electronic transport calculations for self-assembled ... · electronic transport calculations for...

PHYSICAL REVIEW B 69, 085324 ~2004!

Electronic transport calculations for self-assembled monolayers of 1,4-phenylene diisocyanideon Au„111… contacts

Robert Dahlke and Ulrich Schollwo¨ckSektion Physik and Center for Nanoscience, LMU Mu¨nchen, Theresienstrasse 37, D-80333 Mu¨nchen, Germany

~Received 8 October 2003; published 27 February 2004!

We report on electronic transport calculations for self-assembled monolayers of 1,4-phenylene diisocyanideon Au~111! contacts. Experimentally one observes more structure~i.e., peaks! within the measured conduc-tance curve for this molecule with two isocyanide groups, compared to measurements with molecules havingthiol groups. The calculations are performed on the semiempiric extended Hu¨ckel level using elastic scatteringquantum chemistry, and we investigate three possible explanations for the experimental findings. Comparingthe experimental and theoretical data, we are able to rule out all but one of the scenarios. The observedadditional peaks are found to be only reproduced by a monolayer with additional molecules perturbing theperiodicity. It is conjectured that the weaker coupling to Au of isocyanide groups compared to thiol groupsmight be responsible for such perturbations.

DOI: 10.1103/PhysRevB.69.085324 PACS number~s!: 73.23.2b, 72.10.2d, 85.65.1h

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I. INTRODUCTION

Within the last decade an increasing interest in molecuelectronics has developed, with the expectation of realizmolecular diodes and transistors. This is based onprogress in manipulation techniques, which now allowcontrolled attachment of atomic scale structures like mecules to mesoscopic leads. With these new devices onable to determine the conductance properties of molecstructures. Explaining and predicting the electronic behavof such devices is an essential step towards their designuse as nanoscale electronic circuits.

To this end a number of theoretical studies have bperformed with the aim of reproducing measuredIV charac-teristics. These studies differ in the way they take the etronic levels of the molecules, their modification by the copling to the leads, and the change of electrostatic potendue to bias into account. Semiempiric methods1–3 have beenused, as well as first principles techniques,4–7 the latter beingrestricted to systems of moderate size.

The wide range of experimentally observed behavior~seeSec. II! suggests that not only the structure of the molecubut also the details of the device fabrication, affect the cduction properties of molecular devices. The crucial stepthe deposition of molecules onto the surface of the lead.this is achieved by self-assembly the amount of adsormolecules and their individual positions cannot be exaccontrolled and therefore remains unknown. A satisfactoryderstanding of the interplay between geometrical alignmof the molecules and measured conductance propertiesthus not yet been achieved~for a recent review, see, e.g., Re8!.

In this paper we study the way in which changing tgeometrical alignment of the monolayer has an influencethe conduction properties of a molecular device. In so dowe can rule out a number of explanations which have preously been considered9 to explain the occurrence of add

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tional structure in the conductance-voltage~CV! characteris-tics.

The outline is the following: first we summarize somethe recent experimental findings. Then the method we~based on elastic scattering quantum chemistry10! for calcu-lating the conductance properties of molecular devicesdiscussed. Calculations for the conductance of a sassembled monolayer, being built of 1,4-phenylene diisocnide ~PDI! and sandwiched between gold leads are then psented. The results for three qualitatively differegeometrical constitutions of the mono-layer are compawith experimental data. By this we can rule out all but oand conclude that the only geometrical alignment, whgives rise to several peaks in the conductance curve,mono-layer with additional molecules perturbing the perioicity.

II. EXPERIMENTAL OVERVIEW

The devices built to study conductance properties of mlecular structures differ not only in amount and chemicstructure of the molecules in use but also in the way theseattached to metallic or semiconducting leads. Single or fmolecules are accessible in mechanically controllable brjunctions ~MCBs! and with the scanning tunneling microscope~STM!. Many molecules are involved in sandwicheself-assembled monolayer~SAM! experiments. The observeproperties depend on the exact geometry of the device.conductance differs in orders of magnitude and the quative voltage dependency of the current ranges from simohmic behavior to negative differential resistance.11

In the past Reedet al.12 have measured the electrical coductance of a self-assembled molecular monolayer bridga MCB at room temperature. Molecules of 1,4-benzedithiol ~i.e., having two thiol groups, which are known tcouple strongly to Au atoms! were used and theCV charac-teristic was found to be symmetric with one peak in tvoltage range of 0–2V. They measured a current of the or

©2004 The American Physical Society24-1

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ROBERT DAHLKE AND ULRICH SCHOLLWOCK PHYSICAL REVIEW B 69, 085324 ~2004!

of 50 nA at a bias voltage of 2 V, which they claim is prduced by transport through one single active molecule.ichert et al.13 also used a MCB with molecules having twthiol groups, but being considerably longer. The measucurrent amplitude was about 500 nA at 1 V, i.e. althoughmolecule was more than twice as long, the current wastimes larger.

With a different setup, where a SAM is sandwiched btween two metallic leads, Chenet al.11 have found negativedifferential resistance, namely one peak at 2V in theIVcurve. The molecule under investigation had one thiol groonly and was attached to Au leads at both ends. The msurements were taken at room temperature and the meacurrent maximum was of the order of 1nA.

Only recently, sandwiched SAM devices at 4.2 K westudied,9,14 where a benzene ring with two isocyanide insteof thiol groups was used~see Fig. 1!. The measurementexhibited currents of the order of 50–400 nA. TheCV char-acteristic for this molecule revealed more structure, in foof three to five peaks within a voltage range of 1 V. Suchbehavior was not observed with previous devices containother molecules.

III. THEORETICAL FORMALISM

In the literature there has been presented quite a numof techniques to calculate nonequilibrium electronic traport through molecular systems attached to mesoscleads. Usually the Landauer formalism is applied, whichscribes current as elastic electron transmission and thererequires the transmission functionT(E). To this end oneneeds a framework that allows for a description of the mlecular device on the atomic level. This involves not only tmolecules themselves, but also the surface and bulk reof the leads. Quantum chemistry provides such a framewand one can choose the level of theory according to theof the system under consideration and the computationafort one is willing to spend.

Using a quantum chemistry method, the transmissfunction can be obtained from an effective one-partiHamiltonian, which is an appropriate description for strocoupling of the molecules to the leads~as in the case ocovalent binding!. The methods differ in the generation othe one-particle Hamiltonian, which might be based on seempirical grounds1–3 or on first-principles and self-consistetechniques.4–7

A different approach,15 taking many-particle effects explicitly into account, uses a master equation with transitrates calculated perturbatively using the golden rule. Tapproach is appropriate for weak coupling.

We use the Landauer formalism, as the molecules aresumed to be chemically bonded to the gold contacts~i.e.,strong coupling!, together with the semiempiric extende

FIG. 1. Molecular structure of 1,4-phenylene diisocyanide.

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Huckel elastic scattering quantum-chemistry~ESQC!method.10 The molecular structure is optimized16 beforehand.This approach, though limited as compared to more sopticated quantum chemistry methods, is yet justified becawe want to gain a qualitative understanding of a many mecule experiment which cannot be described by firprinciple techniques, as the number of atoms involved isyond the practical limitations of to-date computer resourc

A. Landauer formalism

According to the Landauer formula, current along a defregion is the result of electron transmission from the souto the drain lead, described by the transmission functT(E). For chemical potentialsm1 andm2 of source and drainlead, shifted with respect to each other by an applied voltm15m21eV, the current reads

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T~E!@ f ~E2m1!2 f ~E2m2!#dE, ~1!

where f (E) is the Fermi function. The Landauer formulavalid under the condition that transport is coherent acrossmolecule, which is plausible as the typical mean free pathelectrons within metals is of the order of 500 nm, while tmolecular gap between source and drain lead is only ab1–5 nm in length.

The system is formally partitioned into three regioS i ,i P$0,1,2%, two of them (S1,2) containing the semi-infinite leads, the third one (S0) being the finite region con-taining all molecules as well as a few surface layers of elead~see Fig. 2!. We use periodic boundary conditions in thdirections perpendicular to the surface normal.

By a tight binding approximation, the infinite-dimensionHamiltonian of the entire system can be composed of qutum chemical one-particle block Hamiltonians of finite dmension:

FIG. 2. ~Color online! Partitioning of the system into three partthe two semi-infinite leadsS1,2 ~surrounded by boxes! and the mo-lecular regionS0 between them.

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ELECTRONIC TRANSPORT CALCULATIONS FOR SELF- . . . PHYSICAL REVIEW B 69, 085324 ~2004!

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The first summation describes the isolated molecular regby an on-site energy« and a hopping term. The indicesi andj run over the orbital basis set within that region. The ntwo summations describe the isolated leads, labeled bd.Layer by layer, starting with the surface layerl 0, the firstterm accounts for intra-layer interactions, while the secoone describes the interaction between layers. Finally theterm describes the coupling between the molecular regand each lead. Note that only the first layerl 0 contributes tothat term and that there is no interaction between differleads. These are only formal restrictions, as parts of elead can be included into the molecular region.

The determination of the transmission function involvtwo steps. First the conduction properties of the isolaleads have to be calculated. Thereby each lead will becomposed into conducting and nonconducting incomingoutgoing channels. These correspond to propagating andnescent solutions moving in one of two possible directiorespectively. In a second step, the channels are connecteach other via the molecular region. This is described byscattering matrix and the transmission function is finally otained by summing up the contribution from each channe

The calculation can be performed either using Greefunction techniques17 or equivalently18 using ESQC,10,19

which is a scattering-matrix approach. We present the deof the calculation in the second scheme, as individual conbutions from each channel to the transmission functionthen be easily studied.

B. Bulk propagator

First we will restrict our attention to the semi-infinite leaHamiltonians, which do not have to be identical. The Hamtonians of Eq.~2! for one leadd, namely,Hll 8

d , are layerindependent, if one assumes periodicity, i.e.,Hll

d 5Hl 0l 0d and

Hl ,l 11d 5Hl 0 ,l 011

d . Using Bloch’s theorem one can reduce t

infinite dimensional system of equations to anN3N-matrixequation (N being the number of orbital basis functionsone layer!

@Md~E!1hd~E!eikD1hd†~E!e2 ikD#g l~k,E!50, ; l ,

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with Md(E)ªHl 0l 0d 2ESl 0l 0

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andSll 8d is the overlap matrix between orbitals in layerl and

layer l 8 of leadd for cases when one does not deal with

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5Id•d l l 8). With D wedenote the lattice spacing and the layer coefficientsg l obeythe relation g l 115eikDg l . Defining lªeikD one caneasily see that Eq.~3! is anN3N quadratic eigenvalue equation. It can be transformed into a 2N32N linear eigenvalueproblem:

Pd~E!S g l

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g l 11D , ~4!

Pd~E!ªF 0 1

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† 2hd21MdG ~5!

~where we have dropped the energy- andk-dependency ofg lfor notational ease!. This layer-to-layer propagatorPd(E)also connects the coefficients of adjacent layers

S g l

g l 11D 5Pd~E!S g l 21

g lD ~6!

and therefore reduces the problem of finding solutions forentire isolated lead Hamiltonian to specifying the wave funtion coefficients at two adjacent layersg l andg l 11 only.

All possible solutions at energyE can be decomposed intindependent channels, by solving for the eigenvalues of~4!. These eigenvalues come in pairs such that for eachgenvaluel. , there exists a corresponding eigenvaluel,

satisfying the relationl.51/l,* , as can be seen by transposing Eq.~3!. Eigenvalues withuluÞ1, i.e., complexk, be-long to exponentially diverging solutions@see Eqs.~6! and~4!#. These are of course non physical, as long as the leainfinite. In semi-infinite leads however~which we are dealingwith!, exponentially decaying coefficients at the boundawill contribute to the surface wave function and must notneglected.

C. Current operator

The contribution from a single channel to the net currecannot directly be seen from Eq.~4!. It depends on the current density associated with a solution to the Schro¨dingerequation i\] tSg5Hg and is obtained via the continuitequation. The probability amplitudeugu2 for a stationary so-lution is constant in time,

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becauseH andSare hermitian. For the probability amplitudat all layers betweenl 1 and l 2 one therefore has

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† h†~E!g l 12H.c.]

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† h~E!g l 21g l 211

† h†~E!g l 22H.c.]

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with the projectors l ug&ªg l . This gives rise to the definition of the current operatorWl for layer l as

Wlªu l ,l 11&i

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h† 0 G ^ l ,l 11u. ~9!

Now let bothw andq be solutions at fixed energyE with theeigenvaluesl1 andl2 respectively. Because the expectativalue forWl is layer independent@Eq. ~8!# one has

^quWl uw&5^quWl 11uw&

5l1l2* ^quWl uw&. ~10!

This equation describes the connection between the cuproperties of a solutionw and its eigenvaluel. We summa-rize the results of a detailed analysis of this equation, whis given in Appendix A. Each channeluw i& can be assignedcurrent valuev i , defined as

v iªIm^w i uWuw i&, ~11!

where we have used the layer independence ofWl in simplywriting W.

Channels with eigenvalue modulusuluÞ1, i.e., evanes-cent waves have zero current value. They therefore docontribute to the current.~Yet they are important at the suface, as already mentioned above.! Only channels with aneigenvalue of modulus 1 (ulu51) contribute to the currentThe sign ofv i determines the direction of charge transpo

Solutions for an isolated lead are linear combinationspropagating waves in opposite directions, with the saamount of current being transported in each direction, tcarrying no net current, and resulting in a standing wave

We now defineL. and L, as the twoN3N diagonalmatrices composed of all incoming and outgoing eigenvalL:ªdiag(l:

i ). The 2N32N matricesU andU21PU havethe following forms:

U21PU5FL. 0

0 L,G , UªF U. U,

U.L. U,L,G .

~12!

D. Scattering matrix

Up to now, we have considered the isolated leads oThese are now assumed to be each coupled to the moledefect region and thereby indirectly coupled to one anotWe are interested in stationary solutions which consist ofincoming propagating wave in one lead, being scatteamong all the accessible outgoing channels~propagating andevanescent ones!. This information is contained in the scatering matrixS, which is shown in Appendix B to be of thform

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It is important to notice that the scattering matrix is aways quadratic, because in each lead there are the samount of incoming and outgoing channels and the scaing matrix connects all outgoing channels to all incomiones. This is opposed to the transfer matrixT, which deter-mines the amplitudes of in- and outgoing waves in the drlead given the in- and outgoing waves of the source leThis matrix is quadratic only if both leads have the sanumber of channels. It then is of the form20

T5F F G†

G F†G , ~14!

and the relation to the scattering matrix is20

S5F2F†(21)G F†(21)

F21 G†F†(21)G . ~15!

Methods calculating the scattering matrix via the transmatrix19 fail, if two types of leads are used, becauseF is thenno longer quadratic and cannot be inverted. Thereforecommonly takes source and drain lead to be identically cstituted. But even in such cases, these methods becomemerically unstable, with increasing distance between the mlecular region and one lead. This is because the maelements ofF andG @in Eq. ~14!# diverge exponentially, withincreasing lead separation. Taking the inverse ofF is there-fore a numerically critical procedure. Both these probleare avoided by the direct calculation of the scattering matwhich we present in Appendix B. This calculation is wedefined without any restrictions to the number of leads atheir composition. Therefore it is not necessary to restricidentical leads. Furthermore it allows a numerically stadetermination of the scattering matrix, even for large leseparations.

E. Transmission function

The transmission function is the sum over the contribtions from each combination of incoming channels in tsource lead to outgoing channels in the drain lead:T(E)5( i , jTj← i . The relation between the scattering matrixS andthese transmission function elements is

Tj← i5u~S21! j← i u2v j

v i, ~16!

whereS21 is that block ofS combining the incoming sourcechannels with the outgoing ones in the drain lead. Tweighting with velocity factors comes about becausescattering matrixS does not relate current densities, bwave amplitudes. The current densities are obtained frthese wave amplitudes by multiplication with the corrsponding velocity factorv j . The factorv i in the denominatornormalizes the transmission function to be exactly oneperfect transmission. Ifv i50 thenTj← i50, because incom-

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ing evanescent waves have zero amplitude at the surfThe total current is made up of the contribution from eachannel:

T~E!5 (i Psource

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Tj← i~E!. ~17!

IV. CALCULATIONS FOR PDI

Low temperature experiments with PDI SAMs sanwiched between two metallic leads show several peaks inCV diagram.9,14 The typical voltage differences of thespeaks are in the range ofDU'0.2 V ~i.e., there are aboufive peaks withinU50 and 1 V!. The commonly adoptedexplanation for the occurrence of such peaks is the folloing. Each molecular orbital that enters the energy windwhich is opened by the applied voltage, enables resontunneling. This increases the conductance and thereforesults in a peak within theCV diagram.

Typically, the energy gap between molecular orbitals isthe range ofDE'1 eV. In other words, for applied voltageup toU51 V there should be only a single accessible orbper molecule, giving rise to only a single peak in theCVdiagram. Therefore the following question arises: are thgeometrical alignments of the molecules such that the ational peaks in theCV diagram can also be explained bresonant tunneling through molecular orbitals?

Influence of changes in the molecular alignmentto the transmission spectrum

During the device fabrication, the step under least expmental control is the adsorption of the molecules ontoleads. Therefore the exact geometrical alignment of thelecular SAM and, at least in the sandwich geometry, alsoatomic shape of the top metallic lead, is not exactly knowOne therefore has to expect not only one specific but raquite a variety of molecular alignments to be produced.one is interested in the conduction properties of the resuldevice, it is important to understand the influence of eatype of geometrical alignment to the transmission functio

To this end, we have investigated three such possalignments, which will be discussed separately. First we loat the influence of metallic clusters within the contact regbetween lead and molecule. Then we investigate the difence between single and many molecule experimentsfinally we consider the case of molecular clusters.

1. Influence of metallic clusters

In the sandwich geometry, first the bottom metallic leadcreated. Then the molecular monolayer is adsorbed on toit by self-assembly. Finally the top metallic lead is buiupon the molecular monolayer. The exact shape of neimetallic surface is known and may be anything but flat aregular. It is likely that the surface atoms of the metalleads build up clusters@as, for example, in Fig. 3~b!#.

The influence of such a Au cluster on the molecular eltronic structure is twofold. First it introduces new electronlevels, and second the existing molecular electronic lev

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FIG. 3. ~a! ~Color online! Structure for a molecule without cluster. ~b! ~Color online! Structure for a molecule with a gold clusteon top. ~c! Transmission functionT(E) for both structures. Theenergy scale is relative to the highest occupied–lowest unoccumolecular orbital gap, such thatE50 corresponds to the middle othe gap.

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FIG. 4. ~Color online! The structure for one, two, three, and four molecules adsorbed within an Au-933 supercell. This setup was useto test the sum rule.

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will be shifted, by an amount which depends on the strenof the coupling between cluster and molecule. The latterfect will result in a shifted peak in the transmission functioonly if the coupling between cluster and molecule is differeto the coupling between top electrode and molecule.clusters similar to the one shown in Fig. 3~b!, this is howevernot the case. The energetic peak positions are identicacan be seen in Fig. 3~c!.

Furthermore, there are no additional peaks, whichmight have expected because of the additional electronicels of the cluster. The explanation for their absence isfollowing: an electronic level gives rise to a peak in ttransmission function only, if the corresponding orbital wafunction overlaps with both the top and bottom electrodThe overlap with the electrode the cluster is attached to~saythe top electrode! is of course large. The overlap with thbottom electrode consists of two parts: the direct overlapthe indirect overlap via the molecule. The direct overlapnegligible due to the large spatial separation. The indiroverlap depends on the molecular orbital wave functionthe energy of the cluster level does not coincide with a mlecular energy level, then there is no indirect overlap. Onltwo levels coincide, the indirect coupling is large, but in thcase, there already exists a transmission peak due to theecule itself.

Therefore if transmission is already suppressed bymolecule~at all off-resonant energies!, it can either be fur-

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ther reduced by off-resonant tunneling through the clusterit can ~at best! be left unchanged by resonant tunnelinthrough the cluster. Under no circumstances can transmsion, once suppressed by the molecule, be afterwardscreased by the cluster. This in turn means that metallic cters cannot give rise to additional peaks in the transmissspectrum.

2. Monolayer vs single molecule

What do we expect the transmission functionTi(E) for iperiodically arranged molecules to look like? As long asintermolecular interactions are small~compared to the in-tramolecular ones! the molecular levels of each molecuwill not be significantly changed. Furthermore, as the molayer consists of only one kind of molecule, all of them whave the same electronic structure. Therefore we expectmolecule to contribute the same amount to the transmisfunction:Tn(E)ª( iT

1(E)5nT1(E), wherei runs over allnadsorbed molecules.

We calculated the transmission function forn51 –4 mol-ecules within a Au supercell of size 933 ~the structures areshown in Fig. 4!. The distance between the moleculeschosen to be a multiple of the closest Au-Au separationa(d55.76 Å52a, with a52.88 Å). To our knowledge, theparameters of the PDI-SAM monolayer have never beentermined experimentally, which is why we have to assu

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the above values. However STM studies21 and also theoreti-cal calculations22 have been performed for alkanethiol monlayers, and these parameters motivated our choice.

Independent of the number of molecules present,transmission functions have the same amount of peaksidentical energetic positions~see Fig. 5!. This result is alsoobtained for all larger distances of the molecules, whereinter-molecular interaction is even smaller. Furthermoresum rule is indeed fulfilled, i.e., the calculated transmissfunctions can well be fitted to the relationTn(E)5a(n,m)Tm(E), where the deviation ofa(n,m) from thetheoretically expected value ofn/m is below 6% for alln,mP$1,2,3,4%. The mere fact that one deals with a monlayer instead of a single molecule does not imply thattransmission function changes qualitatively.

3. Influence of molecular clusters

We now investigate cases where the molecular intetions are not negligible. This occurs for example whenperiodic structure of the monolayer is perturbed by an adtional molecule, such that a molecular cluster is formed. Isufficient to study the transmission function of an isolacluster only, because we have already seen that moleculthe periodic SAM arrangement do not influence each otThe sum of the transmission function for the periodic SAand the transmission function for the molecular clusterdue to the sum rule, the total transmission function fordefect and SAM.

We study the influence of a shorter distance between tthree, and four molecules on the transmission spectrumrelate it to the discrete energies of the isolated molecuThe molecules are now separated byd52.88 Å, which cor-responds to the Au-Au atom spacing. The atomic structfor this calculation is shown in Fig. 6~a!, the resulting trans-

FIG. 5. Transmission function for one, two, three, and four Pmolecules.

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mission functions in Figs. 6~b! and 6~c!.By reducing the molecular separation the transmiss

function qualitatively changes. The number of peaks rougdoubles and the new peak positions are different fromones we obtained in the previous calculations. This time,peak positions do depend on the number of moleculesvolved. This is an important point, because if there are seral molecular clusters with different molecular distancthen they all give rise to peaks at different energy valuThe resulting transmission function is the sum of the invidual functions and will thus contain far more peaks ththe transmission function for the nonperturbed periodic lay

The additional peaks are a result of the intermolecuinteractions, which split the former degenerate energy levof the individual molecules, as can be seen in Fig. 7 whwe have again plotted the transmission function for threefour molecules, this time together with the discrete enelevels of the corresponding molecular cluster~shown aspoints along the transmission function! obtained by diagonal-izing the molecular Hamiltonian in the absence of all leaEach of the transmission peaks is related to at least onecrete energy value. But, in turn, not all energy values canrelated to a peak in the transmission function, becausecorresponding energy level of the molecule does not coustrong enough to the leads.

Finally we show that the additional peak structure in ttransmission function for a scenario with an increased inmolecular interaction gives rise to a number of steps inIV-curve. Figure 8 contains anIV calculation for a molecularstructure containing all three molecular clusters shownFig. 6~a!. In this calculation the bias voltageVb enters as ashift of the Fermi levels for source and drain lead:m15m21eVb . The molecular energy has been set toEm5m12dEm2heV, wheredEm is the zero bias displacement othe molecular levels andh50.5, because of the symmetrcoupling to the leads.

Compared to the experiments9,14 the number of steps inthe IV curve is well reproduced by our calculation. The otained current is at least one order of magnitude larger tthe experimental values.14 This is a phenomenon common tall theoretical methods based on the Landauer formula.3,23Asatisfactory explanation for this discrepancy as well asthe broad range of experimentally observed current valhas not yet been found.

V. DISCUSSION

We have shown that the peak structure of the transmisfunction is robust against changes in the number of adsormolecules, as long as the distance between molecules issiderably large (d*6 Å). And also does the exact shapethe top metallic lead not influence the qualitative structurethe transmission function. Only if the distance between mecules becomes so small that inter-molecular interactionsno longer negligible~which is below 6 Å in our case!, doesthe transmission function undergo a qualitative chanNamely an additional peak structure occurs.

How does this finding compare to the experimental daAs we have pointed out in Sec. II, only in devices usi

I

4-7

fted withch alters the


FIG. 6. ~a! ~Color online! Two, three, and four molecules with a shorter intermolecular distance.~b! The transmission functions for twoand three molecules.~c! The transmission functions for three and four molecules. In contrast to all previous cases, the peaks are shirespect to each other and there are also additional peaks. These changes are due to the increase in intermolecular interaction, whielectronic levels.

o

arre

olreoldin

poi.elu

so-mo-

tersto

up,ure.

o

molecules with two isocyanide groups a more or less randpeak structure was observed in theCV characteristic.9,14 Inother devices, molecules with at least one thiol grouptypically used. These show significantly less peak structu

We therefore give the following interpretation: The thigroup is known to bind strongly to Au atoms. It is therefolikely that thiol-based monolayers stably adsorb to gleads. Resulting periodic structures are then robust agadistortions. The conductance of such structures is protional to the corresponding single molecule conductance,the number of molecules involved changes the abso

08532

m

e.

str-.,

te

value of the current only, not the peak structure.The randomlike peak structure in devices made up of i

cyanide based molecules suggests that there are somelecular clusters present in the monolayer. These clusmight occur, because the binding of an isocyanide groupAu is considerably weaker compared to that of a thiol groand weaker binding results in a less robust periodic struct

ACKNOWLEDGMENTS

We would like to thank Xavier Bouju, as well as UdBeierlein for helpful discussions.

4-8

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es


APPENDIX A: CONNECTION BETWEEN EIGENVALUESAND CURRENT VALUES

For ease of notation we transform into the diagonal rresentation of the propagatorP @Eq. ~5!#, i.e., U21PU5diag(l i). The current properties of each channeli can nowbe related to the corresponding eigenvaluel i . We start fromEq. ~10!:

^cuWj uf&5^cuWj 11uf&

5l1* l2^cuWj uf&.

FIG. 7. Magnification of the transmission functions for thrand four closely spaced molecules. Additionally the discrete enelevels of the system without leads are plotted as points alongtransmission function. To each peak there belongs at least onecrete energy level. A detailed discussion is given in the main tInset: Transmission function~original scale! for three and fourclosely spaced molecules@identical to Fig. 6~c!#.

FIG. 8. IV calculation for a molecular region containing all thremolecular clusters shown in Fig. 6~a!. There are three distinct stepwithin the voltage range of 1 V.

08532

-

Let us first consideruc&5uf&, i.e., l15l2, i.e., ^cuWj uf&5ulu^cuWj uf&. For each channel with eigenvalueuluÞ1one then must havecuWj uc&50, i.e., this channel does noitself carry any current. This is consistent with our terminoogy of an evanescent wave. If, however,ul i u51, then^cuWj uc& is purely imaginary, becauseWj is an anti-Hermitian operator. We can therefore define the velocity opropagating wave to bev iªIm^cuWj uc&.

Now we consider the case of two different solutionsuc&Þuf& and definev1,2ª^cuWj uf&. If their eigenvalues donot satisfy l1l2* 51, then the current between these twsolutions is zerov1,250. So let us assumel151/l2* . Be-cause iful1u.1 thenul2u,1, a current can flow between aevanescent left going wave and an evanescent right gwave. But if we restrict ourselves to solutions with finiamplitudes in a semi-infinite lead, then either the left or riggoing wave amplitude must be zero. Therefore evaneswaves do neither carry a current themselves nor do theychange current with other channels, that is they do not acontribute to the net current.

Finally we are left with the casel151/l2* , with ul1u5ul2u51. This is equivalent tol15l2, i.e., the case ofdegenerate eigenvalues. Therefore propagating waves togenerate eigenvalues may exchange current. Howewithin the degenerate eigenvalue subspace ofP, we can per-form an additional rotation, i.e., we can chooseU such thatthe anti-Hermitian operatorW is also diagonal with purelyimaginary eigenvalues.

Summarizing we have shown that the transformationUdiagonalizing the propagatorP ~i.e., U21PU) can be chosensuch that the transformationU†WU of the current operator isdiagonal in the subspace of propagating waves with puimaginary diagonal elements. All the other diagonal entrare zero and the only nonzero nondiagonal elements beto evanescent waves in opposite directions.

APPENDIX B: CALCULATION OF THE SCATTERINGMATRIX

The part of the Hamiltonian containing the molecular rgion and its coupling to the leads can be written as

~H2ES!uc&5F h1 M1 0 0 t1†

0 0 h2 M2 t2†

0 t1 0 t2 M0

G uc&50.

~B1!

~Using this order for the coefficients it is straightforwardextend all formulas to the general case of more thanleads.! The indices 1 and 2 indicate source and drain lesurface layers, while the index 0 is used for the molecuregion. t1,2 are the coupling matrices from source/drainthe molecules.

We now transform into the basis of incoming and outging channels@Eq. ~12!#, i.e., we apply

U5FU1 0 0

0 U2 0

0 0 1G with Ui5F U.

i U,i

U.i L.

i U,i L,

i G

yeis-t.

4-9

thinn

am-pliteing

t,


from the right to Eq.~B1!:

~H2ES!U5F A.1 A,

1 0 0 t1†

0 0 A.2 A,

2 t1†

B.1 B,

1 B.2 B,

2 M0

G , ~B2!

with

A:i 5hiU:

i 1MiU:i L:

i ,

B:i 5t iU:

i L:i .

The first and third columns act on the surface layer ofincoming channels, the second and fourth act on outgoones, while the fifth column, acting on the molecular regioremains unchanged.

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ro

y

r,

08532

eg,

The scattering matrix expresses the outgoing channelplitudes in terms of the incoming ones. Therefore we sthe matrix of Eq.~B2! into two parts, one containing thoutgoing columns, the other one containing the incomones as well as the molecular column:

MoutªF A,1 0 t1

†

0 A,2 t2

†

B,1 B,

2 M0

G , M inªF A.1 0

0 A.2

B.1 B.

2G .

The first matrixMout is a square matrix and by inverting iwe obtain the scattering matrix

S52Mout21

•M in . ~B3!

nd

R.ett.

s.

u,

m.

1S. Datta, W. Tian, S. Hong, R. Reifenberger, J.I. Henderson,C.P. Kubiak, Phys. Rev. Lett.79, 2530~1997!.

2M. Magoga and C. Joachim, Phys. Rev. B56, 4722~1997!.3E.G. Emberly and G. Kirczenow, Phys. Rev. B64, 235412

~2001!.4P.A. Derosa and J.M. Seminario, J. Phys. Chem. B105, 471

~2001!.5P. Damle, A.W. Ghosh, and S. Datta, Chem. Phys.281, 171

~2002!.6M. Brandbyge, J.L. Mozos, P. Ordejon, J. Taylor, and K. Stokb

Phys. Rev. B65, 165401~2002!.7P.S. Krstic, X.G. Zhang, and W.H. Butler, Phys. Rev. B66,

205319~2002!.8A. Nitzan and M.A. Ratner, Science300, 1384~2003!.9C.J.F. Dupraz, U. Beierlein, and J.P. Kotthaus, Chem. Ph

Chem.4, 1247~2003!.10P. Sautet and C. Joachim, Chem. Phys. Lett.185, 23 ~1989!.11J. Chen, M.A. Reed, A.M. Rawlett, and J.M. Tour, Science286,

1550 ~1999!.12M.A. Reed, C. Zhou, C.J. Muller, T.P. Burgin, and J.M. Tou

Science278, 252 ~1997!.

d

,

s.

13J. Reichert, R. Ochs, D. Beckmann, H.B. Weber, M. Mayor, aH.v. Lohneysen, Phys. Rev. Lett.88, 176804~2002!.

14J-O. Lee, G. Lientschnig, F. Wiertz, M. Struijk, R.A.J. Janssen,Egberink, D.N. Reinhoudt, P. Hadley, and C. Dekker, Nano L3, 113 ~2003!.

15M.H. Hettler, W. Wenzel, M.R. Wegewijs, and H. Schoeller, PhyRev. Lett.90, 076805~2003!.

16M.J. Frischet al., Gaussian 98~Revision A.7!, Gaussian, Inc.,Pittsburgh PA, 1998.

17S. Datta,Electronic Transport in Mesoscopic Systems~CambridgeUniversity Press, Cambridge, UK, 1995!.

18D.S. Fisher and P.A. Lee, Phys. Rev. B23, 6851~1981!.19P. Sautet and C. Joachim, Phys. Rev. B38, 12238~1988!.20D. Rosh, Ph.D. thesis, University of Cambridge, 1992.21C. Zeng, B. Li, B. Wang, H. Wang, K. Wang, J. Yang, J.G. Ho

and Q. Zhu, J. Chem. Phys.117, 851 ~2002!.22Y. Yourdshahyan and A.M. Rappe, J. Chem. Phys.117, 825

~2002!.23C.W. Bauschlicher, A. Ricca, N. Mingo, and J. Lawson, Che

Phys. Lett.372, 723 ~2003!.

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