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Thesis

Reference

Quantum transport in nano-scale conductors: entanglement entropy, waiting time distributions, and dynamical Coulomb blockade

THOMAS, Konrad Herrmann

Abstract

In this thesis, we study quantum transport in microscopic electronic conductors in which quantum effects are visible. This is an interesting field of research where much progress has been made in the last two decades, both experimentally and theoretically. We first focus on entanglement entropy as a way to quantify entanglement in condensed matter systems. After that, we discuss the distribution of waiting times between successive transmission events in a nano-scale conductor. This quantity has been suggested as a means to characterize the short-time physics of a stochastic transport process, complementary to the full counting statistics of transferred charge. Finally, we demonstrate that the coupling to environmental modes in scanning tunneling spectroscopy experiments can have a visible effect in the current characteristic of nano-scale systems.

THOMAS, Konrad Herrmann. Quantum transport in nano-scale conductors:

entanglement entropy, waiting time distributions, and dynamical Coulomb blockade. Thèse de doctorat : Univ. Genève, 2014, no. Sc. 4754

URN : urn:nbn:ch:unige-459460

DOI : 10.13097/archive-ouverte/unige:45946

Available at:

http://archive-ouverte.unige.ch/unige:45946

Disclaimer: layout of this document may differ from the published version.

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UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES

Section de physique Professeur E. V. Sukhorukov Département de physique théorique Docteur C. Flindt

Quantum Transport in Nano-Scale Conductors

Entanglement Entropy, Waiting Time Distributions, and Dynamical Coulomb Blockade

THÈSE

présentée à la faculté des sciences de l'Université de Genève pour obtenir le grade de Docteur ès sciences, mention Physique

par

Konrad Herrmann Thomas de

Ebersdorf (Allemagne)

Thèse N

4754 GENÈVE

Atelier de reproduction ReproMail

2015

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List of publications

(I) C. Brun, K. H. Müller (now K. H. Thomas), I-P. Hong, F. Patthey, C. Flindt, and W.-D. Schneider, Dynamical Coulomb blockade Ob- served in Nanosized Electrical Contacts, Phys. Rev. Lett. 108, 126802 (2012).

(II) K. H. Thomas and C. Flindt, Electron waiting times in non- Markovian quantum transport, Phys. Rev. B 87, 12405(R) (2013).

(III) K. H. Thomas and C. Flindt, Waiting time distributions of non- interacting fermions on a tight-binding chain, Phys. Rev. B 89, 245420 (2014).

(IV) K. H. Thomas and C. Flindt, Entanglement entropy in dynamic quantum-coherent conductors, arXiv:1411.3940, submitted to Phys.

Rev. B.

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Abstract

In this thesis, we study quantum transport in microscopic electronic conduc- tors in which quantum eects are visible. This is an interesting eld of research in which much progress has been made in the last two decades, both experimentally and theoretically. We rst focus on entanglement entropy as a way to quantify en- tanglement in condensed matter systems. After that, we discuss the distribution of waiting times between successive transmission events in a nano-scale conductor. This quantity has been suggested as a means to characterize the short-time physics of a stochastic transport process, complementary to the full counting statistics of trans- ferred charge. Finally, we demonstrate that the coupling to environmental modes in scanning tunneling spectroscopy experiments can have a visible eect in the current characteristic of nano-scale systems.

In the rst part we discuss the quantication of entanglement in many-body systems and explore the relation between the entanglement entropy and the charge transfer distribution for non-interacting systems. Specically, we investigate the time evolution of the entanglement entropy between two fermionic leads coupled by a quantum point contact in a non-interacting tight-binding model. We rst con- sider the quantum point contact as a switch and calculate the entanglement that is produced by modulating its transmission. We then tune it to half transmission and investigate the partitioning of noiseless excitations known as levitons on top of the Fermi sea, demonstrating that elementary entanglement processes can be realized in electronic systems. For both of these examples we discuss how well the entan- glement entropy can be reconstructed from the rst few cumulants of the charge transfer distribution.

In the next chapter we discuss waiting times between successive transmission events in a quantum conductor. We present a novel way to calculate the waiting time distribution for one-channel conductors based on a tight-binding approach that may form the basis for future investigations of interacting systems. We demonstrate the agreement of this method with existing approaches by comparing the waiting time distribution for dierent nano-structures. We then focus on transport through quantum dot systems and investigate the eect of the measurement process on the waiting time distribution. We nish the chapter by deriving the waiting time distri- bution for systems described by a non-Markovian master equation, and consider the example of a double quantum dot coupled to a thermal bath.

In the last part of the thesis we investigate experimental data obtained by scan-

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ning tunneling spectroscopy measurements on small metallic islands grown on top of dierent substrates. We explain the reduced dierential conductance at small volt- ages by the theory of dynamical Coulomb blockade, modeling the island-substrate contact as an ohmic resistor in parallel with a capacitor. This interpretation is supported by the area dependence of the contact capacitance and resistance, demon- strating the importance of the eect for scanning tunneling measurements of nano- structures.

This thesis sheds light on dierent aspects of quantum transport. On one hand, we show the potential of the tight-binding approach to model transport in many- body systems. On the other hand, we investigate electronic transport in systems coupled to a thermal bath, inducing both dissipation and non-Markovian dynamics.

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Résumé

Dans cette thèse, nous étudions le transport quantique dans les conducteurs élec- troniques microscopiques, dans lesquels des eets quantique sont visible. Il s'agit d'un champ de recherche intéressant, qui a progressé énormément dans les vingt dernières années grâce à de nombreuses innovations, au niveau expérimental ainsi qu'au niveau théorique. D'abord nous abordons le sujet de l'entropie d'intrication qui est un outil pour quantier l'intrication dans les systèmes de matière condensée.

Ensuite nous considérons la distribution de temps d'attente entre les éventements de transmission consécutifs. Cette quantité a été proposée an de caractériser la statis- tique d'un processus de transport stochastique sur une échelle temporelle courte, complémentaire à la distribution de charge transmise. Finalement, nous montrons que le couplage avec des modes environnementaux dans une expérience de spectro- scopie à eet tunnel peut avoir un eet visible sur la caractéristique de courant d'un système à l'échelle nano.

Dans la première partie, nous discutons de l'intrication dans les systèmes à plusieurs particules et nous explorons la relation entre l'entropie d'intrication et la statistique de charge transmise à l'absence d'interactions. En particulier, nous examinons l'évolution dans le temps de l'entropie d'intrication pour deux réservoirs de fermions liés par un point contact quantique, dans le cadre d'un modèle de liaison forte. Dans un premier temps, nous considérons le point contact quantique en tant qu'interrupteur et nous calculons l'intrication produite en modulant sa transmission.

Ensuite, nous l'ajustons en position moitié ouverte et nous examinons la partition d'excitations sans bruit, qui sont appelées levitons, à la surface de la mer de Fermi, montrant que des processus élémentaires d'intrication peuvent être réalisés dans un système électronique. Pour ces deux exemples, nous discutons à quel degré l'entropie d'intrication peut être reconstruite à partir des premiers cumulants de la distribution de charge transmise.

Dans le chapitre suivant, nous examinons les temps d'attente entre les évènements de transmission consécutifs dans les conducteurs quantiques. Nous présentons une nouvelle méthode pour calculer la distribution de temps d'attente pour les conduc- teurs avec un seul canal de conduction. La méthode est basée sur l'approche de liaison forte et elle pourrait servir comme point de départ pour les recherches futures des systèmes avec interactions. Nous montrons que cette méthode est en accord avec des approches existantes en comparant la distribution de temps d'attente pour plusieurs structures à l'échelle nano. Ensuite, nous nous concentrons sur le transport dans

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les systèmes de points quantiques et nous examinons l'eet de l'appareil de mesure sur la distribution de temps d'attente. Nous concluons ce chapitre en dérivant la distribution de temps d'attente pour les systèmes décrits par une équation maîtresse non-Markovienne, et nous considérons l'exemple d'un double point quantique couplé avec un bain thermique.

Dans la dernière partie de la thèse, nous examinons des données obtenues par des mesures de spectroscopie à eet tunnel sur des îles métalliques crues sur des substrats diérents. Nous expliquons la conductance diérentielle réduite aux tensions faibles par la théorie de blocage de Coulomb dynamique, en modelant l'interface entre le substrat et les îles par une résistance en parallèle avec une capacitance. Cette inter- prétation est soutenue par la dépendance des paramètres à la supercie de l'île, ce qui montre l'importance de cet eet pour des mesures à l'eet tunnel des structures à l'échelle nano.

Cette thèse vise à éclaircir diérents aspects du transport quantique. D'un coté, nous montrons le potentiel de la méthode de liaison forte dans la modélisation de transport dans les systèmes à plusieurs particules. De l'autre coté, nous examinons le transport dans les systèmes électroniques couplés à des bains thermiques, ce qui induit de la dissipation et une dynamique non-Markovienne.

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Kurzdarstellung

In dieser Arbeit wird Quantentransport in mikroskopischen elektrischen Leiter un- tersucht. Auf diesem Feld wurden in den letzten zwei Jahrzehnten grosse Fortschritte gemacht, sowohl im experimentellen Bereich als auch bei der theoretischen Beschrei- bung. Wir beschäftigen uns zuerst mit der Verschränkungsentropie als Mass der Verschränkung in Festkörpersystemen. Ausserdem diskutieren wir die Verteilung von Wartezeiten zwischen aufeinanderfolgenden Transmissionsereignissen in einem nano- skopischen Leiter. Diese Grösse wurde zur Charakterisierung der Kurzzeitstatistik stochastischer Transportprozesse vorgeschlagen, in Ergänzung zur Ladungstransfer- verteilung. Abschliessend zeigen wir, dass die Wechselwirkung mit Umgebungsmoden in Tunnelspektroskopie-Experimenten einen sichtbaren Einuss auf die Stromkennli- nie von Nanosystemen haben kann.

Im ersten Teil untersuchen wir Verschränkung in Vielteilchensystemen und er- forschen die Beziehung zwischen der Verschränkungsentropie und der Ladungstrans- ferstatistik in nichtwechselwirkenden Systemen. Wir berechnen die Zeitentwicklung der Verschränkungsentropie zwischen zwei elektronischen Leitern, die mit einem Quantenpunktkontakt verbunden sind, mit Hilfe des Models stark gebundener Teil- chen. Zuerst betrachten wir den Quantenpunktkontakt als einen Schalter, dessen zeitabhängige Durchlässigkeit Verschränkung erzeugt. Anschliessend stellen wir ihn auf halbdurchlässig und untersuchen die Partition von rauschfreien Anregungen, die Levitonen genannt werden, an der Oberäche des Fermi-Sees. Wir zeigen, dass auf diese Weise elementare Verschränkungsprozesse in einem elektronischen System re- alisiert werden können. Für beide Beispiele diskutieren wir die Rekonstruktion der Verschränkungsentropie durch die Kumulanten der Ladungstransferverteilung.

Im nächsten Kapitel untersuchen wir die Verteilung von Wartezeiten zwischen aufeinanderfolgenden Transmissionsereignissen in einem Quantensystem. Wir präsen- tieren eine neue, auf dem Ansatz stark gebundener Teilchen basierende Methode um die Wartezeitenverteilung für einen Leitungskanal zu berechnen, die die Grundlage für die Untersuchung von wechselwirkenden Systemen bilden könnte. Wir zeigen die Übereinstimmung unserer Methode mit bereits bekannten Ansätzen indem wir die Wartezeitenverteilung für verschiede Nanostrukturen vergleichen. Anschliessend wenden wir uns dem Transport in Quantenpunktsystemen zu und untersuchen den Einuss des Messprozesses auf die Wartezeitenverteilung. Am Ende des Kapitels bes- timmen wir die Wartezeitenverteilung für Systeme, die durch eine nicht-Markovsche Ratengleichung beschrieben werden, und veranschaulichen das Ergebnis am Beispiel

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eines Doppelquantenpunkts, der an ein thermisches Bad gekoppelt ist.

Im letzten Teil der Arbeit untersuchen wir experimentelle Daten, die mithilfe von Tunnelspektroskopie an kleinen metallischen Inseln auf verschiedenen Substraten gewonnen wurden. Wir erklären die verminderte elektrische Leitfähigkeit bei kleinen Spannungen mithilfe der Theorie der dynamischen Coulomb-Blockade, indem wir den elektrischen Kontakt zwischen der Insel und dem Substrat als einen ohmschen Widerstand parallel zu einem Kondensator modellieren. Dieses Model wird durch die Abhängigkeit der Parameter von der Inseläche bestätigt, was zeigt, dass dieser Eekt für Tunnelexperimente an Nanostrukturen von Bedeutung ist.

Diese Arbeit unterstreicht verschiedene Aspekte des Quantentransports. Einer- seits zeigen wir das Potential der Methode stark gebundener Teilchen für die Simu- lation von Transport in Vielteilchensystemen. Andererseits untersuchen wir Trans- port in elektronischen Systemen, die mit einem thermischen Bad wechselwirken, was sowohl zu Dissipation als auch zu nicht-Markovscher Dynamik führen kann.

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Contents

1 Introduction 1

2 Quantum transport in nano-structures 5

2.1 Tight-binding description . . . 6

2.1.1 Tight-binding Hamiltonian . . . 6

2.1.2 Full counting statistics . . . 8

2.1.3 Example: quantum point contact . . . 10

2.2 Master equation approach . . . 13

2.2.1 Full counting statistics . . . 13

2.2.2 Metallic island . . . 15

2.3 Dynamical Coulomb blockade . . . 18

2.3.1 Ohmic environment . . . 20

3 Entanglement entropy in dynamic quantum-coherent conductors 23 3.1 Entanglement entropy and full counting statistics . . . 25

3.1.1 Two particles . . . 27

3.1.2 Non-interacting particles . . . 29

3.2 Quantum switch . . . 30

3.3 Levitons . . . 35

4 Waiting time distributions in electronic transport 39 4.1 Waiting time distributions for stationary processes . . . 39

4.2 Tight-binding approach . . . 41

4.2.1 Results . . . 43

4.3 Master equation approach . . . 50

4.3.1 Detector models . . . 54

4.4 Non-Markovian systems . . . 60

4.4.1 Dissipative double quantum dot . . . 62

5 Dynamical Coulomb blockade in nano-sized electrical contacts 69 5.1 Experiment . . . 69

5.2 Interpretation . . . 71

6 Conclusion 75

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A Appendices for Chapter 3 77 A.1 Time evolution operator . . . 77 A.2 Zero-frequency noise . . . 77 A.3 Creation of levitons . . . 78

B Appendices for Chapter 4 81

B.1 Scattering approach to waiting time distributions . . . 81 B.2 Transmission amplitudes for tight-binding systems . . . 82 B.3 Transition rates in the master equation approach . . . 83

Bibliography 85

List of appreviations 100

List of symbols 102

Acknowledgements 104

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Chapter 1 Introduction

Quantum transport is a part of condensed matter physics in which quantum eects that occur in particle transport through electronic conductors are studied.

The eld has experienced tremendous progress in recent years due to advances in experimental, as well as sample fabrication techniques, and has yielded important results for quantum computing [1], spintronics [2], and thermoelectrics [3].

One important subeld of quantum transport is mesoscopic physics, which is concerned with the study of small conductors [4, 5, 6, 7]. The propagation of charge carriers in mesoscopic samples is coherent, so that it is described in terms of quantum mechanical waves rather than classical trajectories. To this end, the size L of the conductor has to be smaller than the phase coherence lengthlφ, which is the average length a carrier propagates before it suers a random phase change of the order of 2π,

Llφ. (1.1)

In high-mobility semiconductors at low temperatures, the phase coherence length can be as large as several micrometers [8]. If the condition Eq. (1.1) is fullled, eects that are due to the wave-like nature of charge carriers can be observed. Such eects include electron interference [9], the Aharonov-Bohm eect [10, 11], and persistent currents in small metal rings [12, 13, 14].

In experiments, samples are typically realized in two-dimensional electron gases [15]. These are conducting planes with high carrier mobility that form at the interface between semiconductors with dierent band gaps due to charge equilibration. Most widely used are AlGaAs/GaAs heterojunctions in which mobilities of up to30×106 cm2/Vs have been achieved [16]. The carrier density in such a system can be modied by applying gate electrodes, providing for the possibility to deplete parts of a sample.

Using this technique, narrow constrictions known as quantum point contacts (QPCs) can be realized [17, 18, 19]. These structures show conductance steps as a function of the applied gate voltage, and are used as electronic beam splitters.

In many experiments, the path carriers take through the sample is controlled by means of well dened conduction channels at the edge of the sample, that are provided by the quantum Hall eect [20, 21]. This regime is achieved by applying a

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suciently strong magnetic eld which causes the electrons to rearrange into Landau levels. The Landau levels are at in the bulk but bend upwards at the edges of the sample. If the Fermi energy is in-between two Landau levels, the bulk of the sample is insulating but electrons can propagate at the edges where the levels cross the Fermi energy. The energy of the levels increases with increasing magnetic eld strength, thus reducing the number of edge channels.

Experiments realized so far include electronic Mach-Zehnder interferometers in which the Aharonov-Bohm eect was detected [22, 23], as well as on-demand single- electron emitters with the aid of which electronic Hong-Ou-Mandel experiments were performed [24, 25]. Recently, noiseless excitations that were predicted by Levitov and co-workers [26, 27, 28], have been demonstrated, opening up the prospect of performing single-particle experiments in electronic systems [29, 30]. At the same time, fueled by the discovery of topological insulators [31, 32], the quest for the discovery of Majorana fermions in condensed matter physics is underway [33, 34].

Electronic transport in coherent conductors is subject to uctuations [35]. This is due primarily to the quantization of charge. Suppose a currentI ows through a given section of an electric circuit. The chargeQ=I∆t that is transferred through the section in the time interval ∆t is a multiple of the elementary charge, Q = ne.

Since this holds for any length of the time interval, the current through the section uctuates on the time scale e/I. In today's experiments currents below1×10−18 A can be measured, so that this time scale can be as long as one second [36].

Even when no average current is owing through a sample, uctuations occur due to thermal excitation [37]. This thermal or Johnson-Nyquist noise is proportional to the electronic temperature and the conductance of the sample, a manifestation of the uctuation dissipation theorem [38, 39]. Moreover, quantum uctuations persist even at zero temperature in systems at equilibrium [26, 27].

Another source of uctuations is shot noise or partition noise in biased conductors.

Consider a process in which particles that are incident on a barrier from one side, are either transmitted with probability D, or reected with probability 1−D. AfterN attempts, n¯ = DN charges will have been transmitted on average, while (1−D)N charge will have been reected. However, due to the probabilistic nature of the process, these quantities are only approached by averaging over many realizations of this process, while the number of transmitted charges in each realization will uctuate by∆n=p

N D(1−D) around the average value.

The probability P(n) to nd n transmitted charges after N attempts in this process is given by a binomial distribution. For a large number of attempts, and ifD is neither too close to 0 nor to 1, the charge distribution can be approximated as being Gaussian according to the central limit theorem. The Gaussian distribution has the property that all cumulants except the mean and the variance are zero, whereas in general this is not the case. Any transfer process involving non-interacting fermions is characterized by generalized binomial statistics, meaning that it can be decomposed into single-particle events of the type described above with dierent transmission probabilities [40, 41]. The study of the charge distribution function is known as

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full counting statistics (FCS), and has been studied extensively in the literature [26, 27, 42, 43]. The FCS of a system is characterized by its cumulants. In recent years, it has become experimentally feasible to go beyond the variance and measure higher cumulants [44, 45, 46, 47]. In quantum dot systems with low transmission rates it is even possible to monitor the occupation in real time [36, 48, 49, 50]. With this method the cumulants of the charge distribution up to the fteenth order have been detected [51].

The FCS is usually considered in the long time limit. In contrast, the distribution of waiting times between successive transmission events has been suggested as a means to characterize electronic transport on short time scales [52, 53, 54, 55]. In this work we develop a method to obtain waiting time distributions (WTDs) for tight- binding systems, laying the foundation for the treatment of interacting systems with density matrix renormalization group techniques [56, 57, 58, 59, 60]. Furthermore, we present dierent detector models and derive the WTD for non-Markovian master equations.

Another topic we address is the study of entanglement entropy in condensed mat- ter systems [61, 62, 63]. Despite immense theoretical interest, the measurement of this quantity remains challenging, since its denition does not directly refer to any physical observable. This has prompted a search for measurement schemes in quan- tum many-body systems [64, 65, 66, 67, 68]. One proposal relates the entanglement entropy between two reservoirs of non-interacting electrons to the FCS of trans- ferred charge [69, 70, 71, 72]. Since charge uctuations in nano-scale electronics are now being detected experimentally, these relations may provide a means to measure the entanglement entropy in mesoscopic conductors. In this work we build on this proposal and investigate the entanglement entropy for dynamic quantum-coherent conductors in a tight-binding description.

In the nal part of this thesis we discuss the eect of dynamical Coulomb blockade (DCB), by which single electrons tunneling through a barrier exchange energy with the electromagnetic environment of the circuit. These inelastic processes leave clear ngerprints in the currentvoltage characteristics of a device and contain detailed information about the impedance of the electrical circuit in which the tunnel barrier is embedded. The eect was discussed theoretically in seminal works [73, 74, 75]. Ex- perimentally, it was observed in carefully engineered systems with a high-impedance environment close to the tunnel barrier [76, 77, 78]. We consider scanning tunneling spectroscopy experiments in which DCB is used to explain the conductance spectra of small metallic islands grown on dierent substrates.

Content of this thesis

In this thesis we investigate quantities that characterize transport in nano-scale conductors. The material presented is taken in part from the publications that are listed on page iii. In Ch. 2 we start by giving an overview of the techniques that are

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used in this work. We introduce our tight-binding model (Sec. 2.1) as well as the master equation technique (Sec. 2.2) and explain how the FCS is obtained within these two approaches before discussing concrete examples. In Sec. 2.3, we present the theory of DCB for a tunnel junction in series with a frequency dependent impedance and show results for the ohmic case.

Chapter 3 is based on publication (IV). We focus on the entanglement entropy for two leads connected by a QPC, and investigate entanglement production through dynamic eects, as well as the relation between the entanglement entropy and the FCS. In Sec. 3.2 we consider the entanglement entropy produced by the opening and closing of the QPC for both zero and nite temperature, as well as with and without an applied bias. In Sec. 3.3 we then apply designed pulses to the leads in order to create noiseless excitations known as levitons on top of the Fermi sea, and we analyze the entanglement entropy created by interfering two levitons on a half-open QPC.

In Ch. 4 we investigate the WTD for spinless fermions in one-channel conductors.

In Sec. 4.2, which is based on publication (III), we introduce a tight-binding method to calculate the WTD and compare it to results obtained by scattering theory for several dierent scatterers. Next, we explain how to calculate the WTD within a master equation approach in Sec. 4.3 and present results for quantum dot systems.

We go on to investigate the eect of the measurement procedure on the WTD. Finally, we consider the WTD for non-Markonvian systems and treat the example of a double quantum dot (DQD) coupled to a bosonic bath in Sec. 4.4, based on publication (II).

In Ch. 5 we discuss experimental results obtained by scanning tunneling spec- troscopy on small metallic islands grown on dierent substrates, as presented in publication (I). The measurements we performed by Christophe Brun, I-Po Hong and François Patthey in Wolf-Dieter Schneider's group at EPFL Lausanne. We ex- plain the measured conductance spectra in terms of the DCB eect and analyze the resistance and capacitances of the islandsubstrate contact as a function of the island size.

We summarize our ndings and give an outlook in Ch. 6.

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Chapter 2

Quantum transport in nano-structures

We investigate electronic transport through nano-structures coupled to large reservoirs. The nano-structures we consider are typical building blocks of today's experiments like quantum point contacts (QPCs), as well as single and double quan- tum dots (DQDs). To describe these systems, we use two dierent techniques: a tight-binding and a master equation approach. With the help of these techniques we calculate two quantities that are becoming important in condensed matter systems:

the entanglement entropy [61, 62, 63] and the electronic waiting time distribution (WTD) [52, 54, 55]. In addition, we discuss scanning tunneling microscopy (STM) experiments in which the coupling to the electromagnetic environment has a notable impact, and we explain the data in terms of dynamical Coulomb blockade (DCB) [75].

The entanglement entropy quanties the degree of entanglement between dierent parts of a quantum system. For non-interacting fermions, it can be reconstructed from the cumulants of the full counting statistics (FCS). We investigate the time evolution of the entanglement entropy for a one-dimensional tight-binding chain in two dierent scenarios, and consider its relation to the FCS within that framework.

The WTD is the distribution of waiting times between two successive particle transmissions in a stochastic transport process. We develop a method to obtain it within a one-dimensional tight-binding description, and compare our results to scat- tering theory and master equation calculations. We also consider dierent detector models in order to investigate the eect of the measurement procedure on the WTD.

Finally, we derive the WTD for non-Markovian systems and discuss the WTD for a DQD coupled to a bosonic bath.

DCB describes the eect of the electromagnetic environment on a quantum cir- cuit. Starting from a description of the environment as a set of harmonic oscillators that can exchange energy with particles in the circuit, inelastic processes leave clear ngerprints in the voltage characteristics of the circuit. We use this eect to explain scanning tunneling spectroscopy experiments.

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In this Chapter we introduce the techniques that are at the foundation of this thesis. First we present our tight-binding approach and explain how we calculate the FCS for a nite system. As an example we consider two leads coupled by a QPC.

Subsequently, we introduce the master equation approach and discuss the example of Coulomb blockade for a metallic nano-island. Finally, we describe the theory of DCB for a single tunnel junction coupled to an electromagnetic environment.

2.1 Tight-binding description

The tight-binding method has been very successful in condensed matter physics to model various types of materials. A famous example is graphene, whose recent discovery was acknowledged with the Nobel prize in 2010 [79]. The band structure of graphene is well described by a non-interacting Hamiltonian with nearest-neighbor hopping [80].

Usually the tight-binding method is used to describe electrons in a solid that are localized at the site of an ion in the lattice [81]. They can jump to another lattice site with an amplitude that depends exponentially on the distance between the ions.

The most common approximation is to neglect any transitions that are not between nearest neighbors. Interactions can be treated within the tight-binding description as well, for example in the Hubbard model which is widely used in the eld of strongly correlated electrons [82].

Here we follow a dierent approach that is to discretize the one-dimensional Schrödinger equation for a system that is considered homogeneous on the atomic scale. This approach has been suggested by Schönhammer as a way to calculate the FCS in nite systems [83, 84, 85]. Since the discretization is independent of the atomic structure, the lattice spacinga is a free parameter in our model. This allows us, in principle, to recover the continuum description by taking the limita →0while increasing the numberM of sites such that the length the system, L=M a, remains constant.

2.1.1 Tight-binding Hamiltonian

We consider spinless non-interacting fermions in one dimension, governed by the single-particle Hamiltonian

Hˆ =− ~2 2me

d2

dx2 +V(x), (2.1)

whereV(x)is the electrostatic potential. In order to solve Eq. (2.1) for an arbitrary potential we discretize the problem on a lattice with spacing a. The single-particle wave functionψ(x) takes the value

ψl =ψ(x=xl) (2.2)

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on lattice site l with xl =la. Similarly, for the potential V(x) we dene

Vl =V(x=xl). (2.3)

The kinetic part of the Hamiltonian is approximated by [4],

− ~2 2me

d2

dx2ψ(x)|x=xl ' − ~2

2mea2l+1−2ψll1), (2.4) from which we can identify

¯t= ~2

2mea2, (2.5)

as the tunneling amplitude between neighboring sites. The discretization in Eq. (2.4) introduces the constant on-site energy 2¯t, which is absorbed into the potential by redening it as Vl+ 2¯t → Vl. We then arrive at a tight-binding Hamiltonian of the form

Hˆ =−¯tX

m

(|mihm+ 1|+h.c.) +X

m

Vm|mihm|, (2.6) having labeled the sites as{|mi}.

In this work, we treat systems that consist of two leads connected to a scatterer.

In the leads the potential is uniform and set to zero. The Hamiltonian of the leads then reads

0 = X

α=L,R

α, (2.7)

where

α =−t¯

MXα1 m=1

|m, αihm+ 1, α|+h.c., (2.8) describes the left (α = L) and right (α = R) lead containing Mα sites labeled as {|m, αi}. Hˆα can be diagonalized by choosing the standing wave-basis

|kαji =

r 2 Mα+ 1

Mα

X

m=1

sin(kαjm)|m, αi, (2.9) kαj = jπ

Mα+ 1, j = 1, . . . , Mα. The corresponding dispersion relation reads

ε(kαj) = −2¯tcos(kαj). (2.10) Knowing the eigenbasis ofHˆ0 we can prepare the leads in an arbitrary state.

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2.1.2 Full counting statistics

The FCS of transferred chargeP(n, t) gives the probability thatn particles have been transferred between a given section of a system and its complement at time t. It is characterized by the moments,

hnm(t)i=X

n

nmP(n, t) = ∂m

∂(iλ)mχ(λ, t)

λ=0, (2.11)

which are generated by the moment generating function (MGF), χ(λ, t) =X

n

P(n, t)eiλn. (2.12)

Alternatively, one may use the cumulants, which are generated by logχ(λ, t), Cm =hhnmii= ∂m

∂(iλ)m logχ(λ, t)

λ=0, (2.13)

to characterizeP(n, t).

In the original proposal by Levitov, Lesovik and Lee, it was suggested that the FCS could be detected by a spin located at the junction between two parts of a system [26, 27]. A charged particle passing the spin induces a precession by a certain angle so that the nal orientation of the spin compared with the initial one, together with a varying coupling strength between the spin and the particle current, would make it possible to measure the FCS.

Instead of detecting particle transitions, we consider a projective measurement to determine the particle number at a given time. We operate within the framework of the tight-binding model outlined above, starting out in an initial state in which the two leads are disconnected at timet0, so that the density matrix factorizes,

ˆ

ρ0 = ˆρL0 ⊗ρˆR0. (2.14) The probability that n particles have been transferred between the leads at a later time t is then given by

P(n, t) = D δ

n−h

Nˆ(t)−N0RiE

, (2.15)

whereN0R is the number of particles in the right lead at t0, the expectation value is taken in the many-body state andNˆ is the particle-number operator for the section under consideration. The corresponding MGF reads

χ(λ, t) = D

eiλ[ ˆN(t)N(tˆ 0)]E

. (2.16)

The probability to nd a given particle in the right lead at time t is obtained by the expectation value of the operatorPˆR(t) = ˆU(t) ˆPRUˆ(t) in the corresponding single-particle state. Here,PˆR is the projection operator on the right lead,

R=

MR

X

m=1

|m, Rihm, R|, (2.17)

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and U(t)ˆ is the time evolution operator given by Uˆ =Ttexp

−i

~ Z t

t0

dt0H(tˆ 0)

(2.18) for a general time-dependent HamiltonianH(t), where Tt stands for time-ordering.

We consider non-interacting particles for which the expectation value of a product of single-particle operators with respect to a Slater determinant can itself be written as a determinant [43]. The particle number operatorNˆ(t)in Eq. (2.12) is then given byPˆ(t)acting simultaneously on every single-particle state in the Slater determinant.

At zero temperature, only states with energy below the chemical potential are occupied. The particle number in the right leads is thus given by the projection of PˆR(t) onto the subspace of initially occupied states,

Mˆ(t) = ˆn0Pˆ(t)ˆn0. (2.19) Herenˆ0 is the Fermi operator in the initial state,

ˆ

n0 = ˆnL0 + ˆnR0 = X

α=L,R N0α

X

j=1

|kαjihkαj|, (2.20)

andN0α is the initial number of particles in leadα=L, R. Since nˆ0 as well as PˆR are projection operators, we can write eMˆ(t) = 1 + (e−1) ˆM(t) such that the MGF reads [83]

χ(λ, t) =eiλN0Rdeth

1 + (e−1) ˆM(t)i

, (2.21)

where the determinant is taken in a single particle basis.

At nite temperatures, every single-particle state is occupied with a nite prob- ability such that the Fermi operator becomes

ˆ

n0 = X

α=L,R Mα

X

j=1

|kαjihkαj| 1 + exp

β

ε(kαj)−µα

. (2.22)

Hereβ = (kT)−1 is the inverse temperature, |kαji and ε(kjα) are the eigenstates and eigenenergies of Hˆα, and the chemical potential µα in lead α is xed by requiring Tr[ˆnα0]=! N0α. It has been shown that the MGF then reads [41, 85]

χ(λ, t) =eiλMRdeth

1 + (e−1) ˆX(t)i

, (2.23)

whereMR is the number of sites in the right lead and X(t) = (1ˆ −ˆn0) +p

ˆ

n0R(t)p ˆ

n0. (2.24)

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QPC 1     2  …     …   …     MR tQPC

t t

ML ... ... ... 2 1

right lead left lead

Figure 2.1: Tight-binding model of two leads connected by a QPC with tunneling amplitude tQPC. Each lead consists of ML/R sites with nearest-neighbor tunneling amplitudet.¯

With the determinant formulas Eqs. (2.21) and (2.23), we can give an expression for the cumulants in terms of Mˆ and X, respectively. Using the matrix identityˆ det[A] = exp(Tr[logA]), the mean particle number follows as

C1(t) = hhn(t)ii=hn(t)i=

Tr[ ˆM(t)]−N0R, T = 0

Tr[ ˆX(t)]−MR, T >0 , (2.25) and for the higher cumulants we nd

Cm(t) = Xm

k=1

c(m)k Tr[ ˆAk(t)], m≥2, (2.26) whereAˆ= ˆM for T = 0, andAˆ= ˆX for T >0, respectively, and the coecients are given by

c(m)k = Xk

l=1

(−1)(l−1)

k−1 l−1

l(m−1). (2.27)

2.1.3 Example: quantum point contact

We now consider the example of a QPC connecting the leads. In our description the QPC is modeled as a weak link with tunneling amplitude tQPC (see Fig. 2.1). It is described by the Hamiltonian

QPC =−tQPC|1, Lih1, R|+h.c., (2.28) which links the rightmost site of the left lead to the leftmost site of the right lead.

In order to investigate particle transport, we prepare the leads in a many-body eigenstate of Hˆ0 at t0 = 0, which is a separated initial state of type Eq. (2.14). We occupy the lowestN0αeigenstates in leadα=L, Rat zero temperature, and establish an imbalance by choosing leads of equal length,ML=MR=M, and N0L ≥N0R. For t >0 we open the QPC and evolve the system with the Hamiltonian

Hˆ = ˆH0+ ˆHQPC, (2.29)

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20 60 80 100 120 160 t/τ0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

I(t)/(GQV)

t1 t2

a)

ML=MR=150, N0L =100, N0R =50

tQPC=1.t tQPC=0.t tQPC=0.25¯t

0 10 20 30 40 50

t/τ0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

C2(t)

b)

ML=MR=150, N0L =100, N0R =50, tQPCt

Fermi sea occupied Only transport window

Figure 2.2: a) Time-dependent current through a QPC opened att= 0, in units of voltage times the conductance quantum GQ = e2/h. Numerical results (solid lines) are shown for three dierent tunneling amplitudestQPC. The horizontal dashed lines show the expected stationary current based on scattering theory, while the vertical dashed lines mark the time frame[t1, t2] during which the current is quasistationary.

b) Second cumulantC2 =hhn2ii=hni2−hn2ifor a fully transmitting QPC calculated including the states in the Fermi sea (blue) and excluding them (green). The time unit isτ0 =~/¯t.

which leads to a particle ow from the left to the right lead. The time-dependent current is given by the time derivative of the rst cumulant,

I(t) = d

dthn(t)i= d

dtTr[ ˆM(t)] =iTrn ˆ

n0[ ˆH,Pˆ(t)]ˆn0

o=Tr[ ˆJ(t)], (2.30)

where the current operator reads

Jˆ=i[ ˆH,PˆR] =i tQPC|1, Rih1, L| −tQPC|1, Lih1, R|

. (2.31)

In Fig. 2.2a we show the time-dependent particle current for three dierent values of the QPC tunneling amplitude. After the connection is established the current uctuates before reaching a quasi-stationary value att1 ≈M/3τ0, whereτ0 =~/¯t is the time unit. It then stays constant until t2 ≈M τ0, when particles that have been reected at the end of the chain arrive back at the QPC. This is because the group velocity

vk= 1

~

∂εk

∂k = 2¯t

~ sin(k) (2.32)

is maximally2/τ0 atk =π/2, i.e. the fastest particles in the system propagate two sites per time unit, and can thus travel from the QPC to the end of the right lead and back in a time of M τ0.

We get identical results for the current if instead of occupyingN0α states in both leads from the bottom of the band, we occupy only states in the left lead that have

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energies in the transport window, ε(kjL)∈[µR, µL]. This shows that particles in the Fermi sea, i.e. with energy below µR, do not contribute to the average current [83].

However, this is not the case for higher cumulants. In Fig. 2.2b we plot the second cumulant C2 = hni2 − hn2i for a fully transmitting QPC with and without taking into account states in the Fermi sea. Both curves show a logarithmic behavior as expected at zero temperature [27], but they dier by an approximately constant value. This demonstrates that the Fermi sea makes a dierence when calculating higher cumulants at nite times.

We now compare our results to scattering theory. In App. B.2, we show that in our tight-binding model the transmission amplitude across the QPC for a wave with wavenumberk reads,

tk = 2itt¯QPCsink

|tQPC|2−¯t2e−2ik. (2.33) At full transmission,tQPC = ¯t, the transmission probability Tk=|tk|2 is unity for all k. We then expect the stationary current to be

hIi=GQV, (2.34)

where GQ = e2/h is the conductance quantum and V = µL−µR is the bias volt- age. This result is shown with a horizontal blue dashed line in Fig. 2.2a and agrees well with the quasi-stationary current obtained from our calculations. For non-unit transmission the stationary current reads

hIi=GQ

Z µL µR

dεT(ε), (2.35)

where T(ε) is the transmission probability as a function of energy. It is obtained from Eq. (2.33) using the dispersion relation Eq. (2.10),

T(ε) = θ2(4−(ε/t)¯2)

1 +θ2(2−(ε/¯t)2) +θ4, (2.36) where θ = tQPC/¯t. Equation (2.35) takes into account that the transmission prob- ability is energy dependent for tQPC < ¯t. Again we nd good agreement with our numerical results (see Fig. 2.2a).

Usually, a QPC is characterized by an energy-independent transparency D that determines the current, hIi = DGQV. By comparison with Eq. (2.35) we nd that Dis given by

D= 1 V

Z µL

µR

dεT(ε) (2.37)

in our model. Since the transmission probability Eq. (2.36) is smooth, for voltages that are small compared to the bandwidth,V 4¯t, D can be approximated by

D ' T

µLR

2

. (2.38)

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2.2 Master equation approach

Master equations have numerous applications describing stochastic processes in physics and beyond [86, 87]. In quantum transport in particular, the master equation approach has been very successful in modeling electron transport in low-transmission structures like quantum dots [88].

The typical setup consists of a small quantum system coupled to two particle reservoirs. The master equation is the equation of motion for the system's density matrix ρˆS which is obtained by tracing out the degrees of freedom of the reservoirs.

In the Markovian limit, the time scales considered are larger than the relaxation time of the reservoirs so that memory eects can be neglected. The master equation is then local in time and given by the Lindblad equation [86]

d

dtρˆS =Lρˆ=i[ ˆH,ρˆS] +X

k

Γk

ˆ

ckρˆSˆck− 1 2

ρˆSˆckk+ ˆckkρˆS

. (2.39)

The rst term on the right hand side describes the unitary evolution of the system, whereas the second term captures dissipative processes involving the reservoirs. The Liouville superoperatorLhas been introduced, andˆck,cˆkare the Lindblad operators.

If, on the other hand, memory eects cannot be neglected, one needs to go beyond the Markov approximation and take into account the history ofρˆS. This is the case considered in Sec. 4.4, where the coupling of an electronic system to a bosonic bath induces non-Markovian correlations.

2.2.1 Full counting statistics

To calculate the FCS, we resolve the density matrix with respect to the number n of particles that have been transferred from the system to one of the particle reservoirs during the time interval[0, t], [89, 90]. For negativen, more particles have come into the system from the reservoir than have gone out.

The n-resolved density matrix ρˆ(n)S obeys the master equation dρˆ(n)S (t)

dt = X n0=−∞

L(nn0)ρˆ(nS0)(t), (2.40) with the initial condition

ˆ

ρ(n)S (0) = ˆρstatS δn,0. (2.41) HereρˆstatS is the stationary solution to Eq. (2.39),

LρˆstatS = 0. (2.42)

Choosing the initial condition Eq. (2.41) amounts to assuming that the system, after having been prepared in some initial state in the past, has relaxed to the stationary state before counting starts at t = 0. L(n) describes all processes that transfer n

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particles from the system to the reservoir under consideration, and the full Liouvillian is given byL=P

nL(n), just as the full density matrix is recovered byρˆS =P

nρˆ(n)S . The FCS is obtained by tracing overρˆ(n)S ,

P(n, t) = Tr[ˆρ(n)S (t)]. (2.43) The MGF then reads

χ(λ, t) =X

n

P(n, t)eiλn =Trn X

n

ˆ

ρ(n)S (t)eiλno

≡Tr[ˆρS(λ, t)]. (2.44) Here we have introduced the transformed density matrix,

ˆ

ρS(λ, t) = X n=−∞

ˆ

ρ(n)S (t)eiλn. (2.45) It obeys the transformed master equation

dρˆS(λ, t)

dt = X

n,n0

L(nn0)ρˆ(nS0)eiλn n00=n−n= 0 X

n0,n00

L(n00)eiλn00ρˆ(nS0)eiλn0

≡ M(λ)ˆρS(λ, t), (2.46)

where the transformed Liouvillian reads M(λ) = X

n

L(n)eiλn. (2.47)

With ρˆS(λ,0) = ˆρstatS this equation is easily solved, ˆ

ρS(λ, t) = eM(λ)tρˆstatS , (2.48) such that the MGF becomes

χ(λ, t) = Tr

eM(λ)tρˆstatS

. (2.49)

From this result the mean particle number follows as hni= ∂

∂(iλ)χ(λ, t)

λ=0 =Tr

"

X

n

nL(n)

! ˆ ρstatS

#

t, (2.50)

where we have used that ˆ

ρS(λ= 0, t) = eM(0)tρˆstatS =eLtρˆstatS = ˆρstatS , (2.51) due to the denition of the stationary solution. We see thathni is linear in time so that the mean current hIi=hn˙i is constant as it should be in the stationary state.

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A common approximation is the sequential tunneling limit, which is valid if the coupling between the reservoirs and the system is so weak that processes with more than one particle tunneling at the same time can be neglected. The transformed Liouvillian then reads

M(λ) = L(0)+eL(1)+eL(1), (2.52) and for the current one has

hIi=Tr

L(1)− L(1) ˆ ρstatS

. (2.53)

2.2.2 Metallic island

We now discuss the example of sequential tunneling through a metallic island coupled to two leads, taken from Ref. [91]. Other studies of the problem can be found in Refs. [92] and [93]. A metallic island is coupled to a left (L) and right (R) lead by rates ΓαN→N±1, α=L, R, as shown in Fig. 2.3a. It is considered big enough that the average level spacing is negligible and the density of states is continuous.

Electrons on the island are conned in all three dimensions so that the Coulomb interaction between them cannot be neglected. A state with N electrons on the island has an energy of

EN = e2

2CN2− Q0e

C N ≡EC

N2− 2Q0

e N

. (2.54)

HereC is the sum of all capacitances between the island and its surroundings, and Q0 is a charge oset that can usually be controlled by applying a gate voltage.

In addition, we have dened the charging energy EC = e2/2C. For an additional electron to tunnel onto the island, the energy

EN+1−EN =EC

2N + 1− 2Q0

e

(2.55) has to be paid. This energy can be provided by the bias voltage V that is applied between the leads. However, if the island is occupied byN electrons andeV is smaller than the energy dierence Eq. (2.55), no current can ow through the system at zero temperature. This phenomenon is called Coulomb blockade.

To describe this behavior quantitatively, we consider a master equation approach.

For sequential tunneling, the master equation describes transitions in whichN changes by one,

˙

pN =−(ΓN→N+1+ ΓN→N−1)pN + ΓN−1→NpN−1+ ΓN+1→NpN+1. (2.56) HerepN is the probability to ndN electrons on the island. Changes inN can occur by processes involving each of the leads so that we have,

ΓN→N±1 = ΓLN→N±1+ ΓRN→N±1. (2.57)

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N Γ

LNN+1

Γ

LNN1

Γ

RNN1

Γ

RNN+1

V

C

L

, R

L

C

R

, R

R

a) b)

Figure 2.3: Model of a metallic island coupled to two leads. a) Particles tunnel between the island and lead α = L, R with the tunneling rate ΓαNN±1, changing the particle numberN on the island. b) Electric circuit model. The tunnel junction between leadα and the island consists of a capacitanceCα in parallel to the tunnel resistance Rα. The island is located between the junctions and the circuit is biased by a voltageV.

Equation (2.56) is a classical or Pauli master equation because the density matrix, ˆ

ρ= diag (. . . , pN1, pN, pN+1. . .), (2.58) is diagonal so that no quantum-mechanical superpositions occur. In fact, Eq. (2.56) describes a classical stochastic process, and the only quantum mechanical property involved is the fact that transitions between the island and the leads happen through tunneling processes. In contrast, master equations that include quantum-mechanical superpositions between states are also called generalized master equations.

We are interested in the stationary solution, p˙N = 0 for all N. In this case, the relation

ΓNN+1pN = ΓN+1NpN+1 (2.59) follows from Eq. (2.56), which means that the ow between probabilities is constant.

Together with the normalization Tr[ˆρ] =P

NpN = 1, the stationary solution of the problem is determined.

The leads are held at dierent chemical potentials µα such that a bias voltage V = (µL−µR)/e is applied across the island. In the sequential tunneling limit the transition rates are obtained from Fermi's golden rule,

ΓαNN±1 = 2π X

fN±1,iN

D

fN±1

TαiNE2WiNδ εfN±1 −εiN

. (2.60)

Here the sum goes over all initial and nal states iN and fN±1 with N and N ±1 particles on the island, respectively. WiN is the probability of stateiN to occur, and

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the delta function provides for energy conservation. The transfer HamiltonianHˆTα is of the standard form

Tα =X

kα

tkαDˆcνDˆckα +h.c., (2.61) where ˆcνD creates a particle in state νD on the island and ˆckα destroys a particle in state kα in lead α.

We assume energy-independent tunneling rates, a constant density of states in the leads and the island, as well as local thermal equilibrium in each of the leads and on the island. The transition rates then read

ΓαNN±1 = 1 e2Rα

f[EN±1−END−µα], (2.62) where

f(E) = E

eβE−1, (2.63)

and β = (kT)−1 is the inverse temperature. Here Rα is the tunnel resistance of junctionα =L, R.

The chemical potentialµD of the island is determined by the properties of the left and right tunnel junctions. To this end, tunneling is neglected, and the junctions are considered as capacitances CL and CR. The voltages VL and VR, dropping between the left lead and the island and between the island and the right lead, are then given by

VL = CR

CL+CR

V and VR = CL

CL+CR

V, (2.64)

such thatµDL−eVLR+eVR.

We can now calculate the transition rates as a function of the bias voltage V, given that the resistances and capacitances of the junctions are known. The current through the system is then given by the current through the left tunnel junction,

IR(V) = eX

N

ΓRN→N−1(V)−ΓRN→N+1(V)

pN, (2.65)

or alternatively by IL(V) = −IR(V) because of current conservation. Note that Eq. (2.65) has the form of Eq. (2.53).

The outlined approach can explain a rich class of phenomena such as Coulomb diamonds and the I-V characteristics of single-electron transistors [94]. In Ch. 5 we apply it to the conductance spectrum of a Pb island grown on a Si substrate covered by 3 monolayers of NaCl. The spectrum was obtained by scanning tunneling spectroscopy measurements with the STM tip and the substrate constituting the leads. The capacitances as well as the residual charge were inferred from steps and peaks in the conductance spectrum as explained in Ref. [95], while the tunnel resistances were given by the overall current amplitude.

The measurement was part of a broader investigation in which Pb islands were separated from the substrate by dierent barriers. For the other samples the tunnel

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V Z

ex

(ω)

C

T

, R

T

Figure 2.4: DCB circuit. A tunnel junction with capacitanceCT and resistance RT

in series with an impedance Z(ω) is biased by the dc voltage V.

resistance between the island and the substrate was of the order of RK = h/e2 ' 26 kΩ, such that the sequential tunneling approximation does not apply. Those spectra were modeled by the DCB eect instead, which we introduce in the following section.

2.3 Dynamical Coulomb blockade

In Sec. 2.2.2 we have considered a metallic island that shows current suppression at small voltages. This eect is called Coulomb blockade because it relies on the repulsion between conned electrons. A modied version of this eect occurs due to the interaction with the electromagnetic environment, even when the electrons are not conned. This phenomenon is called DCB and has been rst explored in Refs. [73, 74, 75]. Experimentally, it was observed in systems that were carefully engineered in order to obtain a high-impedance environment close to the tunnel barrier [76, 77, 78, 96, 97, 98, 99, 100]. The present discussion is based on Ref. [75].

We consider a circuit consisting of a voltage source in series with a tunnel junction and a frequency-dependent impedance Zex(ω) as shown in Fig. 2.4. Without the impedance, the I-V characteristic of the junction is ohmic. However, a non-zero impedance induces dissipation and causes a non-linearity in the current spectrum for small voltages. In quantum mechanics dissipation can be treated by introducing a thermal bath and tracing out its degrees of freedom [101, 102]. In our case the bath consists of a set of harmonic oscillators that constitute the electromagnetic environment of the tunnel junction.

The charge on the junction capacitor at a given time is Q = CTU, where U is the voltage that drops across the junction. It is dierent from V in general because tunneling events reduceQby an elementary charge and cause it to uctuate in time.

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To characterize these uctuations, the phase ϕ(t) = e

~ Z t

−∞

U(t0)dt0 (2.66)

is introduced. Q and ϕ are conjugate quantum mechanical operators analogous to position and momentum. This becomes clear from the time derivative

˙ ϕ= e

~ Q CT

, (2.67)

where the capacitance plays the role of the mass.

The bath consists of N harmonic oscillators with frequencies ωn =√ CnLn, Hˆbath = Q˜2

2CT

+ XN n=1

"

q2n 2Cn

+ ~

e 2

( ˜ϕ−ϕn)2 2Ln

#

. (2.68)

The phaseϕn of each oscillator couples to the phase uctuation of the junction,

˜

ϕ=ϕ−eV t/~, (2.69)

which implies that a tunnel event excites the environmental modes. Correspondingly, the charge uctuation of the junction is given by

Q˜ =Q−CTV. (2.70)

The quasiparticles in the left (L) and right (R) lead of the junction are described by

leads = ˆHL+ ˆHR=X

k,σ

+eV)ˆc+X

q,σ

εˆcˆc, (2.71) and the Hamiltonian connecting both leads reads

T = X

k,q,σ

tk,qeiϕ˜+h.c. (2.72) Here the operatoreiϕ˜ reduces the charge on the junction by one,

eiϕ˜Qe˜ −iϕ˜ = ˜Q−e, (2.73) which follows from the commutation relation

[ ˜ϕ,Q] =˜ ie. (2.74)

A Fermi's golden rule calculation leads to the tunneling rate ΓLR(V) = 1

e2RT Z

dEf(E)P(E+eV) (2.75)

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