Electrical transmission of molecular bridges
Introduction
Understanding the routing of electrons through molecules is necessary for a variety of potential applications in microelectronics [1]. For example, nanocomputing [2], [3] requires that a molecule acts as a switch that can be turned on or off. Molecular rectifiers [4] are similar in their requirements. For biosensors [5], [6] a bridge that connects the redox protein to the electrode is needed. In these and other applications, the molecule acts as an electron relay. There is therefore a good deal of activity directed at the study of molecular wires [2]. In the background to much of this activity is the conventional wisdom that conjugated molecules will be better conductors. This expectation can be traced back to studies of electron transfer between inorganic ions bridged by different ligands [7].
In line with the experimental studies there has been an extensive theoretical effort aimed at the understanding of electron transmission through molecules [8], [9], [10], [11], surface layers [12] and nanodevices [13].
Over several years we have sought to understand electron routing through planar arrays of metallic quantum dots [14], [15]. These arrays offer the theorist a fertile testing ground because they can be experimentally studied under conditions where the correlation of electrons plays only a secondary role so that a one-electron approximation is sufficient to delineate the major effects. This is unlike electron migration in smaller molecules, e.g., short peptides, where correlation plays a much more major role [16], [17]. In the present study, we direct attention to small molecular bridges but seek to retain a key advantage of the one electron picture, an advantage that allows us to discuss the conduction in familiar quantum chemical terms, as is discussed below. On the other hand, it is necessary to allow as accurately as possible for what solid-state physicists know as the Coulombic blocking effect and quantum chemists call electron–electron repulsion. This effect merits special mention for problems of charge migration because the electron needs to move into regions that already have their equilibrium share of electrons. In this study, we include the electrostatic repulsion by one of two ways. First is the traditional (Hartree–Fock) approach of computing the mean field in which the electron moves. The other route is by using density functional theory. Either way allows us to retain an orbital picture, a picture that simplifies the understanding of the route of conduction.
That a one-electron picture offers a useful picture is readily seen when we compare the study of electron transmission to photoelectron spectroscopy [18]. When we ionize a molecule with a short wavelength photon what we observe is that the outgoing electron comes out with fairly discrete different kinetic energies. By subtracting these from the energy of the photon we obtain a set of ionization potentials (IPs) that we interpret as orbital energies. The lowest IP is the energy of the highest occupied molecular orbital (HOMO) while higher IPs correspond to removal of an electron from orbitals that are below the HOMO in their energy. The same is true for the transmission function at the limiting case of a low voltage. Each peak in the transmission identifies an orbital. In fact, the transmission function gives us even more because what we described so far is ‘hole migration’. If the electrode injects an electron into the molecule it must go into an unoccupied molecular orbital. So this mechanism of conduction maps the empty orbitals. However, for many molecules the unoccupied orbitals are sufficiently high in energy that hole conduction is more important.
There is an important sense in which conduction is not an unbiased sampling of the molecular orbitals. This is that the transmission function is a weighted density of states and this weighting can have dramatic effects in terms of the relative contribution of different molecular orbitals. To contribute to the transmission, an orbital needs to have a significant weight at both ends of the molecule. This is because the coupling to the electrode even when strong, is short ranged. (It requires overlap of wavefunctions.) So unless a molecular orbital spans the molecule, its contribution to conduction will be small. In this connection note the quantal effect that an orbital need not have a significant weight all along the molecular backbone. It is sufficient if it has a weight at both ends. In the absence of inelastic scattering it needs to be the same orbital that has a weight at both ends. In addition, that orbital must be strongly coupled to the electrodes. The transmission scales as the fourth power of this coupling (square of the coupling on the left end of the molecule times square of the coupling on the right end). It is best if the molecule is chemically tethered to the electrodes and chemists know this and use thiols since these bind to gold.
In this Letter, we compare the transmission function of different molecular bridges bound to a gold trimer at both ends via a thio-gold bond. We investigated the conduction properties of both conjugated and unconjugated (alkane) bridges. Details about the ab initio computations carried out to characterize the equilibrium geometry and the electronic structure of the different bridges are given in Section 2. Transmission functions and the resulting I–V curves for the molecular bridges are discussed in 3 The transmission function, 4 Current vs. voltage. We show that even for a molecule not distorted by the applied voltage, the current has a region where it increases exponentially with the applied voltage. In addition, because of the rather limited spatial extent of small molecular bridges it is found that even quite low voltages applied between the two ends of the molecule give rise to fields strong enough to significantly distort the electronic charge distribution. In view of the results, it appears that gating effects will be important in optimizing the transport properties.
Section snippets
Ab initio computations
We have investigated five different conjugated dithiol bridges involving aromatic rings: S–C6H4–S, S–C6H4–C6H4–S, S–C6H4–C2–C6H4–S, S–C6H4–(CH)2–C6H4–S and S–CH2–C6H4–CH2–S. They are compared to three dithioalkane bridges of increasing length: S–(CH2)3–S, S–(CH2)6–S and S–(CH2)9–S. In order to explore the effect of binding to a metallic atom, here a gold, on the electronic structure of the bridge, for each molecular dithiol bridge, a gold trimer is attached on the S atom at either end.
The ab
The transmission function
The coupling of the molecule to the continuum of electronic states of the electrodes endows the energies of the molecular orbital with a ‘width’ that measures the rate of charge transfer to the bulk. We have carried out computations of charge migration where the width was included [16]. In the present study, the energies of the MOs are computed for an electrode that is mimicked as a cluster of three Au atoms. Therefore the ab initio results yield purely real energies. We therefore compute the
Current vs. voltage
The computed current is shown as a function of the voltage in Fig. 4. Fig. 4a shows a linear scale and the order of conductance of the different bridges. This order can essentially be read already from Fig. 2. When the Fermi window is convoluted with Eq. (1) what determines the current is what orbital energies fall within the window. Because the Fermi energy of gold is at −5.53 eV, the current at low voltages is completely determined by the energy and density of the HOMO. The only exception is
Concluding remarks
Microelectronic applications such as sensors require the nanowiring of a selectively active site to an electrode [5], [6]. Different molecules can be used as the bridge that establishes the electrical communication. We report computational results for the current carried by molecules as a function of the voltage. Rates of charge migration ranging all the way from tens of electrons per second to currents in the nanoamperes are obtained for different molecules and voltages. For the case studied
Acknowledgements
We thank Professor I. Willner and Dr. E. Katz for many insightful discussions about their experimental results and for making unpublished results available to us. The work of F.R. is partly supported by Région Wallonne (RW.115012) and A.R.C. This work used the computational facilities provided by SFB 377 (Hebrew University) and NIC and FRFC 2.4562.03 (University of Liège). The final stages of this work were supported by the United States-Israel BiNational Science Foundation.
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Maı̂tre de Recherches, FNRS (Belgium).