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Thermoelectric Transport in Quantum Point Contacts and Chaotic Cavities: Thermal Effects and Fluctuations Adel Abbout

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Thermoelectric Transport in Quantum Point Contacts and Chaotic Cavities: Thermal Effects and Fluctuations Adel Abbout Thermoelectric transport in quantum point contacts and chaotic cavities: thermal effects and fluctuations Adel Abbout To cite this version: Adel Abbout. Thermoelectric transport in quantum point contacts and chaotic cavities: thermal effects and fluctuations. Mesoscopic Systems and Quantum Hall Effect [cond-mat.mes-hall]. Université Pierre et Marie Curie - Paris VI, 2011. English. tel-00793816 HAL Id: tel-00793816 https://tel.archives-ouvertes.fr/tel-00793816 Submitted on 23 Feb 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Université de Paris 6 Années 2008-2011 Transport thermoélectrique dans des contacts quantiques ponctuels et de cavités chaotiques : Effets thermiques et fluctuations. Service de Physique de l’État Condensé CEA Saclay 21 Décembre 2011 Adel Abbout Directeur de thèse : Jean-Louis PICHARD Jury : Doucot BENOIT Klaus FRAHM Jean-Louis PICHARD (Directeur de thèse) Marc SANQUER Dima SHEPELYANSKY (rapporteur) Dietmar WEINMANN (rapporteur) 2 . 3 . To my parents 4 Acknowledgement I would like to thank my thesis advisor Jean Louis Pichard for all the work achieved together. I record my gratitude to him for his supervision, advice, guidance and specially his presence during all this years. His patience and kindness are well appreciated. I gratefully acknowledge Khandker Abdul Muttalib for our collaboration and for introducing me to random matrix theory. I appreciated all the time working together. A huge thank you goes to Geneviève Fleury for the various advices I benefited from her. I am much indebted to her for reading, correcting my thesis and all her valuable suggestions. I enjoyed all the time we spent together. I thanks all my friends in the SPEC for the interesting discussions and the time we spent together. The team of Rodolfo Jalabert and Dietmar Weinmann are acknowledged for the fruitful collaboration and interesting discussions. I am fully indebted to my family which I thank for being myself. My immense thoughts go to them. I studied equations, languages and gestures but none is sufficient to express what I owe them. For the moment I may just say : to the family, the son is grateful. Table of contents 0.1 Introduction and overlook . 11 0.2 Experimental motivation . 11 0.3 This thesis . 12 0.3.1 Part one . 12 0.3.2 Part two . 12 1 Introduction to scanning gate microscopy 13 Summary of chapter 1 . 14 1.1 Two dimensional electronic gases (2DEG) . 15 1.1.1 2DEG in Heterojunctions . 15 1.1.2 Fabrication of 2DEG . 15 1.1.3 Density of states . 16 1.1.4 advantages of 2DEG . 16 1.2 Quantum Point Contact in 2DEG . 17 1.2.1 Conductance quantization . 17 1.2.2 0.7 anomaly . 17 1.3 Models for quantum point contacts . 19 1.3.1 Adiabatic constriction . 20 1.3.2 Saddle-point constriction . 20 1.3.3 Hyperbolic model . 21 1.3.4 Wide-Narrow-Wide model . 22 1.4 Scanning Gate Microscopy . 23 1.4.1 SGM on Quantum Point Contacts . 24 1.4.2 The charged Tip in SGM techniques . 25 1.4.3 Modelization of the charged tip . 25 1.4.4 Questions on experiment . 25 2 Numerical tools for quantum transport 27 Summary of chapter 2 . 28 2.1 Quantum transport . 29 2.1.1 Characteristic lengths . 29 2.1.2 Quantum transport and scattering matrix . 30 2.1.3 Quantum conductance . 31 2.1.4 The Green’s function formalism . 32 2.1.5 Application to quasi 1D wire . 36 2.2 Dyson equation and recursive Green’s function . 37 2.2.1 Dyson equation . 37 2.2.2 Recursive Green’s function . 38 2.3 Including the charged tip . 39 3 Quantum transport and numerical simulation 43 Summary of chapter 3 . 44 3.1 Zero temperature conductance change . 45 3.1.1 Comparing different QPC models . 45 3.1.2 Conductance change as a function of the tip position . 46 3.2 The charged tip effect . 48 5 6 TABLE OF CONTENTS 3.2.1 Dirac delta potential tip model . 54 3.3 Short range impurity . 56 4 Resonant level model and analytical solution for electron transport through na- noconstrictions 59 Summary of chapter 4 . 60 4.0.1 The conclusions of numerical simulations . 61 4.0.2 Toy Model: 2D resonant level model . 61 4.1 The 2D lead self energy in the absence of the charged tip . 64 4.1.1 Presentation of the lattice model . 64 4.1.2 Method of mirror images and self energy of a semi-infinite lead . 66 4.1.3 Expansion of the self energy in the continuum limit . 68 4.2 Self energy of a 2D semi-infinite lead in the presence of a charged tip . 68 4.3 Decay law of the fringes in scanning gate microscopy. 70 4.3.1 Decay law of the fringes when T ◦ < 1 ...................... 71 4.3.2 Decay law of the fringes when T ◦ = 1 ...................... 72 4.3.3 Change in the density of state . 72 4.3.4 Semi-classical approach of the determination of ∆G .............. 73 4.4 Thermal enhancement of the fringes in the interference pattern of a quantum point contact . 73 4.4.1 Temperature dependence of the RLM conductance . 74 4.4.2 Thermal enhancement of the fringes in a Realistic QPC. 78 4.5 Thermal effect in SGM of highly opened QPCs . 80 5 Scanning Gate Microscopy of Thermopower in Quantum Point Contacts 83 Summary of chapter 5 . 84 5.1 Introduction . 85 5.2 Thermoelectric quantum transport and linear response theory . 85 5.2.1 Onsager matrix . 85 5.2.2 Wiedemann-Franz Law . 86 5.2.3 Sommerfeld expansion and the Cutler-Mott formula . 86 5.3 Scanning gate microscopy and thermopower of quantum point contacts . 87 5.4 Focusing effect and the change in the self energy of a 2D lead . 89 5.4.1 Half filling limit: E = 0 .............................. 90 5.4.2 Continuum limit: E ∼ −4 ............................. 90 5.5 Decay law of the fringes of thermopower change . 93 5.5.1 Thermopower change and the resonant level model RLM . 93 5.5.2 Case of fully open QPC: T ◦ = 1.......................... 94 5.5.3 Case of half-opened QPC: T ◦ = 0:5 ....................... 95 6 Thermoelectric transport and random matrix theory 97 Summary of chapter 6 . 99 6.1 Introduction . 100 6.2 Gaussian ensembles and symmetries . 102 6.2.1 Gaussian distribution . 102 6.3 Random matrix theory of quantum transport in open systems: . 107 6.3.1 Circular ensemble: . 107 6.4 Quantum transport fluctuations in mesoscopic systems . 108 6.5 Hamiltonian vs Scattering approach . 112 6.5.1 Eigenvalues distribution of Lorentzian ensembles . 113 6.5.2 Lorentzian distribution characteristics . 114 6.5.3 Decimation-renormalization procedure . 114 6.5.4 Comparison between Gaussian and Lorentzian distributions . 115 6.5.5 Simple model solution . 116 6.5.6 Poisson kernel distribution . 118 6.5.7 Transmission and Seebeck coefficient distribution . 120 6.6 Generating Lorentzian ensembles . 126 TABLE OF CONTENTS 7 6.7 Time delay matrix . 131 6.7.1 Time delay matrix . 133 6.8 Distribution of the Transmission derivative . 134 6.9 Decimation procedure implications . 135 A Hamiltonian of a slice and perfect leads 139 B Green’s function recursive procedure 143 C The self Energy of a semi-infinite leads 145 D Fresnel inegral 147 E The wave number expressions 149 F Bloch and Wannier representations of a one dimensional semi-infinite lead 151 8 TABLE OF CONTENTS résumé Dans cette thèse, on s’intéresse au transport quantique des électrons dans des nano-systèmes et des cavités chaotiques . En particulier, on apporte dans un premier temps la base théorique qui permet d’expliquer les expériences de microscopie à effet de grille dans des contacts quantiques ponctuels. Dans ces expériences, on étudie la conductance d’un contact quantique ponctuel (QPC).À l’aide d’une pointe chargée d’un AFM, quelques nanomètres au dessus d’un gaz bidimensionnel d’électrons, on crée une région de déplétion de la charge électronique. Cette déplétion modifie la conductance du QPC et permet de tracer en changeant la position de la pointe chargée, des figures de franges d’interférence espacées de la moitié de la longueur d’onde de Fermi. En se basant sur des.
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