vibrationally coupled electron transport through single-molecule junctions

VIBRATIONALLY COUPLED

ELECTRON TRANSPORT

THROUGHSINGLE-MOLECULE JUNCTIONS

(Schwingungsgekoppelter Elektronentransport

durch Einzelmolekulkontakte)

Der Naturwissenschaftlichen Fakultatder Friedrich-Alexander-Universitat Erlangen-Nurnberg

zurErlangung des Doktorgrades Dr. rer. nat.

vorgelegt vonDipl.-Phys. Rainer Hartleaus Nordlingen, Bayern

Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultat derUniversitat Erlangen-Nurnberg

Tag der mundlichen Prufung: 26. April 2012

Vorsitzender derPromotionskommission Prof. Dr. Rainer Fink

Erstberichterstatter Prof. Dr. Michael Thoss

Zweitberichterstatter Prof. Dr. Florian Marquardt

Do what you can, with what you have, where you are.

Theodore Roosevelt

Abstract

Single-molecule junctions are among the smallest electric circuits. They consist of amolecule that is bound to a left and a right electrode. With such a molecular nanocontact,the flow of electrical currents through a single molecule can be studied and controlled.Experiments on single-molecule junctions show that a single molecule carries electricalcurrents that can even be in the microampere regime. Thereby, a number of transportphenomena have been observed, such as, for example, diode- or transistor-like behav-ior, negative differential resistance and conductance switching. An objective of this field,which is commonly referred to as molecular electronics, is to relate these transport phe-nomena to the properties of the molecule in the contact. To this end, theoretical modelcalculations are employed, which facilitate an understanding of the underlying transportprocesses and mechanisms. Thereby, one has to take into account that molecules areflexible structures, which respond to a change of their charge state by a profound reor-ganization of their geometrical structure or may even dissociate. It is thus importantto understand the interrelation between the vibrational degrees of freedom of a single-molecule junction and the electrical current flowing through the contact.

In this thesis, we investigate vibrational effects in electron transport through single-molecule junctions. For these studies, we calculate and analyze transport characteristicsof both generic and first-principles based model systems of a molecular contact. To thisend, we employ a master equation and a nonequilibrium Green’s function approach. Bothmethods are suitable to describe this nonequilibrium transport problem and treat theinteractions of the tunneling electrons on the molecular bridge non-perturbatively. Thisis particularly important with respect to the vibrational degrees of freedom, which maystrongly interact with the tunneling electrons. We show in detail that the resulting vibra-tional effects have a profound influence on the transport characteristics of a single-moleculecontact and play therefore a fundamental role in this transport problem.

Our findings demonstrate that vibrationally coupled electron transport through a molec-ular junction involves two types of processes: i) transport processes, where an electrontunnels through the molecular bridge from one lead to the other, and ii) electron-holepair creation processes, where an electron tunnels from one of the leads onto the molec-ular bridge and back to the same lead again. Transport processes directly contributeto the electrical current flowing through a molecular contact and involve both excita-tion and deexcitation processes of the vibrational modes of the junction. Electron-holepair creation processes do not directly contribute to the electrical current and typicallyinvolve only deexcitation processes. Nevertheless, they constitute a cooling mechanismfor the vibrational modes of a single-molecule junction that is as important as coolingby transport processes. As the level of vibrational excitation determines the efficiencyof electron transport processes, they have an indirect influence on the electrical currentflowing through the junction. As we show, however, this influence can be substantial, inparticular, if the molecule is coupled asymmetrically to the leads. Accounting for all theseprocesses and their complex interrelationship, we analyze a number of intriguing trans-

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viii Abstract

port phenomena, including rectification, negative differential resistance, anomalous peakbroadening, mode-selective vibrational excitation and vibrationally induced decoherence.Moreover, we show that higher levels of vibrational excitation are obtained for weakerelectronic-vibrational coupling. Thus, based on physical grounds, we establish a relationbetween the weak electronic-vibrational coupling limit and the limit of large bias voltages,where the level of vibrational excitation in a molecular junction increases indefinitely.

Zusammenfassung

Einzelmolekulkontakte gehoren zu den kleinsten elektrischen Schaltungen. Sie beste-hen aus einem Molekul, das an zwei Elektroden gebunden ist. Mit Hilfe eines solchenmolekularen Nanokontakts kann der Fluss elektrischer Strome durch ein einzelnes Molekulstudiert und kontrolliert werden. Experimente an Einzelmolekulkontakten zeigen, dassein einzelnes Molekul elektrische Strome leitet, die sogar im Mikroamperebereich liegenkonnen. Hierbei wurde eine Vielzahl von Transportphanomenen nachgewiesen, die de-nen in einer Diode oder in einem Transistor ahnlich sind, aber auch negative differen-zielle Leitwerte und das Schalten zwischen verschiedenen Leitungszustanden umfassen.Eine zentrale Aufgabenstellung dieses Gebietes, das als molekulare Elektronik bezeichnetwird, ist es, diese Transportphanomene auf die Eigenschaften des kontaktierten Molekulszuruckzufuhren. Theoretische Modellrechnungen dienen dabei dem Verstandnis der rele-vanten Transportprozesse und -mechanismen. In diesen Rechnungen muss jedoch beruck-sichtigt werden, dass Molekule flexible Strukturen darstellen, die auf eine Anderung ihresLadunszustandes reagieren, indem sich ihre Geometrie andert oder indem sie dissoziieren.Deswegen ist es wichtig, das Wechselspiel zwischen den Schwingungsfreiheitsgraden einesEinzelmolekolkontakts und dem elektrischen Strom, der durch den Kontakt fließt, zu ver-stehen.

In dieser Doktorarbeit untersuchen wir die Rolle von Schwingungseffekten im Elektronen-transport duch Einzelmolekulkontakte. Dafur berechnen und analysieren wir die Trans-porteigenschaften von Einzelmolekulkontakten mit Hilfe von generischen Modellen sowievon Modellen, deren Parameter mit Hilfe von ”first-principles”-Methoden bestimmt wur-den. Zur Berechnung dieser Transporteigenschaften verwenden wir dabei einen Masterglei-chungs- und einen Nichtgleichgewichtsgreensfunktionsformalismus. Beide Methoden sinddazu geeignet Nichtgleichgewichtssysteme zu beschreiben und behandeln die Wechsel-wirkungen der tunnelnden Elektronen auf der molekularen Brucke nicht-storungstheo-retisch. Das ist besonders hinsichtlich der Wechselwirkung der tunnelnden Elektronenmit den Schwingungsfreiheitsgraden des molekularen Kontakts von Bedeutung. Wie wirim Detail aufzeigen, konnen die resultierenden Schwingungseffekte die Transporteigen-schaften eines Einzelmolekulkontakts stark beeinflussen und sind deshalb von grundle-gendem Interesse.

Unsere Ergebnisse zeigen, dass schwingungsgekoppelter Elektronentransport durch einenEinzelmolekulkontakt im Wesentlichen durch zwei Arten von Prozessen bestimmt ist:i) Transportprozesse, bei denen ein Elektron durch den molekularen Kontakt von einerElektrode zur anderen tunnelt, und ii) Elektron-Loch-Paarerzeugungsprozesse, bei de-nen ein Elektron von einer der Elektroden auf die molekulare Brucke tunnelt und wiederzu derselben Elektrode zuruckkehrt. Transportprozesse tragen direkt zum elektrischenStrom bei, der durch den molekularen Kontakt fließt. Dabei konnen die Schwingungsfrei-heitsgrade angeregt oder abgeregt werden. Elektron-Loch-Paarerzeugungsprozesse tragennicht direkt zum Elektronentransport durch einen molekularen Kontakt bei. Sie laufen inder Regel nur durch die Abregung von Schwingungsfreiheitsgraden ab. Dadurch leisten

x Zusammenfassung

sie einen wesentlichen Beitrag zur Kuhlung der Schwingungsfreiheitsgrade eines Einzel-molekulkontakts, der ebenso wichtig ist wie der Beitrag transportinduzierter Prozesse.Da die An- bzw. Abregung von Schwingungsfreiheitsgraden die Effizienz von Transport-prozessen beeinflußt, uben Elektron-Loch-Paarerzeugungsprozesse einen indirekten Ein-fluss auf den elektrischen Strom aus, der durch den Kontakt fließt. Wir zeigen, dassdieser Einfluss trotzdem substanziell sein kann, vor allem in Kontakten, in denen dasMolekul asymmetrisch an die Elektroden gebunden ist. Eine Vielzahl von Transporteigen-schaften und -phanomenen kann auf diese Prozesse und deren komplexes Wechselspielzuruckgefuhrt werden. Dazu gehoren zum Beispiel die Gleichrichtung von elektrischenStromen, negative elektrische Widerstande, anormal breite Resonanzen im differenziellenLeitwert eines Einzelmolekulkontakts, die Moglichkeit modenselektiver Anregungen undschwingungsinduzierte Dekoharenz. Daruber hinaus erklaren wir die physikalischen Grun-de dafur, dass eine starkere Schwingungsanregung mit einer schwacheren Kopplung zwi-schen elektronischen und Schwingungsfreiheitsgraden zusammenhangt. Damit kann einZusammenhang zwischen dem Grenzwert einer veschwindenden Elektron-Schwingungs-Kopplung und dem Grenzwert hoher Spannungen hergestellt werden, in dem der Anre-gungsgrad der Schwingungsmoden eines Einzelmolekulkontakts unbegrenzt anwachst.

Contents

Units, Definitions and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Introduction 3

2 Survey of Experimental Techniques 7

2.1 How to Contact a Single Molecule . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 From Spectroscopy to Function . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Theoretical Methodology 13

3.1 Hamiltonian of a Single-Molecule Junction . . . . . . . . . . . . . . . . . . 13

3.1.1 Modeling the Molecular Bridge . . . . . . . . . . . . . . . . . . . . 14

3.1.2 Coupling to Electrodes: Fermionic and Bosonic Baths . . . . . . . . 17

3.1.3 Diagonalization by the Small Polaron Transformation . . . . . . . . 20

3.2 Master Equation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Density Matrices and the Liouville-von Neumann Equation . . . . . 23

3.2.2 Nakajima-Zwanzig Equation . . . . . . . . . . . . . . . . . . . . . . 24

3.2.3 Markovian Master Equation / Redfield Theory . . . . . . . . . . . . 26

3.2.4 Explicit Representation of the Master Equation . . . . . . . . . . . 27

3.2.5 Observables of Interest . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.6 Vibration in Thermal Equilibrium . . . . . . . . . . . . . . . . . . . 33

3.3 Nonequilibrium Green’s Function Approach . . . . . . . . . . . . . . . . . 34

3.3.1 Single-Particle Green’s Function Gmn . . . . . . . . . . . . . . . . . 35

3.3.2 Dyson and Keldysh Equation . . . . . . . . . . . . . . . . . . . . . 38

3.3.3 Separation of Time Scales in the (Anti-)Adiabatic Regime . . . . . 40

3.3.4 Cumulant Expansion for the Vibrational Part of Gmn . . . . . . . . 41

3.3.5 Electronic Green’s Function Gel,mn and Self-Energy Σmn . . . . . . 42

xii Contents

3.3.6 Vibrational Green’s Function Dαα and Self-Energy Παα . . . . . . 45

3.3.7 Self-Consistent Solution Scheme . . . . . . . . . . . . . . . . . . . . 48

3.3.8 Observables of Interest . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Landauer Theory of Coherent Electron Transport . . . . . . . . . . . . . . 53

3.5 Survey of Other Theoretical Approaches . . . . . . . . . . . . . . . . . . . 55

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Processes and Mechanisms in Vibrationally Coupled Transport 57

4.1 Non-Resonant Transport Processes / Co-Tunneling . . . . . . . . . . . . . 58

4.1.1 Basic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1.2 Multimode Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Resonant Transport Processes / Sequential Tunneling . . . . . . . . . . . . 65

4.2.1 Basic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.2 Vibrational Effects in Transport through Multiple Electronic States 71

4.2.3 Multimode Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.4 Co-Tunneling Assisted Electron Tunneling (CoSET) . . . . . . . . . 79

4.3 Local Heating and Cooling in a Molecular Junction . . . . . . . . . . . . . 82

4.3.1 Role of Resonant Electron-Hole Pair Creation Processes . . . . . . . 83

4.3.2 Off-Resonant Electron-Hole Pair Creation Processes . . . . . . . . . 87

4.3.3 Cooling in the Presence of Multiple Electronic States . . . . . . . . 88

4.3.4 Coulomb Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.5 Pumping High-Frequency with Low-Frequency Modes . . . . . . . . 92

4.3.6 Coupling to a Thermal Bath: Vibrational Excitation due to PolaronFormation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 Vibrational Instabilities in the Resonant Transport Regime 97

5.1 Large Bias Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 Weak Electronic-Vibrational Coupling . . . . . . . . . . . . . . . . . . . . 100

5.2.1 Equivalence to the Limit of Large Bias Voltages . . . . . . . . . . . 100

5.2.2 Cooling by Resonant Electron-Hole Pair Creation Processes . . . . 103

5.2.3 Resonant Absorption Processes in Systems with Multiple ElectronicStates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Contents xiii

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 The Role of Electronic-Vibrational Coupling for the Electrical TransportProperties of a Molecular Junction and Possible Applications 107

6.1 Negative Differential Resistance . . . . . . . . . . . . . . . . . . . . . . . . 108

6.1.1 The Effect of Narrow Conduction Bands . . . . . . . . . . . . . . . 108

6.1.2 Suppression of Electrical Current due to an Increase in VibrationalExcitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.1.3 Breakdown of Electrical Current due to a Blocking State . . . . . . 110

6.1.4 Successive Decrease of Electrical Current in the Presence of a Lo-calized Vibrationally Coupled Electronic State . . . . . . . . . . . . 112

6.2 Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.1 The Importance of an Asymmetric Molecule-Lead Coupling . . . . 114

6.2.2 Rectification by Electronic-Vibrational Coupling . . . . . . . . . . 116

6.2.3 Spectroscopy of Molecular Levels: How Vibrational Side Peaks Dis-appear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3 Mode-Selective Vibrational Excitation . . . . . . . . . . . . . . . . . . . . 119

6.3.1 Molecular Bridge Asymmetrically Coupled to the Leads . . . . . . . 120

6.3.2 Molecular Bridge with Asymmetric Molecular Orbitals . . . . . . . 123

6.4 Anomalous Peak Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.5 Quantum Interference Effects in Single-Molecule Junctions . . . . . . . . . 129

6.5.1 Basic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.5.2 Decoherence due to Inelastic Sequential Tunneling . . . . . . . . . . 135

6.5.3 Decoherence due to Inelastic Co-Tunneling . . . . . . . . . . . . . . 139

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7 First-Principles Based Models of Single-Molecule Junctions 143

7.1 Vibrational Signatures in Benzenedibutanedithiolate Molecular Junctions . 145

7.2 Biphenylacetylene Molecular Junctions: Control of Quantum InterferenceEffects due to Anomalous Temperature Dependence . . . . . . . . . . . . . 150

7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Summary and Outlook 159

A Validity of the Elastic Co-Tunneling Approximation 165

xiv Contents

B The Structure of the Vibrational Self-Energy 169

C The Role of Coherences in a Master Equation Approach 173

Bibliography 177

List of Publications 199

Curriculum Vitae 201

Acknowledgment 201

Units, Definitions and Acronyms

In this work, we give energy values in units of electron-volts (eV), charges in Coulomb(C), electrical currents in Ampere (A) and voltages in Volt (V).

In all the equations and formulas we set ~ = 1 for notational convenience.

The charge of an electron is denoted by −e = −1.6022 · 10−19 C.

The Fermi level of all the systems that we investigate is set to F

= 0 eV.

We define the Fourier transformation of a function f(t) as f() =R∞−∞ dt f(t) eit, where

the respective inverse transformation is given by f(t) =R∞−∞

d2π

f() e−it.

In equations and formulas we use the following symbols:

[A, B] = AB −BA[A, B]

+

= AB + BA

In the text we abbreviate long or frequently used terms by the following acronyms:

BADT = biphenylacetylenedithiolateBDBT = benzenedibutanedithiolateEOM = equation of motionHOMO = highest occupied molecular orbitalHOMO-1 = second highest occupied molecular orbital (and analogously for HOMO-n)IETS = inelastic electron tunneling spectroscopyLUMO = lowest unoccupied molecular orbitalLUMO+1 = second lowest unoccupied molecular orbital (and analogously for LUMO+n)ME = master equationMCBJ = mechanically controllable break junctionNDR = negative differential resistanceNEGF = nonequilibrium Green’s functionPES = potential energy surfaceSTM = scanning tunneling microscope

2 Units, Definitions and Acronyms

Chapter 1

Introduction

Electrical currents are both fascinating and useful. Understanding and controlling elec-trical currents represents a key issue in many scientific disciplines. While a classicaldescription often suffices, the phenomenon of conductance quantization strikingly revealstheir quantum nature [1–3]. Conductance quantization is observed, for example, in elec-tron transport through atomic point contacts [4,5], where the electrical current is carriedby a single atom or a chain of atoms [6,7]. Based on the study of these atomic junctions,physicists and chemists started to investigate how electrical currents are flowing througha single molecule and if the diverse toolbox of molecular synthesis may be used to controlcurrents on a molecular scale. Research on such single-molecule junctions is motivatedby both fundamental and technological aspects and commonly subsumed under the termmolecular electronics [8–18].

Fig. 1.1 shows a typical example of a single-molecule junction [19–28]. It comprises asingle molecule that is covalently bound to a left and a right electrode. By applying apotential bias at the electrodes, which results in an electric field along the molecular bridge,electrons flow from one electrode to the other through the molecule. The respective charge

Figure 1.1: Single-molecule contact, where a Fe2+-bispyrazolylpyridine molecule iscovalently bound to two gold electrodes via thiolate linker groups (Figure courtesy ofS. Ballmann).

4 1. Introduction

transport processes typically result in a nonequilibrium state of the molecular bridge. Thisstate can be controlled not only by the polarity and the magnitude of the potential bias,but also by varying the position of the electrodes [29–35], a third terminal or gate electrode[36–38] or various other external control parameters [39–41]. Single-molecule contacts thusrepresent a unique testbed for investigating a molecule under controllable nonequilibriumconditions.

The electrical current flowing through a single-molecule junction is determined by boththe intrinsic properties of the molecular bridge and its contact to the electrodes. A varietyof electrical transport mechanisms have been theoretically proposed [42, 43] and experi-mentally verified [44–46]. This includes, for example, diode- or transistor-like behavior[47–51], negative differential resistance [52–57] or switching [58–64]. Single-moleculejunctions thus attracted the attention of many theorists and experimentalists, envisioningmolecules as the building blocks for electronic device applications (molecular electronics).Such devices would constitute an ultimate step in miniaturizing electronic circuits [65].

In contrast to macroscopic systems, however, two important aspects have to be consid-ered on the molecular scale: the wave nature of the electron [66–73] and the interactionof the electronic and the vibrational degrees of freedom of the molecular bridge [74–83].Although these aspects are also important in electron transport through other nanostruc-tures, such as, for example, quantum dots [84–86] or resonant tunneling diodes [87–89],they are particularly important in single-molecule junctions due to their small size andmass. Especially electronic-vibrational coupling and the respective current-induced vibra-tional excitation influences the transport characteristics of a molecular junction profoundly[36,90–99].

In this thesis, we investigate the role of electronic-vibrational coupling in electron trans-port through a single-molecule contact. To this end, we calculate and analyze transportcharacteristics of both generic and first-principles based models of a molecular junction.Thereby, the generic model systems allow us to elucidate specific aspects of vibrationallycoupled transport, while the first-principles based models represent the full complexity ofthis nonequilibrium transport problem, which typically comprises a few electronic statesand several tens of vibrational modes. The challenge is to account for the complex inter-play of all these electronic and vibrational degrees of freedom that may be strongly coupledwith each other. As we will show, even the simplest model of vibrationally coupled elec-tron transport through a single-molecule junction, which comprises a single vibrationalmode and a single electronic state, shows a rich spectrum of vibrational effects that havenot been considered in detail yet. Multimode vibrational effects and effects due to thecoupling of electrons in different molecular orbitals have, in addition, a profound influenceon the transport characteristics of a molecular junction. It should be noted that in vibra-tionally coupled electron transport, electron-electron interactions are not only the resultof repulsive Coulomb interactions but are mediated by the electronic-vibrational couplingand, thus, may be influenced by the level of vibrational excitation of the molecular bridge.Moreover, quasi-degeneracies among the electronic states and the vibrational modes arecharacteristic for a single-molecule contact and may lead to unexpected or nontrivial ef-

1. Introduction 5

fects. For example, we have found the electrical current flowing through a single-moleculecontact to be strongly enhanced in the presence of electronic-vibrational coupling. To un-derstand these intriguing effects and phenomena, we give a detailed description of all theprocesses and mechanisms that are relevant in vibrationally coupled electron transport,and outline how these mechanisms are interrelated. Moreover, we show how vibrationaleffects may be used to control the electrical transport properties of a molecular junctionand consider vibrational effects with respect to possible applications of single-moleculejunctions. For these investigations, we employ two complementary transport theories,which are based on master equations and nonequilibrium Green’s functions, respectively.Both methods allow to describe this quantum-mechanical nonequilibrium problem and toperform comprehensive model studies for a large variety of different molecular junctions.Thereby, the results obtained by the two approaches show good agreement, although thecoupling between the molecule and the leads is treated on different footings. Furtherinsight into the relevant transport mechanisms can thus be obtained whenever the twomethods show differences.

Throughout this thesis, we consider transport characteristics with respect to a DC biasvoltage. Other interesting aspects, which are not discussed in this thesis, include molec-ular junctions under the influence of time-periodic fields [100–102], optical phenomena[103–106] and the response of a molecular junction to short bias pulses [107]. The latteris particularly important in the context of conductance switching in molecular junctions[58–64]. Magnetic fields may also be used to control electron transport through a single-molecule junction. Such fields are important, for example, in the context of molecularAharonov-Bohm interferometers [108,109], single-molecule magnets [110–113] or in molec-ular junctions exhibiting Kondo correlations [114–116], but are also beyond the scope ofthis work. Throughout the thesis, we employ models of molecular junctions with unpo-larized metallic electrodes and consider no coupling between different spin states of thejunction. Spin-polarized leads [117–120] may be used, for example, in molecular junctionsoperated as spintronic devices [121,122], where effects induced by spin-orbit and spin-spininteractions play an important role [123,124].

The thesis is organized as follows: In Chap. 2, we give an overview of the experimentaltechniques that have been used to contact a single molecule and outline the quantitiesand the electrical transport properties characterizing a single-molecule contact. The theo-retical methodology that we use to calculate transport characteristics of a single-moleculecontact is outlined in Chap. 3. This includes a description of the underlying model Hamil-tonian and the two transport theories used. We also give an overview of other theoreticalapproaches and outline the relation between the nonequilibrium Green’s function approachand Landauer theory, which represents the exact result for a non-interacting nanoscaleconductor. Chap. 4 gives a comprehensive description of the basic processes and mech-anisms in vibrationally coupled electron transport. In Chap. 5, we complement theseinvestigations by studying specific limits of vibrationally coupled electron transport, suchas the large bias limit and the weak electronic-vibrational coupling limit. This facilitates,for example, an understanding why higher levels of vibrational excitation can be obtained

6 1. Introduction

for weaker electronic-vibrational coupling. The transport characteristics of a real molecu-lar junction, however, are typically determined by a manifold of these basic processes andmechanisms. We therefore elucidate in Chap. 6 how the interrelation between these pro-cesses and mechanisms may lead to nontrivial transport properties. In order to focus onthe relevant degrees of freedom, we employ generic model systems in the first chapters ofthis thesis. In Chap. 7, however, we apply the methodology to analyze transport charac-teristics of first-principles based models of specific single-molecule junctions, in particular,a benzenedibutanedithiolate and a biphenylacetylenedithiolate molecular junction. Bothof these junctions exhibit several of the effects and properties that are outlined in Chap.4 – 6 and are representative for many other molecular junctions. We conclude the thesiswith a summary and give an outlook to possible research projects that may be pursuedon the basis of this work.

Chapter 2

Survey of Experimental Techniques

2.1 How to Contact a Single Molecule

The first single-molecule junctions [19] were fabricated using the mechanically controllablebreak junction (MCBJ) technique [125–127]. This technique is schematically depicted inFig. 2.1. While these break junctions were built from notched gold wires clamped onflexible substrates, more advanced and reliable techniques are employed by now, wheresuch wires are structured by means of electron-beam lithography, evaporation and a dry-etching process [116, 128–130]. As the wire in a such a break junction is prepared free-standing, it can be broken upon bending the substrate. At the point where the wirebreaks, naturally only a few atoms are present (cf. Fig. 2.1b). Thus, breaking the wire,an atomic-sized contact can be established. Before the wire ultimately breaks, distinctsteps in the conductance of such an atomic point contact are observed [4,5]. These stepsare associated with the valence orbitals of the atoms in the respective material [6, 7],which provide conductance channels for electrons tunneling through the junction. As thenumber of atoms connecting both ends of the wire decreases one by one upon breakingthe wire, the associated conductance also decreases step by step. For a gold wire, thevery last step in such a conductance trace corresponds to the quantum of conductance,2e2/h ≈ (12.9 kΩ)−1 [1–3], indicating that at this point the current is carried by just asingle gold atom (or equivalently, a linear chain of gold atoms).

Such an atomic point contact can be used as the precursor of a single-molecule junction.Thereby, one uses the measurement of the conductance quantum to ensure that an atomic-sized contact has been established. After this control measurement, the contact is brokenand molecules are applied onto it from a dilute solution. Eventually, one of these moleculesbinds to the tips of the broken wire. Evaporating the solvent, and closing the breakjunction again, a single molecule may then bridge the gap between the right and the leftend of the broken wire (cf. Fig. 2.1c). That way, a single-molecule can be contacted andits conductance-voltage characteristic studied [19,22–24,34,38,49,99]. According to Ref.[133], the yield of this method is ≈20%.

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8 2. Survey of Experimental Techniques

(a) (b) (c)

Figure 2.1: Schematic representation of the MCBJ technique. Panel a: Experimentalsetup used to operate a MCBJ [129,131]. The substrate (yellow) holds a notched wire. Itcan be bent by moving the pushing rod upwards. Panel b: Atomic-sized point contact, asit appears before the wire breaks [132]. Only a few atoms bridge the gap between the twoends of the wire. Panel c: Single-molecule junction fabricated with the MCBJ technique[33]. The conductance of the molecule can be measured by applying a potential bias atthe broken wire. Thereby, varying the position of the pushing rod, the molecule can bestretched or compressed.

Electromigration constitutes another route to establish a single-molecule contact (cf. Fig.2.2). There, the initial setup is also a break junction. But instead of bending the substrate,a high current-density is used to break the wire. Thereby, gold atoms are driven out ofthe contact by momentum transfer such that a nanogap forms at the weakest point of thewire. Before this electromigration step, the wire is coated with a layer of molecules. Oneof these molecules may bridge the nanogap formed by electromigration. A single-moleculecontact is thus established more or less by chance. However, since several hundreds ofthese junctions can be fabricated and analyzed in parallel [133], it is very likely to findsingle-molecule contacts that give stable and reliable results [20,21,36,57,90,95,97,134].Another advantage of the electromigration technique is that it facilitates the design ofthree-terminal devices [20, 21, 36, 37, 57, 115], while for mechanically controllable breakjunctions, a gate electrode has been established only very recently [38,133].

A different strategy for bridging an electromigrated nanogap is based on click chemistry[135, 136]. Thereby, the molecular bridge is established by applying several chemicalreaction steps. Thereby, in a first reaction step, a molecular species is used that bindsto the material of the wire. Next, using click chemistry, another molecular species isattached to the species of the preceding reaction step. Thus, reaction step by reactionstep a molecular wire is ’clicked’ together, which, after a final step, bridges the nanogapon the shortest route. Each of these steps can be controlled and monitored by Ramanspectroscopy [135,136] (cf. Fig. 2.2b).

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configuration of electrodes?

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can you calculate current-density?how control the nanogap?

2.1. How to contact a single molecule 9

(a) (b)

Figure 2.2: Panel a: Scanning electron micrograph of a nanogap fabricated by electro-migration of a thin gold wire. Before the electromigration step, the gold wire is coated withferrocene-containing molecules. Eventually, one of these molecules bridges the electromi-grated nanogap (as shown in the inset) [137]. Panel b: Scheme of the click-chemistryapproach, which is also based on electromigration. Several chemical reaction cycles, bywhich molecules are attached species by species to the surface of the nanogap, are used tosynthesize a molecular bridge within the nanogap. Each of these cycles can be monitoredby Raman-spectroscopy [138].

Scanning Tunneling Microscopes (STMs) are also used to contact single molecules [29,31,41, 46, 47, 58, 62, 64, 96, 98, 139–144]. Fig. 2.3 provides a schematic representation of thistechnique. Thereby, one approaches molecules that are adsorbed on a substrate with thetip of a STM. Because molecules are much stronger coupled to the substrate than to thetip of the STM, this approach typically results in strongly asymmetric molecular junctionswith asymmetric current-voltage profiles. Thereby, the strong coupling of the molecule tothe substrate may prevent a clear separation between signatures of the molecule and thesubstrate. This, however, can be avoided with an insulating layer on top of the substrate,e.g. a NaCl layer on a copper substrate [64,96]. The STM technique is a very sensitive andfine-tunable tool to establish a single-molecule contact, in particular with respect to thepositioning of the ’tip’-electrode. The STM-tip may also be used to manipulate a molecule(cf. Fig. 2.3). It allows, for example, to study the conductance-voltage characteristic ofa molecule under mechanical stress1, by pealing a molecule off the substrate [29, 31, 32]or inducing step-wise dehydrogenation of the adsorbed molecule [41]. Very robust STMtechniques [27,145] allow to establish hundreds of single-molecule contacts per minute (atroom temperature). However, with a STM only a single molecular contact can be studiedat a time. In contrast to electromigration, this method may thus not be suitable for thelarge-scale production of single-molecule junctions. Similar to the fixed tip of a STM,conducting-probe Atomic Force Microscopes (AFM) [39, 58, 146] are also being used toestablish a single-molecule contact.

1 Note that in a MCBJ molecules can also be stretched or compressed [34]

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Figure 2.3: Graphical representation of a STMmolecular junction. In this specific experiment, the tip(blue atoms) is used to peal off a polymer from a goldsurface (red atoms) [31,147]. During the procedure theconductance of the polymer can be continuously mon-itored. Similar experiments have been performed, forexample, with perylene molecules adsorbed on a silversurface [29,32].

Other approaches to contact single molecules are based on crossed wires [148], mercurydroplets [149], or gold nanoparticles [150]. These techniques, however, which employself-assembled monolayers of molecules, are more likely to result in few-molecule contactsrather than in single-molecule contacts. Thereby, effects due to the neighboring moleculesmay significantly alter the respective conductance-voltage characteristics [151]. To date,single-molecule contacts have been established with a variety of molecules, ranging fromthe simplest hydrogen molecule [26,92,152], to small and intermediate organic molecules[19, 23, 31, 49, 54, 153], metal-coordinated complexes [21, 34, 57], Carbon-60 buckyballs[20,143] or macromolecules like carbon nanotubes [134,154].

2.2 From Spectroscopy to Function

The central characteristic of a single-molecule junction is its current-voltage characteristic,where the corresponding conductance-voltage characteristic, or even higher derivatives ofthe current [155,156], greatly facilitates the analysis of this observable. An example for aconductance-voltage characteristic of a molecular junction, in which a Fe2+-bispyrazolyl-pyridine molecule is clamped between gold electrodes, is shown in Fig. 2.4 [34]. Theconductance properties of such a junction are mainly determined by two factors: i) theway the molecular bridge is coupled to the leads and ii) the level structure of the molecularbridge, including both electronic and vibrational degrees of freedom. Single-moleculecontacts can therefore be used to obtain spectroscopic information about a molecularconductor.

At low bias voltages (Φ . 0.5V ), for example, the vibrational spectrum of a molecularjunction can be probed employing inelastic electron tunneling spectroscopy (IETS) [58,90,98,128,157–162]. Thereby, propensity rules [76, 163,164] determine the active part ofthe vibrational spectrum. With the fine-tunable tip of a STM, vibrational signatures ofa single molecule can be resolved even spatially [142,165], as the tunnel current may flowthrough a specific part of the molecule only. IETS is usually employed in the off-resonanttransport regime, where the molecular bridge represents a tunnel barrier for electrons.Spectroscopic information, complementary to this, can also be extracted in the resonanttransport regime [34,36,37,57,96,99]. In this regime, a molecular junction behaves more

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2.2. From Spectroscopy to Function 11

-80 -40 0 40 800

25

505.2 mV

bias voltage [mV]!

8.3 mV

conduct

ance

g [nA

/V]

Figure 2.4: Conductance-voltage characteristic recorded for the Fe2+-bispyrazolylpyridine molecular junction depicted in Fig. 1.1, using the MCBJ technique[34, 116]. The broad peaks at Φ w 45 mV and at Φ w −70 mV, respectively, indicate theonset of the resonant transport regime.

like a double barrier, where resonances, i.e. quasi-bound states of the molecular bridge,represent pathways for electrons tunneling through the junction. This allows to probe, inaddition, the electronic level structure of the molecular bridge. Thereby, a gate electrodegreatly facilitates the analysis, as it allows to vary the position of the electronic energylevels with respect to the Fermi level of the junction [13,36–38,57,133]. Using the spatialresolution of a STM operated in the resonant transport regime, one can even visualizethe electronic structure of specific molecular orbitals [96]. Another route to assess atleast the HOMO-LUMO gap of a molecular conductor is transition-voltage spectroscopy[39,148,166–168], which probes the cross-over between the non-resonant and the resonanttransport regime.

While the current through a single-molecule junction indicates the number of availabletransport channels, the corresponding current-current correlation function contains addi-tional information about the underlying transport mechanisms. This observable comprisesthe sum of the thermal and the shot noise, and can be measured, for example, employinglock-in techniques [56,152,153,169]. If the shot noise S of a molecular junction is as largeas the current, that is if S = 2eI, electrons tunnel through the junction in statisticallyuncorrelated events. A lower noise level, S < 2eI, indicates that electrons hinder eachother from tunneling through the molecular bridge, for example due to Pauli-blocking. Ahigher noise level, S > 2eI, occurs, if a tunneling electron opens the doorway for severalother electrons to tunnel through the junction. The latter is also referred to as avalanche-like transport [156, 170]. Recent experimental work accessed this regime [34], indicatingextremely high shot-noise levels, S 2eI, at the onset of resonant transport through amolecular junction. In addition, the shot-noise signal allows to unambiguously identifythe conductance channels of a molecular junction [152].


Another important observable is the level of vibrational excitation of a molecular junc-tion. Recent advances in Raman spectroscopy (see Fig. 2.2b) have allowed to measurethis observable [171–173]. It can be recorded in parallel with the corresponding current-voltage characteristic. This allows to monitor electron transport and vibrational excita-tion/deexcitation processes in a molecular junction simultaneously. Since electron trans-port involves not only charge exchange processes, but also energy exchange processeswith the leads, the parallel measurement of current and vibrational excitation is crucialto characterize the corresponding transport mechanisms. Besides Raman spectroscopy,the force required to break a single-molecule contact has also been used to determine thelevel of vibrational excitation, or equivalently, the effective temperature of the vibrationalmodes in a biased molecular junction [28].

In a single-molecule junction, molecules can be studied under controllable nonequilibriumconditions. While this on its own motivates a fundamental research interest, anotherdriving force in this field originates from the vision to use single molecules as the build-ing blocks for molecular electronic devices [174–178]. To this end, classes of molecularjunctions need to be identified that can be used for basic electronic operations. For ex-ample, molecules exhibiting rectification or diode-like behavior have already been found[45, 47, 49, 50], and consequently, the principle of a single-molecule transistor has beendemonstrated [48, 51]. In addition, negative differential resistance has been observed fora number of single-molecule junctions [52–58]. This transport property is relevant, forexample, for the design of analog-digital converters [179], high-frequency oscillators [180]and logic circuits [181,182]. Another intriguing aspect with respect to molecular electronicdevices is the possibility to switch the conductance of a single-molecule contact betweenan ”on-” and an ”off”-state [13, 17]. Such single-molecule switches can be operated ei-ther mechanically [30], optically (in a network of nanoparticles interconnected by singlemolecules) [40], or by applying short bias pulses [37,58–64].

Chapter 3

Theoretical Methodology

In this chapter, we outline the theoretical methodology used to describe electron transportthrough a single-molecule junction. The basis for this description is the Hamiltonianthat we introduce in Sec. 3.1. Thereby, we account for the molecular character of theproblem and define the single-particle states and the vibrational modes with respect to aspecific reference state. In particular, we use the neutral ground state of the molecularbridge. While this is necessary for the description of a molecular system, where anionic orcationic states of the isolated molecule may not even exist, it is a less decisive point for thedescription of larger nanostructures, such as, for example, quantum dots [84–86,183–185].

The transport theories, which we use to calculate the transport characteristics of a molec-ular contact, are presented in Secs. 3.2 and 3.3, where we employ a master equation (ME)and a nonequilibrium Green’s function (NEGF) approach, respectively. These methodsallow an efficient description of many-body systems that may be in a nonequilibrium state,such as it is the case in vibrationally coupled charge transport through a single-moleculejunction. The relation between the nonequilibrium Green’s function method and the Lan-dauer theory for a non-interacting conductor is established in Sec. 3.4. Sec. 3.5 concludeswith a survey of other theoretical approaches.

3.1 Hamiltonian of a Single-Molecule Junction

In this section, we derive the model Hamiltonian of a single-molecule contact in two steps.First, we motivate and outline a description of the molecular bridge in Sec. 3.1.1 using thesecond quantization formalism to represent the Hamiltonian. We extend this descriptionby the electrodes in Sec. 3.1.2 and discuss the coupling of the molecular bridge to boththe electrodes and other environmental degrees of freedom. In Sec. 3.1.3, we show howthe model Hamiltonian can be prediagonalized by a small polaron transformation (Sec.3.1.3). This is a necessary precursor for the theoretical approaches, which will be outlinedin the following sections, Secs. 3.2 and 3.3.

14 3. Theoretical Methodology

3.1.1 Modeling the Molecular Bridge

To describe the electronic states and the vibrational modes located on the molecularbridge (M) of a single-molecule contact, we follow the scheme outlined in Refs. [186–192].The Hamiltonian H

M

of the (isolated) molecular bridge is given by

HM

= Hnuc

+ Hel

, (3.1)

Hnuc

= −X

a

1

2Ma

∆a + Vnuc

(R), (3.2)

Vnuc

(R) =1

4π0

X

a<b

ZaZbe2

|Ra −Rb| , (3.3)

Hel

(R) = −X

i

1

2me

∆i +1

4π0

X

i<j

e2

|ri − rj| −1

4π0

X

ai

Zae2

|Ra − ri| , (3.4)

where Hnuc

describes the nuclear and Hel

(R) the electronic degrees of freedom of themolecule. The nuclear part of the Hamiltonian, H

nuc

, includes the kinetic energy of thenuclei and the Coulomb repulsion between them, V

nuc

(R). Similarly, Hel

(R) represents thekinetic energy of the electrons, the Coulomb repulsion between them, and the Coulombattraction between the electrons and the nuclei. Thereby, the vector R summarizes theposition vectors Ra of the nuclei and ri denotes the position vector of the ith electron.The charge and the mass of the nuclei are given by eZa and Ma, respectively, while themass of an electron is denoted by me. The dielectric constant of the vacuum is given by0

.

In the following, we derive an approximate expression for this Hamiltonian in secondquantization. To this end, we consider the electronic ground state of the unchargedmolecule as a reference state and determine the level structure of the molecular bridge,the frequencies of the vibrational modes and the corresponding coupling strengths withrespect to this state. The reference state represents a solution of the electronic Schrodingerequation

Hel

(R)Ψref

(r;R) = Eref,el

(R)Ψref

(r;R) (3.5)

and depends only parametrically on the nuclear coordinates R. This is indicated bythe semicolon in Ψ

ref

(r;R). Thereby, the vector r represents the position vectors of theelectrons ri. Typically, an exact solution of this many-body Schrodinger equation cannotbe obtained, and approximate schemes have to be used. Employing, for example, theHartree-Fock approximation, the state Ψ

ref

(r;R) is given by a single Slater determinant

|Ψref

(R) =Y

m∈occ.

c†m(R)|0. (3.6)

This Slater determinant involves single-particle states or molecular orbitals ψm(ri;R),which are occupied (m ∈ occ.) in the reference state Ψ

ref

(r;R). Using the creation

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3.1. Hamiltonian of a Single-Molecule Junction 15

operators c†m(R) and the corresponding destruction operators cm(R) for these single-particle states, the electronic part of the Hamiltonian can be rewritten in the occupationnumber representation [186,187]

Hel

(R) =X

m

m(R)c†m(R)cm(R) (3.7)

+1

2

X

mnop

Umnop(R)c†m(R)c†n(R)cp(R)co(R)

−X

mn

X

o∈occ.

(Umono(R)− Umoon(R)) c†m(R)cn(R)

with orbital energies m(R) and electron-electron interaction matrix elements

Umnop(R) =1

4π0

Zd3r

1

Zd3r

2

e2

|r1

− r2

|ψ∗m(r

1

;R)ψ∗n(r

2

;R)ψo(r1

;R)ψp(r2

;R).

(3.8)

Note that similar expressions for the Hamiltonian Hel

(R) can be derived employing othereffective single-particle theories, such as, for example, density functional theory.

In electron transport through a single-molecule junction, electrons continuously tunnelon and off the molecular bridge. The molecular bridge may therefore be in a number ofdifferent charge states. For each of these states, we could define a new set of single-particlestates or molecular orbitals, including the effect of electron-electron interactions Umnop(R).We adopt, however, a simpler approach to describe these interactions and approximate,similar to a Hubbard model, the electron-electron interactions matrix elements by

Umnop(R) ≈ Umnop(R)(δmoδnp − δmpδno), (3.9)

and define Umn(R) = Umnmn(R) − Umnnm(R). Within this approximation, we describethe effect of electron-electron interactions by an additional charging energy Umn(R) of themolecular bridge, if it is doubly charged such as, for example, in a dianionic (m,n /∈ occ.)or a dicationic state (m,n ∈ occ.). Moreover, we account for a different strengthof electron-electron interactions, if the molecular bridge is in a neutral excited state(m ∈ occ., n /∈ occ.). But we do not consider a direct coupling of the single-particlestates, ∼ c†m(R)c†n(R)cp(R)co(R) or c†m(R)co(R) (m, n = o, p), due to electron-electroninteractions in the different charge states of the molecular junction. We thus consider theelectronic Hamiltonian

Hel

(R) = Eref,el

(R) +X

m

m(R)(c†m(R)cm(R)− δm)

+X

m<n

Umn(R)(c†m(R)cm(R)− δm)(c†n(R)cn(R)− δn), (3.10)

where the parameters δm are defined as

δm =

Ω1, m ∈ occ.0, m /∈ occ. . (3.11)


In the reference state Ψref

(r,R), the motion of the nuclei is determined by the adiabaticpotential energy surface (PES)

Vad

(R) = Vnuc

(R) + Eref,el

(R). (3.12)

To characterize the nuclear or vibrational degrees of freedom of the molecular bridge, weapproximate the motion of the nuclei as harmonic. We therefore replace

Hnuc

+ Eref,el

(R) ≈X

α

Ωα

µa†αaα +

1

2

∂+ E

ref,el

(Req

) (3.13)

in the Hamiltonian describing the molecular bridge and obtain

HM

=X

α

Ωα

µa†αaα +

1

2

∂+

X

m

m(R)(c†m(R)cm(R)− δm) (3.14)

+X

m<n

Umn(R)(c†m(R)cm(R)− δm)(c†n(R)cn(R)− δn).

Thereby, we employ the normal modes α of the junction. The frequency of the har-monic mode α is denoted by Ωα, and a†α/aα are the respective ladder operators. Thevector R

eq

represents the equilibrium position of the nuclei in the reference state. Ac-cordingly, we define the vibrational displacement operator by Qα = a†α + aα and thecorresponding momentum operator by Pα = −i

°aα − a†α

¢. These are the dimension-

less analogs of the vibrational position xα =q

1

2mα

Ω

α

°aα + a†α

¢and momentum operator

pα = −iq

mα

Ω

α

2

°aα − a†α

¢.

In line with the harmonic approximation for the PES, we expand the orbital and correla-tion energies, m(R) and Umn(R), to first order in the vibrational coordinates Qα:

m(R) ≈ m(Req

) +X

α

Qα∂m(R

eq

)

∂Qα

≡ m +X

α

Qαλmα, (3.15)

Umn(R) ≈ Umn(Req

) +X

α

Qα∂Umn(R

eq

)

∂Qα

≡ Umn +X

α

QαWmnα, (3.16)

where λmα and Wmnα denote coupling strengths between the vibrational and the electronicdegrees of freedom of the molecular bridge. Moreover, we drop the parametric dependenceof the single-particle operators cm(R) on the nuclear coordinates and define cm(Req) ≡ cm.Neglecting this dependence, we discard, in particular, off-diagonal non-adiabatic couplingterms, ∼ c†mcn and ∼ c†mcnc†ocp with m = n, p. These terms are relevant if level crossingsbecome significant [96] (for example in the vicinity of conical intersections [193]). Droppingan irrelevant constant, we thus summarize the expansion of H

nuc

+ Hel

(R) in Qα as

HM

=X

α

Ωαa†αaα +X

m

m(c†mcm − δm) +X

m<n

Umn(c†mcm − δm)(c†ncn − δn) (3.17)

+X

mα

λmαQα(c†mcm − δm) +X

m<n,α

WmnαQα(c†mcm − δm)(c†ncn − δn).


Note that the fixed parameters δm reflect the fact that, by definition, there is no couplingbetween the normal modes and the molecular orbitals of the reference state Ψ

ref

(r,R).In the following, we do not consider vibrationally dependent electron-electron interac-tions (Wmnα = 0). In contrast to the electronic-vibrational coupling constants λmα, theseinteractions originate from the R dependence of the cm(R) only, and thus, representanother class of non-adiabatic interactions. They have been examined, for example, inthe bachelor thesis of Erpenbeck [194], where it was shown that such vibrationally de-pendent electron-electron interactions can be described by effective population-dependentelectronic-vibrational coupling constants λmα = λmα +

Pn=m Wmnαc†ncn. They are thus

not only relevant for electronic states that are close in energy but also if the electroniclevels are well separated in energy.

The effect of electronic-vibrational coupling can be demonstrated by the change of thePES of a molecular bridge due to charging or decharging. Examples for a PES along asingle vibrational coordinate are depicted in Fig. 3.1. Therein, the black line representsthe PES for the molecular bridge in its reference state, that is, the neutral molecule in itsground state (M0). Panel a in Fig. 3.1 depicts the transition from the neutral molecule tothe anion, where the green line represents the PES of the anion (M−). Panel b representsa transition from the neutral molecule to the cation, where the red line shows the PESof the cation (M+). Charging (decharging) the molecular bridge, the corresponding PESis shifted to higher (lower) energies. Due to electronic-vibrational coupling, however, thePES is also shifted towards a new equilibrium position at Q

eq.

= −2 λΩ

(Qeq.

= 2 λΩ

).This reorganization of the structure of the molecular bridge results also in a shift of therespective PES to lower (higher) energies, which is given by −λ2

Ω

(λ2

Ω

) and adds to thevertical transition energies m. This reorganization energy is also referred to as polaron-shift.

3.1.2 Coupling to Electrodes: Fermionic and Bosonic Baths

A single-molecule junction is an open quantum system, where both the charge and theenergy of the molecular bridge are fluctuating due to the coupling to the electrodes.Thereby, the left (L) and the right (R) lead provide macroscopic reservoirs of electrons,which we assume to be unaffected by the molecular bridge. Both leads can thus bedescribed in a grand-canonical (thermal) equilibrium state, which involves a continuumof electronic quasiparticle states,

HL/R

=X

k∈L/R

kc†kck, (3.18)

with energies k and corresponding creation/annihilation operators c†k/ck. The densityoperator describing the left/right lead is thus given by

ρL/R

=1

tre−(HL/R−µL/RNL/R)/(kBT )e−(HL/R−µL/RNL/R)/(kBT ) (3.19)


(a) (b)

Q=a+a+

εm

2λ/ΩM-

M0εm-λ

2/Ω

Q=a+a+εm+λ

2/Ω

2λ/Ω

M+

M0

εm

λ2/Ω

Figure 3.1: PESs along a single vibrational coordinate Q. Panel a (b) shows the PES forboth the neutral molecule and the anion (cation). Due to electronic-vibrational coupling,the equilibrium position of the vibrational mode changes upon charging the molecule. Thereorganization energy, or polaron-shift, associated with this new equilibrium position isnegative (positive). This is due to the fact that the vertical transition energies m aredefined with respect to the neutral molecule as a reference state.

with NL/R

=P

k∈L/R

c†kck, T the temperature in the leads and kB

the Boltzmann constant.At zero bias, eΦ = µ

L

− µR

= 0, the chemical potentials in the left lead µL

and the rightlead µ

R

are at the same level as the Fermi energy of the system F

= 0, while at Φ = 0they are different. To model this difference, we use the concept of a voltage divisionfactor η. Thereby, according to the discussion given in Refs. [195, 196], we assume thepotential bias to drop symmetrically at the contacts, that is we use µ

L/R

= ηeΦ = eΦ/2and µ

L/R

= (1 − η)eΦ = −eΦ/2, even for an inherently asymmetric molecular junction.Note that a more detailed description of the potential profile in a molecular junction canbe obtained for models, where the specific (contact-)geometry is known. In that case, theelectrostatic potential profile can be obtained employing the Poisson equation [12,49,197].Another strategy to describe the potential drop in a molecular junction is based on anequivalent circuit model of a molecular junction [198–201].

We express the coupling between the molecular bridge and the leads in terms of a simplemodel, where the electronic states located at the molecular bridge are coupled to thecontinuum of states in the leads by interaction matrix elements Vmk,

HML/MR

=X

k∈L/R,m∈M

Vmkc†kcm + V ∗

mkc†mck. (3.20)

The transition matrix elements Vmk, or molecule-lead coupling strengths, depend on thespecific contact geometry between the bridge and the leads. This geometry may differ fromjunction to junction due to the specific fabrication process or due to external mechanicalcontrol (e.g. in a MCBJ [34] or a STM [29, 31, 32]). The corresponding level-width


functions

ΓL/R,mn() = 2π

X

k∈L/R

V ∗mkVnkδ(− k) (3.21)

describe the broadening of the energy levels on the molecular bridge due to this coupling.The coupling to the leads, however, does not only induce broadening. It also mediatesan interaction between the electronic states of the molecular bridge. These interactionsare encoded in the off-diagonal level-width functions, Γ

L/Rmn() with m = n. They are,in addition, crucial for a correct description of quantum interference effects in molecularjunctions [66, 68, 70, 72, 73, 108, 202] (cf. Sec. 6.5). At this point, it should be noted thatwe do not consider a direct coupling between the left and the right electrodes. This iscommonly justified by the assumption that the barrier for a direct tunneling of electronsfrom one lead to the other is typically much larger than the barrier for tunneling throughthe molecular bridge [203]. In general, this assumption has to be handled with care,especially in the presence of partially screened Coulomb interactions1. Moreover, directtunneling is important in the context of transition voltage spectroscopy [39, 148, 166–168, 205], which probes the transition from electron tunneling to field-induced electronemission from the leads.

As the contact geometry between the molecular conductor and the leads may be influencedby the vibrational motion of the molecular bridge, the interaction matrix elements Vmk

are, in principle, functions of the vibrational coordinates Qα. Such position-dependentmolecule-lead couplings have been considered, for example, to describe quantum shuttles[206–209] or contact vibrational motion [210]. In the present work, however, we donot consider such contact-vibrational coupling. In addition, image charges due to thetunneling of electrons are also not included in our description. It has been shown thatthese charges have a significant influence on the vacuum tunneling of electrons [211,212],e.g. in a gold-vacuum-gold MCBJ [168] (i.e. without a bridging molecule), and may also beimportant in electron transport through single-molecule junctions. While the static effectof image charges can, in principle, be accounted for by first-principles methods [205] anddescribed by bias-dependent model parameters, the dynamical effect of transport-inducedpolarizations in the leads may also affect the line-shape of resonances in the resonanttunneling regime [213–215].

Besides charge, the molecular bridge and the leads are also exchanging energy. In ourdescription, such energy exchange is the result of electronic-vibrational coupling and me-diated by inelastic electron tunneling processes [80,83,216,217]. However, it may also bethe result of a direct coupling between the vibrational modes of the molecular conductorand the phononic degrees of freedom in the leads [209] or other environmental degrees offreedom, such as, e.g., in an electrochemically gated molecular junction [13]. Moreover,intramolecular vibrational energy redistribution [218–221], which typically occurs in largemolecules due to the coupling of reactive modes to inactive modes or due to anharmoniceffects, can also induce strong relaxation of a vibrationally excited state of the molecular

1 see, for example, the discussion on p. 146 in Ref. [204]


bridge. To account for these dissipative mechanisms, we employ a linear response model[216,222–224]

Hbath

=X

αβα

ωβα

b†βα

bβα

+X

αβα

ηαβα

Qαqβα

, (3.22)

where each vibrational mode is coupled to a thermal (heat) bath of secondary vibrationalmodes with frequencies ωβ

α

. Thereby, we assume a coupling between the bath and thevibrational modes, which is linear in both the coordinates of the primary modes Qα

and the coordinates of the secondary modes qβα

=≥b†β

α

+ bβα

¥. The respective coupling

strengths are denoted by ηαβα

. The effect of the thermal bath on the dynamic propertiesof the junction can be characterized by the spectral densities

Jα(ω) =X

βα

|ηαβα

|2 δ(ω − ωβα

). (3.23)

In this thesis, we employ ohmic spectral densities[222,225]

Jα(ω) =ζ2

α

ωC,α

ωe− ω

ω

C,α . (3.24)

Summing up, the Hamiltonian H that we use to describe a single-molecule contact isgiven by

H = HM

+ HL

+ HR

+ HML

+ HMR

+ Hbath

. (3.25)

The parameters of this Hamiltonian can be determined, for example, employing first-principles methods (cf. Chap. 7 and Refs. [189–192] or [76, 163, 165, 189–191, 226–232]),or fitted to experimental data [13,24,29,31,32,34,36–39,49,57,99,167,168].

3.1.3 Diagonalization by the Small Polaron Transformation

In the following, we outline theoretical approaches that treat electronic-vibrational cou-pling in electron transport through a single-molecule contact non-perturbatively. Theseapproaches are based on the small polaron transformation [233–236]. Employing thistransformation, the Hamiltonian H

M

can be diagonalized exactly. The small polarontransformation is a canonical transformation, where an operator O is transformed accord-ing to the formula

O = eSOe−S = O + [S,O] +1

2[S, [S,O]] + ..., (3.26)

S = −iX

mα

λmα

Ωα

Pα

°c†mcm − δm

¢. (3.27)


Since S† = −S, the small polaron transformation represents a unitary transformation.The single-particle operators that appear in the Hamiltonian H (cf. Eq. (3.25)) transformunder the small polaron transformation as

cm = cmXm ≡ cmexp

√iX

α

λmα

Ωα

Pα

!, (3.28)

ck = ck,

aα = aα −X

m

λmα

Ωα

°c†mcm − δm

¢,

bβα

= bβα

.

Since

eSf(ABC)e−S = f(ABC) (3.29)

for any analytic function f(x) [234], we replace cm by cm = cmXm and aα by aα =aα −

Pm

λmα

Ω

α

°c†mcm − δm

¢in H and obtain

H = eSHe−S (3.30)

=X

m

m

°c†mcm − δm

¢+

X

m<n

Umn

°c†mcm − δm

¢ °c†ncn − δn

¢

+X

k

kc†kck +

X

k∈L,R,m∈M

≥Vmkc

†kcmXm + H.c.

¥

+X

α

Ωαa†αaα +X

αβα

ωβα

b†βα

bβα

+X

αβα

ηαβα

Qαqβα

−X

mαβα

2λmαηαβα

Ωα

qβα

°c†mcm − δm

¢.

The transformed Hamiltonian H includes polaron-shifted molecular orbital energies

m = m + (2δm − 1)X

α

λ2

mα

Ωα

, (3.31)

vibrationally induced electron-electron interactions

Umn = Umn − 2X

α

λmαλnα

Ωα

, (3.32)

and a molecule-lead coupling term, which is renormalized by the shift operators Xm.Moreover, it contains an electronic-vibrational coupling term, which originates from thecoupling of the vibrational modes to the thermal bath. In order to remove these remainingcoupling terms, we apply another canonical transformation with

S = 2iX

mα

λmαηαβα

Ωαωβα

Pβα

°c†mcm − δm

¢(3.33)


and Pβα

= −i(bβα

− b†βα

). The doubly transformed Hamiltonian

H =X

m

m

°c†mcm − δm

¢+

X

m<n

Umn

°c†mcm − δm


¢(3.34)

+X

k

kc†kck +

X

k∈L,R,m∈M

≥Vmkc

†kcmXmYm + H.c.

¥

+X

α

Ωαa†αaα +X

α

X

βα

ωβα

b†βα

bβα

+X

α

X

βα

ηαβα

Qαqβα

+4X

mα

X

βα

λmα |ηαβα

|2Ωαωβ

α

Qα

°c†mcm − δm

¢,

with

m = m + (2δm − 1)X

α

λ2

mα

Ωα

√1 + 4

X

βα

|ηαβα

|2Ωαωβ

α

!, (3.35)

Umn = Umn − 2X

α

λmαλnα

Ωα

√1 + 4

X

βα

|ηαβα

|2Ωαωβ

α

!, (3.36)

Ym = e−2i

Pmα

λ

mα

η

αβ

α

Ωα

ω

β

α

Pβ

α , (3.37)

still contains a direct electronic-vibrational coupling term. However, for a weak coupling

between the vibrational modes and the thermal bath modes, in particular if 4P

βα

|ηαβ

α

|2Ω

α

ωβ

α

<

1, the strength of the direct electronic-vibrational coupling term in H is smaller than inthe original Hamiltonian H. Repeating similar transformation cycles, one can thereforesuccessively remove the direct electronic-vibrational coupling term and obtains

H = HS

+ HB

+ HSB

, (3.38)

HS

=X

m

m

°c†mcm − δm

¢+

X

α

Ωαa†αaα (3.39)

+X

m<n

Umn

°c†mcm − δm


¢,

HB

=X

k

kc†kck +

X

αβα

ωβα

b†βα

bβα

, (3.40)

HSB

=X

k∈L,R,m∈M

≥Vmkc

†kcmXm + H.c.

¥+

X

αβα

ηαβα

Qαqβα

, (3.41)

where HS

comprises the degrees of freedom of the molecular bridge, HB

the electronicreservoirs in the leads and the secondary vibrational modes, and H

SB

the coupling be-tween the molecular bridge and its environment, in particular, the leads. In the latterexpressions we have disregarded all bath-induced renormalizations of the energies m,

3.2. Master Equation Approach 23

the electron-electron interactions Umn and the molecule-lead couplings VmkXm. Since

4P

βα

|ηαβ

α

|2Ω

α

ωβ

α

< 1, these contributions are typically small. Discarding them is also consis-

tent with the principle that the thermal bath is not to induce dynamical effects in theprimary system but relaxation.

3.2 Master Equation Approach

Density matrix theory is a powerful tool to describe quantum systems [222, 237–241].Electron transport through single molecules has been successfully described employing thismethodology [74, 77, 100, 242–247]. In the following, we outline an approach to computetransport characteristics of a single-molecule contact using density matrix theory [74,83,103,104,106,217,224,248–250]. To this end, we give a brief introduction to density matrixtheory in Sec. 3.2.1 and introduce the concept of a reduced density matrix in Sec. 3.2.2.Sec. 3.2.2 includes the derivation of an exact equation of motion (or master equation)for this reduced density matrix, which is referred to as Nakajima-Zwanzig equation. Anexpansion of the Nakajima-Zwanzig equation with respect to the molecule-lead coupling isoutlined in Sec. 3.2.3. This expansion is carried out to second-order in the molecule-leadcoupling. Employing the eigenstates of the Hamiltonian H (cf. Eq. (3.38)), we give anexplicit representation of the resulting equations in Sec. 3.2.4. Sec. 3.2.5 concludes thischapter, where we show how the observables of interest, in particular the population ofthe electronic states, the average level of vibrational excitation and the electrical current,can be computed in terms of this ME approach.

Other ME approaches to solve the Nakajima-Zwanzig equation are based on higher-orderexpansions in the coupling of the molecular bridge to the leads [242, 246, 251, 252], dia-grammatic real-time techniques [235, 253–256], full-counting statistics [170, 257–259] orother advanced approximation schemes [247, 248, 260]. Besides the stationary transportregime of a molecular junction, time-dependent phenomena, such as transient currents,the effect of AC voltages or optical phenomena, are also considered employing masterequation methodologies [100–104, 106, 107, 261, 262]. A comprehensive overview and dis-cussion of these methods is given for example in Refs. [245, 263]. Even principally exactmethodologies, based on the Nakajima-Zwanzig equation, are being discussed [264,265].

3.2.1 Density Matrices and the Liouville-von Neumann Equa-tion

Master equation approaches are based on the density matrix or density operator ofa quantum-mechanical system. Once the density matrix of a system is known, theexpectation value of any observable O can be obtained from the trace

O = trO =X

a

a| O |a =X

ab

abOba, (3.42)


where ab = a||b denotes the elements of the density matrix. Thereby, |a representsa complete set of orthonormal basis functions that span the Hilbert space of the system.

The time evolution of the density matrix (t) is determined by the Liouville-von Neumannequation

∂(t)

∂t= −i[H, (t)] ≡ −iL(t), (3.43)

where L denotes the Liouville operator L ≡ [H, ]. For a time-independent HamiltonianH, it is given by

(t) = e−iLt(0), (3.44)

where (0) represents the initial state of the system at time t = 0.

The expectation value of an operator O is invariant under a canonical transformation

O = trO = treSe−SeSOe−S = trO, (3.45)

in particular, the small polaron transformation. It is thus equivalent to employ the trans-formed density matrix instead of , and the respective expectation values according toEq. (3.45). The time evolution of is determined by the transformed Hamiltonian H (cf.Eq. (3.38))

∂(t)

∂t= −i[H, (t)] ≡ −iL(t), (3.46)

(t) = e−iLt(0), (3.47)

where L denotes the transformed Liouville operator L ≡ [H, ]. Similar Liouville op-erators L

K

can also be defined with respect to specific parts of the Hamiltonian, HK ,K ∈ S, B, SB.

3.2.2 Nakajima-Zwanzig Equation

To describe an open quantum system, such as a single-molecule junction, it is expedientto employ the reduced density matrix of the system ρ(t). The reduced density matrix isobtained by taking the trace of the full density matrix (t) with respect to the degrees offreedom of the reservoirs (or baths (B))

ρ(t) = trB

(t). (3.48)

Following Nakajima and Zwanzig [266,267], a formally exact equation of motion (EOM)for the reduced density matrix ρ(t) can be derived. To this end, a projection operator

P(t) = ρB

trB

(t) ≡ ρB

ρ(t), (3.49)


together with its orthogonal complement Q = 1 − P , is defined. Here, ρB

denotes thethermal equilibrium density matrix of the reservoirs

ρB

= Z−1e−(HB−µLNL−µRNR)/(kBT ), (3.50)

Z = trB

e−(HB−µLNL−µRNR)/(kBT ). (3.51)

For the two complementary projections of the full density matrix, P(t) and Q(t), EOMscan be derived from the Liouville-von Neumann equation, Eq. (3.43):

P∂(t)

∂t=

∂P(t)

∂t= −iPL(t) = −iPL(P + Q)(t), (3.52)

Q∂(t)

∂t=

∂Q(t)

∂t= −iQL(t) = −iQL(P + Q)(t). (3.53)

To uncouple these EOMs, the latter of these is integrated in time

Q(t) = e−itQLQ(0)− i

Z t

0

dτ e−iτQLQLP(t− τ), (3.54)

and the resulting expression used in the EOM for P(t). We thus obtain the Nakajima-Zwanzig equation

∂P(t)

∂t= −iPLP(t)− PLQ

Z t

0

dτ e−iτLQQLP(t− τ), (3.55)

where, in addition, we assume that there are no initial correlations between the molecularbridge (S) and the leads and/or environment, i.e. Q(0) = 0. The Nakajima-Zwanzigequation simplifies significantly, if the conditions tr

B

HSB

ρB

= 0 and LB

ρB

= 0 arefulfilled. Since this is the case for the given molecule-lead coupling term, H

SB

, and leaddensities, ρ

B

, the following operator identities,

PLSB

P = 0, (3.56)

PLS

= LS

P, (3.57)

PLB

= LB

P = 0, (3.58)

PLQ = PLSB

Q, (3.59)

QLP = QLSB

P, (3.60)

can be used to rewrite Eq. (3.55) as

∂ρ(t)

∂t= −iL

S

ρ(t)−Z t

0

dτ trB

LSB

Qe−iτLQQLSB

ρB

ρ(t− τ). (3.61)

Eqs. (3.55) or (3.61) represent exact equations of motion for the reduced density matrixρ(t). For practical applications, however, approximations are required to solve theseequations of motion. An example for such an approximation scheme is outlined in thenext section.


3.2.3 Markovian Master Equation / Redfield Theory

In order to relate the transport characteristics of a molecular junction to intrinsic prop-erties of the molecular bridge, weak coupling between the molecule and the leads is expe-dient [268]. While this is not fulfilled for every molecular junction, an expansion of theNakajima-Zwanzig equation (3.61) to second order in H

SB

,

∂ρ(t)

∂t= −iL

S

ρ(t)−Z t

0

dτ trB

LSB

Qe−iτ(LS+LB)QQLSB

ρB

ρ(t− τ), (3.62)

often constitutes a useful description of a molecular junction, especially in the resonanttransport regime [80,250]. In line with this second order expansion in H

SB

, we furthermoreapproximate ρ(t− τ) by

ρ(t− τ) ≈ eiLSτρ(t). (3.63)

This approximation is part of the so-called Markov approximation, which involves also ashift of the integration limit:

R t

0

→ R∞0

. Employing the Markov approximation, we obtainthe master equation [74,100,242,269]

∂

∂tρ(t) = −i[H

S

, ρ(t)]−Z ∞

0

dτ trB

[HSB

, [HSB

(τ), ρ(t)ρB

]] (3.64)

with

HSB

(τ) = e−i(HS+HB)τHSB

ei(HS+HB)τ . (3.65)

Due to the shift in the integration limit, the master equation (3.64) is only valid for timeslonger than the correlation time of the bath [262,269]. This is the case in the steady-statetransport regime considered in this work, where only the long-time limit ρ(t→∞) ≡ ρ isrelevant. An account of non-Markovian effects is given for example in Refs. [257,270–272].Taken in the basis of the eigenstates of the system Hamiltonian H

S

, the master equation(3.64) corresponds to the Redfield equation [237, 240, 273]. In the steady-state transportregime, the above equation of motion reduces to an algebraic set of homogeneous equations

0 = −i[HS

, ρ]−Z ∞

0

dτ trB

[HSB

, [HSB

(τ), ρ ρB

]], (3.66)

which can be solved by standard linear algebra techniques. Thereby, the normalizationconstraint tr

S

ρ = 1 ensures a unique solution.

Due to the expansion to second order in HSB

, the master equation (3.66) cannot describetunneling in the non-resonant transport regime. Especially co-tunneling processes [79,216,253–255,274], Kondo physics [235,275–279] and the broadening of resonances due to thecoupling of the molecule to the leads are neglected [80,224]. Except for these deficiencies,however, the ME approach provides a rather accurate description of vibrationally coupledelectron transport in the resonant transport regime. This has been demonstrated recentlyby comparison with results of NEGF theory [80,224,250].


3.2.4 Explicit Representation of the Master Equation

To solve the master equation (3.66), we need to evaluate the corresponding operators, likeρ or HS, in a specific basis. As a basis we use product functions, |a|ν, which span thesubspace of the electronic and the vibrational degrees of freedom, respectively. Thereby,the electronic basis functions, |a, are given in the occupation number representation i.e.|a = |n

1

n2

.., where nm ∈ 0, 1 denotes the population of the mth electronic state. Sincethe presented master equation approach is numerically very demanding for systems withmultiple vibrational modes, we focus, in the following, on results for a single vibrationalmode. For the vibrational part of our basis, we therefore use harmonic oscillator basisfunctions |ν, where ν ∈

0

stands for the excitation number of the vibrational mode.The coefficients of the reduced density matrix can be written in this basis as

ρν1ν2a,b ≡ a|ρν1ν2|b ≡ a|ν

1

|ρ|ν2

|b, (3.67)

where superscript indices refer to states of the vibrational mode and subscript indicesrepresent the electronic part of the respective Hilbert space. Note that, in the following,we do not consider coupling between the vibrational mode and a thermal bath (ηβ1 = 0).

Evaluating Eq. (3.66) first between the vibrational states ν1

| and |ν2

, we obtain thefollowing set of equations

− i

πν

1

|[HS

, ρ]|ν2

= (3.68)X

mnν3ν4k

VmkV∗nk

£fkX

ν1ν3m X†,ν3ν4

n cmc†nδ(Eν3ν4n,k )ρν4ν2 − (1− fk)X

ν1ν3m X†,ν4ν2

n cmρν3ν4c†nδ(Eν4ν2m,k )

§

+X

mnν3ν4k

VmkV∗nk

£fkX

ν3ν4m X†,ν4ν2

n ρν1ν3cmδ(Eν4ν3m,k )c†n − (1− fk)X

ν1ν3m X†,ν4ν2

n cmδ(Eν3ν1m,k )ρν3ν4c†n

§

+X

mnν3ν4k

V ∗mkVnk

£(1− fk)X

†,ν1ν3m Xν3ν4

n c†mcnδ(Eν4ν3n,k )ρν4ν2 − fkX

†,ν1ν3m Xν4ν2

n c†mρν3ν4cnδ(Eν2ν4n,k )

§

+X

mnν3ν4k

V ∗mkVnk

£(1− fk)X

†,ν3ν4m Xν4ν2

n ρν1ν3c†mδ(Eν3ν4m,k )cn − fkX

†,ν1ν3m Xν4ν2

n c†mδ(Eν1ν3m,k )ρν3ν4cn

§

with

Eν1ν2m,k = m − k +

X

n=m

Umn(c†ncn − δn) + Ω1

(ν1

− ν2

),

X(†),ν1ν2m = ν

1

|X(†)m |ν

2

.Here, fk denotes the Fermi distribution function of the left (k ∈ L) or the right lead(k ∈ R), evaluated at the energy k. δ(x) stands for the Dirac-delta function, where e.g.δ( + Uc†

1

c1

)|11 = δ( + U)|11. In Eq. (3.68), we have neglected all principal valueterms. These terms describe the renormalization of the molecular energy levels due tothe molecule-lead and/or mode-bath coupling [243], which, in line with the weak-couplingassumption, is negligible for the results discussed below. Next, we evaluate Eq. (3.68)with respect to the electronic basis functions, which is outlined in the following for asingle and for two electronic states.


Explicit master equation for a single electronic state coupled to a single vi-brational mode

For transport through a single electronic state, the density matrix ρ comprises four dif-ferent kinds of elements, ρν1ν2

0,0 , ρν1ν21,0 , ρν1ν2

0,1 , ρν1ν21,1 , corresponding to the two charge states |0

and |1. To determine these elements, we evaluate Eq. (3.68) first between the chargestates 0| and |0 and obtain

2iΩ1

(ν1

− ν2

)ρν1ν20,0 = (3.69)

X

ν3ν4K

[(1− fK(Eν4ν2))ΓK,11

(Eν4ν2) + (1− fK(Eν3ν1))ΓK,11

(Eν3ν1)] Xν1ν31

X†,ν4ν21

ρν3ν41,1

−X

ν3ν4K

fK(Eν4ν3)ΓK,11

(Eν4ν3)hXν3ν4

1

X†,ν4ν21

ρν1ν30,0 + Xν1ν4

1

X†,ν4ν31

ρν3ν20,0

i

with Eν1ν2 = 1

+ Ω1

(ν1

− ν2

). Thereby, the distribution function in lead K (K ∈ L,R),

fK() =1

1 + e(−µK

)/(kBT )

, (3.70)

is determined by the respective chemical potential µK . The corresponding 1|..|1 com-ponent reads

2iΩ1

(ν1

− ν2

)ρν1ν21,1 = (3.71)

X

ν3ν4K

[fK(Eν2ν4)ΓK,11

(Eν2ν4) + fK(Eν1ν3)ΓK,11

(Eν1ν3)] Xν4ν21

X†,ν1ν31

ρν3ν40,0

−X

ν3ν4K

(1− fK(Eν3ν4))ΓK,11

(Eν3ν4)hXν4ν2

1

X†,ν3ν41

ρν1ν31,1 + Xν4ν3

1

X†,ν1ν41

ρν3ν21,1

i.

From Eqs. (3.69) and (3.71) it is evident that the diagonal elements ρν1ν20,0 and ρν1ν2

1,1 arenot related to the coherences between different charge states ρν1ν2

0,1 and ρν1ν21,0 . This is a

result of the strict second order expansion employed in the Markovian master equation(3.66). Therefore, the 0|..|1 and the 1|..|0 components of Eq. (3.68) do not need tobe considered in the following, as they are also not important for the computation of thecurrent or the respective level of vibrational excitation (cf. Sec. 3.2.5).

Explicit master equation for two electronic states coupled to a single vibra-tional mode

For transport through two electronic states, the density matrix ρ consists of 16 differentkinds of elements, ρν1ν2

a,b with a, b ∈ 00, 1, 2, 11, corresponding to the four charge states|00, |01, |10 and |11. As for the scenario with a single electronic state, coherencesbetween different charge states, ρν1ν2

00,a , ρν1ν2a,00

, ρν1ν2b,11

and ρν1ν211,b with a ∈ 1, 2, 11 and b ∈

00, 1, 2, are not related to the elements ρν1ν200,00

, ρν1ν211,11

and ρν1ν2a,b with a, b ∈ 1, 2, nor to


the observables of interest (cf. Sec. 3.2.5). Therefore, these elements do not need to beconsidered in the following. For notational convenience, we introduce the symbols 1 and2 at this point. They are defined by 1 = 2 and 2 = 1. Thus, if we evaluate Eq. (3.66)between the electronic states 00| and |00, we obtain

2iΩ1

(ν1

− ν2

)ρν1ν200,00

= (3.72)X

mnν3ν4K

£ΓK,nm(Eν4ν2

n,− )(1− fK(Eν4ν2n,− )) + ΓK,nm(Eν3ν1

m,− )(1− fK(Eν3ν1m,− ))

§Xν1ν3

m X†,ν4ν2n ρν3ν4

m,n

−X

mν3ν4K

ΓK,mm(Eν4ν3m,− )fK(Eν4ν3

m,− )£Xν3ν4

m X†,ν4ν2m ρν1ν3

00,00

+ Xν1ν4m X†,ν4ν3

m ρν3ν200,00

§

with Eν1ν2m,− = m − U

12

δm + Ω1

(ν1

− ν2

) and m,n ∈ 1, 2. Evaluation of Eq. (3.66) inbetween the states a| and |b, where a, b ∈ 10, 01, gives

−2i (Ω1

(ν1

− ν2

) + o − p) ρν1ν2o,p = (3.73)

X

mν3ν4K

ΓK,mo(Eν3ν4m,+ )fK(Eν3ν4

m,+ )Xν1ν3o X†,ν3ν4

m ρν4ν2m,p (−1)1+δ

om

+X

nν3ν4K

ΓK,on(Eν4ν3n,− )(1− fK(Eν4ν3

n,− ))Xν3ν4n X†,ν1ν3

o ρν4ν2n,p

−X

ν3ν4K

ΓK,po(Eν3ν1o,+ )(1− fK(Eν3ν1

o,+ ))Xν1ν3o X†,ν4ν2

p ρν3ν411,11

(−1)1+δop

−X

ν3ν4K

ΓK,po(Eν4ν2p,+ )(1− fK(Eν4ν2

p,+ ))Xν1ν3o X†,ν4ν2

p ρν3ν411,11

(−1)1+δop

−X

ν3ν4K

£ΓK,op(E

ν1ν3o,− )fK(Eν1ν3

o,− ) + ΓK,op(Eν2ν4p,− )fK(Eν2ν4

p,− )§Xν4ν2

p X†,ν1ν3o ρν3ν4

00,00

+X

mν3ν4K

ΓK,pi(Eν4ν3m,+ )fK(Eν4ν3

m,+ )Xν3ν4m X†,ν4ν2

p ρν1ν3o,m (−1)1+δ

pm

+X

mν3ν4K

ΓK,mp(Eν3ν4m,− )(1− fK(Eν3ν4

m,− ))Xν4ν2p X†,ν3ν4

m ρν1ν3o,m

with Eν1ν2m,+ = m + U

12

(1 − δm) + Ω1

(ν1

− ν2

) and m,n, o, p ∈ 1, 2. The respective11|..|11 component finally reads

2iΩ1

(ν1

− ν2

)ρν1ν211,11

= (3.74)X

mnν3ν4K

£ΓK,mn(Eν2ν4

n,+ )fK(Eν2ν4n,+ ) + ΓK,mn(Eν1ν3

m,+ )fK(Eν1ν3m,+ )

§Xν4ν2

n X†,ν1ν3m ρν3ν4

m,n (−1)1+δmn

−X

mν3ν4K

ΓK,mm(Eν3ν4m,+ )(1− fK(Eν3ν4

m,+ ))£Xν4ν2

m X†,ν3ν4m ρν1ν3

11,11

+ Xν4ν3m X†,ν1ν4

m ρν3ν211,11

§.

The description of more than two electronic states and/or multiple vibrational modes,where each mode can be coupled to a thermal bath of secondary vibrational modes, is


straightforward. Such transport scenarios, however, are not treated by the ME methodin this work. Note that the numerical effort to solve the respective equations increasesvery rapidly with an increasing number of degrees of freedom. This is in contrast tothe NEGF approach that will be introduced in Sec. 3.3. Especially due to the harmonicapproximation involved, it scales much better with respect to the number of electronicstates and/or vibrational modes.

We finally remark that the above scheme reduces to a rate equation approach [74, 77,244], if the coherences of the density matrix are disregarded, that is, if one enforcesρν1ν2

a,b ∼ δabδν1ν2 . Test calculations show that this is justified for the mechanisms and modelsystems studied in this thesis by the ME method. In general, however, coherences cannotbe neglected in the description of molecular junctions with quasidegenerate levels (see thediscussion in appendix C).

3.2.5 Observables of Interest

To characterize electron transport through a single-molecule junction, we analyze threedifferent observables of interest as functions of the applied bias voltage: i.) the popula-tion of the electronic states, ii.) the average vibrational excitation, and iii.) the electricalcurrent flowing through the junction. In this section, we outline explicitly how these ob-servables are computed, once the coefficients of the density matrix ρ have been determinedby the scheme developed in Secs. 3.2.1 – 3.2.4.

Electronic Population

The diagonal elements of the density matrix, ρννa,a, encode the probability of finding the

molecular bridge in the product state |a|ν. Hence, for a single electronic state on themolecular bridge, the average electronic population is given by

n1

= c†1

c1

H = c†1

c1

H (3.75)

= trS+B

c†1

c1

= trS

ρc†1

c1

=X

ν

ρνν1,1.

Accordingly, we find

n1

= c†1

c1

H =X

ν

ρνν11,11

+ ρνν10,10

, (3.76)

n2

= c†2

c2

H =X

ν

ρνν11,11

+ ρνν01,01

,

for the population of two electronic states located on the molecular bridge. Here, and inthe following, the subscript H/H indicates, which Hamiltonian is used to evaluate therespective expectation value.


Vibrational Excitation

The average excitation of the vibrational mode involves a sum over all electronic degreesof freedom. For a transport scenario with a single electronic state on the molecular bridge,the average vibrational excitation is given by

a†1

a1

H = a†1

a1

H +λ2

11

Ω2

1

(n1

− 2δ1

n1

+ δ1

) =X

νa

νρννa,a +

λ2

11

Ω2

1

(n1

− 2δ1

n1

+ δ1

),(3.77)

with a ∈ 0, 1 and respectively, for a transport scenario with two electronic states, by

a†1

a1

H =X

νa

νρννa,a +

λ2

11

Ω2

1

(n1

− 2δ1

n1

+ δ1

) +λ2

21

Ω2

1

(n2

− 2δ2

n2

+ δ2

)

+2λ

11

λ21

Ω2

1

(X

ν

ρνν11,11

− δ2

n1

− δ1

n2

+ δ1

δ2

)

−2X

ν

√ν + 1Re[

λ11

Ω1

ρνν+1

10,10

+λ

21

Ω1

ρνν+1

01,01

+λ

11

+ λ21

Ω1

ρνν+1

11,11

].

with a ∈ 00, 1, 2, 11. Thereby, we use that Q1

H = 0 in the steady-state transportregime of a molecular junction. This can be shown considering the time dependence ofthe expectation values

0steady state

= i∂tP1

H = −iΩ1

Q1

H − 2iX

β1

ηβ1qβ1H , (3.78)

0steady state

= i∂tPβ1H = −iωβ1qβ1H − 2iηβ1Q1

H , (3.79)

→ 0 =

√1− 4

X

β1

η2

β1

ωβ1Ω1

!Q

1

H , (3.80)

→ 0 = Q1

H , (3.81)

where the latter conjecture follows from the condition 4P

β1(η2

β1/ωβ1Ω1

) < 1 (cf. Sec.

3.1.3). For a single electronic state, this argument can be extended to Q1

c†1

c1

H = 0,since

0 = i∂t

X

k

P1

c†kckH =X

mk

VmkP1

c†kcmXmH −X

mk

V ∗mkP1

c†mX†mckH

= − i

2eP

1

IL

H +i

2eP

1

IR

H . (3.82)

Notice that in the latter equation we have used Q1

c†kckH = 0 and qβ1c†kckH = 0, which

can be derived from Q1

H = 0 and the fact that the expectation value ..H is evaluatedwith the product density matrix ρ

B

ρ. Similar to Eqs. (3.78), using Eq. (3.82), it thus


follows that

0 = i∂tP1

c†1

c1

H = −iΩ1

Q1

c†1

c1

H − 2iX

β1

ηβ1qβ1c†1

c1

H , (3.83)

0 = i∂tPβ1c†1

c1

H = −iωβ1qβ1c†1

c1

H − 2iηβ1Q1

c†1

c1

H , (3.84)

→ 0 =

√1− 4

X

β1

η2

β1

ωβ1Ω1

!Q

1

c†1

c1

H , (3.85)

→ 0 = Q1

c†1

c1

H . (3.86)

Accordingly, terms involving vibrational coherences, ∼ Re[ρνν+1

a,a ], do not explicitly appearin the expression for vibrational excitation in transport through a single electronic state(Eq. (3.77)). For more than one electronic state, however, such terms enter the expres-sion for vibrational excitation (Eq. (3.78)), as the partial fluxes

Pk(VmkP1

c†kcmXmH −V ∗

mkP1

c†mX†mckH) may differ from zero. This is important, for example, in transport

through quasidegenerate molecular levels (cf. appendix C), where vibrational coherencesare significant. Note that the relations Q

1

H = 0 and Q1

c†1

c1

H = 0 are also beingused in the derivation of the NEGF approach (cf. Sec. 3.3). We therefore considered amode-bath coupling ηβ1 at this point.

Current

The current through lead K, IK , is determined by the number of electrons entering orleaving the lead per unit time (K ∈ L,R)

IK = IKH = −2ed

dt

X

k∈K

c†kckH (3.87)

= 2ie

"X

mk

Vmkc†kcmXmH −X

mk

V ∗mkc†mX†

mckH#

.

Here, the factor 2 accounts for spindegeneracy. The specific structure of the currentoperator requires the determination of Q = (1 − P ), since tr

S+B

PIK = 0. Thisprojection of the full density matrix (cf. Eq. (3.54)) has already been used to derive theEqs. (3.55) and (3.61) [266, 267]. Using Eq. (3.54) to evaluate the expression for thecurrent (3.87) and employing the same approximations that we used for the derivationof the master equation (3.66), we obtain the following expression for the current throughlead K [74,100,242,243]

IK = −i

Z ∞

0

dτ trS+B

[HSB

(τ), ρB

ρ]IK. (3.88)

In the numerical calculations, Eq. (3.88) is further evaluated using the basis functionsintroduced in Sec. 3.2.4. Thereby, the current through a single electronic state, which is


coupled to a single vibrational mode, can be represented as

IK = −eX

ν1ν2ν3

(1− fK(Eν2ν1)) ΓK,11

(Eν2ν1)Xν1ν21

X†,ν3ν11

ρν2ν31,1 (3.89)

+eX

ν1ν2ν3

fK(Eν1ν2)ΓK,11

(Eν1ν2)Xν3ν11

X†,ν1ν21

ρν2ν30,0

+eX

ν1ν2ν3

fK(Eν3ν2)ΓK,11

(Eν3ν2)Xν2ν31

X†,ν3ν11

ρν1ν20,0

−eX

ν1ν2ν3

(1− fK(Eν2ν3)) ΓK,11

(Eν2ν3)Xν3ν11

X†,ν2ν31

ρν1ν21,1 ,

and accordingly, for transport through two electronic states as

IK = −eX

mnν1ν2ν3

(1− fK(Eν2ν1m )) ΓK,nm(Eν2ν1

m )Xν1ν2m X†,ν3ν1

n ρν2ν3mn (3.90)

−eX

m,ν1,ν2,ν3

°1− fK(Eν2ν1

m,+ )¢ΓK,mm(Eν2ν1


m ρν2ν311,11

+eX

mnν1ν2ν3

(−1)1+δmn fK(Eν1ν2

m,+ )ΓK,mn(Eν1ν2m,+ )Xν3ν1

n X†,ν1ν2m ρν2ν3

mn

+eX

mν1ν2ν3

fK(Eν1ν2m )ΓK,mm(Eν1ν2


m ρν2ν300,00

+eX

mν1ν2ν3

fK(Eν3ν2m )ΓK,mm(Eν3ν2


m ρν1ν200,00

+eX

mnν1ν2ν3

(−1)1+δmn fK(Eν3ν2

m,+ )ΓK,nm(Eν3ν2m,+ )Xν2ν3

m X†,ν3ν1n ρν1ν2

nm

−eX

mν1ν2ν3

°1− fK(Eν2ν3

m,+ )¢ΓK,mm(Eν2ν3


m ρν1ν211,11

−eX

mnν1ν2ν3

(1− fK(Eν2ν3m )) ΓK,mn(Eν2ν3

m )Xν3ν1n X†,ν2ν3

m ρν1ν2nm .

In Eqs. (3.89) and (3.90), principal value terms, as for the computation of the reduceddensity matrix, are disregarded. Note that this scheme is current conserving, i.e., I

L

=−I

R

= I.

3.2.6 Vibration in Thermal Equilibrium

For the analysis of vibrational nonequilibrium effects, it is often instructive to compareresults, where the full current-induced nonequilibrium state of the vibrational mode isemployed, to results, where the vibrational mode is kept in a known equilibrium state (forexample its ground state). To this end, we employ the commonly used approximation[47,74,83,210,216,236,280–282] that after every electron transmission event the vibrational


mode returns to its thermal equilibrium state characterized by an effective temperaturek

B

T . Note that within this approximation the vibrational mode can be excited as wellas deexcited during an electron transmission event. An electron tunneling through thejunction in a subsequent transmission event, however, finds the vibrational mode in thesame equilibrium state as the electron that was tunneling through the junction before.This behavior of the vibrational mode can be enforced by evaluating the vibrational partof the reduced density matrix ρ according to

ρν1ν2 = ρel

δν1ν2(1− e−Ω1/(kBT ))e−ν1Ω1/(kBT ), (3.91)

where only the electronic part of the reduced density matrix ρel

needs to be determinedaccording to Eq. (3.66).

3.3 Nonequilibrium Green’s Function Approach

Alternatively to the ME approach outlined in the previous section, we use a NEGFapproach [18, 234, 283–286] to describe vibrationally coupled electron transport througha single-molecule junction. This approach will be outlined in this chapter. To this end,we give first a brief introduction to nonequilibrium Green’s function theory in Secs. 3.3.1and 3.3.2, before we outline the particular method in Secs. 3.3.3 – 3.3.7. This methodwas originally proposed by Galperin et al. [236] and facilitates a non-perturbative de-scription of electronic-vibrational coupling. It is based on a factorization of electronic andvibrational time scales in a molecular junction and employs self-consistent second-orderperturbation theory with respect to the coupling of the molecule to the leads. We ex-tend this method to account for multiple vibrational modes and multiple electronic states[73,80,216,224], including a non-perturbative description of electron-electron interactionsin terms of the elastic co-tunneling approximation [80, 285, 287–289]. In Sec. 3.3.8, weshow how the observables of interest (the population of the electronic states, the averagelevel of vibrational excitation and the electrical current flowing through the junction) canbe computed in terms of this NEGF approach. Numerical issues of the approach areaddressed in Appendix B.

Other nonequilibrium Green’s function approaches are based on perturbation theory[74, 75, 230, 289–292], advanced equation of motion techniques [277], projection operatortechniques [280, 293–296], full-counting statistics [81, 297, 298] or other non-perturbativeschemes [299,300]. While most of these methods address the stationary state of a molec-ular junction, there are also a number of approaches that describe transient behavior,optical phenomena and/or the effect of AC voltages [105, 301–307]. A comparison ofresults obtained from nonequilibrium Green’s function theory to other methods can befound, for example, in Refs. [80,224,250,308].

3.3. Nonequilibrium Green’s Function Approach 35

3.3.1 Single-Particle Green’s Function Gmn

The central quantity of NEGF theory is the single-particle Green’s function

Gmn(t, t) = −iT cm(t)c†n(t) = −iΨ0

|T cm(t)c†n(t)|Ψ0

. (3.92)

This function is a measure for the correlation between the annihilation of an electron instate m at time t and the creation of an electron in state n at time t. Thereby, theoperators cm(t) and c†n(t) obey the following equations of motion

i∂tcm(t) = [cm(t), H], (3.93)

i∂tc†n(t) = [c†n(t), H]. (3.94)

In the present context, the expectation value, Ψ0

|..|Ψ0

, is evaluated employing the sta-tionary state of a molecular junction. This is the state the junction acquires in thesteady-state transport regime.

The order of the operators cm(t) and c†n(t) in Eq. (3.92) is crucial. This order is uniquelydefined by the time-ordering operator,

T A(t)B(t) = Θ(t− t)A(t)B(t)±Θ(t − t)B(t)A(t), (3.95)

where the ’-’ sign applies, if A and B are fermionic operators. Hence, Gmn(t, t) is calledthe time-ordered Green’s function. Accordingly, the anti-time ordering operator,

T A(t)B(t) = Θ(t − t)A(t)B(t)±Θ(t− t)B(t)A(t), (3.96)

defines the anti-time ordered Green’s function,

Gmn(t, t) = −iT cm(t)c†n(t). (3.97)

There are four other possibilities for ordering these operators, corresponding to the Green’sfunctions

Gr

mn(t, t) = −iΘ(t− t)[cm(t), c†n(t)]+

, (3.98)

Ga

mn(t, t) = iΘ(t − t)[cm(t), c†n(t)]+

, (3.99)

G<mn(t, t) = ic†n(t)cm(t), (3.100)

G>mn(t, t) = −icm(t)c†n(t), (3.101)

which are referred to as retarded, advanced, lesser and greater Green’s function, respec-tively.

As any single-particle operator O can be represented by an annihilation and a creationoperator, e.g. O =

Pmn Omnc†mcn, all single-particle observables can be computed with the

lesser and the greater Green’s functions G</>mn (t, t). Similar information can be extracted

from the retarded and the advanced Green’s functions Gr/a

mn(t, t) due to the identity

Gr

mn(t, t)−Ga

mn(t, t) = G>mn(t, t)−G<

mn(t, t) ≡ −iA(t, t). (3.102)


The spectral density of a system A(ω), which characterizes the level density of a system,is given by the Fourier transform of equation (3.102). It can thus be obtained from boththe difference between the retarded and the advanced Green’s functions or the differencebetween the greater and the lesser Green’s function. In contrast to the lesser and thegreater Green’s function, however, the retarded and the advanced Green’s function do notinclude further information about the population of the levels in a system. The analyticproperties of the retarded and the advanced Green’s function, i.e. their pole-structure, arethus much simpler, as is their evaluation [285]. A system in (thermal) equilibrium, wherethe population of the levels is determined by the statistical properties of the respectivedegrees of freedom, is therefore typically described in terms of retarded and advancedGreen’s functions. The time-ordered Green’s function Gmn may also be represented as asum or difference of lesser and greater Green’s functions

Gmn(t, t) = Θ(t− t)G>mn(t, t) + Θ(t − t)G<

mn(t, t). (3.103)

The time-ordered Green’s function distinguishes itself by a systematic perturbative ex-pansion [285,286,309]. This expansion was originally formulated for systems in (thermal)equilibrium [234,283], but has also been extended to systems in nonequilibrium [285,286].

The main difficulty in evaluating the single-particle Green’s function, Eq. (3.92), is to finda relation between the state |Ψ

0

and a known state. For a complex, interacting systemthis is typically achieved by propagating the ground state of a simpler, non-interactingsystem, |ψ

0

, in time from −∞ to 0. Thereby, the interaction, which distinguishes thesimple non-interacting from the complex interacting system, is adiabatically switched on.This approach is formally expressed in terms of the Gell-Mann and Low theorem [285]

|Ψ0

= S(0,−∞)|ψ0

, (3.104)

employing the S-matrix S(0,−∞). The corresponding dual state

Ψ0

| = ψ0

|S(∞, 0) (3.105)

is well defined, if the system evolves to a uniquely defined thermal equilibrium state|Ψ

0

. Accordingly, expectation values like Ψ0

|O|Ψ0

are also well defined. Thereby, theS-matrix facilitates a systematic perturbative expansion [234, 283, 285, 286]. In nonequi-librium, however, due to statistical irreversible processes, it is not clear to which state

the system evolves, Ψ0

|S(0,∞)?

= ψ0

|, once the force driving it out of equilibrium isadiabatically switched off. One therefore resorts to a different strategy, which allows torelate not only the state |Ψ

0

but also its dual counterpart Ψ0

| to the same well-definedground state |ψ

0

at t→ −∞.

To this end, one employs complex times τ , and defines a contour C in the complex timedomain [285,309,310]. The simpler non-interacting system is then propagated along thiscontour. This allows both to propagate the system forward in time and to return to theinitial state. Thereby, one uses the so-called Keldysh contour, an example of which isdepicted in Fig. 3.2. Such a contour consists of two branches C

1

and C2

. The upper


t’t

C1

C2

Figure 3.2: Graphical representation of a Keldysh contour C in the complex time do-main, where the time t is passed by the contour before the time t. This choice of thecontour corresponds to a lesser function, for example G<

mn(t, t).

branch C1

runs along the real time axis, including a vanishingly small shift to positiveimaginary parts, while the lower branch C

2

elongates accordingly in the lower half-plane,following the real-time axis in the opposite direction. The two branches are connected bya turning point at max(t, t), where the contour crosses the real-time axis. Thereby, thepoints t and t are passed by the Keldysh contour C = C

1

+ C2

exactly once. Note thatthe turning point of the Keldysh contour can also be defined at later (real) times. For ourpresent purposes, where initial correlations of the system are not important, the contourcan be considered to start in the infinite past. Initial correlations are of importance forthe transient dynamics of a molecular junction, for example when the bias voltage isswitched on or when the molecule is contacted with the leads, or if there is more than onestationary state (a discussion about the existence of a steady state can be found in Ref.[311] and references therein).

On the Keldysh contour, a contour-ordered Green’s function can be defined,

GCmn(t, t) = −iTCcm(t)c†n(t), (3.106)

where the contour-ordering operator TC orders the operators cm(t) and c†n(t) with respectto the position of the times t and t on the Keldysh contour. The contour-ordered Green’sfunction can thus be used to describe the time-ordered, Gmn, the lesser, G<

mn, the greater,G>

mn, and the anti-time ordered Green’s functions, Gmn, as

GCmn(t, t) =

8>><

>>:

Gmn(t, t), t, t ∈ C1

,G<

mn(t, t), t ∈ C1

, t ∈ C2

,G>

mn(t, t), t ∈ C1

, t ∈ C2

,Gmn(t, t), t, t ∈ C

2

.

(3.107)

The retarded and the advanced Green’s function are implicitly included as

Gr

mn(t, t) = Θ(t− t) (G>mn(t, t)−G<

mn(t, t)) , (3.108)

Ga

mn(t, t) = Θ(t − t) (G<mn(t, t)−G>

mn(t, t)) . (3.109)


So far, we have focused on a specific example for a single-particle Green’s function, Gmn.Analogous considerations, however, do also apply for any other two-point correlationfunction, such as, for example, the vibrational Green’s function,

Dαα(t, t) = −iTt

Qα(t)Qα(t). (3.110)

Correlation functions with four (or even more) time variables are needed to compute two-particle observables such as the shot-noise characteristic of a molecular junction. Thecomputation of such observables, however, is not considered in this work. Note that fornon-interacting systems such correlation functions can be often represented in terms ofsingle-particle Green’s functions [312,313].

3.3.2 Dyson and Keldysh Equation

For a complex interacting system, which is in a thermal equilibrium state, the S-matrix(cf. Eq. (3.104)) facilitates a systematic perturbative expansion [234, 283, 285, 286]. Thismeans, for example, that the time-ordered Green’s function Gmn(t, t) of this system canbe computed from the Dyson equation

Gmn(t, t) = G0

mn(t, t) +X

op

Z ∞

−∞dt

1

Z ∞

−∞dt

2

G0

mo(t, t1)Σop(t1, t2)Gpn(t2

, t),

(3.111)

which relates this Green’s function to the time-ordered Green’s function of a simple non-interacting system G0

mn(t, t) and a self-energy Σmn(t, t).

Similarly, a S-matrix can be constructed for a complex interacting system, which is ina nonequilibrium state [285, 309]. This S-matrix, however, does not describe the timepropagation along the real time axis but along the complex Keldysh contour. It can alsobe used for a systematic perturbative expansion of the contour-ordered Green’s function,GC

mn(t, t). The corresponding Dyson equation [285,286]

GCmn(t, t) = GC,0

mn(t, t) +X

op

Z

C

dτ1

Z

C

dτ2

GC,0mo (t, τ

1

)ΣCop(τ1

, τ2

)GCpn(τ

2

, t)

(3.112)

is thus formally the same as in the original formulation for equilibrium systems. Thereby,GC,0

mn(t, t) denotes the single-particle Green’s function of the respective non-interactingsystem, while the self-energy ΣC,0

mn(t, t) comprises the interactions, which distinguishesthe simple non-interacting system from the complex interacting one. The time integralsin the Dyson equation are evaluated along the Keldysh contour C, where the complextimes τ

1

and τ2

are subject to the contour-ordering operators in GC,0mn, ΣC

mn and GCmn.


Instead of evaluating the integral kernel at complex times τ1/2

, however, ones uses analyticcontinuation and projects the kernel of the Dyson equation (3.112) onto the real-time axis.To this end, one deforms the contour C such that one can identify Keldysh contours withrespect to pairs of time variables (t, τ

1

), (τ1

, τ2

) and (τ2

, t). This allows, in particular, toidentify lesser and greater projections of the self-energy ΣC

mn(τ1

, τ2

). The correspondingrules for this projection are summarized in the Langreth theorem [285, 286, 314]. Forexample, a contour-ordered function

FC(t, t) =

Z

C

dτ1

AC(t, τ1

)BC(τ1

, t), (3.113)

which can be represented by an integration over the product of the contour-ordered func-tions AC(t, t) and BC(t, t) along a Keldysh contour C, has the lesser/greater real-timeprojection

F</>(t, t) =

Z ∞

−∞dt

1

£Ar(t, t

1

)B</>(t1

, t) + A</>(t, t1

)Ba(t1

, t)§. (3.114)

For products of complex-valued functions, which, for example, may represent the self-energy of a contour-ordered Green’s function, the lesser/greater projection is given by

FC(τ, τ ) = AC(τ, τ )BC(τ, τ ) −→ F</>(t, t) = A</>(t, t)B</>(t, t), (3.115)

FC(τ, τ ) = AC(τ, τ )BC(τ , τ) −→ F</>(t, t) = A</>(t, t)B>/<(t, t). (3.116)

Applying these Langreth rules, the lesser/greater projection of the Dyson equation (3.112)is given by

G</>mn (t, t) = G<,0

mn(t, t) (3.117)

+X

op

Z ∞

−∞dt

1

Z ∞

−∞dt

2

Gr,0mo(t, t1)Σ

r

op(t1, t2)G</>pn (t

2

, t)

+X

op

Z ∞

−∞dt

1

Z ∞

−∞dt

2

Gr,0mo(t, t1)Σ

</>op (t

1

, t2

)Ga

pn(t2

, t)

+X

op

Z ∞

−∞dt

1

Z ∞

−∞dt

2

G</>,0mo (t, t

1

)Σa

op(t1, t2)Ga

pn(t2

, t).

If initial correlations in the infinite past do not influence the stationary state of a system,this equation greatly simplifies

G</>mn (t, t) =

X

op

Z ∞

−∞dt

1

Z ∞

−∞dt

2

Gr

mo(t, t1)Σ</>op (t

1

, t2

)Ga

pn(t2

, t), (3.118)

and is referred to as the Keldysh equation. An additional benefit of this real-time pro-jection is that one does not need to compute complex-time valued but only real-timevalued quantities, which are much easier to handle in practical applications. This be-comes evident by Fourier transforming the lesser and the greater Green’s function. Since


G</>mn (t, t) = G</>

mn (t−t) in the stationary state, the Keldysh equation for G</>mn () reduces

to an algebraic equation,

G</>mn () =

X

op

Gr

mo()Σ</>op ()Ga

pn(), (3.119)

in contrast to the integral equation (3.118). For completeness, we also give the retardedprojection of the Dyson equation (3.112)

Gr

mn(t, t) = Gr,0mn(t, t) +

X

op

Z ∞

−∞dt

1

Z ∞

−∞dt

2

Gr,0mo(t, t1)Σ

r

op(t1, t2)Gr

pn(t2

, t). (3.120)

The respective Fourier transform, in the stationary state, reads

Gr

mn() = Gr,0mn() +

X

op

Gr,0mo()Σ

r

op()Gr

pn(). (3.121)

3.3.3 Separation of Time Scales in the (Anti-)Adiabatic Regime

Having introduced the general concepts of nonequilibrium Green’s function theory inSecs. 3.3.1 and 3.3.2, we now employ this methodology for the description of vibrationallycoupled electron transport through a single-molecule contact. To this end, we recall thatthe Hamiltonian H

M

, describing the molecular bridge, can be diagonalized by the smallpolaron transformation (cf. Sec. 3.1.3 or Refs. [16, 83, 216, 233, 234]) and use this exactsolution of the molecular problem to derive an approach that treats electronic-vibrationalcoupling non-perturbatively. The single-particle Green’s function transforms under thistransformation as follows (cf. Eqs. 3.28)

Gmn(t, t) = −iT cm(t)c†n(t)H (3.122)

= −iΨ0

|T e−SeScm(t)e−SeSc†n(t)e−SeS|Ψ0

H= −iT cm(t)Xm(t)c†n(t)Xn(t)H .

As in section 3.2.5, the subscript H/H denotes the Hamiltonian used to evaluate therespective expectation value. If the dynamics of the nuclear and the electronic degreesof freedom decouple from each other, one can approximate the single-particle Green’sfunction by the product [73,80,216,236]

Gmn(t, t) ≈ −iT cm(t)c†n(t)HT Xm(t)X†n(t)H ≡ G

el,mn(t, t)T Xm(t)X†n(t)H .

(3.123)

Thereby, Gel,mn(t, t) denotes the ’electronic’ part of the single-particle Green’s function

Gmn(t, t),

Gel,mn(t, t) = T cm(t)c†n(t)H . (3.124)


The ’vibrational’ part is given by the correlation function of the shift operators

T Xm(t)X†n(t)H = T exp(i

X

α

λmαPα(t))exp(−iX

α

λnαPα(t))H . (3.125)

Note that in the limit Vmk → 0, this factorization of the single-particle Green’s functioninto an electronic and a vibrational part is exact [234]. For weak coupling to the leads,when electron tunneling events occur on much longer time scales than the nuclei need toadjust to the respective charge fluctuations, this factorization is also justified. This weak-coupling regime defines the antiadiabatic regime of a molecular junction. In the oppositelimit of strong coupling to the leads, that is the adiabatic regime, electron tunnelingevents occur on much shorter time scales such that the nuclei have no time to respond tosingle charge fluctuations. They move rather in a background potential induced by thecontinuous flow of electrons. In this regime, the above factorization of the single-particleGreen’s function corresponds to the adiabatic approximation. For intermediate couplingstrengths to the leads, where the residence time of the electrons on the molecular bridge iscomparable to the oscillation period of a vibrational mode, Eq. (3.123) does not representa good approximation. In the following we consider the antiadiabatic regime, assumingweak coupling between the molecule and the leads.

3.3.4 Cumulant Expansion for the Vibrational Part of Gmn

In this section, we derive an expression for the correlation function of the shift operators,T Xm(t)X†

n(t)H , in terms of the vibrational single-particle Green’s function, Eq. (3.110).To this end, we employ a cumulant expansion to second order in the electronic-vibrationalcoupling strengths λmα/Ωα [80,216,224,236]. Thereby, the cumulant Φ(t, t) of the shift-operator correlation function,

T Xm(t)X†n(t)H ≡ eΦ(t,t), (3.126)

is expanded in a Taylor series

Φ(t, t) ≈ 1 +X

α

µλmα

Ωα

φ1

m,α(t, t) +λnα

Ωα

φ1

n,α(t, t)

∂(3.127)

+X

αα

µλmαλmα

2ΩαΩαφ2

m,αα(t, t) +λnαλnα

2ΩαΩαφ2

n,αα(t, t) +λmαλnα

ΩαΩαφ2

mn,αα(t, t)

∂,

which is terminated at O(λ2

mα). It involves the following derivatives of the cumulant

φ1

m,α(t, t) = ∂λmα

Φ(t, t), (3.128)

φ2

m,αα(t, t) = ∂λmα

∂λmα

Φ(t, t), (3.129)

φ2

mn,αα(t, t) = ∂λmα

∂λnα

Φ(t, t). (3.130)


We determine these derivatives using an expansion of the shift-operator correlation func-tion to second order in the electronic-vibrational coupling strengths

T Xm(t)X†n(t)H = T ei

Pα

λ

mα

Ωα

Pα

(t)e−iP

α

λ

nα

Ωα

Pα

(t)H (3.131)

≈ 1 +X

αα

λmαλnα

ΩαΩαT Pα(t)Pα(t) (3.132)

−X

αα

λmαλmα

2ΩαΩαT Pα(t)Pα(t) −

X

αα

λnαλnα

2ΩαΩαT Pα(t)Pα(t).

Thereby, we have used that Pα(t)H = Pα(t)H and Pα(t)Pα(t)H = Pα(t)Pα(t)Hin the steady-state transport regime. We require that the expression eΦ(t,t) has the samesecond order expansion with respect to the dimensionless electronic-vibrational couplingstrengths λmα/Ωα and, thus, obtain

T Xm(t)X†n(t)H ≈ e

iP

αα

λ

mα

λ

nα

Ω

α

Ωα

D

αα

(t,t)−iP

αα

λ

mα

λ

mα

+λ

nα

λ

nα

2Ω

α

Ωα

D

αα

(t,t). (3.133)

It should be noted that this expansion yields the exact result in the limit of an isolatedmolecule, Vmk → 0 [234].

3.3.5 Electronic Green’s Function Gel,mn and Self-Energy Σmn

As a next step, we outline the computation of the electronic part of the single-particleGreen’s function, G

el,mn(t, t). This Green’s function, and the respective self-energy, isneeded in the derivation of the vibrational self-energy Παα(t, t), which will be introducedin following section, Sec. 3.3.6.

We compute the Green’s function Gel,mn(t, t), employing an equation of motion technique

[285]. The equation of motion of this function is determined by the time dependence ofthe time-ordering operator T and the Heisenberg equations of motion of the annihilationand creation operators cm and c†m

i∂tcm = [cm, H] = mcm +X

o=m

Umocmc†oco +X

k

V ∗mkckX

†m, (3.134)

i∂tc†m = [c†m, H] = −mc†m −

X

o=m

Umoc†mc†oco −

X

k

Vmkc†kXm. (3.135)

Due to the small polaron transformation, these EOMs involve the transformed Hamilto-nian H. The derivative of G

el,mn(t, t) with respect to time t is therefore given by

∂tGel,mn(t, t) = −iδ(t, t)[cm, c†n]+

H − T [cm(t), H]c†n(t)H , (3.136)

where the δ(t, t)-term originates from the derivative of the time-ordering operator. Em-

ploying the inverse operator°G0

el,m

¢−1

= i∂t− m +P

o =m Umoδo, we rewrite this equation


of motion as°G0

el,m

¢−1

Gel,mn(t, t) = δ(t, t)δmn − i

X

k

V ∗mkT ck(t)X

†m(t)c†n(t) (3.137)

−iX

o=m

UmoT cm(t)c†o(t)co(t)c†n(t).

Similarly, we apply the operator°G0

el,n

¢−1

= −i∂t − n +P

p=n Unpδp to the right handside of Eq. (3.137) and obtain

°G0

el,m

¢−1

Gel,mn(t, t)

°G0

el,n

¢−1

= (3.138)

δ(t, t)δmn

°G0

el,n

¢−1 − iX

o =m,p=n

UmoUnpT cm(t)c†o(t)co(t)c†n(t)c†p(t

)cp(t)

+δ(t, t)Umncmcn+ δ(t, t)δmn

X

o=m

Umoc†oco − iX

kk

V ∗mkVnkT ck(t)X

†m(t)c†k(t)Xn(t)

−iX

k,p=n

V ∗mkUnpT ck(t)X

†m(t)c†n(t)c†p(t

)cp(t) − i

X

k,o=m

UmoVnkT cm(t)c†o(t)co(t)c†k(t

)Xn(t),

where the derivative ∂t acts to the left. Due to the electron-electron interaction term inH

M

, this equation of motion includes a variety of different correlation terms. In the fol-lowing, we outline how we treat these correlations in an approximate but non-perturbativeway, that is, the elastic co-tunneling approximation [80,285,287–289].

To this end, we disregard electron-electron interactions for the time being and continueto evaluate the EOM (3.138) up to second order in the molecule-lead coupling (O(V 2

mk)):

°G0

el,m

¢−1

Gel,mn(t, t)

°G0

el,n

¢−1

= δ(t, t)δmn

°G0

el,n

¢−1

(3.139)

+X

k

V ∗mkVnkgk(t, t

)T X†m(t)Xn(t),

where gk(t, t) denotes the free Green’s function of lead state k. This defines an approxi-mate expression for the electronic self-energy Σmn(t, t),

Σmn(t, t) = ΣL,mn(t, t) + Σ

R,mn(t, t), (3.140)

ΣL/R,mn(t, t) =

X

k∈L/R

V ∗mkVnkgk(t, t

)T X†m(t)Xn(t), (3.141)

which describes the coupling of the electronic states located on the molecular bridge tothe continuum of electronic states in the left and the right lead. Using this self-energyexpression, we integrate the equation of motion (3.139) in time, and obtain

Gel,mn(t, t) = G0

el,mn(t, t) +X

op

Z ∞

−∞dt

1

Z ∞

−∞dt

2

G0

el,mo(t, t1)Σop(t1, t2)G0

el,pn(t2

, t),

(3.142)


where G0

el,mo(t, t1) denotes the single-particle Green’s function of the isolated molecularbridge. This integral equation for G

el,mn(t, t) is valid to O(V 2

mk). Higher-order terms areeasily generated by replacing one of the G0

el,mn(t, t) in the kernel of the time integral byG

el,mn(t, t), for example,

Gel,mn(t, t) = G0

el,mn(t, t) +X

op

Z ∞

−∞dt

1

Z ∞

−∞dt

2

G0

el,mo(t, t1)Σop(t1, t2)Gel,pn(t2

, t).

(3.143)

The lesser, greater and retarded projection of the corresponding contour-ordered self en-ergy ΣC

mn(t, t) determine the lesser/greater Green’s function via the Keldysh equation,Eq. (3.119),

G</>el,mn() =

X

op

Gr

el,mo()Σ</>op ()Ga

el,pn(), (3.144)

and the retarded/advanced Green’s function via the Dyson equation, Eq. (3.121),

Gr/a

el,mn() = G0,r/a

el,mn() +X

op

G0,r/a

el,mo()Σr/a

op ()Gr/a

el,pn(). (3.145)

Since the tunnel couplings, HML

and HMR

, are bilinear in c†k and cm, this replacement yieldsthe exact result in the non-interacting limit, where neither electron-electron interactions(Umn = 0) nor electronic-vibrational coupling (λmα = 0) is present.

At this point, we include electron-electron interactions Umn again. To this end, we employthe exact result for the retarded Green’s function G0,r

el,mn of the isolated molecular bridge(Vmk = 0). In order to derive this expression, we consider first a unit-cube in N

el

-dimensions, where N

el

corresponds to the number of the electronic states located at themolecular bridge. One corner of this cube coincides with the origin, while all other cornersare located in the positive coordinate space. There are 2Nel vectors, pNel

α , pointing to thecorners of this unit-cube. All coefficients of these vectors are either 0 or 1. The exactGreen’s function, G0,r

el,mn, can be written in terms of these vectors as

G0,rel,mn(t, t) = −iΘ(t− t)δmn

X

α=1..2Nel

e−i(m

−P

o=m

Umo

(pNelα

)

o

)(t−t) (3.146)

·Y

n

(1− nn)1−(pNelα

)

n n(pNelα

)

n

n ,

G0,rel,mn() = δmn

X

α=1..2Nel

Qn(1− nn)1−(p

Nelα

)

n n(pNelα

)

n

n

− m −P

o=m Umo(pNelα )o + i0+

, (3.147)

where i0+ represents a vanishingly small positive imaginary part and the nm denote thepopulations of the electronic states m. This expression can easily be verified evaluating Eq.


(3.138) in the limit Vmk → 0. For a single-molecule contact (Vmk = 0), the populationsnm are determined with respect to the stationary state of the junction. The retardedGreen’s function Gr

el,mn is obtained according to the Dyson equation (3.145), using theexact expression for the retarded zeroth-order Green’s function (3.147) and the retardedprojection of the self-energy Σmn(t, t) (cf. Eq. (3.140)). Using that Ga

el,mn =°Gr

el,nm

¢∗and

the lesser projection of Σmn(t, t), the corresponding lesser and greater Green’s functions

G</>el,mn are determined according to the Keldysh equation (3.144).

This approximate approach treats electron-electron interactions non-perturbatively buton a mean-field level. It is referred to as the elastic co-tunneling approximation [80, 285,287–289]. For non-degenerate electronic states, that is, if the off-diagonal elements of theself-energy Σmn can be disregarded, this approach corresponds effectively to the Hartree-Fock approximation (as outlined, for example, in Ref. [285]). The mean-field characterof the approach needs to be considered with care. For example, Kondo physics cannotbe accounted for [285]. Moreover, the approach can fail to describe transport throughquasidegenerate electronic states, for which the off-diagonal elements of the self-energyare needed. This is particularly important in the context of quantum interference ef-fects (cf. section 6.5 and appendix A). Apart from these deficiencies, however, the elasticco-tunneling approximation allows a rather accurate description of electron-electron inter-actions. This becomes evident by comparison with the ME approach (see Ref. [80]), whereto the given order in the molecule-lead coupling, O(V 2

mk), electron-electron interactionsare exactly accounted for.

3.3.6 Vibrational Green’s Function Dαα and Self-Energy Παα

For the computation of the vibrational Green’s function Dαα , we also employ the equationof motion technique [285], using the Heisenberg equation of motion of the vibrationalmomentum operator

i∂tPα = −iΩαQα − 2iX

βα

ηαβα

qβα

. (3.148)

Similar to the equation of motion for Gel,mn, we thus obtain an EOM for the time-ordered

vibrational Green’s function

i∂tDαα(t, t) = −ΩαT Qα(t)Pα(t)H − 2X

βα

ηαβα

T qβα

(t)Pα(t)H . (3.149)

In contrast to the electronic Green’s function, however, the inverse of the non-interactingvibrational Green’s function,

°D0

α

¢−1

= − 1

2Ωα

°∂2

t + Ω2

α

¢, (3.150)

involves two derivatives with respect to the time t. In order to derive a Dyson-like equationof motion for Dαα , as for G

el,mn in Sec. 3.3.5, we use the differential operator (D0

α)−1 in


the following and obtain

°D0

α

¢−1

Dαα(t, t) = δααδ(t, t)− i1

Ωα

X

βα

ηαβα

ωβα

T Pβα

(t)Pα(t)H (3.151)

+X

mk

λmα

Ωα

VmkT c†k(t)cm(t)Xm(t)Pα(t)H −X

mk

λmα

Ωα

V ∗mkT c†m(t)ck(t)X

†m(t)Pα(t)H .

Thereby, we use the Heisenberg equation of motion

i∂tQα = iΩαPα − 2X

mk

λmα

Ωα

Vmkc†kcmXm + 2

X

mk

λmα

Ωα

V ∗mkc

†mckX

†m, (3.152)

i∂tqβα

= iωβα

Pβα

, (3.153)

for the vibrational displacement operators Qα and qβα

, respectively. Applying (D0α)

−1 =− 1

2Ω

α

(∂2

t − Ω2

α) from the right, where the derivative ∂2

t acts to the left, we obtain thefollowing equation of motion for the vibrational Green’s function

°D0

α

¢−1

Dαα(t, t)°D0

α¢−1

= δααδ(t, t)

√°D0

α¢−1

+2

Ω2

α

X

βα

η2

αβα

ωβα

!(3.154)

−iX

βα

βα

ηαβα

ηαβα

ωβα

ωβα

ΩαΩαT Pβ

α

(t)Pβα

(t)

+X

mkβα

ηαβα

ωβα

λmα

ΩαΩα

≥VmkT Pβ

α

(t)c†k(t)cm(t)Xm(t)+ V ∗

mkT Pβα

(t)c†m(t)ck(t)X†

m(t)¥

+X

mkβα

ηαβα

ωβα

λmα

ΩαΩα

≥VmkT c†k(t)cm(t)Xm(t)Pβ

α

(t)+ V ∗

mkT c†m(t)ck(t)X†m(t)Pβ

α

(t)

¥

+iX

mnkk

λmαλnα

ΩαΩαVmkVnkT c†k(t)cm(t)Xm(t)c†k(t)cn(t)Xn(t)

−iX

mnkk

λmαλnα

ΩαΩαVmkV

∗nkT c†k(t)cm(t)Xm(t)c†n(t)ck(t)X†

n(t)

−iX

mnkk

λmαλnα

ΩαΩαV ∗

mkVnkT c†m(t)ck(t)X†m(t)c†k(t)cn(t)Xn(t)

+iX

mnkk

λmαλnα

ΩαΩαV ∗

mkV∗nkT c†m(t)ck(t)X

†m(t)c†n(t)ck(t)X†

n(t).

We further evaluate the correlation functions on the right hand side of this equationto second order in the molecule-lead coupling (O(V 2

mk)) and in the mode-bath coupling(O(η2

αβα

)). We thus arrive at the following approximate expression for the vibrational


self-energy Παα(t, t) (similar to Eqs. (3.138) and (3.139))

°D0

α

¢−1

Dαα(t, t)°D0

α¢−1 ≈ δααδ(t, t)

°D0

α¢−1

+ Παα(t, t), (3.155)

Παα(t, t) = Πbath,αα(t, t) + Π

el,αα(t, t), (3.156)

Πbath,αα(t, t) = −i

X

βα

η2

αβα

T Pβα

(t)Pβα

(t), (3.157)

Πel,αα(t, t) = −i

X

mn

λmαλnα

ΩαΩα(Σnm(t, t)G

el,mn(t, t) + Σmn(t, t)Gel,nm(t, t)) . (3.158)

Thereby, we employ the non-crossing approximation and disregard terms mixing themolecule-lead and mode-bath couplings, ∼ ηαβ

α

Vmk. This is justified as both couplingsare ideally weak such that these terms represent only a small contribution of O(η2

αβα

V 2

mk)to the self-energy. In addition, we employ the approximation ω2

αβα

/Ω2

α ≈ 1 in the expres-sion for Π

bath,αα(t, t). This is not a decisive step, as the spectral densities Jα(ω) mayabsorb this factor (see Eq. (3.24)). The renormalization of the frequencies Ωα due tothe coupling of the vibrational modes to the thermal bath is also not accounted for inour considerations, which is in line with the notion that a thermal bath induces no otherdynamical effects but relaxation.

As for the electronic Green’s function, we use the retarded projection of the self-energies,Πr

bath,αα() and Πr

bath,αα(), to compute the retarded projection of the vibrational Green’sfunction, Dr

αα() according to the Dyson equation

Dr

αα() = D0,rαα() +

X

α1α2

D0,rαα1

()°Πr

bath,α1α2() + Πr

el,α1α2()

¢Dr

α2α(). (3.159)

Accordingly, the Keldysh equations

D</>αα () =

X

α1α2

Dr

αα1()

≥Π</>

bath,α1α2() + Π</>

el,α1α2()

¥Da

α2α() (3.160)

give the lesser and the greater projections of the vibrational Green’s function. Similarequations apply in the time domain. Having determined the vibrational Green’s functions,we can readily compute the correlation function of the shift operators according to Eq.(3.133). Note that a description of the vibrational degrees of freedom ’in thermal equi-librium’, as outlined in Sec. 3.2.6 for the ME method, can be achieved in the frameworkof this NEGF method by using Π

el,αα(t, t) = 0 instead of the expression given by Eq.(3.158).


3.3.7 Self-Consistent Solution Scheme

As we have seen in Sec. 3.3.6, the vibrational self-energy Παα(t, t) (Eq. (3.156)) dependson both the electronic self-energy Σmn(t, t) (Eq. (3.140)) and the electronic Green’s func-tion G

el,mn(t, t) (Eq. (3.144)). The corresponding vibrational Green’s function Dαα(t, t),as outlined in Sec. 3.3.4, enters the correlation function of the shift operators Xm and X†

n

(Eq. (3.133)). This correlation function, in turn, is required to determine the electronicself-energy Σmn(t, t) and the electronic Green’s function G

el,mn(t, t) (cf. Eqs. (3.144) and(3.145)). It is therefore not possible to determine these Green’s functions and self-energiesstraightforwardly. The solution of this closed set of equations requires a self-consistentsolution scheme. The specific steps of such a scheme are detailed in this section.

The starting point is the electronic self-energy Σ(0)

mn, which represents the self-energy of anon-interacting molecular bridge. In the stationary state of a molecular junction, wheretime-translational invariance can be employed, the non-interacting electronic self-energyis given by

Σ(0)

L/R,mn(t) =X

k∈L/R

V ∗mkVnkgk(t). (3.161)

It has the Fourier transformed real-time projections

Σ(0),rL/R,mn() =

X

k∈L/R

V ∗mkVnkg

r

k() ≡ ∆L/R,mn()− i

2Γ

L/R,mn(), (3.162)

Σ(0),<L/R,mn() = if

L/R

()ΓL/R,mn(), (3.163)

Σ(0),>L/R,mn() = −i(1− f

L/R

())ΓL/R,mn(). (3.164)

Accordingly, a first estimate for the population of the electronic states can be given by

nm =Γ

L,mm(m)fL

(m) + ΓR,mm(m)f

R

(m)

ΓL,mm(m) + Γ

R,mm(m). (3.165)

Next, we compute the Fourier transforms of the real-time projections of the non-interactingvibrational Green’s function

°D(0),r()

¢−1

αα = δαα

µ1

− Ωα + i0+

− 1

+ Ωα + i0+

∂−1

− Πr

bath,αα , (3.166)

D(0),<,>αα () =

X

α1α2

D(0),rαα1

()Π<,>bath,α1α2

D(0),aα2α(), (3.167)

using the self-energy contribution Πbath,αα due to coupling to the thermal bath only.

At this point, we employ a self-consistent cycle, including the following steps:

1. We compute the lesser and greater projection of the electronic self-energy in thetime domain

Σ</>mn (t) = Σ(0),</>

mn (t)ei

Pαα

λ

nα

λ

mα

Ω

α

Ωα

D

>/<

αα

(−t)−iP

αα

λ

mα

λ

mα

+λ

nα

λ

nα

2Ω

α

Ωα

D

>/<

αα

(0)

, (3.168)


using Eq. (3.133) and the Langreth rule (3.116). If we perform this step for the

first time, we use D(0),</>αα (t) instead of D</>

αα (t). The Fourier transformed lesserand greater projections of the electronic self-energy are then used to calculate thecorresponding retarded and advanced projections,

Σr

mn(E) = Σ>mn(E)− Σ<

mn(t), (3.169)

Σa

mn(E) = (Σr

nm(E))∗ . (3.170)

2. The electronic populations nm are used to determine the retarded Green’s functionG0,r

el,mn(E) according to Eq. (3.147). This result, in conjunction with the self-energiesdetermined in the preceding step, are used to compute the retarded projection ofthe electronic Green’s function, Gr

el,mn(E), according to the Dyson equation (3.145).

The respective Keldysh equation (3.144) gives G</>el,mn(E).

3. With the Fourier transforms of the lesser and the greater projections of the electronic

Green’s, G</>el,mn(E)

FT→ G</>el,mn(t), and the self-energy functions Σ</>

mn (E)FT→ Σ</>

mn (t),we determine the vibrational self-energy Παα(t) from Eqs. (3.156).

4. We update the real-time projections of the vibrational Green’s function, Dαα(E),with the Fourier transforms of the real-time projections of the self-energies Παα(t)that we obtained in step 3, employing the Dyson- and the Keldysh equations (3.159)and (3.160), respectively. The Fourier transforms of the thus updated vibrational

Green’s functions D</>αα (t) is used in step 1 of the next cycle.

5. In the last step, we recalculate the electronic populations nm = Im£G<

el,mm(t = 0)§

from the lesser Green’s functions G<el,mm(t) that are obtained in step 3. We terminate

the self-consistent cycle, if the difference between the electronic populations obtainedin two subsequent cycles is smaller than 10−7.

Thus, we obtain a self-consistent solution for the single-particle Green’s functions, Gel,mn

and Dαα , and the corresponding self-energies, Σmn and Παα .

3.3.8 Observables of Interest

As already mentioned in Sec. 3.2.5, we characterize electron transport through a single-molecule junction by three different transport characteristics, that is the population ofthe electronic states, the average level of vibrational excitation and the electrical currentflowing through the junction as functions of the applied bias voltage Φ. In this section, weoutline how these observables are computed, once the single-particle Green’s functions,G

el,mn and Dαα , and the respective self-energies, Σmn and Παα , have been determinedaccording to the nonequilibrium Green’s function approach outlined in Secs. 3.3.3 – 3.3.7.


Electronic Population

The lesser and greater real-time projections of the single-particle Green’s function aredirectly related to physical observables. For example, the population nm of the electroniclevels of a molecular junction are given by the lesser Green’s function G<

el,mn:

nm = c†mcmH = c†mcmH = Im£G<

el,mm(t = 0)§

=

Z ∞

−∞

d

2πIm

£G<

el,mm()§. (3.171)

Vibrational Excitation

In contrast to the electronic populations, the average level of vibrational excitation can-not be directly obtained from the vibrational Green’s function Dαα. Indeed, the lesserprojection of the vibrational Green’s function,

Im [D<αα(t = 0)] = −a†αa†αH − aαaαH + 2a†αaαH + 1, (3.172)

contains the expectation value of the vibrational occupation number operator a†αaα, butalso two other terms. The same terms do also appear in the expectation value of theproduct of the displacement operators Qα and Qα,

QαQαH = a†αa†αH + aαaαH + 2a†αaαH + 1, (3.173)

but with a different sign as in Eq. (3.172). Therefore, we obtain the average level ofvibrational excitation, summing up Eqs. (3.172) and Eq. (3.173):

a†αaα =1

4(QαQα+ Im [D<

αα(t = 0)])− 1

2. (3.174)

We are thus left with the computation of the expectation value QαQαH . To this end,we consider the time derivative of the expectation values

0 = i∂tQαPαH = iΩαPαPαH − iΩαQαQαH − 2iX

βα

ηαβα

Qαqβα

H , (3.175)

0 = i∂tQαPβα

H = iΩαPαPβα

H − iωβα

Qαqβα

H − 2iηαβα

QαQαH , (3.176)

0 = i∂tPαqβα

H = iωβα

PαPβα

H − iΩαQαqβα

H − 2iX

βα

ηαβα

qβα

qβα

H , (3.177)

which vanish in the steady-state transport regime. Thereby, we have used that

X

k

VmkPα/βα

c†kcmXmH −X

k

V ∗mkPα/β

α

c†mX†mckH = (3.178)

X

k

≥Vmkc†kcmXmH − V ∗

mkc†mX†mckH

¥Pα/β

α

H = 0,


which reflects the factorization approximation employed in Eq. (3.123) and conservationof the partial currents through each of the electronic states

0 = i∂tc†mcmH =X

k

≥V ∗

mkc†mX†mckH − Vmkc†kcmXmH

¥. (3.179)

We continue to evaluate these equations to second order in the mode-bath coupling,O(η2

αβα

), and derive from Eqs. (3.176) and (3.177)

Qαqβα

H = (2ηαβα

Ωαqβα

qβα

H − 2ηαβα

ωβα

QαQαH)P 1

ω2

βα

− Ω2

α

, (3.180)

where P stands for the principal value. This result can be used to derive the followingrelation from Eq. (3.175)

QαQαH = PαPαH − 4X

βα

P η2

αβα

ω2

βα

− Ω2

α

qβα

qβα

H + 4X

βα

ωβα

Ωα

P η2

αβα

ω2

βα

− Ω2

α

QαQαH

≈√

1 + 4X

βα

ωβα

Ωα

P η2

αβα

ω2

βα

− Ω2

α

!PαPαH − 4

X

βα

P η2

αβα

ω2

βα

− Ω2

α

qβα

qβα

H .

(3.181)

The average level of vibrational excitation can thus be calculated according to

a†aH = a†aH +X

mn

λmαλnα

Ω2

α

(c†mcm − δm)(c†ncn − δn)H (3.182)

≈ −µ

Aα +1

2

∂Im [D<

αα(t = 0)]−µ

Bα +1

2

∂(3.183)

+X

mn

λmαλnα

Ω2

α

(c†mcm − δm)(c†ncn − δn)H ,

Aα =X

βα

ωβα

Ωα

P η2

αβα

ω2

βα

− Ω2

α

, (3.184)

Bα =X

βα

P η2

αβα

ω2

βα

− Ω2

α

(1 + 2Nβ) . (3.185)

In contrast to the master equation approach, due to the factorization employed in thisNEGF scheme (cf. Sec. 3.3.3), terms ∼ Qαc†mcmH are not present in this expression (cf.Sec. 3.2.5).


Current

The electrical current flowing through the junction

IK = 2ie

"X

k∈K,m

Vmkc†kcmXmH −X

k∈K,m

V ∗mkc†mX†

mckH#

(3.186)

= −4eRe

"X

k∈K,m

V ∗mkG

<I,km(t, t)

#(3.187)

can be determined with the lesser projection of the Green’s function,

GI,km(t, t) = −iT ck(t)c†m(t)X†

m(t)H . (3.188)

To relate this Green’s function to the single-particle Green’s function Gmn, and the re-spective self-energy Σmn, we employ the equation of motion

g−1

k GI,km(t, t) =X

n

VnkT cn(t)Xn(t)c†m(t)X†m(t)H =

X

n

VnkGnm(t, t), (3.189)

where we use the inverse Green’s operator g−1

k = i∂t − k. Integrating this equation ofmotion, the time-ordered Green’s function GI,km(t, t) can be expressed in terms of thesingle-particle Green’s functions Gnm and gk:

GI,km(t, t) =X

n

Vnk

Zdt

1

gk(t, t1)Gnm(t1

, t). (3.190)

This expression is formally the same for the corresponding contour-ordered Green’s func-tion (cf. Sec. 3.3.2, or Ref. [285])

GCI,km(t, t) =

X

n

Vnk

Z

C

dτ1

gCk (t, τ

1

)GCnm(τ

1

, t). (3.191)

Projection of the integral kernel on the real-time axis (with t, t ∈ C1

of the KeldyshContour, cf. Fig. 3.2), gives the corresponding lesser Green’s function

G<I,km(t, t) =

X

n

Vnk

Z ∞

−∞dt

1

(gr

k(t, t1)G<nm(t

1

, t) + g<k (t, t

1

)Ga

nm(t1

, t)) , (3.192)

which is required for the computation of the current

IK = −4eRe

"X

mnk

V ∗mkVnk

Z ∞

−∞dt

1

(gr

k(t, t1)G<nm(t

1

, t) + g<k (t, t

1

)Ga

nm(t1

, t))

#. (3.193)

3.4. Landauer Theory of Coherent Electron Transport 53

In the steady-state transport regime, we employ the Fourier transform of Gmn and gk,

IK = −2eX

mnk

V ∗mkVnk

Z ∞

−∞

d

2π(gr

k()G<nm() + g<

k ()Ga

nm()) (3.194)

+2eX

mnk

VmkV∗nk

Z ∞

−∞

d

2π(ga

k()G<mn() + g<

k ()Gr

mn()) ,

and use that Ga

nm() = (Gr

mn())∗ and (g<k ())∗ = −g<

k (). With relation (3.102), thisexpression for the current can thus be further simplified to

IK = 2eX

mnk

V ∗mkVkn

Z ∞

−∞

d

2π(g<

k ()G>nm()− g>

k ()G<nm()) (3.195)

≈ 2eX

mn

Z ∞

−∞

d

2π

°Σ<

L,mn()G>el,nm()− Σ>

L,mn()G<el,nm()

¢, (3.196)

where according to the antiadiabatic regime we employ the same separation of electronicand vibrational time scales as in section 3.3.3, that is, we use the factorized expressionfor the Green’s function Gmn, Eq. (3.123).

3.4 Landauer Theory of Coherent Electron Trans-port

In this section, we derive the widely-used Landauer formula, which describes the electricalcurrent flowing through an ideal, that is, non-interacting conductor, and introduce theconcept of a transmission probability and/or amplitude. To this end, we use the resultsderived in Sec. 3.3, employing nonequilibrium Green’s function theory.

Due to the simple vertex structure of the molecule-lead couplings HML

and HMR

, theNEGF method gives the exact result2 in the non-interacting limit (that is without elec-tronic-vibrational coupling, λmα = 0, and without electron-electron interactions, Umn = 0).

The real-time projections of the electronic self-energies, Σ(0),rmn (E) and Σ(0),</>

mn (E), corre-sponding to this exact result, are already summarized in Eqs. (3.162). The respectiveelectronic Green’s functions read

°G(0),r()

¢−1

mn= δmn

µ1

− m + i0+

∂−1

− Σ(0),rmn (), (3.197)

G(0),</>mn () =

X

op

G(0),rmo ()Σ(0),</>

op ()G(0),apn (). (3.198)

Using this expression in the formula for the current, Eq. (3.196), we obtain the current of

2 With electronic-vibrational coupling, λiα = 0, and electron-electron interactions, Uij = 0, the NEGFmethod is also exact in the limit Vik → 0.


a molecular junction in the non-interacting limit

I(0) = 2eX

mn

Z ∞

−∞

d

2π

≥Σ(0),<

L,mn()G(0),>el,nm()− Σ(0),>

L,mn()G(0),<el,nm()

¥(3.199)

= 2eX

mn

Z ∞

−∞

d

2πt() (f

L

()− fR

()) . (3.200)

It can also be expressed in terms of the transmission function

t() =X

mnop

ΓL,mn()G(0),r

el,no()ΓR,op()G(0),ael,pm(). (3.201)

Here, we have used the eigenstates of the molecular bridge to represent the transmissionfunction. As it is a positive definite quantity, it can also be represented as

t() ≡ trΛ()Λ†() (3.202)

with the transmission amplitude matrix

Λmn() =X

op

ΓL,mo()G

(0),rel,op()Γ

†R,pn() (3.203)

and ΓL/R,mn() =

Po Γ†

L/R,mo()ΓL/R,on() [66,315]. Employing the eigenvalues of Λ()Λ†(),the transmission function can be rewritten as a simple sum

t() =X

j

Ξj, (3.204)

as can be the current

I = 2eX

j

Z+∞

−∞

d

2πΞj (f

L

()− fR

()) . (3.205)

Eq. (3.199) or (3.205) are commonly referred to as the Landauer formula [203, 284, 316]for the current through an ideal, or non-interacting, conductor. For small bias voltagesand temperatures, this formula can be further reduced to the sum

I =e2

πΦ

X

j

Ξj, (3.206)

where Φ denotes the applied bias voltage. The corresponding expression for the conduc-tance g = dI/dΦ represents the conductance quantization of an ideal conductor [284].Conductance quantization was experimentally verified first for point-contacts in a two-dimensional electron gas [1–3] and later for atomic point-contacts [4–7].

3.5. Survey of Other Theoretical Approaches 55

3.5 Survey of Other Theoretical Approaches

To conclude this chapter about the theoretical methodology, we give a brief overviewof other methods in this section that have been used to describe vibrationally coupledelectron transport through a single-molecule junction. There is, for example, a variety ofapproaches [9,209,210,317–321], where this nonequilibrium transport problem is treatedon the same footing as an electron-molecule scattering experiment. Thereby, the molecularbridge is described as a scattering region, where an electron incoming from one of theleads scatters elastically or inelastically on its way to the other lead. Electron-moleculescattering is a longstanding and successful field of chemical physics. Consequently, there isa variety of advanced methods, which describe this scattering problem efficiently and oftenalso numerically exactly. These methods have also been applied to electron transport inmolecular junctions. They are, however, designed to describe the scattering process on asingle-electron level, which means that the electrical current is described by these methodsas a sum of independent electron scattering events. Thus, changes of the state of themolecular bridge due to inelastic processes are not taken into account. Scattering theoryis therefore very useful, if these state changes do not influence the next electron tunnelingthrough the junction, for example, due to fast relaxation processes. In addition, scatteringtheory yields the exact result for coherent transport through a non-interacting molecularbridge (cf. the discussion of the Landauer formula in Sec. 3.4). Based on scattering theory,a flux-correlation approach [322] has also been developed. This approach distinguishesitself by describing long-lived correlations in the transient currents of a molecular junctionefficiently.

In addition to the approximate schemes mentioned so far, there are also a number of (nu-merically) exact methods. A straightforward concept is, for example, to directly addressthe many-body wave function of a molecular contact [323–327], where efficient solutionschemes, for example based on a multiconfigurational wave function expansions [328,329],allow to treat numerous active degrees of freedom. Real-time path integral simulations[330–332], quantum Monte-Carlo simulations [333,334] or methods based on exact diago-nalization [279] constitute another route to obtain exact results. However, the statisticaland/or systematic error inherent of these methods increases with the times simulated suchthat long-lived correlations, which may occur in molecular junctions [32, 144], representa challenge for these approaches. Numerical renormalization-group approaches are, inprincipal, also exact but strictly apply only at zero bias voltage, while an extension ofthese methodologies to finite bias voltages is currently being discussed [335–341]. Thisis similar to the variational approaches outlined in Refs. [300, 342], where the groundstate energy is minimized at zero bias. Typically, all these methods are numerically verydemanding. They provide, however, valuable benchmark calculations for comparativestudies [308,329].

Other approaches, where the electronic and the vibrational degrees of freedom are treatedon different theoretical levels, have also been developed. For example, approaches basedon Langevin equations [244,343,344], Navier-Stokes equations [345] or molecular dynamics


simulations [346–348] indeed account for the quantum mechanical nature of the electronicdegrees of freedom but consider vibrational motion on a classical level. These approachesaddress the adiabatic regime of vibrationally coupled electron transport, where Ω Γ(cf. the discussion given in Sec. 3.3.3). Note that transport through other nanostructures,such as quantum dots [84–86, 183–185] or quantum point contacts [4–7, 158], can bedescribed by the same methodology used for single-molecule junctions. These systems,however, have also been successfully described in terms of semiclassical approaches, forexample based on Boltzmann equations [285], or linear response methods, such as theKubo formalism [18].

3.6 Summary

In this chapter, we have outlined the theoretical methodology that we use to describe asingle-molecule contact. It is based on the model Hamiltonian (3.25). We have derivedthis Hamiltonian from microscopic principles, employing single-particle states or molecularorbitals of the molecular bridge and the respective normal modes. Thereby, we have usedthe ground state of the neutral molecular bridge as a reference state. We have alsooutlined how the molecular part of the model Hamiltonian, H

M

, can be diagonalizedby the small polaron transformation. This facilitates a non-perturbative description ofelectronic-vibrational coupling.

We have also introduced the master equation approach and the nonequilibrium Green’sfunction approach that we use to calculate transport characteristics of a single-moleculejunction. To this end, we have given a brief introduction to both density matrix andnonequilibrium Green’s function theory and outlined the specific approximations thatwe use in both approaches. In particular, the master equation approach is based ona strict second order expansion of the Nakajima-Zwanzig equation (3.61). It thereforecaptures all resonant transport processes. The nonequilibrium Green’s function approachemploys a factorization scheme that is valid in the antiadiabatic regime, that is the regime,where the nuclei of the molecular bridge can follow the state changes induced by thetunneling electrons. This approach captures both resonant and non-resonant transportprocesses. Both approaches treat electronic-vibrational and electron-electron interactionsnon-perturbatively, but on a different footing.

We concluded the chapter by pointing out the relation between the nonequilibrium Green’sfunction approach and the Landauer theory. Both theories are capable to describe a non-interacting nanoscale conductor exactly. Moreover, we have given a brief overview ofother transport theories that have been used to describe electron transport through asingle-molecule junction.

Chapter 4

Processes and Mechanisms inVibrationally Coupled Transport

In this chapter we give a comprehensive overview of the processes and mechanisms thatare relevant in vibrationally coupled electron transport through a single-molecule junc-tion. Thereby, we distinguish transport and electron-hole pair creation processes. Thecurrent-voltage characteristic of a molecular junction and the corresponding populationof the electronic states can be readily understood in terms of transport processes, atleast on a qualitative level (vide infra). We therefore study these processes and trans-port characteristics in the first part of this chapter, Secs. 4.1 and 4.2. Electron-hole paircreation processes do not directly contribute to the current flowing through a molecularjunction but to the respective level of vibrational excitation. As the efficiency of trans-port processes is, inter alia, determined by the level of vibrational excitation, pair creationprocesses have an indirect influence on the current-voltage characteristics and the corre-sponding population of the electronic states. Electron-hole pair creation processes andvibrational excitation characteristics are therefore the main subject of the second part ofthis chapter, Sec. 4.3.

In addition, we distinguish processes in the non-resonant (Sec. 4.1) and the resonanttransport regime of a molecular junction (Sec. 4.2). Non-resonant electron transportprocesses are typically important at low bias voltages. There, electrons can tunnel throughthe molecular bridge only by virtually occupying states of the molecular bridge. Asthe corresponding tunnel current is rather low (pA – nA), this type of tunneling is alsoreferred to as co-tunneling. In contrast, resonant electron transport is characterized bylarger currents (nA – µA). In this regime, an electron tunnels through the junction intwo sequential tunneling events, populating intermediately an eigenstate of the molecularbridge. Due to the strict second order expansion in the molecule-lead coupling, the MEapproach does not account for co-tunneling processes. In contrast, the NEGF methodincludes both co-tunneling and sequential tunneling processes. The comparison of resultsobtained by the two methods thus reveals the influence of non-resonant transport processeson the transport characteristics of a molecular contact.

58 4. Processes and Mechanisms in Vibrationally Coupled Transport

4.1 Non-Resonant Transport Processes / Co-Tunnel-ing

In this section, we discuss current-voltage and conductance-voltage characteristics1 of amolecular junction in the non-resonant transport regime. This regime is defined as therange of bias voltages, where the electronic levels of the molecular bridge are located out-side the bias window, that is, for bias voltages, where the energies of the electronic statesare either above the chemical potentials in the leads,

1

> µL/R

, or below, 1

< µL/R

. Inthis regime, only direct tunneling processes from the left to the right lead, or co-tunnelingprocesses, occur. Examples of such co-tunneling processes are depicted in Fig. 4.1.

In the first part of this section, Sec. 4.1.1, we discuss basic co-tunneling processes employ-ing a minimal model of a molecular junction. This model involves a single electronic stateand a single vibrational mode. In the second part, Sec. 4.1.2, we extend our studies tomultiple vibrational modes and describe multimode vibrational effects in the non-resonanttransport regime. These effects are crucial for an understanding of the IETS-spectra of amolecular junction, which typically involves tens of vibrational modes.

4.1.1 Basic Processes

The first model for a molecular junction that we consider comprises a single electronicstate and a single vibrational mode (model E1V1). The electronic state is located

1

−F

=0.6 eV above the Fermi level of the junction and coupled to a left and a right lead withcoupling strength ν

L/R,1 = 0.1 eV. Each lead is modeled by a semi-elliptic conductionband with a band width that is determined by the internal hopping parameter γ = 3.0 eV[216, 319]. Accordingly, the level-width functions of the left and the right lead are givenby

ΓL/R,mn() =

νL/R,m

νL/R,n

γ2

q4γ2 − °

− µL/R

¢2

, (4.1)

where m = n = 1 corresponding to a single electronic state (in general m,n ∈ 1, .., Nel

for N

el

electronic states). The electronic state is also coupled to the vibrational modewith coupling strength λ

11

= 0.06 eV. The frequency of the mode is given by Ω1

=0.1 eV. Thus, the antiadiabatic condition Γ

L/R,11

< Ω1

is fulfilled. Moreover, the electronicstate is located well above the Fermi level of the junction, that is by several units ofΩ

1

. Therefore, this model system allows to study vibrational effects in the non-resonanttransport regime. The parameters of this model molecular junction are also summarizedin Table 4.1. They represent typical values for molecular junctions as they are foundfor example in experiments [13, 24, 29, 31, 32, 34, 36–39, 49, 57, 99, 167, 168] or employingab-initio calculations [76,163,165,189–191,226–232]. The same applies for the parametersused to describe all other model systems in this thesis (see Tabs. 4.1 – 7.2).

1 Note that we consider the differential conductance g = dI/dΦ.

4.1. Non-Resonant Transport Processes / Co-Tunneling 59

(a) (b) (c) (d)

ε1

RL M

Ω1

ε1

RL M

Ω1

ε1

RL M

Ω2

Ω1

ε1

RL M

Figure 4.1: Examples of co-tunneling processes that occur in the non-resonant transportregime of a molecular junction. Panel a depicts direct tunneling of an electron from the leftto the right lead. Panel b and c show inelastic co-tunneling processes, where the tunnelingelectron excites and deexcites a vibrational degree of freedom, respectively. Panel d depictsa co-tunneling process, which involves the simultaneous excitation of a vibrational modeand the deexcitation of another.

Fig. 4.2 represents current-voltage and conductance-voltage characteristics of this junc-tion. The three lines shown in this figure correspond to three different transport scenarios:electronic transport, where we do not consider electronic-vibrational coupling, vibronictransport, where we take into account electronic-vibrational coupling, and thermally equi-librated transport, where the vibrational mode is enforced to relax to its thermal equilib-rium state after every electron transmission event (cf. Sec. 3.2.6). This thermal equilibriumstate is characterized by the same temperature as used in the leads, that is 10K. Thus,the vibrational mode returns effectively to its ground state, even if it has been excited inthe course of an electron tunneling event. We use the comparison between the vibronicand the thermally equilibrated transport scenario to highlight vibrational nonequilibriumeffects.

In the non-resonant transport regime, electronic transport (solid purple lines) is charac-terized by an almost linear increase of the current with increasing bias voltage2. This canbe understood by the number of electrons that can directly tunnel from the left to theright lead (cf. Fig. 4.1a for an example of such a co-tunneling process). This number alsoincreases almost linearly with the applied bias voltage Φ. The respective conductanceis therefore almost constant. In this regime, tunneling electrons populate the electronicstate only virtually such that it remains essentially unoccupied. Only for larger bias volt-ages, where the chemical potential in the left lead approaches the electronic level of the

2 In general, there is, besides a linear dependence, also an exponential dependence of the current I on theapplied bias voltage Φ (see for example the Simmons formula for tunneling through a potential barrier[349, 350]). For low bias voltages and 1 F, however, this exponential dependence is typically notsignificant.


model 1

νL,1 ν

R,1 γ Ω1

Ω2

λ11

λ12

kB

Tbath

E1V1 0.6 0.1 0.1 3 0.1 - 0.06 - 0.1E1V2 1.0 0.1 0.1 2 0.1 0.25 0.06 0.15 0.1INST 0.15 0.1 0.1 3 0.1 - 0.06 - -BAND 0.15 0.02 0.02 0.2 0.1 - 0.06 - -REC 0.6 0.1 0.03 3 0.1 - 0.06 - -

RECBAND 0.15 0.02 0.006 0.2 0.1 - 0.06 - -

Table 4.1: Model parameters for molecular junctions with a single electronic state (en-ergy values are given in eV). The temperature in the leads, k

B

T = 10 K, the characteristicfrequencies of and the coupling strengths to the thermal bath, ωC,α = 1 eV and ζα = 0, arethe same for all these model molecular junctions.

molecular bridge, the conductance through the junction increases slightly. This is becausethe electronic state exhibits broadening due to the coupling to the leads and, thus, facil-itates (resonant) tunneling of electrons even before the electronic state has entered thebias window (cf. Sec. 4.2.1).

Including electronic-vibrational coupling (solid black and dashed gray lines), the electronicstate is polaron-shifted to lower energies,

1

= 0.6 eV→ 1

= 0.564 eV, according to Eq.(3.35). As an electron thus tunnels through a virtual state of the junction more easily,one expects a larger current-/conductance. On the other hand, however, the equilibriumposition of the vibrational mode is also shifted, ∆Q

1

= −2λ11

/Ω1

. This shift results in asuppression of electronic co-tunneling events (cf. Fig. 4.1a), because different charge statesof the molecular bridge exhibit effectively less overlap (cf. Fig. 3.1). This can be quantifiedby the respective Franck-Condon (FC) matrix element |X00

1

|2 = e−λ211/Ω

21 ≈ 0.7, which is

significantly smaller than one. Therefore, at least for low bias voltages, eΦ < Ω1

, whereelectronic co-tunneling processes are dominant, the vibronic and thermally equilibratedtransport scenario give a lower current than the electronic transport scenario.

At higher bias voltages, however, electronic-vibrational coupling facilitates inelastic co-tunneling processes (cf. Fig. 4.1b), where an electron emits one, two, or more vibrationalquanta upon tunneling through the junction. Since these processes require electronswith sufficiently high energies, these processes become active one by one at bias voltageseΦ = nΩ

1

(n ∈ ), that is multiples of the vibrational frequency Ω1

. These processesresult in an increased transmission probability for electrons with an energy >

F

+nΩ

1

/2. Accordingly, as the conductance for these electrons is also larger, the respectiveconductance-voltage characteristic exhibits a step-like increase at eΦ = nΩ

1

.

In fact, inelastic co-tunneling processes become active at slightly different bias voltagesor energies, because the vibrational frequencies Ωα are renormalized due to the couplingof the molecule to the leads. The corresponding frequency shifts are described, for exam-ple, by the real parts of the vibrational self-energy matrix Π

el,αα . In the antiadiabatic


(a)

NEGF electronic

NEGF th. equ.NEGF vibronic

0 0.1 0.2 0.3 0.40

1

2

3

bias voltage ! !V"

currentI!nA"

(b)

NEGF electronic


0 0.1 0.2 0.3 0.40

5

10

15

bias voltage ! !V"

conductanceg!nA#V"

Figure 4.2: Current-voltage and conductance-voltage characteristics of a model molec-ular junction in the non-resonant transport regime. This junction (model E1V1, cf. Tab.4.1) comprises a single electronic state that is coupled to a single vibrational mode. Dis-regarding electronic-vibrational coupling (electronic transport), the current through thismodel molecular junction increases almost linearly with the applied bias voltage Φ. Includ-ing electronic-vibrational coupling (vibronic transport), the current-voltage characteristicof this junction exhibits kinks at multiples of the vibrational frequency Ω

1

. These kinkstranslate to steps in the respective conductance-voltage characteristic and are associatedwith the onset of inelastic co-tunneling processes (cf. Fig. 4.1). Current-induced heatingof the vibrational mode is less important in this regime, as the comparison of the vibronicand the thermally equilibrated current-/conductance-voltage characteristic shows.


regime (Γ Ωα), however, this renormalization of the vibrational frequencies is notvery pronounced, and therefore, it is neglected whenever we study vibrational effects inthe non-resonant transport regime. That way, we avoid numerical errors that spoil theanalysis of the corresponding transport characteristics (cf. the discussion in appendix B).

The height of the steps due to the onset of inelastic co-tunneling processes correlates withthe respective FC factors for a transition from the vibrational ground to its nth excitedstate, |X0n

1

|2 = e−λ211/Ω

21(λ2n

11

/n!). For λ11

/Ω1

< 1, these transition matrix elements be-come gradually smaller with an increasing number of vibrational quanta n involved inthe respective co-tunneling processes. While for eΦ . 2Ω

1

the vibronic and the thermalconductance is smaller than the electronic one, they slightly exceed the electronic con-ductance for higher bias voltages. This behavior is a result of both the polaron-shift ofthe broadened electronic level and the fact that electrons entering the bias window atthese bias voltages do not suffer from the reduced FC factors for electronic co-tunnelingprocesses (cf. Fig. 4.1a), as in parallel a number of excitation and deexcitation chan-nels (cf. Figs. 4.1b and 4.1c) are active. Steps in the conductance translate to kinksin the corresponding current-voltage characteristic such that the vibronic and the ther-mal current-voltage characteristic approach the electronic current-voltage characteristicat larger bias voltages, Φ ≈ 0.5 V.

The comparison of the solid black and the dashed gray line, which correspond to vibronicand thermally equilibrated transport, respectively, reveals that heating of the vibrationalmode due to vibrational excitation processes (cf. Fig. 4.1b) is less significant in the non-resonant transport regime. In nonequilibrium, however, the current (solid black line) isslightly larger due to additional vibrational deexcitation processes (cf. Fig. 4.1c), whichare suppressed in the thermally equilibrated transport scenario.

Note that in the adiabatic regime, Γ > Ω1

, the opening of inelastic channels at eΦ =nΩ

1

does not result in a step-wise increase of the conductance but rather in a step-wisedecrease, if the transmission probability for an electron t() is larger than ≈ 0.8 [81,292,297, 298, 351]. While in this regime the contribution of inelastic co-tunneling channels isstill present, they also result in an effective Pauli-blocking of elastic co-tunneling processes[352]. Thus, instead of increasing the conductivity of the junction, these processes resultin a decrease of the conductivity. This intriguing phenomenon has been found and verifiedin a number of experiments [22,56,158], and theoretical studies [75,81,153,229,292,297,298,351,352].

4.1.2 Multimode Effects

As we have seen in Sec. 4.1.1, inelastic co-tunneling processes result in a step-wise increaseof the conductance-voltage characteristic of a single-molecule junction at multiples of thevibrational frequencies Ωα. Therefore, one can use these steps to determine the energy ofthe vibrational levels in a molecular junction. This is commonly referred to as InelasticElectron Tunneling Spectroscopy (IETS). Thereby, the multitude of vibrational modes


in a molecular junction gives a characteristic fingerprint of the molecule bridging thegap between the left and the right lead. IETS of multiple vibrational modes, however,is typically rather complex due to mixed processes that involve not one but multiplevibrational modes. These processes and the corresponding multimode vibrational effectsare the subject of this section.

To discuss these effects, we consider a model system comprising two vibrational modesthat are coupled to a single electronic state (model E1V2). The electronic state of thisjunction is located 1.0 eV above the Fermi level, and symmetrically coupled to the left andthe right lead with coupling strengths ν

L/R,1 = 0.1 eV. As before, the leads are modeledas semi-elliptic conduction bands with an internal hopping parameter γ = 2 eV (cf. Eq.(4.1)3. The two modes with frequencies Ω

1

= 0.1 eV and Ω2

= 0.25 eV are coupled tothe electronic state with coupling strengths λ

11

= 0.06 eV and λ12

= 0.15 eV, respectively.We also consider coupling of the vibrational modes to a thermal bath. Thereby, we usea coupling strength of ξ

1/2

= 0.01 and assume a temperature of Tbath

= 1000 K≈ Ω1

/kB

for the secondary vibrational (bath) modes. In the present context, the coupling to sucha ’hot’ bath represents a tool to control the excitation of the two vibrational modesexternally due to energy transfer processes from the ’hot’ bath to the vibrational modesof the junction. This facilitates the analysis of vibrational nonequilibrium effects in thissection. Note that the coupling to a thermal bath is typically used in another way,that is to describe dissipative processes, for example due to intramolecular vibrationalenergy redistribution or energy transfer to the environment (e.g. phononic excitation ofthe electrodes) [13,100,209,353,354].

The electronic current and the respective conductance of this model molecular junctionare depicted by the solid purple lines in Fig. 4.3. As for the previous model system, weobserve a linear increase of the electronic current due to an increasing number of elec-trons in the left lead that are located within the bias window. Accounting for electronic-vibrational coupling, we obtain the solid black, the dashed gray and the dashed greenlines of Fig. 4.3, which correspond to the vibronic, the thermally equilibrated and the ’hotbath’ transport scenario of this junction. Note that for the latter transport scenario weconsider coupling of the two vibrational modes to the ’hot’ thermal bath. As discussedin Sec. 4.1.1, electronic-vibrational coupling leads to a suppression of electronic transportprocesses (cf. Fig. 4.1a), and thus, to a suppression of current and conductance at lowbias voltages, Φ . 0.3V. While for low bias voltages electronic processes are dominant,inelastic co-tunneling processes (cf. Figs. 4.1b-d) become active one by one if the biasvoltage exceeds multiples of the vibrational frequencies, eΦ = nΩ

1

and/or eΦ = nΩ2

.Accordingly, kinks/steps appear in the corresponding current-/conductance-voltage char-acteristics depicted in Fig. 4.3. However, due to the coupling of the electronic state totwo vibrational modes, steps appear also at bias voltages, where a co-tunneling electronexcites both vibrational modes (see, for example, the pronounced step at eΦ = Ω

1

+ Ω2

).

3 The electronic current of this model is almost the same as for the model system studied in Sec. 4.1.1.The non-resonant transport regime of the present model system, however, extends over a broader rangeof bias voltages, as the electronic state is located farther from the Fermi level. This facilitates thediscussion of multimode vibrational effects.


(a)

NEGF electronic


NEGF hot bath

0 0.1 0.2 0.3 0.4 0.50

0.5

1.

1.5

2.

2.5

3.

bias voltage ! !V"

currentI!nA"

(b)

NEGF electronic


NEGF hot bath

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

bias voltage ! !V"

conductanceg!nA#V"

Figure 4.3: Current-voltage and conductance-voltage characteristics of a model molec-ular junction in the non-resonant transport regime (eΦ 2

1

). This junction (modelE1V2, cf. Tab. 4.1) involves a single electronic state coupled to two vibrational modes.In contrast to the characteristics shown in Fig. 4.2, where only a single vibrational modeis involved, kinks/steps appear in these current-/conductance-voltage characteristics notonly at multiples of the vibrational frequencies eΦ = nΩ

1/2

but also at eΦ = nΩ1

+ mΩ2

(n,m ∈ ). Thereby, negative integers, n or m, correspond to deexcitation processes (cf.Fig. 4.1c) that become active either at higher bias voltages (eΦ > max(Ω

1

, Ω2

)) and/ordue to coupling of the vibrational modes to a ’hot’ thermal bath.

4.2. Resonant Transport Processes / Sequential Tunneling 65

In principle, a step appears also at eΦ = Ω2

− Ω1

, which corresponds to an inelasticco-tunneling process, where mode 1 and mode 2 are simultaneously deexcited and excited(see Fig. 4.1d). These processes, however, require a finite level of excitation for mode 1.Therefore, the corresponding step is only visible in the conductance-voltage characteristicdepicted by the dashed green line, where the coupling to the ’hot’ thermal bath (ζα = 0.01)leads to a finite level of excitation of both vibrational modes. Consequently, deexcitationprocesses (Fig. 4.1c) are active in this case already at low bias voltages (eΦ < Ω

1

). Thecurrent represented by the dashed green line is thus larger than for the vibronic and thethermally equilibrated transport scenario (black and dashed gray line), where both vibra-tional modes are almost not excited. Another example for such a nonequilibrium effect isthe kink/step at Φ = 0.4 V. It appears in the solid black and the dashed green line, butnot in the dashed gray one. This indicates that the main contribution to this step is notco-tunneling processes, which involves the simultaneous excitation of mode 1 by four vi-brational quanta, but co-tunneling processes, which involve the simultaneous deexcitationof mode 1 by one vibrational quanta and excitation of mode 2 by two vibrational quanta.

4.2 Resonant Transport Processes / Sequential Tun-neling

In Secs. 4.1.1 and 4.1.2, we have analyzed current-voltage and conductance-voltage char-acteristics in the non-resonant transport regime of a molecular junction. In the following,we address the resonant transport regime of a molecular junction, and study the cor-responding population of the electronic states as a function of the applied bias voltage.Thereby, the complexity of our model systems increases section by section. In particular,we consider resonant transport within a minimal model in Sec. 4.2.1, which involves asingle electronic state and a single vibrational mode. Effects that occur due to electronic-vibrational coupling to multiple electronic states are studied in Sec. 4.2.2, while Sec.4.2.3 addresses multimode vibrational effects. In the last section, Sec. 4.2.4, we discussvibrational effects that involve both non-resonant and resonant transport processes.

4.2.1 Basic Processes

In this section, we discuss vibrational effects in resonant electron transport through amolecular junction described by the minimal model E1V1. The non-resonant transportregime of this model system (Φ < 0.6V) has already been analyzed in Sec. 4.1.1. Itcomprises a single electronic state and a single vibrational mode (respective parametersare detailed in Tab. 4.1). Corresponding current-voltage and conductance-voltage char-acteristics are shown in Fig. 4.5. As in Sec. 4.1.1, the solid purple and the solid black linerepresent results obtained by the NEGF method for electronic and vibronic transport,respectively. In addition, we also show results obtained by the ME approach, where thesolid yellow line corresponds to electronic and the solid red line to vibronic transport.


(a) (b) (c) (d)

ε1

RL M

Ω1

ε1

RL M

Ω1

ε1

RL M

Ω1

ε1

RL M

Figure 4.4: Schematic representation of example processes for sequential tunneling ina molecular junction. Panel a shows sequential tunneling of an electron from the leftlead to the right lead that involves two consecutive tunneling processes: from the left leadonto the molecular bridge, and from the molecular bridge to the right lead. In panelb/ c the latter of the two tunneling processes is accompanied by an excitation/deexcitationprocess, where due to electronic-vibrational coupling the vibrational mode is singly ex-cited/deexcited. While the processes depicted by Panel a – c become active at the samebias voltage, that is for eΦ ≈ 2

1

, resonant excitation processes like the one depicted byPanel d require higher bias voltages, eΦ & 2

1

+ Ω1

.

The two electronic current-voltage characteristics in Fig. 4.5a (solid purple and yellowlines) exhibit a single step at eΦ w 2

1

. This step indicates the onset of resonant transport,where electrons in the left lead have enough energy to pass through the molecular junctionby two sequential tunneling processes (as depicted by Fig. 4.4a). At higher bias voltages,the electrical current remains more or less at the same level, as the number of electrons inthe left lead, which can resonantly tunnel through the molecular resonance, is also moreor less constant4. Due to the coupling of the molecule to the leads, however, the energyof the electronic level is not well-defined, or equivalently, the electronic level exhibitsbroadening due to the molecule-lead coupling. As the ME method does not account forsuch broadening, the width of the corresponding step (solid yellow line) is given by thethermal broadening in the leads only (k

B

T ≈ 1meV). In contrast, the NEGF approachincludes such broadening and the corresponding step in the solid purple line is significantlybroader than in the solid yellow one. Here, the width is given by Γ

L/R,11

+ kB

T ≈ 7meV.Apart from broadening, however, the results obtained by both methods agree very well.

Electronic-vibrational coupling triggers a variety of steps in the current-voltage charac-teristic of this junction (see the solid black and the solid red line in Fig. 4.5a). The first ofthese steps at eΦ w 2

1

indicates, as for electronic transport, the onset of resonant trans-port processes. Due to the polaron-shift of the electronic level (cf. Sec. 3.1.3), however,

4 This statement is based on the fact that, in the given range of bias voltages, the density of states inboth leads is not significantly influenced by the applied bias voltage.


(a)

NEGF electronicNEGF vibronicME electronicME vibronic

0 0.5 1. 1.5 2. 2.50

0.5

1.

1.5

bias voltage ! !V"

currentI!"A"

(b)


0 0.5 1. 1.5 2. 2.50

10

20

30

bias voltage ! !V"

conductanceg!"A#V"

Figure 4.5: Current-voltage and conductance-voltage characteristics for a model molec-ular junction in the resonant transport regime. This junction comprises a single electronicstate that is coupled to a single vibrational mode (model E1V1, cf. Tab. 4.1). The onsetof resonant transport processes leads to a step-wise increase of the current or peaks in therespective conductance-voltage characteristic. While the electronic current-voltage char-acteristic exhibits just a single step at eΦ = 2

1

, electronic-vibrational coupling triggersa number of additional steps at eΦ = 2(

1

+ nΩ1

), which are associated with inelasticsequential tunneling processes (like the one depicted by Fig. 4.4d). ME and NEGF givevery similar current-/conductance-voltage characteristics, where minor deviations resultfrom both the broadening of the molecular levels and the renormalization of the vibrationalfrequency due to the coupling of the molecular conductor to the leads.



0 0.5 1. 1.5 2. 2.50

0.2

0.4

bias voltage ! !V"

electronicpopulation

Figure 4.6: Population characteristics of the electronic state corresponding to thecurrent-/conductance-voltage characteristics shown in Fig. 4.5. Due to the symmetryof the molecule-lead coupling the average population of the electronic level is ≈ 1/2 forlarge bias voltages Φ.

this step appears at lower bias voltages. In addition, the height of this step is reducedcompared to the one in the electronic current-voltage characteristic. This is related to thesuppression of electronic transport processes (cf. Fig. 4.4a) due to the different equilib-rium positions of the vibrational mode in different charge states of the molecular bridge,∆Q

1

= −2λ11

/Ω1

(cf. Sec. 4.1). This shift leads to a reduced overlap between the chargestates (cf. Fig. 3.1) involved in these processes, and can be quantified by the FC-factor|X00

1

|2 = e−λ211/Ω

21 ≈ 0.7 (cf. Sec. 4.1.1). This suppression of the current at the onset

of the resonant transport regime is also referred to as Franck-Condon blockade, and isparticularly pronounced for large coupling strengths λ

11

[104, 281, 355]. Steps that ap-pear at higher bias voltages, eΦ w 2(

1

+ nΩ1

) (n ∈ ), indicate the onset of resonantexcitation processes (cf. Fig. 4.4d), where an electron with energy

1

+ nΩ1

tunnels reso-nantly onto the molecular bridge exciting the vibrational mode by n vibrational quanta.The respective step heights can be qualitatively understood by the Franck-Condon matrix

elements, |X0n1

|2 = 1

n!

≥λ11Ω1

¥2n

exp(−λ211

Ω

21), which correspond to a transition from the vi-

brational ground to the nth excited state. For a quantitative analysis, however, the exactnumber of sequential tunneling processes and the associated level of vibrational excita-tion needs to be taken into account [83]. Note that at the onset of the resonant tunnelingregime (eΦ = 2

1

) not only electronic transport processes (Fig. 4.4a) become active, butalso a number of vibrational excitation and deexcitation processes (such as the exampleprocesses shown in Figs. 4.4b and 4.4c).


Peaks in the conductance-voltage characteristics appear (Fig. 4.5b) at bias voltages, wherethe corresponding current-voltage characteristics (Fig. 4.5a) exhibit a step-wise increase.Analyzing the position of these peaks reveals that the NEGF method (solid black line)and the ME approach (solid red line) give different peak positions. This difference canbe understood as an effect of the renormalization of the vibrational frequency Ω

1

due tothe coupling of the molecule to the leads. The ME approach does not account for thisrenormalization, as we have neglected the contribution of principle-value terms in Eq.(3.68). The NEGF method, however, accounts for these effects by the real part of therespective vibrational self-energy, Π

el,11

. Except for the renormalization of Ω1

and thebroadening of the molecular levels, NEGF and ME give almost the same results.

The onset of resonant transport is not only accompanied by a steep increase in the elec-trical current flowing through this molecular junction, but also by a steep increase in theaverage population of the electronic level. Fig. 4.6 shows the population characteristics ofthe electronic state corresponding to the current-voltage and conductance-voltage char-acteristics depicted in Fig. 4.5. As sequential tunneling is the dominant contribution tothe current, and because such resonant tunneling involves the intermediate populationof the electronic state, the average population of the electronic level and the respectivecurrent-voltage characteristic are highly correlated, that is, both characteristics exhibit avery similar resonance (step) structure. Thereby, in the resonant transport regime, theaverage population of the electronic level is ≈ 1/2. This value indicates that the electroniclevel is populated on the same time scales, as it gets depopulated. This is closely relatedto the symmetric molecule-lead couplings, ν

L

= νR

, in this model system.

Considering thermally equilibrated transport, we obtain the current-voltage characteristicshown in Fig. 4.7, where the dashed gray and the dashed brown line correspond to resultsobtained by NEGF and ME, respectively. For comparison, we also show the respectivevibronic current-voltage characteristic (solid black and solid red line). Recall that inthermally equilibrated transport the vibrational mode is effectively reset to its thermalequilibrium state after each electron transmission process. At the effective temperatureof 10K used in these calculations, this state corresponds essentially to the ground stateof the vibrational mode. Electrons tunneling through the junction are thus not affectedby inelastic processes that occurred in previous tunneling events. This is in contrast tovibronic transport, where current-induced vibrational excitation is accounted for. There,the efficiency of electron tunneling processes depends on inelastic processes induced bypreviously tunneling electrons. The thermal current-voltage characteristic, as the vibroniccurrent-voltage characteristic, show a multitude of steps at eΦ = 2(

1

+ nΩ1

) (n ∈0

),which are associated with the onset of resonant transport processes (cf. Fig. 4.4a-d). Inthermally equilibrated transport, however, the respective currents reach the electroniccurrent level (≈ 1.6µA) in a few steps, while the vibronic currents show significantlylower current levels in the resonant transport regime. This indicates a suppression of thecurrent through a molecular junction due to (current-induced) vibrational excitation.

This is a rather general phenomenon in molecular junctions, and can be qualitativelyunderstood by the following analysis. If, for example, a positive bias voltage is applied to



ME th. equ.ME vibronic

0 0.5 1. 1.5 2. 2.50

0.5

1.

1.5

bias voltage ! !V"

currentI!"A"

Figure 4.7: Comparison of vibronic and thermally equilibrated current-voltage charac-teristics for a model molecular junction with a single electronic state coupled to a singlevibrational mode (model E1V1, cf. Tab. 4.1) in the resonant transport regime. Current-induced excitation of the vibrational mode, which is not present in thermally equilibratedtransport, results in a significant suppression of the current. This is a rather generalphenomenon in vibrationally coupled electron transport through a molecular junction.

a molecular junction and allows for m = mod°

eΦ2

− 1

, Ω1

¢resonant excitation processes

(cf. Fig. 4.4d) and if the molecular bridge is in its ground state, the transition probabilityfor an electron tunneling from the left lead onto the bridge can be written as

Pmn=0

|X0n1

|2.This transition probability converges typically in a few steps to unity with increasing m, orbias voltage Φ, and explains why the thermally equilibrated current-voltage characteristicreaches the electronic current level in a few steps. For this transition probability, we donot consider the second tunneling process of the corresponding sequential tunneling eventsbecause the number of the relevant tunneling processes is not restricted by the bias voltage,at least as long as the electronic state is located above the Fermi level by several units ofΩ

1

. If the vibrational mode, however, is in a nonequilibrium state, where the populationof the lth vibrational level is given by αl = δ

0l (l ∈0

), the transition probabilityassociated with the tunneling of an electron from the left lead onto the molecular bridge isdetermined by

Pl=0..∞ αl

Pl+mn=0

ØØX ln1

ØØ2. This transition probability includes all excitationand deexcitation processes that occur, if the vibrational mode is in its ground state (l = 0),its first excited state (l = 1), or in an higher excited state (l > 1). It is, in general, smallerthan the transition probability

Pmn=0

|X0n1

|2 for a thermally equilibrated vibrational mode(at 10K). Accordingly, vibronic transport exhibits lower current levels than thermallyequilibrated transport through this molecular junction.


4.2.2 Vibrational Effects in Transport through Multiple Elec-tronic States

In a next step, we extend the scope of our considerations to vibrational effects in resonantelectron transport through molecular junctions with multiple electronic states. To thisend, we employ a model for a molecular junction (model E2V1) with two electronic states:a lower- and a higher-lying electronic state located

1

= 0.15 eV and 2

= 0.8 eV above theFermi level of the junction, respectively. Both states are coupled to a single vibrationalmode with coupling strengths λ

11

= 0.06 eV and λ21

= −0.06 eV5. Due to the different signin the electronic-vibrational coupling strengths, electron-electron interactions, which areeffectively induced in this model molecular junction by the coupling of the two electronicstates to the same vibrational mode (cf. Sec. 3.1.3), are repulsive: U

12

= −2λ11

λ21

/Ω1

> 0.All parameters of this model molecular junction are summarized in Tab. 4.2. Respectivecurrent-/conductance-voltage characteristic are shown in Fig. 4.9.

The electronic current-/conductance-voltage characteristic of this model molecular junc-tion is depicted by the solid yellow (ME) and the solid purple line (NEGF). Accordingto the analysis given for a single electronic state, the two steps in this current-voltagecharacteristic, at eΦ = 2

1

and eΦ = 22

, indicate the onset of resonant transport pro-cesses through state 1 and state 2, respectively. Since both states are coupled to theleads with the same coupling strengths, and because the two states are non-degenerate,|

1

− 2

| ΓL/R,mn, these steps, as well as the peaks in the corresponding conductance-

voltage characteristic, are very similar in height and shape.

The solid black and the solid red lines of Fig. 4.9 depict the current-/conductance-voltagecharacteristic associated with vibronic transport through this model molecular junction,where the NEGF and the ME approach have been employed, respectively. In contrast toelectronic transport, where the analysis given for a single electronic state (one state→ onestep) readily applies to two electronic states (two states → two steps), this characteristiccannot be understood solely in terms of vibrationally coupled electron transport througha single electronic state (cf. Sec. 4.2.1).

Indeed, in accordance with the analysis given in Sec. 4.2.1 for a single electronic state, wefind steps/peaks in the current-/conductance-voltage characteristic of this model molecu-lar junction at eΦ = 2(

1

+nΩ1

) and eΦ = 2(2

+nΩ1

) (n ∈0

). These are associated withthe onset of resonant transport processes through state 1 and state 2, respectively (cf.,for example, Fig. 4.8a, which represents a vibrational excitation process with respect tostate 1). However, the steps/peaks associated with state 2 are much less pronounced thanthe ones associated with state 1, although both states are coupled to the leads and to thevibrational mode with coupling strengths that have the same absolute value. Moreover,additional steps at eΦ = 2(

2

+U12

+nΩ1

) indicate a third electronic resonance at 2

+U12

.These findings can be explained by (vibrationally induced) electron-electron interactions,

5 This is a quite common electronic-vibrational coupling scenario, as can be inferred, for example, fromTab. 7.2, which shows the electronic-vibrational coupling strengths obtained for a first-principles basedmodel of a biphenylacetylene molecular junction.


model 1

2

U12

νL,1 ν

L,2 νR,1 ν

R,2 γ Ω1

λ11

λ21

E2V1 0.15 0.8 0 0.1 0.1 0.1 0.1 3 0.1 0.06 -0.06BLOCK 0.15 0.4 0.5 0.1 0.1 0.1 0.01 3 0.1 0.03 -0.03LOCAL 0.15 0.4 0.5 0.1 0.01 0.1 0.01 3 0.1 0.03 -0.03SPEC 0.15 0.8 0 0.1 0.1 0.03 0.03 3 0.1 0.06 -0.06

Table 4.2: Model parameters for molecular junctions with two electronic states (energyvalues are given in eV). The temperature in the leads, k

B

T = 10 K, the characteristicfrequencies of and the coupling strengths to the thermal bath, ωC,α = 1 eV and ζα = 0, arethe same for all these model molecular junctions.

(a) (b) (c)

ε1

ε2

RL M

Ω1

Ω1

ε1

ε2

RL M

Ω1

ε1

ε2

RL Mε2+U12

Figure 4.8: Examples of sequential tunneling processes involving two electronic states.Panel a depicts a sequential tunneling process, which involves a resonant excitation processwith respect to state 1. Such processes occur also in the presence of a single electronicstate. Examples of sequential tunneling processes that involve deexcitation processes withrespect to state 2 are depicted in Panels b and c, where the lower-lying electronic state isunoccupied and occupied, respectively.


(a)

!2" #######################################$212 % 2&$11$21

'1

!2" #########$212

'1


0 0.5 1. 1.5 2. 2.50

1

2

3

bias voltage ( !V"

currentI!)A"

(b)


0 0.5 1. 1.5 2. 2.50

10

20

30

40

bias voltage ! !V"

conductanceg!"A#V"

Figure 4.9: Current-voltage and conductance-voltage characteristics for a model molec-ular junction in the resonant transport regime (model E2V1, cf. Tab. 4.2). The twoelectronic states of this model molecular junction are coupled to a single vibrationalmode. While the current-/conductance-voltage characteristic associated with electronictransport can be understood in terms of transport through a single electronic state, thecurrent-/conductance-voltage characteristic associated with vibronic transport cannot. In-deed, there are steps/peaks in the vibronic current-/conductance-voltage characteristic ateΦ = 2(

1

+ nΩ1

) and eΦ = 2(2

+ nΩ1

) (n ∈ ), which are associated with resonanttransport processes through state 1 and state 2, respectively. However, (vibrationallyinduced) electron-electron interactions and current-induced vibrational excitation couplecharge transport through the two electronic states, which leads to a variety of additionalsteps/peaks at eΦ = 2(

2

− nΩ1

) and eΦ = 2(2

+ U12

− nΩ1

).


U12

, which become effective once the applied bias voltage allows for sequential tunnelingthrough both electronic states, eΦ > 2

2

. Due to these interactions, sequential tunnelingdepends on the population of the electronic states. For example, electrons with an energyof

2

can resonantly tunnel through state 2 only if state 1 is unoccupied, while electronswith an energy of

2

+U can resonantly tunnel through state 2 only if state 1 is occupied.This results effectively in a splitting of resonances associated with state 2, since the cor-responding tunneling processes become active at different bias voltages and because state1 is neither empty nor fully occupied but n

1

≈ 1/2 for eΦ w 22

(see Fig. 4.10 for therespective population characteristics of the two electronic states)6. While steps/peaks inthe current-/conductance at eΦ = 2(

2

+ nΩ1

) are thus suppressed by a factor 1−n1

, thesteps/peaks at eΦ = 2(

2

+nΩ1

+U12

) are linearly correlated with the population of state1, n

1

. That way, electron transport processes through the two electronic states do notoccur independently from each other, but are coupled by electron-electron interactions.This applies for electron transport processes through molecular junctions with multipleelectronic states in general.

Another source for such coupling is current-induced vibrational excitation. This can bedemonstrated comparing results for vibronic and thermally equilibrated transport. Fig.4.11 provides such a comparison in terms of the respective current-voltage characteristics,where the dashed gray and the dashed brown line show results for thermally equilibratedtransport, for which the NEGF and the ME approach have been employed, respectively.As in Sec. 4.2.1, vibronic transport is characterized by a suppression of current due tovibrational excitation. This trend is observed at the onset of current eΦ w 2

1

as wellas for higher bias voltages eΦ & 2

1

. In the intermediate voltage regime, however, wherestate 2 is close to, but still outside the bias window, vibronic transport yields a largercurrent level than thermally equilibrated transport. This is a result of current-inducedvibrational excitation, which is induced by resonant excitation processes with respect tostate 1 (cf. Fig. 4.8a). The vibrational energy thus generated facilitates resonant tunnelingof electrons through state 2 by vibrational deexcitation processes, even though this stateis still outside the bias window (cf. Figs. 4.8b and 4.8c). In addition to the steps/peaksanalyzed in the latter paragraph, these sequential tunneling processes result in steps ateΦ = 2(

2

−nΩ1

) and eΦ = 2(2

+U12

−nΩ1

) (n ∈ ). While transport processes throughthe two electronic states are thus coupled in thermally equilibrated transport by electron-electron interactions only, they are coupled in vibronic transport by both electron-electroninteractions as well as vibrational excitation.

6 For even stronger (repulsive) electron-electron interactions, where for example 1 < 2 < 1 + U12 <2 + U12 holds, electron transport through either of the electronic states depends on the population ofthe other state. Consequently, resonances with respect to both states exhibit splitting due to electron-electron interactions.


(a)


0 0.5 1. 1.5 2. 2.50

0.2

0.4

bias voltage ! !V"

electronicpopulationn 1

(b)

!2" #######################################$212 % 2&$11$21

'1

!2" #########$212

'1


0 0.5 1. 1.5 2. 2.50

0.2

0.4

bias voltage ( !V"

electronicpopulationn 2

Figure 4.10: Population characteristics of state 1 and state 2 corresponding to thecurrent-voltage and conductance-voltage characteristics shown in Fig. 4.5. Due to thesymmetry of the molecule-lead coupling, and the weak electron-electron interactions,2

− 1

< U12

, the average population of state 1 is ≈ 1/2 in the resonant transport regime.State 2 exhibits a similar population but at higher bias voltages. However, state 1 reachesits maximal population after a few steps, while state 2 is successively populated in manymore steps. These steps are a result of both current-induced vibrational excitation, whichfacilitate population of this state even before it has entered the bias window, and electron-electron interactions, which induce splitting of the steps associated with resonant transportprocesses through state 2.


!2" #######################################$212 % 2&$11$21

'1

!2" #########$212

'1


ME th. equ.ME vibronic

0 0.5 1. 1.5 2. 2.50

1

2

3

bias voltage ( !V"

currentI!)A"

Figure 4.11: Comparison of vibronic and thermally equilibrated current-voltage char-acteristics for a model molecular in the resonant transport regime (model E2V1, cf. Tab.4.2). This junction comprises two electronic states coupled to a single vibrational mode.Current-induced excitation of the vibrational mode, which is discarded in thermally equili-brated transport, results in a significant suppression of the current at the onset of resonanttransport through state 1, that is for eΦ w 2

1

, and for larger bias voltages eΦ > 22

. Inthe intermediate voltage regime, however, vibronic transport yields a larger current levelthan thermally equilibrated transport, as current-induced vibrational excitation facilitatesresonant tunneling through state 2, even if this state is located outside the bias window.

4.2.3 Multimode Effects

As we have seen in Sec. 4.2.2, the interplay of multiple electronic states induces a num-ber of intriguing transport phenomena in the resonant transport regime of a molecularjunction. Electronic-vibrational coupling to multiple vibrational modes does also inducea variety of interesting effects and phenomena. In this section, we study these effectsemploying the model molecular junction E1V2, which we have already used in Sec. 4.1.2to describe multimode vibrational effects in the non-resonant transport regime. Due tothe unfavorable scaling of the ME approach with respect to the number of vibrationalmodes, however, we focus in this section on results obtained by the NEGF method.

Fig. 4.13 represents current-/conductance-voltage characteristics of this junction in theresonant transport regime (1 V<Φ<3V). The electronic current-/conductance-voltagecharacteristic of this model molecular junction (solid purple line) is very similar to thecurrent-/conductance-voltage characteristic shown in Fig. 4.5 of Sec. 4.2.1, where elec-tronic transport through a single electronic state was already discussed (by model E1V1).


Ω2

Ω1

ε1

RL M Figure 4.12: Schematic representa-tion of a sequential tunneling process,where simultaneously a high-frequencymode is excited and a low-frequencymode is deexcited. Similar processesdo also occur in the non-resonanttransport regime (cf. Fig. 4.1d). Theseprocesses become active at bias volt-ages eΦ = 2(m + Ω

2

− Ω1

).

Vibronic transport in this model molecular junction is governed by a number of steps/peaksin the respective current-/conductance-voltage characteristic (solid black lines), whichcannot be explained by the interaction of a single electronic state with a single vibra-tional mode. Indeed, in line with the analysis given in Sec. 4.2.1, steps/peaks appear ateΦ = 2(

1

+nΩ1

) and at eΦ = 2(1

+nΩ2

), which are associated with the onset of sequentialtunneling processes through the electronic state, where mode 1 and mode 2, respectively,are excited n-times. Besides these processes, however, there are also processes, wherea tunneling electron interacts with both vibrational modes simultaneously. An exampleprocess for such an interaction is schematically represented in Fig. 4.12, where a tunnelingelectron excites mode 2 and deexcites mode 1 simultaneously. Accordingly, steps/peaks inthe current-/conductance-voltage characteristic appear at eΦ = 2(

1

+ nΩ1

+ mΩ2

) withn,m ∈ . Thereby, negative values for n and/or m correspond to deexcitation processes(Fig. 4.12). Note that these processes require a finite level of (current-induced) vibra-tional excitation. Therefore, processes that involve deexcitation of the vibrational modesdo not appear in the respective thermally equilibrated transport characteristics (dashedgray lines), for which steps/peaks appear only at eΦ = 2(

1

+nΩ1

+mΩ2

) with n,m ∈ .Renormalization of the vibrational frequencies Ω

1

and Ω2

, which is a result of the couplingbetween the molecular bridge and the leads, leads to slightly different positions of thesesteps/peaks in the vibronic and the thermal current-/conductance-voltage characteristic(as was already outlined in Sec. 4.2.1).

As was already discussed in Secs. 4.2.1 and 4.2.2, the population of the electronic level,given in Fig. 4.14, is very similar to the respective current-voltage characteristics (Fig.4.13). At higher bias voltages, however, the population of the electronic state slightlyincreases, while the respective current level decreases7. This is a result of the reducedband width in the leads and the high bias voltages considered, where the semi-ellipticconduction bands in the left and the right lead exhibit a reduced overlap, Γ

L,11

(1

)ΓR,11

(1

),at the resonance energy

1

. For electronic transport this overlap determines the current,I ≈ Γ

L,11

(1

)ΓR,11

(1

)/(ΓL,11

(1

) + ΓR,11

(1

)), while the population of the electronic levelis determined by the ratio Γ

L,11

(1

)/(ΓL,11

(1

) + ΓR,11

(1

)) (for positive bias voltages).

7 Note that this increase of the electronic population is also present in the model system E1V1, whichinvolves only a single vibrational mode. In this system, however, this effect is less pronounced, as therespective electronic state is located closer to the Fermi level of the junction.


(a)

NEGF electronic


1. 1.5 2. 2.5 3.0

1

2

bias voltage ! !V"

currentI!"A"

(b)

NEGF electronic


1. 1.5 2. 2.50

10

20

bias voltage ! !V"

conductanceg!"A#V"

Figure 4.13: Current-voltage and conductance-voltage characteristics of a model molec-ular junction in the resonant transport regime (model E1V2, cf. Tab. 4.1). This junctioncomprises a single electronic state that is coupled to two vibrational modes. Steps/peaks inthe current-/conductance-voltage characteristic appear due to the onset of resonant tun-neling processes at eΦ = 2(

1

+ nΩ1

+ mΩ2

) with n, m ∈ . Thereby, negative values of nand m are associated with deexcitation processes, which are accounted for in vibronic butnot in thermally equilibrated transport.


NEGF electronic


1. 1.5 2. 2.5 3.0

0.1

0.2

0.3

0.4

0.5

bias voltage ! !V"

electronicpopulation

Figure 4.14: Population characteristics of the electronic state corresponding to thecurrent-/conductance-voltage characteristics shown in Fig. 4.13. As the coupling to theleads is symmetric, the population of the electronic level is ≈ 1/2 for large bias voltagesΦ. Increasing the bias voltage, however, the population of the electronic level increasescontinuously, since the width of the conduction bands in the left and the right lead is finite.

This ratio increases continuously in the given range of bias voltages. As can be seen inFigs. 4.13 and 4.14, electronic-vibrational coupling tends to soften these effects, whichare induced by the electronic structure of the leads. This is because vibronic transport(as well as thermally equilibrated transport) involves resonant tunneling of electrons at avariety of different energies, whereas electronic transport probes the conduction band inthe leads only at a single point. This is also a rather general phenomenon in vibrationallycoupled electron transport, and will appear at several places throughout this thesis (forexample in Secs. 6.1 or 7.1).

4.2.4 Co-Tunneling Assisted Electron Tunneling (CoSET)

In Secs. 4.1 and 4.2.1 – 4.2.3, we have discussed transport characteristics of a molecularjunction in the non-resonant and the resonant transport regime, respectively. Often,however, there is no clear separation between these regimes. This will be discussed in thissection, where we analyze conductance-voltage characteristics of a molecular junction atthe cross-over between the non-resonant and the resonant transport regime. In particular,we discuss the origin of the small steps/peaks that appear in the vibronic (but not in thethermal) current-/conductance-voltage characteristics at intermediate bias voltages, thatis at eΦ = 2(

1

− nΩ1

− mΩ2

) (n,m ∈ ). Moreover, we address the question, whichprocesses generate the necessary (current-induced) level of vibrational excitation.


Figure 4.15: Scheme for co-tunneling assisted sequential elec-tron tunneling (CoSET). This is asequence of: 1. Excitation of thevibrational mode upon an inelasticco-tunneling process and 2. De-excitation of the mode in a subse-quent sequential electron tunnelingprocess. Note that these processesdo not have to occur directly oneafter the other.

(1.) (2.)

Ω1

ε1

RL M

Ω1

ε1

RL M

For example, in electronic transport the transition from the non-resonant to the resonanttransport regime is determined by the broadening of the electronic levels due to molecule-lead coupling. This is evident in the electronic transport characteristics shown in Fig. 4.16.In this figure, the conductance-voltage characteristics of both model E1V1 (Fig. 4.16a) andE1V2 (Fig. 4.16b) are depicted for bias voltages eΦ < 2

1

. Both electronic conductance-voltage characteristics (solid purple lines) show a smooth transition from the non-resonant(co-tunneling) regime to the resonant transport regime (sequential tunneling). While aclear separation between both regimes is therefore not possible, one may qualitativelydistinguish between both regimes at eΦ ∼ 2(

1

−ΓL,11

−ΓR,11

). Note that the ME approach,due to the strict second-order expansion used, does not account for the broadening dueto the molecule-lead coupling. We therefore consider only results in this section that areobtained by NEGF.

In contrast to electronic transport, electrons can resonantly tunnel through these molecu-lar junctions not only at a single but a variety of different energies if electronic-vibrationalcoupling is present (cf., for example, the discussion in Secs. 4.2.1 and 4.2.3). The transitionbetween the non-resonant and the resonant transport regime thus extents over an evenbroader range of bias voltages in vibronic transport. This can be seen, for example, in thesolid black lines of Fig. 4.16, which represent the vibronic conductance-voltage characteris-tics of model E1V1 and E1V2 in the cross-over regime between non-resonant and resonanttransport. At eΦ = 2(

1

−Pα nαΩα) < 2

1

with nα ∈ , resonant deexcitation processesfacilitate sequential tunneling of electrons through the electronic state, even before theelectronic level has entered the bias window. This results in peaks in the conductanceat these bias voltages. Such deexcitation processes occur due to vibrational excitation,which is generated in the course of inelastic co-tunneling processes. This is referred toas co-tunneling assisted sequential electron tunneling (CoSET). An example for a corre-sponding sequence of tunneling processes is depicted in Fig. 4.15. Due to the nature ofthese processes, a strict separation between the resonant and the non-resonant transportregime is even more involved for vibronic transport than it is for electronic transport. Sincethese processes require both inelastic co-tunneling as well as (current-induced) vibrationalexcitation, they appear neither in electronic nor in thermally equilibrated transport.

4.3. Local Heating and Cooling in a Molecular Junction 81

(a)

NEGF electronic


0.6 0.7 0.8 0.9 1.0

0.1

0.2

0.3

bias voltage ! !V"

conductanceg!"A#V"

(b)

NEGF electronic


1.2 1.3 1.4 1.5 1.60

0.2

0.4

0.6

0.8

1.

1.2

bias voltage ! !V"

conductanceg!"A#V"

Figure 4.16: Conductance-voltage characteristics in the intermediate bias voltage regimebetween non-resonant and resonant transport. Panel a shows results for model E1V1,while Panel b depicts results for model E1V2 (cf. Tab. 4.1 for the specific parameters).Small peaks in the conductance appear even before the electronic state enters the biaswindow. These peaks are associated with co-tunneling assisted sequential electron tun-neling (cf. Fig. 4.15). Due to these processes, which consist of both a non-resonant anda resonant transport process, the transition between the non-resonant and the resonanttransport regime extents over a range of bias voltages that is determined by the vibrationalfrequencies Ωα.


4.3 Local Heating and Cooling in a Molecular Junc-tion

In an inelastic transport process (cf. Figs. 4.1, 4.4 and 4.8), the molecular bridge exchangessimultaneously an electron and quanta of vibrational energy with the leads. Vibrationallycoupled electron transport thus involves both charge and energy exchange processes withthe leads. While in the first part of this chapter we have discussed in detail the role ofcharge exchange processes for the current-voltage characteristics of a molecular junctionand the respective population of the electronic states, in this section we focus on the roleof energy exchange processes (local heating and cooling [16,74,77,80,83,143,216,217,244,250,345,348,356–360]) and the corresponding vibrational excitation characteristic.

To this end, we consider another class of processes, that is electron-hole pair creationprocesses (see Fig. 4.17). In such a process, an electron tunnels from one lead ontothe molecular bridge and back again to the same lead. Thereby, the electron absorbsvibrational energy from the molecular bridge. Thus, effectively, an electron-hole pairis created in the respective lead. Note that in symmetric systems such pair creationprocesses occur with the same probability as corresponding transport process (for examplethe deexcitation process shown in Fig. 4.4c is as probable as the electron-hole pair creationprocess depicted in Fig. 4.17a). These processes, in combination with inelastic transportprocesses, constitute all possible energy exchange processes between the molecular bridgeand the leads that involve electron tunneling8.

Electron-hole pair creation processes are crucial to understand the vibrational excitationcharacteristic of a molecular junction. This is demonstrated, for example, in Sec. 4.3.1,where we show that electron-hole pair creation processes facilitate an understanding why ahigher level of vibrational excitation is obtained for weaker electronic-vibrational coupling.The role of off-resonant pair-creation processes, in particular at high bias voltages, iselucidated in Sec. 4.3.2. Extending our considerations to transport through multipleelectronic states (Secs. 4.3.3 and 4.3.4), we study the bias dependence of local heatingand cooling, which is highly correlated with the location of the electronic states withrespect to the Fermi level of the junction. Thereby, deexcitation processes involving bothelectron-hole pair creation and transport processes are discussed. In Sec. 4.3.5, we studyasymmetric electronic-vibrational coupling of multiple vibrational modes. This can leadto local cooling in molecular junctions due to vibrational energy transfer between stronglycoupled high-frequency modes and weakly coupled low-frequency modes [244]. Finally, inSec. 4.3.6, we consider local cooling by a thermal bath of secondary vibrational modes,where we distinguish vibrational excitation due to inelastic transport processes and dueto the nonequilibrium population of electronic levels (’polaron-formation’) [224,250,329].

8 Note that a direct coupling between the vibrational modes of the molecular bridge and the phononmodes in the leads [209] constitutes another mechanism for energy exchange with the leads (cf. Sec.3.1.2).


(a) (b) (c) (d)

Ω1

ε1

RL M

Ω1

Ω1

ε1

RL M

Ω1

ε1

ε2

RL MΩ1

ε1

RL M

Figure 4.17: Examples of electron-hole pair creation processes. Panel a shows a res-onant electron-hole pair creation process, where in two sequential tunneling events anelectron tunnels from the left lead onto the molecular bridge and back again to the leftlead. Thereby, the electron takes up a quantum of vibrational energy. Panel b depicts asimilar process with respect to the right lead. As in this configuration the chemical poten-tial in the right lead is located further away from the electronic level, electron-hole paircreation requires the absorption of several vibrational quanta. This, however, is typicallyunfavorable and results in less efficient cooling. A pair creation process with respect to ahigher-lying electronic state is shown in Panel c. Panel d represents an off-resonant paircreation processes. These processes constitute the pair creation analog of deexcitation viainelastic co-tunneling (cf. Fig. 4.1c).

Although electron-hole pair creation processes do not directly contribute to the currentflowing through a molecular junction, they have a significant influence on the respectivecurrent-/conductance-voltage and the population characteristic of the electronic states(vide infra). Electron-hole pair creation processes are well known from spectroscopic[46,361–363] and theoretical studies [364–366] of molecular adsorbates at metal surfaces.However, they have not been analyzed in detail in the context of nonequilibrium transportin molecular junctions yet [143,224,250,359]. Different mechanisms for electron-hole paircreation, which involve an electronically excited instead of a vibrationally excited state ofthe junction, are discussed, for example, in Refs. [367,368].

4.3.1 Role of Resonant Electron-Hole Pair Creation Processes

The most fundamental mechanisms of local heating and cooling in a molecular junctioncan be studied by the minimal model E1V1. The corresponding current-/conductance-voltage and the electronic populations characteristics have been analyzed in the non-resonant (Sec. 4.1.1) and in the resonant transport regime (Sec. 4.2.1). In this and in thefollowing section, we discuss the respective vibrational excitation characteristic. Fig. 4.18


0 0.5 1. 1.50

1

2

ME vibronicNEGF vibronic

0 0.5 1. 1.5 2. 2.5 3.0

10

20

30

40

bias voltage ! !V"

vibrationalexcitation

Figure 4.18: Vibrational excitation characteristic for a model molecular junction witha single electronic state coupled to a single vibrational mode (model E1V1, cf. Tab. 4.1).Steps at eΦ = 2(

1

+ nΩ1

) (n ∈ ) indicate the onset of inelastic sequential tunnelingprocesses (Fig. 4.4b), but also the suppression of electron-hole pair creation processes (Fig.4.17a). Inelastic co-tunneling processes generate small but significant levels of vibrationalexcitation in the non-resonant transport regime (highlighted in the inset of this figure).Resonant electron-hole pair creation processes, in conjunction with CoSET processes (cf.Sec. 4.2.4), result in a step-wise decrease of this co-tunneling induced level of vibrationalexcitation, once the electronic state approaches the bias window.

represents the vibrational excitation characteristics of this junction, where the solid blackand the solid red line represent results obtained by the NEGF and the ME approach,respectively. Distinct steps appear in this characteristic for both the solid black and thesolid red line at eΦ = 2(

1

+nΩ1

) (n ∈0

). Similar steps appear in the respective current-voltage characteristic at the same bias voltages (cf. Fig. 4.5). These are associated withthe onset of resonant excitation processes, where electrons tunneling from the left leadonto the molecular bridge excite the vibrational mode by n vibrational quanta. Sucha resonant excitation process is shown, for example, in Fig. 4.4d, which represents aninelastic sequential tunneling process including both a resonant excitation process andan electronic tunneling event. Whenever such an excitation channel becomes active, thelevel of vibrational excitation increases. As these processes involve an increasing number ofvibrational quanta, the relative step heights in the vibrational excitation characteristic areexpected to be larger than the relative step heights in the current-voltage characteristic.

So far, the vibrational excitation characteristic of this molecular junction can be un-derstood on the same footing as its current-voltage characteristic. There is, however,a qualitative difference between the relative step heights in both characteristics. While


the step heights in the current-voltage characteristic become smaller with increasing biasvoltage, the step heights in the vibrational excitation characteristic become larger. Thiscan only be understood, if we take into account that the level of vibrational excitationis determined by the number and the efficiency of both excitation and deexcitation pro-cesses, and not only by the number and the efficiency of transport processes, such as thecurrent. We therefore need to consider both inelastic transport and electron-hole paircreation processes (cf. Fig. 4.17). Thereby, low levels of vibrational excitation indicatethat deexcitation processes are dominant, while high levels of vibrational excitation areobtained, if the probability for exciting and deexciting the vibrational mode are very simi-lar. In that case, the system probes the ladder of vibrational states like in a ’random-walk’scenario. As a result, the vibrational states are populated with very similar probabilities,leading to high levels of vibrational excitation. Note that deexcitation of the vibrationalmode, due to the harmonic approximation, is always more favorable than exciting it, atleast in steady state and for finite bias voltages.

Similar to inelastic transport processes, electron-hole pair creation processes are morelikely to occur in this molecular junction the less vibrational quanta they involve. Thus, atthe onset of the resonant transport regime, cooling by electron-hole pair creation processesis very efficient, as electron-hole pairs can be created in the left lead by the absorptionof just a single vibrational quantum (cf. Fig. 4.17a). This leads to a rather low level ofvibrational excitation at eΦ ≈ 2

1

, although the number and the efficiency of inelastictransport processes, where the vibrational mode is excited (Fig. 4.4b) and deexcited (Fig.4.4c), is very similar.

In contrast to inelastic transport processes, the number of resonant electron-hole pair cre-ation processes does not increase at higher bias voltages but decreases. At Φ > 2(

1

+nΩ1

),for example, resonant electron-hole pair creation processes involving n vibrational quantaare suppressed. As a result, the probability for deexciting the vibrational mode decreasesat these bias voltages. In addition, resonant excitation channels (Fig. 4.4d) become active,which increase the probability for exciting the vibrational mode. Accordingly, the level ofvibrational excitation increases at these bias voltages due to both the onset of resonantexcitation channels and the suppression of electron-hole pair creation processes. Thereby,the level of vibrational excitation increases by increasingly larger steps, which is relatedto the fact that processes involving less vibrational quanta are more likely to occur thanprocesses involving more vibrational quanta.

This can be demonstrated considering a very similar model system, which differs fromjunction E1V1 only by a weaker electronic-vibrational coupling λ

11

= 0.03 eV . The re-spective vibrational excitation characteristics is shown in Fig. 4.19. Comparing the levelsof vibrational excitation in both systems reveals that higher levels of vibrational ex-citation are obtained for weaker electronic-vibrational coupling. This counterintuitivephenomenon can be understood in terms of the analysis we have given before. Forsmaller electronic-vibrational coupling, the electron-hole pair creation processes that in-volve a smaller number of vibrational quanta are even more important than for strongerelectronic-vibrational coupling. The same holds true for the corresponding resonant ex-


NEGF vibronic !11"0.06eVNEGF vibronic !11"0.03eVME vibronic !11"0.06eVME vibronic !11"0.03eV

0 0.5 1. 1.5 2. 2.50

5

10

15

bias voltage # !V"


Figure 4.19: Vibrational excitation characteristics for a model molecular junction witha single electronic state coupled to a single vibrational mode (model E1V1, cf. Tab. 4.1).Shown are results for two different electronic-vibrational coupling strengths λ

11

. Forsmaller electronic-vibrational coupling, the relative step heights in the vibrational exci-tation characteristic increase, although the associated excitation processes (cf. Fig. 4.4a)are less probable. This can be explained in terms of electron-hole pair creation processes,which are suppressed one by one with increasing bias voltage Φ.

citation channels. Thus, at Φ > 2(1

+ nΩ1

), the level of vibrational excitation increasesfaster for a weaker electronic-vibrational coupling, because the probabilities for excitingand deexciting the vibrational mode faster increase and decrease, respectively.

We thus arrive at the conclusion that the vibrational excitation characteristic of a single-molecule junction can only be understood, if we account for all excitation and deexcitationprocesses, including electron-hole pair creation processes. This is in contrast to the corre-sponding current-/conductance-voltage and/or population characteristic of the electronicstates, which can be understood, at least qualitatively, in terms of transport processes. Aquantitative analysis of these transport characteristics, however, can only be achieved, iflocal cooling of the vibrational degrees of freedom by electron-hole pair creation processesis taken into account (vide infra).

Note that the arguments given above strictly apply only for weak to intermediate electronic-vibrational coupling strengths, λ

11

/Ω1

. 1. For larger coupling strengths the relative stepheights in the vibrational excitation characteristic show deviations from the behaviordescribed above. Nevertheless, electron-hole pair creation processes are also crucial tounderstand the vibrational excitation characteristic of these systems.


Electron-hole pair creation processes do also result in low levels of vibrational excitation inthe non-resonant transport regime (eΦ < 2

1

). The corresponding vibrational excitationis highlighted by the inset of Fig. 4.18. In this regime (cf. the discussion in Sec. 4.1),inelastic co-tunneling processes lead to a level of vibrational excitation that is not as largeas in the resonant transport regime but nevertheless significant. At the onset of inelasticco-tunneling, that is, at eΦ = Ω

1

, vibrational excitation increases almost linearly withthe applied bias voltage Φ. This behavior is very similar to the behavior of the respectivecurrent-voltage characteristic. Resonant electron-hole pair creation processes, which arealready active in this regime, control these levels of excitation. They become active at thesame bias voltage, as the corresponding CoSET processes, that is, at eΦ = 2(

1

−nΩ1

) (cf.Sec. 4.2.4). Since both resonant electron-hole pair creation and CoSET involve resonantabsorption of vibrational energy by tunneling electrons, the respective levels of vibrationalexcitation are rather low in this regime. Moreover, they lead to a stepwise decrease ofvibrational excitation at eΦ = 2(

1

− nΩ1

), where the electronic state gets close to thebias window (see inset of Fig. 4.18).

4.3.2 Off-Resonant Electron-Hole Pair Creation Processes

If the electronic state is located far away from the chemical potential in the leads, resonantelectron-hole pair creation processes (cf. Sec. 4.3.1) or CoSET (cf. Sec. 4.2.4) are not veryefficient. Nevertheless, we observe rather low levels of vibrational excitation in the non-resonant transport regime of a molecular junction. In this section, we thus present anothermechanism for local cooling, which is important in the non-resonant transport regime and,as will we see, for large bias voltages. To this end, we consider the vibrational excitationcharacteristics shown in Figs. 4.18 and 4.19, which are calculated using both the ME andthe NEGF method.

The comparison of results that are obtained by NEGF and ME theory allows to eluci-date the role of higher-order processes, for example, co-tunneling processes. Since theseprocesses are described by the NEGF approach but not by the ME method, NEGF givesa finite current (see Sec. 4.1) and a finite level of vibrational excitation (see inset ofFig. 4.18) in the non-resonant transport regime of a molecular junction, while the MEapproach does not. Resonant and non-resonant transport processes thus influence thetransport characteristics of a molecular junction in a different way. Similarly, resonantand off-resonant electron-hole pair creation processes affect the transport characteristicsof a molecular junction in their own, characteristic way. An example for an off-resonantelectron-hole pair creation process is depicted in Fig. 4.17d. Since these processes do notdirectly contribute to the current through the junction but to the corresponding aver-age level of vibrational excitation, their effect is most suitably studied by the respectivevibrational excitation characteristics.

Naturally, off-resonant electron-hole pair creation processes become important, wheneverresonant electron-hole pair creation processes are inefficient. This is, for example, thecase for high bias voltages, where the resonant creation of an electron-hole pair requires


the absorption of many vibrational quanta. In contrast, off-resonant electron-hole paircreation processes can always occur upon the absorption of a single vibrational quantum,(almost) irrespective of the applied bias voltage9. The levels of vibrational excitationthat are obtained by the NEGF and the ME approach therefore deviate more stronglyfor large bias voltages: |eΦ|− m Ωα. Thereby, due to cooling by off-resonant electron-hole pair creation processes the level of excitation obtained by NEGF is smaller thanthe one obtained by the ME approach. This behavior can be observed, for example,in the vibrational excitation characteristic of model E1V1 (Fig. 4.18) for bias voltagesΦ & 2 V, but also in Fig. 4.19, where the vibrational excitation characteristic of thisjunction is shown for two different electronic-vibrational coupling strengths λ

11

. For yetsmaller electronic-vibrational coupling, results obtained by NEGF and ME exhibit thesedeviations for even smaller bias voltages (), as cooling by resonant pair creation processesis also less efficient at these bias voltages.

In the non-resonant transport regime, off-resonant electron-hole pair creation processesplay a similar role as resonant pair creation processes in the resonant transport regime(cf. Sec. 4.3.1). While excitation and deexcitation by inelastic co-tunneling processeswould also lead to a ’random-walk’ through the ladder of vibrational states and, thus, tohigh levels of vibrational excitation, as these processes occur with similar probabilities,off-resonant electron-hole pair creation processes constitute another cooling mechanismthat leads to rather low levels of vibrational excitation. However, the average vibrationalexcitation in the non-resonant transport regime is much smaller than in the resonantone. The reason for this behavior is twofold. First, CoSET and resonant pair creationprocesses can diminish the level of vibrational excitation in this regime significantly (asoutlined in Secs. 4.2.4 and 4.3.1). Second, the number of inelastic co-tunneling processesincreases linearly with the applied bias voltage Φ (cf. Sec. 4.1.1), while the number ofoff-resonant electron-hole pair creation processes is the same for all bias voltages Φ. Thisis in contrast to resonant pair creation processes, which become blocked one by one inthe resonant transport regime. Thus, local heating by inelastic co-tunneling processes isefficiently counterbalanced by cooling due to off-resonant pair creation processes for allbias voltages Φ.

4.3.3 Cooling in the Presence of Multiple Electronic States

So far, we have discussed local cooling with respect to a single electronic state. While, inprinciple, processes that occur in transport through a single electronic state do also occurwith respect to multiple electronic states, the interplay of these processes is non-trivial.For example, the current-voltage characteristic of a junction with multiple electronic statesis not simply given by the sum of currents that is obtained for each of these states sep-arately. As discussed in Sec. 4.2.2, local heating of the vibrational degrees of freedomwith respect to lower-lying electronic states of a molecular junction facilitates transportthrough the respective higher-lying electronic states, even before these states have entered

9 At Φ→∞, however, off-resonant pair creation processes are also suppressed.


NEGF !2"0.8eVNEGF !2"0.2eVME !2"0.8eVME !2"0.2eV

0 0.5 1. 1.5 2.0

2

4

6

8

10

bias voltage # !V"


Figure 4.20: Vibrational excitation characteristics for a model molecular junction withtwo electronic states coupled to a single vibrational mode (model E2V1, cf. Tab. 4.2).Shown are results for two different positions of the higher-lying electronic state

2

. If thetwo electronic states are close in energy, that is,

1

= 0.15 eV and 2

= 0.2 eV for thesolid gray and orange lines, the respective vibrational excitation characteristic increasescontinuously with the applied bias voltage Φ similar to inelastic transport through a singleelectronic state. If the two electronic states, however, are separated by several units of thevibrational energy Ω

1

, that is, 1

= 0.15 eV and 2

= 0.8 eV for the solid black and red lines,resonant deexcitation processes with respect to the higher-lying electronic state facilitatelocal cooling of the vibrational mode in the intermediate bias regime 2

1

< eΦ < 22

.Accordingly the slope of the respective excitation characteristic can be even negative inthis bias regime.

the bias window. Steps in the current-voltage characteristic, which are associated withhigher-lying electronic states, are therefore much less pronounced than the steps associ-ated with lower-lying electronic states. This behavior has already been analyzed for thecurrent-voltage characteristics of the two-state model molecular junction E2V1 (cf. Fig.4.9, Sec. 4.2.2).

The corresponding excitation characteristic is shown in Fig. 4.20, where the solid blackand solid red line have been obtained using the NEGF and the ME approach, respectively.In addition, the solid gray (NEGF) and the solid orange (ME) line of this figure showthe vibrational excitation characteristic of a model molecular junction that deviates frommodel E2V1 just by the position of the higher-lying electronic state, that is

2

= 0.2 eVinstead of

2

= 0.8 eV. For small bias voltages, Φ < 1V, one observes a continuousstep-wise increase of vibrational excitation in these junctions, which is a result of localheating by inelastic transport processes with respect to the lower-lying electronic state


(1

= 0.114 eV). Thereby, the solid gray and orange lines exhibit additional steps, whichare associated with local heating by the higher-lying electronic state at

2

= 0.164 eV.Apart from these additional steps, the respective levels of vibrational excitation are verysimilar to the one of model E2V1 for Φ < 1V, and even to the one including just a singleelectronic state (cf. for example, the excitation characteristic of model E1V1 depictedin Fig. 4.18). This demonstrates that the level of vibrational excitation in a molecularjunction is determined by the ratio between the number of excitation and deexcitationprocesses, which is almost the same for these three cases, rather than by the total numberof these processes.

For larger bias voltages, Φ > 1V, the levels of vibrational excitation that are obtained forboth systems strongly deviate from each other. This is due to the onset of resonant deexci-tation processes with respect to sequential tunneling through (Fig. 4.8b) and electron-holepair creation at the higher-lying electronic state (Fig. 4.17c). These processes result ina significantly lower level of vibrational excitation in junction E2V1, where the slopeof the respective excitation characteristic can be even negative in the intermediate biasregime 2

1

< eΦ < 22

. Although similar processes are also active for the junction with2

= 0.2 eV, the efficiency of these processes is much larger in junction E2V1. This is dueto the fact that these processes require much less vibrational quanta in junction E2V1 atbias voltages Φ > 1V. Thus, the higher-lying electronic state in this molecular junctionfacilitates efficient cooling mechanisms for the vibrational degrees of freedom. These cool-ing mechanisms extend over a broad range of bias voltages, or in other words, stabilizethe molecular junction with respect to vibrationally induced dissociation. These coolingor stabilization mechanisms are quite general phenomena in electron transport throughmolecular junctions [80]. They do also occur, for example, if the two electronic states arelocated below the Fermi level of the junction, such as for the benzenedibutanedithiolatemolecular junction considered in Sec. 7.1. In general, these mechanisms are facilitated byresonant deexcitation processes with respect to electronic states that are located fartheraway from the Fermi level of the junction.

4.3.4 Coulomb Cooling

In this section, we discuss the effect of (repulsive) electron-electron interactions on thevibrational excitation characteristics of model E2V1. As we have seen in the previoussection, the corresponding level of vibrational excitation shows a strong dependence on theposition of the higher-lying electronic state of this junction. Electron-electron interactions,U

12

= 0, result in a splitting of the corresponding resonance into an anionic resonance at2

and a dianionic resonance at 2

+ U12

(cf., for example, the discussion in Sec. 4.2.2).These interactions may therefore have a strong effect on the efficiency of excitation anddeexcitation processes in this molecular junction.

To study this effect, we compare the vibrational excitation characteristic of this modelmolecular junction with the vibrational excitation characteristic that we obtain for thisjunction including an additional electron-electron interaction term U

12

= 0.5 eV. These


ME U12!0eVME U12!0.5eV

NEGF U12!0eVNEGF U12!0.5eV

0 0.5 1. 1.5 2. 2.50

5

10

15

20

25

bias voltage " !V"


Figure 4.21: Vibrational excitation characteristics for a model molecular junction withtwo electronic states coupled to a single vibrational mode (model E2V1, cf. Tab. 4.2).Shown are results with (U

12

= 0.5 eV) and without Coulomb interactions (U12

= 0).Including Coulomb interactions, the corresponding vibrational excitation characteristicshows less pronounced variations in the level of vibrational excitation than without theseinteractions. This indicates that resonant excitation and deexcitation processes are ef-fective over a broader range of bias voltages, which is due to a splitting of the respectiveresonances induced by electron-electron interactions. As a result, at high bias voltages, thelevel of vibrational excitation is much lower in the presence of (repulsive) electron-electroninteractions (Coulomb Cooling).

additional electron-electron interactions may represent, for example, repulsive Coulombinteractions. The corresponding excitation characteristics are depicted in Fig. 4.21. Theeffect of repulsive electron-electron interactions on the corresponding level of vibrationalexcitation is twofold. First, deexcitation processes with respect to the dianionic resonancedo not contribute to the local cooling of the vibrational mode in the intermediate biasregime 2

1

< eΦ < 22

, because this resonance is shifted to higher energies (2

+U12

). Thereduction of vibrational excitation that is observed for the model system E2V1 in this biasregime is thus less pronounced if Coulomb interactions are taken into account. Second, athigher bias voltages, Φ & 2V, resonant deexcitation processes with respect to the shiftedresonance are more efficient than without additional electron-electron interactions, sincethey require the absorption of less vibrational quanta at these bias voltages. This resultsaccordingly in significantly lower levels of vibrational excitation. Thus, the excitationcharacteristic of model E2V1 exhibits less pronounced variations in vibrational excitationif additional electron-electron interactions are considered and a significantly lower levelof vibrational excitation for high bias voltages. We therefore conclude that (repulsive)


electron-electron interactions stabilize this molecular junction, because deexcitation pro-cesses with respect to a higher-lying electronic state are more effective over a broaderrange of bias voltages. Similar effects may occur in many other molecular junctions.

4.3.5 Pumping High-Frequency with Low-Frequency Modes

In junctions with multiple vibrational degrees of freedom, local heating of low-frequencymodes triggers inelastic transport processes involving high-frequency modes. Such pro-cesses, where a low- and a high-frequency mode is deexcited and excited simultaneously,is depicted in Fig. 4.12. The role of these processes in local cooling of a molecular junctionis studied in this section.

To this end, we employ the two-mode model molecular junction E1V2, which current-/conductance-voltage characteristics have already been analyzed in the non-resonant andthe resonant transport regime in Secs. 4.1.2 and 4.2.3, respectively. The correspondingvibrational excitation characteristics of the two modes are plotted in Fig. 4.22a. They arevery similar to the excitation characteristic of the single vibrational mode in model systemE1V1 (cf. Fig. 4.18). In contrast, however, processes that involve both vibrational modes(such as the one shown in Fig. 4.12) result in a number of additional steps. The excitationcharacteristic of model system E1V2 thus shows a richer spectrum of resonances, but atthe same time also smoother transport characteristics than for example E1V1. Moreover,mode 1 shows a much higher level of (current-induced) excitation than mode 2, whichreflects the fact that the frequency of mode 1 is lower than that of mode 2 (note thatλ

11

/Ω1

= λ12

/Ω2

= 0.6).

In addition, the solid gray and dashed gray lines in Fig. 4.22a depict results, where thecoupling of mode 1 to the electronic state is reduced from λ

11

= 0.06 eV to λ11

= 0.01 eV.According to the argument given in Sec. 4.3.1, a much higher level of excitation for mode1 is thus expected. This, however, is not observed due to cooling by processes, wherethe excitation of mode 2 is accompanied by a deexcitation of mode 1 (see Fig. 4.12 foran example process). Thus, the level of vibrational excitation for mode 1 steeply risesat eΦ = 2(

1

+ Ω1

) due to the blocking of resonant electron-hole pair creation processesand the onset of resonant excitation processes that involve a single vibrational quantumof this mode. However, it also drops drastically at eΦ = 2(

1

+ Ω2

−Ω1

), where electronscan resonantly tunnel from the left lead onto the molecular bridge by the simultaneousexcitation of mode 2 and deexcitation of mode 1. Note that at these bias voltages suchmixed tunneling processes are the only resonant tunneling processes that are cooling mode2 by the absorption of a single vibrational quantum. Due to the low coupling strengthλ

11

these processes constitute the most important cooling mechanism for mode 1. As aresult, the corresponding average vibrational energy,

Pα Ωαa†αaα, is reduced accordingly

at the onset of these processes (cf. Fig. 4.22b). While the efficiency of these processes islarge compared to the other processes involving mode 1, it is rather low compared tothose involving mode 2. Pumping of the high-frequency mode (mode 2) due to a weaklycoupled low-frequency mode, as proposed in Ref. [244], is, therefore, very inefficient.


(a)

NEGF mode 1 !11"0.06eVNEGF mode 2 !11"0.06eVNEGF mode 1 !11"0.01eVNEGF mode 2 !11"0.01eV

1. 1.5 2. 2.5 3.0

3

6

9

bias voltage # !V"


(b)

NEGF !11"0.06eVNEGF !11"0.01eV

1. 1.5 2. 2.5 3.0

0.5

1.

1.5

2.

bias voltage # !V"

vibrationalenergy!eV"

Figure 4.22: Panel a: Vibrational excitation characteristics for a model molecularjunction with a single electronic state coupled to two vibrational modes (model E1V2,cf. Tab. 4.1). Panel b: Average vibrational energy,

Pα Ωαa†αaα, corresponding to the

excitation characteristics in panel a. Local cooling of the low-frequency mode (mode 1) byprocesses, where the high-frequency mode (mode 2) is excited and the low-frequency modeis deexcited, can be so efficient that the average vibrational energy of a molecular junctiondecreases at the onset of these processes. This, however, requires a weak coupling of thelow-frequency mode to the electronic degrees of freedom. Pumping of high-frequency modesdue to these processes [244], however, is very inefficient.


0 0.5 1.0

0.5

1.

NEGF vibronic !"0NEGF vibronic !"0.03

0 0.5 1. 1.5 2. 2.50

5

10

15

bias voltage # !V"


Figure 4.23: Vibrational excitation characteristics for a model molecular junction witha single electronic state coupled to a single vibrational mode (model E1V1, cf. Tab. 4.1).The black line depicts results, where no coupling of the vibrational mode to a thermalbath is considered, while the blue line is computed including such a coupling (ζ

1

= 0.03).Vibrational relaxation induced by the mode-bath coupling reduces the level of current-induced vibrational excitation drastically, except at eΦ ∼ 2

1

, where vibrational excitationoriginates to a large extent from polaron-formation. In the presence of a thermal bath(blue line), vibrational excitation does saturate for high bias voltages, while it does not,if cooling of the vibrational mode is facilitated by electron tunneling processes only (blackline).

4.3.6 Coupling to a Thermal Bath: Vibrational Excitation dueto Polaron Formation

In this last section, we discuss local cooling of a vibrational mode due to coupling to abath of secondary modes. Such modes may represent phononic degrees of freedom of theleads [209] or of the solvent surrounding the molecular bridge in an electrochemicallygated molecular junction [13]. To this end, we employ the model system E1V1, includingcoupling of the vibrational mode to a ’cold’ thermal bath (T

bath

= 10 K) with couplingstrength ζ

1

= 0.03. The corresponding relaxation time of the vibrational mode is ≈ 3 ps.

The respective excitation characteristic is represented by the solid blue line in Fig. 4.23.Overall, the average level of vibrational excitation in this molecular junction is much lowerin the presence of a mode-bath coupling than without it (solid black line). This is dueto vibrational energy transfer from the primary mode of the junction to the bath modes,or equivalently, dissipation of current-induced vibrational excitation by the thermal bath.Thereby, much lower levels of excitation are found in both the non-resonant transport


regime which is highlighted in the inset of this figure, and in the resonant transportregime for large bias voltages. At the onset of the resonant transport regime, eΦ ∼ 2

1

,however, there is almost no reduction of vibrational excitation due to the presence ofthe thermal bath. There, vibrational excitation is strongly related to the formation of apolaronic state, that is, to the (nonequilibrium) population of the electronic state. Thiscan be readily inferred from Eq. (3.77) or Eq. (3.182), which comprise two contributionsof the average vibrational excitation: a current-induced part, a†

1

a1

H , and a part that is

correlated with the population of the electronic state, λ211

Ω

21c†

1

c1

H . Since the population of

the electronic level is determined by the chemical potentials in the leads, this contributionto the excitation characteristic remains almost unchanged wether the vibrational mode iscoupled to a thermal bath or not.

Another observation is that the vibrational excitation characteristic in the presence ofa thermal bath (blue line) saturates. This means that the level of current-induced vi-brational excitation increases for high bias voltages by continuously smaller steps. Thisis in contrast to the excitation characteristic of model E1V1, where no coupling to athermal bath is considered (black line). Without a bath, the successive suppression ofresonant electron-hole pair creation processes and the successive onset of resonant exci-tation processes lead to an increase of vibrational excitation by continuously larger steps(cf. Sec. 4.3.1). As bath induced cooling is not affected by the applied bias voltage, it pre-vails over cooling by electron-hole pair creation and electron transport processes at highbias voltages, and thus, leads to the observed saturation of (current-induced) vibrationalexcitation.

4.4 Conclusions

In this chapter, we have presented the most important transport mechanisms that are rel-evant in vibrationally coupled electron transport through a single-molecule junction. Thisincludes both non-resonant and resonant processes. As we have seen, the current-voltagecharacteristics of a molecular junction and the corresponding population of the electronicstates can be qualitatively understood in terms of transport processes, that is processes,where an electron tunnels from one lead to the other through the molecular bridge10. Fora quantitative analysis, however, and to understand the respective vibrational excitationcharacteristics, electron-hole pair creation processes need to be taken into account.

We have shown that inelastic processes have a profound influence on the transport charac-teristics of a single-molecule contact and that they result in manifold resonance structures.Thereby, multimode processes and the interplay of multiple electronic states, either dueto repulsive Coulomb interactions or due to the coupling of these states to the same vi-

10Note that this statement applies for symmetric molecular junctions, which are studied in this chapter.In asymmetric molecular junctions, which are the subject of Chap. 6, asymmetries in the current-voltagecharacteristic or the population of the electronic levels cannot be solely understood in terms of transportprocesses, but requires to account for the effect of electron-hole pair creation processes.


brational degree of freedom, play a significant role. Besides electron transport, we havefocused on local heating and cooling effects of the vibrational modes of a molecular junc-tion. While these effects determine, on one hand, the efficiency of electron tunnelingprocesses, they are also important for the stability of a molecular contact. We have foundthat the corresponding levels of vibrational excitation do not necessarily monotonouslyincrease but may also decrease with the applied bias voltage. The latter occurs when-ever an electronic state enters the bias window, for example, at the transition betweenthe resonant and the non-resonant transport regime, or if a higher-lying electronic stateenters the bias window in the resonant transport regime. Moreover, we have found mul-timode effects as well as (repulsive) Coulomb interactions to enhance the stability of asingle-molecule junction.

While we have employed generic model systems to discuss these mechanisms, they areall likely to occur in real molecular junctions. This is demonstrated, for example, inChap. 7, where we discuss models of molecular junctions that are based on first-principlescalculations. Each of these model systems exhibits several of the effects that we have beendiscussing in this chapter. However, not all of them are typically equally pronounced.

Chapter 5

Vibrational Instabilities in theResonant Transport Regime

The level of vibrational excitation as a function of a specific parameter can increaseindefinitely in a molecular junction, that is the junction exhibits a vibrational instability.Such an instability occurs, in a trivial sense, in the limit T →∞, where the temperature inthe leads is the largest energy scale of a molecular junction. In this limit, inelastic electrontunneling processes between the leads and the molecular bridge transfer the temperatureof the leads to the vibrational degrees of freedom of the junction. As a result, the effectivetemperature, or level of vibrational excitation, increases indefinitely as T → ∞. Similararguments apply in the static limit, where Ωα → 0.

In this chapter, we study other nontrivial vibrational instabilities in vibrationally cou-pled electron transport through a single-molecule junction. In particular, we address thequestion if vibrational excitation in a molecular junction increases indefinitely with in-creasing bias voltage Φ (Sec. 5.1) or decreasing electronic-vibrational coupling λ

11

(Sec.5.2). These studies reveal and underline the importance of resonant electron-hole paircreation processes, and corroborate the arguments given in Chap. 4.

As these vibrational instabilities occur in the resonant transport regime, we focus inthe following primarily on results obtained by the ME approach. In contrast to the self-consistent solution scheme of the NEGF approach, which includes both resonant and non-resonant processes, the ME methodology allows a more straightforward analysis of theselimits. Moreover, we employ the wide-band approximation for the level-width functions,Γ

L/R,mn(E) ≈ 2(νL/R,mν

L/R,n)/γ = ΓL/R,mn, in order to study these limits irrespective of

the specific level structure in the leads.

98 5. Vibrational Instabilities in the Resonant Transport Regime

5.1 Large Bias Voltage

In the resonant transport regime, the average vibrational excitation of a single-moleculejunction typically increases rapidly with the applied bias voltage Φ [80,83,216]. An exam-ple for this behavior is the vibrational excitation characteristic of junction E1V1, which isshown in Fig. 4.18. As we have already outlined in Sec. 4.3.1, vibrational excitation in thismodel molecular junction increases in steps at bias voltages eΦ = 2(

1

+ nΩ1

) (n ∈ ).The height of these steps becomes continuously larger with increasing bias voltage. Thisbehavior is typical for single-molecule junctions (cf. Sec. 4.3) and results from both thesuppression of electron-hole pair creation processes (Fig. 4.17) and the opening of inelastictransport channels (Fig. 4.4). Thus, the question arises wether the level of vibrationalexcitation saturates in the limit Φ→∞, or not.

To prove that vibrational excitation increases indefinitely in the limit Φ→∞, we assumein the following that vibrational excitation is finite and derive a contradiction to thisassumption. To this end, we evaluate Eq. (3.66) for a single electronic state that iscoupled to a single vibrational mode (cf. Sec. 3.2.4). We further evaluate the resultingequation in between the basis functions 0|ν

1

| and |ν1

|0 and obtain

X

Kν3

fK(1

+ Ω1

(ν3

− ν1

))ΓK,11

|X1,ν1ν3|2 ρν1ν1

0,0 = (5.1)

X

Kν3

(1− fK(1

+ Ω1

(ν3

− ν1

)))ΓK,11

|X1,ν1ν3|2 ρν3ν3

1,1 .

Furthermore, we multiply this equation by ν1

, and add all the resulting equations. Thisyields

X

Kν1ν3

ν1

fK(1

+ Ω1

(ν3

− ν1

))ΓK,11

|X1,ν1ν3|2 ρν1ν1

0,0 = (5.2)

X

Kν1ν3

ν1

(1− fK(1

+ Ω1

(ν3

− ν1

)))ΓK,11

|X1,ν1ν3|2 ρν3ν3

1,1 .

If we assume vibrational excitation a†1

a1

to converge in the limit Φ → ∞, so must theleft hand side and the right hand side of the above equation. Hence, we take the limitΦ→∞ on both sides in Eq. (5.2), replacing f

L

() by 1 and fR

() by 0. This results in

X

ν1ν3

ν1

ΓL,11

|X1,ν1ν3|2 ρν1ν1

0,0 =X

ν1ν3

ν1

ΓR,11

|X1,ν1ν3|2 ρν3ν3

1,1 . (5.3)

Applying the sum ruleP

ν3|Xν1ν3|2 = 1 and

Pν1

ν1

|Xν1ν3|2 = λ2

11

/Ω2

1

+ ν3

to the left andthe right hand side, respectively, gives

X

ν1

ν1

ΓL,11

ρν1ν10,0 =

X

ν3

(ν3

+ λ2

11

/Ω2

1

)ΓR,11

ρν3ν31,1 . (5.4)

5.1. Large Bias Voltage 99

ME !"1.5VME !"2.0VME !"2.5V

0 10 20 30 400

0.05

0.1

vibrational level #

populationofvibrationallevels

Figure 5.1: Population of vibrational levels, pν, in a model molecular junction with asingle electronic state coupled to a single vibrational mode (model E1V1, cf. Tab. 4.1) fordifferent bias voltages Φ. Increasing the bias voltage, the population characteristic of thevibrational levels becomes continuously broader and exhibits equal population of the lower-lying vibrational levels. The level of vibrational excitation, a†

1

a1

, therefore increasesindefinitely as Φ → ∞. Local cooling by resonant electron-hole pair creation processes iscompletely suppressed in this limit.

Analogously, using the 1|ν1

|...|ν1

|1-projection of Eq. (3.66), we obtain

X

ν3

(ν3

+ λ2

11

/Ω2

1

)ΓL,11

ρν3ν30,0 =

X

ν1

ν1

ΓR,11

ρν1ν11,1 . (5.5)

Subtracting Eq. (5.5) from Eq. (5.4) leads to the following equation

−λ2

11

/Ω2

1

X

ν3

ΓL,11

ρν3ν30,0 = λ2

11

/Ω2

1

X

ν3

ΓR,11

ρν3ν31,1 , (5.6)

which is a contradiction, since the left hand side of Eq. (5.6) is negative while its righthand side is positive if λ

11

= 0. Hence, for finite electronic-vibrational coupling λ11

,vibrational excitation must diverge as Φ→∞.

To illustrate this analytic finding, we show in Fig. 5.1 numerical results for the populationof the vibrational levels pν =

Pa ρνν

a,a in junction E1V1 at different bias voltages Φ. It isseen that the population of the vibrational levels becomes broader with increasing biasvoltage and exhibits equal population of the lower vibrational states for large bias voltages.This leads to a continuous increase in the average vibrational excitation with increasingbias voltage, which does not saturate.


We attribute this behavior to the lack of cooling by electron-hole pair creation processes(cf. Fig. 4.17), which are completely blocked in the high-bias limit. As a consequence,the number of excitation and deexcitation processes upon electron transport through themolecule is equal. The respective stationary state is given by a vibrational distributionfunction, where all vibrational levels are equally populated (cf. Fig. 5.1). This can berationalized considering the population of the νth vibrational level pν after a transportprocess in this limit. Since there are as many excitation as deexcitation processes, therespective population is given by the sum

P∞ν

=1

pν |Xνν |2 with pν the correspondingpopulation before such a transport process. The only nonequilibrium state, which isinvariant under these conditions, i.e. pν = pν , is that, where all vibrational levels areequally occupied since

P∞ν

=1

|Xνν|2 = 1. The molecular bridge can thus be understoodto undergo a random walk through the ladder of vibrational states, leading to an equalpopulation of all vibrational states. Note that the current flowing through the molecularjunction in this limit is given by 2eΓ

L,11

ΓR,11

/(ΓL,11

+ ΓR,11

), which is the same result asobtained by Gurvitz et al. [323] without electronic-vibrational coupling.

5.2 Weak Electronic-Vibrational Coupling

For a finite bias voltage that fulfills eΦ > 2(1

+Ω1

), the vibrational excitation of junctionE1V1 may also diverge in the limit of vanishing vibronic coupling λ

11

→ 0. This counter-intuitive phenomenon was reported before [74,82,217,244,360] and analyzed in some detail.In Sec. 5.2.1, we give a short overview of the phenomenon and a physical interpretation interms of electron-hole pair creation processes. This way, we establish the relation betweenthe limit of a weak vibronic coupling and the high-bias limit discussed in the previoussection. In the two subsequent sections, we investigate the limit λ

11

→ 0 for lower biasvoltages (Sec. 5.2.2), where an exact expression for the population of the vibrational levelspν can be derived, and in the presence of a second higher-lying electronic state that isvibrationally coupled (Sec. 5.2.3).

5.2.1 Equivalence to the Limit of Large Bias Voltages

The solid black line in Fig. 5.2 illustrates the phenomenon of vibrational instability inthe limit λ

11

→ 0. It represents the vibrational excitation due to vibrationally coupledtransport through a single electronic state (model INST, cf. Tab. 4.1) as a function ofthe electronic-vibrational coupling strength λ

11

at a fixed bias voltage eΦ = 0.55 eV> 2(

1

+Ω1

). Upon reducing λ11

, the level of vibrational excitation monotonously increases[74,80,82,83,217,244,360] with a slope that gets larger for smaller values of λ

11

. Moreover,the respective vibrational distribution function (cf. Fig. 5.3) becomes broader and showsan increasing number of vibrational levels, starting from the ground state, that are equallypopulated. This occurs only if the associated bias voltage eΦ exceeds 2(

1

+Ω1

). For suchbias voltages electron-hole pair creation processes (cf. Fig. 4.17a) are suppressed to lowest

5.2. Weak Electronic-Vibrational Coupling 101

ME single state !"0.45eVME single state !"0.55eVME two states !"0.55eV

0 0.01 0.02 0.03 0.04 0.05 0.060

5

10

electronic#vibrational coupling $11 !eV"


Figure 5.2: Vibrational excitation for a model molecular junction with a single electronicstate coupled to a single vibrational mode (model INST, cf. Tab. 4.1) as a function of theelectronic-vibrational coupling strength λ

11

. The black line represents the phenomenon ofvibrational instability, which is outlined in Sec. 5.2.1. The gray line shows the vibrationalexcitation obtained at a bias voltage 2(

1

+ Ω1

) > eΦ > 21

, reflecting the results of Sec.5.2.2. The dashed blue line shows that a higher-lying electronic state, which is coupledto the vibrational mode, prevents the vibrational instability depicted by the black line (cf.Sec. 5.2.3).

order in λ11

. In this sense, the limit λ11

→ 0 is equivalent to the high-bias limit, where alsoall relevant electron-hole pair creation processes become blocked and the correspondinglevel of vibrational excitation increases indefinitely as Φ→∞.

These numerical findings demonstrate the general phenomenon of vibrational instabilityin the limit λ

11

→ 0. They are, nevertheless, limited by the number of basis functionsused in the calculations. In the following, we analyze this instability in an analytic andmore general way. To second order in λ

11

, with eΦ > 2(1

+ Ω1

) > 2Ω1

, we obtain fromthe master equation (3.66) the following set of equations (ν ≥ 1)

0 = ΓL,11

ρνν0,0 − Γ

R,11

ρνν1,1 − Γ

R,11

°νλ2

11

ρν−1ν−1

1,1 − (2ν + 1)λ2

11

ρνν1,1 + (ν + 1)λ2

11

ρν+1ν+1

1,1

¢,

(5.7)

0 = ΓL,11

ρνν0,0 − Γ

R,11

ρνν1,1 + Γ

L,11

°νλ2

11

ρν−1ν−1

0,0 − (2ν + 1)λ2

11

ρνν0,0 + (ν + 1)λ2

11

ρν+1ν+1

0,0

¢.

(5.8)

From these equations one infers that the difference ΓL,11

ρνν0,0 − Γ

R,11

ρνν1,1 is of second order

in λ11

. Therefore, we can replace the terms ΓR,11

ρνν1,1 by Γ

L,11

ρνν0,0 in Eq. (5.7) (or vice versa


ME !11"0.06eVME !11"0.01eVME !11"0.005eV

0 10 20 30 400

0.05

0.1

vibrational level #

populationofvibrationallevels

Figure 5.3: Population of vibrational levels, pν, in a model molecular junction witha single electronic state coupled to a single vibrational mode (model INST, cf. Tab. 4.1)for different electronic-vibrational coupling strengths λ

11

. The bias voltage Φ is fixed to0.55 V > 2(

1

+ Ω1

) such that the lowest order (O(λ2

11

)) resonant electron-hole pair cre-ation processes are blocked. For weaker electronic-vibrational coupling λ

11

, the vibrationaldistribution function becomes broader and exhibits equal population of the lower-lying vi-brational levels.

in Eq. (5.8)). Subtracting Eq. (5.7) from Eq. (5.8) thus gives

0 = (2ν + 1)ρννa,a − (ν + 1)ρν+1ν+1

a,a − νρν−1ν−1

a,a , (5.9)

where a ∈ 0, 1. The recurrence relation defined by Eq. (5.9) leads to divergent popula-tions ρνν

a,a → ±∞ in the limit ν →∞ if ρ11

a,a− ρ00

a,a = 0. Thus, the only physically relevantsolution1 is the one, where all vibrational levels are equally occupied, ρν+1ν+1

a,a − ρννa,a = 0.

This corresponds to an infinite vibrational excitation or a vibrational instability in thelimit λ

11

→ 0.

It is noted that rigorous divergence of the vibrational excitation is only found for anisolated molecular vibration. Vibrational relaxation processes, introduced, e.g., by cou-pling of the vibrational mode to a thermal bath of secondary vibrational modes, wouldrestrict the vibrational excitation in this model molecular junction to a finite value (cf.the discussion in Sec. 4.3.6).

As in the high-bias limit, the current obtained at these bias voltages, eΦ > 2(1

+ Ω1

),in the limit λ

11

→ 0 is also given by 2eΓL,11

ΓR,11

/(ΓL,11

+ ΓR,11

). For weak electronic-vibrational coupling, inelastic transport processes do not contribute to the current, asthey occur on time scales that are much longer than for electronic transport processes.

1 Note that this solution has to give, for example, the correct electrical current.


5.2.2 Cooling by Resonant Electron-Hole Pair Creation Pro-cesses

For bias voltages in the range 2(1

+ Ω1

) > eΦ > 21

> Ω1

, the vibrational excitation ofa mode coupled to a single electronic state remains finite in the limit λ

11

→ 0. This isillustrated by the gray line in Fig. 5.2, which has been obtained for the same parametersas considered above (black line) but with a smaller bias voltage Φ = 0.45 V. Intriguingly,for λ

11

→ 0, the level of vibrational excitation approaches a value that is non-zero.

This result can be derived from the master equation (3.66). For voltages 2(1

+ Ω1

) >eΦ > 2

1

> Ω1

, we obtain the following set of equations

∆ν1 = ΓL,11

(ν1

+ 1)λ2

11

ρν1ν10,0 + (ν

1

+ 1)λ2

11

ΓL,11

ρν1+1ν1+1

1,1 (5.10)

+ΓR,11

(ν1

+ 1)λ2

11

ρν1+1ν1+1

1,1 − ΓR,11

(2ν1

+ 1)λ2

11

ρν1ν11,1 + Γ

R,11

ν1

λ2

11

ρν1−1ν1−1

1,1 ,

∆ν1 = ΓL,11

ν1

λ2

11

ρν1ν11,1 − Γ

L,11

°−(2ν1

+ 1)λ2

11

ρν1ν10,0 + (ν

1

+ 1)λ2

11

ρν1+1ν1+1

0,0

¢,

with

∆ν1 ≡ ΓL,11

ρν1ν10,0 − Γ

R,11

ρν1ν11,1 . (5.11)

From Eqs. (5.10), which are valid to second order in λ11

, we further deduce

0 = (ν1

(2 + ΓL,11

/ΓR,11

) + ν1

+ 1) ρν1ν1a,a (5.12)

−(ν1

+ 1)λ2

11

(ΓL,11

/ΓR,11

+ 2) ρν1+1ν1+1

a,a − ν1

λ2

11

ρν1−1ν1−1

a,a ,

where a ∈ 0, 1. The solution of this equation is given by

ρννa,a = ρν−1ν−1

a,a / (2 + ΓL,11

/ΓR,11

) , (5.13)

pν = ρνν0,0 + ρνν

1,1 (5.14)

=Γ

L,11

+ ΓR,11

ΓL,11

+ 2ΓR,11

µΓ

R,11

ΓL,11

+ 2ΓR,11

∂ν

,

which corresponds to an average vibrational excitation of

a†1

a1

= ΓR,11

/ (ΓL,11

+ ΓR,11

) , (5.15)

that is, 1/2 for ΓL,11

= ΓR,11

. Thus, vibrational excitation approaches a finite value asλ

11

→ 0, because in this case, the leading-order electron-hole pair creation processes (Fig.4.17a) are active. For negative bias voltages −2(

1

+ Ω1

) < eΦ < −21

< Ω1

, we obtain avery similar result

a†1

a1

= ΓL,11

/ (ΓL,11

+ ΓR,11

) . (5.16)

Note that for yet lower bias voltages, |eΦ| < 21

, the current and the respective current-induced vibrational excitation obtained by the ME method vanish.


5.2.3 Resonant Absorption Processes in Systems with MultipleElectronic States

In this section, we address the question if resonant absorption processes with respect toanother electronic state provide a cooling mechanism [80] that prevents the vibrational in-stability we observed for a single electronic state (Sec. 5.2.1). Model calculations shown bythe dashed blue line in Fig. 5.2 support this conjecture. The current-induced vibrationalexcitation of a vibrational mode that is coupled to a lower- and a higher-lying electronicstate remains finite in the limit λ

11

→ 0. The parameters of these calculation are the sameas those for the black line considered in Sec. 5.2.1, but include a higher lying-electronicstate at

2

= 0.8 eV that is coupled to the vibrational mode with a coupling strength ofλ

21

= −0.06 eV and to the leads in the same way as the lower-lying state (model E2V1,cf. Tab. 4.2).

This behavior can be rationalized by the master equation (3.66). To zeroth order in λ11

,we obtain for bias voltages 2

2

> eΦ > 2(1

+ Ω1

)

0 = ΓR,11

ρν1ν110,10

− ρν1ν100,00

ΓL,11

− ρν1ν100,00

X

Kν3

ΓK,22

|X2,ν1ν3|2 fK(

2

+ Ω1

(ν3

− ν1

)), (5.17)

0 = ΓL,11

ρν1ν100,00

− ρν1ν110,10

ΓR,11

− ρν1ν110,10

X

Kν3

ΓK,22

|X2,ν1ν3|2 fK(

2

+ Ω1

(ν3

− ν1

)), (5.18)

0 =X

Kν3

ΓK,22

X†2,ν1ν3

X2,ν3ν1ρ

ν3ν300,00

fK(2

+ Ω1

(ν1

− ν3

)), (5.19)

0 =X

Kν3

ΓK,22

X†2,ν1ν3

X2,ν3ν1ρ

ν3ν310,10

fK(2

+ Ω1

(ν1

− ν3

)). (5.20)

Thereby, the populations ρνν01,01

and ρνν11,11

are treated as second order contributions . λ2

11

,because the population of the higher-lying state requires a preceding resonant emissionprocess with respect to the lower-lying state (∼ λ2

11

). Since Eqs. (5.19) and (5.20) involveterms that are either positive or zero, these equations can only be fulfilled, if the pop-ulations ρνν

00,00

and ρνν10,10

vanish for values of ν, where fK(2

+ Ω1

(ν − ν3

)) = 0. Thus,vibrational levels with a quantum number larger than (

2

− µL/R

)/Ω1

are not populatedin the limit λ

11

→ 0. Vibrational excitation is therefore finite in this limit, if a sec-ond higher-lying electronic state couples to the vibrational mode with a finite couplingstrength λ

21

.

Note that for these considerations we have assumed the energy 2

of the second electronicstate to be larger than

1

+ Ω1

. Similar results would be obtained, however, if the secondelectronic state is located below the Fermi level but outside the bias window,

2

< −(1

+Ω

1

). If the second electronic state is located within the bias window considered, |2

| <|

1

+ Ω1

| < eΦ, we would obtain the level of vibrational excitation, which results fromtransport through this state only, because the other electronic state decouples from thevibrational mode in the limit λ

11

→ 0.


5.3 Conclusions

In this chapter, we have investigated various limits of vibrationally coupled electron trans-port. Thereby, we have focused on vibrational instabilities, that is limits, where the levelof vibrational excitation increases indefinitely. We have shown, for example, that the limitof large bias voltages, Φ→∞, exhibits a vibrational instability and pointed out that thisis related to the fact that electron-hole pair creation processes are completely blockedin this limit. Similarly, cooling by electron-hole pair creation processes is suppressed forvanishing electronic-vibrational coupling, λ

11

→ 0, if the magnitude of the applied biasvoltage does not allow for resonant electron-hole pair creation processes involving just asingle vibrational quantum, that is for eΦ > 2(

1

+ Ω1

). As a result, the level of vibra-tional excitation also increases indefinitely in this limit. These findings complement thediscussion that we have given in Sec. 4.3.1, where we have demonstrated that the levelof vibrational excitation in a molecular junction increases in steps that become largerfor both increasing bias voltages and weaker electronic-vibrational coupling. We havealso investigated the weak electronic-vibrational coupling limit for lower bias voltageseΦ < 2(

1

+ Ω1

) and demonstrated that the corresponding levels of vibrational excitationare finite but may not necessarily vanish.

Since all the phenomena described in this chapter occur in the resonant transport regime ofa molecular junction, we have employed the master equation approach. Thereby, we havediscussed both numerical and analytical results for the vibrational distribution function.While the master equation approach captures all resonant processes in this nonequilib-rium transport problem, it does not account for higher order effects like co-tunnelingor off-resonant electron-hole pair creation processes. The role of these processes for thevibrational instabilities discussed in this chapter may be the subject of future research.


Chapter 6

The Role of Electronic-VibrationalCoupling for the Electrical TransportProperties of a Molecular Junctionand Possible Applications

In this chapter, we focus on vibrational effects that influence the electrical transport prop-erties of a molecular junction and discuss their relevance for technological applications.Thereby, we increase the complexity of our considerations systematically section by sec-tion. In particular, we discuss negative differential resistance (Sec. 6.1) and rectification(Sec. 6.2) in single-molecule junctions. These transport properties are crucial for the de-sign of electronic devices and/or logic circuits. Possible applications of single-moleculejunctions in the context of mode-selective chemistry are outlined in the following section,Sec. 6.3, where we demonstrate the possibility to selectively excite vibrational modes ina single-molecule contact. Another effect involving multiple vibrational modes is dis-cussed in Sec. 6.4, where we show that anomalous peak broadening, which is observed inmany experiments, can be explained by electronic-vibrational coupling to a number of low-frequency modes. Quantum interference effects can result in similar peak broadening, but,as is shown in Sec. 6.5, are strongly quenched in the presence of electronic-vibrational cou-pling. Such vibrationally induced decoherence is of both fundamental and technologicalinterest, as novel device applications in nanoelectronics are currently derived exploitingthe wave nature of the electron, such as, for example, the quantum interference effecttransistor [369] or the wave function extension transistor [89].

108 6. The Role of Electronic-Vibrational Coupling

6.1 Negative Differential Resistance

Negative Differential Resistance (NDR) is defined as a decrease of electrical current uponan increase of the applied bias voltage Φ. It is relevant for the design of numerouselectronic devices, such as analog-digital converters [179], high-frequency oscillators [180],or logic circuits [181, 182]. NDR in nanostructures was reported decades ago for double-barrier heterostructures [87,88,370,371] and does also occur in resonant electron transportthrough single molecules [52–58]. In this section, we study various mechanisms for NDR insingle-molecule junctions. In particular, we discuss NDR related to the level structure inthe leads (Sec. 6.1.1), vibrational excitation (Sec. 6.1.2) and electron-electron interactionson a molecular bridge, where for the latter we distinguish junctions with a blocking (Sec.6.1.3) and a centrally localized electronic state (Sec. 6.1.4).

6.1.1 The Effect of Narrow Conduction Bands

One of the simplest model systems exhibiting NDR is the resonant tunneling model,where narrow conduction bands in the leads are considered [370, 371]. The current-voltage characteristic of such a model molecular junction is shown in Fig. 6.1. Thisjunction (model BAND, cf. Tab. 4.1) comprises a single electronic state that is coupledsymmetrically to a left and a right lead. These leads are modeled as semi-infinite tight-binding chains with a relatively small internal hopping parameter γ = 0.2 eV. The widthof the corresponding conduction bands is thus very narrow, 4γ = 0.8 eV.

The solid purple line (NEGF) in Fig. 6.1 represents the electronic current-voltage char-acteristic of such a molecular junction. For bias voltages Φ . 2

1

, the respective currentincreases continuously, as the electronic state approaches and enters the bias window. Inthe resonant transport regime, however, that is for eΦ > 2

1

, the current flowing throughthe junction decreases relatively fast, and finally, at eΦ = 2(2γ −

1

), drops to zero. Thisis due to the overlap of the electronic state and the conduction bands in the leads, whichbecomes continuously smaller with increasing bias voltage Φ. A measure for this overlapis the level-width functions Γ

L/R

(1

) evaluated at the energy of the electronic state. Itultimately vanishes for eΦ > 2(2γ −

1

), where also the electrical current flowing throughthe junction breaks down. Thereby, the slope of the electronic current-voltage character-istic is always negative, which means that due to the narrow band width in the leads, 4γ,the junction exhibits strong NDR in the whole resonant transport regime.

Including electronic-vibrational coupling, we obtain the current-voltage characteristicsdepicted by the solid black (NEGF) and the solid red line (ME) in Fig. 6.1. The dashedgray line is obtained from a NEGF calculation, where the vibration is restricted to itsthermal equilibrium state at 10K. As already discussed in Sec. 4.2, electronic-vibrationalcoupling induces a shift of the resonant transport regime to lower bias voltages, eΦ = 2

1

<2

1

, which is related to the polaron-shift of the electronic level. It also triggers a numberof additional steps, which are associated with inelastic sequential tunneling processes

6.1. Negative Differential Resistance 109

0. 0.2 0.4 0.60

5

10 NEGF electronicNEGF vibronicNEGF th. equ.ME vibronic

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1.

bias voltage ! !V"

currentI!"A"

Figure 6.1: Current-voltage characteristics of a model molecular junction with a singleelectronic state coupled to a single vibrational mode (model BAND, cf. Tab. 4.1). The in-set shows the corresponding vibrational excitation characteristic. Due to the narrow bandwidths in the left and the right lead, the current decreases relatively fast in the resonanttransport regime. Such NDR is diminished in the presence of electronic-vibrational cou-pling, since inelastic transport processes tend to probe the conduction bands in the leadsnot at a single but a number of different energies. This results in an effective averagingof the electronic band structure in the leads. Moreover, NDR at eΦ = 2(

1

+ Ω1

) ap-pears only, if current-induced vibrational excitation is taken into account. This is evidentby comparison of the solid black (vibronic transport) and the dashed gray line (thermallyequilibrated transport).

(cf. Fig. 4.4d). In the present context, however, it is more important to realize thatthese current-voltage characteristics exhibit resonant transport over a much wider rangeof bias voltages than the respective electronic current-voltage characteristic. Electronic-vibrational coupling thus results in an effective widening of the conduction bands of thismolecular junction. This is because the current in the presence of inelastic transportprocesses is determined by the respective level-width functions evaluated at a numberof different energies, Γ

L/R,11

(1

+ nΩ) (n ∈ ), and not at a single resonance energy, asfor the electronic current. Inelastic transport processes thus average the electronic bandstructure in the leads. Features in the transport characteristics of a molecular junction,which are induced by the band structure in the leads, are thus less pronounced, or mayeven disappear in the presence of electronic-vibrational coupling. This is a rather generalphenomenon in single-molecule junctions, and was reported before, for example, in Ref.[189]. There, NDR was found in the electronic but not in the vibronic current-voltagecharacteristic of a benzenediethanedithiolate molecular junction.


Comparison of the solid black (NEGF) and the solid red line (ME) in Fig. 6.1 showsthat higher-order processes, such as, for example, co-tunneling processes, tend to inducea similar averaging of the electronic band structure. The broadening of molecular levelsdue to coupling of the molecule to the leads thus softens the steps in the solid black line,for which NDR is consequently less pronounced than in the solid red line.

6.1.2 Suppression of Electrical Current due to an Increase inVibrational Excitation

In contrast to the NDR mechanism outlined in Sec. 6.1.1, which is less pronounced inthe presence of electronic-vibrational coupling, we focus in this section on NDR, whichrequires electronic-vibrational coupling, in particular, the corresponding current-inducedvibrational excitation. To this end, we compare calculations, where current-induced vibra-tional excitation is taken into account, to results, where we consider thermally equilibratedtransport. For example, the current-voltage characteristics depicted by the solid black anddashed gray line in Fig. 6.1 show overall a very similar behavior. At eΦ = 2(

1

+ Ω1

),however, the dashed gray line exhibits a step-wise increase of the current, while the solidblack line including the effect of current-induced vibrational excitation exhibits a step-wise decrease (NDR). This can be understood as follows. At this bias voltage, sequentialtunneling processes become active, where an electron tunneling from the left onto themolecular bridge excites the vibrational mode by a quantum of vibrational energy (Fig.4.4d). This additional transport channel results in a larger current if thermally equi-librated transport is considered (dashed gray line). In addition, resonant electron-holepair creation processes involving just a single vibrational quantum (Fig. 4.17a) are alsosuppressed. The respective level of vibrational excitation thus increases significantly (cf.the solid black and the solid red line in the inset). As a result of this increased levelof vibrational excitation the corresponding vibronic current is suppressed. NDR in thismodel molecular junction, therefore, appears not only due to the narrow band width inthe leads (cf. Sec. 6.1.1) but also when the level of current-induced vibrational excitationincreases. Similarly, NDR occurs in asymmetric molecular junctions, where blocking ofelectron-hole pair creation processes with respect to the stronger coupled electrode leadsto substantially higher levels of vibrational excitation, and thus, to lower currents flowingthrough these junctions [83].

6.1.3 Breakdown of Electrical Current due to a Blocking State

The effects and mechanisms discussed so far involve a single electronic state. In transportthrough multiple electronic states, NDR may also be the result of (repulsive) electron-electron interactions. Such systems are studied in this and in the following section. Here,we consider a model for a molecular junction with two electronic states (model BLOCK,cf. Tab. 4.2), where the lower-lying state is symmetrically and the higher-lying state


!2. !1. 0 1. 2.0

0.5

1.

NEGF electronicNEGF vibronicNEGF th. equ.ME vibronic

!2 !1 0 1 2!1.5!1.!0.50

0.51.1.5

bias voltage " !V"

currentI!#A"

Figure 6.2: Current-voltage characteristics of a model molecular junction with two elec-tronic states coupled to a single vibrational mode (model BLOCK, cf. Tab. 4.2). Thelower-lying electronic state is coupled symmetrically to the leads, while the higher-lyingstate is coupled more strongly to the left than to the right lead. This state constitutesa blocking state. Its population characteristic is shown in the inset. Once the blockingstate is populated at eΦ = 2

2

, the current through this molecular junction is stronglysuppressed, as resonant tunneling through the lower-lying electronic state requires an ad-ditional charging energy U . Electronic-vibrational coupling is less significant for this NDRmechanism.

asymmetrically coupled to the leads. Such an asymmetry in the coupling of the electronicstates to the leads reflects, for example, an asymmetric structure of the molecular bridge(as is outlined, for example, in Refs. [372–374]). Fig. 6.2 represents current-voltage char-acteristics of this model molecular junction, where the inset depicts the correspondingpopulation characteristics of the higher-lying electronic state. Note that in the followingwe refer to the higher-lying electronic state as the blocking state.

As the blocking state is more strongly coupled to the left than to the right lead, νL,2 > ν

R,2,it is populated much faster for positive bias voltages than it is depopulated, and therefore,almost fully occupied for eΦ & 2

2

. For negative bias voltages, however, the time scalesfor populating and depopulating this state are reversed, such that it remains almostunoccupied. This asymmetric population of the blocking state has a dramatic effecton the respective current-voltage characteristics. While for negative bias voltages thecurrent-voltage characteristics of this molecular junction is not influenced by the blockingstate but dominated by the lower-lying electronic state, it is strongly influenced by thehigher-lying electronic state for positive bias voltages. Especially at eΦ = 2

2

, where thehigher-lying electronic state gets almost fully occupied, transport through the lower-lying


electronic state is strongly suppressed. This is because electrons in the left lead have nolonger enough energy to resonantly tunnel through the lower-lying electronic state at thisbias voltage. Due to population of the blocking state such tunneling requires an additionalcharging energy U . Thus, the current breaks down, and the junction exhibits strong NDR.The current remains at this level until the bias voltage is large enough (eΦ > 2(

1

+U)) toallow for resonant tunneling processes through the lower-lying electronic state, which atthese bias voltages can occur irrespective of the population of the blocking state. Note thattransport processes involving the higher-lying electronic state, due to the weak couplingto the right lead, ν

R,2 = 0.01 eV, do not significantly contribute to the current.

Because of the different signs in the electronic-vibrational coupling strengths of this modelmolecular junction, λ

11

= −λ21

, for which U < U , electronic-vibrational coupling leadsto a slight widening of the bias range, 2

1

< eΦ < 2(1

+ U), in which population of theblocking state leads to a suppression of the current through this junction. Thereby, as thecomparison of the solid black and the dashed gray line shows, current-induced vibrationalexcitation facilitates population of the blocking state by resonant deexcitation processes(as depicted, for example, in Fig. 4.8b) even if this state is located outside the bias window.This leads to a pronounced suppression of current in the bias range 2(

2

−Ω1

) < eΦ < 22

.Furthermore, at the onset of resonant transport through the lower-lying state, eΦ = 2

1

,as well as through the respective dianionic complement, eΦ = 2(

1

+ U), vibrationalexcitation does also lead to a suppression of the current flowing through the junction (cf.the discussion at the end of Sec. 4.2.1). As a result, the main NDR feature at eΦ =2

2

appears less pronounced in the presence of electronic-vibrational coupling. Overall,however, electronic-vibrational coupling influences this NDR mechanism only weakly.

6.1.4 Successive Decrease of Electrical Current in the Presenceof a Localized Vibrationally Coupled Electronic State

In this section, we discuss another mechanism for NDR that is induced by electron-electroninteractions, but, in contrast to the previously discussed mechanism (see Sec. 6.1.3), morestrongly influenced by electronic-vibrational coupling. To this end, we consider a modelsystem with a centrally localized electronic state (model LOCAL, cf. Tab. 4.2). Thelocalization of this state on the central part of the molecular bridge is reflected by theweak coupling strengths ν

L/R,2 = 0.01 eV of this state to the left and the right lead.Apart from the coupling strength of the higher-lying electronic state to the left lead, ν

L,2,however, the parameters of this model molecular junction are the same as for the modelsystem considered in Sec. 6.1.3.

In contrast to a junction with a blocking state, such a junction is L↔R symmetric. Thecurrent-voltage characteristic (see Fig. 6.3) is therefore antisymmetric with respect to theapplied bias voltage Φ. The electronic and thermal current-voltage characteristics exhibitpronounced NDR at eΦ = 2

2

. This is the bias voltage at which the centrally localizedelectronic state enters the bias window and, thus, is populated. Similar to the previously


0 1. 2.0

0.25

0.5

NEGF electronicNEGF vibronicNEGF th. equ.ME vibronic

0 0.5 1. 1.5 2.0

0.5

1.

1.5

bias voltage ! !V"

currentI!"A"

Figure 6.3: Current-voltage characteristics of a model molecular junction with twoelectronic states coupled to a single vibrational mode (model LOCAL, cf. Tab. 4.2). Bothstates are coupled symmetrically to the leads, but with different coupling strengths. Inparticular, the higher-lying electronic state is significantly weaker coupled to the leads, asit is assumed to be stronger localized on the central part of the molecular bridge. Due toelectronic-vibrational coupling this state is not populated in a single step, but in a numberof steps. As a result, the current through the lower-lying electronic state is also suppressedstep by step, as an additional charging energy U is required for the respective tunnelingprocesses if the higher-lying electronic state is populated. NDR in this model molecularjunction thus extends over a range of bias voltages, from eΦ = 2(

2

−Ω1

) to eΦ = 2(1

+U).

discussed model system with a blocking state, population of the higher-lying electronicstate blocks sequential tunneling through the lower-lying electronic state, as tunnelingthrough the lower-lying state requires an additional charging energy U . The electricalcurrent flowing through the junction is thus significantly reduced at this bias voltage.However, as the bias voltage is increased and exceeds a value of 2(

1

+ U), electronsin the left lead have enough energy to resonantly tunnel through the lower-lying stateirrespective of the population of the higher-lying state. The current level of the junction,therefore, rises again to its previous value, which was obtained at Φ ∼ 0.5 V already.

In the electronic and the thermal current-voltage characteristics, effects induced by localheating of the vibrational mode are disregarded. Including these effects, one obtainsthe current-voltage characteristics depicted by the solid black (NEGF) and the solid red(ME) line in Fig. 6.3. These vibronic characteristics show NDR not at a single pointbut rather over a range of bias voltages that spans from 2(

2

− Ω1

) to 2(1

+ U). Thisbehavior is based on two effects. First, resonant deexcitation processes with respect tothe higher-lying electronic state (Fig. 4.8b) lead to population of the centrally localized


electronic state even before it is located inside the bias window. This leads to a step-wisesuppression of the corresponding electrical current, which occurs at lower bias voltagesbut is also less pronounced as in electronic or thermally equilibrated transport. Second, ifthe higher-lying electronic state is located inside the bias window, vibrational excitationhinders population of this state, as is evident by comparison of the solid black and thedashed gray line. Similarly, the population of a single electronic state is suppressed byvibrational excitation (as was already outlined at the end of Sec. 4.2.1). Therefore, thecentrally localized electronic state is not populated in a single but a number of steps fromeΦ = 2(

2

−Ω1

) to 2(1

+U). Accordingly, NDR in this model molecular junction appearsnot at a single point of the current-voltage characteristic, but extends over a range of biasvoltages that spans over several multiples of the vibrational frequency Ω

1

.

6.2 Rectification

Rectification represents another important transport property for the design of (molecular)electronic devices, as, for example, diodes or transistors. The main characteristic of adiode is an asymmetric current-voltage characteristic, that is a substantial reduction ofthe current flow if the polarity of the applied bias voltage is reversed, e.g. I(Φ) > −I(−Φ).Single-molecule junctions with an asymmetric structure of the molecular bridge and/oran asymmetry in the contacting electrodes typically exhibit highly asymmetric current-voltage characteristics [47,49,375]. Such diodes are the precursor for (molecular) transistordevices, which allow the control of electrical currents by electric fields. As molecules aremicroscopically small, such single-molecule transistor devices may represent the ultimatestep in the miniaturization of nanoelectronic devices [48, 51]. In the following sections,we study two different mechanisms for diode-like behavior in single-molecule junctions.First, in Sec. 6.2.1, we describe rectification that is induced by the electronic structureof the leads and, second, in Sec. 6.2.2, we show that asymmetries in the current-voltagecharacteristic are likely to be the result of current-induced vibrational excitation. Finally,in Sec. 6.2.3, we outline how an asymmetry in single-molecule junctions facilitates thespectroscopy of molecular energy levels.

6.2.1 The Importance of an Asymmetric Molecule-Lead Cou-pling

Asymmetric coupling of the molecular bridge to the leads is a prerequisite to observean asymmetric current-voltage characteristic, or diode-like behavior, of a single-moleculejunction. It is, however, not sufficient1. This is demonstrated by the solid purple and thedashed blue line in Fig. 6.4, which show electronic current-voltage characteristics for twoasymmetrically coupled molecular junctions: model REC and RECBAND (cf. Tab. 4.1).

1 At least, if effects induced by a voltage division factor η = 1/2 are disregarded, as is done in this work.Such effects are considered, for example, in Refs. [373,376].

6.2. Rectification 115

!2 !1 0 1 2051015

NEGF electronicNEGF vibronicNEGF th. equ.ME vibronicNEGF electronic !BAND"

!2 !1 0 1 2!0.3

!0.2

!0.1

0

0.1

0.2

0.3

bias voltage " #V$

currentI##A$

Figure 6.4: Current-voltage characteristics of an asymmetric molecular junction witha single electronic state coupled to a single vibrational mode (model REC, cf. Tab. 4.1).The inset shows the corresponding vibrational excitation characteristic. The dashed blueline refers to model RECBAND (cf. Tab. 4.1), where the asymmetry ratio in the molecule-lead coupling is given by ν

R,1/νL,1 = 0.3. Although the asymmetry ratio is the same forboth model systems, only the electronic current-voltage characteristic of the model sys-tem with the narrow conduction band (model RECBAND) shows an asymmetric current-voltage profile. Including electronic-vibrational coupling, the asymmetrically coupled junc-tion REC also shows an asymmetric current-voltage characteristic, although the respectiveelectronic current-voltage characteristic is almost antisymmetric with respect to the appliedbias voltage Φ. Current-induced vibrational excitation greatly enhances this mechanism forrectification (vibrational rectification).

Thereby, model RECBAND differs from model REC only by a narrower conduction band.

While the electronic current-voltage characteristic of model REC is almost antisymmetricwith respect to the applied bias voltage Φ, that is I(Φ) ≈ −I(−Φ), the electronic current-voltage characteristic of model RECBAND is not. On the contrary, for positive biasvoltages this current-voltage characteristic shows current levels that are roughly given byhalf of the current flowing through the junction at negative bias voltages. This asymmetryin the current flow is not just the result of an asymmetric molecule-lead coupling, sincethe asymmetry ratio ν

R,1/νL,1 = 0.3 is the same for both model systems. It is, in addition,the result of the narrower conduction bands in model RECBAND and the position of theelectronic state with respect to the Fermi level of the junction2.

2 Note that both model systems, despite their asymmetry in the molecule-lead coupling, would giveperfectly antisymmetric current-voltage characteristics, if the electronic state is located next to theFermi level, that is for 1 ∼ 0


This can be understood as follows. In the given range of bias voltages, |Φ| < 2 V, theconduction bands in junction REC appear rather flat and unstructured. As a result,electrons that are sequentially tunneling from the left to the right lead at positive biasvoltages see the same two tunneling barriers as electrons that are sequentially tunnelingfrom the right to the left lead at negative bias voltages. For junction RECBAND, however,due to the narrow conduction bands, this is not the case. As the electronic state is locatedabove the Fermi level of the junction, the left barrier is more transparent for positivethan for negative bias voltages. This is due to the fact that for positive bias voltagesthe maximum of the respective conduction band is located closer to the position of theelectronic state. Similar arguments hold for the right barrier. As the left lead is morestrongly coupled to the molecular bridge than the right lead, the electronic state appearsbroader for positive than for negative bias voltages. Similarly, the asymmetry in thetransparency of the two tunneling barriers is larger for positive than for negative biasvoltages, which results in a reduced current level for positive bias voltages.

6.2.2 Rectification by Electronic-Vibrational Coupling

In the previous section, we have discussed electronic transport (λ11

= 0) through theasymmetrically coupled molecular junction REC. Despite the asymmetry in the couplingto the leads, the electronic current-voltage characteristic of this junction (solid purple linein Fig. 6.4) is almost antisymmetric with respect to the applied bias voltage Φ.

Including electronic-vibrational coupling, however, one obtains a current-voltage charac-teristic for this molecular junction that is no longer anti- but asymmetric with respect tothe polarity of the applied bias voltage Φ. The thermally equilibrated current-voltage char-acteristic (dashed gray line in Fig. 6.4), for example, is strongly asymmetric at |eΦ| ≈ 2

1

.This can be understood solely in terms of tunneling processes with respect to the rightlead, which, due to the weak coupling of the electronic state to this lead, represent thebottle neck for electron transport in this junction. As a result of electronic-vibrationalcoupling, the number of resonant tunneling processes with respect to the right lead varieswith the applied bias voltage Φ. For example, for positive (negative) bias voltages eΦ = 2

1

(eΦ = −21

), there are mod(|1

−µR|, Ω1

)+1 = 12 (mod(|1

−µR|, Ω1

)+1 = 1) sequentialtunneling processes available, which correspond to the excitation of 0 – 11 (0) vibrationalquanta (cf. Fig. 4.4b). At the onset of the resonant transport regime, the current ateΦ = 2

1

is thus larger than for eΦ = −21

, because at eΦ = 21

all relevant tunnelingprocesses with respect to the right lead become active at the same bias voltage, whileat eΦ = −2

1

only a single of these processes is active. For yet smaller negative biasvoltages eΦ < −2

1

, these processes become active one by one. This results in a step-wiseincrease of the current flow for negative bias voltages, which converges in a few steps tothe electronic current level, while only a single step is observed for positive bias voltages.

Taking into account current-induced vibrational excitation, this asymmetry in the current-voltage characteristic (solid black and red lines) is much more pronounced. While forpositive bias voltages the current still increases in a single step and is as large as the

6.2. Rectification 117

respective electronic current, it is strongly suppressed for negative bias voltages and sig-nificantly lower than the electronic current. Vibrational excitation thus results in a per-sistent rectification of the current flowing through this asymmetrically coupled molecularjunction (vibrational rectification). Accordingly, the corresponding vibrational excitationcharacteristic depicted in the inset of Fig. 6.4 does also exhibit a strong asymmetry withrespect to the applied bias voltage Φ and shows much higher levels of vibrational exci-tation for negative than for positive bias voltages. This behavior can be understood interms of electron-hole pair creation processes (cf. Fig. 4.17). Due to the asymmetry inthe coupling of the electronic state to the leads, electron-hole pair creation processes withrespect to the left lead are the most important processes in this junction. They are veryefficiently cooling the vibrational mode, if the chemical potential in the left lead is locatedclose to the position of the electronic level. This is the case for positive, but not for neg-ative bias voltages. The asymmetry in the efficiency of cooling of the vibrational modeby electron-hole pair creation processes is thus the origin of the asymmetries observed inboth the vibrational excitation and the current-voltage characteristic.

As we will discuss in the following section, understanding these mechanisms of vibra-tionally induced rectification in asymmetric molecular junctions is crucial for the spec-troscopy of molecular energy levels. Another perspective on this phenomenon is thatcurrent-induced vibrational excitation can be controlled by an external potential bias Φ.The possibility to control vibrational excitation in asymmetric junctions with a singlevibrational mode readily points to selective excitations of specific vibrational modes injunctions with multiple vibrational degrees of freedom, that is, mode-selective vibrationalexcitation (see Sec. 6.3).

6.2.3 Spectroscopy of Molecular Levels: How Vibrational SidePeaks Disappear

At this point, we discuss the usability of single-molecule junctions as spectroscopic tools.For example in Sec. 4.2.2, we have discussed a model for a molecular junction with twoelectronic states (model E2V1). The respective electronic current-voltage characteristic(cf. Fig. 4.9) shows two distinct steps at bias voltages that correspond to the energy ofthese states: eΦ = 2

1/2

. It is thus possible to extract spectroscopic information fromthe current-voltage characteristic of a single-molecule junction. However, if electronic-vibrational coupling is included for this molecular junction, the current-voltage character-istic exhibits only a single step at the onset of (eΦ = 2

1

) and an almost linear increasein the resonant transport regime (eΦ > 2

1

). This behavior has already been explainedin Sec. 4.2.2 in terms of current-induced vibrational excitation, which triggers resonantdeexcitation processes with respect to the higher-lying electronic state (cf. Figs. 4.8b and4.8c). This facilitates electron tunneling through this state even before it has entered thebias window. As a result, the position of the lower-lying electronic level,

1

, may be easilydetermined in the presence of electronic-vibrational coupling, while the position of thehigher-lying electronic state,

2

, may not.


!2 !1 0 1 20

10

20

NEGF vibronicNEGF vibronic !symm."ME vibronicME vibronic !symm."

!2 !1 0 1 2!0.6

!0.4

!0.2

0

0.2

0.4

0.6

bias voltage " #V$

currentI##A$

Figure 6.5: Current-voltage characteristics of an asymmetric molecular junction withtwo electronic states coupled to a single vibrational mode (model SPEC, cf. Tab. 4.2).For comparison, the solid gray (NEGF) and the solid orange lines (ME) show results ofthe corresponding symmetric molecular junction E2V1 rescaled by a factor of 1/5. Theinset shows the corresponding vibrational excitation characteristic. The asymmetry in themolecule-lead coupling of this model molecular junction enhances/attenuates cooling of thevibrational mode by resonant electron-hole pair creation processes for positive/negativebias voltages. As a result, resonant deexcitation processes with respect to the higher-lying electronic state are less/very effective. Therefore, only a single step / a numberof closely-lying steps is observed at eΦ 2

2

/ eΦ −22

. Asymmetric molecule-leadcoupling thus facilitates the spectroscopy of molecular energy levels. Moreover, it resultsalso in a different number of distinguishable steps/peaks in the current-/conductance-voltage characteristic for different polarities of the applied bias voltage Φ.

Due to vibrational rectification, this is different for asymmetric molecular junctions. Fig.6.5 shows the current-voltage characteristic of an asymmetric molecular junction. Thismodel system (model SPEC, cf. Tab. 4.2) differs from junction E2V1 only by weakercouplings of the two electronic states to the right lead, ν

R,1/2

= 0.03eV. The current-voltage characteristic of the symmetric junction E2V1 is also depicted in Fig. 6.5. Incontrast to this, the vibronic current-voltage characteristic of the asymmetrically coupledsystem shows two distinct steps at positive bias voltages, eΦ = 2

1

and eΦ = 2(2

+ U).For negative bias voltages, however, both characteristics do not exhibit two but onlyone significantly pronounced step. This can be explained as follows. If the chemicalpotential in the left lead, which is the lead both states are most strongly coupled to, islocated close to the position of the electronic states, cooling of the vibrational mode byelectron-hole pair creation processes is very efficient. This is the case for positive but

6.3. Mode-Selective Vibrational Excitation 119

not for negative bias voltages. The corresponding vibrational excitation characteristic isrepresented in the inset of Fig. 6.5. It shows a very low level of vibrational excitationfor positive bias voltages, which even drops almost to zero, when the second electronicstate enters the bias window. Resonant deexcitation processes with respect to the higher-lying electronic state, which lead to a strong fragmentation of the associated step inthe transport characteristics of model E2V1, are thus strongly suppressed. For negativebias voltages, the levels of vibrational excitation obtained for the asymmetrically coupledmolecular junction are even larger than for the symmetrically coupled junction. Resonantdeexcitation processes are therefore even more pronounced.

An asymmetric coupling to the leads, due to bias-dependent cooling of the vibrationalmodes by electron-hole pair creation processes, thus facilitates the spectroscopy of molec-ular energy levels. Moreover, it leads to a different number of distinguishable steps (peaks)in the current-voltage (conductance-voltage) characteristic of a molecular junction for dif-ferent polarities of the applied bias voltage Φ. Note that in our model studies the asym-metry ratio, ν

R,m/νL,m = 0.3, is not very large such that this phenomenon may not only

occur in experiments with an inherently strong asymmetry in the molecule-lead coupling,such as in STM experiments, but also in more symmetric setups, such as MCBJs [130].

6.3 Mode-Selective Vibrational Excitation

As we have seen in Sec. 6.2, the level of vibrational excitation in an asymmetric molecularjunction can be controlled by the polarity and the magnitude of an external bias voltageΦ. In this section, we generalize this bias-controlled vibrational excitation to multiple,that is, two vibrational modes. Thereby, we study vibrational modes that have differentfrequencies Ωα. Naturally, one expects the mode with a lower frequency to be higherexcited, corresponding to a statistical (Boltzmann) distribution of vibrational energy in amolecular junction. We, however, show that the level of excitation of a specific vibrationalmode can be higher than the level of excitation of another mode, irrespective of thefrequency of the two modes. Thereby, the polarity of the applied bias voltage can be usedto select the mode that is higher excited. This is, for example, important in the contextof mode-selective chemistry [377–380], where one seeks to break a specific (not necessarilythe weakest) chemical bond in a molecule.

To this end, we study two generic models for asymmetric molecular junctions that exhibitsuch mode-selective vibrational excitation (MSVE). The first model (Sec. 6.3.1) comprisestwo electronic states, one above and one below the Fermi level of the junction. Thereby,the two states are asymmetrically coupled to the leads, reflecting an asymmetric structureof the electrodes and/or the contact geometry of the junction, as it is typical, for example,in STM experiments. The second model (Sec. 6.3.2) does also involve two electronic states.They are located above the Fermi level of the junction and are also asymmetrically coupledto the leads. In contrast to the first model, however, the asymmetric coupling of thesestates to the leads reflects an asymmetric structure of the molecular bridge.


6.3.1 Molecular Bridge Asymmetrically Coupled to the Leads

In the following, we consider a model for a molecular junction with two electronic statesthat are asymmetrically coupled to the leads (model MSVE-A, cf. Tab. 6.1). One of thesestates is located below the Fermi level of the junction. This state may be associatedwith the HOMO of the molecular bridge. The other state is located above the Fermilevel of the junction, and thus, may be associated with the respective LUMO. Both statesare coupled to two vibrational degrees of freedom. Thereby, the electronic-vibrationalcoupling strengths reflect a different electronic structure of the HOMO and the LUMO.In particular, we assume λmα ∼ δmα, that is, each mode couples predominantly to a specificelectronic state. Such an asymmetry in the electronic-vibrational coupling strengths, isrequired to observe the MSVE phenomenon. Thereby, the specific choice λmα ∼ δmα,where off-diagonal coupling λ

12

= λ21

= 0 is completely disregarded, gives the mostpronounced effect.

Fig. 6.6a shows current-voltage characteristics of this model molecular junction, where theinset represents the corresponding population characteristics of the two electronic states.In addition, the excitation characteristics of the two vibrational modes are depicted in Fig.6.6b. While the vibronic current-voltage characteristic of this junction is, more or less,antisymmetric with respect to the applied bias voltage Φ, the corresponding populationand excitation characteristics are not. In particular, for positive bias voltages, mode 2 ismuch higher excited than mode 1, while the excitation levels of the two modes are almostreversed for negative bias voltages. It is thus possible to control the level of excitation ofthe two vibrational modes by the polarity of the external bias voltage Φ (MSVE). Thisbehavior can be explained in terms of the effects and phenomena discussed in Sec. 6.2.2.Since there is no off-diagonal electronic-vibrational coupling, the subsystem comprisingstate 1 and mode 1 is almost completely decoupled3 from the subsystem consisting ofstate 2 and mode 2. The population and excitation characteristics of the two subsystemsthus correspond to the population and excitation characteristic of a single electronic stateasymmetrically coupled to the leads and to a single vibrational degree of freedom.

MSVE in this model for a molecular junction is based on both an asymmetry in themolecule-lead coupling, ν

L,m & 3νR,m, as well as weak off-diagonal electronic-vibrational

coupling, λmα ∼ δmα. Thereby, due to the asymmetry in the molecule-lead coupling,electron-hole pair creation processes with respect to the left lead are the most importantcooling mechanism in this junction. The efficiency of these processes is, however, alsostrongly influenced by the distance between the position of the electronic states andthe chemical potential in the left lead, m − µ

L

. For positive bias voltages, state 1 islocated much closer to the chemical potential in the left lead than state 2. Accordingly,resonant electron-hole pair creation processes that involve state 1 are more efficient forpositive bias voltages than electron-hole pair creation processes that involve state 2. Thus,for positive bias voltages, mode 1 is much more efficiently cooled by electron-hole pair

3 Note that the coupling of the two electronic states to the leads induces, in principle, correlations betweenthe two subsystem. However, as the electronic states are energetically well separated, 2−1 ΓK,mm,these correlations are not very pronounced.


(a)

!2 !1 0 1 20

0.5

1.

NEGF electronicNEGF vibronic

n2

n1

!2 !1 0 1 2!0.9

!0.6

!0.3

0

0.3

0.6

0.9

bias voltage " !V"

currentI!#A"

(b)

NEGF mode 1NEGF mode 2

!2 !1 0 1 20

5

10

bias voltage " !V"


Figure 6.6: Current-voltage and vibrational excitation characteristics of an asymmetri-cally coupled molecular junction (model MSVE-A, cf. Tab. 6.1). This junction comprisestwo electronic states that are coupled differently to two vibrational modes. The inset showsthe population characteristics of the two electronic states. Due to the asymmetry in themolecule-lead (ν

R,m = 0.3νL,m) as well as the electronic-vibrational coupling (λmα ∼ δmα)

electron-hole pair creation processes with respect to the left lead cool mode 1 very efficientlyfor positive bias voltages. As a result, mode 2 is much higher excited than mode 1 (viceversa for negative bias voltages). It is thus possible to control the level of excitation of thetwo modes by the polarity of an external bias voltage Φ (MSVE).


model 1

2

νL,1 ν

L,2 νR,1 ν

R,2 Ω1

λ11

λ21

Ω2

λ12

λ22

MSVE-A 0.65 -0.575 0.1 0.1 0.03 0.03 0.15 0.09 0 0.2 0 0.12MSVE-B 0.65 0.575 0.1 0.03 0.03 0.1 0.15 0.09 0 0.2 0 0.12

Table 6.1: Model parameters for asymmetric molecular junctions exhibiting MSVE(energy values are given in eV). The temperature in the leads, k

B

T = 10 K, and thecorresponding internal hopping parameter, γ = 2 eV, are the same for all these modelmolecular junctions. Moreover, we do not consider electron-electron interactions in thesemodel systems, U

12

= 0.

NEGF mode 1 !"#2.0VNEGF mode 2 !"#2.0VNEGF mode 1 !"$2.0VNEGF mode 2 !"$2.0V

0 0.5 1. 1.50

2

4

6

8

electron$electron interaction strength U12 !eV"


Figure 6.7: Vibrational excitation of mode 1 and mode 2 in model molecular junctionMSVE-A (cf. Tab. 6.1) as a function of the electron-electron interaction strength U

12

.Thereby, the bias voltage Φ is fixed to 2 V for the black and to −2 V for the gray lines.Although MSVE is both enhanced (at U

12

≈ 0.5 eV) and attenuated (at U12

≈ 1.5 eV), itis maintained over a wide range of electron-electron interaction strengths.


Ω1/2

ε2

ε1

RL Mε1/2+U12

Figure 6.8: Schematic representation of an electron-holepair creation process involving the dianionic resonance

1/2

+U

12

. Since the bias voltage is such that this resonance is lo-cated close to the chemical potential in the left lead µ

L

, itfacilitates the creation of an electron-hole pair in the respec-tive lead by absorption of just a single quantum of vibrationalenergy (Ω

1/2

). This is in contrast to the anionic resonancesat

1/2

, for which the creation of an electron-hole pair in theleft lead requires much more vibrational energy at this biasvoltage.

creation processes than mode 2, as mode 1 is more strongly coupled to state 1 (vice versafor negative bias voltages). In the presence of off-diagonal electronic-vibrational coupling,mode 2 would also be efficiently cooled for positive bias voltages. Off-diagonal electronic-vibrational coupling would therefore counteract MSVE in this molecular junction. Testcalculations, however, show that MSVE can be observed in this molecular junction even inthe presence of small to moderate off-diagonal electronic-vibrational coupling strengths,that is for λ

12

= λ21

. λ11/22

/4 [250].

Besides off-diagonal electronic-vibrational coupling, the two subsystems (state 1, mode 1)and (state 2, mode 2) may also be coupled by electron-electron interactions. Fig. 6.7 showsthe levels of excitation of the two modes as functions of the electron-electron interactionstrength U

12

at a fixed bias voltage, that is Φ = 2 V for the black and Φ = −2V forthe gray lines. For the negative bias polarity, Fig. 6.7 reveals a very weak dependenceof vibrational excitation on U

12

, as both electronic states are almost unpopulated. Forpositive bias voltage, however, both states are almost fully occupied, and accordingly, thelevels of vibrational excitation show a stronger dependence on U

12

. In particular, if thedianionic resonance at

1

+ U12

(2

+ U12

) gets close to the chemical potential in the leftlead, mode 1 (mode 2) is very efficiently cooled by electron-hole pair creation processeswith respect to this resonance (cf. Fig. 6.8). This results in substantially reduced levelsof excitation of mode 1 at U

12

≈ 0.5 eV and of mode 2 at U12

≈ 1.5 eV. For yet largerelectron-electron interaction strengths, U

12

> 2 eV, when the energy difference betweenthe chemical potential in the left lead and the dianionic resonances

1/2

+ U12

exceedsseveral units of the vibrational frequency Ω

2

, as is the case for U12

= 0, the excitationlevels of the two modes become as large as for U

12

= 0. While MSVE is thus maintainedover a wide range of electron-electron interaction strengths, it is enhanced for U

12

≈ 0.5 eVand attenuated for U

12

≈ 1.5 eV, respectively.

6.3.2 Molecular Bridge with Asymmetric Molecular Orbitals

In this section, we discuss MSVE for another model of a molecular junction (model MSVE-B, cf. Tab. 6.1). MSVE in this system is also based on an asymmetry in the coupling of themolecule to the leads and weak off-diagonal electronic-vibrational coupling. In contrast


(a)

!2 !1 0 1 20

0.5

1.

NEGF electronicNEGF vibronic

n1n2

!2 !1 0 1 2!0.9

!0.6

!0.3

0

0.3

0.6

0.9

bias voltage " !V"

currentI!#A"

(b)

NEGF mode 1NEGF mode 2

!2 !1 0 1 20

5

10

bias voltage " !V"


Figure 6.9: Current-voltage and vibrational excitation characteristics of an asymmet-rically coupled molecular junction with two electronic states coupled to two vibrationalmodes (model MSVE-B, cf. Tab. 6.1). The inset shows the population characteristics ofthe two electronic states. Due to the asymmetry in the molecule-lead (ν

R,1 = 0.3νL,1 and

νL,2 = 0.3ν

R,2) as well as the electronic-vibrational coupling (λmα ∼ δmα), electron-holepair creation processes with respect to the left lead cool mode 1 very efficiently for positivebias voltages, while mode 2 is very efficiently cooled for negative bias voltages. As a result,mode 2 is much higher excited than mode 1 for positive bias voltages, despite the fact thatboth modes have different frequencies (Ω

1

= 0.15 eV, Ω2

= 0.2 eV).


NEGF mode 1 !"#2.0VNEGF mode 2 !"#2.0VNEGF mode 1 !"$2.0VNEGF mode 2 !"$2.0V

0 0.5 1. 1.50

4

8

electron$electron interaction strength U12 !eV"


Figure 6.10: Vibrational excitation of mode 1 and mode 2 in model molecular junctionMSVE-B (cf. Tab. 6.1) as a function of the electron-electron interaction strength U

12

.Thereby, the bias voltage Φ is fixed to 2 V for the black and to −2 V for the gray lines.Although MSVE is attenuated at U

12

≈ 0.5 eV, it nevertheless occurs for a rather widerange of electron-electron interaction strengths.

to the model system previously discussed in Sec. 6.3.1, however, the respective electronicstates are located above the Fermi level of the junction. Thereby, state 1 is more stronglycoupled to the left than to the right lead, and vice versa for state 2. Such an asymmetry inthe molecule-lead coupling may reflect an inherent asymmetric structure of the molecularbridge, where, for example, state 1 (state 2) is more strongly localized on the left (right)hand side of the molecular bridge.

As before (Sec. 6.3.1), the vibronic current-voltage characteristic of junction MSVE-B(shown in Fig. 6.9a) is also almost antisymmetric with respect to the applied bias voltageΦ, while the respective population (see the inset of Fig. 6.9a) and vibrational excitationcharacteristics (Fig. 6.9b) of the two electronic states and the two vibrational modes, re-spectively, are very different for different polarities of the applied bias voltage Φ. Thereby,the level of excitation of mode 2 is much higher than the level of excitation of mode 1 forpositive, and vice versa for negative bias voltages, that is junction MSVE-B also exhibitsmode-selective vibrational excitation. This is, similar to junction MSVE-A, a result of theefficiency of electron-hole pair creation processes, which is different for the two vibrationalmodes at different polarities of the applied bias voltage Φ. Overall, MSVE in junctionMSVE-B is as pronounced as in the model system MSVE-A (cf. Fig. 6.6b).

As can be inferred from Fig. 6.10, however, the MSVE phenomenon in junction MSVE-Bis less stable with respect to electron-electron interactions. Due to the asymmetry inthe molecule-lead coupling strengths ν

L/R,m state 1 is almost fully occupied for positive


model 1

νL,1 ν

R,1 γ η kB

T Ωα λmα

PEAKEL 27 2.1 2.57 1000 0.6 0.1 – –PEAKVIB 39 2.1 2.57 1000 0.6 0.1 2.05+α0.45 3

Table 6.2: Model parameters to describe the Fe2+-bispyrazolylpyridine molecular junc-tion depicted in Fig. 1.1 (energy values are given in meV).

bias voltages Φ = 2 V (cf. inset of Fig. 6.9), while state 2 is almost unpopulated. Theexcitation level of vibrational mode 2, which is mainly determined by inelastic tunnelingevents with respect to state 2, therefore shows a strong dependence on the interactionstrength U

12

. In contrast, the excitation level of mode 1 is almost constant with respectto U

12

. For negative bias voltages, Φ = −2V, the population of the electronic states isreversed, as is the behavior of the excitation levels of the two modes with respect to theelectron-electron interaction strength U

12

. Especially if the dianionic resonance at 2

+U12

for Φ = 2 V (1

+ U12

for Φ = −2V) is close to the chemical potential in the left lead µL

,cooling of mode 2 (mode 1) by electron-hole pair creation processes (Fig. 6.8) becomesvery efficient. This results in a suppression of MSVE at U

12

≈ 0.5 eV. This is in contrastto junction MSVE-A, where MSVE is attenuated only for significantly larger interactionstrengths, U

12

≈ 1.5 eV. Apart from this, however, MSVE occurs in this junction also fora wide range of electron-electron interactions strengths.

6.4 Anomalous Peak Broadening

In this section, we study another multimode vibrational effect. In particular, we investi-gate the phenomenon of anomalous peak broadening. We relate this phenomenon to thelow frequency modes, which are typical for a polyatomic single-molecule junction.

For example, the Fe2+-bispyrazolylpyridine molecular junction, which we already consid-ered in Sec. 2.2 (see Fig. 1.1 for its geometrical structure and Fig. 2.4 for a respectiveconductance-voltage characteristic, which was measured in a MCBJ experiment [34]),exhibits more than 200 vibrational degrees of freedom4. The frequencies of the corre-sponding normal modes range from 0.5 meV to 400 meV. In the following, we discuss howthis multitude of vibrational modes influences the current-/conductance-voltage charac-teristic of this molecular junction. Thereby, we focus on the width of the broad peaksthat appear at positive, Φ w 45 mV, and negative bias voltages, Φ w −70mV. Similaranomalously broad peaks have been found in the conductance-voltage characteristics ofnumerous molecular junctions, including different molecular species [34].

4 The corresponding normal mode analysis for the isolated molecule was performed by O. Rubio-Pons.Thereby, in order to simulate the effect of the junction, the acetaldehyde groups, which in the junctionare replaced by gold electrodes (cf. Fig. 1.1), were not included in the analysis.

6.4. Anomalous Peak Broadening 127

!80 !40 0 40 80!2

0

2

NEGF electronicNEGF th. equ.

currentI!nA"

!80 !40 0 40 800

20

40

60

80

bias voltage " !mV"

conductanceg!nA#V"

Figure 6.11: Conductance-voltage characteristics for different models of the Fe2+-bispyrazolylpyridine molecular junction depicted in Fig. 1.1 (model PEAKEL and modelPEAKVIB, cf. Tab. 6.2). Model PEAKEL (solid purple line) suits to describe the positionand the height of the steps in the corresponding current-voltage characteristic, but fails todescribe the width of these steps. These anomalously broadened peaks can be explained con-sidering the low-frequency modes of this system, which are included in model PEAKVIB(dashed gray line). The width of these peaks is determined by the respective reorganiza-tion energy,

Pα λ2

mα/Ωα = 22 meV, rather than the coupling of the molecule to the leads,Γ

L,11

+ ΓR,11

∼ 0.02 meV, or the thermal broadening in the leads, kB

T . 0.1 meV.

Employing a simple model (model PEAKEL, cf. Tab. 6.2), where no vibrational degreesof freedom are included, these peaks may be associated with resonant transport processesthrough a single electronic state (cf. Sec. 4). Thereby, the positions of the conductancepeaks at different polarities of the bias voltage Φ indicate, in correspondence to an equiva-lent circuit model [198–201], a voltage-division factor of η = 0.6, that is µ

L

= ηeΦ = 0.6eΦ.Note that this value is close to the voltage division factor η = 0.5, which is used for all othermodel systems considered in this thesis. Such a potential drop can be associated with anasymmetry in the molecule-lead coupling: η = Γ

R,11

/(ΓL,11

+ΓR,11

). The respective valuesfor the corresponding level-width function, Γ

L,11

(µL

) = 8.8µeV and ΓR,11

(µR

) = 13.2µeV,are determined by both the voltage-divison factor η and the height of the steps in therespective current-voltage characteristic (data not shown).

Considering these model parameters, we obtain the conductance-voltage characteristicthat is depicted by the solid purple line in Fig. 6.11. The position of the conductancepeaks and the height of the steps in the respective current-voltage characteristic (seethe inset of Fig. 6.11) are correctly described. However, the width of these resonancesis strongly underestimated. Neither the thermal broadening, k

B

T ∼ 0.1meV, nor the


broadening due to the coupling of the molecule to the leads, encoded in the level-widthfunctions, Γ

L,11

+ ΓR,11

∼ 0.02meV, can explain the observed width of the conductancepeaks, which is about 22meV.

To understand this phenomenon, we take into account the vibrational degrees of free-dom of this junction, in particular, its low-frequency modes. We find more than tenvibrational modes in this junction with a frequency lower than 10meV. In order todemonstrate the generality of anomalous peak broadening due to such low frequencymodes, we describe these modes in terms of a simplified model with equidistant frequen-cies, Ωα = 2.05 meV+α 0.45 meV, and uniform electronic-vibrational coupling strengths,λmα = 3 meV (model PEAKVIB, cf. Tab. 6.2). The respective reorganization energy,P

α λ2

mα/Ωα = 22 meV, shifts the energy of the electronic state from 1

= 39 meV to1

≈ 17meV.

The solid gray line in Fig. 6.11 represents the corresponding current-/conductance-voltagecharacteristic of this model5. As before, both the position of the conductance peaks andthe height of the steps in the respective current-voltage characteristic (shown in the insetof this figure) are well described. However, in contrast to the previous model, the widthof these peaks has the correct magnitude (≈ 22meV). A detailed analysis of the resonanttransport processes that occur in this molecular junction reveals the mechanism for thisanomalous peak broadening. At eΦ = 5

1

/3 ≈ 28meV, a small peak in the conductance-voltage characteristic indicates the onset of resonant transport through the electronicstate (Fig. 4.4a). Peaks at eΦ = 5/3(

1

+ Ωα) appear at higher bias voltages and canbe associated with resonant excitation processes, where mode α is singly excited (Fig.4.4d). In contrast to the single peak at eΦ = 5

1

/3 ≈ 28 meV, these peaks cannot bedistinguished and overlap with each other, as the mode frequencies Ωα are rather close inenergy. Resonant excitation processes that involve even more vibrational quanta of thevarious modes show up at even higher bias voltages. Their overlap is even larger suchthat the multitude of these peaks merges to a single broad peak at Φ ≈ 0.05V. Thereby,the width of this peak is determined neither by the coupling to the leads nor by thetemperature in the leads but by the reorganization energy of the electronic level due tocoupling to the low-frequency modes. A similar analysis applies for negative bias voltages.Due to the asymmetry in the voltage drop, the peak at this bias polarity appears to beeven broader.

Overall, the agreement of these results with the experimental data (Fig. 2.4) is very good,and even the fine structure that appears on top of the broad peaks is well described. Notethat anharmonic effects, the coupling to phonon modes in the leads or vibrational nonequi-librium effects would result in further broadening of the peaks in the conductance-voltagecharacteristic of this junction. Another mechanism, which may cause a ”mismatch” be-tween the current level and the width of the respective resonances, is quantum interference(see the following section).

5 For simplicity, we describe this multitude of vibrational modes in thermal equilibrium.

6.5. Quantum Interference Effects in Single-Molecule Junctions 129

6.5 Quantum Interference Effects in Single-MoleculeJunctions

Another fundamental aspect of electron transport in single-molecule junctions is quan-tum interference. It is, however, also interesting from a technological point of view, as isevident by the recently issued patent for a quantum interference effect transistor [369].Quantum interference effects are important in electron transport through quasidegenerateelectronic states of a molecular junction [66–68,70,72,120,381]. Such states are typicallyrelated to the symmetry of a junction. This can be a simple L↔R symmetry or a morecomplex symmetry, such as the D

6h-symmetry of a benzene molecule [68, 72, 120, 381].The electrodes of a single-molecule contact often break these symmetries. While, there-fore, exactly degenerate electronic states are very unlikely to be observed in a molecularjunction, quasidegenerate electronic states are rather common.

In this section, we study quantum interference effects by generic models of a molecularjunction, which involve two electronic states. Employing different coupling scenarios forthese states we give a comprehensive account of the basic quantum interference phenom-ena in Sec. 6.5.1. Thereby, we show that quantum interference effects may also induceanomalous peak broadening (cf. Sec. 6.4). In Secs. 6.5.2 and 6.5.3, we take into accountcoupling of these states to vibrational degrees of freedom and demonstrate how electronic-vibrational coupling leads to a quenching of quantum interference effects in a molecularjunction. The resulting electrical currents can thus be substantially increased as com-pared to the case, where no vibrations are taken into account (Sec. 6.5.2). We showthat this quenching is typically enhanced by (current-induced) vibrational excitation, orequivalently, by a larger effective temperature of the vibrational degrees of freedom.

6.5.1 Basic Phenomena

A simple model for a molecular junction exhibiting quantum interference effects is depictedin Fig. 6.12a. It comprises two degenerate states with energy

0

. Each of these statesis localized on a different end of the molecular bridge. As the respective wave functionsoverlap, these states are coupled to each other and can be described by the Hamiltonian

H =

µ0

αα

0

∂, (6.1)

where α denotes the coupling strength between the two localized states. The eigenstatesof H are symmetric and antisymmetric combinations of these states and are representedin Fig. 6.12b. While the symmetric eigenstate is symmetrically coupled to the leads,ν

L,1 = νR,1 = ν, the coupling strengths of the antisymmetric state, ν

L,2 = −νR,2 = ν,

differ by a phase of π.


(a) (b) (c) (d)

ε0 ε0

α

RL M

ν/√2ν/√2 ε0+|α|ε0-|α|

νν

RL M

-νν

ε0ε0

α

RL M

ν/√2ν/√2 ε0+|α|ε0-|α|

νν

RL M

νν

Figure 6.12: Schematic representation of model molecular junctions exhibiting quantuminterference. Panel a depicts a junction with two electronic states (sites) localized on theleft- and the right hand side of the molecular bridge, respectively. The correspondingeigenstates, which are symmetric and antisymmetric combinations of these states, aredepicted in panel b. These eigenstates correspond to bonding and antibonding states ofthe molecular bridge. Quantum interference also occurs in junctions with a localized and adelocalized state (panel c). Such a scenario describes, for example, a branched molecularconductor [67,382,383]. The respective eigenstates shown in panel d are both symmetricwith respect to the L↔R-symmetry of the junction, and therefore, symmetrically coupledto the leads.

model 1

2

νL,1/2

νR,1 ν

R,2 Ω1

λ11

λ21

INTEL 0.4 0.404 0.1 0.1 -0.1 – – –INTVIB 0.4 0.44 0.1 0.1 -0.1 0.1 0 0.06SINTEL 0.4 0.404 0.1 0.1 0.1 – – –SINTVIB 0.4 0.44 0.1 0.1 0.1 0.1 0 0.06

Table 6.3: Model parameters for molecular junctions exhibiting quantum interferenceeffects (energy values are given in eV). The temperature in the leads, k

B

T = 10 K, andthe corresponding internal hopping parameter, γ = 3 eV, are the same for all these modelmolecular junctions. Moreover, we do not consider electron-electron interactions in thesemodel systems, U

12

= 0.


If the overlap between the two localized states is small, that is for a weak inter-site couplingα, the respective eigenstates are (quasi-)degenerate (

1/2

= 0

± |α|). As a result, theyprovide, similar to a double slit experiment [384,385], two different pathways for electronstunneling through the junction. Due to the different symmetry of these eigenstates, thetwo wave functions emerging from these two tunneling-pathways differ by sign, and thus,destructively interfere. This interference naturally results in a strong suppression of therespective tunnel-current.

The dashed purple line in Fig. 6.13a depicts the current-voltage characteristic of a molec-ular junction, which comprises a pair of such electronic states (model INTEL, cf. Tab.6.3). These states are quasidegenerate6 and coupled to the leads symmetrically and anti-symmetrically, respectively. The solid purple line represents the electrical current througha very similar molecular junction (model INTVIB, cf. Tab. 6.3), where, in contrast to theprevious junction, the position of the antisymmetric electronic state is higher in energysuch that the two electronic states of this junction are not degenerate7. Quantum inter-ference effects are thus much weaker in this junction. Accordingly, it exhibits a currentthat is almost an order of magnitude larger than the current through junction INTEL,where the current flow is substantially reduced by quantum interference effects. Note thatthis suppression of the current gives a ’mismatch’ between the width and the height ofthe step in the corresponding current-voltage characteristic. Quantum interference effectsmay thus constitute another mechanism for anomalous peak broadening complementaryto the one discussed in Sec. 6.4.

Analysis of the phenomenon by the corresponding transmission function t() (cf. Eq.(3.202) in Sec. 3.4) reveals the underlying interference mechanism. The transmissionfunction t() of this junction can be expressed in terms of two transmission amplitudes

t() = |Λ1

()− Λ2

()|2 , (6.2)

Λ1/2

() =Γ

− 1/2

+ iΓ, (6.3)

where, for simplicity, we employ the wide-band approximation:

ΓL/R,mn() ≈ Γ

L/R,mn(µL/R

) =2ν

L/R,mνL/R,n

γ, Γ ≡ |Γ

L/R,mn(µL/R

)|. (6.4)

The minus sign in Eq. (6.2) is associated with the antisymmetry of the higher-lying elec-tronic state, or equivalently, with the coupling strength of this state to the right lead,ν

R,2 = −ν. The transmission function t() can be split into two parts: an incoherentterm, |Λ

1

()|2 + |Λ2

()|2, which describes electron transport through the two states as anincoherent sum of transmission events, and an interference term, −Λ∗

1

()Λ2

()−Λ1

()Λ∗2

(),which accounts for the respective quantum interference effects. The transmission function

6 The energy difference of the two states, 2− 1 = 4 meV, is smaller than the broadening induced by thecoupling of these states to the leads, ΓL,11 + ΓR,11 = 15 meV.

7 The energy difference of the two states, 2− 1 = 40 meV, is larger than the broadening induced by thecoupling of these states to the leads, ΓL,11 + ΓR,11 = 15 meV.


(a)

NEGF INTVIBNEGF INTEL

0 0.5 1. 1.5 2. 2.50

1

2

3

bias voltage ! !V"

currentI!"A"

(b)

NEGF SINTVIBNEGF SINTEL

0 0.5 1. 1.5 2. 2.50

1

2

3

bias voltage ! !V"

currentI!"A"

Figure 6.13: Electronic current-voltage characteristics for molecular junctions exhibitingquantum interference effects. Panel a shows current-voltage characteristics for the modelsystems INTEL and INTVIB (cf. Tab. 6.3). Due to destructive quantum interference, thecurrent through junction INTEL is almost an order of magnitude smaller than the corre-sponding current through junction INTVIB. Panel b represents electronic current-voltagecharacteristics for junctions SINTEL and SINTVIB (cf. Tab. 6.3), where in contrast tothe models INTEL and INTVIB all electronic states are symmetrically coupled to the leads.Quantum interference effects appear less pronounced in the current-voltage characteristicsof these junctions.


incoherent terminterference termtransmission t!!"

electronic SINTEL

electronic INTEL

offset"!#0.025,$2.5"

0.35 0.4 0.45 0.5

#1

0

1

2

3

4

energy ! #eV$

transmissiont!!"

Figure 6.14: Transmission functions for the model molecular junctions INTEL andSINTEL, where no vibrational degrees of freedom are considered (cf. Tab. 6.3). Thesejunctions comprise two quasidegenerate electronic states that provide two equivalent path-ways for electrons tunneling through the junction. Quantum interference in these junctionscan thus be visualized by plotting the incoherent sum of the respective transmission ampli-tudes (|Λ

1

()|2 + |Λ2

()|2, dashed orange lines) as well as the overlap of these amplitudes,which we refer to as interference term (±(Λ∗

1

()Λ2

() + Λ∗2

()Λ1

()), solid green lines).Negative values for the interference term indicate destructive interference, which governs,for example, transport through junction INTEL. In contrast, the symmetrically coupledjunction SINTEL exhibits constructive interference. For energies

1

< < 2

, however,the interference term changes its sign for junction SINTEL, indicating destructive inter-ference effects in this energy range. As a result, the respective transmission function dropsto zero at = (

2

− 1

)/2.

t() of the model molecular junction INTEL, including the incoherent and the interferenceterm, are depicted in Fig. 6.14. It is seen that the interference term is almost as large asthe incoherent sum |Λ

1

()|2 + |Λ2

()|2 such that the total transmission t() is indeed verysmall (destructive interference). This is different for the electronic transmission functionof junction INTVIB (cf. Fig. 6.15a). Overall, the interference term is much weaker forthis junction such that the corresponding transmission function is almost as large as therespective incoherent term. For energies

1

< < 2

, where the interference term exhibitsa sign change (constructive interference), the electronic transmission is even larger thanthe incoherent term.

Such a change from destructive to constructive interference can also be observed if boththe lower and the higher-lying electronic state are symmetrically coupled to the leads,ν

L/R,1/2

= 0.1 eV (cf. Fig. 6.12d). According to our previous considerations, electrons


(a)


INTVIBelectronic

INTVIBth. equ.T"10K

INTVIBth. equ.

T"5000K

offset"#2.7

offset"$2.5

0.3 0.4 0.5 0.6

#2

0

2

4

energy ! #eV$

transmissiont!!"

(b)


SINTVIBelectronic

SINTVIBth. equ.T"10K

SINTVIBth. equ.T"5000K

offset"!0.03,#2.9"

offset"!0.05,$2.9"

0.3 0.4 0.5 0.6

#2

0

2

4

energy ! #eV$

transmissiont!!"

Figure 6.15: Transmission functions for the model molecular junctions INTVIB (panela) and SINTVIB (panel b), where a single vibrational degree of freedom is taken intoaccount (cf. Tab. 6.3). For simplicity, this mode is coupled to one of the two quaside-generate electronic states only. Inelastic processes thus give which-path information onan electron tunneling through the junction. As a result, interference effects arise in thevicinity of the electronic transmission peaks, that is for ≈

1/2

, but not in the vicinityof the satellite peaks at ≈

1/2

± Ω1

(n ∈ ). The more important these satellite peaksare, for example if the vibrational modes is highly excited, the less important are quantuminterference effects in electron transport through these junctions.


tunneling through this junction interfere constructively. This would result in an enhance-ment of the electrical current flowing through this junction. This is, however, not thecase, as can be seen by the dashed purple line in Fig. 6.13b. This line represents theelectronic current-voltage characteristic of the symmetric junction SINTEL (cf. Tab. 6.3),which differs from junction INTEL only by the coupling of the higher-lying electronicstate to the right lead, ν

R,2 = ν. The respective current can be very well described by anincoherent sum of the partial currents through each of the two electronic states. Thus, theelectrical current of this system is not enhanced by constructive interference effects. Thiscan be understood by an analysis of the respective transmission function, in particular,the corresponding interference term.

The transmission amplitudes describing electronic transport through junction SINTEL(in the wide-band approximation) are given by

Λ1/2

() =Γ

− 1/2

+ iΓ2−1−2−2/1

, (6.5)

t() = |Λ1

() + Λ2

()|2 . (6.6)

The corresponding interference term, Λ∗1

()Λ2

() + Λ1

()Λ∗2

(), is represented by the solidgreen line in Fig. 6.14, which for better comparison is offset in both the value of thetransmission function (+2.5) and energy (−25meV). For energies far away from theelectronic resonances, <

1

and > 2

, the interference term is indeed constructive,and gives a positive contribution to the total transmission probability that is as large asthe contribution of the incoherent term. For energies

1

< < 2

, however, it changes,similar as in junction INTVIB, its sign. The respective transmission function thereforevanishes at = (

1

− 2

)/2. Such an antiresonance in the transmission function, whichindicates strong destructive interference effects, is characteristic for transport through twoquasidegenerate electronic states with similar symmetry properties. These resonances arealso referred to as Fano resonances [18]. If the transmission function is integrated oversuch a Fano- or antiresonance, constructive and destructive interference effects cancel eachother. Therefore, the electrical current of this system can be described as an incoherentsum of the partial currents flowing through each of these states, except for bias voltageseΦ w 2

1/2

, where the chemical potential in one of the leads is located close to theantiresonance.

6.5.2 Decoherence due to Inelastic Sequential Tunneling

In this section, we study quantum interference effects in the resonant transport regimeof a molecular junction, where inelastic sequential tunneling due electronic-vibrationalcoupling is significant. To this end, we employ model INTVIB (cf. Tab. 6.3) and considerthe antisymmetric eigenstate of this junction8 to be coupled to a vibrational degree of

8 Note that we obtain almost the same results if the vibrational mode would be coupled to the symmetriceigenstate (λm1 ∼ δm1).


freedom, λm1

∼ δm2

. This specific choice for the electronic-vibrational coupling strengthsλm1

is not crucial for the decoherence mechanism discussed in the following, but simplifiesthe analysis of the phenomena. Similar effects are obtained if the other state is alsocoupled to the vibrational mode (except for the special case λ

11

= λ21

).

Due to electronic-vibrational coupling in junction INTVIB, the energy level of the higher-lying electronic state is shifted from

2

= 0.44 eV to 2

= 0.404 eV. The (shifted) elec-tronic energy levels of junctions INTEL and INTVIB are therefore effectively the same.Consequently, one may expect similar quantum interference effects in both junctions. De-structive quantum interference effects, however, do not significantly reduce the currentthrough junction INTVIB, as is the case in junction INTEL (cf. Sec. 6.5.1). The re-spective current-voltage characteristic is depicted by the solid black line in Fig. 6.16a. Itshows several steps in the resonant transport regime, which are associated with inelasticsequential tunneling processes (cf. Sec. 4.2). While the first of these steps at eΦ = 2

1/2

is somewhat reduced due to destructive quantum interference, the current at higher biasvoltages even exceeds the electronic current-voltage characteristic of this system (solidpurple line in Fig. 6.14a). We therefore conclude that electronic-vibrational couplingleads to a strong quenching of quantum interference effects in this junction (vibrationallyinduced decoherence).

The corresponding vibrational excitation characteristic is shown in the inset of Fig. 6.16a.While the level of vibrational excitation is rather low in the non-resonant as well as at theonset of the resonant transport regime, it increases very rapidly for larger bias voltages.The influence of this current-induced heating on the current-voltage characteristic canbe demonstrated by comparing it to results, where we evaluate the vibrational mode inthermal equilibrium. For example, the dashed gray and the solid red line show current-voltage characteristics, where we describe the vibrational mode in thermal equilibrium at10K and 5000K, respectively. These temperatures simulate a junction without vibrationalexcitation and a junction, in which the thermal excitation is as large as the current-inducedvibrational excitation would be. As is evident by comparison of the dashed gray and redline, an increase of the effective temperature of the vibrational mode results in a significantenhancement of vibrationally induced decoherence. As a result, the dashed red and thesolid black line show a current, which is approximately the same as the incoherent sumof the currents through each of the two electronic states.

Similar to Sec. 6.5.1, this decoherence mechanism can be analyzed in terms of a trans-mission function t(). In the presence of electronic-vibrational coupling, however, thedefinition of such a transmission function, as for example given by Eq. (3.202), is more in-volved. To this end, we employ the wide-band approximation, a thermal equilibrium statefor the vibration and a high bias voltage eΦ 2

1/2

such that for all relevant energies

the electronic self-energies Σ</>L/R,mn() can be approximately evaluated as

Σ<L,mn() = iΓ, Σ>

L,mn() = 0,

Σ<R,mn() = 0, Σ>

R,mn() = i(1− 2δmn)Γ.


(a)

0.5 1.50

5

10

15

NEGF vibronicNEGF th. equ.NEGF th. equ. T!5000K

0 0.5 1. 1.5 2. 2.50

1

2

3

bias voltage " !V"

currentI!#A"

(b)

0.5 1.50

5

10

15

NEGF vibronicNEGF th. equ.NEGF th. equ. T!5000K

0 0.5 1. 1.5 2. 2.50

1

2

3

bias voltage " !V"

currentI!#A"

Figure 6.16: Current-voltage characteristics for molecular junctions exhibiting quan-tum interference effects (cf. Tab. 6.3). Panel a shows current-voltage characteristics forthe model system INTVIB, while panel b refers to junction SINTVIB. In contrast to theelectronic current-voltage characteristics of these junctions, which are represented in Fig.6.13, the vibronic current-voltage characteristics of these junctions do not exhibit pro-nounced quantum interference effects. This is due to vibrationally induced decoherence,that is quenching of quantum interference effects due to inelastic electron transport pro-cesses. The comparison of the solid black with the dashed gray and red lines reveals thata higher level of vibrational excitation strongly enhances this decoherence mechanism.


For our specific model system, we obtain [73,386]

t() = tinc

() + tint

(), (6.7)

tinc

() =Γ2

|− 1

+ iΓ|2 + A∞X

l=−∞

Il(x)e(lΩ1)/(2kBT )

Γ2

|− 2

− lΩ1

+ iΓ|2 , (6.8)

tint

() = −2A Re

∑Γ2

(− 1

+ iΓ)(− 2

− iΓ)

∏, (6.9)

where the prefactor A = e−(λ221/Ω

21)(2Nvib+1) is determined by the average vibrational exci-

tation Nvib

= (eΩ1/(kBT ) − 1)−1 and the temperature T , and

Il(x) = Il(2(λ2

21

/Ω2

1

)p

Nvib

(Nvib

+ 1)) (6.10)

denotes the lth modified Bessel function of the first kind. Fig. 6.15a shows the trans-mission function (solid black lines), the respective incoherent (dashed orange lines) andinterference term (solid green lines) of this system for three different scenarios: i) withoutelectronic-vibrational coupling, ii) including electronic-vibrational coupling to a vibra-tional mode in its ground state (i.e. with an effective temperature for the vibrationalmode of 10K) and iii) with electronic-vibrational coupling to a highly excited mode (i.e.with an effective temperature of 5000 K). For better comparison, the results obtained forthese three different scenarios are offset with respect to each other.

Electronic-vibrational coupling has a profound influence on the transmission function andits two components t

inc

() and tint

(). First, in the vicinity of = (2

− 1

)/2 the polaron-shift of the second electronic state results in a transition of the interference term fromconstructive to destructive interference. At low temperatures (10K), interference effectsare thus greatly enhanced and comparable to the ones observed in junction INTEL. Theyare, however, significantly less pronounced in junction INTVIB. This is due to resonantexcitation and deexcitation processes (cf. Fig. 4.4b – Fig. 4.4d). As these processes in-volve exclusively the higher-lying but not the lower-lying electronic state, they provideeffectively which-path information. Electrons interacting with the vibrational mode arethus no subject to quantum interference. Accordingly, the interference term does notcounterbalance the side peaks in the incoherent term that appear at energies ≈

2

+nΩ1

(n ∈ ). These side peaks are associated with inelastic sequential tunneling processes(cf. Fig. 4.4d) and become more pronounced the higher excited the vibrational modes is.Moreover, the electronic peak at ≈ (

2

− 1

)/2 exhibits a pronounced suppression dueto the prefactor A. This prefactor appears in both the incoherent and the interferenceterm and is significantly reduced if the average level of vibrational excitation N

vib

is high.While the electronic peak in the incoherent term is thus only halved at high levels ofvibrational excitation, the electronic peak in the interference term completely vanishes.

Similar arguments apply for the symmetric junction SINTVIB. While quantum interfer-ence effects in the corresponding current-voltage characteristics (see Fig. 6.16b) are not aspronounced as in the current-voltage characteristics of junction INTVIB (see Fig. 6.16a),


quenching of quantum interference effects due to electronic-vibrational coupling occursalso in this junction. This can be illustrated, again, analyzing the interference and theincoherent term of the respective transmission function t() = t

inc

() + tint

(). Employingthe same approximations as for the expressions given in Eq. (6.7), these terms are givenby

tinc

() =Γ2

°|− 2

+ iΓ|2 − AΓ2

¢

|(− 1

+ iΓ)(− 2

+ iΓ) + AΓ2|2 (6.11)

+A∞X

l=−∞

Il(x)elΩ1/(2kBT )

Γ2

°|− 1

− lΩ1

+ iΓ|2 − AΓ2

¢

|(− 1

− lΩ1

+ iΓ)(− 2

− lΩ1

+ iΓ) + AΓ2|2 ,

tint

() = 2AΓ2 ((−

1

)(− 2

)− Γ2(1− A))

|(− 1

+ iΓ)(− 2

+ iΓ) + AΓ2|2 . (6.12)

They are depicted in Fig. 6.15b for the same three scenarios that we considered in Fig.6.15a, including the corresponding transmission function. Constructive interference, whichgoverns the transmission function of this junction if no coupling to the vibrational mode isconsidered, is quenched in the presence of electronic-vibrational coupling in the same wayas destructive interference in junction INTVIB. This quenching is also related to which-path information provided by the interaction of a tunneling electron with the vibration,and is also strongly enhanced, if the vibrational mode is highly excited. Strong destructiveinterference does appear in the vicinity of the antiresonance at =

2

−1

/2, even for largeeffective temperatures of the vibrational mode. For yet larger temperatures, however, itdoes also completely vanish.

6.5.3 Decoherence due to Inelastic Co-Tunneling

So far, our analysis of quantum interference effects in molecular junctions focused onthe resonant transport regime, where sequential electron tunneling is dominant and canbe understood in close analogy to a double-slit experiment. In this section, we discussquantum interference effects in the non-resonant transport regime, where co-tunneling isdominant. The respective quantum interference effects can also be understood in termsof a double-slit experiment.

For example, if we compare the electronic current-voltage characteristics of the modelmolecular junctions INTEL and INTVIB in the non-resonant transport regime, that isfor eΦ < 2

1/2

(cf. the purple lines in Fig. 6.17a), we observe the same suppression of theelectrical current due to destructive quantum interference effects as before in the resonanttransport regime (cf. the purple lines in Fig. 6.13a). Including electronic-vibrational cou-pling, these effects are quenched and a higher current level is obtained (solid black anddashed gray lines).

For bias voltages eΦ < Ω1

, where the vibrational mode cannot be excited by inelastictransport processes, this quenching of quantum interference is due to a suppression of


electronic co-tunneling processes (Fig. 4.1a), which involve the vibrationally coupled elec-tronic state. This suppression has already been described, for example, in Sec. 4.1.1,where we have outlined that the corresponding suppression factor is given by the FC-factor |X00

1

|2 = e−λ221/Ω

21 ≈ 0.7. As a result, the two states no longer provide equivalent

pathways for electrons (co-)tunneling through this molecular bridge. Quantum interfer-ence effects are therefore less pronounced. Note that, in general, this effect can eitherenhance or decrease quantum interference effects. For example, if the lower-lying elec-tronic state of this junction would be coupled to the leads with a weaker coupling strengthν

L/R,1 ≈ |X00

1

|2 νL/R,2, quantum interference effects would not be quenched but more pro-

nounced by this effect.

There is, however, another aspect of vibrationally induced decoherence in the non-resonanttransport regime that emerges for bias voltages eΦ > Ω

1

. At these bias voltages, inelasticco-tunneling processes occur (cf. Figs. 4.1b – 4.1d). Corresponding to the effect of inelasticsequential tunneling processes in the resonant transport regime, they give which-pathinformation about the tunneling electron, and, therefore, lead to a significant quenchingof quantum interference effects in the non-resonant transport regime. As the number ofinelastic co-tunneling processes increases almost linearly with the applied bias voltageΦ, this mechanism for vibrationally induced decoherence also becomes more effective forlarger bias voltages. Accordingly, the vibronic current-voltage characteristic of junctionINTVIB (see Fig. 6.17a) exhibits a kink at eΦ > Ω

1

, which indicates the onset of inelasticco-tunneling processes. Thereby, current-induced heating of the vibrational mode furtherincreases the number of inelastic co-tunneling channels (cf. Fig. 4.1c). Thus, the vibroniccurrent-voltage characteristic (solid black line) increases faster with Φ as the correspondingthermally equilibrated current-voltage characteristic (see Fig. 6.17a).

Similar arguments apply for the model molecular junction SINTVIB. However, the effectof electronic-vibrational coupling in the non-resonant transport regime of this junction isreversed to the effect in junction INTVIB. This is due to the fact that in this junctionconstructive instead of destructive interference effects are quenched. This can be seen bycomparison of the electronic current-voltage characteristics of junction SINTVIB (solidpurple line in Fig. 6.17b) and junction SINTEL (dashed purple line in Fig. 6.17b). Dueto a higher degree of quasidegeneracy between the electronic levels of junction SINTELand the correspondingly stronger constructive quantum interference effects, the electroniccurrent-voltage characteristic of this junction shows higher current levels than the elec-tronic current-voltage characteristic of junction SINTVIB. Including electronic-vibrationalcoupling, the quenching of these constructive interference effects results in a reduction ofthe corresponding tunnel current. The opening of inelastic co-tunneling channels is there-fore much less pronounced than, for example, in junction INTVIB. Moreover, a higherlevel of vibrational excitation does not result in a higher but a lower current level, as canbe inferred by comparison of the vibronic and the thermally equilibrated current-voltagecharacteristic shown in Fig. 6.17b.

Finally we note that decoherence due to electronic-vibrational coupling is an intrinsic prop-erty of a single-molecule junction. Thereby, the polaron-shift of the electronic levels as well


(a)

NEGF electronicNEGF electronic INTELNEGF vibronicNEGF th. equ.

0 0.05 0.1 0.150

0.2

0.4

0.6

bias voltage ! !V"

currentI!nA"

(b)

NEGF electronicNEGF electronic SINTELNEGF vibronicNEGF th. equ.

0 0.05 0.1 0.150

5

10

15

bias voltage ! !V"

currentI!nA"

Figure 6.17: Current-voltage characteristics of junction INTVIB (panel a) and junctionSINTVIB (panel b) in the low bias regime. For comparison, the electronic current-voltagecharacteristics of junction INTEL (dashed purple line in panel a) and junction SINTEL(dashed purple line in panel b) are also shown. For eΦ < Ω

1

, destructive (constructive)quantum interference effects in junction INTVIB (SINTVIB) are less pronounced in thepresence of electronic-vibrational coupling due to a suppression of electronic co-tunnelingprocesses involving the vibrationally coupled state (state 2). For eΦ > Ω

1

, these quantuminterference effects are additionally quenched due to inelastic co-tunneling events. This issimilar to decoherence induced by inelastic sequential tunneling processes (cf. Sec. 6.5.2).


as the ’FC-blockade’ of electronic tunneling events constitute ’static’ effects, which maylead either to an increase or a decrease of quantum interference effects [387]. In contrast,inelastic co- and sequential tunneling processes induce a dynamic decoherence mechanism,which is solely quenching quantum interference effects. The phenomenon is based on anasymmetry in the electronic-vibrational coupling constants, that is λmα = λna for m = n,such that interactions between the vibrational modes and the electrons tunneling throughthe junction provide effectively which-path information. Since single-molecule junctionstypically exhibit multiple vibrational degrees of freedom that are strongly and differentlycoupled to the electronic states of the molecular bridge, we expect vibrationally induceddecoherence to be relevant in electron transport through most molecular junctions. This isshown by a first-principles based model of biphenylacetylenedithiolate molecular junctionin Sec. 7.2.

6.6 Conclusions

In this chapter, we have discussed the role of electronic-vibrational coupling for vari-ous electrical transport properties and possible applications of single-molecule junctions.This includes electrical transport phenomena, such as, for example, negative differen-tial resistance or rectification. These phenomena play a crucial role in electronic deviceapplications and, thus, constitute key issues of molecular electronics. We have also out-lined the possibility to selectively excite vibrational modes of a molecular junction. Thisis a more chemically oriented application and may be relevant in the context of mode-selective chemistry. Thereby, we have shown that electron-hole pair creation processesplay a crucial role. In addition, we have elucidated the role of the low frequency modes inelectron transport through a molecular junction. Considering electronic-vibrational cou-pling of these modes, we have been able to explain the phenomenon of anomalous peakbroadening, which has been observed for a number of different single-molecule junctions.Moreover, we have considered quantum interference effects in single-molecule junctionsand demonstrated that electronic-vibrational coupling, in particular due to inelastic elec-tron tunneling processes, induces a strong quenching of these effects. This vibrationallyinduced decoherence can lead to substantially larger electrical currents in the presence ofelectronic-vibrational coupling than compared to the electrical current that is obtaineddisregarding the effect of vibrations.

As in Chap. 4, we have employed generic model system to investigate these effects andphenomena. In general, however, single-molecule junctions exhibit a manifold of the effectsand phenomena that have presented in this chapter. This has been demonstrated in thischapter, for example, by comparison to experimental data for a Fe2+-bispyrazolylpyridinemolecular junction, and will be elaborated in the next chapter, Chap. 7, by first-principlesbased models of specific molecular junctions.

Chapter 7

First-Principles Based Models ofSingle-Molecule Junctions

So far, we have employed generic model systems to study effects and phenomena in vi-brationally coupled electron transport through molecular junctions. In this chapter, wecomplement these general studies by investigating transport characteristics of specificmolecular systems. In particular, we analyze the transport characteristics of a benzene-dibutanedithiolate (BDBT) single-molecule junction in Sec. 7.1 and of a molecular con-tact based on biphenylacetylenedithiolate (BADT) in Sec. 7.2. Thereby, we consider themolecular bridge to be contacted by gold electrodes, as it is typical for break-junctionexperiments [19, 22–24, 27, 38, 49, 99, 145]. We show that a multitude of electronic andvibrational degrees of freedom needs to be taken into account in order to obtain a realisticdescription of these molecular contacts.

A realistic description, however, is not the only motivation to consider all the degrees offreedom of a single-molecule contact. The analysis of their intricate interplay often re-veals new effects and phenomena. For example, the effect of local cooling via higher-lyingelectronic states, which we have introduced in Sec. 4.2.2, was first observed analyzing thevibrational excitation characteristic of the BDBT molecular junction (cf. Sec. 7.1 or Ref.[80]). Besides, this junction exhibits a number of other vibrational effects, which we havealready discussed in Chaps. 4 and 6 by generic model systems. Similarly, the comparisonof the electronic and the vibronic current-voltage characteristic of the BADT molecularjunction revealed the mechanism of vibrationally induced decoherence in molecular junc-tions (cf. Sec. 6.5). The resulting temperature dependence of the corresponding electricalcurrent may be a rather general phenomenon and used to control quantum interferenceeffects in experiments on this or other single-molecule contacts (cf. Sec. 7.2 or Ref. [73]).

To obtain the model parameters for these investigations, we1 have employed density func-tional theory (DFT) performed with the TURBOMOLE package (v6.1) [388]. Thereby,we have obtained the extended structures shown in Fig. 7.1. These include, besides the

1 In particular, C. Benesch and O. Rubio-Pons have conducted these calculations.

144 7. First-Principles based Models for Single-Molecule Junctions

(a) (b)

Figure 7.1: Panel a: Geometry of the BDBT single-molecule junction. To reduce thecomputational effort of the corresponding first-principles calculations, we have assumedthat this junction has a center of inversion symmetry (C

i

). Panel b: Geometry of theo-biphenylacetylenedithiolate single-molecule junction, exhibiting a strict left L↔ R sym-metry (C

s

). Figure courtesy of O. Rubio-Pons.

molecular bridge, a cluster of gold atoms, representing the tip of the gold electrodes con-sidered. This allows to account for the effect of both the electrodes on the electronicstructure of the molecular bridge and the specific contact geometry on the coupling be-tween the molecular bridge and the electrodes. To obtain the geometrical structure ofthe thus extended molecular bridge, we have used the B3-LYP hybrid functional [389]in combination with the SV(P) basis set, where, in addition, the pseudopotential ECP-60-MWB [390] has been employed to describe the atoms in the gold clusters. The goldclusters represent the very end of macroscopic electrodes, which have been further mod-eled by adding the surface self-energy of a Au(111)-surface to the Hamiltonian of theextended molecular bridge. The extended structure has then been partitioned into threeparts: the molecular bridge and the tip of the left and the right lead. In a final step, theHamiltonians associated with each of these three subspaces have been diagonalized. Thus,we have obtained the electronic structure of both the molecular bridge and the contacts,accounting for their mutual hybridization.

To characterize the nuclear degrees of freedom of the molecular bridge, we performeda normal-mode analysis, where the positions of the gold atoms were fixed in order toseparate the nuclear motion in the molecular bridge from that in the gold clusters. Theelectronic-vibrational coupling constants λmα were obtained from the numerical gradientsof the energies m(Qα) with respect to the dimensionless normal coordinates Qα,

λmα = (m(∆Qα)− m(−∆Qα))/(2∆Qα). (7.1)

To this end, two DFT calculations with molecular geometries that are elongated alongthe normal coordinates by ±∆Qα = ±0.1 from the equilibrium geometry were performed.Thereby, we considered only the totally symmetric normal modes, since all other normalmodes no linear electronic-vibrational coupling ∼ Qαc†mcn (n = m), except for linearnonadiabatic coupling ∼ Qαc†mcn (n = m) [76, 163, 190]. Note that a more detaileddescription of this methodology is given in Refs. [189] and [190].

7.1. Vibrational Signatures in BDBT Molecular Junctions 145

7.1 Vibrational Signatures in Benzenedibutanedithi-olate Molecular Junctions

In this section, we study transport characteristics of a BDBT single-molecule junction[80,189,190,192]. This junction comprises two butanethiolate groups, which are attachedto a benzene ring at opposite positions (para). The sulfur atoms of these butanethiolategroups establish strong covalent bonds to the gold electrodes of this junction. The specificgeometry of the system is depicted in Fig. 7.1a. We expect pronounced vibronic effects inthe resonant transport regime of this junction, as the butanethiolate linkers insulate thedelocalized π-system of the benzene ring from the gold electrodes. The dwell time of anelectron on the benzene ring is thus much longer than, for example, in a benzenedithiolatemolecular junction [189,190,192]. Accordingly, the probability that an electron interactswith the vibrational degrees of freedom of this junction during a sequential (resonant)tunneling event is enhanced.

This idea is supported by Fig. 7.2a, which shows the electronic transmission function t()of this junction (solid purple line). There are two narrow peaks at = −1.38 eV and = −1.77 eV. The width of these peaks, which is less than 2meV, corresponds to thelifetime of the electrons tunneling through the respective electronic resonances. Theselifetimes are much longer than the oscillation period of a vibrational mode (except formodes with very low frequencies Ωα . 5meV). This corresponds to the antiadiabaticregime of vibrationally coupled electron transport (cf. Sec. 3.3.3) and indicates strongelectronic-vibrational coupling in this system.

The peaks at = −1.38 eV and = −1.77 eV are associated with the HOMO and theHOMO-1 of the molecular bridge, respectively. This is demonstrated by comparisonwith the dashed yellow and the dashed turquoise line, which show transmission functionsconsidering a single electronic state only. The peaks in these transmission functionsalmost perfectly coincide with the peaks seen in the transmission function including allstates of this system. The small shift of these peaks with respect to energy is related tothe renormalization of the electronic energy levels due to an effective interaction of theelectronic states that is mediated by their coupling to the leads. The two relevant orbitals,HOMO and HOMO-1, are depicted in Fig. 7.3.

As is shown in the thesis of C. Benesch [192], the current through this BDBT single-molecule junction can be very well described considering these two levels. Moreover, thefour most strongly coupled vibrational modes were considered (cf. Tab. 7.1 for the relevantparameters). To this end, scattering and density matrix theory were employed [192]. Wecomplete these studies employing the NEGF approach. This allows to study the current-induced levels of excitation of the four vibrational modes on the same footing, which isneither possible in terms of scattering theory (cf. Sec. 3.5) nor by density matrix theoryas it may be used to describe one or two of these modes but not all of them.

Fig. 7.4 shows current-voltage characteristics of this system in the resonant transportregime, i.e. for Φ & 2.5V. In accordance with the electronic transmission function,


(a)

only HOMOonly HOMO!1

all states

!1.8 !1.6 !1.40

0.5

1.

energy " !eV"

transmissiont#"$

(b)

HOMO!1off!diagonal

HOMO

!3. !2.5 !2. !1.5 !1. !0.5 0

!1.

!0.5

0

0.5

energy " !eV"

level!widthfunction!meV"

Figure 7.2: Panel a: Electronic transmission function t() for the BDBT single-moleculejunction. The two narrow peaks are associated with the HOMO and the HOMO-1 of themolecular bridge (see Figs. 7.3a and 7.3b). For comparison we also show the transmissionfunction, where we consider only the HOMO (HOMO-1) level. They are depicted by thedashed yellow (turquoise) line. Panel b: Level-width functions Γ

L,mn() for the electronicstate corresponding to the HOMO (m =HOMO,n =HOMO), the HOMO-1 (m =HOMO-1,n =HOMO-1) and the respective off-diagonal component (m =HOMO,n =HOMO-1,which is the same as m =HOMO-1,n =HOMO). At zero bias the level-width functionsassociated with the left and the right lead are the same: Γ

L,mn() = ΓR,mn(). For positive

bias voltages, the level-width functions ΓL,mn() are shifted towards higher and Γ

R,mn()towards lower energies, and vice versa for negative bias voltages.


(a) HOMO

(b) HOMO-1

Figure 7.3: Molecularorbitals relevant forelectron transportthrough the BDBTmolecular junction.Due to the butanethi-olate linkers, whichinsulate the benzenering of the molecularbridge from the elec-trodes, these orbitalsshow almost no overlapwith the electrodes.Figure courtesy of O.Rubio-Pons.

the electronic current (solid purple line) shows two distinct steps at Φ = 2.76V andΦ = 3.54 V. These steps correspond to the onset of resonant transport processes throughthe HOMO and the HOMO-1, respectively. Due to the energy dependence of the cor-responding level-width functions Γ

L/R,mn() (see Fig. 7.2b), the efficiency of transportthrough the two levels changes with the applied bias voltage Φ. This leads to variationsin the electrical current even though the number of resonant transport channels doesnot change upon increasing the bias voltage Φ. For example between the two steps atΦ = 2.76V and Φ = 3.54V, the current increases in a nonuniform way. Moreover, atΦ = 3.5, 3.6 and 3.7V, the energy dependence of the level-width function leads to NDR(a similar mechanism for NDR was already discussed in Sec. 6.1.1).

The solid black line in Fig. 7.4a shows the vibronic current-voltage characteristic of thissystem. In addition, we also show the thermally equilibrated current-voltage characteris-tic (dashed gray line in Fig. 7.4a), where we employ an effective temperature of 10 K forthe four vibrational modes. The two steps, which appear at Φ = 2.54V and Φ = 3.29V inboth characteristics, correspond to the two steps in the electronic current-voltage charac-teristic. They are also associated with the onset of resonant transport processes throughthe HOMO and the HOMO-1, but appear at lower bias voltages due to the polaron-shift of the respective electronic states (cf. Sec. 3.1.3). Due to vibrationally inducedelectron-electron interactions, U

12

= U21

= −0.203 eV, there is another pronounced stepat Φ = 2.89V. This step corresponds to the onset of resonant transport through thedicationic resonance, where both electronic states are unoccupied (this is similar to thedianionic resonance discussed in Sec. 4.2.2). These three steps are highlighted in Fig. 7.4aby dashed vertical lines.

There is, however, a number of additional steps, which are associated with the onset of res-onant excitation and deexcitation processes (cf. Sec. 4.2). Thereby, multimode processes(such as the one depicted in Fig. 4.12) play an important role [216]. As outlined in Sec.


(a)

!"HOMO#1$U"""

!"HOMO#1

!"HOMO

NEGF electronicNEGF vibronicNEGF th. equ.

2.6 2.8 3. 3.2 3.4 3.6 3.80

0.1

0.2

0.3

0.4

bias voltage % !V"

currentI!&A"

(b)

2.5 3. 3.50

1

2

NEGF mode 1NEGF mode 2NEGF mode 3NEGF mode 4

Evib!eV"

2.6 2.8 3. 3.2 3.4 3.6 3.80

1

2

3

4

bias voltage ! !V"


Figure 7.4: Current-voltage and vibrational excitation characteristic of the BDBTsingle-molecule junction in the resonant transport regime (Φ & 2.5 V). Electronic-vibrational coupling leads in this system to effects that are very similar to the effectsdiscussed in Sec. 4.2 (in particular Sec. 4.2.2) and Sec. 6.1.1.


state m m

HOMO -1.38 -1.27HOMO-1 -1.77 -1.65

mode Ωα λHOMO α λHOMO-1 αλHOMO α

Ω

α

λHOMO-1 α

Ω

α

1 0.070 0.021 0.049 0.30 0.702 0.149 0.052 0.037 0.35 0.253 0.153 0.039 0.080 0.25 0.524 0.208 0.120 0.093 0.58 0.45

Table 7.1: Model parameters for the BDBT single-molecule junction (energy values aregiven in eV). The temperature in the leads is given by k

B

T = 10 K. In this model system,we do not consider electron-electron interactions, U

12

= 0 (but U12

= −0.203).

4.2.2, resonant deexcitation processes facilitate transport through electronic levels evenif these levels are outside the bias window. This leads effectively to a broadening of thesteps in the current-voltage characteristic that appear at higher bias voltages. Current-induced vibrational excitation leads to an almost ohmic increase of the electrical currentfor Φ > 2.6V and quenches NDR that is observed in the electronic current-voltage char-acteristic of this BDBT molecular junction. This is similar to our findings in Sec. 6.1.1.

The corresponding current-induced levels of vibrational excitation are shown in Fig. 7.4bfor each of the four vibrational modes. The excitation levels of the four modes increasealmost linearly with the applied bias voltage Φ. Only the mode with the lowest frequencyshows a pronounced decrease in its excitation characteristic at Φ ∼ 2.7V, which is dueto resonant deexcitation processes with respect to the dicationic resonance (note thatλHOMO-1 α ≈ 2λHOMO α, α = 1). Overall, this mode also shows the highest level of excitation.In contrast, the mode with the highest frequency is not necessarily the mode with thelowest excitation level. This is a result of the specific electronic-vibrational couplingstrengths in this system (cf. the discussion in Secs. 4.3.1 and 5.2, where we outline thatweak electronic-vibrational coupling can lead to higher levels of vibrational excitation).The overall vibrational energy E

vib

=P

α Ωα

≠a†αaα

Æ, is depicted in the inset of this figure.

It increases steadily with the applied bias voltage Φ.

In the non-resonant transport regime, i.e. for Φ . 2.4V, the levels of vibrational excitationare rather low. Therefore, the vibronic and the thermally equilibrated current-voltagecharacteristics agree very well in this regime, as can be seen in Fig. 7.5. This figure showsthe respective conductance-voltage characteristics for bias voltages Φ . 0.6V. Steps inthese characteristics indicate the onset of inelastic co-tunneling processes. A number ofthese steps is associated with multimode vibrational effects (cf. Sec. 4.1.2). Around zerobias, the comparison of the electronic and the vibronic characteristics reveals a pronouncedsuppression of the electrical current in the presence of electronic-vibrational coupling. This


NEGF electronicNEGF vibronicNEGF th. equ.

0 0.1 0.2 0.3 0.4 0.50

10

20

30

bias voltage ! !V"

conductanceg!pA#V"

Figure 7.5: Conductance-voltage characteristics for the BDBT single-molecule junctionin the non-resonant transport regime. Vibrational nonequilibrium effects are less impor-tant, as the corresponding levels of vibrational excitation are rather low (data not shown).

is related to the reduced overlap between vibrational states of the PES of the neutralmolecule and the cation (cf. Sec. 4.1.1). However, as the number of inelastic co-tunnelingprocesses increases with the applied bias voltage Φ, larger tunnel currents are observed.This effect is also enhanced by the polaron-shift of the electronic levels towards the Fermilevel of this system. The BDBT single-molecule junction thus reveals a complex interplayof the various transport phenomena that we have already discussed in Chap. 4 – Chap. 6.

7.2 Biphenylacetylene Molecular Junctions: Controlof Quantum Interference Effects due to Anoma-lous Temperature Dependence

In this section, we analyze electron transport through the BADT molecular junctiondepicted in Fig. 7.1b. Thereby, we focus on quantum interference effects. These effectsare important in electron transport through quasidegenerate states (cf. Sec. 6.5). Thisrequires small energy differences between the electronic states and/or a strong coupling ofthe molecular conductor to the leads. Both conditions are met by the BADT molecularjunction. The thiolate linker groups establish strong covalent bonds between the π-systemof the molecular bridge and the gold electrodes. In addition, the L↔R (mirror) symmetryof the junction supports quasidegeneracies. Such a L↔R symmetry can be found in manymolecular junctions.

7.2. BADT Molecular Junctions 151

(a)all statesonly HOMOonly HOMO!1only HOMO!4HOMO & HOMO!1

!2. !1.5 !1. !0.5 0 0.50

0.5

1.

energy "!"F !eV"

transmissiont#"$

(b)

HOMO!4off!diagonal

HOMO

!2. !1.5 !1. !0.5 0 0.5!0.5

!0.4

!0.3

!0.2

!0.1

0

energy " !eV"

level!widthfunctions!eV"

Figure 7.6: Panel a: Electronic transmission function t() for the BADT single-moleculejunction. There are three peaks in this transmission function, which can be associated withelectronic states corresponding to the HOMO, the HOMO-1 and the HOMO-4 (cf. Figs.7.7a, 7.7b, and 7.7c). For comparison we also show transmission functions, where we con-sider transport through the HOMO, the HOMO-1 and the HOMO-4 level only, depictedby the dashed yellow, turquoise and red lines. The dashed blue line depicts the trans-mission function that we obtain, if we consider transport through two electronic levels,corresponding to the HOMO and the HOMO-1. Panel b: Level-width functions Γ

L,mn()for the HOMO (m =HOMO,n =HOMO), the HOMO-4 (m =HOMO-4,n =HOMO-4)and respective off-diagonal component (m =HOMO,n =HOMO-4). As the HOMO andthe HOMO-1 are symmetric and antisymmetric combinations of the same localized or-bitals, the off-diagonal component with m =HOMO-1,n =HOMO-4 differs from the onewith m =HOMO,n =HOMO-4 just by a minus sign. For the same reason, the level-widthfunctions of the HOMO and the HOMO-1 are the same. The respective off-diagonal com-ponents (m =HOMO,n =HOMO-1) are given by the negative of the level-width functionsof the HOMO (or the HOMO-1).


The electronic transmission function of this junction is represented in Fig. 7.6a (solid pur-ple line). It shows three distinct peaks, which can be associated with resonant transportthrough the HOMO, the HOMO-1, and the HOMO-4 of the molecular bridge. This isdemonstrated by transmission functions, where we consider transport through the stateassociated with the HOMO, the HOMO-1 or the HOMO-4 only (dashed yellow, turquoiseand red lines). These orbitals are depicted in Figs. 7.7, and the energies of the correspond-ing electronic states are summarized in Tab. 7.2. As these states are coupled to the sameleads, they are also effectively coupled with each other. This leads to a reorganization ofthe resonances in the electronic transmission function (solid purple line) with respect tothe resonances seen in the dashed lines. This includes shifts of the respective peak posi-tions (as already discussed in Sec. 7.1 for the BDBT molecular junction), but may alsoalter the shape and/or the appearance of resonances overlapping with each other. For ex-ample, the resonances associated with the HOMO and the HOMO-1 almost vanish, if oneconsiders transport through these orbitals only (dashed blue line). As the HOMO and theHOMO-1 are symmetric and antisymmetric combinations of the same localized orbitals,this can be understood in terms of destructive quantum interference effects (see Sec. 6.5.1).However, if the other electronic states, especially the one associated with the HOMO-4,are also taken into account, the degree of quasidegeneracy between the HOMO and theHOMO-1 decreases due to a renormalization of the respective energy levels. As a result,quantum interference effects are less pronounced and two peaks emerge at ≈ −1.4 eVand ≈ −1.5 eV, which can be associated with transport through the HOMO and theHOMO-1. Note that the energies of these levels m, due to the strong coupling to the leads(cf. the respective level-width functions in Fig. 7.6b), are not directly correlated with theposition of these peaks. Thus, the peak at higher energies ≈ −1.0 eV is associated withthe HOMO-4, although the corresponding state is located further away from the Fermilevel than the HOMO and the HOMO-1. Similar effects occur in transport through theHOMO-2 and the HOMO-3. Renormalization of these levels, however, does not lead to asignificant quenching of the corresponding quantum interference effects such that trans-port through the corresponding electronic states does not significantly contribute to theelectrical current of this system. All other states show either similar quantum interferenceeffects or are located far from the Fermi level such that they are irrelevant for transport(at least for bias voltages < 5V, as test calculations show).

The effect of destructive quantum interference can also be seen in the electronic current-voltage characteristic depicted by the solid purple line in Fig. 7.8. It shows significantlylower current levels than the currents that are obtained considering transport through asingle electronic state only (dashed yellow, turquoise and red lines). There are two pro-nounced steps in the electronic current-voltage characteristic that appear at Φ ≈ 1.65Vand Φ ≈ 2.05 V. While the step at higher bias voltage corresponds to transport through theHOMO-4, the step at lower bias voltages is associated with the quasidegenerate HOMOand HOMO-1. This contradicts the assignment of the peaks in the electronic transmis-sion function of Fig. 7.6a. The reason is that this assignment refers to the electronictransmission functions at zero bias. As the corresponding level-width functions show a


state m m Um HOMO Um HOMO-1 Um HOMO-4

HOMO 0.38 -0.29 0 -0.08 -0.08HOMO-1 0.37 -0.27 -0.08 0 -0.19HOMO-4 -1.01 -0.83 -0.08 -0.19 0

mode Ωα λHOMO α λHOMO-1 α λHOMO-4 αλHOMO α

Ω

α

λHOMO-1 α

Ω

α

λHOMO-4 α

Ω

α

1 3.2 1.0 1.4 10.2 0.30 0.43 3.152 6.2 -0.8 -1.1 -7.0 -0.13 -0.18 -1.133 7.0 -16.8 -17.8 -6.7 -2.42 -2.56 -0.974 17.6 -13.0 -12.3 -2.8 -0.74 -0.70 -0.165 18.1 6.3 6.7 -2.7 0.35 0.37 -0.156 31.8 -2.1 -2.3 0.4 -0.07 -0.07 0.017 32.6 -21.9 -22.1 -35.8 -0.67 -0.68 -1.108 36.1 -3.6 -4.3 -3.6 -0.10 -0.12 -0.109 42.2 -15.6 -16.0 -2.3 -0.37 -0.38 -0.0610 51.5 26.6 26.4 -16.9 0.52 0.51 -0.3311 56.8 -3.6 -4.2 -3.2 -0.06 -0.07 -0.0612 61.7 0.0 0.8 13.3 0.00 0.01 0.2213 74.2 6.0 6.4 3.1 0.08 0.09 0.0414 83.4 -2.1 -2.1 -3.8 -0.03 -0.03 -0.0515 88.4 3.3 4.0 3.9 0.04 0.04 0.0416 98.5 2.0 1.3 -38.9 0.02 0.01 -0.4017 101.2 4.9 5.7 6.8 0.05 0.06 0.0718 115.8 -1.4 -1.8 -1.0 -0.01 -0.02 -0.0119 116.7 -1.3 -1.4 -1.2 -0.01 -0.01 -0.0120 125.0 -6.6 -7.2 -18.8 -0.05 -0.06 -0.1521 125.9 -3.1 -3.5 -2.6 -0.02 -0.03 -0.0222 136.3 -3.0 -3.2 -4.1 -0.02 -0.02 -0.0323 138.4 -7.7 -8.8 -11.4 -0.06 -0.06 -0.0824 145.6 3.2 4.0 -39.3 0.02 0.03 -0.2725 147.1 9.6 10.4 -7.8 0.07 0.07 -0.0526 161.6 -5.4 -5.8 -12.9 -0.03 -0.04 -0.0827 167.6 9.8 10.4 2.9 0.06 0.06 0.0228 177.6 -4.5 -4.5 6.2 -0.03 -0.03 0.0429 186.3 -0.9 -0.9 -4.1 0.00 -0.01 -0.0230 199.5 -0.1 -0.1 -4.1 0.00 0.00 -0.0231 202.7 -3.4 -4.0 42.9 -0.02 -0.02 0.2132 286.2 0.9 0.7 105.8 0.00 0.00 0.3733 393.1 3.8 4.3 3.3 0.01 0.01 0.0134 394.8 8.3 9.0 8.2 0.02 0.02 0.0235 396.6 0.6 0.5 1.9 0.00 0.00 0.0036 396.9 -8.5 -9.5 -14.2 -0.02 -0.02 -0.04

Table 7.2: Model parameters for the BADT single-molecule junction (energy values aregiven in eV). The temperature in the leads is set to k

B

T = 10 K. In this model system,we do not consider electron-electron interactions, U

12

= 0.


Figure 7.7: Molecular orbitalsrelevant for electron transportthrough the BADT molecularjunction. Due to the strong cou-pling between the molecular bridgeand the leads, electron transportthrough these orbitals is governedby quantum interference effects.Thereby, the different L ↔ Rsymmetries of the orbitals play acrucial role. The strong couplingto the leads results in a localizationof the HOMO and the HOMO-1at the two ends of the molecularbridge. Figure courtesy of O.Rubio-Pons.

(a) HOMO

(b) HOMO-1

(c) HOMO-4

pronounced energy dependence (cf. Fig. 7.6b), the corresponding transmission functionsexhibit an accordingly strong bias dependence. Fig. 7.8b shows the same transmissionfunctions as depicted in Fig. 7.6a at a higher bias voltage Φ = 1.6V. At this bias voltage,the electronic transmission function shows two peaks at ≈ −1.1 eV and ≈ −0.9 eV,where the peak at higher energy corresponds to transport through both the HOMO andthe HOMO-1. It is also associated with the first step in the electronic current-voltage char-acteristic. In addition, the strong bias dependence of the transmission function inducesalso strong variations in the electronic current of this system. This leads, for example, toa continuous increase of the electronic current for Φ & 2V.

Thermally equilibrated current-voltage characteristics of this system are depicted by thedashed gray and black lines in Fig. 7.8a. They show the effect of electronic-vibrationalcoupling, where an effective temperature of 10 K (dashed gray line) and 300 K (dashedblack line) for the vibrational modes is taken into account. There are 36 active, i.e.totally symmetric, normal modes in this system. The respective electronic-vibrationalcoupling strengths are summarized in Tab. 7.2. Due to these couplings the electronic stateassociated with the HOMO-4 is shifted by ≈ 0.18 eV towards the Fermi level of the system.The step that appears at Φ = 2.08 V in the electronic current-voltage characteristic is thusshifted to lower bias voltages (Φ ≈ 1.95 V). However, the polaron-shift of the HOMO-4does not directly correlate with the shift of this step. This is due to interactions betweenthe HOMO, the HOMO-1 and the HOMO-4, which lead to an increased width of this stepand an additional step at Φ ≈ 2.15 V that is the result of vibrationally induced electron-electron interactions (see Tab. 7.2 for the respective values of Umn). In contrast, the stepthat appears in the thermally equilibrated current-voltage characteristic depicted by thesolid gray line at Φ ≈ 0.6 V does directly correlate with the polaron-shifted energy levels ofthe HOMO and the HOMO-1. Since this step is not visible in the electronic current-voltage


(a)

electronic

electronic, only HOMO!4electronic, only HOMO!1electronic, only HOMO

th. equ.th. equ. T"300K

0 0.5 1. 1.5 2. 2.5 3.0

4

8

12

bias voltage # !V"

currentI!$A"

(b)

all statesonly HOMOonly HOMO!1only HOMO!4HOMO & HOMO!1

"#1.6V

!2. !1.5 !1. !0.5 0 0.50

0.5

1.

energy $!$F !eV"

transmissiont#$$

Figure 7.8: Panel a: Current-voltage characteristics of the BADT single-molecule junc-tion. The electronic current through this system is strongly suppressed due to destructivequantum interference effects, as can be seen by comparison with the current-voltage char-acteristics depicted by the dashed yellow, turquoise and red lines. Panel b: Electronictransmission functions, as they are also shown in Fig. 7.6a, but here they are evaluatedat Φ = 1.6 V.


characteristic, its appearance indicates a significant quenching of destructive quantuminterference effects. As discussed in Sec. 6.5.2, this quenching is associated with electronic-vibrational coupling.

In Sec. 6.5.2 it was also shown that higher levels of vibrational excitation enhance vi-brationally induced decoherence. Thus, the thermally equilibrated current-voltage char-acteristics, where we consider a higher effective temperature (dashed black line) and thecorresponding level of excitation for the vibrational modes, shows significantly higher cur-rents than the thermally equilibrated current-voltage characteristics, where we considera very low effective temperature for the vibrational modes. As the effective temperatureof the vibrational modes may be controlled by the temperature in the leads (for exam-ple at low bias voltages and/or in the presence of strong vibrational relaxation, wherecurrent-induced vibrational excitation is rather low), one may control quantum interfer-ence effects in this junction by varying the temperature in the leads. Vibrationally induceddecoherence thus results in a rather strong temperature dependence of the electrical cur-rent. As the underlying interference mechanism is based on a simple L↔R symmetry andthe electronic-vibrational coupling strengths of this junction are rather typical, such ananomalous temperature dependence of the electrical current may be observed in electrontransport through a number of different molecular junctions.

7.3 Conclusions

As we have seen in this chapter, single-molecule junctions exhibit a manifold of the vi-brational effects that we have discussed in Chaps. 4 – 6 by generic model systems. Thiswas exemplified by first-principles based models for a benzenedibutanedithiolate and abiphenylacetylenedithiolate molecular junction.

The transport characteristics of the biphenylacetylenedithiolate molecular junction, forexample, exhibit pronounced vibrational signatures. This is due to the butanethiolatelinker groups of this molecule, which insulate the π-system of the benzene ring fromthe electrodes. The residence time of an electron tunneling through this single-moleculecontact is thus long enough to allow for an effective interaction between the tunnelingelectron and the vibrational degrees of freedom of this junction. Thereby, multimodevibrational effects and the effect of local cooling due to the presence of a lower-lyingelectronic state, which is associated with the HOMO-1, needed to be taken into accountfor both a qualitatively and a quantitatively correct analysis of these characteristics.

In contrast, the transport characteristics of the biphenylacetylenedithiolate molecularjunction show less pronounced vibrational signatures. As we have seen, electron trans-port through this junction is dominated by quantum interference effects and vibrationallyinduced decoherence. This is due to the strong coupling of the π-system of this molecularjunction to the leads. The respective molecular orbitals show a strong hybridization withthe leads, and thus, provide several pathways for electrons tunneling through the junc-tion. Thereby, the different symmetry of the orbitals lead to strong destructive quantum


interference effects, which suppress the electrical current flowing through this molecularjunction. The vibrational degrees of freedom of this junction couple similarly to the cor-responding electronic states. Vibrationally induced decoherence associated with each ofthese modes is thus rather weak but in total leads to a significant quenching of destruc-tive quantum interference effects. This includes, in particular, the low frequency modesof this junction. The electrical current may thus exhibit an anomalous dependence on thetemperature in the leads.


Summary and Outlook

In this thesis, we have investigated vibrational effects in electron transport through single-molecule junctions. We have shown that these effects play an important role in thisnonequilibrium transport problem and related this to the fact that molecules, due to theirsmall size and mass, respond to changes of their charge state by a profound reorganizationof their geometrical structure. This is in contrast to other nanostructures, such as, forexample, quantum dots or resonant tunneling diodes, where, due to the size of thesesystems and the surrounding environment, electronic and vibrational degrees of freedomare typically much weaker interrelated. To this end, we have employed comprehensivemodel calculations, representing a large variety of different molecular junctions. Thesestudies reveal that the vibrational degrees of freedom of a molecular junction can behighly excited. Considering the resulting vibrational nonequilibrium effects, we haveanalyzed the corresponding transport characteristics both qualitatively and quantitativelyand described a number of different transport phenomena in electron transport througha single molecule.

Although vibrational effects in single-molecule junctions have been extensively investi-gated over the past decade (see, for example, Refs. [16, 18, 74–79, 81, 82, 204, 210, 244,246,391] and references therein), we have presented some important aspects in this workthat have not been considered before [34, 73, 80, 83, 216, 217, 224, 250]. In particular, wehave demonstrated the importance of electron-hole pair creation processes. These pro-cesses are well known from spectroscopic [46, 361–363] and theoretical studies [364–366]of adsorbates at metal surfaces, but have not been considered in detail in the context ofsingle-molecule junctions yet. Our studies reveal that they constitute a cooling mecha-nism for the vibrational degrees of freedom of a molecular junction that is as importantas cooling by inelastic electron transport processes. Thus, we have been able to show thathigher levels of vibrational excitation are obtained in vibrationally coupled charge trans-port for a weaker coupling between the electronic and the vibrational degrees of freedom.While this counterintuitive phenomenon had been reported before [74, 82, 244, 360], wehave given a physical explanation taking into account the effect of cooling by electron-holepair creation processes. In addition, the study of this phenomenon allowed us to show theequivalence between the high-bias and the weak electronic-vibrational coupling limit.

160 Summary and Outlook

Another important aspect of electron-hole pair creation processes is that they involveeither the left or the right lead. This is in contrast to transport processes that involveboth leads. For this reason, pair creation processes can directly translate an asymmetryin the coupling to the leads to the transport characteristics of a molecular junction. Ouranalysis of asymmetrically coupled molecular junctions thus revealed that the spectroscopyof molecular energy levels can be more easily conducted in asymmetric than in symmetricsystems. On the same basis we have explained the appearance of negative differentialresistance and rectification in molecular junctions.

In addition, we have shown the importance of multimode vibrational effects in single-molecule junctions [34,216]. These include co-tunneling processes, which are relevant forinelastic electron tunneling spectroscopy, and multimode sequential tunneling processes,which have to be considered when analyzing vibrational signatures in the resonant trans-port regime of a molecular contact. We have also discussed the pumping of high-frequencyby low-frequency modes and presented a mechanism for anomalous peak broadening thatis based on the electronic-vibrational coupling of the low-frequency modes of a molecularcontact. Another effect that involves multiple vibrational modes occurs in molecular junc-tions, where the electronic states of the molecular bridge couple differently to both theleads and the vibrational degrees of freedom. This is a rather general coupling scenarioand results in a strong bias dependence of cooling by electron-hole pair creation processes.We have shown that this bias dependence facilitates control of the excitation levels of thedifferent vibrational modes and, thus, may provide a route to mode-selective vibrationalexcitation in a single-molecule junction [224,250].

The level of vibrational excitation in a molecular contact does not necessarily increasewith the applied bias voltage. It may also decrease, if an electronic level of the molecularbridge is located close to the chemical potential in one of the leads. We have related thisto the fact that resonant deexcitation processes with respect to such a state constitutean efficient absorption mechanism for vibrational energy. This is particularly importantin the resonant transport regime, where higher-lying anionic or lower-lying cationic res-onances facilitate such cooling effects and, thus, stabilize the junction. Electron-electroninteractions may enhance this stabilization mechanism, because they lead to a splittingof the respective resonances and, thus, to efficient cooling of the vibrational modes in amolecular junction over a broader range of bias voltages.

In transport through multiple electronic states, in particular if they are quasidegenerate,quantum interference effects need to be considered. We have demonstrated that electronic-vibrational coupling, in particular due to inelastic excitation and deexcitation processes,quenches these interference effects. We refer to this phenomenon as vibrationally induceddecoherence. Thereby, a higher level of vibrational excitation enhances this quenching. Asa result, the corresponding electrical current exhibits a strong dependence on the effectivetemperature of the vibrational modes in a single-molecule contact. This (anomalous)temperature dependence may be used in experiments to control quantum interferenceeffects in molecular junctions.


For our theoretical studies, we have employed both generic and first-principles basedmodels of a molecular junction. While we have chosen the parameters of the genericmodels as simple as possible, in order to highlight the important degrees of freedom andspecific effects, the parameters of the first-principles based models have been obtained fromdensity functional theory calculations. We have calculated the transport characteristicsof these model molecular junctions with both a master equation and a nonequilibriumGreen’s function approach.

The master equation approach has been used by a number of different authors [74, 77,100,242–247]. It thus provides a valuable benchmark for comparison with other methodsand for the discussion of vibrational effects. The respective equation of motion for thereduced density matrix (or master equation) is derived from the exact Nakajima-Zwanzigequation by employing a strict second-order expansion in the coupling of the molecularbridge to the leads. Using this method, we have obtained both numerical and analyticalresults. In particular, we have given a proof by contradiction, which shows that the levelof vibrational excitation in a molecular junction increases indefinitely with increasing biasvoltage. Moreover, we have derived an analytic expression for the vibrational distributionfunction in the limit of vanishing electronic-vibrational coupling and showed that thecorresponding level of vibrational excitation does not necessarily vanish in this limit butmay even increase indefinitely.

The nonequilibrium Green’s function approach is an extension of the work of Galperinet al. [236]. It is based on a factorization of the single-particle Green’s function, whichis similar to the adiabatic approximation. Instead of the adiabatic regime, however, wehave considered the antiadiabatic regime, where the timescales for changes of the elec-tronic degrees of freedom are much longer than the oscillation period of the respectivevibrational modes. The factorized single-particle Green’s function is evaluated using anequation of motion technique [285] based on second-order (self-consistent) perturbationtheory in the molecule-lead coupling. We have extended this approach to account for mul-tiple vibrational [216] and multiple electronic degrees of freedom [80]. Thereby, the majorchallenge was to account for electron-electron interactions in an efficient way, especiallysince (resonant) electron transport through molecular junctions typically involves severalelectronic states. To this end, we have employed the elastic co-tunneling approximation[80, 285, 287–289]. Comparison to the master equation approach shows that this approx-imation gives the correct number, location and strength of the resonances. It shouldbe noted, though, that it treats electron-electron interactions on a mean-field level, andtherefore, has to be applied with care in the presence of quantum interference effects(cf. the discussion given in App. A). Moreover, it misses higher-order effects, such as,for example, the Kondo effect [285]. Apart from these deficiencies, it provides a usefuldescription of electron-electron interactions in many molecular junctions.

Both the master equation and the nonequilibrium Green’s function approach are non-perturbative with respect to interactions on the molecular bridge, that is electron-electroninteractions and electronic-vibrational coupling. However, they treat the coupling of themolecular bridge to the leads on a different footing. While the master equation approach


thus accounts only for sequential tunneling events or resonant transport processes, thenonequilibrium Green’s function approach also includes higher-order processes. As a re-sult, the broadening of the molecular energy levels, co-tunneling processes and off-resonantelectron-hole pair creation processes are described by the nonequilibrium Green’s functionscheme but not by the master equation approach. The role of co-tunneling processes hasalready been outlined before [79, 216, 253–255, 274], whereas we have demonstrated theimportance of off-resonant electron-hole pair creation processes [250]. To this end, wehave compared results obtained by the master equation and the nonequilibrium Green’sfunction method and showed that these processes are particularly important, wheneverresonant pair creation processes are suppressed, for example, in the non-resonant transportregime or at high bias voltages. On the other hand, the nonequilibrium Green’s functionapproach involves approximations with respect to both electron-electron interactions andelectronic-vibrational coupling in a single-molecule contact, while the master equation ap-proach treats these interactions exactly, that is, within the strict second order expansionof the Nakajima-Zwanzig equation. The two methods are therefore complementary. Theresults obtained by the two approaches, however, show good agreement.

The work presented in this thesis offers the possibility for a variety of further develop-ments and extensions. For example, the nonequilibrium Green’s function scheme may beimproved to account for the Kondo effect. Thereby, the scheme of Ref. [392], employ-ing a slave-boson technique, may be straightforwardly implemented, because it employssimilar equations and approximations. Such a scheme may be used to study vibrationalnonequilibrium effects in the Kondo regime of a molecular junction. An extension of themaster equation approach to fourth and higher orders in the molecule-lead coupling hasalready been given in the literature [246, 251, 252, 265]. These approaches may be usedto study the effect of non-resonant processes in the weak electronic-vibrational couplinglimit, in particular, the contribution of off-resonant electron-hole pair creation processes.Moreover, we did not consider spin degrees of freedom explicitly. Indeed, test calculationsshowed that spin degrees of freedom, due to electron-electron interactions, result in a fur-ther splitting of resonances. But apart from this splitting, we did not observe qualitativelynew effects and phenomena. However, if different spin states of the molecular bridge arecoupled [123, 124], or if spin-polarized leads [117–120] are used, the spin of the tunnel-ing electrons may have a strong influence on the transport characteristics of a molecularjunction. Future research may also address more elaborate electronic-vibrational couplingscenarios. As we have outlined in the motivation of the model Hamiltonian, for example,electronic-vibrational coupling originates not only from the dependence of the molecu-lar orbital energies on the vibrational coordinates but also from the electron-electroninteraction terms. Preliminary studies [194] show that these non-adiabatic interactionsresult effectively in population-dependent electronic-vibrational coupling constants. Thispopulation dependence may significantly alter the transport characteristics of molecularjunction, where transport is governed by more than one electronic state. As a next step,one can go beyond the linear expansion of the molecular orbital energies and electron-electron interaction terms with respect to the vibrational coordinates. The next term


in this expansion has already been considered, employing a master equation approach[391]. It comprises a shift of the vibrational frequencies, which may result, for example,in strong negative differential resistance. Other electronic-vibrational coupling scenariosmay also be investigated, such as, for example, Jahn-Teller coupling [393–395] and/oranharmonic effects [249,282,320,396–398]. Master equation approaches are most suitableto describe such interactions, but also projection-operator based nonequilibrium Green’sfunction methods may be used [280, 293–296]. Finally, it would be desirable to identifymolecular species that exhibit mode-selective vibrational excitation in a single-moleculecontact. This research may be guided by similar findings in electron-molecule scatter-ing [399, 400] and may pave the way for new applications in catalysis or mode-selectivechemistry [377,401].


Appendix A

Validity of the Elastic Co-TunnelingApproximation

In this appendix, we discuss the validity of the elastic co-tunneling approximation in thepresence of quantum interference effects. We have introduced and discussed this approxi-mation [80,285,287–289] in Sec. 3.3.5, where we derived a non-perturbative approximatescheme to describe electron-electron interactions in terms of the nonequilibrium Green’sfunction approach. Our results show (cf., e.g., Sec. 4.2.2) that this approximation gives asatisfactory description of electron-electron interactions with respect to the number andthe position of the relevant transport channels as well as the corresponding transmissionprobability. Moreover, it allows to treat multiple electronic states simultaneously and onthe same footing. This is important for the description of a single-molecule contact, whichtypically involves multiple electronic states (cf. Sec. 7).

In the presence of quantum interference effects, however, the mean-field character of theelastic co-tunneling approximation can lead to unphysical, spurious effects. This can bedemonstrated by comparing results that are obtained by the NEGF approach to exactresults that are obtained by the scheme of Gurvitz and Prager [323, 324]. To this end,we consider transport through the model molecular junction INTEL (cf. Tab. 6.3), em-ploying the wide-band approximation and an additional electron-electron interaction termU

12

= 0.15 eV. Due to the wide-band approximation, the electrical current flowing throughthe junction remains constant for bias voltages Φ & 2V, since all relevant transport chan-nels are already inside the bias window and the respective transmission probabilities areconstant with respect to the applied bias voltage Φ. This current as a function of thestate energy

2

is shown in Fig. A.1, where the solid purple line has been calculated usingthe NEGF scheme and the solid green line according to Ref. [324], that is according tothe formula

I(2

) = 4eΓ

L,11

ΓR,11

ΓL,11

+ ΓR,11

(1

− 2

)2

(1

− 2

)2 + 4ΓL,11

ΓR,11

. (A.1)

Note that this formula represents the exact expression for the current in the limit Φ→∞.

166 A. Validity of the Elastic Co-Tunneling Approximation

NEGF electronicexact electronic

0.2 0.4 0.60

1

2

3

energy !2 !eV"

currentI!"A"

Figure A.1: Current flowing through the model molecular junction INTEL at Φ = 2.0 Vas a function of the state energy

2

, where an additional electron-electron interaction termU

12

= 0.15 eV is also considered (cf. Tab. 6.3 for the other model parameters). Theside dips, which appear in the solid purple line at

2

= 1

± U12

, and the width of thedip at

2

= 1

indicate that the elastic co-tunneling approximation, due to its mean-fieldcharacter, leads to spurious effects in the presence of quantum interference effects.

If the two electronic states are not degenerate, the current through this junction is given by3.2µA. In contrast, for

1

= 2

, where the two electronic states are degenerate, the currentvanishes due to destructive quantum interference effects (as outlined in Sec. 6.5.1). Thisbehavior of the current is captured by both methods. The two approaches, however, givequalitatively different results, if the two electronic states are not exactly but quasidegen-erate, that is for 0 < |

1

− 2

| < ΓL,11

≈ 7meV. They also differ, if the anionic resonancesare quasidegenerate with respect to the dianionic resonances, that is if

2

≈ 1

± U12

.This can be understood as an artifact of the mean-field nature of the elastic co-tunnelingapproximation, where anionic and dianionic transport channels are treated as equivalentpathways for electrons tunneling through the junction. However, both channels differ bythe electronic population of the molecular bridge. Thus, the elastic co-tunneling approx-imation gives quantum interference effects which are unphysical. This results in the sidedips that appear in the solid purple line at

2

≈ 1

±U12

. Similarly, interference effects aresuppressed at

2

≈ 1

. This leads to a decrease of the electrical current due to destructivequantum interference that occurs in a smaller range of energies than it is obtained fromthe exact expression (A.1) [323,324].

This limits the validity of the elastic co-tunneling approximation in the presence of quan-tum interference effects. The approximation applies, for example, whenever the electron-electron interaction strengths are considerably smaller than the level-width functions,

A. Validity of the Elastic Co-Tunneling Approximation 167

Umn < ΓL/R,mn, but can also be used to describe systems with strong Coulomb interac-

tions, Umn > ΓL/R,mn, where, however, the corresponding anionic and dianionic resonances

must not be (quasi-)degenerate. Note that the findings presented in this appendix havebeen outlined first in the diploma thesis of M. Butzin [386].

168 A. Validity of the Elastic Co-Tunneling Approximation

Appendix B

The Structure of the VibrationalSelf-Energy

The numerical bottleneck of the NEGF scheme is the evaluation of the vibrational Green’sfunctions Dr

αα(). We will demonstrate this in the following by a discussion of the func-tional structure of the respective self-energies Πr

el,αα() and the physical processes asso-ciated with them. To this end, we consider the vibrational self-energy Πr

el,11

() (cf. Eq.(3.156)) for the model molecular junction E1V1 at a bias voltage of Φ = 0.3 V.

The real and the imaginary part of this self-energy are depicted in Fig. B.1 by the solidgreen and the solid black line, respectively. As can be seen, the imaginary part of Πr

el,11

()is rather small at energies || ≈ Ω

1

= 0.1 eV. At these energies, the non-interactingvibrational single-particle Green’s function,

D0,r11

() =1

− Ω1

+ i0+

− 1

+ Ω1

+ i0+

, (B.1)

has two poles, which via the Dyson equation (3.159) translate to two peaks in correspond-ing interacting Green’s function Dr

11

(). The numerical representation of these peaks,which are rather narrow due to the small value of Πr

el,11

(±Ω1

), is typically the most ex-pensive part of the NEGF scheme.

This is due to the specific structure of the vibrational self-energy

Πel,11

= −iλ2

11

Ω2

1

(Σ11

(t, t)Gel,11

(t, t) + Σ11

(t, t)Gel,11

(t, t)) .

It comprises a sum of products, where each involves two propagators: one for the propa-gation of an electron (or hole) on the molecular bridge, G

el,11

(t, t), and a second for thepropagation of an electron (or hole) in one of the leads, Σ

11

(t, t) =P

k∈L,R

|V1k|2gk(t, t).

As in each of these propagators the order of the times t and t is different, the respec-tive products describe the propagation of an electron-hole pair. This can be graphicallyrepresented by the Feynman diagram shown in Fig. B.2, where a vibrational quantum

170 B. The Structure of the Vibrational Self-Energy

Re!!el,11""#$Im !!el,11""#$

#1.5 #1. #0.5 0. 0.5 1. 1.5

#4

#2

0

2

energy " !eV$

self#energycontributions!meV$

Figure B.1: Real and imaginary part of the vibrational self-energy Πr

el,11

(). For thisspecific calculation we considered the model molecular junction E1V1 (cf. Tab. 4.1) and abias voltage of Φ = 0.3 V. The dashed vertical lines indicate the energies = ±|

1

−µL/R

|.If these lines are located within the energy range || < Ω

1

, the electron-hole pair creationprocesses, which are encoded in the self-energy Πr

el,11

() and depicted by the Feynmandiagram shown in Fig. B.2, can occur resonantly, while otherwise they cannot. This is thecase at the onset of the resonant transport regime but not in the non-resonant transportregime or at high bias voltages.

decays into an electron-hole pair that recombines at some later time. In the non-resonanttransport regime, in particular for Φ = 0.3V, or at high bias voltages, these processescan occur only virtually, and thus, give only a small decay rate for the vibrational degreeof freedom, or equivalently, very narrow peaks in the corresponding vibrational Green’sfunction Dr

11

(). Only if the chemical potential in one of the leads is close to the energyof the electronic state, that is for |µ

L/R

− 1

| . Ω1

, a vibrational quantum can resonantlydecay into an electron-hole pair. Accordingly, the broadening due to the vibrational self-energy Πr

el,11

() is significantly larger. This behavior is highlighted in Fig. B.1 by dashedvertical lines, which mark the energies = ±|

1

− µL/R

|. At these energies, the absolutevalue of the imaginary part of the vibrational self-energy changes drastically, indicatingthe cross-over from non-resonant to resonant electron-hole pair creation. Note that, incontrast to the vibrational Green’s functions, Dr

αα(), the electronic Green’s functions,Gr

el,mn(), typically exhibit much broader peaks, as the associated tunneling processes canalways occur resonantly (except for the edges of the conduction band).

While the representation of the narrow peaks in the vibrational Green’s functions Dr

αα()requires, on one hand, a sufficient numerical resolution, the convergence and stability ofour self-consistent solution scheme (see Sec. 3.3.7) can be greatly enhanced if the real

B. The Structure of the Vibrational Self-Energy 171

e-

h

Ω1 Ω1

Figure B.2: Example of an electron-hole pair cre-ation process as it is encoded in the expression of thevibrational self-energy, Eq. (3.156). The respectiveFeynman diagram is also referred to as polarizationbubble.

part of the vibrational self-energy Πr

el,αα() is neglected. This is particularly importantin the non-resonant transport regime, where one is not only interested in the current-voltage characteristic of a single-molecule contact but also in higher derivatives of thischaracteristic with respect to the applied bias voltage Φ. This behavior can also beunderstood by inspection of Fig. B.1. The real part of the vibrational self-energy leadsto a renormalization of the peak positions in the vibrational Green’s function. Thus, atevery step of our self-consistent cycle both the width and the position of the peaks inthe vibrational Green’s function change, while only one of these quantities changes if thereal part of Πr

el,αα() is neglected. If the renormalization of the vibrational frequenciesdue to the molecule-lead coupling is very small, for example, due to a weak coupling ofthe molecule to the leads, or for large vibrational frequencies Ωα, the neglect of these realparts represents an appealingly simple compromise to obtain numerically stable results.This is particularly important for the description of more than one vibrational degree offreedom, where numerical errors are more likely to occur due to the matrix inversionsinvolved in the Dyson equation (3.159).

172 B. The Structure of the Vibrational Self-Energy

Appendix C

The Role of Coherences in a MasterEquation Approach

In this section, we investigate the role of coherences in the description of a single-moleculecontact by the ME approach. Coherences are important, whenever the eigenstates |a|ν(a ∈ 0, 1 or a ∈ 00, 01, 10, 11) of the molecular bridge exhibit (quasi-)degeneracies.This is quite common in realistic systems, which involve multiple electronic states andnumerous vibrational modes (cf. Chap. 7). For a single electronic state and a singleharmonic mode, the molecular states are only quasidegenerate if the broadening of thelevels exceeds the level spacing set by the frequency Ω

1

. But in this regime, where Ω1

.Γ

L/R,mn, a perturbative expansion in Vmk may not be appropriate. Therefore, we startour discussion with a model system comprising two electronic states.

We expect coherences to play a major role for degenerate or quasidegenerate electronicstates, i.e. for |

1

−2

| < ΓL/R,mn [243]. As long as the couplings of the two quasidegenerate

electronic states to the leads are symmetric, we find the same results with and withoutcoherences. This can be understood by inspection of Eqs. (3.72) and (3.74). For symmetricjunctions with

1

≈ 2

and with U ≈ 0, where ρ00,00

≈ ρ11,11

holds, Eqs. (3.72) and (3.74)have almost the same structure. The only difference is the sign by which coherences enterthese equations. For that reason, the coherences ρ

1,2 = ρ∗2,1 cancel in these equations

and, as a result, do not influence the respective transport characteristics. Only for non-symmetric couplings to the leads we find a significant effect of coherences on the transportcharacteristics of this system. The most pronounced effect appears, if one of the molecule-lead couplings differs by sign, e.g., for ν

L,1/2

= νR,1 = −ν

R,2 and if the two states aredegenerate

2

= 1

. If no electronic-vibrational coupling is considered λ1α = λ

2α = 0, sucha system can be identically transformed to a system with two orthogonal states that arenot interacting with each other and that are coupled to one of the leads only (either leftor right). Hence, the current through this system is zero for any bias voltage (cf. Sec.6.5.1) [86,184]. However, if we disregard electronic coherences, we obtain a finite currentthat corresponds to the current of two states symmetrically coupled to the leads. Thus,electronic coherences must be accounted for to obtain physically correct results.

174 C. The Role of Coherences in a Master Equation Approach

(a)

!1 0 100.30.60.9n1 n2

w!o coherenceswith coherences

!1. !0.5 0 0.5 1.

!50

0

50

bias voltage " "V#

currentI"nA#

(b)

w!o coherenceswith coherences

!1. !0.5 0 0.5 1.0

0.5

1.

1.5

2.

bias voltage " "V#


Figure C.1: Current-voltage and vibrational excitation characteristics for a model sys-tem similar to the one of Fig. 6.5. Here, the energy of the second electronic state ischosen such that the νth vibrational level of the electronically excited state of the anionis degenerate with respect to the (ν + 1)th vibrational state of the anionic ground state:2

= 1

+Ω1

. The solid red line is obtained disregarding all coherences of the reduced den-sity matrix, while the solid blue line is obtained taking all coherences of ρ into account.The inset shows the respective population of the electronic levels, where the dashed linesrefer to the population of state 1 (n

1

) and the solid lines to the one of state 2 (n2

).

C. The Role of Coherences in a Master Equation Approach 175

The role of vibrational coherences can be studied employing a similar model system withtwo electronic states, which are coupled to a single vibrational mode and which energiesdiffer by the frequency of the mode, i.e.

2

= 1

+ Ω. Systems with 2

= 1

+ nΩ andn ≥ 2 display similar but less pronounced effects, as the impact of coherences decreasesthe further they are located from the diagonal of the density matrix [243]. Again, forsymmetric molecule-lead couplings νK,i, we do not observe a significant influence of coher-ences. Only for asymmetric transport scenarios we find coherences to play a significantrole for the transport characteristics. In Fig. C.1 we present the current-voltage and vi-brational excitation characteristics for the asymmetric model system SPEC (see Tab. 4.2)with the energy of the higher-lying state adjusted to

2

= 1

+ Ω = 0.25 eV. Thereby,the red line represents a calculation where all coherences are disregarded, while the blueline represents the results for a calculation where all coherences are taken into account.Only in the vicinity of eΦ = 2

1

to eΦ = 2(2

+ U) vibrational coherences influence thecurrent-voltage characteristic. For positive bias voltages, the first step in the blue lineis diminished, since coherences result in a small population of the second higher-lyingelectronic state (cf. the inset of Fig. C.1a), which is thus blocking transport throughthe lower-lying electronic state due to vibrationally induced repulsive electron-electroninteractions U = −2λ

11

λ21

/Ω. This blocking disappears for higher bias voltages, whenelectrons in the left lead have enough energy, i.e. more than

1

+U . Vibrational excitationis enhanced as well, because there is an additional resonant emission process for tunnelingfrom the higher-lying electronic state to the right lead. Similarly, coherences result in asomewhat larger current for bias voltages 2

2

< eΦ < 2(2

+ U), where the higher-lyingelectronic state enters the bias window. Again, the population of the second electronicstate is increased by vibrational coherences, although repulsive electron-electron interac-tions U block the population of, and similarly, transport through this state. Coherencessoften this blocking, resulting in a larger current and vibrational excitation.

We expect systems with more than one vibrational degree of freedom to display graduallymore quasidegenerate levels, and hence, coherences to play a gradually more importantrole. Moreover for anharmonic potentials that are describing e.g. molecular motors [249],coherences are crucial to characterize the actual motion of the molecule.

176 C. The Role of Coherences in a Master Equation Approach

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List of Publications

Part of this work has already been published.

First-Author Publications

1. R. Hartle, C. Benesch and M. ThossMultimode vibrational effects in single-molecule conductance: A nonequilibrium Green’sfunction approachPhys. Rev. B 77, 205314 (2008)

2. R. Hartle, C. Benesch and M. ThossVibrational Nonequilibrium Effects in the Conductance of Single Molecules with Mul-tiple Electronic StatesPhys. Rev. Lett. 102, 146801 (2009)

3. R. Hartle, R. Volkovich, M. Thoss and U. PeskinCommunication: Mode-selective vibrational excitation induced by nonequilibriumtransport processes in single-molecule junctionsJ. Chem. Phys. 133, 081102 (2010)

4. R. Hartle and M. Thoss

Resonant electron transport in single-molecule junctions: Vibrational excitation, rec-tification, negative differential resistance, and local cooling

Phys. Rev. B 83, 115414 (2011)

5. R. Hartle and M. Thoss

Vibrational instabilities in resonant electron transport through single-molecule junc-tions

Phys. Rev. B 83, 125419 (2011)

6. R. Hartle, M. Butzin, O. Rubio-Pons and M. Thoss

Quantum Interference and Decoherence in Single-Molecule Junctions: How Vibra-tions Induce Electrical Current

Phys. Rev. Lett. 107, 046802 (2011)

200 List of Publications

Co-Author Publications

1. C. Benesch, M.F. Rode, M. Cizek, R. Hartle, O. Rubio-Pons, M. Thoss and A. L. SobolewskiSwitching the Conductance of a Single Molecule by Photoinduced Hydrogen TransferJ. Phys. Chem. C 113, 10315 (2009)

2. O. Rubio Pons, R. Hartle, J. Li and M. Thoss

Theoretical Study of Electron Transfer and Electron Transport Processes in Molec-ular Systems at Metal Substrates

pp. 613-626 in ’High Performance Computing in Science and Engineering’ISBN 978-3-642-13871-3, Springer-Verlag Berlin Heidelberg (2010)

3. D. Secker, S. Wagner, S. Ballmann, R. Hartle, M. Thoss and H.B. Weber

Resonant Vibrations, Peak Broadening, and Noise in Single Molecule Contacts: TheNature of the First Conductance Peak

Phys. Rev. Lett. 106, 136807 (2011)

4. R. Volkovich, R. Hartle, M. Thoss and U. Peskin

Bias-Controlled Selective Excitation of Vibrational Modes in Molecular Junctions:A Route Towards Mode-Selective Chemistry

Phys. Chem. Chem. Phys. 13, 14333 (2011)

5. H. Wang, I. Pshenichnyuk, R. Hartle and M. Thoss

Numerically exact, time-dependent treatment of vibrationally coupled electron trans-port in single-molecule junctions

J. Chem. Phys. 135, 244506 (2011)

Curriculum Vitae

PERSONAL DATA

Name Rainer HartleAddress Heinrich-Hertz Str. 15

D-91058 Erlangen, GermanyTelephone +49-(0)9131-85-28816E-Mail [email protected] of birth 18.09.1980Nationality German

EDUCATION

Graduate studies – Ph.D.

Topic Vibrational Nonequilibrium Effects in the Conductanceof Single Molecules

Supervisor Prof. Dr. M. Thossgraduation in progresssince 2009 Universitat Erlangen-Nurnberg

Theoretical Solid State PhysicsChair: Prof. Dr. O. Pankratov

2007 – 2008 Technische Universitat MunchenTheoretical ChemistryChair: Prof. Dr. W. Domcke

Undergraduate studies – Diploma

Topic Antikaon Nuclear Bound States

Supervisor Prof. Dr. W. WeiseApplied Quantum Field Theory

Field of study Physics

Special subjects Quantum Field TheoryTheoretical Particle and Nuclear Physics

2001 – 2006 Technische Universitat Munchen

202 Curriculum Vitae

Acknowledgments

At this point, I would like to express my gratitude to all the people that have been directlyor indirectly contributing to this work. All of them have greatly enriched the content ofthis thesis as well as this period of my life.

First of all, I would like to thank Prof. M. Thoss for supervising my thesis. He provided avery calm and productive working atmosphere and kept me going by many interesting andinspiring discussions. He also offered me the chance to work on many different subjectsnot only related to research but also to teaching and what else is needed to be doneat a university (like the installation of an IT infrastructure). I also appreciate that heintegrated me in his research network by giving me the opportunity to visit various otherresearch groups and to attend many conferences. Apart from the great scientific exchange,he thus allowed me to explore many interesting spots of the world.

In that context, I would like to give special thanks to Prof. U. Peskin. He was hostingme more than once and very nicely at the Technion in Haifa and taught me a lot ofthings. It was a great pleasure to work with him and his PhD-students, especially withDr. R. Volkovich. I would also like to acknowledge productive discussions with Dr. M.Caspary-Toroker, D. Brisker and Dr. S. Klaiman at the Gregs or other nice places aroundthe Technion.

Prof. M. Cızek was hosting me at the Charles University in Prague, which I also enjoyedpretty very much. I would like to thank him for sharing his enthusiastic approach to solvechallenging physical problems and his former PhD-student Dr. I. Pshenichnyuk for hisgreat companionship in Prague.

Moreover, I would like to thank Prof. H. Wang for reminding me that a theorist shouldalways seek for exact solutions and for the great discussions we had in Telluride.

In Erlangen, I am appreciating the very fruitful discussions with Prof. H. Weber and hisgroup. In particular, I would like to thank Dr. S. Wagner, Dr. D. Secker and S. Ballmannfor confronting me with their experimental data, for explaining me the challenges theyface in their labs and for inviting me to countless cups of coffee. In addition, I wouldlike to thank Dr. B. Kubala not only for very interesting scientific discussions but also forsharing his other experiences in academic life.

I would also like to acknowledge the support of Prof. W. Domcke and Prof. O. Pankratovfor hosting me at their institutes. It has been a pleasure for me to be part of their groups.

204 Acknowledgments

During my PhD, I enjoyed co-supervising the theses of M. Butzin, S. Leitherer, C. Schin-abeck and A. Erpenbeck. The continuous discussions with these great and dedicatedstudents taught me a lot of things. Moreover, they greatly increased the scope of myresearch interests.

Furthermore, I would like to acknowledge the nice working atmosphere with my colleaguesDr. C. Benesch, Dr. D. Egorova, Dr. M. Gelin, Dr. V. Timmakondu, Dr. M. Bounouar,Dr. O. Vieuxmaire, Dr. I. Craig, Dr. J. Li, D. Opalka, M. Banck, A. Samsonyuk, M.Bauer and L. Zimmerman-Sharp in Munich, as well as with Dr. O. Rubio-Pons, Dr. P.Brana-Coto, PD Dr. M. Bockstedte, Dr. I. Pshenichnyuk and C. Hofmeister in Erlangen.Especially, I appreciate the help of C. Scheurer, G. Schwarz and M. Siegmund with mysystem administration duties. For their assistance with all the bureaucratic stuff, I wouldlike to thank R. Mosch and U. Graupner.

For correcting and proof-reading my thesis I would like to thank K. Turschmann, A.Erpenbeck, C. Schinabeck, M. Kolb and PD Dr. M. Bockstedte.

I would also like to give thanks to Dr. P. Raab and Dr. U. Wurstbauer for their compan-ionship and support.

Last but not least I would like to thank my parents. They have supported me in the bestway they could and encouraged me to follow my own path. This thesis would not existwithout them.

Eigenstandigkeitserklarung

Hiermit erklare ich, diese Arbeit selbstandig angefertigt und keine anderen als die zuge-lassenen Hilfsmittel verwendet zu haben.

Erlangen, 17. November 2011

Rainer Hartle

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vibrationally coupled electron transport through single-molecule junctions

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