Molecular Origins of Conduction Channels Observed in Shot-Noise

NANO LETTERS

Molecular Origins of Conduction Channels Observed in Shot-Noise Measurements

2006 Vol. 6, No. 11 2431-2437

Gemma C. Solomon,† Alessio Gagliardi,‡,§,| Alessandro Pecchia,⊥ Thomas Frauenheim,| Aldo Di Carlo,⊥ Jeffrey R. Reimers,*,† and Noel S. Hush†,# School of Chemistry, The UniVersity of Sydney, NSW 2006, Australia, Institute for Theoretical Physics, UniVersity of Paderborn, D-33098 Paderborn, Germany, PaSCO Graduate School, UniVersity of Paderborn, D-33098 Paderborn, Germany, Bremen Center for Computational Materials Science, UniVersity of Bremen, 28359 Bremen, Germany, INFM- Dipartimento di Ingegneria Elettronica, UniVersita´ di Roma “Tor Vergata”, Via del Politecnico 1, Roma 00133, Italy, and School of Molecular and Microbial Biosciences and the School of Chemistry, The UniVersity of Sydney, NSW 2006, Australia Received June 23, 2006; Revised Manuscript Received August 25, 2006

ABSTRACT Measurements of shot noise from single molecules have indicated the presence of various conduction channels. We present three descriptions of these channels in molecular terms showing that the number of conduction channels is limited by bottlenecks in the molecule and that the channels can be linked to transmission through different junction states. We introduce molecular-conductance orbitals, which allow the transmission to be separated into contributions from individual orbitals and contributions from interference between pairs of orbitals.

The promise of molecular electronics is not just in the small size of molecular components but also in tunable electrical characteristics. If molecular properties can control conduction characteristics, then these could be as varied as the immense range of chemical substitution allows. In the devices developed today, it is not always clear that it is the molecular characteristics, as opposed to properties of the junction, that dominate the transmission.1,2 However, increasingly sophisticated experimental techniques are allowing more information to be obtained about the conducting entity than just simple current-voltage characteristics. Inelastic electron tunneling spectroscopy measurements3 have confirmed that the current is carried by the molecule in question, as opposed to a metal filament for example, and calculations have shown how the relative intensities of peaks in the spectra can indicate certain molecular binding geometries.4 More recently, shot-noise measurements on a single molecule were reported,5 which made it possible to distinguish between proposed binding geometries because of the differing numbers of con* Corresponding author. E-mail: [email protected]. † School of Chemistry, The University of Sydney. ‡ Institute for Theoretical Physics, University of Paderborn. § PaSCO Graduate School, University of Paderborn. | University of Bremen. ⊥ Universita ´ di Roma “Tor Vergata”. # School of Molecular and Microbial Biosciences and the School of Chemistry, The University of Sydney. 10.1021/nl0614516 CCC: $33.50 Published on Web 09/27/2006

© 2006 American Chemical Society

duction channels that were active in each case. The conduction channels observed in shot-noise measurements must have a molecular origin; however, it has not been clear how to describe them in terms of the properties of the molecule. Qualitatively, it is clear that resonant transport occurs at energies close to that of the molecular orbitals of the molecular component of the system.6 However, molecular conduction has, so far, eluded quantitative description in chemical terms. It was shown by Bu¨ttiker that it is possible to separate transmission into components from individual channels7 whose transmission coefficient lies between zero and one. This has been applied to molecular systems8 and necked metallic wires,9-12 but it is not clear how they may be quantitatively described in terms of molecular properties, for example the molecular orbitals, of the system. In the case of single-atom metallic wires, it has been shown both experimentally13 and theoretically11 that the number of channels relates to the valence of the atom; however, this needs to be extended to describe how conduction channels operate in molecular systems. This paper will show how conduction channels can be linked to molecular properties. First, we show that only a small number of conduction channels are expected because of the nature of chemical binding of molecules to electrodes. Second, we show alternatively how conduction channels can be understood in terms of the

connections that they provide between incoming and outgoing junction states. Finally, we show that conduction channels can be described in molecular terms, specifically in terms of molecular-conductance orbitals of the system. Using the ubiquitous Landauer-Bu¨ttiker equation,14 the current (I) through a molecule can be expressed as a function of bias (V) as I(V) )

2e h

∫∞-∞ g(E,V)(fL - fR) dE

(1)

where e is the magnitude of the charge on the electron, E is the electron energy, and f is the Fermi function for each electrode. The quantum conductance, g(E,V) (the sum of the transmission probabilities through all available channels) is defined as g(E,V) ) Tr[ΓL(E,V) Gr(E,V) ΓR(E,V) Gr†(E,V)]

(2)

g(E,V) )

∑i Ti(E,V)

(8)

where Ti are the resulting eigenvalues and 0 e Ti e 1. Performing this analysis for a model system, 1,4-benzenedithiol illuminates a number of interesting features. The molecule is chemisorbed at the bridge site of two gold electrodes as shown in Figure 1, with partitioning of H into its HM, HL, and HR components occurring across the goldsulfur bonds and the calculation is performed using the gDFTB (Green’s function density functional tight-binding) method.15 The first feature of this system is that there are very few eigenchannels with calculable (above 10-14) transmission. Figure 2 shows the eigenchannel transmissions for this system separated into the symmetry components of the molecular-conductance point group.16 The number of eigenchannels is formally the dimension of the molecule (for our minimal basis set this is 46); however, the vast majority of these channels do not contribute to the total transmission. The small number of nonzero eigenchannel transmissions

where the Γ matrices embodying the molecule-electrode interactions. These are defined in terms of the self energies as ΓL(R)(E,V) ) -2Im

∑L(R)(E,V) r

(3)

Gr, the in situ retarded Green’s function for the molecule as modified by the self energies can be determined as Gr(E,V) ) lim [zS - (HM(V) Im zf0+

∑L (E,V) r ∑R (E,V))]-1 r

(4)

where Re z ) E, S is the overlap matrix, and HM is the oneelectron Hamiltonian operator for the molecular part of the system. Finally,

∑L(R) (E,V) ) J†L(R)(V) GrL(R)(E,V) JL(R)(V)

(5)

GrL(R)(E,V) ) lim [zS - HL(R)(V)]-1

(6)

r

Figure 1. Binding geometry of the chemisorbed 1,4-benzenedithiol used in numerical calculations herein.

where

Im zf0

The one-electron Hamiltonians, HM, HL, and HR, as well as the electrode-molecule coupling matrices, JL and JR, are partitioned from the Hamiltonian of the full electrodemolecule-electrode system as

[

HL(V) JL(V) 0 H(V) ) J†L(V) HM(V) J†R(V) JR(V) HR(V) 0

]

(7)

As shown by Bu¨ttiker, the contributions from individual eigenchannels can be determined by diagonalizing ΓL(E,V) Gr(E,V) ΓR(E,V) Gr†(E,V) from eq 2 to yield7 2432

Figure 2. Contributions to the transmission from individual conduction channels separated by symmetry group for clarity. In each symmetry group, the primary conduction channel (dominant transmission at the Fermi energy) is shown in red, the secondary channel is shown in blue, and only in the A1 case is a third channel visible in green. Nano Lett., Vol. 6, No. 11, 2006

can be understood by considering the nature of the chemical binding to the electrodes and the impact this has on the matrices that generate the transmission. Expanding the matrix product used to generate ΓL(R)(E,V) we can write ΓL ij ) -2

r JL kj ∑k JL ik Im G Lkk

ΓR ij ) -2

r JR kj ∑k JR ik Im G Rkk

(9)

If only one electrode orbital k couples with the molecule, then the off-diagonal elements of ΓL(R)(E,V) can be written as ΓL ij ) - xΓL ii ΓL jj ΓR ij ) - xΓR ii ΓR jj

(10)

Matrices of this form have rank 1, that is, only 1 nonzero eigenvalue, independent of their dimension. This will result in any system with one of ΓL(E,V) or ΓR(E,V) taking the form of eq 9 having only one open channel, irrespective of the size of the molecule. Similarly, if the electrode only couples with a small number of molecular states then there will be many near zero elements in JL(R)(E,V). As a result, rows and columns of near-zero elements appear in ΓL(R)(E,V), which in turn leads to eigenchannels with negligible transmission. This is an intuitive picture of transmission; the number of open eigenchannels will not be greater than those supported by the smallest bottleneck in the system. It has long been recognized to apply to necked junctions in atomic wires11 but here it is shown to be generally relevant to bridged-conductor conductivity. Mathematically, this process is readily identified when the bottleneck occurs at the sites at which HM, HL, and HR are partitioned, as described above. However, such a partitioning is not acceptable in many quantitative calculations for which it is necessary to include a number of “electrode” atoms in the “molecule” part of the partitioned system. When such “extended molecules” are used, the number of nonzero matrix elements in JL(R) grows rapidly and the above mathematical arguments no longer apply. We examine this situation numerically, including 25 gold atoms on each side in HM. This results in a modified transmission curve as the influence of the electrodes is treated more precisely through the inclusion of explicit gold atoms in the molecular part compared to when the effect of the electrodes is limited to the self energies, ∑rL(E,V) and ∑rR(E,V).2 However, the number of eigenchannels found with transmission above 10-10 is inVariant to this enhancement. The internal coupling elements between the gold and sulfur atoms within HM are the dominant factor controlling the number of open channels by effectively reducing the rank of the product ΓL(E,V) Gr(E,V) ΓR(E,V) Gr†(E,V). This is an important result: it is the physical and chemical nature of the system, not the choice of partitioning in the calculation, that determines the number of conduction channels with significant transmission. The number of channels with significant transmission is determined by the size of the bottleneck that the connection Nano Lett., Vol. 6, No. 11, 2006

through the sulfur allows and is not determined by the size of the matrix HM, which determines the theoretical limit for the number of perceived eigenchannels. However, through judicious choice of the partitioning, the nature of the conductivity may be described using simple models10,12 or molecular symmetry.16 The number of eigenchannels with nonzero transmission, as well as the total transmission, is dependent on the size of the basis set used for the atoms at the bottleneck. In the above calculations, the basis set used for the sulfur atom has s, p, and d functions. If the d functions are removed, then the A2 channel shown in Figure 2 disappears completely in addition to the changes in shape that come from the total transmission being modified. A result similar to this was obtained in the work of Cuevas et al.11 for tunneling through atomic wires. Significantly, in that application the electronic coupling elements were assumed to connect only nearest-neighbor atoms, resulting in the size of the bottleneck being exactly the dimension of the number of basis functions on a single metal atom. For molecules in comprehensive treatments, it is not always reasonable to assume that only nearest-neighbor coupling elements are nonzero. At the molecule-electrode junction, for example, the accuracy of such an assumption will be dependent on the nature of the binding group. In the case of chemisorbed 1,4-benzenedithiol, direct Au-S and S-C couplings are the dominant terms but an extensive calculation will also give small but nonzero Au-H and Au-C couplings. This means that the number of open conduction channels will not be precisely determined by the Au-S interaction. Beyond the technicalities of the calculations, the significance of this result is that the number of open conduction channels arises from a chemical property of the system. The smallest bottleneck in the system can be controlled by chemical substitution, be it by multiple binding groups between the molecule and the electrode or by different backbone structures for molecular wires. This understanding allows for appropriate synthetic targets to be envisaged without any unforseen constraints in conductivity dominating the conductance characteristics. Additionally, when the bottleneck occurs at the binding site, as was the case for the example system above, the size of this bottleneck, that is, the number of observable conduction channels, will change with changes in the binding geometry. For example, if the molecule was tilted on the surface direct Au-H and Au-C coupling would be more significant and may increase the number of conduction channels observed. Conversely, if the terminal sulfur atom sat directly above a super-surface gold atom or at the apex of a single atom on a gold tip, then this effect may reduce the coupling with the sulfur orbitals of π symmetry and thereby reduce the number of channels observed. This means that molecules such as chemisorbed 1,4-benzenedithiol can be used as probes for the nature of thiol binding on gold surfaces as the number of channels observed in shot-noise measurements will be sensitive to the geometry of the junction region. Understanding the cause of the small number of open eigenchannels does not, however, provide any further il2433

cross, a situation that cannot occur for Hermitian operators, and the eigenvectors are not continuous over this region with the meeting point coinciding with the discontinuity. Also, the eigenvectors are not guaranteed to be orthogonal (zero overlap) and this point actually coincides with the maximum in the overlap between the two eigenvectors. An example of where this unusual behavior occurs is the meeting of two channels around -13 eV in the A1 symmetry transmission. This region is shown in high resolution in Figure 3 with the coefficients on the sulfur atom at the charge-injecting electrode as a function of energy shown below for the primary and secondary conduction channels shown above, the discontinuity being readily apparent around -13.15 eV, whereas the overlap peaks at 0.8, indicating nearly parallel eigenvectors. In fact, the determinant of the eigenvector matrix, which is 1 for an orthogonal transformation, takes a maximum value as a function of energy of just 10-32, indicating a high degree of eigenvector linear dependency. As mentioned previously, the eigenvectors of the Bu¨ttiker channels reflect only the site of charge injection and do not capture the full symmetry of through-molecule conductivity. A means by which this symmetry can be perceived is through pre-diagonalization of the two sets of molecule-electrode interactions ΓL(R)(E,V) as suggested by Troisi and Ratner.17 We ignore the orbital overlap during this process, introducing the (real diagonal) eigenvalue and (real orthogonal) eigenvector matrices Γ′′L(R)(E,V) and DL(R)(E,V), respectively, through Γ′′L(R)(E,V) ) D†L(R)(E,V) ΓL(R)(E,V) DL(R)(E,V) (11) Figure 3. Unusual properties of the eigenvectors associated with the primary and secondary channel of A1 symmetry at the point where they meet but do not cross. The maximum in the overlap and the discontinuities in the coefficients on the sulfur at the charge injecting electrode atom illustrate the unusual characteristics of the eigenvectors associated with conduction channels.

lumination of the relationship between conduction channels and molecular orbitals. Indeed, a striking feature of Figure 2 is that each conduction channel spans a range that encompasses many molecular orbitals further complicating any quantitative description of conduction through a single orbital. As has been reported elsewhere,10 when the eigenvectors are computed in the atomic orbital basis they are difficult to interpret. In fact, the details of the unusual character of the eigenvectors of ΓL(E,V) Gr(E,V) ΓR(E,V) Gr†(E,V) are important. The eigenvectors at all transmission energies E are dominated by coefficients on the sulfur atom at the charge-injecting electrode. The charge injecting electrode (L in the notation used herein) is distinguished by the asymmetry in ΓL(E,V) Gr(E,V) ΓR(E,V) Gr†(E,V), which allows the total transmission to be calculated through the knowledge of the transmission across the interface between the charge injecting electrode and the molecule. Where two channels cross (akin to avoided crossings between molecular potential energy surfaces) the identity of each channel can, in principle, be determined from the eigenvectors. However, there are energies at which two channels meet but do not 2434

If the two junctions are symmetrically related, then Γ′′L(E,V) ) Γ′′R(E,V) ) Γ′′(E,V) so the current from eq 2 can be written without approximation as g(E,V) ) Tr[Γ′′(E,V) Gr′′(E,V) Γ′′(E,V) (Gr′′(E,V))†] ) Γ′′ii(E,V) Γ′′jj(E,V)|Grij′′(E,V))|2 (12)

∑ij

where r Gr′′(E,V) ) D-1 L (E,V) G (E,V) DR(E,V)

(13)

This expression for the transmission suggests that the tunneling current can be viewed as entering one electrode through the input junction channel i, being transferred 2 through the molecule with probability |Gr′′ ij (E,V)| and then leaving through the exit junction j. In this way, the total current can be considered to be decomposed into the sum of n2 independent channels, where n is the number of atomic orbitals of the molecule. This interpretation is formally applicable whenever the eigenvalues Γ′′L(R)(E,V) are guaranteed to be all positive, as is the case when the basis functions do not overlap.18 In general, the arguments used earlier to interpret the Bu¨ttiker conduction channels may again be applied, however, concluding that most of the eigenvalues should be zero with just a few large and positive contributions Nano Lett., Vol. 6, No. 11, 2006

Table 1. Nonzero Eigenvalues of Γii at -5.0 eV Separated into Symmetry-Block Contributions A1 1 2 3 4 5 6 7

A2 10-1

1.9462 × 1.0196 × 10-2 1.5149 × 10-3 7.6416 × 10-5 4.9025 × 10-8 3.6017 × 10-9 -1.1494 × 10-12

1.2691 ×

facilitating current flow. Hence this interpretation, though only formally exact for a restricted problem, is expected to be rather useful in characterizing through-molecule conductivity. For the case of 1,4-benzenedithiol symmetrically chemisorbed between two gold contacts as studied earlier using full treatment of orbital overlap, numerical calculations reveal only 15 junction eigenvalues Γii that are nonzero (i.e., |Γii| > 10-16 within numerical precision); these are listed in Table 1. Of them, only one is negative corresponding to a junction path of A1 symmetry with Γii ) -1.1 × 10-12. Hence, the interference contributions to eq 13 are at least 12 orders of magnitude weaker than the direct contributions and so the current may indeed be perceived in terms of simple contributions from each junction linked through the molecule. From Table 1, it can be seen that a significant number of the junction eigenvalues fall within 3 orders of magnitude of the most prolific junction channel. On the basis of the very small number of Bu¨ttiker channels depicted in Figure 2, a smaller number of significant junction channels could be expected. The change is due to the neglect of the orbital overlap in eq 11, a feature required in order to obtain the ′′ desired |Gijr (E,V))|2 dependence of the molecular contribution in eq 13. The contributions of the dominant terms ′′ from the individual Γ′′ii(E,V) Γ′′jj(E,V)|Gijr (E,V))|2 terms in eq 3 are shown in Figure 4. For each symmetry the con′′ tribution from the Γ′′11(E,V) Γ′′11(E,V)|G11r (E,V))|2 term is in general the most important one. The energy dependence of these probabilities reflects largely the nature of the conducting molecule and is independent of the path that the tunneling electrons takes through the entry and exit channels.

B1 10-3

B2 10-2

8.1665 × 10-3 3.5221 × 10-4 4.7558 × 10-10 9.4243 × 10-14

3.4940 × 4.4071 × 10-3 8.7315 × 10-8

The Bu¨ttiker channels focus on the conductivity bottlenecks and associated shot noise, whereas the Troisi-Ratner channels focus on the nature of both junctions and the net way in which the molecule passes the current, but neither of these illuminates the means by which the molecule facilitates conductivity. To determine this, we take a third approach. Equation 2 may be expanded and written as a four-index sum as g(E,V) )

∑i ∑j ∑k ∑l ΓL ij(E,V) Grjk(E,V) ΓR

kl(E,V)

r† G1i (E,V)] (14)

This form illuminates the complexity of this equation and the difficulties in extracting components of transmission through something akin to a molecular orbital. However, if we diagonalize Gr(E,V) to yield molecular-conductance orbital coefficients C(E,V) given the overlap matrix S it is possible to determine the contributions to the conductance from these individual orbitals. These orbitals are closely related to the molecular orbitals of the system (the orbitals that would be obtained by diagonalizing HM) because HM is the main component of Gr(E,V). However, the molecular conductance orbitals also contain the contributions from ∑Lr(E,V) and ∑Rr(E,V), which introduce the effects of the electrodes and, along with the explicit zS term from eq 4, makes the molecular-conductance orbital coefficients complex as well as dependent on energy and voltage. It is also possible to link the molecular conductance orbitals to the underlying molecular orbitals, this results in a loss of information about the energy and voltage dependence of the self energies; however, it allows transmission features to be understood in terms the common descriptors of molecular systems. We transform the components of eq 2 into the basis of molecular-conductance orbitals with the following transformations: Gr/(E,V) ) C-1(E,V) S-1Gr(E,V) C(E,V) † -1

-1

(G (E,V)) ) (C(E,V)) (G (E,V)) S (C (E,V)) r/

†

†

Γ/L(E,V) Γ/R(E,V) Figure 4. First three conduction channels obtained through the Troisi-Ratner method for each system. In each case the primary channel is shown in green, the secondary channel is shown in blue, and the tertiary channel is shown in yellow. Where visible, the red curve represents the total transmission for each symmetry. Nano Lett., Vol. 6, No. 11, 2006

r

) C (E,V) SΓL(E,V) SC(E,V) †

-1

-1

) C (E,V) ΓR(E,V) (C (E,V))

(15) †

(16) (17)

†

(18)

This somewhat unconventional series of transformations have the unique property that both Gr/(E,V) and (Gr/(E,V))† are diagonal matrices simplifying eq 10 to give 2435

g(E,V) ) Tr[Γ/L(E,V) Gr/(E,V) Γ/R(E,V) (Gr/(E,V))†] )

/ (E,V)|Gr/ii (E,V)|2 + ∑i Γ Lii/ (E,V) Γ Rii / / Γ Lij (E,V) G/ii(E,V)|Γ Rjj (E,V)(Gr/jj (E,V))* ∑i ∑ j*i

(19)

This form can now be readily interpreted in terms of conductance through individual molecular-conductance orbitals. The first term in eq 19 gives the transmission through individual molecular conductance orbitals and the second term gives the interference between pairs of molecular-conductance orbitals. Figure 5 shows the total transmission for each symmetry component of the system as well as the contribution through individual molecular-conductance orbitals (green) and the interference between pairs of orbitals (red). Below each plot of the transmission, Figure 5 also shows |Gr/ii(E,V)|, which is the most rapidly varying quantity in eq 15 as a function of E and hence provides a realistic description of the resonance at each molecular-conductance orbital. Significantly, when conductance is expressed in terms of these two contributions it is evident that at many energies the magnitude of the transmission through individual orbitals is almost exactly matched by that of the interference term. This property is the reason that a simple description of conduction in terms of transmission through single orbitals alone is impossible. Together these plots show the nature of the contributions that different molecular-conductance orbitals make to the transmission. These can be divided into three classes. First, there are molecular conductance orbitals that form sharp resonances in |Gt/ii(E,V)| but do not contribute at all to the transmission through the system. These can be seen in the A2 and B2 plots. The molecular orbitals that underlie molecular-conductance orbitals of this kind are orbitals that do not couple with the electrodes, an example of an orbital of this kind is shown in Figure 6a. Second, there are molecular-conductance orbitals with broader resonances in |Gr/ii(E,V)|, indicating coupling with the electrodes but these

Figure 5. Transmission through individual molecular-conductance orbitals (green) and the interference between pairs of molecularconductance orbitals (red) for each of the four symmetry groups. Shown below in each case is the |Gr/ii(E,V)| giving a description of the resonance at each molecular-conductance obital. 2436

Figure 6. Form of a number of molecular orbitals underlying various molecular-conductance orbitals. Molecular-conductance orbitals arising from orbitals such as a and b result in no net transmission, whereas those such as c contribute to the transmission.

still do not contribute to the total transmission as the transmission through these orbitals is exactly matched by the interference with transmission through another orbital. An example of molecular orbitals underlying this class are the sulfur lone pair orbitals, which do not couple with the π system, the features from these states can be seen in the B2 plot around -6 eV. This class of molecular-conductance orbitals arise from degenerate pairs of delocalized orbitals, the aforementioned pair of lone pair orbitals are shown in Figure 6b. Finally, the third class of molecular-conductance orbitals are those which have broad resonances in |Gr/ii(E,V)| and result in a net increase in transmission; the transmission through these orbitals is greater than the interference terms between them. These may also come from pairs of orbitals, for example the sulfur lone pair orbitals which do couple with the π system as shown in Figure 6c; however, these pairs are not degenerate as the through-molecule coupling induces an energy gap between them. For a pair of orbitals such as those shown in Figure 6b and c, interference plays a very important role. The exact magnitude of the transmission through molecular-conductance orbitals of this kind is controlled precisely by the energy gap between them. Any error in this energy gap can result in either a large increase, if degenerate orbitals are artificially split, or decrease, if the energy gap between a split pair is reduced, in the magnitude of the transmission. This effect has been shown previously in the case of pairs of orbitals split by errors in the electronic structure method.19 Nano Lett., Vol. 6, No. 11, 2006

In that case, orbitals that should have formed degenerate pairs with the transmission through the orbitals being exactly canceled by the interference terms between them were found to have spurious large net transmission. The orbitals in question were in fact localized on gold atoms that were part of the extended molecule and the large net transmission occurred as a closed shell electronic structure method would artificially split the pairs of orbitals resulting in the interference terms not being sufficient to cancel the transmission. The quantitative description of conduction in terms of molecular-conductance orbitals provides a powerful tool for predicting conduction characteristics on the basis of traditional chemical understanding. Chemical substitution on a system such as 1,4-benenedithiol will have a well-understood effect on the electronic structure. By separating the transmission into components arising from individual molecularconductance orbitals and the interferences between them, it is possible for one to predict the effects of chemical substitution on conduction properties simply from the changes in electronic structure. In conclusion, by separating molecular transmission in a variety of ways into, for example, non-interacting or sometimes-interacting components it is possible to isolate the roles of the junction region and the bridging molecule in determining conductivity properties. First, the small number of rigorously defined eigenchannels with nonzero transmission can be understood to arise from the size of the smallest bottleneck in the system. This, in turn arises, from the chemical structure of the molecule or the interaction of the binding group with the electrodes and consequently can be controlled by synthetic modifications. Further, it means that, with careful selection of target molecules, shot-noise measurements can be used to probe the details of molecular binding geometry. Second, by pre-diagonalizing ΓL/R, conduction can be described in terms of molecular conductivity channels linking welldefined junction states. Although this process is only approximate, the interference terms found manifest in our model calculations of conductivity through 1,4-benzenedithiol were 12 orders of magnitude smaller than the primary contributions, making this an excellent practical method for understanding the role of the junction in the process. Indeed, this type of analysis is important for gaining insight into problems such as the operation of inelastic electron tunneling spectroscopy.17,20 Third, the means by which the bridging molecule controls the conductivity can be made manifest by transforming the problem into the basis of molecular-conductance orbitals. This allows for the transmission to be separated into contributions from individual orbitals and the interferences between pairs of orbitals. Significantly, the large contributions that can arise from in particular near-degenerate pairs of orbitals show why a single orbital picture of molecular conduction proves illusory. Interference terms are found to be of critical importance when localized orbitals interact strongly with the alternate electrodes but communicate poorly through the molecule, allowing the localized/delocalized nature of molecular states to be made manifest in an intuitive way. Together these features make it possible to quantitatively describe molecular conduction in terms of atomistic descripNano Lett., Vol. 6, No. 11, 2006

tors of the conducting system. This work significantly extends previous results obtained for conductivity through atomic wires to include general bridged-conductor conductivity. Acknowledgment. G.C.S., J.R.R., and N.S.H. acknowledge the Australian Research Council for funding and the Australian Partnership for Advanced Computing and the Australian Centre for Advanced Computing and Communications for computing resources. A.G. and T.F. acknowledge PaSCo Research Training Group 692 “Scientific Computation”. References (1) (a) Lau, C. N.; Stewart, D. R.; Williams, R. S.; Bockrath, M. Nano. Lett. 2004, 4, 569. (b) Basch, H.; Cohen, R.; Ratner, M. A. Nano. Lett. 2005, 5, 1668. (c) Ke, S.-H.; Baranger, H. U.; Yang, W. J. Chem. Phys. 2005, 123, 114701. (2) Solomon, G. C.; Reimers, J. R.; Hush, N. S. J. Chem. Phys. 2005, 122, 224502. (3) (a) Kushmerick, J. G.; Lazorcik, J.; Patterson, C. H.; Shashidhar, R.; Seferos, D. S.; Bazan, G. C. Nano. Lett. 2004, 4, 639. (b) Wang, W.; Lee, T.; Kretzschmar, I.; Reed, M. A. Nano Lett. 2004, 4, 643. (4) Solomon, G. C.; Gagliardi, A.; Pecchia, A.; Frauenheim, T.; Di Carlo, A.; Reimers, J. R.; Hush, N. S. J. Chem. Phys. 2006, 124, 094704. (5) Djukic, D.; van Ruitenbeek, J. M. Nano. Lett. 2006, 6, 789. (6) (a) Kemp, M.; Roitberg, A.; Mujica, V.; Wanta, T.; Ratner, M. A. J. Phys. Chem. 1996, 100, 8349. (b) Tian, W.; Datta, S.; Hong, S.; Reifenberger, R.; Henderson, J. I.; Kubiak, C. P. J. Chem. Phys. 1998, 109, 2874. (c) Hall, L. E.; Reimers, J. R.; Hush, N. S.; Silverbrook, K. J. Chem. Phys. 2000, 112, 1510. (7) Buttiker, M. IBM J. Res. DeV. 1988, 32, 63. (8) (a) Wu, X.; Li, Q.; Yang, J. Phys. ReV. B 2005, 72, 115438. (b) Heurich, J.; Cuevas, J. C.; Wenzel, W.; Schon, G. Phys. ReV. Lett. 2002, 88, 256803. (9) (a) Brandbyge, M.; Sorensen, M. R.; Jacobsen, K. W. Phys. ReV. B 1997, 56, 4956. (b) Brandbyge, M.; Kobayashi, N.; Tsukada, M.; Phys. ReV. B 1999, 60, 17064. (c) Dreher, M.; Pauly, F.; Heurich, J.; Cuevas, J. C.; Scheer, E.; Nielaba, P. Phys. ReV. B 2005, 72, 075435. (d) Jelı´nek, P.; Pe´rez, R.; Ortega, J.; Flores, F. Phys. ReV. B 2003, 68, 085403. (e) Rubio-Bollinger, G.; Heras, C. D. L.; Bascones, E.; Agraı¨t, N.; Guinea, F.; Vieira, S. Phys. ReV. B 2003 67, 121407(R). (f) Lee, H.-W.; Kim, C. S. Phys. ReV. B 2001, 63, 075306. (10) Jacob, D.; Palacios, J. J. Phys. ReV. B 2006, 73, 075429. (11) Cuevas, J. C.; Yeyati, A. L.; Martin-Rodero, A. Phys. ReV. Lett. 1998, 80, 1066. (12) Bagrets, A.; Papanikolaou, N.; Mertig, I. Phys. ReV. B 2006, 73, 045428. (13) Scheer, E.; Joyez, P.; Esteve, D.; Urbina, C.; Devoret, M. H. Phys. ReV. Lett. 1997, 78, 3535. (14) (a) Landauer, R. Philos. Mag. 1970, 21, 863. (b) Buttiker, M.; Imry, Y.; Landauer, R.; Pinhas, S. Phys. ReV. B 1985, 31, 6207. (15) (a) Pecchia, A.; Di Carlo, A. Rep. Prog. Phys. 2004, 67, 1497. (b) Frauenheim, T.; Seifert, G.; Elstner, M.; Niehaus, T.; Koehler, C.; Amkreutz, M.; Sternberg, M.; Hajnal, Z.; Di Carlo, A.; Suhai, S. J. Phys.: Condens. Matter 2002, 14, 3015. (c) Elstner, M.; Porezag, D.; Jugnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G. Phys. ReV. B 1998, 58, 7260. (d) Porezag, D.; Frauenheim, T.; Kohler, T.; Seifert, G.; Kaschner, R. Phys. ReV. B 1995, 51, 12947. (e) Frauenheim, T.; Seifert, G.; Elstner, M.; Hagnal, Z.; Jungnickel, G.; Porezag, D.; Suhai, S.; Scholz, R. Phys. Status Solidi B 2000 217, 41. (16) Solomon, G. C.; Gagliardi, A.; Pecchia, A.; Frauenheim, T.; Di Carlo, A.; Reimers, J. R.; Hush, N. S., submitted for publication, 2006. (17) Troisi, A.; Ratner, M. A. Nano. Lett. 2006, 6, 1784. (18) Bagrets, A.; Papanikolaou, N.; Mertig, I. cond-mat/0510073. Unpublished work. (19) Solomon, G. C.; Reimers, J. R.; Hush, N. S. J. Chem. Phys. 2004, 121, 6615. (20) Gagliardi, A.; Solomon, G. C.; Pecchia, A.; Frauenheim, T.; Di Carlo, A.; Reimers, J. R.; Hush, N. S., to be submitted for publication, 2006.

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