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Conductance of Single Electron Devices from Imaginary–Time Path Integrals

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Conductance of Single Electron Devices from Imaginary–Time Path Integrals Conductance of Single Electron Devices from Imaginary–Time Path Integrals Dissertation zur Erlangung des Doktorgrades der Fakult¨at f¨ur Mathematik und Physik der Albert–Ludwigs–Universit¨at, Freiburg im Breisgau vorgelegt von Christoph Theis aus Bernkastel-Kues Freiburg, April 2004 Dekan : Prof. Dr. R. Schneider Leiter der Arbeit : Prof. Dr. H. Grabert Referent : Prof. Dr. H. Grabert Koreferent : Tag der m¨undlichen Pr¨ufung: 26. Mai 2004 Contents 1 Introduction and Overview 1 2 Concepts of Transport in Nanoscopic Structures 5 2.1 Resonant Tunneling through Discrete Levels . ........... 5 2.2 CoulombBlockadeofTransport. ...... 8 2.3 KondoEffectinQuantumDots . 11 I Transport Properties from Imaginary-Time Path Integrals 15 3 Path Integrals for Fermions 17 3.1 Introduction: The Feynman Path Integral . ......... 17 3.2 SecondQuantization .............................. 19 3.3 GrassmannAlgebra ................................ 20 3.3.1 Motivation and Definition of the Grassmann Algebra . ........ 21 3.3.2 Calculus for Grassmann Variables . ..... 22 3.3.3 Important Integration Formulas . ...... 23 3.4 FermionCoherentStates. ..... 25 3.4.1 Definition of Fermion Coherent States . ..... 25 3.4.2 PropertiesofFermionCoherentStates . ....... 26 3.5 CoherentStatePathIntegral . ...... 28 3.6 Example: Non–Interacting Fermions . ........ 29 3.6.1 ThePartitionFunction . 29 3.6.2 TheThermalGreen’sFunction . 30 4 Path Integral Monte Carlo 33 4.1 BasicsofMonteCarloIntegration. ........ 33 4.2 Importance Sampling and Markov Processes . ......... 34 4.2.1 Reduction of Statistical Errors by Importance Sampling.......... 34 4.2.2 Markov Processes and the Metropolis Algorithm . ........ 35 4.3 Statistical Analysis of Monte Carlo Data . .......... 37 4.3.1 Estimates for Uncorrelated Measurements . ........ 37 4.3.2 Correlated Measurements and Autocorrelation Time . .......... 38 4.3.3 Binning Analysis of the Monte Carlo Error . ...... 39 4.4 Systematic Errors and Trotter Extrapolation . ........... 39 4.4.1 Approximations for the Short–Time Propagator . ......... 40 4.4.2 TrotterErrorofExpectationValues . ...... 40 4.4.3 TrotterExtrapolation . 42 i ii CONTENTS 4.5 Non-Positive Actions and the Sign Problem . ........ 42 5 Correlation Functions and Inverse Problems 45 5.1 Time Correlation Functions and Linear Response . ........... 45 5.1.1 Real–Time Correlation Functions . ...... 45 5.1.2 Linear Response Theory and Fluctuation–Dissipation Theorem . 46 5.1.3 The Kubo Formula for the Conductance . 48 5.1.4 Imaginary–Time Correlation Functions . ........ 49 5.2 LinearInverseProblems . ..... 50 5.2.1 Definition and Examples of Inverse Problems . ...... 50 5.2.2 Ill–Posedness and Regularization . ....... 52 5.3 The Singular Value Decomposition (SVD) . ........ 54 5.3.1 Formal Solution for Linear Inverse Problems . ........ 54 5.3.2 Regularization of the Solution . ...... 55 5.3.3 Additional Constraints . 57 5.4 The Maximum Entropy Method (MEM) . 60 5.4.1 BayesianInference . .. .. .. .. .. .. .. 60 5.4.2 The Maximum Entropy Functional . 61 5.4.3 Determination of the Regularization Parameters . .......... 63 5.5 TestoftheSVDMethod.............................. 64 5.5.1 AnExactlySolvableModel . 64 5.5.2 ApplicationoftheSVDMethod . 71 5.5.3 Comparison of SVD and MEM Results . 74 II Applications 85 6 The Metallic Single Electron Transistor 87 6.1 Single Electron Tunneling through a Metallic Island . ............. 87 6.1.1 Experimental Realizations and Model Parameters . ......... 87 6.1.2 ChargingModel ................................ 88 6.2 PathIntegralFormulation . ...... 89 6.2.1 PathIntegralAnsatz. 89 6.2.2 TheCoulombAction.............................. 90 6.2.3 Coherent State Path Integral and Source Terms . ........ 92 6.3 Effective Action of the Single Electron Transistor . ............ 93 6.3.1 Exact Integration of Quasi-Particle Baths . ........ 93 6.3.2 TheTunnelAction............................... 94 6.3.3 The Current Autocorrelation Function . ....... 97 6.4 Monte Carlo Calculation of the Correlation Function . ............. 97 6.4.1 Discretization of the Path Integral . ....... 97 6.4.2 Details of the Monte Carlo Simulation . 100 6.4.3 Results for the Cosine Correlation Function . ......... 102 6.5 Results for the Conductance . 104 6.5.1 Inverse Problem for the Conductance . 104 6.5.2 Coulomb Oscillations of the Conductance . ....... 108 6.5.3 Temperature Dependence of the Conductance . ....... 110 6.5.4 Dependence on the Tunneling Strength . 111 CONTENTS iii 7 Semiconductor Quantum Dots 113 7.1 Band Diagram of Semiconductor Heterostructures . ........... 114 7.1.1 Band Structure of GaAs and AlGaAs . 114 7.1.2 BandProfileofaHeterostructure. 116 7.2 Electrostatics of Gated Quantum Dots . ........ 119 7.2.1 The Constant Interaction Model and its Limitations . .......... 119 7.2.2 Electrostatic Energy and Work of the Power Sources . ......... 120 7.2.3 Green’s Function for a Vertical Quantum Dot . ....... 121 7.3 TheoreticalModel ................................ 123 7.3.1 ModelHamiltonian.............................. 123 7.3.2 ActionandSourceTerms . 124 7.3.3 DecouplingoftheInteraction . 125 7.4 EffectiveAction................................... 126 7.4.1 Integration over the Lead Fermions . 126 7.4.2 Integration over the Quantum Dot Fermions . 127 7.5 DiscussionoftheResults. 128 7.5.1 GeneralDiscussion . 128 7.5.2 Outlook: Stationary Phase Approximation . ....... 128 8 Summary and Conclusions 133 III Appendices 137 A Properties of Correlation Functions 139 B Linear System of de Villiers’ SVD Method 141 C The Damped Harmonic Oscillator 143 C.1 Influence Functional for a Linearly Coupled Harmonic Bath ............ 143 C.2 ClassicalDynamicalFrictionKernel . ......... 145 C.3 Correlation Function for the Tagged Oscillator . ............ 146 D Representation of Operators 147 D.1 TheChargeShiftOperator . 147 D.2 TheCurrentOperator .............................. 147 E Electrostatics of Quantum Dots 151 E.1 FormalSolutionoftheDirichletProblem . ......... 151 E.2 Green’sFunctionforaCylindricalDot . ......... 152 Bibliography 156 Chapter 1 Introduction and Overview The continuing progress in miniaturization of electronic circuits has reduced the length of a single transistor down to the nanometer scale. Not only does this imply that the size of the fundamental building blocks approaches that of the chemical units of the material but also that quantum mechanical effects play a very important role in their operation. On the one hand this poses new problems as we are reaching a fundamental limit of miniaturization where noise and quantum mechanical interference effects reduce the reliability of ”classical” transistors as logic units. On the other hand new possibilities open up that can be summarized under the keywords ”molecular electronics” and ”quantum computing”. In this thesis we will examine two model systems that are important for the understanding of the relevant concepts of molecular electronics and that are currently under investigation for applications in quantum computing. The metallic single electron transistor (SET) [1] shown on the reflection electron microscope (REM) picture in fig. 1.1 consists of a small Al island (with linear dimension L 500nm and ≈ capacitance C) coupled to Al leads via tunnel barriers formed by an oxide layer. The Al island is also coupled electrostatically to gate electrodes via a gate capacitance Cg. The SET is an important model system for the study of the Coulomb blockade effect which is responsible for a suppression of the source drain current for voltages V V with V [0, e/C] depending ≤ th th ∈ on the gate voltage Ug. For the linear response conductance it leads to oscillations with period e/Cg as a function of the gate voltage. gate source island drain gate Figure 1.1: REM picture of a four junction SET. In the Coulomb blockade measurements both gates as well as the two source and the two drain electrodes are connected in parallel (from [2]). 1 2 CHAPTER 1. INTRODUCTION AND OVERVIEW Among other possible applications [3] it can be used as an ultra sensitive electrometer [4] and it represents a building block in the so called ”quantronium” circuit [5] which is a promising candidate for a qubit, i.e. the basic unit of information in quantum computing. Single–atom transistors [6], single–molecule transistors [7] or carbon nanotube single electron transistors [8] in which gold electrodes are used and the central island is replaced by a molecule or carbon nanotube are applications of the concept of the single electron transistor for molecular electronics research. Semiconductor quantum dots which are sometimes also referred to as artificial atoms [9, 10] are based on the realization of a two–dimensional electron gas (2DEG) formed in a semiconductor heterostructure. Using electrostatic gates or lithographic techniques (or a combination of both) to create a confinement potential in the plane of the 2DEG one forms two–dimensional ”atoms” containing between one and several hundred electrons. The quantum dot can be contacted either laterally or from above and below by n–doped GaAs layers. Examples for both geometries are shown in fig. 1.2. Figure 1.2: Subfigure a) shows the schematic layout of a vertical quantum dot consisting of a InGaAs layer sandwiched between AlGaAs tunnel barriers and contacted from above and below (from [10]). Subfigure b) displays an electron micrograph showing the electrostatic gates defining a lateral quantum dot. The 2DEG is situated 190 nm below the surface of the sample. In the left part of b) one can see another
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