Gateway state-mediated, long-range tunnelling in molecular wires

If the factors controlling the decay in single-molecule electrical conductance G with molecular length L could be understood and controlled, then this would be a significant step forward in the design of high-conductance molecular wires. For a wide variety of molecules conducting by phase coherent tunneling, conductance G decays with length following the relationship G = Aexp-\b{\eta}L. It is widely accepted that the attenuation coefficient \b{\eta} is determined by the position of the Fermi energy of the electrodes relative to the energy of frontier orbitals of the molecular bridge, whereas the terminal anchor groups which bind to the molecule to the electrodes contribute to the pre-exponential factor A. We examine this premise for several series of molecules which contain a central conjugated moiety (phenyl, viologen or {\alpha}-terthiophene) connected on either side to alkane chains of varying length, with each end terminated by thiol or thiomethyl anchor groups. In contrast with this expectation, we demonstrate both experimentally and theoretically that additional electronic states located on thiol anchor groups can significantly decrease the value of \b{eta}, by giving rise to resonances close to EF through coupling to the bridge moiety. This interplay between the gateway states and their coupling to a central conjugated moiety in the molecular bridges creates a new design strategy for realising higher-transmission molecular wires by taking advantage of the electrode-molecule interface properties.


Introduction
Understanding electron transport in metal−molecule−metal (MMM) junctions and identifying molecular wires whose conductance decays only slowly with length is important for the advancement of molecular electronics. The critical factors which determine conductance in a MMM junctions are the metal−molecule contacts and the structure of the molecular backbone. 1 While a wide variety of molecular backbones can be synthesised, the nature of the anchor groups that act as connectors to the metallic leads is limited by the strength of their interaction with the metal. As gold is the most widely used electrode material in molecular electronics, the choice of anchor can be made from moieties that can form X-Au covalent bonds, such as thiols 2,3 and carbodithioates, 4 moieties that react to give a C-Au bond, such as organostannanes 5 or diazonium salts, 6 and moieties that interact with gold with a coordination bond, such thiomethyls, [7][8][9] amines, 7,10 pyridines, [11][12][13] and phosphines 10 .
Tunnelling theory predicts that conductance across a nanojunction should decay exponentially with its length, following a relationship G = Ae -βL , where L is the junction length and A is a preexponential factor dependent on junction contacts and nature of metallic leads. The nature of the molecular wire bridging the two metallic leads has a strong effect on the exponential attenuation respectively. Extremely low values of β were found in systems such as meso-to-meso bridged oligoporphyrins 13,17,18 (0.040 ± 0.006 Å -1 ), axially-bridged oligoporphyrins 19 (0.015 ± 0.006 Å -1 ), oligoynes 20 (0.06 ± 0.03 Å -1 ), carbodithioate-capped oligophenylene-ethynylene 4 (0.05 ± 0.01 Å -1 ), and extended viologens 21 (0.006 ± 0.004 Å -1 ). Oligothiophenes, on the other hand, showed a more complex behaviour, with unusual conductance decay with the number of thiophene rings [22][23][24] and, in the case of longer oligothiophenes with alkylthiol linkers, water-dependent conductance and conductance decay. 25 A hopping charge-transport mechanism could explain the low value of β in some of these systems, but it is generally believed that tunnelling is dominant in short molecular wires. Transition to tunnelling to hopping has been observed in various systems, at a critical length ranging from 5 nm (oligonaphtalenefluoreneimine) 16 to 8 nm (oligothiophene). 26 The length-dependent conductance of alkanedithiols (as archetypal saturated molecular wires) has been the subject of investigation by several research groups. Li et al. 27 reported exponential decrease of the conductance with molecular length with β ≈ 0.84 Å -1 for N(CH2) < 7. Other studies with longer alkanedithiols showed that the conductance decay is less pronounced for shorter molecules (N(CH2) < 8), whereas conductance decay is more rapid for longer lengths (N(CH2) > 8). 28,29 Another study reports experimental decay constants β ≈ 0.94 -0.96 Å -1 . 30 Inclusion of heteroatom in the aliphatic alkyl chain to give oligoethers or oligothioethers resulted in negligible effect on β, with reported values of 1.11 (per atom unit) for alkanedithiols, 1.19 for oligoethers and 1.17 for oligothioethers. 31 The above comprehensive experiments, combined with detailed modelling and material-specific transport calculations, take into account complex features introduced by metal-molecule contact, [32][33][34] orbital resonances, and other quantum mechanical effects that can strongly affect molecular conductance. 14,23,35-40 They have improved our understanding of the conductance decay with length in MMM junctions, but the effect of the molecular wire structure on the value of β is still not completely understood. An important feature in the transport characteristics of alkanedithiols is the presence of a broad resonance, called in previous studies a "gateway state" 41 or "contact-level", 30 close in energy to the Fermi level of the metallic leads. In a systematic study of alkane molecular junctions with gold electrodes, Kim et al, reported a small peak close to the Fermi energy and a broad one about -1 eV from the Fermi energy, and they showed that the resonances are due to molecular orbitals localized on sulphur at these energies. 42 This peak is also present in the calculations of Hüser et al in the case of thiol end-groups connected to a single gold tip atom. 43 We found that the presence of a central group attached via thiol-terminated alkane linkers to Au electrodes will magnify the effect of the resonance peak close to the Fermi energy, and we attribute this feature to atomic wave functions localised on sulfur atoms bound to the leads. The phenomenon is not limited to thiol contacts, and it has also been observed in MMM junctions with covalent, highly conducting C-Au contacts. 44 In what follows, we reveal the peculiar effect of these gateway orbitals on the decay constant β. In the conductance-length relationship G = Ae -βL , the attenuation coefficient β(EF) is a property of the backbone and the value of the electrode Fermi energy EF relative to the frontier orbitals of the molecule, which determines the tunnelling gap for electrons passing from one electrode to the other. On the other hand, for a given EF, it is often 6 assumed that the coupling between the anchor groups and electrodes contributes to the prefactor A only. This assumption is surely correct in the asymptotic limit of large L, providing transport takes place by phase-coherent tunnelling. However, β is usually obtained experimentally from the slope of plots of ln(G) versus L, for limited values of L, and the question of whether these values are sufficiently large is usually not addressed. However, there are some exceptions. For instance, Xie et al. have shown that, for junctions formed using conducting atomic force microscopy (C-AFM) measurements on monolayers of short oligophenyl molecules, the value of β depended upon whether mono-or dithiols were employed, 45 being smaller in the latter case. In what follows, we refer to the slopes of such graphs as pseudo-attenuation coefficients and denote them β'. The main reason for doing this is that here we add methylene groups at either side of the central moiety, rather than having a homologous series incrementing by monomer units, the latter being the most widely used for the determination of attenuation factors β. The assumption that the coupling between the anchor groups and electrodes contributes to the prefactor A only has restricted the range of proposed strategies for manipulating β to those which mainly rely on tuning EF by electrochemical or electrostatic gating, or by doping the backbone with electron donors or acceptors. The aim of the present paper is to demonstrate the counterintuitive result that the coupling between anchor groups and electrodes can contribute to both the β' and the prefactor A.
In this work we demonstrate this dependence, both experimentally and theoretically, by studying the length-dependent conductance of molecules containing a central conjugated moiety connected on either side to alkane chains of varying length. The nature of the central unit has been demonstrated to have a strong effect on molecular conductance, 46,47 and therefore we chose these three different moieties on the basis of their extent of conjugation and electron density, going from a poorly-conjugated, electron-deficient moiety such as a viologen salt (with a break in conjugation due to inter-ring torsion in its dication state 48 ) to the well-conjugated, electron-rich α-terthiophene. Intuitively, one might expect the value of β' for these molecular wires to approach the value determined for alkanedithiols (β ≈ 1 Å -1 ).
Surprisingly, in what follows we shall demonstrate that the presence of a conjugated moiety in the alkyl tunnelling barrier strongly affects β', due to transport through "gateway states" and "coupling states", the magnitude of which depends on the nature of the conjugated system.

Results and Discussion
The series of molecular wires shown in Figure 1 were synthesised and characterised using common synthetic laboratory techniques (see ESI for synthetic procedures). The STM-based I(z) technique 49 (details in the Methods section) was used to measure the conductance of the molecular wires presented in this work, and the more widely used STM-BJ technique 11 was used as comparison for the most conductive molecular wire (more information in the ESI). In brief, a gold tip is moved towards a gold surface with a sub-monolayer of the target molecule and then retracted to yield current (I) -distance (z) traces that show a number of features characteristic of MMM junctions, such as steps and plateaux. Hundreds of such conductance-distance traces are collected and subsequently compiled in histograms bearing a distribution of conductance values. Peaks in the histograms were fitted to a Gaussian distribution to determine the most probable conductance, expressed in nS. Data for the X[Ph]X class of molecular wires was taken after Brooke et al., 41 and data for the alkanedithiol series was taken from Haiss et al. 29 The experimentally determined conductance values were then plotted as ln(G) vs length, and a linear fitting was used to obtain the β' attenuation and its standard deviation is used as error.  Example of results of single-molecule conductance measurements for the molecular wires capped with protected thiol functions are presented in Figure 2a and      (Figure 3a), a broad resonance is present in the HOMO-LUMO gap, at about EF = -1 eV (labelled "Gs" in Figure 3a). This feature has been previously assigned to "gateway" or "Au-S" states located on the sulfur anchors either by theoretical 30,37,43 or spectroscopic means. 56 Additional smaller but sharper resonances arise in our calculations very near to EF = 0 (labelled "Cs" in figure 3a), and we attribute these to the strong coupling between the two gateway states, through the molecular backbone ("coupling states"; see local density of states plots in Figure 3e and 3f). These two features have already been discussed in the literature, 42,43 and are present in all the calculations performed on the compounds presented in this study with thiol contacts, but their combined magnitude is more pronounced in the presence of central moieties comprising a α-terthiophene (Figure 3b) or a phenyl ring (Figure 3c) and, as shown in Figure 4, they lead to a low β' value for these molecules.    To confirm the theoretical findings, we synthesised the series of molecular wires bearing a phenyl central unit and alkyl spacers of varying length with thiomethyl contacts (X[Ph]X-SMe), and measured their conductance (Figure 5c). The results confirmed the theoretical prediction, with an increased attenuation factor β' = 0.50 ± 0.04 Å -1 (0.56 ± 0.05 per methylene unit) upon removal of the gateway/coupling states (Figure 5d). Thiomethyl is not the only contact group that increases the β' value in these dialkyl benzene compounds, and an even higher value of > 1 per methylene unit has been reported, for instance, in carboxylic acid-capped molecular wires. 57 The role of these additional states can be further described from an analytical perspective, by using a simple theory which captures their effect in terms of two dimensionless parameters. In the lowvoltage and low-temperature limit, the electrical conductance of a single molecule is given by  14 Now consider the case of a molecule containing two gateway orbitals of equal energy , coupled to each other by a tunnel barrier such as an alkyl chain represented by a tunnelling matrix element , which decays exponentially with the length of the chain (Figure 6b). Each of the orbitals is connected separately to electrodes 1 and 2. Since the energy spacing between these orbitals is zero (and therefore less than Γ1 + Γ2 ), equation (1) cannot be used and the mathematical description of ( ) is more complex (equation S1 in the ESI). 58 For simplicity, we consider only the case of a symmetric junction, for which the formula of equation S1 reduces to Like equation (2), this expression also involves two dimensionless parameters, namely and .
However, as shown in Figure 7, unlike equation (2), when plotted against , equation (3) possesses two maxima when | | > 1. These maxima occur at = ±( − 1) / , which are associated with bonding and anti-bonding combinations of the two gateway orbitals, induced by the coupling . Since decreases with the length of the alkyl bridge, two maxima (the "coupling states") are present for short molecules (i.e. for larger ) and merge into a single maximum for longer molecules. This splitting, for instance, also appears in DFT calculations of short ADT ( Figure S24 of the ESI). 43

Conclusions
The above results demonstrate that when a conjugated central unit is sandwiched between two insulating alkyl chains the decay in conductance with the length of the chains is much shallower than that of alkyl chains alone. For example, the beta factor of an alkanedithiol is β = 0.9 Å -1 , whereas in the presence of a phenyl ring central unit, this decreases to β' = 0. 18  close to the gold substrate at a defined setpoint current and under constant bias, so that junctions can form, and rapidly retracted (40 nm s -1 ), while a current (I) vs. distance (z) curve is recorded.
The process is repeated many times, and hundreds of such junction making and breaking curves are analysed statistically in histograms to yield a distribution of conductance values. Spurious traces with no evidence of junction formation (plateaux and steps) were discarded to avoid ambiguity and reduce noise. The average hit rate (percentage of scans showing evidence of junction formation) is 10-15 %, depending on the molecular wire. Plateaux in current-distance curves result in peaks in the histogram, and a Gaussian fit was used to determine the most probable conductance value.
Theoretical calculations: The Hamiltonian of the structures described in this paper were obtained using density functional theory as described below or constructed from a simple tight-binding model with a single orbital per atom of site energy ε = 0 and nearest neighbour couplings γ = −1.