the fringes in scanning gate microscopy
4.3.1 Decay law of the fringes when T < 1
4.3.2 Decay law of the fringes when T = 1
4.3.3 Change in the density of state
4.3.4 Semi-classical approach of the determination of G
4.4 Thermal enhancement of the fringes in the interference pattern of a quantum point contact
4.4.1 Temperature dependence of the RLM conductance
4.4.2 Thermal enhancement of the fringes in a Realistic QPC
4.5 Thermal effect in SGM of highly opened QPCs
5 Scanning Gate Microscopy of Thermopower in Quantum Point Contacts
Summary of chapter 5
5.1 Introduction
5.2 Thermoelectric quantum transport and linear response theory
5.2.1 Onsager matrix
5.2.2 Wiedemann-Franz Law
5.2.3 Sommerfeld expansion and the Cutler-Mott formula
5.3 Scanning gate microscopy and thermopower of quantum point contacts
5.4 Focusing effect and the change in the self energy of a 2D lead
5.4.1 Half filling limit: E = 0
5.4.2 Continuum limit: E 4
5.5 Decay law of the fringes of thermopower change
5.5.1 Thermopower change and the resonant level model RLM
5.5.2 Case of fully open QPC: T = 1
5.5.3 Case of half-opened QPC: T = 0:5
6 Thermoelectric transport and random matrix theory
Summary of chapter 6
6.1 Introduction
6.2 Gaussian ensembles and symmetries
6.2.1 Gaussian distribution
6.3 Random matrix theory of quantum transport in open systems:
6.3.1 Circular ensemble:
6.4 Quantum transport fluctuations in mesoscopic systems
6.5 Hamiltonian vs Scattering approach
6.5.1 Eigenvalues distribution of Lorentzian ensembles
6.5.2 Lorentzian distribution characteristics
6.5.3 Decimation-renormalization procedure
6.5.4 Comparison between Gaussian and Lorentzian distributions
6.5.5 Simple model solution
6.5.6 Poisson kernel distribution
6.5.7 Transmission and Seebeck coefficient distribution
6.6 Generating Lorentzian ensembles
6.7 Time delay matrix
6.7.1 Time delay matrix
6.8 Distribution of the Transmission derivative
6.9 Decimation procedure implications
A Hamiltonian of a slice and perfect leads
B Green’s function recursive procedure
C The self Energy of a semi-infinite leads
D Fresnel inegral
E The wave number expressions
F Bloch and Wannier representations of a one dimensional semi-infinite lead
