Dynamics of Shuttle Devices

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by  Andrea Donarini

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. Dynamics of Shuttle Devices
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Ph.D. Thesis
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Foreword
This thesis is submitted in candidacy for the Ph.D. degree within the
Physics Program at the Technical University of Denmark. The thesis de-
scribes part of the work that I have carried out under supervision of Antti-
Pekka Jauho from the Department of Micro and Nanotechnology.
I’m grateful to my supervisor for the very stimulating atmosphere that
he has been able to create and maintain within his group and also for his
gentle but firm guidance. I want to thank all the members of the Theoretical
Nanotechnology Group at the MIC Department and in particular Dr. Tom´aˇs
Novotny´ andChristian Flindt for the close andintense collaborationthat has
generated a large part of the results presented in this thesis.
I would like to thank Prof. Timo Eirola for having introduced me to
the subject of the iterative numerical methods and for his precious help in
the solution of some of the numerical problems I encounter in my research. I
want alsotothankDr. TobiasBrandesandNeillLambertforthenicephysics
discussions I could have with them during the period I spent collaborating
with them in Manchester. I want to thank Christian Flindt also for the
enthusiasm and the very positive attitude that he has been always able to
spread within the “Shuttle Group”. He has also been an indefatigable reader
of the proofs and deserves many thanks for that. Finally I also want to
thank my family for their far but constant support and all the Italian friends
in Copenhagen for being very patient and encouraging me during the writing
period.
Lyngby, September 3, 2004
Andrea Donarini

Preface
Much interest has been drawn in recent years to the concept and realiza-
tionofNanoelectromechanicalsystems(NEMS).NEMSarenanoscaledevices
that combine mechanical and electrical dynamics in a strong interplay. The
shuttle devices are a particular kind of Nanoelectromechanical systems. The
characteristiccomponentthatgivesthenametothesedevicesisanoscillating
quantum dot of nanometer size that transfers electrons one-by-one between
a source and a drain lead. The device represents the nano-scale analog of an
electromechanical bell in which a metallic ball placed between the plates of
a capacitor starts to oscillate when a high voltage is applied to the plates.
This thesis contains the description and analysis of the dynamics of two
realizations of quantum shuttle devices. We describe the dynamics using the
Generalized Master Equation approach: a well-suited method to treat this
kind of open quantum systems. We also classify the operating modes in three
different regimes: the tunneling, the shuttling and the coexistence regime.
The characterization of these regimes is given in terms of three investigation
tools: Wigner distribution functions, current and current-noise. The essen-
tial dynamics of these regimes is captured by three simplified models whose
derivation from the full description is possible due to the time scale separa-
tion of the particular regime. We also obtain from these simplified models a
more intuitive picture of the variety of different dynamics exhibited by the
shuttle devices.
Lyngby, October 10, 2004
Andrea Donarini

Contents
1 Introduction 11
1.1 NEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 A new transport regime . . . . . . . . . . . . . . . . . . . . . 12
1.3 Experimental implementations . . . . . . . . . . . . . . . . . . 13
1.4 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 The models 19
2.1 Single-Dot Quantum Shuttle . . . . . . . . . . . . . . . . . . . 19
2.2 Triple-Dot Quantum Shuttle . . . . . . . . . . . . . . . . . . 22
3 Generalized Master Equation 25
3.1 Coherent dynamics of small open systems . . . . . . . . . . . 27
3.2 Quantum optical derivation . . . . . . . . . . . . . . . . . . . 27
3.2.1 Interaction picture . . . . . . . . . . . . . . . . . . . . 28
3.2.2 Initial conditions . . . . . . . . . . . . . . . . . . . . . 29
3.2.3 Reformulation of the equation of motion . . . . . . . . 29
3.2.4 Average over the bath variables . . . . . . . . . . . . . 30
3.2.5 Weak coupling . . . . . . . . . . . . . . . . . . . . . . 30
3.2.6 Markov approximation . . . . . . . . . . . . . . . . . . 30
3.3 Derivation “`a la Gurvitz” . . . . . . . . . . . . . . . . . . . . 31
3.3.1 Many-body basis expansion . . . . . . . . . . . . . . . 33
3.3.2 Recursive equation of motion for the coefficients . . . . 35
3.3.3 Injection and ejection rates . . . . . . . . . . . . . . . . 36
3.3.4 The reduced density matrix . . . . . . . . . . . . . . . 38
3.3.5 Generalized Master Equation . . . . . . . . . . . . . . 39
3.3.6 Spin and strong Coulomb blockade . . . . . . . . . . . 43
3.3.7 Coherencies and double-dot model . . . . . . . . . . . 47
3.4 GME for shuttle devices . . . . . . . . . . . . . . . . . . . . . 50
3.4.1 Single Dot Quantum Shuttle . . . . . . . . . . . . . . . 51
3.4.2 Triple Dot Quantum Shuttle . . . . . . . . . . . . . . . 61
3.5 The stationary solution: a numerical challenge . . . . . . . . . 64
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CONTENTS
3.5.1 A matter of matrix sizes . . . . . . . . . . . . . . . . . 64
3.5.2 The Arnoldi scheme . . . . . . . . . . . . . . . . . . . 66
3.5.3 Preconditioning . . . . . . . . . . . . . . . . . . . . . . 68
4 Wigner function distribution 71
4.1 A quantum phase-space distribution . . . . . . . . . . . . . . . 71
4.2 Klein-Kramers equations for the SDQS . . . . . . . . . . . . . 75
4.2.1 Coherent Liouvillean . . . . . . . . . . . . . . . . . . . 75
4.2.2 Driving Liouvillean . . . . . . . . . . . . . . . . . . . . 76
4.2.3 Damping Liouvillean . . . . . . . . . . . . . . . . . . . 78
5 Investigation tools 81
5.1 Phase space distributions: dots, rings and bananas . . . . . . 81
5.1.1 Shuttling instability . . . . . . . . . . . . . . . . . . . . 82
5.1.2 Classical vs. Quantum Shuttle . . . . . . . . . . . . . . 84
5.1.3 Different electronic processes in the TDQS . . . . . . . 85
5.2 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.1 Calculation of the stationary current . . . . . . . . . . 87
5.2.2 Current characteristics of the SDQS . . . . . . . . . . . 89
5.2.3 Electromechanical resonances of the TDQS . . . . . . . 90
5.3 Current noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.1 The MacDonald formula . . . . . . . . . . . . . . . . . 93
5.3.2 Current noise in the SDQS . . . . . . . . . . . . . . . . 95
5.3.3 The Fano factor . . . . . . . . . . . . . . . . . . . . . . 98
6 The three regimes 103
6.1 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Shuttling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 Simplified models 109
7.1 Renormalized resonant tunneling . . . . . . . . . . . . . . . . 109
7.1.1 Electrical rates . . . . . . . . . . . . . . . . . . . . . . 111
7.1.2 Phase-space distribution . . . . . . . . . . . . . . . . . 114
7.1.3 Current . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.1.4 Current-noise . . . . . . . . . . . . . . . . . . . . . . . 117
7.2 Shuttling: a classical transport regime . . . . . . . . . . . . . 119
7.2.1 Equation of motion for the relevant variables . . . . . . 120
7.2.2 Stable limit cycles . . . . . . . . . . . . . . . . . . . . . 122
7.3 Coexistence: a dichotomous process . . . . . . . . . . . . . . . 129
7.3.1 Dichotomous process between current modes . . . . . . 129
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CONTENTS
7.3.2 Effective potential . . . . . . . . . . . . . . . . . . . . 134
7.3.3 Switching Rates . . . . . . . . . . . . . . . . . . . . . . 146
7.3.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . 152
8 Conclusions 157
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.2 Some open questions . . . . . . . . . . . . . . . . . . . . . . . 160
Bibliography 163
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