# Resonant level model and analytical solution for electron transport through nanoconstrictions

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## Dyson equation and recursive Green’s function

Although we reduced the size of the problem of electrons transport to the size of the scattering  region, it is still hard to get its Green’s function by direct matrix inversion. Moreover, we only need the Green’s function submatrix of the sites on the surfaces connected to the leads, and all the other elements all useless to the conductance calculation. For this reason, we proceed recursively and construct the scattering region slice by slice (or sometimes site by site. See[25]). The tool to do this recursion is based on the Dyson equation.

### Zero temperature conductance change

The algorithms we use to calculate the conductance are very easy to implement: The new method of expressing the conductance by means of the Green’s function using leads with tip(See previous chapter), uses simple mathematical operations, restricted to inverse and product of matrices of a small size. The way the recursive procedure is fulfilled to spatially map the zero-temperature conductance change shows no complexity and doesn’t necessitate neither huge amount of memory nor long time of calculation, if the procedures detailed in the previous chapter and in the appendix B are followed.

#### Comparing different QPC models

In Fig. (3.1), we applied the method of recursive Green’s function to two different QPC models: Figure (a) shows the change in the zero-temperature conductance for a wide-narrow-wide model(see Fig 1.7). Figure (b) gives the same quantity with a more realistic model, where the QPC is given by the following potential[32]: V (x; y) = a1 2y2(1 􀀀 3( 2x l )2 􀀀 2j 2x l j3)2 if j2xj < l 0 if j2xj > l (3.1) a is a constant which, for a given Fermi energy EF , gives the number N of conducting modes in the QPC .

#### Conductance change as a function of the tip position

We focused the discussion on the comparison of the results for different QPC models without taking into account the details of the model itself. Here, we discuss the results for the potential model V (x; y) defined previously in Eq. (3.1).
For each picture we obtain, we need to specify the conductance of the system in the absence of the tip noted generally G. Of course, we need to study the effect of the tip for different values of G that is to say, for different QPC openings.

Effect of the charged tip on the higher conducting modes

In this discussion, we mainly talked about the first mode of conduction. It is easy to understand that most of the arguments apply for the higher modes. However, the 2D maps of the conduction changes have a different angular dependence. For example, Fig. (3.5) shows the results of numerical simulation for the zero-temperature conductance change of the SGM of the second conducting mode. The system without tip is transparent for the two first modes of conduction (second plateau). We still obtain an interference pattern of fringes spaced by half the Fermi wavelength. Moreover, the effect of the tip is mostly to reduce the conductance (G < 0) because we can not open a new mode. Nevertheless, the effect of the tip on the second plateau of conduction is different from the first plateau as it can be seen just by comparing the two Fig. (3.5) and (3.1 b): Two relevant directions seem to emerge in the case of two modes of conduction: 45. In the direction of these two angles, the effect of the tip is higher than in other directions. This behaviour is similar to what was obtained experimentally in [33]. We can do the same, just by changing the parameters of the potential in Eq. (3.1) 2, to obtain the zero-temperature change for the third plateau of conduction or even for higher modes. Finally, may be we should remind that the figures obtained so far are the simulation of SGM experiments using an on-site potential for the tip. This means that the tip is less than a short range potential: it is only non-zero on a unique site. This is not a bad approximation and the numerical results seem pretty good and enough to explain all the physical aspects of the experiments. In fact, this assumption is valid since the wavelength of the electrons (around 37 nm) is bigger than the range of the potential induced by the charged tip. In other words, the electron do not see the details of lengths smaller than half the Fermi wavelength that is way the tip of small range can be considered as an on-site delta potential. I present more details on the effect of the tip in the following section.

Dirac delta potential tip model

The comparison of the figures of interference pattern obtained with realistic models both for the QPC and the tip with the figures obtained using a simpler model for the charged AFM tip suggests that it is enough to consider the Dirac delta function when we do not seek a very good precision in the numerical results. Indeed, if only the behavior behind the physics of the experiment is needed, the simpler model for the tip reproduces the main results and explains well the quantum effects induced in Scanning Gate Microscopy (SGM) experiments.
The Dirac delta function modeling the tip is a perturbation which is non-zero only on one site: V (x; y) = Vtipjx0; y0ihx0; y0j (3.5).
Where x0 and y0 are the coordinates of the tip and Vtip is the value of the scattering potential. The advantage of using the local on-site scatterer is to avoid the task of inverting matrices in the algorithm of recursive Green’s function. In fact, the Dyson equation expressing the Green’s function G of the system with the tip using the the Green’s function G of the system without the tip reads: G = G + GV G.

READ  LEARNING, CONSTRUCTIVISM, CONSTRUCTIONISM, AND CCAIML

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
1.2 Quantum Point Contact in 2DEG
1.2.1 Conductance quantization
1.2.2 0.7 anomaly
1.3 Models for quantum point contacts
1.3.3 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

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