Theoretical Study of Long-Range Electron Transport in Molecular

(6) Their experimental results disagree with the conventional theory of exponential distance dependence in the ET rate.(4) Long-range ET is known to p...
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J. Phys. Chem. C 2008, 112, 17408–17415

Theoretical Study of Long-Range Electron Transport in Molecular Junctions Daijiro Nozaki, Yvan Girard, and Kazunari Yoshizawa* Institute for Materials Chemistry and Engineering, Kyushu UniVersity, Fukuoka 819-0395, Japan ReceiVed: July 31, 2008; ReVised Manuscript ReceiVed: September 10, 2008

A study of long-range electron transport through a series of molecular junctions comprised of π-conjugated molecules is presented. A theoretical model is built by using a quantum chemistry based Landauer theory combined with Green’s function formalism and the D’Amato-Pastawski model to describe the electronic conduction in the presence of incoherent scattering effects. Effects of incoherent transport on the total conductance under the nonbiased, stationary, and nondissipative conditions are investigated. Calculation results with the Newns-Anderson approximation show that the distance dependence of electron conduction through molecular wires smoothly interpolates the transition of the dominant transporting mechanism from coherent tunneling with the exponential decay in short distance to incoherent tunneling associated with sequential hopping with weak distance dependence. I. Introduction Interest in molecular electronics has been growing over the past few decades1 since Aviram and Ratner proposed the first molecule-based diode to illustrate the possibility of making molecular components with the same functionality as semiconductor devices.2 Because of synthetic advances and the development of measuring techniques, it is now possible to measure current flow through single molecules or small molecular groups attached between two electrodes. Single-molecule devices showing interesting properties such as bistable switching, negative differential resistance, memory effects, and conformational switching applicable to the components of molecular electronics have been demonstrated by many groups.3 For the further development of molecular electronics, it is important to elucidate the relationships between the conductance of individual molecules in molecular junctions and influential factors such as electronic properties and coupling with contact. In this study, we focus on the relationship between the conductance of molecules and their lengths. Several experimental results show that the conductance of molecular wires with a large energy gap decreases exponentially. The damping behavior can be predicted by standard electron transfer (ET) theory4 since the electron transport through molecular junctions resembles ET in molecular systems. On the other hand, there has been renewed and growing interest in long-range intramolecular ET in macromolecules5 since the report of photoinduced long-range ET through DNA bases by Barton’s group.6 Their experimental results disagree with the conventional theory of exponential distance dependence in the ET rate.4 Long-range ET is known to play an important role in biological processes such as repair of DNA from damage7 and photosynthesis.8 Thus, the interpretation of the experimental results is important since it has wide-ranging implications for DNA regulation of biological process,9 biological applications for diagnostic tests of specific DNA sequences, and industrial applications for catalytic processes. Many theoretical studies have been focused on these issues of long-range ET in DNA6,10 and donor-acceptor systems5 using mainly the density matrix * To whom correspondence should be addressed. E-mail: kazunari@ ms.ifoc.kyushu-u.ac.jp.

formalism. Many theoretical reports have illustrated the crossover of ET rate from exponential decay in short-distance ET to algebraic distance dependence in long-range ET.11-13 Although a large number of experimental and theoretical studies have been made so far on long-range ET in molecular systems,14 few experimental reports are known about long-range electron transport through a molecular junction,15 especially molecular junctions having a small band gap.16 A few theoretical studies have been made on the estimation of long-range quantum transport in molecular junctions at the atomic models including the dephasing effects.13,17 From the mechanical analogy between ET in molecular systems and quantum transport in molecular junctions, it seems natural to consider that the transition of the dominant ET mechanisms in molecular systems also appears in molecular junctions. Thus, when a molecule-based integrated circuit is built by combining molecular devices, we should pay attention not only to coherent transport but also to the dephased incoherent transport since it should play a major role in longrange transport. The aim of this paper is to investigate the influence of the dephasing process on overall conductance and its length dependence. The main results of this paper are presented as follows. In section II, we remark on general aspects about theoretical approaches to address long-range electron transport through molecular junctions and describe a model adopted in this work. In section III, we apply the model to a series of molecular junctions consisting of π-conjugated systems, and in section IV we discuss the results about various properties and roles of incoherent transport in molecular junctions. Finally, we summarize results obtained from this paper in section V. II. Theoretical Background The length dependence of electron transport through molecular junctions is easily derived by classical particle description assuming molecular junctions as linear conductors connected in series.18 Although this simple model can explain the transition of conductance from the exponential decay in Landauer’s model to the algebraic length dependence of Ohm’s law, the effect of phase interference is ignored. Thus, this model should be improved in terms of quantum mechanics for a more realistic description of the length dependence of electron transport in

10.1021/jp806806j CCC: $40.75  2008 American Chemical Society Published on Web 10/11/2008

Electron Transport in Molecular Junctions

J. Phys. Chem. C, Vol. 112, No. 44, 2008 17409

Vk2 Σk(E) ) ) Vk2gk(E), E - Rk - δk

k ) L, R, 1, 2, ..., N

(3)

where Vk is the coupling strength between the kth site of the molecular wire and the left lead, the right lead, or the phasebreaking probe: Vk ) VL(R) if k ) L(R), and Vk ) VDi if k ) 1, 2, ..., N. E is the energy of electron. The term

Figure 1. Schematic description of the molecular junction modeled in this work. The molecular wire is connected to the electron reservoirs via conductive leads. Each lead described by a one-dimensional tightbinding chain is treated within the Newns-Anderson model. The fictitious dephasing probes connect all or some sites to the external electron reservoirs.

molecular junctions including partially dephasing transport. The approach to improve this model is in (1) the use of a nonequilibrium statistical mechanism in conjunction with density matrix and equation of motion,12,13,19 (2) the use of the irregular nature of Franck-Condon overlaps,20 and (3) the use of the Bu¨ttiker probe with scattering theory.21 In this work, we have adopted the D’Amato-Pastawski model, which is the extension of Bu¨ttiker’s approach enabling us to calculate the conductance of disordered chains in the presence of dephasing effects by means of Green’s function formalism.22-24 Figure 1 describes the molecular junction used in this paper. The molecular wire consisting of site 1 to N couples to two electron reservoirs via conductive leads. In the orthogonal basis presentation, the Hamiltonian of the molecular wire is given by

δk )

{

}

1 θ - i 4β 2 - θk2 ; 2 k √ k

k ) L, R, 1, 2, ..., N

(4)

is the self-energy correction due to the coupling of an electrode or a phase-breaking probe within the Newns-Anderson approximation. The parameters are defined as θk ) E - Rk; Rk is the site energy of the electron at the kth site for the left lead, the right lead, or the phase-breaking probe; βk is the nearestneighbor transfer integral for the left electrode, the right electrode, or the phase-breaking probe. Theoretical details of Green’s function, gk(E), for semi-infinite single-band onedimensional tight-binding chains are shown in appendix of ref 26. The problem is to obtain transmission probability of charge injected from the left reservoir to the right reservoir including the partially phase-breaking process. By taking µL > µR and assuming that the temperature is low enough, the effective current flowing through the sample including incoherent transport is given by the D’Amato-Pastawski model in terms of the difference between the chemical potentials µL and µR as

I)

2e T (µ - µR) h eff L

(5)

where

[HM]i,j ) εicˆi+cjδi,j + Vi,j

cˆi+cj + cˆj+ci

](1 - δi,j)

[

(1)

where the first term is the site energy of an electron at the ith site for all electronic states and the second term is the electronic coupling between the ith and jth sites for all possible sites. To make the discussion simple, we assumed that the molecule is connected to the one-dimensional electrodes consisting of orthogonal s-like orbitals, as depicted in Figure 1. We deal with the electrodes within the Newns-Anderson approximation,25 which is a convenient way to parametrize the electrode fundamental parameters: density of states (DOS), energy shift, and escape rate through the self-energy within the Fermi Golden rule.23 The Newns-Anderson approximation remains the issue for the loss of the correct information of the contact and the molecule and causes the quantitative error. However, our essential conclusion can be derived well beyond this simplification by describing more realistic three-dimensional electrodes with multiple channels and bindings. In order to account for the partially phase-breaking process, external electron reservoirs are conceptually introduced and connected to the every site from site 1 to site N via fictitious probes. These phase-breaking probes are also treated likewise with electrodes. Thus, the correction of the molecular Hamiltonian due to the coupling of the phase-breaking probes and the electrodes is given by

Heff ) HM + ΣL + ΣR +

∑

N

Teff ) TL,R +

∑ TR,iWij-1Tj,L

Here Ti,j is the transmission probability from channel j to i (i, j ) 1, 2, ..., N, L, R) and W-1 is the inverse matrix of W, whose matrix elements are given by22-24

Wi,j ) [(1 - Ri,i)δij - Ti,j(1 - δij)]

(2)

k

where ΣL, ΣR, and Σk are, respectively, the self-energies for the left lead, right lead, and phase-breaking probes. Each self-energy term is given by

(7)

This effective transmission Teff has a clear physical interpretation. The first term is the probability of a coherent tunneling through the system, while the second term is the probability of electrons that have already suffered at least once from phase randomizing scatterings at the lateral reservoirs. Consequently, eq 8 defines the effective conductance of the system in the presence of dephasing effects.

Geff )

2e2 T h eff

(8)

To obtain the effective conductance, all transmission probabilities Ti,j from channel j to i between any pairs among all leads and probes have to be required. The transmission probability Ti,j is calculated from

Ti,j(E) ) Trace[GR(E)Γi(E)GA(E)Γj(E)] Σk

(6)

i,j)1

(9a)

where GR(E) (GA(E)) and Γi (Γj) represent retarded (advanced) Green’s functions of the system and the broadening function of the ith (jth) position in the molecule connected to dephasing probe, respectively.22,23 Note that setting i ) L and j ) R in eq 9a leads to the original Fisher-Lee expression,27

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TL,R(E) ) Trace[GR(E)ΓL(E)GA(E)ΓR(E)]

Nozaki et al.

(9b)

Retarded and advanced Green’s functions of this system are given by

GR(E) ) [GA(E)]† ) [(E + i0+)I - Heff]-1

(10)

where 0+ denotes positive infinitesimal. The broadening functions are defined as

Γk ) i[Σk - Σ†k ],

k ) L, R, 1, 2, ..., N

(11)

and related to the scattering rate by the following relation

ki ) 2Γi/p ) 1/τi

(12)

where ki is the scattering rate. Thus, the influence of incoherent transport on total conductance can be analyzed by parametrizing the scattering rate ki. This view was proposed by D’Amato and Pastawski. They adopted a discrete or tight-binding description of the spatial variables at each point (or orbital) in the real space.22,23,28 Indeed, this dephasing model can be used to describe various processes involving power dissipations such as phonons, rotation of phenyl rings, or solvents that can be represented through the Fermi Golden rule.23 However, in this study, we restrict our discussion under nonbiased conditions, zero-temperature approximation, and elastic dephasing, where no energy is exchanged between the transporting electrons and external reservoirs. III. Calculations Figure 2 shows the molecular wires we studied in this work: polyene dithiols (PDs) having (a) even- and (b) odd-numbered carbon atoms and (c) oligo(1,4-phenylene ethynylene) dithiols (OPEDs). We have chosen thiol-terminated molecules as molecular wires and modeled molecular junctions with gold leads. These choices for the terminal sulfur atoms and leads are preferable because it is well-known that the sulfur atom possesses strong affinity for the gold atom in the molecular junctions.29 In order to include the coupling effect of gold leads to the molecular wires, we have calculated the equilibrium geometry of the complexes consisting of molecular wire and two gold clusters using Gaussian 0330 with the B3LYP density functional31 and the LANL2DZ basis set.32 The calculations of the molecules having an odd-number of carbon atoms in Figure 2b were performed using the unrestricted-B3LYP density functional method in the spin doublet and charge neutral state. The coupling strength of the molecular wire to the lead, Vk ) VL(R), was set to -4.898 eV, which corresponds to the

Figure 2. The molecules used in this work: polyene dithiols (PDs) consisting of (a) even- and (b) odd-numbered carbon atoms and (c) oligo(1,4-phenylene ethynylene) dithiols (OPEDs).

coupling between the gold 6s orbital and the sulfur 3p orbital separated by 2.39 Å in the extended Hu¨ckel method.33 This distance was chosen from the ab initio geometry optimization study of thiols on gold surface.34 There are also experimental results supporting this setting.35 We refer to this as the optimal configuration at the molecule/lead interface and adopt it throughout this work. The site energy, RL(R), and nearestneighbor transfer integral, βL(R), for each lead were set to -10.92 and -2.67 eV, respectively. These values are, respectively, the site energy of the 6s orbital of gold atom and transfer integral between two 6s orbitals in the gold atomic chain separated by 2.88 Å at the extended Hu¨ckel level of theory.33 The site energy, RD, and nearest-neighbor transfer integral, βD, for the phasebreaking probes were, respectively, set equal to that of the left and right leads. Because of the numerical efficiency, dephasing proves attached to the hydrogen atoms were omitted. The effective Hamiltonian and overlap matrices were calculated at the extended Hu¨ckel theory,33 which has an advantage of intuitive clarity. From chemisorption studies of small clusters and other metal systems, it is known to give qualitatively correct predictions. Under these conditions, coherent and incoherent transmissions were calculated. IV. Results and Discussion Coherent Transport Characteristics. The left part of Figure 3a,b,c shows the transmission spectra as a function of energy E for PD-based molecular junctions having (a) even or (b) odd number of carbon atoms and (c) OPED-based molecular junctions calculated within the coherent tunneling regime, respectively.36 In Figure 3a,c, the spectra consist of a series of valleys and peaks: the valence peaks formed below -11.3 eV, the conduction peaks formed above -10.5 eV, and the valleys between them, where significant reduction of transmissions is present. In Figure 3b, striking peaks are present in the band gaps. Each peak corresponds to the molecular energy level. Because of the interaction with the leads, each peak is broadened and slightly shifted compared with the bare molecular orbital level. The energy levels of the bare molecular wires and those with small gold clusters are listed in Figure S1 in the Supporting Information. In the PD-based molecular junctions having an even number of carbon atoms in Figure 3a, the HOMO-LUMO gap reduces with the increment of the bridging unit. This is caused by the strong π-π orbital interaction between the incremental bridging unit and the origin. The OPED-based molecular junctions in Figure 3c also show similar characteristics but slightly elusively, representing weaker π-π orbital interactions between the bridging units. We will see later that this orbital interaction strength between unit cells plays an important role in the characteristic length dependence of the incoherent transport. In general, the conductivity of the molecular wires consisting of organic molecules is evaluated by their length dependence of their conductance. This distance dependence is expressed by the Simmons model in the low bias regime: G ) G° exp(-βd), where G° is the quantum conductance, 2e2/h, d is the distance between the two electrodes, and β is a damping factor.37 In this regime, the off-resonant coherent electron tunneling at the Fermi surface, where the Fermi energy of the system lies between HOMO and LUMO, mostly in the middle of the HOMO-LUMO gaps, that is, at midgap, governs electrical conduction.38 In this work, we set the Fermi energy at the midgap as reference energy and evaluate β values for the simplicity. Although each molecular junction has similar but different midgap energy, here the midgap energies were set equal to Emid ) -10.90 eV in

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Figure 3. Transmission spectra (left side) and DOS (right side) of PD-based junctions having (a) even and (b) odd number of carbon atoms and (c) OPED-based junctions. The spectra are divided into three regions in panels a and c: the valence peaks below -11.3 eV, the conduction peaks above -10.5 eV, and the gap between them. In panel b, striking peaks are present between valence peaks and conduction peaks.

Figure 3a and Emid ) -10.80 eV in Figure 3c for simplicity. The estimated damping factors in Figure 3a,c are 0.076 and 0.267 Å-1, respectively. The comparison with the value obtained at a higher level of theory is satisfactory, agreement within a factor 4 nevertheless of simple approximations.38,39 The transmission spectra of PD-based junctions having oddnumbered carbon atoms in Figure 3b show striking resonant conduction peaks at the midgap. The electron transmission does not decay with distance, that is, β ≈ 0.0 at E ) -10.90 eV in Figure 3b. This feature is caused by the soliton formation40 and delocalization of nonbonding π-orbitals throughout the molecular wire in the SOMO level. Figure 4 shows an optimized geometry of PD having odd-numbered carbon atoms, in particular, eleven carbon atoms, with its SOMO. It is known

Figure 4. Optimized geometry of polyene dithiol with two gold atoms having eleven carbon atoms and its SOMO.

that dimerization and soliton formation are two closely related processes, which have been extensively studied, first in the context of insulator-metal transition where dimerization is known as the Peierls distortion41 and later in the field of conducting polymers.42 These two processes arise from the combined effect of electronic correlation and electron-phonon

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Figure 5. The left side shows molecular length dependence of total conductance including incoherent scattering effects for (a) PD-based junctions having even-numbered carbon atoms and (b) OPED-based junctions. Coherent components are depicted in solid lines for comparison. On the right side incoherent components and coherent components are depicted in dashed lines and solid line, respectively. The electron injection energy is chosen at the midgap: (a) E ) -10.90 eV and (b) E ) -10.80 eV. Dephasing strength parameters Γ are shown in the insets corresponding to changing the distances between the dephasing probes and atomic orbitals from 3.0 to 3.6 Å in panel a and from 3.5 to 4.5 Å in panel b at intervals of 0.1 Å.

interaction and play an essential role in quantum transport in quasi-one-dimensional systems. These two effects dramatically modify the DOS and conductivities of molecular junction.40 The dimerization effect creates an energy gap in the molecular band and reduces the coherent transmission, while the soliton formation, seen in a system having odd-numbered sites, creates a new state and resonant channel at the midgap.40 The siteprojected DOS can be easily calculated by means of Green’s function as a function of energy E:

Fk(E) )

|

|

1 Im[Gkk(E)] π

for

k ) 1, 2, ..., N

(13)

and the total DOS at energy E is given by N

D(E) )

∑ Fk(E)

(14)

k )1

The right part of Figure 3 shows the total DOS for three different types of molecular junctions shown in Figure 2 as a function of energy E. From the geometry optimizations, the distance differences between single bonds and double bonds get smaller and lead to an equidistant one-dimensional chain.43 This satisfies the condition required for the resonant tunneling where the current does not decay with distance, as discussed by Mujica et al.40,44 These features are in good agreement with the result obtained by the simple tight-binding description reported by Olson et al.40

Local Environmental Effect on Molecular Conduction. It has been reported that the conductivity of molecular wires differs several orders of magnitude per single molecular wire depending on the local environment.46 The result presented in this work gives insight into the qualitative understanding of the local environmental effect on molecular conduction. We discuss this effect taking the OPED-based junctions as an example. As shown in Figure 5, the crossing point between the coherent and incoherent curve is in conjunction with the dephasing strength. If OPED-based junctions are created on the local environment, for instance, in a densely packed SAM,16 where the free rotation of the phenyl rings is prevented, the dephasing strength is considered to be small for the following reasons: The energy barrier for the rotation of the phenyl rings in a Tour-wire is known to be low enough to enable free rotation.47 Once the rotation of phenyl rings is initiated, the electronic coupling between the neighboring bridging units is weakened and the coherent tunneling is reduced significantly compared with the incoherent tunneling, as shown in Figure 6. Thus, the vibrational modes, which twist the phenyl rings, activate the incoherent tunneling associated with the hopping mechanism. When these twisting fluctuations are suppressed, as mentioned above, the escape rate, that is, the frequency for which the transporting electron is dephased and conceptually trapped into the external reservoir due to interaction with the environment, is reduced. This small dephasing energy or high barrier for initiating

Electron Transport in Molecular Junctions

Figure 6. Length dependence of the (a) coherent and (b) incoherent transport in PD-based junctions with the broadening strength 8.57 × 10-1 eV, artificially scaled in terms of orbital interaction between bridging units by a factor from 1.0 to 0.6 at an interval of 0.1.

incoherent transport could result in the failure to observe a transition of the carrier transporting mechanism and leads to the exponential decay of the total conductance. On the other hand, if the OPED-based junctions are fabricated on the local environment where the transporting electrons will be more frequently dephased than the previous rigid case, the incoherent transport becomes more efficient than the direct coherent tunneling resulting in the crossover of the predominant charge transporting mechanisms. To better understand the environmental effects on carrier conduction through molecular junctions, additional calculations are necessary with respect to the ignored effects in this work such as the phonon effect, the dynamic effect, and the thermal effect including energy dissipation. However, the large discrepancies of molecular conduction per single molecular wire depending on the local environments can be explained from the interpretation in terms of dephasing strength or escape rate qualitatively. The viewpoint given in this work will provide a guideline for the construction of molecular based integrated circuits. Incoherent Transport Characteristics. The left parts of Figure 5a,b show the total conductance (coherent plus incoherent) of PD-based junctions having even-numbered carbon atoms and OPED-based junctions as a function of molecular length at the midgap energy, respectively. Each dashed plot is param-

J. Phys. Chem. C, Vol. 112, No. 44, 2008 17413 etrized in terms of the resulting broadening energy (inset). The coherent components are plotted in solid lines for comparison. Incoherent and coherent components are plotted in the right part of Figure 5. The dephasing strength is controlled by the distance between each end of the dephasing probe and the orbital of molecular wire. Corresponding broadening energies were calculated from the overlap between these orbitals. Details of dephasing strength are listed in Table 1. We can see that the effective transmission probabilities smoothly interpolate the transition of the predominant electron transporting mechanics from coherent tunneling with exponential distance dependence to incoherent tunneling associated with the hopping mechanism that exhibits weak distance dependence. This feature resembles the length dependence of photoinduced ET rate observed in molecular systems. The features of incoherent transports in PD-based junctions are different from the features in OPED-based junctions. The incoherent transport in PD-based junctions at first increases and after that decreases,45 while in the OPED-based junctions, it monotonically decreases with molecular length. This difference is attributed to the number of bridging units and the strength of the orbital interaction between bridging units. On one hand, the increment of bridging unit reduces the incoherent transport. On the other hand, the orbital interaction between the bridging units makes the barrier of the bridging states lower and enhances the transmission since the injected electron energetically comes close to a resonant tunneling state. The length dependence in the incoherent transport depends on the balance of the gain and loss by these two aspects. Figure 6 shows length dependences of the coherent and incoherent transport in PD-based junctions with broadening strength of 8.57 × 10-1 eV, artificially scaled in terms of orbital interaction between the bridging units by a factor from 0.6 to 1.0 at an interval of 0.1. As the orbital interaction between bridging units gets weaker, the characteristics of the incoherent transport leads to monotonic decrease from the estimated feature, at first to increase and then to decrease. Calculations with ab Initio Approach. In this paper, all transmission calculations have been performed by using the extended Hu¨ckel method with the molecular configurations optimized from ab initio calculations. The extended Hu¨ckel theory has the advantage of intuitive clarity and gives qualitatively correct results. However, this may not be accurate as one needs more quantitative pictures. Hopefully, we feel that the qualitative results will be obtained by performing calculations with higher basis including multibonding at the molecule/ electrode interfaces with more accurate Green’s function of three-dimensional electrodes. Although this is beyond the scope of this work, we show a few preliminary transmission calculations of PD-based junctions based on an ab initio approach and that the results calculated on ab initio approach with the Newns-Anderson approximation also lead to the crossing of coherent and incoherent transports seen above in the extended Hu¨ckel approach. The geometry of molecular wires and the coupling strength between the molecular wires and leads were taken from the aforementioned optimal configuration. The coupling strength was set as -2.470 eV corresponding to the ppπ coupling between outermost p-orbital on the gold atom and outermost p-orbitals on the sulfur atom separated by 2.39 Å. The parameters for each lead, the site energy RL (or RR) and nearest-neighbor transfer integral βL (or βR), were set as -1.031 eV and -0.890 eV, respectively. These values are, respectively, the site energy of the outermost p-orbital on the gold atom and the transfer integral for ppπ coupling between the two outermost

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TABLE 1: Dephasing Parameters between s-like Orbitals at the Terminal of Phase-Breaking Probes and Carbon 2p Orbitals in a Molecular Wirea interval of escape process, τ (s)

length between atom and probe (Å)

resonant integral (eV)

3.0 3.1 3.2 3.3 3.4 3.5 3.6

-1.584 -1.336 -1.112 -0.932 -0.773 -0.637 -0.523

(a) Dephasing Parameters in PD-Based Junctions 8.57 × 10-1 3.74 × 10-3 6.09 × 10-1 2.66 × 10-3 4.15 × 10-1 1.82 × 10-3 2.95 × 10-1 1.29 × 10-3 2.04 × 10-1 8.90 × 10-4 1.39 × 10-1 6.06 × 10-4 9.32 × 10-2 4.07 × 10-4

1.30 × 1015 9.21 × 1014 6.28 × 1014 4.58 × 1014 3.08 × 1014 2.10 × 1014 1.41 × 1014

7.72 × 10-16 1.09 × 10-15 1.59 × 10-15 2.24 × 10-15 3.25 × 10-15 4.77 × 10-15 7.10 × 10-15

3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5

-0.637 -0.523 -0.424 -0.348 -0.282 -0.227 -0.183 -0.146 -0.117 -0.0932 -0.0740

(b) Dephasing Parameters in OPED-Based Junctions 3.63 × 10-3 1.40 × 10-1 9.41 × 10-2 2.44 × 10-3 6.04 × 10-2 1.57 × 10-3 4.16 × 10-2 1.09 × 10-3 2.74 × 10-2 7.16 × 10-4 1.79 × 10-2 4.63 × 10-4 1.14 × 10-2 2.97 × 10-4 7.38 × 10-3 1.89 × 10-4 4.47 × 10-3 1.22 × 10-4 2.95 × 10-3 7.65 × 10-5 1.87 × 10-3 4.86 × 10-5

2.13 × 1014 1.43 × 1014 9.18 × 1013 6.32 × 1013 4.16 × 1013 2.72 × 1013 1.73 × 1013 1.12 × 1013 6.80 × 1012 4.48 × 1012 2.85 × 1012

4.69 × 10-15 7.00 × 10-15 1.09 × 10-14 1.58 × 10-14 2.40 × 10-14 3.68 × 10-14 5.78 × 10-14 8.93 × 10-14 1.47 × 10-13 2.23 × 10-13 3.51 × 10-13

broadening, Γ (eV)

energy shift (eV)

escape rate, τ-1 (s-1)

a Escape rate and interval of each escape process are determined from eq 12. Dephasing strengths in PD-based junctions and OPED-based junctions are parameterized from (a) 3.0 to 3.6 Å and from (b) 3.5 to 4.5 Å at an interval of 0.1 Å, respectively.

p-orbitals in the one-dimensional gold atomic chain separated by 2.88 Å calculated at B3LYP/LANL2DZ. The parameters for phase-breaking probes, the site energy RD and nearest-neighbor transfer integral βD, were, respectively, also set equal to those of the left and right leads. Figure S1 of the Supporting Information shows the coherent transmission spectra between the two outermost p-orbitals on the sulfur atoms in PD-based junctions having even-numbered carbon atoms calculated with the Newns-Anderson approximation. Due to the interaction between the leads and molecule, each peak is broadened and shifted compared with the bare molecular orbital level. The energy levels of the bare molecular wires and those with small gold clusters are listed in Table S2 of the Supporting Information. We set the injection energy in the middle of HOMO-LUMO gaps and investigated the decay of coherent conductance through the molecular junctions as a function of molecular length. The decay is exponential (solid line) in Figure S1, and the estimated damping factor is 0.106 Å-1. Figure S2 of Supporting Information shows the length dependence of the incoherent transport through the molecular wires. In each calculation, electron is injected from the midgap. The results calculated from ab initio calculations also have shown the weak length dependence of incoherent transport and the crossing of coherent and incoherent transport.

nential decay in short distance to incoherent tunneling associated with sequential hopping with weak distance dependence, just as ET in macromolecular systems. These results presented in this work could capture the paradigm shift between the exponential decay in nanoscopic molecular junctions and the Ohmic behavior of macroscopic resistors. The relationship among the off-resonant tunneling, the resonant tunneling, and other closely related effects were also revealed on the static analysis. Further elucidation of several transporting mechanisms including disregarded effects in this work will provide deeper understanding of molecular conduction in molecular electronics and a guideline for the establishment of the molecule-based integrated circuits. Acknowledgment. The authors are grateful to Aleksandar Staykov and Tomofumi Tada for useful discussion. K.Y. acknowledges Grants-in-Aid (Nos. 18350088, 18GS02070005, and 18066013) for Scientific Research from Japan Society for the Promotion of Science (JSPS) and the Ministry of Culture, Sports, Science and Technology of Japan (MEXT), the Nanotechnology Support Project of MEXT, the Joint Project of Chemical Synthesis Core Research Institutions of MEXT, and CREST of Japan Science and Technology Cooperation for their support of this work. D.N. and Y.G. thank JSPS for graduate and postdoctoral fellowships, respectively.

V. Conclusions In this paper, we have discussed the long-range electron transport through molecular junctions considering elastic incoherent conduction under the dephasing environment applying the D’Amato-Pastawski model in the tight-binding description. By means of this model, various properties of molecular junctions such as DOS and conductivities are estimated under stationary, equilibrium, and nondissipative conditions. Numerical calculation results with the Newns-Anderson approximation show that the distance dependence of off-resonant electron conduction through molecular wires, injected from midgap energy, smoothly interpolates the transition of the dominant transporting mechanism from coherent tunneling with expo-

Supporting Information Available: Complete ref 30 and two figures and three tables giving the results from ab initio calculations. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Cuniberti, G.; Fagas, G.; Richter, K., Eds.; Introducing Molecular Electronics; Springer-Verlag: Berlin, Heidelberg, 2005. (2) Aviram, A.; Ratner, M. A. Chem. Phys. Lett. 1974, 29, 277. (3) (a) Reed, M. A. Future Electron DeVices Journal 2001, 11, 57. (b) Carroll, R. L.; Gorman, C. B. Angew. Chem., Int. Ed. 2002, 41, 4379. (4) (a) Marcus, R. A. J. Chem. Phys. 1956, 24, 966. (b) McConnell, H. J. Chem. Phys. 1961, 35, 508. (c) Marcus, R. A. Annu. ReV. Phys. Chem. 1964, 15, 155. (d) Marcus, R. A. J. Chem. Phys. 1965, 43, 679.

Electron Transport in Molecular Junctions (5) Isied, S. S.; Ogawa, M. Y.; Wishart, J. F. Chem. ReV. 1992, 92, 381. (6) Murphy, C. J.; Arkin, M. R.; Jenkins, Y.; Ghattia, N. D.; Bossmann, S. H.; Turro, N. J.; Barton, J. K. Science 1993, 262, 1025. (7) (a) Fromherz, P.; Rieger, B. J. Am. Chem. Soc. 1986, 108, 5361. (b) Saito, I.; Takayama, M.; Sugiyama, H.; Nakatani, K. J. Am. Chem. Soc. 1995, 117, 6406. (c) Hall, D. B.; Holmlin, R. E.; Barton, J. K. Nature 1996, 382, 731. (d) Hall, D. B.; Barton, J. K. J. Am. Chem. Soc. 1997, 119, 5045. (8) Szacilowski, K.; Macyk, W.; Drzewiecka-Matuszek, A.; Brindell, M.; Stochel, G. Chem. ReV. 2005, 105, 2647. (9) Kalodimos, C. G.; Boelens, R.; Kaptein, R. Chem. ReV. 2004, 104, 3567. (10) (a) Henderson, P. T.; Jones, D.; Hampikian, G.; Kan, Y.; Schuster, G. B. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 8353. (b) Kelley, S. O.; Barton, J. K. Science 1999, 283, 375. (c) Giese, B.; Amaudrut, J.; Kohler, A. K.; Spormann, M.; Wessely, S. Nature 2001, 412, 318. (11) (a) Jortner, J.; Bixon, M.; Langenbacher, T.; Michel-Beyerle, M. E. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 12759. (b) Okada, A.; Chernyak, V.; Mukamel, S. J. Phys. Chem. A 1998, 102, 1241. (c) Petrov, E. G.; May, V. J. Phys. Chem. A 2001, 105, 10176. (12) Segal, D.; Nitzan, A.; Davis, W. B.; Wasielewski, M. R.; Ratner, M. A. J. Phys. Chem. B 2000, 104, 3817. (13) Segal, D.; Nitzan, A. Chem. Phys. 2001, 268, 315. (14) Jortner, J.; Bixon, M., Eds.; Electron Transfer from Isolated Molecules to Biomolecules; John Wiley & Sons Inc: New York, 1999; Parts 1 and 2. (15) York, R. L.; Nguyen, P. T.; Slowinski, K. J. Am. Chem. Soc. 2003, 125, 5948. (16) Blum, A. S.; Ren, T.; Parish, D. A.; Trammell, S. A.; Moore, M. H.; Kushmerick, J. G.; Xu, G. L.; Deschamps, J. R.; Pollack, S. K.; Shashidhar, R. J. Am. Chem. Soc. 2005, 127, 10010. (17) (a) Segal, D.; Nitzan, A. J. Chem. Phys. 2002, 117, 3915. (b) Zhang, H. Y.; Li, X. Q.; Han, P.; Yu, X. Y.; Yan, Y. J. J. Chem. Phys. 2002, 117, 4578. (c) Xue, Y. Q.; Ratner, M. A. Appl. Phys. Lett. 2003, 83, 2429. (18) (a) Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, U.K., 1995. (b) Jortner, J.; Nitzan, A.; Ratner, M. A. Lect. Notes Phys. 2005, 680, 13. (c) Datta, S. Quantum Transport: Atom to Transistor; Cambridge University Press: Cambridge, U.K., 2005. (19) (a) Segal, D.; Nitzan, A.; Ratner, M. A.; Davis, W. B. J. Phys. Chem. B 2000, 104, 2790. (b) Segal, D.; Nitzan, A. Chem. Phys. 2002, 281, 235. (20) (a) Bixon, M.; Jortner, J. J. Chem. Phys. 1997, 107, 1470. (b) Bixon, M.; Jortner, J. J. Chem. Phys. 1997, 107, 5154. (21) Buttiker, M. Phys. ReV. Lett. 1986, 57, 1761. (22) D’Amato, J. L.; Pastawski, H. M. Phys. ReV. B 1990, 41, 7411. (23) Pastawski, H. M.; Foa Torres, L. E. F.; Medina, E. Chem. Phys. 2002, 281, 257. (24) (a) Buttiker, M. IBM J. Res. DeV. 1998, 32, 317. (b) Buttiker, M. Phys. ReV. B 1986, 33, 3020. (25) (a) Anderson, P. W. Phys. ReV. 1961, 124, 41. (b) Newns, D. M. Phys. ReV. 1969, 178, 1123. (26) Emberly, E. G.; Kirczenow, G. J. Phys.: Condens. Matter 1999, 11, 6911. (27) Fisher, D. S.; Lee, P. A. Phys. ReV. B 1981, 23, 6851.

J. Phys. Chem. C, Vol. 112, No. 44, 2008 17415 (28) Foa Torres, L. E. F.; Pastawski, H. M.; Medina, E. Europhys. Lett. 2006, 73, 164. (29) (a) Andres, R. P.; Bein, T.; Dorogi, M.; Feng, S.; Henderson, J. I.; Kubiak, C. P.; Mahoney, W.; Osifchin, R. G.; Reifenberger, R. Science 1996, 272, 1323. (b) Tour, J. M. Chem. ReV. 1996, 96, 537. (c) Zhou, C.; Deshpande, M. R.; Reed, M. A.; Jones, L.; Tour, J. M. Appl. Phys. Lett. 1997, 71, 611. (30) Frisch, M. J.; et al. Gaussian 03; Gaussian, Inc: Wallingford, CT, 2004. (31) Becke, A. D. J. Chem. Phys. 1998, 98, 5648. (32) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 299. (33) Hoffmann, R. J. Chem. Phys. 1963, 39, 1397. (34) Sellers, H.; Ulman, A.; Shnidman, Y.; Eilers, J. E. J. Am. Chem. Soc. 1993, 115, 9389. (35) (a) Strong, L.; Whitesides, G. M. Langmuir 1988, 4, 546. (b) Whitesides, G. M.; Laibinis, P. E. Langmuir 1990, 6, 87. (36) For the results shown in Figure 3, each geometry of the molecule is slightly shifted from the optimized structure, which is an almost planar structure as shown in Figure 4, to planar structure in order to avoid the noise peaks due to the division error caused in calculating Green’s function. Preliminary calculations using original structures do not show significant changes in the results. (37) Vilan, A. J. Phys. Chem. C 2007, 111, 4431. (38) Tomfohr, J. K.; Sankey, O. F. Phys. ReV. B 2002, 65, 245105. (39) (a) Tomfohr, J. K.; Sankey, O. F. J. Chem. Phys. 2004, 120, 1542. (b) Tada, T.; Nozaki, D.; Kondo, M.; Hamayama, S.; Yoshizawa, K. J. Am. Chem. Soc. 2004, 126, 14182. (40) Olson, M.; Mao, Y.; Windus, T.; Kemp, M.; Ratner, M. A.; Leon, N.; Mujica, V. J. Phys. Chem. B 1998, 102, 941. (41) Peierls, R. E. Quantum Theory of Solids; Clarendon Press: Oxford, U.K., 1955. (42) Conwell, E. M. Semiconductors and Semimetals; Academic Press: Dordrecht, The Netherlands, 1988. (43) Li, Y. W.; Zhao, J. W.; Yin, X.; Yin, G. P. J. Phys. Chem. A 2006, 110, 11130. (44) (a) Mujica, V.; Kemp, M.; Ratner, M. A. J. Chem. Phys. 1994, 101, 6856. (b) Mujica, V.; Kemp, M.; Ratner, M. A. J. Chem. Phys. 1994, 101, 6849. (45) Cresti, A.; Grosso, G.; Parravicini, G. P. J. Phys.: Condens. Matter 2006, 18, 10059. (46) (a) Selzer, Y.; Cabassi, M. A.; Mayer, T. S.; Allara, D. L. Nanotechnology 2004, 15, S483. (b) Selzer, Y.; Cai, L. T.; Cabassi, M. A.; Yao, Y. X.; Tour, J. M.; Mayer, T. S.; Allara, D. L. Nano Lett. 2005, 5, 61. (47) (a) Abramenkov, A. V.; Almenningen, A.; Cyvin, B. N.; Cyvin, S. J.; Jonvik, T.; Khaikin, L. S.; Romming, C.; Vilkov, L. V. Acta Chem. Scand. Phys. Inorg. Chem. 1998, 42, 674. (b) Sipachev, V. A.; Khaikin, L. S.; Grikina, O. E.; Nikitin, V. S.; Traetteberg, M. J. Mol. Struct. 2000, 523, 1. (c) Levitus, M.; Schmieder, K.; Ricks, H.; Shimizu, K. D.; Bunz, U. H. F.; Garcia-Garibay, M. A. J. Am. Chem. Soc. 2001, 123, 4259. (d) Yang, J. S.; Yan, J. L.; Hwang, C. Y.; Chiou, S. Y.; Liau, K. L.; Gavin Tsai, H. H.; Lee, G. H.; Peng, S. M. J. Am. Chem. Soc. 2006, 128, 14109. (e) Greaves, S. J.; Flynn, E. L.; Futcher, E. L.; Wrede, E.; Lydon, D. P.; Low, P. J.; Rutter, S. R.; Beeby, A. J. Phys. Chem. A 2006, 110, 2114. (f) James, P. V.; Sudeep, P. K.; Suresh, C. H.; Thomas, K. G. J. Phys. Chem. A 2006, 110, 4329.

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