OPE3: A model system for single-molecule transport

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OPE3:

A MODEL SYSTEM FOR

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OPE3:

A MODEL SYSTEM FOR

SINGLE

-

MOLECULE TRANSPORT

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op 22, Januari, 2016 om 10:00 uur

door

Riccardo F

RISENDA

natuurkundig doctorandus geboren te Casorate Primo, Italië.

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Dit proefschrift is goedgekeurd door de promotor: Prof. dr. Ir. Herre S. J. van der Zant

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. Ir. Herre S. J. van der Zant, Technische Universiteit Delft, promotor

Prof. dr. Juan Carlos Cuevas, Universidad Autonoma Madrid

Prof. dr. Mogens Brondsted Nielsen, Copenhagen University Prof. dr. Antonio Cassinese, Universita Federico II Napoli Prof. dr. Peter Steeneken, Technische Universiteit Delft Prof. dr. Jan M. van Ruitenbeek, University of Leiden

Prof. dr. Yaroslav Blanter, Technische Universiteit Delft

Keywords: Single-Molecule, Electrical Transport, Nanotechnology, Break-Junction, Molecule-Metal Interface

Printed by: Gildeprint - Enschede

Front & Back: Artistic depiction of one of the single-molecule junctions investi-gated in this thesis. Art and design by Riccardo Frisenda.

An electronic version of this dissertation is available at

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C

ONTENTS

1 Introduction 1 1.1 Single-molecule studies. . . 2 1.2 Molecular-scale electronics. . . 3 1.3 Dissertation outline. . . 5 References. . . 6 2 Theory 9 2.1 The molecular hamiltonian. . . 10

2.1.1 Electronic structure of molecules . . . 10

2.2 Molecule-metal interaction. . . 12

2.3 Transport through a single-molecule . . . 14

References. . . 18

3 Experimental 21 3.1 Mechanically controlled break junction. . . 22

3.2 Set-up. . . 23

3.2.1 Electronics. . . 24

3.3 Measurements . . . 25

3.3.1 Gold break junctions. . . 25

3.3.2 Conductance histograms . . . 27

3.4 Calibration of the attenuation factor . . . 27

References. . . 30

4 Room temperature transport in OPE3 33 4.1 Introduction . . . 34

4.2 Conductance histograms. . . 35

4.3 I-V characteristics of single-molecule junctions. . . 38

4.4 Ab-initio calculations. . . 40

4.5 Conclusions. . . 42

References. . . 43 vii

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viii CONTENTS

5 Conductance and mechanical stability: the influence of the anchoring

groups 45

5.1 Introduction . . . 46

5.2 Results and Discussion . . . 47

5.2.1 Conductance histograms . . . 47 5.2.2 Current-voltage characteristics . . . 50 5.2.3 Self-breaking. . . 55 5.3 Theoretical calculations. . . 57 5.4 Conclusions. . . 59 References. . . 60

6 Conjugation and interference: the role of the backbone 63 6.1 Quantum interference in a benzene ring . . . 64

6.1.1 Conductance of para and meta configurations. . . 65

6.1.2 Theoretical calculations . . . 66

6.2 Quantum interference in donor-acceptor compounds . . . 67

6.2.1 Conductance of D-A molecules . . . 69

6.2.2 Discussion. . . 72

References. . . 74

7 Strong to weak coupling transition in OPE3 77 7.1 Introduction . . . 78

7.2 Low-temperature I-Vs. . . 78

7.3 Discussion . . . 82

References. . . 86

8 Local environment: inelastic current and vibrations 91 8.1 Introduction . . . 92

8.2 Results . . . 93

8.2.1 Time dependence of IETS . . . 94

8.2.2 Distance dependence of IETS . . . 97

References. . . 102

9 Long-range interactions: off-bond transport 105 9.1 Introduction . . . 106

9.2 Conductance ofπ-stacked junctions . . . 107

9.3 Conductance fine structure. . . 111

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CONTENTS ix

A Appendix A 119

A.1 Molecular deposition. . . 119

References. . . 123 B Appendix B 125 B.1 Molecules investigated . . . 125 Summary 127 Samenvatting 131 Curriculum Vitæ 135 List of Publications 137 Acknowledgements 139

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1

I

NTRODUCTION

In this chapter we introduce the field of molecular-scale electronics. After a brief comparison between ensemble experiments and single-molecule experiments, we discuss the historical aspects of the field together with the contemporary start of the art.

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2 1.INTRODUCTION

1.1. S

INGLE

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MOLECULE STUDIES

Molecular-scale electronics aims at the study of the electrical properties of single molecules with a focus both on applications in electronics and at the fundamental aspects. This field can be considered part of the larger community of the single-molecule experiments (SME) that proposes to gain additional and new informa-tion about molecules by studying them one at a time, in comparison to more tradi-tional ensemble experiments (EE) where a large number of molecules is simultane-ously probed. Figure1.1shows an example of a SME compared to an EE where the energy spectrum of a certain molecular species is investigated. In EE the observ-ables are constituted by ensemble-averaged quantities and the statistical average introduces broadening of the spectral features, represented by the broad distribu-tion for the outcome of the EE in figure1.1. This typically hinders the possibility of investigating the behavior of individual molecules and makes very difficult the deconvolution of collective effects from local effects and intrinsic molecular ef-fects. On the contrary, SME with their ability of addressing individual molecules, provide different information about the molecular properties that are inaccessible in EE. Figure1.1shows the outcome of a SME performed on the same molecule at three different times. Small changes in the electrostatic environment or in the structure of the molecule between different experiments introduce variability in the outcome as shown by the sharp lines at different energies, in the plot in fig-ure1.1. Typically thousands of these separated measurements are performed and collected together. A data-analysis performed with statistical methods can give re-sults that are comparable to the ones used in EE. On the other hand, the amount of information contained in SME goes beyond the statistical behavior and can reveal details about processes happening at the atomic level. For example, one could

fol-Figure 1.1: In ensemble experiments (left) a large number of molecule is simultaneously measured and the result, in this example the energy of a certain electronic molecular level, is constituted by a broad distributions (center) originating from the large number of different configurations probed at the same time. In single-molecule experiments (right) one molecule is measured at a time, and the results from the different runs of the same experiment are usually compiled together in a histogram (middle).

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1.2.MOLECULAR-SCALE ELECTRONICS

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low the time dynamic of a molecule jumping between two stable binding sites on a surface extracting information about the molecule/surface interaction or the local topology of the surface. In this case the SME allows access to a time process that would be not accessible to a traditional EE.

Among the single-molecule techniques one can recognize three general ap-proaches. According to the detection scheme and the type of probe we identify: fluorescence based methods, force-based detection and electrical transport mea-surements. The first two methods are typically used to study large biological molecules such as DNA (l > 5nm), while the electric measurements are mostly used to ad-dress smaller molecules (l < 2nm), with the smallest example being an hydro-gen molecule (H2) [1]. Molecular-scale electronics deals with the study of single

molecules focusing especially on charge transport; this dissertation deals only with this aspect of SME. The focus is on the investigation of charge transport properties of single-molecule junctions, like the one depicted in figure1.2.

1.2. M

OLECULAR

-

SCALE ELECTRONICS

Molecular-scale electronics sees his birth in the experiments in the early 1970’s from Hans Kuhn on the tunneling of electrons through cadmium salt of fatty acid molecules sandwiched between two metallic electrodes. The first real proposal of a single-molecule electrical device came in 1974 from Aviram and Ratner [2] who, by elaborating on ideas on charge transfer complexes advanced by Mulliken and Foster in the 1950s and 60s [3], proposed a donor-σ-acceptor molecule that could work as a rectifier for the current when contacted by two electrodes. Given the hard experimental task of fabricating molecular-scale electrodes, the first di-rect electrical measurement of a single molecule arrived twenty years later in 1995 from Joachim and Gimzewsky [4]. The authors used a scanning tunneling

micro-Figure 1.2: Artistic depiction of a molecule bridging two electrodes in a single-molecule junction. Sim-ilar molecules are in the proximity but only bridging one electrode at a time.

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1

4 1.INTRODUCTION

scope (STM) in ultra high vacuum to contact a single C60molecule and measure

its current-voltage (I-V) characteristics. This experiment was followed, a few years later, by a study from Reed and Tour et al. that using mechanically controlled break junction (MCBJ) gold electrodes, contacted (1,4)benzene-dithiol molecules and measured reproducible I-Vs at room temperature with features interpreted as molecular features. The development of the break-junctions (BJ) techniques, me-chanically controlled, scanning probe microscopy (SPMBJ) and electromigrated (EMBJ), gave a big fuel to the field of transport through a single molecule and in the early 2000s important reviews summarized the early efforts [5,6].

Some internal critics to the field and to some of the claims of the first gen-eration of experiments arrived mainly due to the lack of a direct imaging of the molecular junction in the BJ technique, that makes hard in a single measurement the identification of the presence of a single or multiple molecules bridging the nanoelectrodes [7–10]. Moreover the eventual presence of contaminants on the electrodes or metallic grains could mimic a molecular electrical behavior. The first measurement with a BJ technique showing direct proofs of the formation of single-molecule junctions arrived in the year 2002 from Smit et al. [1]. The authors used platinum electrodes to contact a single hydrogen molecule. They monitored the vibrational features present in the current-voltage characteristics of the metal-molecule-metal junctions and observed an isotopic shift after replacing the hydro-gen with deuterium.

In the following years the research, in various active groups in the world, fo-cused on various aspects of single-molecule junctions [11–14]. On one side new measurement techniques have been introduced [15] and, apart from electrical sig-nals, additional signals can be nowaday probed at the single-molecule level, like Raman radiation [16,17], force profiles [18,19] or thermal transport [20,21]. An-other part of the research has focused on the exploration of molecular systems suitable for electronics measurements and eventually applications [22,23]. Re-garding this aspect, different works focused either on the binding of certain molec-ular families to the metallic electrodes or on the modification of the molecmolec-ular core [24]. Many different anchoring groups are known nowaday and are used to anchor molecules to metallic (especially gold) electrodes [25]. The molecular core has been also throrughly investigated and the most used chemical units in the core areπ-conjugated or saturated chains [26]. Typical studies focused on the depen-dence of the single-molecule junction transport characteristics, such as the low-bias conductance or the current-voltage characteristics, as a function of a chem-ically tailorable parameter, like the molecular length [27–29] or the torsion angle between two benzene rings [30].

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1.3.DISSERTATION OUTLINE

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1.3. D

ISSERTATION OUTLINE

In this dissertation, charge-transport through individual organic molecules is in-vestigated. The single molecules are contacted with two-terminal mechanically controllable break junction gold electrodes and their electrical and mechanical be-havior studied at room and low temperature.

After this introductory chapter, chapter2introduces the theoretical basis needed to understand charge-transport at the nanoscale and in particular in single molecules, while chapter3explains the experimental methods used throughout this thesis. The three subsequent chapters are dedicated to room-temperature measurements on oligo(phenylene ethynylene) (OPE3) molecules looking at the influence on the conductance and on the electrical and mechanical properties of the molecular core or backbone, chapter4, of the different anchoring groups for the electrodes, chapter5, and of quantum interference effects, chapter6. Chapter7present cryo-genic temperature measurements of OPE3, where we discuss a transition from strong coupling to weak coupling between molecule and electrodes as a conse-quence of the electrodes stretching. Chapters8describe the inelastic contribution to the transport through OPE3 and how information about the local environment of a single-molecule can be extracted from it. The final chapter9reports on a study of asymmetrically anchored OPE3 junction, where transport take place through molecular dimers held together by long-range forces.

Figure 1.3: Artistic depiction of a molecule bridging two electrodes in a single-molecule junction. Sim-ilar molecules are in the proximity of the molecular junction but they bridge only one electrode at a time and thus do not contribute to the electrical transport.

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1

6 REFERENCES

R

EFERENCES

[1] R. H. M. Smit, Y. Noat, C. Untiedt, N. D. Lang, M. C. van Hemert, and J. M. van Ruitenbeek,

Mea-surement of the conductance of a hydrogen molecule, Nature 419, 906 (2002).

[2] A. Aviram and M. A. Ratner, Molecular rectifiers, Chemical Physics Letters 29, 277 (1974).

[3] R. S. Mulliken, Overlap integrals and chemical binding1, Journal of the American Chemical Society

72, 4493 (1950).

[4] C. Joachim, J. K. Gimzewski, R. R. Schlittler, and C. Chavy, Electronic transparence of a single c60

molecule, Physical Review Letters 74, 2102 (1995).

[5] J. K. Gimzewski and C. Joachim, Nanoscale science of single molecules using local probes, Science

283, 1683 (1999).

[6] A. R. I. Aviram and M. Ratner, Preface, Annals of the New York Academy of Sciences 852, ix (1998).

[7] R. F. Service, Molecular electronics. next-generation technology hits an early midlife crisis, Science

302, 556 (2003).

[8] J. R. Heath, J. F. Stoddart, and R. S. Williams, More on molecular electronics, Science 303, 1136 (2004).

[9] A. H. Flood, J. F. Stoddart, D. W. Steuerman, and J. R. Heath, Chemistry. whence molecular

elec-tronics? Science 306, 2055 (2004).

[10] H. Choi and C. C. M. Mody, The long history of molecular electronics: Microelectronics origins of

nanotechnology, Social Studies of Science 39, 11 (2009).

[11] A. Nitzan and M. A. Ratner, Electron transport in molecular wire junctions, Science 300, 1384 (2003).

[12] J. R. Heath and M. A. Ratner, Molecular electronics, Physics Today (2003).

[13] C. Joachim and M. A. Ratner, Molecular electronics: Some views on transport junctions and beyond, Proceedings of the National Academy of Sciences of the United States of America 102, 8801 (2005).

[14] J. R. Heath, Molecular electronics, Annual Review of Materials Research 39, 1 (2009).

[15] S. V. Aradhya and L. Venkataraman, Single-molecule junctions beyond electronic transport, Nature Nanotechnology 8, 399 (2013).

[16] D. R. Ward, N. J. Halas, J. W. Ciszek, J. M. Tour, Y. Wu, P. Nordlander, and D. Natelson,

Electromi-grated nanoscale gaps for surface-enhanced raman spectroscopy, Nano Letters 8, 919 (2007).

[17] D. R. Ward, D. A. Corley, J. M. Tour, and D. Natelson, Optical rectification and field enhancement

in a plasmonic nanogap, Nature Nanotechnology 6, 33 (2010).

[18] B. Q. Xu, X. Y. Xiao, and N. J. Tao, Measurements of single-molecule electromechanical properties, Journal of the American Chemical Society 125, 16164 (2003).

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REFERENCES

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[19] M. Frei, S. V. Aradhya, M. Koentopp, M. S. Hybertsen, and L. Venkataraman, Mechanics and

chem-istry: single molecule bond rupture forces correlate with molecular backbone structure, Nano

Let-ters 11, 1518 (2011).

[20] P. Reddy, S.-Y. Jang, R. A. Segalman, and A. Majumdar, Thermoelectricity in molecular junctions, Science 315, 1568 (2007).

[21] J. Widawsky, P. Darancet, J. Neaton, and L. Venkataraman, Simultaneous determination of

con-ductance and thermopower of single molecule junctions, Nano Letters 12, 354 (2012).

[22] M. Ratner, A brief history of molecular electronics, Nature Nanotechnology 8, 378 (2013).

[23] E. Lortscher, Wiring molecules into circuits, Nature Nanotechnology 8, 381 (2013).

[24] A. Salomon, D. Cahen, S. Lindsay, J. Tomfohr, V. B. Engelkes, and C. D. Frisbie, Comparison of

electronic transport measurements on organic molecules, Advanced Materials 15, 1881 (2003).

[25] E. Leary, A. La Rosa, M. T. Gonzalez, G. Rubio-Bollinger, N. Agrait, and N. Martin, Incorporating

single molecules into electrical circuits. the role of the chemical anchoring group, Chemical Society

Reviews 44, 920 (2015).

[26] M. L. Perrin, E. Burzuri, and H. S. J. van der Zant, Single-molecule transistors, Chemical Society Reviews 44, 902 (2015).

[27] X. L. Li, J. He, J. Hihath, B. Q. Xu, S. M. Lindsay, and N. J. Tao, Conductance of single alkanedithiols:

Conduction mechanism and effect of molecule-electrode contacts, Journal of the American

Chem-ical Society 128, 2135 (2006).

[28] G. Sedghi, K. Sawada, L. J. Esdaile, M. Hoffmann, H. L. Anderson, D. Bethell, W. Haiss, S. J. Hig-gins, and R. J. Nichols, Single molecule conductance of porphyrin wires with ultralow attenuation, Journal of the American Chemical Society 130, 8582 (2008).

[29] B. Kim, S. H. Choi, X. Y. Zhu, and C. D. Frisbie, Molecular tunnel junctions based on pi-conjugated

oligoacene thiols and dithiols between ag, au, and pt contacts: effect of surface linking group and metal work function, Journal of the American Chemical Society 133, 19864 (2011).

[30] L. Venkataraman, J. E. Klare, C. Nuckolls, M. S. Hybertsen, and M. L. Steigerwald, Dependence of

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T

HEORY

The question "how is the charge transported in a single-molecule junction?" is a dif-ficult one and the answer requires both a chemical and a physical understanding of the problem. Different theoretical methods have been developed in the years, since the first calculation of a current-rectifying molecule from Aviram and Rat-ner [1] more than 40 years ago. In respect to those times, the accuracy of density functional theory (DFT) calculations of the electronic structure of molecules has im-proved enormously especially thanks to the introduction of new functionals that go beyond the local density approximation (LDA). Quantum chemistry calculations of single molecule-junctions are nowadays mostly done with DFT and the transport properties are calculated with the help of non equilibrium Green’s function (NEGF) and the Landauer formalism. In the following chapter we introduce the theoreti-cal basis to understand the phenomenon of charge-transport through an individual molecule.

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10 2.THEORY

2.1. T

HE MOLECULAR HAMILTONIAN

An isolated molecule, "in gas-phase", is essentially a collection of two or more atoms bound together by chemical bonds between them. The general non-relativistic Hamiltonian of a molecule containing n electrons and N nuclei can be written as:

nuclei. The first and second terms of the equation depend only on the nuclear coordinates and correspond to the kinetic energy of all the nuclei and to the mu-tual electrostatic repulsion between the nuclei. The third and fourth terms are the kinetic energy of all the electrons and the Coulombic repulsion between the elec-trons. The last term is the attractive interaction between electrons and nuclei. The static Schrödinger equation [2] is:

H_Ψ(R1...RN, r1...rn) = EΨ(R1...RN, r1...rn). (2.2)

By solving the previous equation one could find the total wavefunction of the molecule Ψ(R1...RN, r1...rn) and from that calculate all the properties of interest, like the

ionization energy or the dipole moment. Unfortunately, for a number of elec-trons larger than one, an analytical solution of Eq.2.2does not exist, because the electron-electron repulsive term, the fourth in the equation, prevents the separa-tion of the variables and gives origin to the electronic correlasepara-tion. Different ap-proaches have been used to approximate the many-body nature of the problem of electronic motion in molecules and to determine their electronic structures.

2.1.1. E

LECTRONIC STRUCTURE OF MOLECULES

We can identify two classes of methods that aim at solving Eq.2.2widely used in the field of single-molecule transport and in many other fields of physics and chemistry. The first class is based on the molecular orbital (MO) theory [3–5] and its most known implementation, the Hartree-Fock theory [6–8] (HF) and the sec-ond class contains the density functional theory [9–11] (DFT) formulated by Ho-nenberg, Kohn and Sham [12,13]. Briefly, the HF method is based on a description of the molecule as a set of molecular orbitals delocalized over the entire molecule and it neglects entirely the correlation part of the problem, but, through the use of Slater determinants [14], it treats exactly the exchange part by producing an an-tisymmetric wavefunction, as required by the fermionic nature of the electrons.

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2.1.THE MOLECULAR HAMILTONIAN

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On the contrary, DFT describes the many-electron molecular system on the basis of the electron density and it approximates both the correlation and the exchange part, incorporating their effects into effective potentials. Efforts to go beyond the mean-field approximation of HF and DFT and treat more explicitly exchange and correlation effects culminated in the years in the development on one side of post-HF methods, like configuration interaction (CI) [15], and from the other side better functionals for DFT like the hybrid functional B3LYP [16] or dispersion corrected functionals [17]. Finally, it is worth to mention the many-body methods based on the Green’s function, with the GW approximation being popular in the recent liter-ature [18,19].

In the theories just mentioned, we can reasonably describe electrons in molecules as organized in discrete single-particle levels called molecular orbitals,φi(ri), which

possess a well defined energy and shape that can be calculated, at different lev-els of approximation and accuracy. Figure2.1b shows the energy configuration of the valence molecular orbitals, based on DFT calculations at the B3LYP//TZ2P level, of three organic molecules depicted in figure2.1a that are of importance for molecular electronics. These three molecules areπ-conjugated systems, where the overlap between p-orbitals of adjacent carbon atoms, with alternating single and double or triple bonds, promotes the delocalization of the electrons contained in

Figure 2.1: a) Chemical schematic of three organic molecules of importance for single-molecule elec-tronics. b) Electronic structure of these molecules in gas phase calculated from DFT B3LYP//TZ2P and iso-surfaces of a few selected orbitals.

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12 2.THEORY

those shells. The first molecule on the left, benzene, contains only carbon and hydrogen atoms while the two on the right, BDT and OPE3 dithiol, contain also sulfur atoms as thiol groups, often used as binding groups for gold electrodes. The sequence of occupation of the molecular orbitals, depicted in Fig.2.1b, is dictated by the aufbau principle which states that the energy levels are filled sequentially with the lowest energy levels filled first. From the figure we see that the highest occupied molecular orbital (HOMO) in each molecule is separated from the low-est unoccupied molecular orbital (LUMO) by an energy gap, called HOMO-LUMO gap. The size of the energy gap gets smaller when going from benzene to the larger OPE3 dithiol because of the less confinement felt by the valence electrons [20].

In Fig.2.1b we plot the iso-surface representation of a few orbitals, withπ sym-metry, for each molecule. Generally inπ-conjugated systems the HOMOs and the LUMOs have a_{π character with contributions from the p}zatomic orbitals (2pzin

the case of carbon atoms). Importantly, for charge transport and molecular elec-tronics the π-orbitals are highly delocalized and can couple well with the elec-trodes, via the overlap of the MOs and the electrodes wave-functions, meaning that they can transport charge more efficiently than more localized orbitals. Addi-tionally, according to the symmetry of the MOs in respect to inversion operations we identify g (gerade) and u (ungerade) orbitals. This is also important for charge transport since the g or u character of two MOs determines whether they interfere constructively or destructively when participating to a transport channel [20].

2.2. M

OLECULE

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METAL INTERACTION

Many aspects of the interaction between a molecule and a metal surface can be understood from the molecular orbitals point of view. We consider the electrons in the metal as an ideal Fermi electron gas characterized by the Fermi-Dirac distri-bution:

f (E ) = 1 1 + exp³E −µ_k_B_T´ ,

(2.3) whereµ is the chemical potential and T is the temperature. The electrons in the metal fill up the available quasi-continuum set of levels up to an energy EF (for

T = 0 K). Figure2.2sketches the different effects happening to the molecular fron-tier orbitals when a molecule approaches a metal starting from gas phase. With the molecule in gas phase, the MOs appear at discrete energies and the density of states of the molecule can be seen as a series of narrow peaks separated in energy. When the molecule starts to approach a metal surface, the first interactions that play a role are due to long-range forces, like van der Waal’s forces. In this regime of physisorption the wave-function of the molecule and the one of the metal do not overlap and the charge transfer between molecule and metal is negligible. The

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2.2.MOLECULE-METAL INTERACTION

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energy of the MOs is renormalized and the HOMO-LUMO gap energy decreases as the interaction between molecule and metal increases (when the molecule gets closer to the surface). The increased screening from the electrons in the metal and the formation of image charges in the electrodes are the main reasons for the reduction of the HOMO-LUMO gap. When the molecule is close enough to the surface, the wave-function of the metal overlaps with the molecule wave-function and chemical bonds between molecule and metal can form. An important con-sequence of the hybridization of the wave-functions is the substantial Lorentzian broadening introduced in the MOs density of states. The electrons life-time in the MOs is reduced because of the possibility for the electrons of escaping in the metal, thus introducing a life-time broadening.

Another important effect of the formation of a bond between molecule and metal is the alignment of the HOMO or of the LUMO toward the metal Fermi en-ergy of the metal, with all the MOs shifting rigidly to higher or to lower energies re-spectively. In a very simplified picture, in order to form a bond two atoms have to share a pair of electrons whose charge, in the case of equal atoms, get distributed equally between the two nuclei. Differences in the electronegativity of the two atomic species can polarize the bond, shifting spatially the charge distribution and promoting the formation of a permanent dipole. Moreover charge can be trans-ferred from metal to molecule or from molecule to metal leaving the electrostatic

Figure 2.2: Schematic drawing of the frontier molecular orbitals in a molecule in respect to the Fermi energy (EF) of the metallic electrodes. The isolated molecule (left) interacts with a metal surface first

only via long-range interactions (middle) and then also with stronger chemical bonds (right). Notice the reduction of the energy gap between the two levels due to screening from the electrodes and the rigid shift of both levels to higher energy due to the formation of a surface dipole and charge transfer.

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14 2.THEORY

environment inside the molecule modified, thus shifting rigidly in energy all the molecular levels.

These effects shift the MOs upward or downward in energy, thus promoting the alignment of the HOMO or of the LUMO to the metal EF, with the sign of the

shift being determined by the nature of the charge transfer and the direction of the electric dipoles. Thiol groups (SH), for example, typically shift the levels upwards in energy when binding to gold, as a consequence of charge transfer from the elec-trodes to the molecule, while pyridine groups (Pyr), on the contrary, shift the levels to lower energies and the charge transfer happens in the opposite direction. Most of the binding groups used in molecular electronics form a covalent bond with one (or more than one) gold atom with bond strengths of the order of 0.5 − 2.0 eV.

In the next section we will discuss the situation in which two metallic elec-trodes bridged by a single molecule are driven out-of-equilibrium in respect to each other, by changing their chemical potentials, allowing charge to flow through the molecular bridge.

2.3. T

RANSPORT THROUGH A SINGLE

-

MOLECULE

In describing electric transport in molecules and in general at the nanoscale, macro-scopic formulas like the Ohm’s law lose validity and quantum effects must be con-sidered, like for example the particle-wave dual nature of the electrons transport-ing charge. There are a few important length scales to consider when model-ing transport through an object: the physical dimensions of the system Li (i =

x, y, z), the mean free path of the electrons le, that is the average distance that an

electron travels between two elastic scattering events, the phase coherence length l_φ, the average distance that an electron travels between two phase-randomizing events, and finally the Fermi wavelength of the electronsλF. The classical limit

of a macroscopic object is Li À le, lφ,λF. In the case of a single-molecule

junc-tion formed with an organic molecule and gold nano-electrodes, schematically de-picted in Fig.2.3, the dimensions are Ly, Lz∼ 0.5 nm, Lmol ∼ 2 nm , typical values

for the mean free paths [21,22] are le> 100 nm and lφ> 100 nm and the Fermi wavelength of gold isλF∼ 0.5 nm. This regime, with le, lφÀ Li∼ λF, is the

quan-tum ballistic regime meaning that the charge-carriers suffer few or no scattering events as they traverse the short molecular bridge. In this regime the conductance comes from the transmission of the system, its upper limit is the quantum of con-ductance, 1 G0=2e

2

h ∼ 77.5 µS, in the case of perfect transmission (T = 1) through

a single quantum transport channel [23].

The scattering approach treats the molecule as a scatterer connected to two metallic leads [24,25], that act as electron reservoirs, as depicted in figure2.3b. The left and right electron reservoirs are characterized by the chemical potentials,µL

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2.3.TRANSPORT THROUGH A SINGLE-MOLECULE

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Figure 2.3: a) Schematic depiction of a single-molecule junction. b) Landauer-Büttiker description of the nanojunction as a scatterer for states propagating to and from the left and right electrodes, which are treated as ideal electron reservoirs. c) Representation of the molecular junction as a set of chemical potential (µ) for addition and removal of charge in between two Fermi electron distributions. ΓLand

ΓRrepresent the coupling of the molecule to the left and right electrode, N represents the number of

electron on the neutral molecule.

andµR, which at T ∼ 0 K coincide with the respective Fermi energies. In order for

charge to flow, the two chemical potentialsµLandµRmust be different from each

other, so the system can be driven out of equilibrium and charge can flow from the higher to the lower chemical potential. This is typically done by applying a voltage difference between the left and right electrode.

When considering the quantum mechanical problem of the states reflected and transmitted by a central scatterer, in the elastic scattering limit, toward the right or left electrode, one arrives to the Landauer formula [26,27] for the conduc-tance of charge carriers with an energy E :

G(E ) = G0

X

n

Tn(E ), (2.4)

where Tn(E ) is the probability that the channel n of the scatterer transmits, from

one electrode to the other, an electron with an energy E . In the Landauer-Büttiker formalism the molecular junction is described by an energy-dependent transmis-sion T (E ) that depends on the molecular density of states D(E ) and typically con-sists in a series of peaks separated in energy, more or less broadened by the interac-tion with the metal electrodes [28]. In the case of weak molecule-metal interaction (coupling), Fig.2.3c shows a typical diagram of the chemical potentials of a molec-ular junctions with the two electrodes described by their Fermi distributions. The different levels sketched correspond to the chemical potentials to add or remove electrons from the molecule and essentially the MOs become conduction chan-nels that can transmit electrons from one electrode to the other. The potential µN →N +1is equal to the energy needed to add an electron to the molecule in the

junction, starting from the neutral state (for N electrons). This process is the anal-ogous of the electron affinity (E A) for a molecule in gas phase, that quantifies the

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2

16 2.THEORY

energy needed to add an electron to an isolated molecule. The levelµ_{N −1→N} in-stead is related to the addition of an electron from the first ionized state (N −1 elec-trons) to the neutral state, analogous to the ionization potential (I P ) in an isolated molecule. It is important to know that the I P (and the E A) is intimately linked by the Koopmans’ theorem [29,30] to the energy of the HOMO (and the LUMO) cal-culated in the Hartree-Fock approximation (an extension of the theorem to DFT exists as well).

In the situation depicted in figure2.3c, there is no difference betweenµLand

µR, since no voltage is applied, and both electrodes keep the number of the

elec-trons in the molecule fixed at N . To discuss the transport of charge through the molecular bridge, when a bias voltage difference is applied between the electrodes, we need to define the electronic couplingΓ between molecule and electrodes. This quantity depends on the molecule/electrode interactions and is equal to the sum of the coupling of the molecule respectively to the left and to the right electrode Γ = ΓL+ ΓR. The coupling to the left (right) electrode is proportional to the

over-lap between the molecular wavefunction and the wavefunction of the left (right) electrode. It’s important to stress that in a MO picture, the wavefunctions of differ-ent molecular orbitals can have very differdiffer-ent values of overlap with the electrodes wavefunctions according, for example, to their degree of (de)localization and their symmetry. The total couplingΓ determines the amount of current that can pass in a molecular junction. In addition to the coupling strength,Γ gives also an extimate for the traversal time of the electrons, that is the time that an electron spends in the molecular bridge given byτel= ħ/Γ. If τelis smaller than the electron-electron

in-teraction time, approximated byτe−e= ħ/U where U is the charging energy due

to the electrostatic repulsion between electrons, then the transport is mainly co-herent. In the following we consider this regime whereΓ >> U and U ≈ 0, called strong coupling regime. Chapter7deals with a molecular junction that shows a transition to a weak coupling configuration, characterized by_{Γ << U.}

Considering only a single Lorentzian broadened level, to describe a molecule in a junction, gives a simple expression for the elastic coherent current. Figure2.4a shows the energy diagram of such a model, where the occupied molecular level is represented by a Lorentzian peak with a density of states:

D(E ) = 1 π

Γ (E − EF)2+ (Γ/2)2

(2.5) The center of the peak lies below the Fermi energy of the electrodes, separated from EFby an energy²0(that quantifies the level alignment or injection barrier).

The half-width at half-maximum is given by the total couplingΓ and the level is equally coupled to the left and to the right electrodes [31],ΓL= ΓR= Γ/2. In the

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2.3.TRANSPORT THROUGH A SINGLE-MOLECULE

2

17

Figure 2.4: a) Schematic representation of a molecular junction consisting of a single level (HOMO) coupled equallly to the left and right electrode,ΓL= ΓR, and Lorentzian broadened with half-width

at half-maximumΓ. The Lorentzian peak is represented in logarithmic scale, for clarity, and the blue shaded area below the peak correspond to the occupation of the peak. When no bias voltage is applied (a), no current can flow. For a small voltage difference between left and right electrode some current can flow due to the non-equilibrium situation (b). The shaded red area under the peak feels a different occupation from the two electrodes, it is filled by the left electrode and emptied by the right electrode. When the bias is large enough to reach the peak (d) a large current flows through the system. For a large negative bias (e) the situation is symmetrical with a large negative current flowing. The transport happens through the same Lorentzian level level as in (d). c) Calculated single-level model current-voltage characteristic, in the low-bias region. The black line is a linear fit of the current done around zero bias. f ) Same I-V as in (c) shown in a larger bias voltage range, the inset is a zoom in the positive step region. by: I (V ) =G0 e Z_+∞ −∞ T (E )£ fL(E ) − fR(E )¤ dE . (2.6)

From the equation above we see that in order for charge to flow through the molecule, there must be a difference in the chemical potentialsµLandµRof the two

elec-trodes. The effect of the bias voltage V (applied symmetrically) is to shift in re-spect to each other the chemical potentials of the two electrodes by a factor eV (Fig.2.4b) according to:

µL= EF+ eV /2,

µR= EF− eV /2.

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2

18 REFERENCES

The transmission of a single level can be expressed from the density of state as T (E ) = 2πD(E)ΓL+ΓR

Γ . By substituting T (E ) in equation2.6and solving the integral

at zero temperature, we find the analytical expression for the current as a function of the voltage through a single molecular level:

I (V ) =G0 e Γ " arctan Ã ²0+eV₂ Γ ! − arctan Ã ²0−eV₂ Γ !# . (2.8)

Figure2.4a schematically shows the energy diagram of a single Lorentzian-broadened HOMO level, at an energy²0from EFand coupled symmetrically to the electrodes

with total strengthΓ. At zero bias voltage the level is in equilibrium with the elec-trodes and the blue shaded area under the peak corresponds to the (zero-bias) occupation of the level and takes a value close to 2 electrons. When a small posi-tive bias voltage is applied between the electrodes (Fig.2.4b), in the limit eV << ²0,

some charge flows due to the difference in occupation in the molecule induced by the left and the right electrodes. Nevertheless, the bias is not enough to include the peak in the bias window and conduction takes place only through the tail of the Lorentzian peak, in the regime called off-resonance. In this low-bias regime the I-V is mostly linear until a bias of ±0.5 V, as shown in figure2.4c, with some non-linearities in the current start to appear for a larger bias voltages [32–34].

When the bias is large enough (Fig.2.4d) for one of the electrodes Fermi energy to align with the center of the Lorentzian peak, a large current can flow through the system and strong non-linearities appear in the current. The opposite situation of a large negative bias (Fig.2.4e) is symmetrical to the positive bias one and a large negative current can flow through the same level. In the I-V of Fig.2.4f we see a step in the current both at negative and positive voltages centered at a bias volt-age eV ' ±²0. The current after the step, eV >> ²0, saturates to a value dependent

only onΓ and independent on ²0. In this model the position of the step in current

gives information about the level alignment while the width of the step about the life-time of the level. Among the implications of the model two are especially im-portant when using it to interpret real experiments: its mean-field character, that breaks down in presence of large correlation between electrons, and the absence of multi-level quantum interference effects.

R

EFERENCES

[1] A. Aviram and M. A. Ratner, Molecular rectifiers, Chemical Physics Letters 29, 277 (1974).

[2] E. Schrodinger, An undulatory theory of the mechanics of atoms and molecules, Physical Review

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[3] R. S. Mulliken, The assignment of quantum numbers for electrons in molecules. i, Physical Review

32, 186 (1928).

[4] R. S. Mulliken, The assignment of quantum numbers for electrons in molecules. ii. correlation of

molecular and atomic electron states, Physical Review 32, 761 (1928).

[5] R. S. Mulliken, Overlap integrals and chemical binding1, Journal of the American Chemical Society

72, 4493 (1950).

[6] D. R. Hartree and W. Hartree, Self-consistent field, with exchange, for beryllium, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 150, 9 (1935).

[7] C. C. J. Roothaan, New developments in molecular orbital theory, Reviews of Modern Physics 23, 69 (1951).

[8] J. C. Slater, The electronic structure of atoms: The hartree-fock method and correlation, Reviews of Modern Physics 35, 484 (1963).

[9] M. Schluter and L. J. Sham, Density functional theory, Physics Today 35, 36 (1982).

[10] K. Burke, Perspective on density functional theory, Journal of Chemical Physics 136, 150901 (2012).

[11] A. Zangwill, A half century of density functional theory, Physics Today 68, 34 (2015).

[12] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Physical Review 136, B864 (1964).

[13] W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys-ical Review 140, A1133 (1965).

[14] J. C. Slater, A simplification of the hartree-fock method, Physical Review 81, 385 (1951).

[15] P. Delaney and J. C. Greer, Correlated electron transport in molecular electronics, Physical Review Letters 93, 036805 (2004).

[16] P. Stephens, F. Devlin, C. Chabalowski, and M. J. Frisch, Ab initio calculation of vibrational

ab-sorption and circular dichroism spectra using density functional force fields, Journal of Physical

Chemistry 98, 11623 (1994).

[17] S. Grimme, S. Ehrlich, and L. Goerigk, Effect of the damping function in dispersion corrected

den-sity functional theory, Journal of Computational Chemistry 32, 1456 (2011).

[18] K. S. Thygesen and A. Rubio, Nonequilibrium gw approach to quantum transport in nano-scale

contacts, Journal of Chemical Physics 126, 091101 (2007).

[19] K. S. Thygesen and A. Rubio, Conserving gw scheme for nonequilibrium quantum transport in

molecular contacts, Physical Review B 77, 115333 (2008).

[20] J. C. Cuevas and E. Scheer, Organic Chemistry; Structure and Function (Palgrave macmillan, 2014) Chap. 14.

[21] G. Bergmann, Weak localization in thin films: a time-of-flight experiment with conduction

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[22] G. Dumpich and A. Carl, Anomalous temperature dependence of the phase-coherence length for

inhomogeneous gold films, Physical Review B 43, 12074 (1991).

[23] J. C. Cuevas, J. Heurich, F. Pauly, W. Wenzel, and S. Gerd, Theoretical description of the electrical

conduction in atomic and molecular junctions, Nanotechnology 14, R29 (2003).

[24] Y. Xue, S. Datta, and M. A. Ratner, First-principles based matrix green’s function approach to

molec-ular electronic devices: general formalism, Chemical Physics 281, 151 (2002).

[25] D. Supriyo, Electrical resistance: an atomistic view, Nanotechnology 15, S433 (2004).

[26] R. Landauer, Spatial variation of currents and fields due to localized scatterers in metallic

conduc-tion, IBM Journal of Research and Development 1, 223 (1957).

[27] Y. Imry and R. Landauer, Conductance viewed as transmission, Review of Modern Physics 71, S306 (1999).

[28] J. C. Cuevas and E. Scheer, Molecular electronics: an introduction to theory and experiment, Vol. 1 (World Scientific, 2010).

[29] T. Koopmans, Uber die zuordnung von wellenfunktionen und eigenwerten zu den einzelnen

elek-tronen eines atoms, Physica 1, 104 (1934).

[30] W. G. Richards, The use of koopmans’ theorem in the interpretation of photoelectron spectra, Inter-national Journal of Mass Spectrometry and Ion Physics 2, 419 (1969).

[31] G. Breit and E. Wigner, Capture of slow neutrons, Physical Review 49, 519 (1936).

[32] I. Baldea, Counterintuitive issues in the charge transport through molecular junctions, Physical Chemistry Chemical Physics 17, 31260 (2015).

[33] I. Baldea, Important issues facing model-based approaches to tunneling transport in molecular

junctions, Physical Chemistry Chemical Physics 17, 20217 (2015).

[34] Z. Xie, I. Baldea, C. E. Smith, Y. Wu, and C. D. Frisbie, Experimental and theoretical analysis of

nanotransport in oligophenylene dithiol junctions as a function of molecular length and contact work function, ACS Nano 9, 8022 (2015).

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3

E

XPERIMENTAL

In this chapter we present the mechanically controllable break junction technique used, in the rest of this thesis, to fabricate the nano-electrodes used to contact single molecules. We first introduce the principle of the technique together with the real implementation in our lab, consisting in the measurements set-up, the sample used and the measurement methods.

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3

22 3.EXPERIMENTAL

3.1. M

ECHANICALLY CONTROLLED BREAK JUNCTION

All the single-molecule experiments presented in this dissertation have been per-formed with the mechanically controlled break junction (MCBJ) technique. This technique, that allows for fast and controlled formation of molecular junctions, finds its origin in the point-contact studies from Yanson and coworkers [1–3] in the 1970s and 80s and in the studies of Moreland and Hansma [4] and Moreland and Ekin [5] who introduced the term ’break’ junction in 1985. Muller and coworkers in the early 1990s developed the MCBJ technique used nowadays in many labora-tories around the world [6,7].

The nano-fabrication of the MCBJ samples has been extensively described in the PhD dissertations of C. Martin [8] and of M. Perrin [9], in the following we describe it only briefly. The MCBJ sample is fabricated with electron-beam lithog-raphy by defining a gold wire on top of a flexible phosphorous bronze substrate coated with an insulating layer of fewµm of polyimide, schematically depicted in figure3.1. The center of the wire consists of a 40 nm wide constriction that is sus-pended after etching the polyimide with an O2-CF4plasma, as shown in the

scan-ning electron microscopy (SEM) pictures in Fig.3.2. The sample is then mounted in the MCBJ setup.

The flexible substrate is clamped between two lateral supports in a three-point bending mechanism and the head of a central pushing rod is connected to a can-tilever that can be driven either by a stepper motor or by a piezoelectric element. Upon bending of the substrate, the gold wire initially gets stretched after which rupture occurs and two nano-sized electrodes are formed whose separation can be adjusted mechanically. Due to the geometrical design, the ratio between the vertical displacement of the pushing rod and the change in electrode separation

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3.2.SET-UP

3

23

Figure 3.2: Colorized scanning electron microscopy images of the four parallel wires of a MCBJ sample and of the central constriction of one of the wires.

is 5 · 10−5_{, which results in a sub-picometer control over the electrodes separation.}

The two broken extremities can be fused again to form the wire by unbending the substrate and approaching the electrodes. This process can be repeated thousands of time without a noticeable aging of the wire making the technique suited for sta-tistical studies. Moreover, the electrodes have an high stability both in respect to mechanical noise and to large electric-fields with accessible voltages up to 3 V.

3.2. S

ET

-

UP

In the following we describe our own (TU Delft) realization of the MCBJ technique. For a detailed description and characterization of the set-up see the two articles by Martin et al. [10,11]. The set-up is based on a dipstick carrying the three-point bending mechanism at the bottom. The bending of the sample can be controlled either by a brushless servo motor (Faulhaber) or by a piezoelectric element (Physik Instrumente - PI). The stepper motor is slow, but has a high-dynamic range and is used for the initial breaking of the wire; the piezoelectric element, on the other hand, is fast and is used during the conductance measurement of molecular junc-tions as then only a limited displacement range is required. Due to a strong ther-mal dependence of its properties, the piezoelectric element is only used at room-temperature.

The servo motor is connected first to a gearbox with attenuation 246:1 and finally to a differential screw with a pitch of 150µm/turn. The rotation of the

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3

24 3.EXPERIMENTAL

motor is thus transferred to the vertical displacement of the three-point bending mechanism elements. The smallest displacement that can be achieved in the ver-tical direction is approximately 0.1µm which translates in a displacement of the nano-electrodes of 5 pm in the horizontal direction. The largest speed that can be achieved is 5µm/s in the vertical direction (0.25 nm/s for the electrodes) and the total range is 3 mm (200 nm). The piezoelectric element is connected to a can-tilever connected to the three-point bending mechanism that amplifies the dis-placement in the vertical direction of a factor 4.6. In the plane of the electrodes, the lowest step that can be achieved when using the piezoelectric element is 0.15 pm while the largest speed is approximately 40 nm/s and the total range is 20 nm. The set-up can be closed with a conical vacuum-tight seal and evacuated to high vac-uum p ≈ 10−6mbar. Once closed, the vacuum seal act also as a shield to protect the the signal measured from electromagnetic interference coming from outside.

3.2.1. E

LECTRONICS

The measurement electronics (developed by Raymond Schouten, TU Delft) is hosted in a shielded rack fed by two batteries. The different modules can be connected to the rack and accessed via optically coupled isolation amplifiers. The electrical isolation of the measurement electronics is ideal to perform low-noise studies of small current signals. In the MCBJ measurements we use a voltage source to apply a voltage between the two-terminal MCBJ sample and either a logarithmic ampli-fier or a linear ampliampli-fier to record the current flowing in the MCBJ circuit. The logarithmic amplifier has a dynamic range of about ten orders of magnitude and its response time at the lowest current (I ' 10 pA) reaches hundreds of millisec-onds. The linear amplifier can be set at a gain from 106V/A to 109V/A and can measure currents with a noisefloor of 5 fA/pHz at the highest gain setting.

The communication with the measurement electronics is achieved via an AD-win Gold (Jäger Computergesteuerte Messtechnik GmbH). The ADAD-win is capable of operating up to a frequency of 40 MHz and is equipped with two 16 bit digital-to-analog converters (DAC) with a range of ±10 V. Moreover, sixteen 16 bit analog-to-digital converters (ADC) are separated in two groups of eight converters, hosted on two different multiplexers. One of the two DAC is used to control the voltage applied by the measurement electronics while the other controls the voltage am-plifier connected to the piezoelectric element. Two ADC, one on each multiplexer, are used to read the outputs of the logarithmic and of the linear amplifier.

The ADwin communicates with the measurement computer through an USB nection while the servo motor communicates with the computer via a serial con-nection. The measurement routines are controlled using home-made Python codes and ADbasic scripts. The fastest acquisition rate that we achieve is approximately

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3.3.MEASUREMENTS

3

25

200 kHz.

3.3. M

EASUREMENTS

In the course of this dissertation different measurement schemes are employed, all based on the measurement of the current as a function of another variable like voltage or time. In this section we discuss the general aspects of the measure-ments, looking at the characterization of a pristine gold sample and at the deposi-tion of the target molecule onto the sample.

3.3.1. G

OLD BREAK JUNCTIONS

A typical MCBJ experiment starts with the characterization of the clean gold elec-trodes at room temperature. One measurement that can be done consist in mea-suring the conductance at a low bias while breaking and reforming the gold wire. In this fast breaking conductance-length measurements we apply a bias voltage of typically 0.1 V to the gold wire and measure the current passing through it while driving, with the piezoelectric element, up and down the pushing rod to mechan-ically break and form the contact. The current is measured with the logarithmic amplifier since a large range of value for the current is probed when going from the metallic conduction of the gold wire (resistance < 10 kΩ) to the tunneling regime of the broken wire (resistance > 10 MΩ). The typical speed of the electrodes is larger than 1 nm/s and the bias applied is around 0.1 V.

The blue and red lines in figure3.3a represent in semilogarithmic scale four fast breaking traces measured on two different MCBJ samples that illustrate the typical breaking behavior of a gold wire. The measurements start from a conduc-tance of 30 G0in the metallic regime, not shown in the figure, that corresponds

to a constriction of about 2 nm of diameter (roughly speaking six gold atoms as a diameter). The stretching the gold wire results in the thinning of the constriction and in the decrease of the wire conductance. The conductance decreases in steps corresponding to atomic rearrangements and below 5 G0becomes quantized (in

the figure the region between 3 G0and 1 G0) resulting in plateaus at values close

to integers values of 1 G0. The final plateau in conductance centered around 1 G0,

visible in all the four traces of figure3.3a, indicates the formation of a single-gold atom contact. This characteristic of gold comes from its 6s1valence electrons, that in bulk gold give rise to a nearly-free electron gas around the Fermi energy, and in quantum transport participate as a single transport channel per atom.

An additional stretching of the wire results in the rupture of the single-gold atom junction visible in the measurements as a sharp drop in conductance from values around 1 G0to values below 10−1G0, in these particular cases and in

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gen-3

26 3.EXPERIMENTAL

Figure 3.3: a) Conductance-length two-dimensional histograms of two different gold samples built from thousands of individual breaking traces. The conductance traces have been logarithmically binned along the conductance axis and linearly binned along the displacement axis. The traces, offset along the x-axis for clarity, are individual breaking traces typical example of the stretching and rupture of a gold wire. b) One-dimensional conductance traces built by integrating along the displacement axis direction the two-dimensional histograms.

eral at room temperature we find a typical value of 10−4G0. The lack of counts

between 10−4 _G

0and 1 G0is typical for gold nano-electrodes at room

tempera-ture and is due to the stress-releasing fast retraction of the first gold atoms after the breaking of the atomic point contact [12,13]. The broken extremities of the gold wire can act as nano-electrodes, eventually to contact molecules. However, in a pristine gold sample no stable junctions can be formed once the electrodes are created and one should observe only traces with exponential tunneling-like decay of the conductance below 1 G0due to the Simmons tunneling between the

separated electrodes. The slope of the exponential tunneling decay, of the traces in figure3.3a, is related to the workfunction of gold and can be analyzed with the Simmons model. Once the conductance falls below the noisefloor of the setup, typically at 2 · 10−7G0with a bias applied of 100 mV, we stretch for an additional

distance of 2 nm and then we release the bending to reform the gold wire. Once a conductance of 30 G0is reached a new breaking trace can start.

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3.4.CALIBRATION OF THE ATTENUATION FACTOR

3

27

3.3.2. C

ONDUCTANCE HISTOGRAMS

The conductance histograms are statistical methods widely used, in molecular scale electronics, to represent large datasets of conductance breaking traces. We will make large use of the histogram representation in this dissertation and we briefly introduce them here in the case of the pristine gold samples. Given a set of N conductance fast breaking traces, where Gn(d ) is the trace n of the dataset

we define the one-dimensional conductance histograms as H1(Gi) and the

two-dimensional conductance-length histogram as H2(dj,Gi). To find H1(Gi) we

di-vide (the logarithm of ) the conductance axis in discrete bins Gi with index i and

then for each conductance trace in the dataset we find the number of data points, Nn

i , that falls in each bin. The histogram is then defined as:

H1(Gi) =

X

n

N_in. (3.1)

The two-dimensional histogram H2(dj,Gi) requires the alignment of all the traces

to a common reference zero in length (usually defined at the final rupture of the 1 G0plateaus). With the additional binning of the length axis dj we find, for each

conductance trace, the number of data points, N_{i , j}n , that falls in each two-dimensional square bin.

H2(dj,Gi) =

X

n

N_{i , j}n . (3.2)

Figure3.3a displays the two-dimensional histograms of two pristine gold samples built from thousands of fast breaking traces each, while figure3.3b shows the cor-responding conductance histograms. The histograms have high counts regions at values around 1 G0and 2 G0due to single-gold atom and dimer junctions

re-spectively. In the plots these atomic configurations show up as flat plateaus. For conductance values below 1 G0, tunneling between the two electrodes appears as

a high-count region, with an onset between 10−3G0and 10−4G0, that decays

ex-ponentially with the distance between the electrodes. In the one-dimensional his-tograms the gold atomic junctions appear as peaks around 1 G0of conductance,

while the tunneling region gives a broad step and a flat background below 10−3_G 0.

As we already mentioned, the lack of counts between 10−3_G

0and 1 G0is due to

the jump-out-of-contact of the contacts terminal atoms.

3.4. C

ALIBRATION OF THE ATTENUATION FACTOR

Until now we have not discussed one crucial aspect of the MCBJ sample, the atten-uation ratio r . This dimensionless quantity is the ratio between the displacement of the pushing rod in the vertical direction, that is known, and the displacement of the electrodes in the horizontal direction, that needs to be found. Among dif-ferent factors the geometry of the sample is the most important in determining r .

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3

28 3.EXPERIMENTAL

The typical value of r for a lithographically fabricated break-junction is between 1 · 10−4_{and 1 · 10}−6_{. In order to find the value of r from the measurements one has}

to rely on some measurable quantity whose distance dependence is known and re-producible. In gold break junctions different processes are known to give a precise signal as a function of the distance between the electrodes. In the following we will consider the average length that a single gold atom junction can sustain the stretching and the vacuum tunneling between the two electrodes after the rupture of the wire.

Figure3.4a shows three examples of room temperature conductance breaking traces measured on pristine gold electrodes plotted in linear scale as a function of the vertical displacement. As already discussed, the appearance of steps at integer values of 1 G0is due to stable few- or single-atoms junction configurations. The

trace on the left and the one on the right, both have a plateau at 2 G0followed by

a plateau at 1 G0, while the trace in the middle displays a transition from 3 G0to

1 G0. The plateau at 1 G0is a very reproducible and robust feature in the stretching

of gold nanowires and in average has a length of 0.25 nm. At low temperature the formation of single-atoms chains increases the chance of observing stable config-urations at multiples of 0.25 nm. The purpose of the length calibration is to find the attenuation factor r and convert the distance scale from the vertical displace-ment (in the order ofµm) to the horizontal displacement of the electrodes (nm).

Figure3.4b displays a histogram built from thousands of individual breaking traces of the length of the 1 G0plateau. From each trace we extract this length

by subtracting the vertical position corresponding to the last conductance point above 1.4 G0from the one corresponding to the last data point above 0.7 G0. By

fitting the length distribution to a single Gaussian peak we find a value of 4µm for the average 1 G0length that gives a value for the calibration of r = (6 ± 0.5) · 10−5

close within the experimental error to the ones obtained earlier.

Another physical process that can be used for calibration purposes is the dis-tance dependence of the tunneling current flowing between the gold nanoelec-trodes of the MCBJ. We do not use it for the calibration of r in this dissertation but we illustrate the principle here and use it as a double check for the value of r just found from the analysis of the 1 G0plateau length. The tunneling current is

expressed in the Simmons model as a function of different variables among which there is the electrodes distance d :

I (V ) = e A 4π2_ħd2[(Φ − eV /2)exp(− 2d ħ p 2m(Φ − eV /2))+ − (Φ + eV /2) exp(−2d ħ p 2m(Φ + eV /2))], (3.3)

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3.4.CALIBRATION OF THE ATTENUATION FACTOR

3

29

Figure 3.4: a) Three example of conductance breaking traces measured at room temperature on a pris-tine gold MCBJ sample plotted in linear scale, the bias voltage is 0.1 V. The driving speed of the piezo-electric element is 300 V/s corresponding to 84µm/s in the vertical direction and 5.0 ± 0.8 nm/s in

the electrodes plane after the calibration. The plateaus at 1 G0are colored in blue and correspond to

single-gold atom junctions while the plateaus between 1.6 G0and 2 G0, corresponding to dimer

con-figurations, are shown in red. b) Histogram of the 1 G0plateaus length together with a Gaussian fit to

estimate the average length. From the Gaussian fit we find the center of the distribution at 0.23 nm. c) Three examples of breaking traces measured at low temperature and represented in logarithmic scale shown in green, the calibration used is the one found from (a). The exponential decay of the con-ductance is due to vacuum tunneling between the gold nanoelectrodes. The black line is a guide-line for the eyes representing the conductance as a function of the distance calculated from the Simmons model of direct tunneling using the work function of gold W.F. = 5.1 eV.

where A is the junction area andΦ is the work function of the gold electrodes. In the limit of a small voltage V∗, such that eV∗<< Φ, the conductance G de-cays exponentially with the increase of the distance d . The rate of the exponential decay (corresponding to the slope of l og (G) versus d ) depends only onΦ and as-suming thatΦ is known (for clean gold the work function in ultra high vacuum is 5.1 eV) the experimental measurement of the decay rate can been used for the calibration of r . Figure3.4c presents three breaking traces measured at low tem-perature and showing an exponential decay of the conductance that we attribute

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3

30 REFERENCES

to vacuum tunneling between the gold electrodes, the attenuation factor used is the one found above r = 6 · 10−5_{. The slope of the three curves agrees well with the}

slope expected by the Simmons model considering a work function in the range between 4.9 eV and 5.1 eV as seen in the comparison with the black curve in the figure generated from Eq.3.3. The observed differences in the onset of the tunnel-ing conductance, rangtunnel-ing from 10−1G0to 10−5G0, can be explained by a spread in

the size of the gap between the electrodes formed at the rupture of the wire (that modifies the absolute value of d ) or, less probably, by a difference in the area of the junctions.

R

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