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Controlling the conductance of molecular wires by defect engineering: a divide et impera

approach

Daijiro Nozaki,1 Horacio M. Pastawski,2 and Gianaurelio Cuniberti1∗1Institute for Materials Science and Max Bergmann Center of Biomaterials,

Dresden University of Technology, D-01069 Dresden, Germany2Instituto de Fısica Enrique Gaviola (CONICET-UNC) and Facultad de Matematica,Astronomıa y Fısica, Universidad Nacional de Cordoba, 5000 Cordoba, Argentina

(Dated: October 17, 2018)

Understanding of charge transport mechanisms in nanoscale structures is essential for the devel-opment of molecular electronic devices. Charge transport through 1D molecular systems connectedbetween two contacts is influenced by several parameters such as the electronic structure of themolecule and the presence of disorder and defects. In this work, we have modeled 1D molecularwires connected between electrodes and systematically investigated the influence of both soliton for-mation and the presence of defects on properties such as the conductance and the density of states.Our numerical calculations have shown that the transport properties are highly sensitive to the po-sition of both solitons and defects. Interestingly, the introduction of a single defect in the molecularwire which divides it into two fragments both consisting of an odd number of sites creates a newconduction channel in the center of the band gap resulting in higher zero-bias conductance thanfor defect free systems. This phenomenon suggests alternative routes toward engineering molecularwires with enhanced conductance.

I. INTRODUCTION

The investigation of quantum transport throughmolecular systems has become an important researchfield in the last few decades.1 Progress in measuring andfabrication techniques2 has led to the continuous minia-turization of electronic devices, which have reached thepoint where quantum effects are important. For exam-ple, semiconductor devices have been reduced in sizeto the nanoscale3,4 and even to the atomic scale.5,6 Inaddition to the report of metal atomic wires,7–11 sta-ble and rigid carbon atomic chains have been reportedrecently.12,13 A key idea behind the advances in the un-derstanding of charge transport through molecular sys-tems is based on the view proposed by Landauer, con-ductance is transmission.14,15

Potential applications of molecular devices range fromnovel computer architectures16,17 to chemical sensors18

and medical diagnostics19. Among the various molecu-lar devices, 1D conductors such as molecular wires,20–22

have been considered to be one of the most fundamentalcomponents for nanotechnology. Due to the reduction ofsize and dimensions of materials, 1D systems show sensi-tive response to external field or intrisic characteristics,which can be exploited for the development of moleculardevices such as biological sensors18 with high sensitivity.

Polymers are one of the most promising materialsfor acting as 1D conductors. Their applicability23

ranges from displays24–26 to thin film transistors,27

photovoltaics,28 and solar cells.29,30 The compatibility ofpolymer materials with light-weight, mechanically flexi-ble plastic substrates and new fabrication methods makethem possible candidates for future electronic devices.

Intrinsic properties of molecules as well as externalperturbations can strongly affect the transport proper-ties of the resulting low dimensional devices. For ex-

ample, the conductance of 1D polymers drastically de-pends on the concentration and position of impurities(dopant)31,32 or defects.33 Other effects which are impor-tant include dimerization and the formation of solitonicdefects. Hence, all of these properties should be consid-ered in the design of 1D devices.

Therefore, in order to facilitate the design of control-lable molecular devices, it is necessary to use a theoreti-cal approach that simplifies the monitoring of how theseintrinsic features affect their transport properties. Suchapproach should allow one to gain understanding anddeep insights into the physical origins of the behavior ofthese materials. The knowledge acquired through thesemodels can be used for the interpretation and elucidationof experimental observations, as guidelines for the plan-ning of experiments, as well as in the design of moleculardevices.

In this paper we model molecular junctions where 1Dmolecular wire is connected between two electrodes usinga tight-binding approach. Then we systematically inves-tigate how the degree of dimerization, and the positionof solitonic and binding defects affect the transport prop-erties of such 1D systems. We calculate the electronicdensity of states (DOS) and the conductance using theLandauer model in terms of the equilibrium Green func-tions.

This article is divided into the following sections. Sec-tion II briefly presents the theoretical framework and sec-tion III presents the effect of solitonic and binding de-fects on quantum transport through molecular systems,respectively. Finally section IV summarizes this article,stressing on the importance of the defect position on thelinear conductor.

The effect of an odd vs. an even number of sites inthe DOS and in the conductance is summarized in Sup-plementary Information A, and the effect of dimerization

2

is summarized in Supplementary Information B. In ad-dition, the relationship between dimerization and lengthdependence of conductance is summarized in Supplemen-tary Information C.

II. THEORETICAL FRAMEWORK

FIG. 1: Tight-binding description of the molecular junc-tion considered in this work. The on-site energy terms,αn;n = 1, 2, · · · , N,L,R, are set to 0 for the simplicity. TheβL/R are hopping integrals for left/right 1D electrode. VL/R

and Vn,n+1;n = 1, 2, · · · , N − 1 are transfer integrals for themolecule/electrode interface in the left/right side and hoppingintegral between nearest neighbors, respectively. The detailof theoretical framework is shown in Appendix A.

We model a molecular junction by connecting a 1Dmolecular wire between two electrodes. We describethis molecular system using a standard tight-bindingHamiltonian34–37 HC that considers only π orbitals.Here, the dimerization is represented by a sequence ofalternating weak and strong bonds while the solitonic de-fect is described by a pair consecutive weak bonds. Forclarity, we model the electrodes as 1D wires with Hamil-tonians HL and HR, which result in a fair representationof infinite reservoirs. Fig. 1 shows the molecular junc-tion considered in this work. The tunneling between thecentral molecule and the left and right electrodes is repre-sented by two matrix elements VL and VR. Note that theelectrode parameters and tunneling amplitude reduced tointo two independent parameters ΓL and ΓR; the escaperates to the left and to the right electrode, respectively.38

The details this of theoretical framework are described inAppendix A.

In principle, we can expand this method to generalmolecular systems which include any type of orbitals. Inthis work, we simplified the model Hamiltonian by onlyconsidering π-orbitals since the contribution from otherorbitals to the conductance near the Fermi energy is neg-ligible in the sp2 carbon systems which we addressed.However, in the case of the molecular wires includingmetal atoms,39,40 where s-, p-, and/or d-orbitals play arole in charge transport, those orbitals have to be takeninto account.

III. RESULT AND DISCUSSION

A. Influence of the position of soliton

Dimerization and soliton formation are closely relatedprocesses which have been extensively studied. Thedimerization has been examined the first in the context ofmetal-insulator transition where dimerization is known asPeierls distortion,41 while the soliton formation has beenexamined in the field of conducting 1D polymers.42 Theseeffects strongly modify the conductance and other elec-tronic properties of 1D polymers. Thus, such effects insingle molecular wires connected between contacts alsoshould be considered. In order to estimate the influ-ence of solitons on transport, we modeled a 1D molec-ular system consisting of an odd number of sites cou-pled with electrodes, and examined the dependence ofthe transmission spectra, total DOS (TDOS) and localDOS (LDOS) on the position of a soliton. For simplicitywe assumed that there is a single soliton on a molecu-lar wire. However, this approach can be also applied tolarger numbers of solitons on molecular wires.

FIG. 2: Schematic description of poly-acetylene-based (PA-based) molecular wires consisting of 19 sites, with single soli-tonic defect, connected between two 1D electrodes. Depend-ing on the position of solitonic defect, 8 cases can be consid-ered. Transfer integrals for double bonds and single bondsare set to Vd = 1.2β and Vs = 0.8β, respectively. Two hop-ping parameters nearby the solitonic defect are set to 1.0β.Coupling constants are set as VL/R = 0.8β. Left/right 1D

electrodes are treated by Newns-Anderson model43,44 withαL/R = 0 and βL/R = β.

Figure 2 shows modeled poly-acetylene-based (PA-based) molecular wires consisting of 19 sites connectedbetween 1D electrodes and allowing for possible 8 soli-ton positions along the molecular wires. Transfer in-tegrals associated with double bonds and single bondsare set to Vd = 1.2β and Vs = 0.8β, respectively. Thetwo hopping parameters on either side of the soliton

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FIG. 3: (a) Transmission spectra and (b) TDOS of the molecular system consisting of 19 sites connected between two leadsparameterized by the positions of a soliton.

FIG. 4: Surface plot of LDOS of molecular wire shown inFig. 2 as a function of energy and site index. The localizedpeaks of LDOS at E = 0 migrate along the wire in conjunctionwith the position of soliton in Fig. 2.

site are set to 1.0β. All on-site energies are set toαn = 0;n = 1, 2, · · · , N . The coupling constants to

the electrodes are set as VL/R = 0.8β. The 1D elec-

trodes are treated by the Newns-Anderson model43,44

with αL/R = 0 and βL/R = β. Fermi energy is set toEF = 0.

Fig. 3 shows calculated transmission spectra andTDOS of the 1D molecular systems with different solitonpositions. Interestingly, depending on the position of thesoliton, the transmission at the Fermi energy (E = EF)changes drastically (See animation in Supplemental In-formation). When the soliton lies near an electrode,the resonances in the transmission spectra and TDOSat E = EF broaden with lower peaks. In this case, inspite of having an odd number of sites, the features ofthe TDOS and transmission resemble those for the PA-based molecular junctions which have an even number ofsites (See Fig. S4(a) or (c) in Supplemental Information).On the other hand, when a soliton lies in the middle ofthe molecular wire, the resonances in the transmissionspectra and TDOS sharpen and get narrower. The valueof the transmission at E = EF approaches its theoreticalmaximum of 1.0.

In order to elucidate the relationship between the po-sitions of a soliton and sharpness of resonant peaks, thedistribution of LDOS along the molecular framework wascalculated. Figure 4 shows the LDOS surface plot as afunction of energy and site index. The LDOS surface plothas peaks near the Fermi energy (E = EF) where the soli-ton is located. The LDOS peaks at E = EF broaden andreduce in height when the soliton is located at the ter-minal of the molecular wire (Fig. 2(a) and (h)), whereasthe LDOS peaks at E = EF sharpen and narrow whenthe soliton is located in the middle of molecule (Fig. 2(d)and (e)).

This feature can be explained as follows. When a soli-ton lies at the terminal of a molecular wire, the local-ized state on the molecular wire strongly interacts withsurface states of the lead broadening the LDOS peaksaround E = EF. This means an electron occupying thesoliton state easily escapes back to the electrode where

4

the electron was injected from, so that the chance of theelectron traveling through to the other electrode is re-duced resulting in low transmission at E = EF. Mean-while, when a soliton lies at the center of a molecularwire, the localized state of the molecular wires interactssymmetrically and weakly with surface states of the twoleads producing sharp LDOS peaks.This conclusion can also be understood in terms of the

generalized symmetry condition.45–48 The transmissionpeak close to the resonance is approximately describedas Tmax(E ∼ E0) = 4ΓLΓR/(ΓL + ΓR)

2, where E0 is theenergy of the molecular orbital associated to the soliton.ΓL/R follows a simple relation, ΓL/R ∝ |VL/R||ΨL/R|2,where ΨL/R is the orbital amplitude of the molecule atthe left/right interface. Thus, symmetric coupling of themolecule with two electrodes, ΓL = ΓR, which corre-sponds to the soliton localization at the center of themolecule in case 4 or 5 in Fig. 2, gives optimal trans-mission probabilities, while asymmetric coupling, as canbe seen in case 1 or 8 in Fig. 2, gives lower transmissionprobabilities.

B. Creation of transmission channels by a defect

We have examined how the position of a soliton mod-ifies the conductance of molecular wires and seen thatthe conductance is greatly reduced when the soliton liesnear electrodes. Here, we investigate the influence of thedefect and demonstrate the counter-intuitive result thatthe introduction of a defect satisfying special conditionscreates a transmission channel leading to higher conduc-tance than in defect free systems.In general, realistic molecular junctions will contain

defects. The defects may induce disorder in the molecu-lar framework. In the presence of defects both the on-siteenergies and the hopping integrals are modified. There-fore, the transport properties of the molecular junctionswill also be changed. Thus we need to consider the effectof defects on quantum transport. We will now examinethe dependence of the transmission and DOS on the po-sition of the defect. Here we only consider fluctuations inthe hopping integrals and introduce a defect as a reducedtransfer integral in a molecular system with an even num-ber of sites. For simplicity we consider a single defect inthe molecular system.Figure 5 shows the schematic description of 1D molecu-

lar wire consisting of 6 sites, with a single defect, coupledto two electrodes. Transfer integrals for nearest neighborsand coupling strength at the left and right interfaces areset to Vn,n+1 = β and VL/R = β/2, respectively. The de-fect is introduced as a reduction of the transfer integralV between the two sites on either side of the defect to avalue of β/2. We considered 5 cases of different positionsof a single defect on a molecular wire and investigatedposition-dependence of the transmission and DOS of themolecular wire.Figure 6 shows the transmission spectra and TDOS of

FIG. 5: Schematic description of a non-dimerized 1D linearmolecule consisting of 6 sites, with a single binding defect,connected between two electrodes. Depending on the posi-tions of the binding defect, 5 cases can be considered (Case1 - 5). Transfer integrals for nearest neighbors and couplingstrength at left/right interface are set to Vn,n+1 = 1.0β; n =1, 2, · · · , N − 1 and VL/R = β/2, respectively. The transferintegral corresponding to the position of the binding defect isreduced to β/2.

the molecular wire shown in Fig. 5. Depending on theposition of the defect, the transport properties changedramatically particularly around E = EF. Fig. 7 showsI-V curves of the molecular wire calculated in the linearresponse regime. Surprisingly, the transmission at E =EF in cases 1, 3, and 5 is higher than for defect freesystems (Fig. 6).

In order to further analyze this behaviour, we calcu-lated the LDOS as a function of energy and site index.Figure 8 shows the surface plot of LDOS as a functionof electronic energy and site index. The position of thedefect is illustrated by changing the colour of the LDOScurve from red to blue on either side. When a defect isintroduced such that the molecular wire is divided intotwo fragments which have an odd number of sites, for ex-ample 3 + 3, the LDOS plot of each fragment, as shownin Fig. 8(c), resembles that of a linear chain having 3sites (N = 3) without defects, as shown in Fig. 3(a).The LDOS in Fig. 8(c) shows non-negligible split peaksaround E = EF. Likewise when a defect is introducedsuch that the molecular wire is divided into fragmentscontaining 1 + 5 or 5 + 1 sites, the LDOS plot of longerfragment resembles that of a linear chain having 5 sites(N = 5) without defects as can be seen in Fig. 3(b).The LDOS in Fig. 8(a) and (e) show non-negligible splitpeaks around E = EF. This is why, despite includingthe single defect, these cases lead to higher transmissionprobabilities and TDOS around E = EF than the defectfree systems shown in Fig. 6 and Fig. 7.

On the other hand, when the defect is introduced suchthat the molecular wire is divided into two fragmentsboth having an even number of sites, for example 2 + 4,the LDOS plot of each fragment resembles that of a linearchain having an even number of sites without defects.The LDOS in Fig. 8(b) and (d) are small around E = EF.Thus, these cases lead to low transmission probabilitiesand TDOS around E = EF as shown in Fig. 6.

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Case 1 & 5Case 2 & 4Case 3 No defect

Case 1 & 5Case 2 & 4Case 3 No defect

(a) (b)

FIG. 6: (a) Transmission spectra and (b) TDOS of the molecular system consisting of 6 sites parameterized by the position ofa defect.

FIG. 7: I-V curves of the molecular wires consisting of 6 siteswith a single defect. The I-V curves of the molecular wireswith a single defect for case 2 and 3 are shown in green andblue, respectively. As a reference, I-V curve for the defect freemolecular wire is shown in black.

IV. CONCLUSION

In summary, in order to elucidate the factors control-ling charge transport through 1D molecules connectedbetween electrodes we have modeled molecular junctionsusing the tight-binding method. Then we calculatedthe the influence of the parameters including molecu-lar lengths, degree of dimerization, odd-even effects, soli-ton formation, and defects on transport properties of themolecular junctions at equilibrium using the Landauerapproach combined with the Green’s function formalism.The numerical calculations have shown that the trans-

port properties at the Fermi energy (EF = 0) dramati-cally change depending on the degree of dimerization andon whether the number of the sites is odd or even. It hasbeen shown that dimerization of the molecular wires as-sociated with Peierls distortion reduces the DOS at the

FIG. 8: Surface plot of LDOS of molecular wires with asingle defect as shown in Fig. 5 as a function of energy andsite index. In each case LDOS plots are depicted in differentcolors across a defect.

Fermi energy, leading to exponential decay of length de-pendence of conductance of the molecular wires. Wealso proved that the damping factor is closely relatedto the degree of dimerization of the molecular wires. In-terestingly, the longer chain system without dimerizationshowed no decay in the conductance at the Fermi energy.In the case of molecular wires with an odd number ofsites, the extra non-bonding state appears in the middleof the band gap contributing to high conductance at lowbias. Additionally, in the molecular wires with an odd

6

number of sites, the conductance at the Fermi energywas independent of the length of the molecular wires.Our calculations also demonstrate that the transport

properties are highly sensitive to the position of solitonsand defects. When a soliton lies near one electrode itstrongly interacts with surface states of that electrodegiving rise to a low, broad peaks in the zero-bias con-ductance, while when a soliton lies in the middle of themolecular wire, the interaction of the localized state withthe electrodes is weaker leading to high conductance atthe Fermi energy with sharp resonant peak. Concerningon the calculation of the influence of defects, we obtaineda counter-intuitive result that in some cases defects in-crease the zero-bias conductance. Our findings are thata defect which divides the molecular wire into two frag-ments both having an odd number of sites creates a newconduction channel and enhances the zero-bias conduc-tance. Since thermal fluctuation naturally creates thissort of defects that, at least for a while, strongly favorscharge transfer, we might infer that this is one of theunderlying mechanisms of transport at room tempera-ture. We believe that the presented systematic studycould be predictive for a wider material (e.g. semicon-ductor/organic interfaces). Our results could guide thesynthesis of novel molecular wires with enhanced conduc-tors.

Acknowledgments

This work was partially funded by the VolkswagenFoundation and by the WCU (World Class University)program through the Korea Science and EngineeringFoundation funded by the Ministry of Education, Sci-ence and Technology (Project No. R31-2008-000-10100-0), and the European Social Funds in Saxony and thecluster of excellence ”ECEMP – European Centre forEmerging Materials and Processes Dresden” within theexcellence initiative of the Free State of Saxony. We ac-knowledge the Center for Information Services and HighPerformance Computing (ZIH) at the Dresden Universityof Technology for computational resources. The workof Horacio M. Pastawski was possible under the EramusMundus Master’s Program at Biotechnologische Zentrum(BIOTEC). We thank Claudia Gomes da Rocha, CormacToher, and Luis Foa Torres for useful discussions.

Appendix A: Theoretical framework

We model a molecular junction by connecting a 1Dmolecular wire between two electrodes. For clarity wemodel the electrodes as 1D wires. We describe this sys-tem using a tight-binding Hamiltonian, considering onlyπ orbitals. The tight-binding parameters used depend ex-plicitly on the interatomic distances in real space. Thus,we can introduce defects or disorder by directly con-trolling on-site energies (diagonal elements) and hopping

integrals (off-diagonal elements) of the electronic tight-binding Hamiltonian matrices.Figure 1 shows the molecular junction considered in

this work. Our starting point is to divide the entire 1Dsystem into three regions; left electrode, right electrode,and central molecular region. We write the electronicHamiltonian as

H = HL + VL +HC + VR +HR,

where HL/R, HC, and VL/R are respectively the Hamil-tonian matrices for the left/right electrodes, for the thecentral molecular region, and the matrix representing thecoupling between the central molecule and the left/rightelectrodes. In this case, we considered only nearest neigh-bor interactions, so that the terms VL/R are scalar val-ues. The matrix elements of HC are given by [HC]m,n =αnδm,n + Vm,n(1 − δm,n), where αn and Vm,n are on-site energies and hopping integral between m-th and n-thatomic orbitals, respectively. In this study, the electrodesare approximated as semi-infinite 1D chains with a single-orbital per site, thus the Hamiltonians of the semi-infiniteelectrodes HL/R have similar forms with HC. For sim-plicity we only consider the interaction between nearestneighbors. The matrix elements of left/right electrodesare given by [HL/R]m,n = αL/Rδm,n + βL/R(1 − δm,n).Hereafter we normalize all hopping integrals, βL/R andVm,n, in terms of β and set all on-site energies as 0,αn = 0;n = 1, 2, · · · , N,L,R.In the orthogonal basis the retarded Green function is

given by

GR(E) = [(E + iη)I −HC − ΣL(E)− ΣR(E)]−1,

where I, η, and ΣL/R(E) are the identity matrix, thepositive infinitesimal, and the self energy in the left/rightelectrode, respectively. The self-energy term ΣL/R(E) is

defined as ΣL/R(E) = VL/RgL/R(E)V †L/R, where gL/R(E)

is the surface Green function of left/right 1D electrode.The left/right electrode consists of equally spaced 1Dsites and coupling between nearest neighbors is set asβL/R = β. The surface Green function of the elec-trode consisting of equally spaced 1D sites consideringonly nearest neighbor interaction can be analytically de-scribed by the Newns-Anderson model43,44 as gL/R(E) =

exp (ik)/βL/R whose derivation49 is summarized in Ap-pendix B.The conductance of a molecular junction in a low bias

is estimated from Landauer’s formula G = G0T (E),

where G0 = 2(for spin) e2

h is the conductance quan-tum and T (E) is the electronic transmission probability.T (E) is obtained from the Fisher-Lee relation50: T (E) =Tr[GR(E)ΓL(E)GA(E)ΓR(E)], where ΓL/R(E) is thebroadening function defined as ΓL/R(E) = i[ΣL/R(E) −Σ†

L/R(E)]. The DOS is calculated from following equa-

tion:

DOS(E) = Im[

G(E)−G†(E)]

/2π.

7

In this study we only consider coherent transport andignore internal scattering effects such as inelastic trans-port or incoherent transport51 associated with electron-phonon coupling.52,53

Appendix B: Analytic solution of surface green

function in semi-infinite 1D electrode

Here we derive the analytic solution of surface greenfunction of a semi-infinite 1D electrode consisting ofequally spaced sites. Consider a semi-infinite 1D elec-trode with lattice constant a as shown in Fig. 9. Each sitehas only one orbital whose position is given by xn = anand its on-site energy is α. We only consider the inter-action between nearest neighbors. The hopping integraland overlap between nearest neighbors are β and S, re-spectively. Energy-dependent coupling between nearestneighbors is defined as βE = β − ES.

FIG. 9: Semi-infinite 1D electrode consisting of equally spacedsites with lattice constant a. α is on-site energy. S and β arehopping integral and overlap matrix between nearest neigh-bors, respectively.

The wave function of semi-infinite electrode can beconstructed from forward- and backward-propagatingplane wave e±ikx:

Ψ(k) =1√2(eikx − e−ikx).

The bra and ket are defined as

|Ψ(k)〉 = 1√2(eikx − e−ikx)|x〉 = 1√

2(2i sin kx)|x〉

〈Ψ(k)| = 1√2(e−ikx − eikx)|x〉 = 1√

2(−2i sinkx)〈x|.

Thus, the wave function at site n is given by:

〈xn|Ψ(k)〉 = 1√2

∑

x

〈xn|Ψ(k)|x〉 = 1√2(2i sinkxn).

This function satisfies the boundary condition of semi-infinite 1D electrode that the wave function on site n = 0is zero since Ψ(x0) = 〈x0|Ψ(k)〉 = 0.

The operator of Green’s function is defined as

GR(E) =∑

k

|Ψ(k)〉〈Ψ(k)|E + iδ − ε(k)

.

The free propagator of the electrode between sites n andn′ is given by

(GR(E))n,n′ =∑

k

〈xn|Ψ(k)〉〈Ψ(k)|xn′〉E + iδ − ε(k)

.

Especially in the case of n = n′ we get

(GR(E))n,n =∑

k

2 sin2 kna

E + iδ − ε(k).

We only need the matrix element at the terminal sitesince this is the only site where the molecular systemcouples, thus putting n = 1 gives

(GR(E))1,1 =∑

k

2 sin2 ka

E + iδ − ε(k).

Hereafter we drop E from (GR(E))1,1. Since the numberof sites is very large thus the summation can be converted

into integral (∑

k → a2π

∫ π/a

−π/a dk) leading to

(GR)1,1 =a

2π

∫ π/a

−π/a

2 sin2 ka

E + iδ − ε(k)dk.

Using E-k relation of 1D electrode consisting of equallyspaced sites (ε(k) = α+ 2βE cos ka), we get

(GR)1,1 =a

π

∫ π/a

−π/a

sin2 ka

E + iδ − α− 2βE cos kadk.

When we define θ = ka, the variable of integral can bechanged as

(GR)1,1 =1

π

∫ π

−π

sin2 θ

E + iδ − α− 2βE cos θdθ.

When we define z = eiθ, then we get (z − 1/z)2 =−4 sin2 θ, (z + 1/z) = 2 cos θ, and dz = izdθ. Substi-tuting these expressions into (GR)1,1, we obtain

(GR)1,1 =1

4iπβE

∮

(z2 − 1)2

z2(z2 − 2pz + 1 + iδ)dz,

where p ≡ E−α2βE = cos θ. The term z2 − 2pz + 1 can be

changed to (z−z1)(z−z2). These z1 and z2 must satisfyfollowing relations: z1z2 = 1 and z1+z2 = 2p. Note thatz1 = 1

z2= eiθ1 satisfies these relations. Then (GR)1,1 is

written as

(GR)1,1 =1

4iπβE

∮

(z2 − 1)2

z2((z − z1)(z − z2) + iδ)dz

=1

4iπβE

∮

f(z)dz.

(B1)

8

We can solve this complex integral using residue theorem:

(GR)1,1 =1

4iπβE2πi

n∑

k=0

Res[f(z)]z=zk .

Figure 10 shows the singular points of function f(z)on complex variable plane. The residual term forRes[f(z)]z=z2 vanishes since the imaginary positive in-finitesimal term iδ move the singular point at z2 outsideof the circle. Therefore, (GR)1,1 is

FIG. 10: Singularity points of the function f(z) on complexvariable plane. The gray zone is surrounded by unit circlez = eiθ. Because of the imaginary term iδ in Eq. (B1), thesingularity points, z1 and z2 move to inside and outside of thecircle, respectively.

(GR)1,1 =1

2βE(Res[f(z)]z=z0 +Res[f(z)]z=z1),

where

Res[f(z)]z=z1 = [(z − z1)f(z)]z=z1 = z1 − z2,

and

Res[f(z)]z=z0 =

[

d

dz

(z2 − 1)2

(z − z1)(z − z2)

]

z=z0

= z1 + z2.

Finally, we get the surface green function of 1D electrodeconsisting of equally spaced sites as

ΣL/R(E) = V EL/R(G

R0 (E))1,1V

E†L/R = V E

L/R

eiθ

βEV E†L/R,

where, V EL/R = VL/R − ESL/R is energy-dependent cou-

pling strength between the electrode and molecule innon-orthogonal description, where SL/R is the overlap be-tween the terminal site in the electrode and the molecule.

∗ Electronic address: g.cuniberti@tu-dresden.de1 Cuniberti G, Fagas G and Richter K, Eds. 2005 IntroducingMolecular Electronics. (Springer-Verlag Berlin Heidelberg)

2 Agraıt N, Levy-Yeyati A and van Ruitenbeek J M 2003Phys. Rep. 377 81

3 Hisamoto D, Lee W-C, Kedzierski J, Takeuchi H, AsanoK, Kuo C, Anderson E, King T-J, Bokor J and Hu C 2000IEEE, Trans. Elec. Dev. 47 2320

4 Chang P-C, Fan Z, Chien C-J, Stichtenoth D, Ronning Cand Lu J G 2006 Appl. Phys. Lett. 89 133113

5 Huang Y, Duan X, Cui Y and Lieber C M 2002 Nano Lett.2 101

6 Singh N, Agarwal A, Bera L K, Liow T Y, Yang R, RustagiS C, Tung C H, Kumar R, Lo G Q, Balasubramanian Nand Kwong D-L 2006 IEEE, Elec. Dev. Lett. 27 383

7 Ohnishi H, Kondoh Y and Takayanagi K 1998 Nature 395,780

8 Yanson A I, Bollinger G R, vand den Brom H E, Agraıt Nand van Ruitenbeek J M 1998 Nature 395 783

9 Smit R H M, Untiedt C, Rubio-Bollinger G, Segers R Cand van Ruitenbeek J M 2003 Phys. Rev. Lett. 91 076805

10 Bettini J, Sato F, Coura P Z, Dantas S O, Galvao D S andUgarte D 2006 Nature Nanotechnol. 1 182

11 Nilius N, Wallis T M and Ho W 2003 Phys. Rev. Lett. 90186102

12 Jin C, Lan H, Peng L, Suenaga K and Iijima S 2009 Phys.Rev. Lett. 102 205501

13 Chuvlin A, Meyer J C, Algara-Siller G and Kaiser U 2009New J. Phys. 11 083019

14 Landauer R 1981 Phys. Lett. 85A 9115 Buttiker M 1986 Phys. Rev. Lett. 57 176116 Ellenbogen J C and Love J C 2000 Proceedings of the IEEE

88 38617 Joachim C, Gimzewski J K and Aviram A 2000 Nature

408 54118 Cui Y, Wei Q, Park H and Lieber C M 2001 Science 293

128919 Patolsky F, Zheng G, Hayden O, Lakadamyali M, Zhuang

X and Lieber C M 2004 Proc. Natl. Acad. Sci. USA 101

1401720 De Cola L Ed. Molecular Wires: From Design to Proper-

ties. Top. Curr. Chem. 257 (Springer-Verlag Berlin Hei-delberg, 2005).

21 Calev Y, Cohen H, Cuniberti G, Nitzan A and Porath D2004 Israel J. Chem. 44 133

22 Lang N D 1997 Phys. Rev. Lett. 79 1357

9

23 Forrest S R 2004 Nature 428 91124 Burroughes J H, Bradley D D C, Brown A R, Marks R

N, McKay K, Friend R H, Burns P L and Holmes A 1990Nature 347 539

25 Friend R H, Gymer R W, Holmes A B, Burroughes J H,Marks R N, Taliani C, Bradley D D C, Dos Santos D A,Bredas J L, Logdlund M and Salaneck W R 1999 Nature397 121

26 Kulkarni A P, Tonzola C J, Babel A and Jenekhe S A 2004Chem. Mater. 16 4556

27 Katz H E, Bao Z and Gilat S L 2001 Acc. Chem. Res. 34359

28 Gratzel M 2001 Nature 414 33829 Gunes S, Neugebauer H and Sariciftci N S 2007 Chem.

Rev. 107 132430 Blom P W M, Mihailetchi V D, Koster L J A and Markov

D E 2007 Adv. Mater. 19 155131 Chiang C K, Fincher Jr. C R, Park Y W, Heeger A J,

Shirakawa H, Louis E-J, Gau S C and MacDiarmid A G1977 Phys. Rev. Lett. 39 1098

32 Chiang C K, Dury M A, Gau S C, Heeger A J, Louis E-J, MacDiarmid A G, Park Y W and Shirakawa H 1978 J.Am. Chem. Soc. 100 1013

33 Gregg B A 2009 J. Phys. Chem. C 113 589934 Su W P, Schrieffer J R and Heeger A J 1979 Phys. Rev.

Lett. 42 169835 Mujica V, Kemp M and Ratner M A 1994 J. Chem. Phys.

101 684936 Mujica V, Kemp M and Ratner M A 1994 J. Chem. Phys.

101 685637 Pastawski H M, Weisz J F and Albanesi E A 1995 Phys.

Rev. B 52 1066538 Pastawski H M and Medina E 2001 Rev. Mex. Fis. 47s1 139 Tyler D R 2007 Frontiers in transition Metal-Containing

Polymers (Wiley-Interscience, Weinheim, Germany)40 Bera J K and Dunbar K R 2002 Angew. Chem. Int. Ed.

41 445341 Peierls R E 1955 Quantum Theory of Solids (Clarendon

Press; Oxford, U.K.)42 Conwell E M 1988 Semiconductors and Semimetals (Aca-

demic Press: Dordrecht, Netherlands)43 Anderson P W 1961 Phys. Rev. 124 4144 Newns D M 1969 Phys. Rev. 178 112345 Ricco B and Azbel M Y 1984 Phys. Rev. B 29 197046 Foa Torres L E F, Pastawski H M and Makler S S 2001

Phys. Rev. B 29 19330447 D’Amato J L, Pastawski H M and Weisz J F 1989 Phys.

Rev. B 39 355448 Levstein P R, Pastawski H M and D’Amato J L 1990 J.

Phys.: Condens. Matter 2 178149 Emberly E G and Kirczenow G 1998 Phys. Rev. B 58 1091150 Fisher D S and Lee P A 1981 Phys. Rev. B 23 685151 Nozaki D, Girard Y and Yoshizawa K 2008 J. Phys. Chem.

C 112 1740852 Ryndyk D A and Cuniberti G 2007 Phys. Rev. B 76 15543053 Frederiksen T, Brandbyge M and Jauho A-P 2004 J.

Comp. Elec. 3 423