Mechanism of the enhanced conductance of a molecular junction under tensile stress
Despite its fundamental importance for nano physics and chemistry and potential device applications, the relationship between atomic structure and electronic transport in molecular nanostructures is not well understood. Thus the experimentally observed increase of the conductance of some molecular nano junctions when they are stretched continues to be counterintuitive and controversial. Here we explore this phenomenon in propanedithiolate molecules bridging gold electrodes by means of ab initio computations and semi-empirical modeling. We show that in this system it is due to changes in Au-S-C bond angles and strains in the gold electrodes, rather than to the previously proposed mechanisms of Au-S bond stretching and an associated energy shift of the highest occupied molecular orbital and/or Au atomic chain formation. Our findings indicate that conductance enhancement in response to the application of tensile stress should be a generic property of molecular junctions in which the molecule is thiol-bonded in a similar way to gold electrodes.
Molecular junctions in which a single molecule forms a stable electrically conducting bridge between two metal electrodes have been studied intensively experimentally and theoretically for more than a decade due to their fundamental interest and potential device applications Geo (). Achieving control over the electronic transport properties of these nanostructures is crucial for the realization of practical single-molecule electronic devices. The electronic conductances of molecular junctions are sensitive to their atomic geometries.Geo () These geometries can be modified by varying the separation between the two metal electrodes of the molecular junction, and this approach has been used to modulate the conductances of a variety of molecular junctions including those in which benzenedithiolate (BDT),Bruot (); Kim () octanedithiolate, Xu (); Huang () bipyridineXu () or propanedithiolate (PDT) Hihath () molecules bridge a pair of gold electrodes. In the case of Au-BDT-Au, conductance changes of more than three orders of magnitude have been achieved in this way.Kim () Thus BDT has been proposed recently as a candidate for applications as a broad range coherent molecular conductor with tunable conductance.Kim () However, despite its great importance for molecular electronics, the present day understanding of the relationship between the structure and electronic transport characteristics of molecular junctions remains far from complete. For example, intuitively, one might expect the conductance of a molecular junction to decrease as the junction is stretched and the interatomic bonds within it are weakened and therefore become less conducting. However, a recent experiment Bruot () on Au-BDT-Au junctions has demonstrated that, surprisingly, this need not always be the case and that, to the contrary, stretching these junctions can result in their conductances increasing by more than an order of magnitude. This increase in the conductance was attributed Bruot () to a strain-induced shift of the highest occupied molecular orbital (HOMO) towards the Fermi level of the electrodes resulting in resonant tunneling, as was predicted by theories of the dependence of the HOMO energy on the molecule-electrode separation, i.e., the Au-S bond length. Andrews (); Ke (); Romaner (); Li3 (); Toher (); Pontes () However, another recent study Kim () has indicated that the conductance of Au-BDT-Au junctions is controlled mainly by the strength of the molecule-electrode coupling rather than by the energy of a molecular electronic level. Most recently it has been proposedFrench () that gold atomic chains formed at the molecule-electrode interfaces when the junction is stretched may be responsible for the large increase observedBruot () in the conductance. Thus the origin of the observed behaviorBruot () remains controversial. Interestingly, the conductances of Au-PDT-Au junctions have also been found experimentally to increase as the junctions are stretched at cryogenic temperaturesHihath (). While the bonding geometries between the molecule and gold electrodes in Au-BDT-Au junctions have not as yet been determined experimentally and remain the subject of conjecture, Geo () experimentalHihath () and theoreticalFiruz1 (); Firuz2 (); Firuz3 () inelastic tunneling spectroscopy (IETS) studies have determined the experimentally realized molecule-gold bonding geometries in Au-PDT-Au junctions. For this reason Au-PDT-Au junctions may be more amenable as model systems for developing a definitive understanding of the counter-intuitive increase of the conductance as the junction is stretched. We therefore explore this phenomenon in Au-PDT-Au junctions theoretically in this Article. We present a comprehensive analysis of the relationship between the molecular configuration, contact geometry, and molecular conductance, as the junction is stretched. We find the calculated conductance to increase by factors consistent with experiment,Hihath () as the junction is stretched. Before the junction ruptures we find the conductance to decrease, also as in the experiment.Hihath () We find the conductance to decrease shortly before and during the formation of a gold atomic chain at a molecule-electrode interface, but that the conductance increase resumes with further stretching once an atomic chain has formed. However, we find the conductance increases for junctions with atomic chains to be somewhat smaller than for those without them. We show the increasing conductance as the junction is stretched to be due mainly to increasing Au-S-C bond angles at the molecule-electrode interfaces and strains in the electrodes (geometrical changes that are expected to occur in many molecular junctions under tensile stress) rather than to atomic chain formation or elongation of the Au-S bonds. We find the latter to be too small to affect the measured conductanceHihath () strongly.
Ii Estimation of the Junction Geometries
We study the evolution of the molecular junction’s geometry as the Au-PDT-Au junction is stretched by carrying out ab initio calculations of the relaxed geometries of extended molecules each consisting of a PDT molecule and two 14-atom gold clusters bound to the PDT sulfur atoms as shown in Fig. 1. Our previous work Firuz1 (); Firuz2 (); Firuz3 () showed the electronic transport properties of Au-PDT-Au extended molecules to have converged sufficiently with increasing gold cluster size that 14-atom gold clusters can serve as a realistic model of the macroscopic gold electrodes that contact the PDT molecule in experiments.Hihath () Our starting point is the relaxed geometry shown in Fig. 1(a) that our IETS studies Firuz1 (); Firuz2 (); Firuz3 () identified as having the gold-sulfur bonding geometry that was most commonly realized in experiments Hihath () on Au-PDT-Au junctions. The other structures that we considered were generated from this one by moving the atom of each gold cluster that was furthest from the molecule slightly further from or closer to the molecule and then freezing the positions of these two outer atoms and relaxing the positions of all of the other atoms of the extended molecule. This procedure was iterated to produce relaxed junction geometries with differing distances between the two outer Au atoms. Examples are shown in Fig. 1. The relaxations were all carried out using the GAUSSIAN 09 package with the B3PW91 functional and Lanl2DZ pseudopotentials and basis sets gaussian (); Pedrew () to locate local minima of the total energy of the extended molecule computed within density functional theory.
Since our approach to calculating the molecular junction geometries is fully quantum mechanical and the total energies of the trial geometries sampled in the relaxation process are computed within density functional theory, our relaxed geometries are expected to be more accurate than those obtained using computationally less demanding methods that employ parameterized interatomic potentials and classical mechanics, such as those in Ref. French, .
Since our geometry relaxations were performed effectively at zero temperature, the relaxed structures that we obtained may be expected to be representative of those realized in the cryogenic temperature experiments of Hihath et al. Hihath () on Au-PDT-Au junctions, especially in view of the fact that our starting molecule-electrode bonding geometry before the junction was stretched was that previously identified as the one most commonly realized in those experiments.Firuz1 (); Firuz2 (); Firuz3 () The fact that the bonding geometry most commonly realized in the experiments on Au-PDT-Au junctionsHihath () is knownFiruz1 (); Firuz2 (); Firuz3 () obviated the need for us to perform relaxations/stretching for a large ensemble of different starting junction geometries such as was necessary in the study in Ref. French, of Au-BDT-Au junctions for which the bonding geometry most commonly realized in experiments has yet to be determined.
Iii Conductance Calculations
We calculate the zero bias conductance at the Fermi energy for the Au-PDT-Au molecular wire at various elongations from the Landauer formula Geo ()
where , is the transmission amplitude through the PDT molecule, is the electronic state of a carrier with velocity that is coming from the left lead, and is the electronic state of a carrier with velocity that has been transmitted to the right lead. To couple the extended molecule to the electron reservoirs, as in previous work Firuz1 (); Firuz2 (); Firuz3 (); Cardamone (); Piva1 (); Piva2 (); Dalgleish1 (); George2 (); Renani (); Saffar (), we attach a large number of semi-infinite quasi-one-dimensional ideal leads to the valence orbitals of the outer gold atoms of the extended molecule. We find the transmission amplitudes by solving the Lippmann-Schwinger equation
where is an electron eigenstate of the th ideal semi-infinite one-dimensional left lead that is decoupled from the extended molecule, is the Green’s function of the decoupled system of the ideal leads and the extended molecule, is the coupling between the extended molecule and the ideal leads, and is the scattering eigenstate of the complete coupled system associated with the incident electron state . The semi-empirical extended Hückel model Geo () with the parameters of Ammeter et al. yaehmop () was used to evaluate the Hamiltonian matrix elements and atomic valence orbital overlaps that enter the Green’s function in Eq. (2). As has been discussed in Refs. Geo, and Cardamone, , this methodology involves no fitting to any experimental data relating to transport in molecular junctions. It is known to yield low bias conductances in reasonably good agreement with experiments for PDT bridging gold electrodesFiruz1 (); Firuz2 () as well as other molecules thiol-bonded to gold electrodes Kushmerick02 (); Cardamone (); Geo (); Datta1997 (); EmberlyKirczenow01 (); EmberlyKirczenow01PRB ().
We note that density functional theory (DFT)-based transport calculations are often used to calculate the conductances of molecular junctions.Geo () For example, transport calculations based on DFT with approximate self-interaction corrections have been used in the recent theoretical studyFrench () of the conductance enhancement of Au-BDT-Au junctions under tensile stress. However, density functional theory is a tool designed specifically for calculating ground state total energies.HK (); KS () Thus it is appropriate to use DFT to calculate molecular junction geometries, as is done in the present work, since these geometries are found by minimizing the total energy. On the other hand, transport is not a ground state property. For this reason DFT-based transport calculations are not well justified at the fundamental level and the results obtained from them in practice are often unreliable.Geo () Attempts are being made to correct for these limitations of DFT,Geo () however, these corrections involve uncontrolled approximations and their effectiveness is uncertain at this time. Moreover, DFT-based transport calculations are especially problematic for the case in which the distances between the molecule and the contacts in a molecular junction are being varied, as has been discussed by Koentopp et al.Koentopp () and others.Geo () For these reasons it is unclear whether any counterintuitive transport effect predicted by a DFT-based theory in a molecular junction under tensile strain is physical or an artifact of the shortcomings of DFT and/or any corrections that have been applied to it. We therefore prefer to use instead the semi-empirical extended Hückel theory-based approach to transport that has been outlined and justified in the preceding paragraph.
The calculated conductances of our relaxed structures are plotted in Fig. 2 vs. the junction length , i.e., the distance between the outermost atoms of the two gold clusters. The structure in Fig. 1 (a) (with Å and labeled U in Fig. 2) is an unstrained junction since it was relaxed without constraints on any atoms.
Fig. 2 shows that when a compressive strain is applied to the unstrained junction, the conductance of the junction decreases (red circles to the left of U in Fig. 2). A larger drop in the conductance occurs when the molecular geometry switches from trans (red circles) to gauche (blue circles) under compressive strain. Lower conductances for gauche than trans conformations of other molecular wires have been predicted previously.Li2 (); Paulsson2009 () Note that, in the present work gauche conformations (blue circles) were formed only for Å; for all other junction lengths only trans molecular conformations were found.
On the other hand, stretching the junction from =21.75 Å (U in Fig. 2) to =29.50 Å results in an increasing junction conductance followed by a decrease at =29.75 Å (labeled c in Fig. 2) which signals that the junction is close to rupturing or forming an Au atomic chain as in Fig. 1 (e). In the junction length interval from =20.00 Å to =25.64 Å, i.e. from a to a in Fig. 2, the nearest gold atoms to the PDT molecule bond to three neighboring gold atoms and the junction geometry for the trans molecular conformation stays qualitatively similar to that in Fig. 1(a). At =25.70 Å, labeled b in Fig. 2, a bond breaks between the Au atom at the tip of the right gold cluster and one of its neighbors, as shown in Fig. 1(b). However, interestingly, this bond breaking does not result in any drop in the conductance which, to the contrary, continues to rise. The nature of the resulting bonding geometry is maintained until =26.25 Å, labeled b. At Å labeled c a bond breaks between the Au atom at the tip of the left cluster and one of its neighbors, resulting in a structure similar to that in Fig. 1(c). However, the conductance continues to rise smoothly as the junction is stretched although with an increasing slope until Å (c) where the conductance declines although the structure remains qualitatively similar to that in Fig. 1(c).
We find further stretching to result in either rupture of the junction or formation of a chain of gold atoms between the molecule and one of the gold clusters, depending on the details of the stretching procedure. Formation of the atomic chain results in the structure shown in Fig. 1(e) and a significant drop in the conductance to the value labeled e in Fig. 2. This conductance drop is reasonable because gold atomic chains have only a single conducting quantum channel as compared to the multiple channels in the gold clusters. Compressing the structure in Fig. 1(e) results in the structure in Fig. 1(d) and a decline in the junction conductance from e to d in Fig. 2. Interestingly, although the structure in Fig. 1(d) involves a gold atomic chain, its junction length Å is smaller than that of c ( Å) although the latter structure has no atomic chain and resembles Fig. 1(c). If the junction in Fig. 1 (d) or (e) is stretched, the conductance resumes its increase until the junction length approaches at e in Fig. 2 where a small decrease in the conductance again heralds the appearance of a Au atomic chain, this time at the right Au cluster (Fig. 1 (f)). The formation of this chain with Å again results in an abrupt drop in the conductance to the value labeled f in Fig. 2. Thus an abrupt drop in conductance when an Au atomic chain is formed appears to be a generic property of this system. Upon further stretching of the junction from Å to Å, i.e. from f to f, the conductance increases and then declines while the number of Au-Au bonds between the tips of the gold clusters and the adjacent gold atoms remains the same; the elongation of the junction results in changes in various bond angles in the system.
Our numerical results show that the responses of the electronic energy eigenvalues and eigenvectors of the extended molecule to stretching of the junction both contribute to the rises in the conductance. The details are complex since several pairs of almost degenerate eigenstates mediate conduction through the molecule and the contributions of the two states of each degenerate pair interfere destructively with each other. The reason for the destructive interference is that the two degenerate states have opposite parity (within the PDT molecule) with respect to the approximate left-right mirror symmetry of the PDT molecules in Fig. 1.
However, we shall show that the physics underlying conductance increases as the junction is stretched can be identified by examining the relationship between the conductance and the geometrical changes that occur in the junction. Stretching results in changes in the chemical bond lengths and bond angles in the junction. For example, when the junction length increases from Å to 32.80 Å (i.e., from d to e in Fig. 2), the S-Au bond lengths increase by only a small amount (0.05 Å), while the C-S-Au bond angles at the left (right) end of the molecule change from 110.15 to 118.60 (108.07 to 119.40), indicating that most of the elongation is due to changes in bond angles. Importantly, during this elongation the conductance of the junction increases by a factor of but we find that only a small conductance change can be attributed to the elongation of the Au-S bonds. We find that much greater elongation of the Au-S bonds (by 0.6 Å) would result in large conductance increases (by factors ) due to a transport resonance approaching the electrode Fermi level, qualitatively similar to the findings of previous theories relating the conductance of Au-BDT-Au junctions to the molecule-electrode separation. Andrews (); Ke (); Romaner (); Li3 (); Toher (); Pontes () However, our ab initio junction geometry calculations show such large Au-S bond elongations to be an order of magnitude larger than those predicted to be realized during the experimental conductance measurements Hihath (), and therefore not relevant to the observed conductance increases.
Thus we turn now to the role of the bond angles in the conductance. Fig. 3 shows the dependence on the junction length of the C-S-Au angles, and also the S-Au-Au and Au-Au-Au angles when an Au atomic chain forms. The corresponding angles at the left and right ends of the molecule (solid and hollow circles) are very similar. The dependence of the C-S-Au bond angles (blue circles) in Fig. 3 on the junction length is strikingly similar to that of the conductance in Fig. 2; the main features of either plot are reproduced in the other almost quantitatively. In contrast, although the S-Au-Au and Au-Au-Au angles depend on and the latter angle even changes by more than 30 in the stretching process, the role of these angles is not crucial in the conductance increase, since conduction through Au atoms is mainly via the isotropic orbitals.
To better understand the role of the bond angles, we chose two junctions marked A and C in Fig. 2 and 3. For these we computed the dependence of the conductance on the C-S-Au angles alone, varying these angles by rotating the gold clusters rigidly about the S atoms, holding the PDT molecule fixed. This yielded the solid curves in the inset of Fig. 3. These plots show that for each structure A or C the dependence of the conductance on the C-S-Au angles accounts for roughly half of the conductance change between a and c in Fig. 2, the remainder being due mainly to the structural differences between the A and C gold clusters. A similar calculation but including only one Au atom from each Au cluster yielded the dashed curves in the inset of Fig. 3. Since the difference between the dashed red and blue curves is small and is due to the stretching of the Au-S bonds and structural changes within the PDT molecule itself in going between structures A and C, we conclude that these effects make at most minor contributions to the conductance increase when the junction is stretched.
As has been discussed above, we find the conductances of Au-PDT-Au junctions to increase by similar amounts as the junction is stretched, whether a gold atomic chain is present at the interface between the molecule and one of the gold electrodes, or at the interfaces between the molecule and both of the gold electrodes, or at neither interface. However, the maximal value of this conductance increase is somewhat larger if there is no gold atomic chain present than in either of the latter two cases. By contrast in Ref. French, it has been predicted that for Au-BDT-Au junctions the conductance increase in response to stretching of the junction is much larger if a gold atomic chain forms than if it does not. The mechanism proposed in Ref. French, for this very large conductance increase is unrelated to the mechanism of conductance enhancement in Au-PDT-Au junctions that we propose in the present Article. The conductance increase predicted in Ref. French, is due to a strong feature in the Au and density of states of the atomic chains near the Fermi energy that results in enhanced electron transmission through the molecular junction. Some caution regarding this prediction is in order since it is the result of DFT-based transport calculations although a self-interaction correction intended to address some of the deficiencies of DFT was included in the calculation.French ()
In summary, we have presented a systematic exploration of the response of the conductance of propanedithiolate molecular nanowires bridging gold electrodes to mechanical stretching of this junction, based on ab initio density functional theory and semi-empirical techniques. Our results demonstrate theoretically that the counterintuitive phenomenon of increasing conductance in response to junction elongation should occur in a gold-alkanedithiolate-gold junction. We showed that three different junction geometries can each show a conductance increase in response to junction elongation followed by a conductance decrease and either rupture or a transition to a different junction geometry. A very similar conductance increase followed by a decrease and subsequent junction rupture has been observed experimentally in an Au-PDT-Au junction and can be seen in Fig. 4a of Ref. Hihath, . We showed the conductance increases to be due primarily to increases in the C-S-Au bond angles at the molecule-electrode interfaces and to atomic rearrangements in the gold electrodes. Conductance increases in response to increasing C-S-Au bond angles are reasonable because of the important role that carbon and sulfur atomic valence -orbitals play in transport through organic molecules and the anisotropic nature of the overlaps between these orbitals on different atoms. For this reason we expect the mechanism of the conductance increase under tensile stress that we have identified to be active in all molecular junctions with molecules thiol bonded in a similar way to gold electrodes. Therefore we predict that this interesting phenomenon is a universal generic property of such junctions. We have also demonstrated that changes in the Au-S bond length and in the structure of the propanedithiolate molecule that occur in response to elongation of the junction play a minor role in the observed conductance increase. Our findings also demonstrate that electron transport through the PDT molecule can be controlled by varying the length of the junction and that both continuous modulation of the conductance and abrupt switching between high and low conductance states can be induced in this way. These remarkable changes in the molecular conductance indicate that alkanedithiol molecular nanowires have potential applications as mechanical switches in nanoscale devices.
This work was supported by NSERC, CIFAR, WestGrid, and Compute Canada.
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