Conductance of single electron devices from imaginary-time path integrals [Elektronische Ressource] / vorgelegt von Christoph Theis
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Conductance of single electron devices from imaginary-time path integrals [Elektronische Ressource] / vorgelegt von Christoph Theis

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Conductance of Single Electron Devicesfrom Imaginary–Time Path IntegralsDissertationzur Erlangung des Doktorgrades derFakult¨at fu¨r Mathematik und PhysikderAlbert–Ludwigs–Universit¨at,Freiburg im Breisgauvorgelegt vonChristoph Theisaus Bernkastel-KuesFreiburg, April 2004Dekan : Prof. Dr. R. SchneiderLeiter der Arbeit : Prof. Dr. H. GrabertReferent : Prof. Dr. H. GrabertKoreferent :Tag der mu¨ndlichen Pru¨fung: 26. Mai 2004Contents1 Introduction and Overview 12 Concepts of Transport in Nanoscopic Structures 52.1 Resonant Tunneling through Discrete Levels . . . . . . . . . . . . . . . . . . . . . 52.2 Coulomb Blockade of Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Kondo Effect in Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11I Transport Properties from Imaginary-Time Path Integrals 153 Path Integrals for Fermions 173.1 Introduction: The Feynman Path Integral . . . . . . . . . . . . . . . . . . . . . . 173.2 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.1 Motivation and Definition of the Grassmann Algebra . . . . . . . . . . . . 213.3.2 Calculus for Grassmann Variables . . . . . . . . . . . . . . . . . . . . . . 223.3.3 Important Integration Formulas. . . . . . . . . . . . . . . . . . . . . . . . 233.4 Fermion Coherent States . . . . . . . . . . . . . . . . .

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Conductance of Single Electron Devices
from Imaginary–Time Path Integrals
Dissertation
zur Erlangung des Doktorgrades der
Fakult¨at fu¨r Mathematik und Physik
der
Albert–Ludwigs–Universit¨at,
Freiburg im Breisgau
vorgelegt von
Christoph Theis
aus Bernkastel-Kues
Freiburg, April 2004Dekan : Prof. Dr. R. Schneider
Leiter der Arbeit : Prof. Dr. H. Grabert
Referent : Prof. Dr. H. Grabert
Koreferent :
Tag der mu¨ndlichen Pru¨fung: 26. Mai 2004Contents
1 Introduction and Overview 1
2 Concepts of Transport in Nanoscopic Structures 5
2.1 Resonant Tunneling through Discrete Levels . . . . . . . . . . . . . . . . . . . . . 5
2.2 Coulomb Blockade of Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Kondo Effect in Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Fermion Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4.1 Definition of Fermion Coherent States . . . . . . . . . . . . . . . . . . . . 25
3.4.2 Properties of Fermion Coherent States . . . . . . . . . . . . . . . . . . . . 26
3.5 Coherent State Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6 Example: Non–Interacting Fermions . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6.1 The Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6.2 The Thermal Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Path Integral Monte Carlo 33
4.1 Basics of Monte Carlo Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Trotter Error of Expectation Values . . . . . . . . . . . . . . . . . . . . . 40
4.4.3 Trotter Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
iii 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 Linear Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4.2 The Maximum Entropy Functional . . . . . . . . . . . . . . . . . . . . . . 61
5.4.3 Determination of the Regularization Parameters . . . . . . . . . . . . . . 63
5.5 Test of the SVD Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5.1 An Exactly Solvable Model . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5.2 Application of the SVD Method . . . . . . . . . . . . . . . . . . . . . . . 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 Charging Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Path Integral Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.1 Path Integral Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.2 The Coulomb Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 The Tunnel Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . 111CONTENTS 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 Band Profile of a Heterostructure . . . . . . . . . . . . . . . . . . . . . . . 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 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3.2 Action and Source Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3.3 Decoupling of the Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Effective Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.4.1 Integration over the Lead Fermions . . . . . . . . . . . . . . . . . . . . . . 126
7.4.2 Integration over the Quantum Dot Fermions . . . . . . . . . . . . . . . . 127
7.5 Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.5.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Classical Dynamical Friction Kernel . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.3 Correlation Function for the Tagged Oscillator . . . . . . . . . . . . . . . . . . . 146
D Representation of Operators 147
D.1 The Charge Shift Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
D.2 The Current Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
E Electrostatics of Quantum Dots 151
E.1 Formal Solution of the Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . 151
E.2 Green’s Function for a Cylindrical Dot . . . . . . . . . . . . . . . . . . . . . . . . 152
Bibliography 156Chapter 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 C . The SET is ang
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] dependingth th
on the gate voltage U . For the linear response conductance it leads to oscillations with periodg
e/C as a function of the gate voltage.g
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]).
12 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
nanotubeareapplicationsoftheconceptofthesingleelectrontransistorformolecularelectronics
research.
Semiconductor quantum dots which are sometimes also referred to as artificial atoms [9, 10]
arebasedontherealizationofatwo–dimensionalelectrongas(2DEG)formedinasemiconductor
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”
containingbetweenoneandseveralhundredelectrons. Thequantumdotcanbecontactedeither
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]). Subfigureb)displaysanelectronmicrographshowingtheelectrostaticgatesdefining
alateralquantumdot. The2DEGissituated190nmbelowthesurfaceofthesample. Intheleft
part of b) one can see another realization of a SET that is used as an electrometer to measure
the charge on the dot (adapted from [11]).
Semiconductor quantum dots are an interesting model system since they provide the possi-
bility to study an electron gas with well–defined contacts that shows atom–like properties which
canbeeasilytunedbyelectrostaticgatesandmagneticfields. Duetotheconfinementonascale
of≈100 nm in the plane of the 2DEG and.10 nm in the perpendicular direction the electrons
inside the quantum dot have a discrete spectrum which is responsible for an important aspect
of electronic transport which is known as resonant tunneling. The existence of a singly occupied
(spin) degenerate level in the quantum dot can also give rise to many–body effects between the
electron gas in the leads and the localized states of the dot that are analogous to the Kondo
effect in a metal containing dilute magnetic impurities. The ”tunable” Kondo effect [12] in
quantum dots has been studied extensively in the last years and constitutes another important
concept of electronic transport through a confined electron system.
Single quantum dots or quantum dot arrays as artificial atoms or molecules represent a step
in the development towards molecular electronics. They (usually) are produced by traditional
top–down approaches and lack the mechanical degrees of freedom and the possibility to undergo3
conformational changes but they already share many of the mechanisms that will be important
for transport through real molecules. With respect to applications in quantum computing in
particulardoubledotsystemsarestudiedextensivelyandhavebeenusedtorealizechargequbits
[13]andspinqubits[14,15]. Likethequantroniumcircuit,(double)quantumdotsarefabricated
from materials that are already well established in information processing and thus can be more
easily incorporated in integrated circuits than other realizations of qubits.
The aim of this thesis is to examine theoretical approaches that allow a quantitative calcu-
lation of charge transport in these important model systems over the range of experimentally
accessible parameters. For the description of the models we use path integral methods that
have been applied successfully in the non–perturbative treatment of tunneling. To avoid the
so–called dynamic sign problem in the (direct) numerical calculation of real–time quantities we
employ imaginary–time path integrals. Imaginary–time methods rely on linear response theory
and schemes for the ”analytical continuation” of numerical data that will be critically examined
for the exactly solvable model of a harmonic oscillator embedded in a harmonic environment.
The path integral formulation for the metallic single electron transistor [16] has proven to be
a promising candidate for a quantitative description of Coulomb blockade effects although a
rigorous comparison with experiment was hampered by the fact that the relevant parameters
of the theory were not (all) accessible to measurement. In a recent experiment Wallisser et al.
employed an improved layout that allows a complete characterization of the sample and enables
us to perform a comparison between theory and experiment without any adjustable parameters.
A quantitative theory for the conductance of semiconductor quantum dots that gives a unified
descriptionoftheKondoeffectandCoulombblockadedoesnotyetexistthoughfirstapproaches
in that direction have been made [17]. Therefore we will derive a realistic model of a (vertical)
semiconductor quantum dot to which we apply the imaginary–time formalism to assess whether
this approach can be generalized to this system.
The first part of this work is devoted to the development of the methods that are used for
our study. As a starting point we take the path integral description of quantum mechanics and
its generalization to many–body systems that will be discussed in detail in chapter 3. For a
non–perturbative treatment of the SET in the regime of strong tunneling and for a realistic
description of the electrostatics of a semiconductor quantum dot, the use of numerical methods
forthecalculationoftheconductanceisrequired. Chapter4describeshowMonteCarlomethods
can be used to evaluate numerically the high–dimensional integrals that result from the path
integral description. Numerical approaches to the calculation of real–time correlation functions
for quantum mechanical many–body systems are plagued by the so called ”sign–problem” that
leads to an exponential decrease of the signal–to–noise ratio with increasing time t. Therefore
we have used an alternative approach based on the calculation of imaginary–time correlation
functions that can be determined with high accuracy by Monte Carlo methods. In chapter 5
we use linear response theory to work out the relations between the conductance and these
correlation functions. For the imaginary–time formalism these relations have the form of an
inverse problem and require special numerical methods for their solution. Since this represents
a crucial step of the calculations we will discuss in detail the tests of their implementation for
an exactly solvable model system.
The second part of the thesis describes the application of the imaginary–time path integral
formalismtothemetallicsingleelectrontransistorandsemiconductorquantumdots. Inchapter
6 the single electron transistor is modeled as a macroscopic charge degree of freedom coupled
to quasiparticle baths in the leads and on the island. We derive the effective action of the SET
by exact integration over the quasiparticle degrees of freedom and use Monte Carlo methods to
evaluatethecurrent–currentcorrelationfunctioninimaginary–timefromwhichtheconductance4 CHAPTER 1. INTRODUCTION AND OVERVIEW
of the SET can be calculated. The results are compared in detail with experimental findings
of Wallisser et al. [2] and Joyez et al. [18]. In chapter 7 we outline the extension of the
imaginary–timepathintegralapproachtothedescriptionofsemiconductorquantumdots. Since
the screening length in this system is larger than in the metallic SET of chapter 6, a realistic
model of a semiconductor quantum dot has to take into account the screened electron–electron
interaction on the quantum dot and the effects of the gate electrode on the confinement in more
detail. We show how the geometry of this electrostatic problem can be incorporated in the
action of the imaginary–time path integral. As in the case of the SET the quasiparticles can
be integrated out exactly and an effective action for the description of semiconductor quantum
dots can be derived. We compare the resulting theory with the path integral approach for the
SET and point out directions for further research in relation with recently published theoretical
results by Bednarek et al. [19].
A part of this thesis has been published in [2].