Molecular Electronics is self-contained and unified in its presentation. It can be used as a textbook on nanoelectronics by graduate students and advanced undergraduates studying physics and chemistry. In addition, included in this new edition are previously unpublished material that will help researchers gain a deeper understanding into the basic concepts involved in the field of molecular electronics.
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Chapter 1: The birth of molecular electronics (1,389 KB)
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https://doi.org/10.1142/9789813226036_fmatter
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How does the electrical current flow through a single molecule? Can a molecule mimic the behavior of an ordinary microelectronics component or maybe provide a new electronic functionality? How can a single molecule be addressed and incorporated into an electrical circuit? How to interconnect molecular devices and integrate them into complex architectures? These questions and related ones are by no means new and, as we shall see later in this chapter, they were already posed many decades ago. The difference is that we are now in position to at least address them in the usual scientific manner, i.e., by providing quantitative experimental and theoretical results. The advances in the last two or three decades, both in nanofabrication techniques and in the quantum theory of electronic transport, allow us now to explore and to understand the basic properties of rudimentary electrical circuits in which molecules are used as basic building blocks. It is worth stressing right from the start that we do not yet have definitive answers for the questions posed above. However, a tremendous progress has been made in recent years and some concepts and techniques have already been firmly established. In this sense, one of main goals of this book is to review such progress, but more importantly, this monograph is intended to provide a solid basis for the new generation of researchers that should take the field of molecular electronics to the next level.
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The discussion of the scattering formalism in the previous chapter has left two basic questions open: (i) How to calculate the elastic transmission of real systems such as atomic and molecular junctions? and (ii) how to generalize Landauer formula to take into account correlation effects and inelastic mechanisms? Indeed, both questions can be answered, at least to a large extent, with the help of Green’s function techniques. For this reason, we initiate here a series of three chapters devoted to this subject.
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In the previous chapter we have seen that the calculation of the zerotemperature Green’s functions of a non-interacting system in equilibrium reduces to solving an algebraic linear system, summarized in Dyson’s equation. This is practically all we need to tackle the problem of the determination of the elastic transmission of realistic systems. However, if we want to go beyond and treat systems where the electron correlations or inelastic interactions play a major role, we need many-body techniques. For this reason, we present in this chapter a systematic perturbative approach for the calculation of zero-temperature equilibrium Green’s functions. This formalism is valid for any type of system and interaction and it constitutes the most general method for the computation of Green’s functions. Moreover, the nonequilibrium formalism introduced in the next chapter follows closely the perturbative approach that we are about to describe.
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So far we have shown how the Green’s function techniques can help us to understand the physics of systems in equilibrium. Since our goal is the analysis of the transport properties of different nanocontacts, we have to generalize those techniques to deal with situations in which the systems are driven out of equilibrium. This is precisely the goal of this chapter in which we shall discuss the so-called nonequilibrium Green’s function formalism (NEGF). This formalism was developed independently by Kadanoff and Baym [259] and Keldysh [260] in the early 1960’s. Here we shall follow Keldysh formulation of this approach and we shall refer to it as the Keldysh formalism. This formalism is a natural extension of the diagrammatic theory that we have presented in the previous chapter. The importance of the Keldysh formalism lies in the fact that it allows us to go beyond the usual linear response in a systematic manner. Since its appearance, it has been used in a great variety of topics (see Refs. [261, 262] and references therein). In particular, it has been applied to the study of electronic transport in many types of nanoscale devices and it constitutes a basic tool that will be used throughout the rest of the book.
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In the previous chapter we showed how the Keldysh formalism can be combined with simple Hamiltonians to compute the current in model systems. In this chapter we shall exploit this technique and derive some general expressions for the electrical current that can be combined with realistic methods for the determination of the electronic structure. To be precise, we shall address three basic issues:
This chapter is rather technical and it can be skipped by those who are not so interested in the algebra behind the current formulas. Anyway, we recommend to read the next section about the derivation of the Landauer formula, since the expression obtained there for the elastic transmission will be frequently used in subsequent chapters.
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In previous chapters we have shown how to compute the transport properties of an atomic-scale junction once the corresponding Hamiltonian is known. Therefore, in order to make our theoretical background selfcontained, at least to a certain extent, we need to discuss how those Hamiltonians are determined in practice. In other words, we have to describe adequate methods for the description of the electronic structure of atomic and molecular junctions. Such methods are based on the standard approaches for the calculation of the electronic structure of atoms, molecules and solids that are used in atomic physics, theoretical chemistry and solid state physics. There is a great variety of electronic structure methods and, obviously, we cannot review all of them here. We shall focus our attention on the two methods that have had the largest impact so far in the field of molecular electronics. First, in this chapter we shall discuss the tightbinding approach, which is a very intuitive empirical or semi-empirical method that has been crucial to elucidate the physics of, in particular, metallic atomic-size contacts. Then, the next chapter is devoted to the density functional theory (DFT), which is the most widely used approach among the so-called ab initio methods.
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This second chapter about electronic structure calculations provides a basic introduction to the density functional theory (DFT). This theory is presently the most successful approach for computing the electronic structure of matter. Its applicability ranges from atoms, molecules and solids to nuclei and quantum and classical fluids. Thus for instance, in chemistry DFT is widely used to predict a great variety of molecular properties: molecular structures, vibrational frequencies, atomization energies, ionization energies, electric and magnetic properties, reaction paths, etc. Originally, DFT was designed to provide the electron density and total energy of the ground state of (non-magnetic) electronic systems. However, meanwhile the theory has been generalized to deal with many different situations: spin polarized systems, multicomponent systems such as nuclei and electron hole droplets, free energy at finite temperatures, superconductors with electronic pairing mechanisms, relativistic electrons, time-dependent phenomena and excited states, bosons, molecular dynamics, etc.
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In order to understand the electrical and thermal conduction through molecular junctions, which is the main goal of this monograph, it is necessary to first understand the corresponding properties of the metallic atomic contacts that are used as electrodes in these nanoscale circuits. The conduction through atomic-scale wires constitutes a field of its own that started at the beginning of the 1990’s and reached maturity in the early 2000’s. Metallic wires of atomic dimensions have turned to be a marvelous playground where many basic concepts of quantum transport have been tested [15]. The physics of these nanocontacts and the progress made in this field until 2003 were reviewed in a comprehensive article by Agraït, Levy Yeyati and van Ruitenbeek [15]. For this reason, we shall not make any attempt to provide a historical revision of this field or to give a complete list of references. Instead, we shall present here a short elementary introduction to some basic aspects that will be useful in subsequent chapters where the physics of molecular transport junctions is discussed.
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The use of the spin degree of freedom of the electron in conventional chargebased electronic devices has lead to the discovery of many fundamental effects and, in some cases, to new technological applications [461, 462]. The emblematic physical effects in this field, known as spintronics, like the giant magnetoresistance (GMR), tunneling magnetoresistance (TMR) or anisotropic magnetoresistance (AMR) stem from the spin-sensitivity of the scattering mechanisms that dominate the transport properties in electronic devices made of magnetic materials. In recent years, a great effort has been devoted to understand how these fundamental effects are modified when the dimensions of a magnetic device are reduced all the way down to the atomic scale. The goal of this chapter is to provide a brief introduction to the transport properties of ferromagnetic atomic-size contacts and to draw the attention to different issues that are currently being investigated in the context of molecular junctions.
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As we have just seen in Part 3, the level of understanding achieved in the field of metallic atomic-size contacts is certainly remarkable. However, it is also clear that such metallic nanowires are not very “flexible” in many respects. Thus for instance, their conductance can hardly be changed with a gate voltage and often their current-voltage characteristics are simply linear, which hinders the implementation of interesting electronic functionalities. Thus, it seems natural to investigate the use of molecules as possible building blocks of nanoscale circuits. Molecules are still small enough to take advantage of their size, and the great variety of their physical properties make them ideal not only to mimic ordinary components of today’s microelectronics, but also to provide new electronic functions. For these reasons, the analysis of the transport properties of molecular junctions is attracting a lot of attention and this will be the subject of the rest of this book.
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In the previous chapter we have learned that the coherent transport through molecular junctions is determined by the strength of the metal-molecule coupling as well as by the intrinsic properties of the molecules, including their length, conformation, the HOMO-LUMO gap, and the alignment of this gap with the metal Fermi level. Moreover, we have shown that in many cases the experimental observations can be explained by means of very simple qualitative arguments. In this chapter we shall go on discussing the coherent transport in single-molecule junctions, but from a more quantitative point of view. Our goal is two-fold. On the one hand, we want to calibrate our present level of understanding and for this purpose, we shall compare different experimental and theoretical results for various test systems. On the other hand, we shall illustrate some of the basic concepts discussed in the previous chapter in more quantitative terms.
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This chapter is devoted to the theoretical description of the vibrational effects detailed in the previous one. In particular, our main goal is to explain the origin of the different signatures summarized in section 16.6 (see also Fig. 16.17).
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In Chapters 15-17 we have discussed how the electronic transport is modified when the quantum coherence is partially destroyed by either Coulomb correlations or the excitation of molecular vibrations. One of the central subjects of this chapter will be the analysis of the charge transport in situations in which this coherence is completely lost. As we explained in previous chapters, this incoherent regime is realized when the tunneling traversal time is considerably larger than the time scales associated to the inelastic interactions. Obviously, this becomes more likely as the molecular length increases. In the extreme case in which the inelastic scattering time is much smaller than the tunneling time, the current is transported by electrons that hop sequentially from one segment of the molecule to another via thermally activated tunneling events in which the quantum coherence is completely lost. For this reason this transport regime is referred to as hopping regime. The discussion of this regime is one of the central goals of this chapter.
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In the previous chapters we have addressed the main transport regimes that are realized in molecular junctions. In our discussion so far, we have focused our attention on the analysis of the electrical conductance. However, there are many other properties that provide valuable information about the charge transport that is not contained in the conductance. A paradigmatic example is the current fluctuations or noise. Its investigation has contributed decisively to our understanding of the transport mechanisms in a great variety of mesoscopic and nanoscale devices [224]. On the other hand, the charge transport is not the only important aspect in the context of conduction in molecular junctions. Thermal transport is also a key issue in the field of molecular electronics from a fundamental as well as from a practical viewpoint. Molecular-scale contacts provide a new territory to study heat conduction in regimes never explored before and, issues like heating have to be investigated and understood, if molecular electronics wants to become a viable technology. Obviously, the study of thermoelectric phenomena in molecular junctions, resulting from the interplay between electrical and thermal transport, can also give a new insight into the physics of these nanocircuits.
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We have discussed so far different ways of controlling the current through a molecular junction such as gating or appropriate chemical synthesis. Another possibility is the use of an external electromagnetic field, which has been widely explored in larger mesoscopic structures [287]. In addition to controlling transport with external radiation, many other issues related to the optical properties of molecular junctions are of interest and have been recently studied [284, 1235–1238]. In this sense, the goal of this chapter is to discuss the physical phenomena that emerge as a result of the interplay between current-carrying molecular junctions and an electromagnetic field.
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At this stage in the development of molecular electronics it is already impossible to cover all the aspects of this multidisciplinary field in a single monograph. Our selection of topics has been biased, as it could not be otherwise, by our own backgrounds and research interests and we are aware of the fact that important issues have been left out. Therefore, we would like to close this manuscript by pointing out some of those topics and suggesting some references where the reader can find information about them.
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All the relevant systems in molecular electronics are composed of many identical particles such as electrons, protons, phonons (or vibrations), etc. As we all know, these particles obey their corresponding quantum statistics depending on whether there are fermions or bosons. This statistics is reflected in the symmetry of the many-particle wave functions. Thus for instance, a fermionic wave function is expressed in the form of Slater determinants to ensure its antisymmetry (Pauli’s exclusion principle).
https://doi.org/10.1142/9789813226036_bmatter
The following sections are included:
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Chapter 1: The birth of molecular electronics (1,389 KB)
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