I V characteristics and differential conductance ...peter/publications/prb2002.pdf · I-V...

PHYSICAL REVIEW B, VOLUME 65, 195419

I -V characteristics and differential conductance fluctuations of Au nanowires

H. Mehrez, Alex Wlasenko, Brian Larade, Jeremy Taylor, Peter Gru¨tter, and Hong GuoCenter for the Physics of Materials and Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8

~Received 5 June 2001; revised manuscript received 4 January 2002; published 10 May 2002!

Electronic transport properties of the Au nanostructure are investigated using both experimental and theo-retical analysis. Experimentally, stable Au nanowires were created using a mechanically controllable breakjunction in air, and simultaneouscurrent-voltage (I -V) and differential conductancedI /dV data were mea-sured. The atomic device scale structures are mechanically very stable up to bias voltageVb;0.6 V and havea lifetime of a few minutes. Facilitated by a shape function data analysis technique which finger printselectronic properties of the atomic device, our data show clearly differential conductance fluctuations with anamplitude.1% at room temperature and a nonlinearI -V characteristics. To understand the transport featuresof these atomic scale conductors, we carried outab initio calculations on various Au atomic wires. Thetheoretical results demonstrate that transport properties of these systems crucially depend on the electronicproperties of the scattering region, the leads, and most importantly the interaction of the scattering region withthe leads. For ideal, clean Au contacts, the theoretical results indicate a linearI -V behavior for bias voltageVb,0.5 V. When sulfur impurities exist at the contact junction, nonlinearI -V curves emerge due to atunneling barrier established in the presence of the S atom. The most striking observation is that even a singleS atom can cause a qualitative change of theI -V curve from linear to nonlinear. A quantitatively favorablecomparison between experimental data and theoretical results is obtained. We also report other results con-cerning quantum transport through Au atomic contacts.

DOI: 10.1103/PhysRevB.65.195419 PACS number~s!: 85.65.1h, 72.80.2r, 73.22.2f

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I. INTRODUCTION

Electron transport through atomic nanocontacts has ban active research area for a decade both experimentallytheoretically. The scientific interest of these systemslargely driven by their peculiar electronic and transport bhavior. Atomic nanocontacts are structures with low atomcoordination number and, as a result, can behave very diently from their bulk counterpart. From a practical pointview, understanding the electronic and structural properof the atomic nanocontacts is an important step towards nodevice fabrication and characterization. The first set ofperiments on nanocontacts focused on their zero-biasductance ~G! using scanning tunneling microscop~STM!,1–17 mechanically controllable break junctions,18–24

and relay contacts~RC’s!.25,26 In these experiments a narroconstriction with a few atoms at the cross section is formAs the electrodes are pulled apart,G is measured and founto change discontinuously forming plateaus with values clto nG0, wheren is an integer andG052e2/h.1/12.9KV isthe conductance quanta. Pioneering experiments9,10 clearlyshowed the correlation between conductance jumps andchanical properties in the nanocontacts. These resultsfirmed earlier predictions27 that the conductance variationare due to abrupt changes of nanostructure cross sectionfunction of wire elongation. Extensive theoretical investigtions on nanostructures have been published recently tolyze these systems. One major focus of theory is to calcuthe zero-bias conductance through a ballistic quantum pcontact. These calculations start by assuming varicontact28–39 geometries, or by using more realistic atompositions derived from molecular-dynamics simulations. Tpotential of the constriction and/or interaction Hamiltonianthen constructed40–52 from which the zero-bias transport co

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efficients are evaluated. While different levels of approximtions were employed in these theoretical analysis, densfunctional theory basedab initio analysis have also beereported,33,36,53which provide self-consistent calculationsatomic nanocontacts.

While zero-bias transport coefficients have receivedgreat deal of attention, one must go beyond this limitunderstand the full nonlinear current-voltage (I -V) charac-teristics of the nanocontacts, as this information is essenfor the understanding of real device operation. For examdue to the small cross section of these systems, theyexposed to substantial current density;108 A/cm2, whichmay result in atomic rearrangement. It is also expectedelectron-electron (e2e) interaction is enhanced due to thstrong lateral confinement possibly leading to Luttingliquid behavior for thequasi-one-dimensionalatomic wires.Separating these different effects is experimentally challeing due to the many variables that can affect the results. Tis probably the origin of the existing controversy in explaiing the experimentally observed nonlinearI -V curves of theatomic scale wires.54–57From a theoretical point of view, thisis also a challenging problem: so far only two computatioally accurate techniques exist, which can treat systems wopen boundariesout of equilibrium due to externabias.33,58,59 In the approach of Tayloret. al.,58,59 realisticatomic leads can be treated and the problem is solvedconsistently within the local-density approximation. Therfore, the leads, the device~scattering region!, and their cou-plings are incorporated without any preconditionparameters.

To further shed light on the physics of quantum transpat molecular scale, we report in this paper our investigaton transport properties of Au nanostructures both experimtally and theoretically. Experimentally, we created stable

©2002 The American Physical Society19-1

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MEHREZ, WLASENKO, LARADE, TAYLOR, GRUTTER, AND GUO PHYSICAL REVIEW B65 195419

FIG. 1. ~a! Schematic of theexperimental setup with a mechanical controllable break junction ~MCBJ! shown at the centerwith Au wires attached.~b! DcconductanceI /V ~lower! and dif-ferential conductancedI /dV ~up-per! plotted vs V, inset shows theI -V curve. The dashed line, withslope.2.2G0, corresponds to lin-ear behavior and helps to recognize nonlinearity. These experimental results are measured over5-voltage sweep from positive~dark! to negative~gray! bias.

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nanowires using the mechanically controllable break jution technique in air, and we simultaneously measure theI -Vcurve and the differential conductancedI /dV. We found thatour atomic scale Au nanocontacts are mechanically vstable up to bias voltageVb;0.6 V and have a lifetime of afew minutes, which is adequate for our measurements. Asare interested in features due to electronic degrees of fdom of the nanocontacts, careful data analysis is neededcause transport data can be affected by many factorsdefining theshape function S5(dI /dV)(V/I ), which fingerprints electronic properties of the atomic device, our dclearly shows differential conductance fluctuations withamplitude .1% at room temperature and nonlinearI -Vcharacteristics. To understand these transport featurescarried outab initio calculations on various Au atomic wirebonded with atomic Au electrodes using the first-principtechnique of Refs. 58 and 59. Our calculations show tpure and perfect Au nanocontacts do not give the nonlinI -V curves as measured in the experiments. However wsulfur impurities are present near the wire-electrode conregion, the nonlinearI -V curves emerge due to the tunnelinbarrier provided by the impurity atoms. The most strikiobservation is that even a single S atom can cause a qutive change of theI -V curve from linear to nonlinear. Aquantitatively favorable comparison between experimedata and theory results is then obtained. Our theoreticavestigation suggests that transport through Au atomic wis strongly affected by the properties of the wire-electrocontacts.

The rest of the paper is organized as follows. In the flowing section, experimental measurement and resultspresented. Section III presents the theoretical results wSec. IV discusses the transmission coefficients in moretail. We also discuss and compare previous works with oin Sec. V, followed by a conclusion.

II. EXPERIMENTAL RESULTS

Atomic scale gold contacts were formed with a mechacally controllable break junction in air at room temperatuOnce suitably stable atomic junctions were formed, a slo

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varying bias voltage was applied~typically a 0.1 Hz trianglewave, 2Vp-p) along with a small modulation voltage@typi-cally 40 kHz, 431023Vrms# across the contact and a loaresistor of 3 kV. Current I and differential conductancedI /dV were measured simultaneouslywith an I -V preamp-lifier and a lock-in amplifier. The experimental setupshown in Fig. 1~a!. A typical measurement through a stabAu nanocontact is presented in Fig. 1~b!. We show our datafor the differential conductance and the dc conductanceG5I /V). In the inset of Fig. 1~b!, we also display a typicalI -Vmeasurement. These data were taken over the course ofsvoltage sweep from positive~dark lines! to negative~graylines! bias voltageV. Both polarities share a common overashape, but seem to vary significantly in the details of thconductance behavior. However, for each polarity, onetices that there seem to be similar details present in theand differential conductance.

An important issue in order to understand these resultthe separation of effects due to atomic rearrangement innanostructure from electronic properties. We addressproblem by writing, very generally,

I ~X,V![g~X! f ~X,V!, ~1!

where the variableX symbolizes the effects of atomic structure and all other nonvoltage parameters including therfluctuations. This parameterX does not implicitly depend onV, but may be affected by the history of the measurem~i.e., howV is ramped! and current-driven electromigrationHenceX is not a proper function ofV. The functionf (X,V)gives the normalized functional form of the voltage depedence of current, and is defined such that in the zero-blimit V→0, f (X,V)5V. Therefore the functiong(X), de-fined by Eq.~1!, becomes the conductance atV→0. At afinite VÞ0, g(X) is simply a coefficient~to be discussedbelow!. We further define a new quantity called the ‘‘shafunction’’ (S),

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I -V CHARACTERISTICS AND DIFFERENTIAL . . . PHYSICAL REVIEW B 65 195419

Figure 2~a! plotsScorresponding to the data of Fig. 1~b!, forboth positive and negative bias voltages. The similaritytween the curves of the shape functions for both polaritincluding fine details, is in striking contrast to the easdistinguishable conductance plots of Fig. 1~b!. By its defini-tion @Eq. ~2!#, S depends only onf, the functional form ofI (X,V), and is independent ofg(X).60 This fact and the usefulness of the shape functionS can be seen by considerinthe following ‘‘Gedanken experiment.’’ Let us assume thwe measured simultaneously IanddI /dV through a variable,ohmic potentiometer as a function of the applied voltaSuppose there were drastic changes in the temperature dthe measurement, and some troublemaker turned the knothe variable potentiometer without telling anyone. Both theuncontrollable parameters~random thermal fluctuations anunknown knob turning! are contained in variableX. Glancingat the measurements ofI (X,V) anddI /dV alone, one mightwrongly conclude that the potentiometer was exhibitingnonlinear behavior. However, a plot ofS would show thatS(V)51 @from Eq. ~2!#, which would allow us to deducethat theI -V curve was actually linear and thus in fact ohmOne would also conclude that the origin of the apparennonlinear behavior was due to a change in the functiong(X),rather than due to a true nonlinearity in voltage.

Using this example, we would also like to stress a subbut important point: our shape function analysis dist

FIG. 2. ~a! The shape function@S, Eq. ~2# vs V calculated fromthe data shown in Fig. 1~b!. ~b! Normalized functional form of thevoltage f 0(V) with a linear dashed line shown to help view thonset of nonlinearity.~c! Unbiased conductanceg(X) vs V. ~d!Normalized conductanced f /dV vs V, with the inset correspondingto a magnified section of the same data to show more clearlydetails. In all graphs, dark~gray! lines correspond to positive~nega-tive! bias.

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guishes between a quantity being a function ofV and a quan-tity being merely voltage driven. For instance, the changetemperature of the atomic contact, and thusg(X), can de-pend on thehistoryof how we ramp up the bias voltage fromzero to a finite valueV at which theI (X,V) is measured: onecould increaseV so slowly that it does not destroy thermequilibrium, or the troublemaker could monitor the applidc voltage while turning the knob so that changes ing(X)were correlated withV. Despite the fact that both of thesvoltage-drivenprocesses would be related toV and lead torepeatableI (X,V) versusV curves, neither would be mistakenly attributed to a functional dependence ofV by our analy-sis. This is a pure mathematical consequence of the definof g(X) and the fact thatX is not a proper function ofV forthese voltage-driven events, which depend on the prochistory. Any functional dependence onV is included inf (X,V) by definition.

Returning to the experimental results shown in Fig. 2~a!,we note the overlap of the two curves ofS for positive andnegative bias voltages.60 This points to a similar functionaform for both bias polarities. Because of this, we deduce tfor this casef (X,2V). f (X,1V). However, the rampingfor positive and negative bias voltages is quite different, tis why f (X,V) should weakly depend onX ~especially atbias voltageVb,0.47 V). We can therefore writef (X,V). f 0(V)}exp(*S/VdV) and function g(X)5I (X,V)/ f 0(V).Hence, as an experimental voltage sweep typically takesec, time dependent atomic rearrangements~changes inX)manifest themselves as stochastic variations ofg(X). Otherdetails of this data analysis technique and further discuson the shape function are not within the scope of this paand can be found elsewhere.61

The normalized functional form of the voltage depedence off 0(V) for the measured data is shown in Fig. 2~b!.We note that the curves for both polarities appear on topeach other and they are indistinguishable. However, aspected, the ‘‘troublemaker’’ in our Gedanken experimeshows up as fluctuations ing(X) shown in Fig. 2~c!, indi-cated by the fluctuations and by the fact that positive anegative biases give different traces ofg(X). We emphasizethat during the course of our bias sweep from initial bvoltage to some bias valueV5V1, the atomic structure habeen fluctuating and changed from what we started withsomething unknown. But if we could find the instantaneostructure at the moment of the measurement and freezemake a new contact and remeasure theI -V curve, g(X)would be the conductance of the new contact atV→0. Inother words, Fig. 2~c! shows a parametric plot of whatg(X)would be atV→0 at the point in time when this voltage wameasured experimentally; henceg(X) could also be viewedas a function of time in this figure. Our analysis shows ththe fluctuations ofg(X) are less than 5% peak-peak over tcourse of the experiment. We attribute this to changes inatomic configuration of the junction. We have also discoered that the power spectrum ofg(X) exhibits a primarily1/f behavior. This is also found for measurements using vage biases as low as 0.05 V. Further analysis ong(X) in-cluding its average value, the standard deviation, and s

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tral distribution at different values ofV @see Fig. 2~c!# mayprovide useful information on the accuracy of the appromation f (X,V). f 0(V).

In Fig. 2~d! we plot thenormalizeddifferential conduc-tance,d f /dV. Examining the curves in Fig. 2~d!, we can seethe same subtle fluctuations ind f /dV as those found inS@Fig. 2~a!#. Both plots show broad and fine details that auncorrelated to the fluctuations ofg(X) @Fig. 2~c!#. Theseplots reveal that it is the fluctuations ofg(X) that dominatethe fluctuation features in the measuredI (X,V) anddI /dV.

Over different voltage ranges (0.2–0.35 V0.1–0.35 V,0.1–0.5 V),d f /dV has a high correlation62 of0.99 between both polarities. After subtracting the oveshape, the correlations reduced to 0.64, 0.58, and 0.56spectively. Over these same ranges,g(X) had a weaker cor-relation (20.5,0.25, and20.2, respectively! between bothpolarities, which changed drastically, even fluctuatingsign, depending on the voltage range. In addition to comping the correlation values, we have also calculated the colations by shifting the voltage of one polarity with respectthe other by610 mV ~the scale of the fine details! in incre-ments of 1 mV. Over all three ranges, the correlationd f /dV had a local maximum for a 0-V shift, with and without the overall shape subtraction. No correlation extrewere found in the case ofg(X). This quantifies how similarthe details in both polarities ofd f /dV are to each other, incontrast to the more easily distinguishable curves forg(X).We thus conclude that the ‘‘wiggles’’ observed ind f /dV areelectronic in nature and they are not as strongly influenby X as the functiong(X), i.e., f (X,V). f 0(V) is a reason-able approximation for the system we considered. We ction, however, that this may not be the case for other strtures and a careful analysis of this approximation is requir

The wiggles@magnified in the inset of Fig. 2~d!# of d f /dVmay actually be much more pronounced than indicatedthese plots. They are smeared out by unavoidable experimtal constraints. Since we must add a modulation signamake our lock-in measurement ofdI /dV, we end up averaging over a range ofV. This leads to a broadening and dcrease in amplitude of these fine details.61,63 This unavoid-able averaging artifact keeps us from making a more premeasurement of the shape, amplitude, and voltage charaistics of this fine structure, but the true features shouldsharper and more pronounced than they are measured tFrom our data, we observe wiggles on the voltage sc,10 mV and with amplitude;1% of the signal. The widthof these fluctuations is comparable to the modulation vage, (431023)Vrms;(1031023)Vp2p ; hence, it is pos-sible to have features on a voltage scale,10 mV with anamplitude that is orders of magnitude larger than thesefluctuations.

If one were to merely examine ourI (V) measurementssimilar results have been observed in air and at ltemperatures with RC~Refs. 25 and 56!, and STMconfigurations.57,64 The fine details exposed by the analyspresented above have not been discussed in the literatuthey tend to be hidden by the fluctuations in the unbiaconductanceg(X) @see Fig. 2~c!#. The data presented her

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represents the behavior of one junction. However, we ephasize that the general features presented in Fig. 2 areperimentally found to be independent of the zero-bias cductanceg(V50) for values between (1 –10)Go . The finedetails ofd f /dV are reproducible for a given stable junctiobut the specific details change for different junctions. Athough there seems to be a voltage scale associatedthese details (,10 mV), Fourier analysis does not indicaany strong periodicity. It should be noted that we have iposed a selection rule on the junction type by studyingvice configurations that are stable on the time scale of mutes. We have also observed junctions that exhibit linbehavior (S51) as have others.25,56,57These junctions, how-ever, are not stable over this long time scale. In comparito their nonlinear counterparts, linear junctions have larfluctuations ing(X).

In the following we will discuss the possible physical orgin of the observedI (V) characteristics. In this aspect wwill investigate the electronic effects rather than the strtural parameters, which will be assumed static. This willlow us to gain valuable insight into the voltage-dependconduction properties of nanoscale electrical contacts.will thus compare our modeling to the normalized functionform of the current,f 0(V), shown in Fig. 2~b!, rather thanthe originalI -V curve presented in the inset of Fig. 1~b!. Wewill also provide an explanation for the origin of the fluctutions in the normalized differential conductance shownFig. 2~d! as well as the effects of temperature and modution signal on their amplitude.

III. AB INITIO ANALYSIS OF THEI -V CHARACTERISTICS

To provide a possible theoretical understanding of theperimental data presented above, we have calculated theI -Vcharacteristics of Au nanocontacts self consistently by cobining the density-functional theory and the Keldysh noequilibrium Green’s functions. The method is based onnewly developedab initio approach for treating open electronic systems under finite bias. For technical details of tmethod we refer interested readers to the original papers58,59

Very briefly, our analysis uses ans,p,d real-space linearcombination of atomic orbital basis set58,59,65and the atomiccores are defined by the standard nonlocal norm conserpseudopotential.66–68 The density matrix of the device iconstructed via Keldysh nonequilibrium Green’s functionand the external biasVb provides the electrostatic boundaconditions for the Hartree potential, which is solved inthree-dimensional real-space grid. Once the density matrobtained, the Kohn-Sham effective potentialVe f f(r ;Vb),which includes contributions from the Hartree, exchancorrelation, potentials and the atomic core, is calculated. Tprocess is iterated until numerical convergence of the sconsistent density matrix is achieved. In this way, we obtthe bias dependent self-consistent effective potenVe f f(r ;Vb), from which we calculate58,59 the transmissioncoefficient T(E,Vb)[T„E,@Ve f f(r ,Vb)#…, where E is thescattering electron energy andT is a function of biasVbthrough its functional dependence onVe f f(r ;Vb).

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In our analysis, the scattering states are defined for enranges between the left and right chemical potentialsmL andmR , respectively. To solve for these states, at a given eneE, we solve an inverse energy-band-structure problem.59 Wethen group all states as left and right propagating statespending on their group velocity. For a scattering state com

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where f L f R is the Fermi function on the left~right! leadevaluated at temperatureT50 K unless otherwise stated. Iaddition to predicting the overall transport properties odevice, our formalism enables us to study transmissthrougheach incoming Bloch state of the leads separatetherefore allowing us to separate effects due to the leadsdue to the scattering region. We have used this formalismcalculate I -V characteristics of structurally different Ananocontacts and compared them with the experimentasults on the normalized functional form of the current, whiis described above. However, structural analysis on thsystems,g(X), goes beyond the scope of this work. Threquiresab initio molecular-dynamics simulations with opeboundaries under nonequilibrium conditions due to the exnal bias. While these calculations could be performeddetermining the force on each atom in the device scatteregion, it is computationally prohibitive due to the largsimulation time required to reach equilibrium.

A. Perfect Au nanocontacts

In a first attempt to model our experiments, we calculathe I -V characteristics of four Au atoms~or a molecule! incontact with Au~100! leads. The structure of the atomic dvice is illustrated in Fig. 3~a!. The scattering region isbonded by two semi-infinite Au leads, which extend to ele

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tron reservoirs at6`, where bias voltage is applied ancurrent is collected. The device scattering region, indicaby D, is described by three Au layers from the left lead, foAu atoms in a chain, andtwo layers of Au from the rightlead. We have also increased the two Au layers on the rside of the chain to four to ensure that convergencereached with respect to the screening length. In this structthe registry of the atomic chain with respect to the lead sface layer can be different. The most common structureswe analyze in this work are the hollow site, where the atomchain faces the vacant position in the lead layer as showthe upper left inset of Fig. 3~b!, and the top site, where thatomic chain and an atom from the lead surface layer f

FIG. 3. I -V characteristics of Au contacts with hollow site reistry ~a! and top-site registry~b!; with zero-bias conductance0.94G0 and 0.8G0 for ~a! and~b!, respectively. The structure of thdevice is illustrated in~a!, where LL,D, and RL correspond to leftlead, effective device, and right lead, respectively. The hollowis shown in~c! and top site in~d!, where chain atoms are illustrateby dark circles. Inset~e! shows the band structure of the lead alothe transport direction, the conducting band is shown by a sline. Inset ~b! shows differential conductance fluctuations of tHollow site as a gray dotted line, dark dotted line, and dark ctinuous line for T50 K, T5300 K and forT5300 K with 431023Vrms modulation voltage taken into account, respectively

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each other as illustrated in the upper right inset of this figuWe note that long and thin gold necks have directly beobserved experimentally.50

Our calculations show that in all cases charge tranbetween the Au chain and the leads is not important, beonly ;0.0720.1 electrons per atom to the Au chain at dferent bias voltages. This corresponds to less than 1% dience in electron population per atom and, therefore, doesplay any significant role in theI -V characteristics of Au contacts. This is in contrast to a binary atomic system suchcarbon chains between Al~100! leads.70 In addition, solvingfor the energy eigenvalues of the four-atom chain givehighest occupied molecular-orbital–lowest unoccupmolecular-orbital gap of 0.68 eV, indicating that the moecule eigenstates should only have a secondary effect ontransport properties. Therefore, the major effect on theI -Vcharacteristics is due to the character of electronic statethe leads and their couplings to the molecule at the chlead interface.

In Fig. 3~a!, we show the results for a system with atomin the hollow site. At small biasVb , we note that current is alinear function ofVb with a slopeG.0.94Go . The linearfunction suggests thatT(E,V)5T0 with a weak voltage de-pendence. In this regard, our self-consistent calculation ga result apparently similar to previous theoretical work28–52

in the low-bias regime. However, we will show later, baddressing the origin of this ‘‘perfect linearity,’’ that thphysical picture of a bias-independent transmission coecient is not valid even for such a simple chain. We also nthat the linearI -V characteristics observed in these systedo not agree with our experimental data.

A major feature of Fig. 3~a! is the huge plateau atVb50.5 V–0.9 V, as well as the fine structures~or sometimesnegative differential resistance! observed for larger voltagesIn the lower inset of Fig. 3~b!, we show the band structure othe Au~100! lead along thez direction ~transport direction!.Even though at a given energyE there are many electronistates that are potential candidates for transporting currour investigation found that forE,0.5 eV there is onlyonestate that is actually conducting~presented by a continuouline in the inset!. Once this state is terminated atE'0.5 eV, the current is saturated resulting in a large platuntil new conducting states emerge at higher bias voltaFor E.0.9 V, transport properties are more complex sinmore states contribute to transmission. Under these circstances, band crossing occurs more frequently, andT(E,Vb)changes over small ranges of bias leading to the small sttures seen in theI -V characteristics of perfect Au contacts.fact these variations inT(E,Vb) are the origin of the fluctuations in the normalized differential conductance shownFig. 2~d!. They thus need to be attributed to the effects ofleads’ band structure. In the lower inset of Fig. 3~a!, we plotthe theoreticaldI /dV. The magnitude of these conductanfluctuations is of the order of 60% at zero temperatuwhich is much larger than the experimental finding. Hoever, these fluctuations are reduced to 15% if the currencalculated using Eq.~3! at a temperatureT5300 K ~thetemperature of our experiments!. In addition, the fluctuationsfurther decrease to;1% when current is averaged over th

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experimental modulation voltage range of 4 mV, completconsistent with our experimental results of Fig. 2~d!.

The effect of site registry is studied by placing the eatom of the Au chain at the top site of the leads. In thsituation the chain atoms are facing one atom of the lsurface layers. This analysis is quite important, becauswas shown46,47 that atoms at the junction change registfrom hollow to top sites resulting in bundle formation jubefore the nanostructure breaks. TheI -V characteristics ofthese systems@shown in Fig. 3~b!# are similar to the previousresults, therefore no change is observed in the transport perties for the top-site registry.

To further investigate the huge and peculiar plateau occring at Vb50.5 V–0.9 V, we have plotted in Fig. 4 thcharge density at a given energyE, r(E,x,z)5*dyr(E,x,y,z). We see clearly that at zero bias,Vb50 V, the device property turns from a perfect conductorE50 eV @Fig. 4~a!# to an insulator atE50.68 eV @Fig.4~c!# due to the termination of the conducting state. Hereinterpret the charge concentration as the effective bondstrength, or conductance probability. Applying a bias voltato the system drives it out of equilibrium, and atVb50.68 V andE50.68 eV, the charge in the molecule rdistributes, but the bonding is still very weak as shownFig. 4~d!, with some molecular regions havingzero chargeand resulting in the large plateau observed in ourI -V curveof Figs. 3~a,b!. This effectively demonstrates the importanof both the energy and the voltage dependence of the trmission coefficientT(E,V). This point will be discussed inmore detail in Sec. V. A further point to notice is the diffeence of zero-bias conductanceG.0.8G0 for the top-site de-vice andG.0.94G0 for the hollow-site device. This differ-ence is due to a change in the coupling between the cend atoms and the surface of the leads. To ensure the snearest-neighbor separation distance for both cases, weup with four nearest neighbors for the hollow-site regisand only one nearest neighbor for the top-site registry. Unthese circumstances, the hollow site has a better couplinthe chain and hence a larger conductance.

A pure and perfect Au nanocontact, as studied in this stion, shows rich and interesting transport properties. It a

FIG. 4. Contour plots of surface charge density at particular bvoltageVb(V) and energyE(eV) indicated on each figure. We usthe same scale for all graphs to compare the conductance probity for each configuration. Dark circles correspond to atom positioalong the constriction for guidance.

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gives a good understanding of the origin of the obserdifferential conductance fluctuations as due to couplingthe Au chain to the leads’ band structure. However, it halinear I -V curve at smallVb,0.5 V, rather than the experimentally observed nonlinearI -V characteristics. How cannanocontact produce a nonlinearI -V curve such as that oFig. 1~b!? The simplest possibility to observe such a phnomenon is to have a tunneling barrier at the molecule-ljunction whose effect gradually collapses as a function ofincreasing bias voltage.71 While a tunneling barrier can bestablished by several means, we will investigate a mowhere it is a result of impurities. Indeed, recent experimenfindings indicate that perfectly linearI -V characteristics werereproducibly found in gold-gold nanocontacts in ultrahivacuum,57 while nonlinear effects emerge when the expement was performed in the air. This suggests that impuriplay an important factor.

B. Au nanocontacts with S impurity

To simulate the effect of an impurity at the contact, whave replaced one of the Au atoms at the interface layer wa sulfur atom. This is presented in the inset of Fig. 5. Tchoice of sulfur is motivated by the fact that in our expemental labs, sulfur is a non-negligible airborne polluta~diesel exhausts!; sulfur atoms bond actively with Au. Wealso note that the bandwidth of sulfur,;10 eV, is muchhigher than that of Au (;1 eV), thus a tunneling barrier iexpected to be provided by the presence of S atoms. In

FIG. 5. I -V characteristics of Au contact doped with S impuri~solid line!. The dashed line has a slope.0.67G0 and is shown tohelp view the onset of nonlinearity; circles correspond to theperimental results obtained fromf 0(V) @Fig. 2~b!# and multipliedby the zero-bias conductance 0.67G0. The inset~a!, shows the at-oms registry, where dark circles show the atomic chain, gray oare Au atoms in the leads’ surface, and light gray is the S atom

19541

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system, charge transfer to the atomic chain is still small,thus inadequate in explaining the experimentally obserI -V characteristics. We note that the S atom suffers fromelectron deficiency;4%. Also due to the presence of theatom, the coupling of the electronic states in the leads todevice scattering region is quite different as compared tomonatomic gold structure. When the S atom is present,calculations found that all Bloch states are coupled toscattering region, with the highest transmitting mode sbeing that corresponding to the conducting mode of the matomic gold system.

The I -V characteristics of the S doped Au nanocontactshown in Fig. 5. We note that theI -V curve for voltages upto 0.5 V is very similar to the experimental values with nolinearity onset atnonzerobias voltage. We also note that thhuge current plateau of the pure Au device has now dimished because more states contribute to electronic transTo compare these results with the experimental measment, we have used the normalized functional form ofvoltage, f 0(V) shown in Fig 2~b!, and multiplied it by thesimulated zero-bias conductance (0.67G0). The result isshown as open circles in Fig. 5. The qualitative and quatative agreement between the theoretical and experimeresults is rather encouraging. In fact, this is the first time texperimental nonlinearI -V characteristics could be compared so well and so directly withab initio self-consistentcalculations. It is also a very surprising result becaussingle S impurity can qualitatively alter transport in thenanocontacts from linear to nonlinear.

The I -V curve, in Fig. 5, still shows the small featuresimilar to those found in pure and perfect Au contacts~pre-sented in Fig. 3!. These fine details of the calculatedI -Vcurve would result in differential conductance fluctuatiosimilar to those shown in the experimental data of Fig. 2~d!and the pure Au device of Fig. 3~b!. However, these finefeatures as calculated are wider and occur at higher biasages than the experimentally observed ones. The formerbe attributed to the zero temperature we used in our calction. The absence of these fine features at smaller voltagattributed to the small size of the leads used in our theoretmodeling: close toEF there are just a few states so thabrupt variation ofT(E,Vb) at smallerVb is less probable,resulting in a smootherI -V curve at lowVb .

To understand other possible factors that can affectI -Vcharacteristics of nanocontacts we have studied the effeca larger number of S impurities, disorder, and their combineffects. The results of these calculations are presented infollowing section.

C. Contacts with several impurities and disorder

Including more S impurities at the contact enhancestunneling barrier and may give rise to a smaller current wa larger nonlinear behavior. The result of replacing twoatoms at the interface by S atoms is plotted in Fig. 6~a!.Indeed, as expected theI -V curve for this system showslarger nonlinear character. The nonlinearity starts atVb;0.17 eV. It is actually more nonlinear than that of thexperimental data; this is mainly due to the high concen

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FIG. 6. I -V characteristics ofAu contacts with two S impurities~a!, a disordered interface~b!, anda disordered interface with oneimpurity ~c!. The insets illustratethe corresponding structures, witS represented as a dark circle; Aatoms in the lead and the chain ashown with gray and light graycircles, respectively.~a!, ~b!, and~c! have zero-bias conductances0.55G0 ,0.915G0, and 0.833G0,respectively.

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tion of S at the interface. Therefore, an experiment that doan Au contact with more S impurity should show an ehancement of theI -V nonlinearity. This also gives a possibexplanation of why experimentally the nonlinearI -V fittingparameters are not universal:54,55 they dramatically dependon the contact structure as well as the impurity concention.

Studying the effect of disorder in nanocontacts is anotimportant problem. To investigate this effect, we have rdomized the contact layer in the left lead of a pure Au nacontact, as shown in the inset of Fig. 6~b!. The distribution ofdisorder leads to a smaller distance between the contactand the Au chain, resulting in a better coupling. The currthrough this device is larger than that of the ideal contactis shown in Fig. 6~b!. For this device, the slope of the curreat zero bias isG;0.915Go , and it slightly increases toG;0.92Go at Vb;0.2 V. The I -V curve shows very weaknonlinear characteristics. This suggests that disorder amay create a tunneling barrier, which is overcome throuthe application of a bias voltage. However, to observeeffect, there need to be conducting states in the scatteregion. For our pure Au device, onlyone single stateis con-ducting and the maximum zero-bias conductance isG5Go . Therefore, atG;0.915Go , the conducting channel ialready open to near its maximum at zero bias, hence it cnot be further enhanced in any significant way by applyinbias. The results in Fig. 6~b! show that disorder is an important factor that allows more Bloch states in the leadscouple with the scattering region and contribute to transpproperties. This is clearly seen when we notice that the hcurrent plateau observed in Fig. 3 essentially vanishesdisordered device.

Combining the effect of disorder and S impurity is alcrucial. The latter enhances the tunneling barrier andformer may enhance the coupling of the device to the leaWe have used the disordered structure studied in theparagraph and replaced one of the Au atoms at the conlayer with an S atom, as shown in the insets of Fig. 6~c!. Thezero-bias conductance for this device isG;0.833Go . Thisnumber is larger than that with only a tunneling barrier~ideal

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contact with S impurity!, but is smaller than that with only adisordered contact~which has better coupling!. The finalconductance is due to a competition between both effeThe I -V curve for this device is presented in Fig. 6~c!, itshows very weak nonlinearI -V characteristics: analysis oour data show that the slope reachesG;0.85G0 at Vb;0.18 V. Therefore, there is still a weak nonlinear behavbut it is small due to the fast saturation of conducting chnel in the device.

From these results we can conclude that theI -V charac-teristics of an atomic junction is a complex phenomenonwhich the leads, eigenstates of the scattering region, asas impurity and disorder play major roles. In particular,tunneling barrier created by impurities can result in nonlinI -V behavior of the nanodevice. However, to be able toserve this effect, conducting channels in the device neebe present, otherwise transmission saturation is reachesmall voltages and a linearI -V characteristics is seen. Fomally, it is the transmission coefficientT(E,Vb) that is ofcrucial importance when analyzing the effect of the eigestates of the leads and the device, as well as the lead-decoupling.

In the following section we determine the behaviorT(E,Vb) and we will address the following questions: Is thvoltage independence ofT(E,Vb) an adequate pictureWhich T(E,Vb) behavior would result in nonlinearI -V char-acteristics? Can the major characteristics ofI (Vb) be quali-tatively estimated from simple arguments or does one alwneed to perform an extensiveab initio simulation?

IV. BEHAVIOR OF TRANSMISSIONCOEFFICIENT T„E,VB…

For all the Au nanocontacts we have investigated theorcally, T(E,Vb50) increases as a function ofE ~for E,0.2 eV). A typical behavior is shown in Fig. 7~a! by adashed line for an ideal top site, pure and perfect Au devFrom this curve, it is clear that there is transmission enhanment as a function ofE, which should result in a nonlineaI -V curve if T(E,VbÞ0) behaves in the same way. In th

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same graph we have also plottedT(E,Vb50.136 V). Onecan observe that the general energy dependence ofE is stillthe same, namely, increasing, but there is a global shift ofcurve downward. Therefore, an increase in the transmiscoefficient as a function ofE is compensated by its decreadue to increased bias voltage. The total effect on the curis to produce alinear I-V curve as shown in Fig. 3~b!. Wealso note that this complete compensation betweenE andVbis not universal and can be different from one systemanother. In fact, even for the same device it can behaveferently at different energy ranges. This results in differefeatures in theI -V curves such as plateaus, wiggles, etc.we have discussed previously.

A similar analysis is done on a device with a S impurity.The results are shown in Fig. 7~b!. For this device it is clearthat the effect ofE on transmission atVb50 is more pro-nounced. This is evidence that tunneling is an important ftor. In addition to this, we note that applying a bias voltagethe system causes a decrease in the transmission coeffibut it is not a global decrease. In particular, a bias voltageVb50.136 V actually increases transmission at small engies as shown in Fig. 7~b!. Therefore, the combined effect oE andVb does not cancel and it gives rise to nonlinearI -Vcharacteristics as seen in Fig. 5. According to this analyone can easily predict some aspects of theI -V curves just bystudying the transmission coefficientT(E,Vb) at two differ-ent bias voltages. Obviously, this helps to predictI (Vb) char-acteristics with much less computational effort.

In Figs. 7~c,d!, we do a similar analysis on a device a widisorder and a device with a disorder plus impurity, resp

FIG. 7. Scaled transmission coefficientT(E) at Vb50 ~dashedline! andVb50.136 V~solid line! for ideal contact structure at thtop site ~a!, one S impurity structure~b!, disordered interface~c!,and disordered interface with one S impurity~d!. The scaling factoris T(E50,Vb50).

19541

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tively. Due to the conductance channel saturation effecthe scattering region, the transmission coefficient has a vweak energy dependence~roughly constant!. This behavioralso emerges in theI -V curve, which shows a very weanonlinear behavior. We conclude that the roughly linearI -Vcurves of Figs. 3~b! and 6~b,c! are due to very different ori-gins. In the latter case it is due to the channel saturaeffect in the scattering region, whereas in the former it is dto a compensation between the effects of increasing enand bias voltage onT(E,Vb).

V. DISCUSSIONS

We have already shown in the preceding sectionstransport at the molecular level is a complex phenomenTo understand these systems, careful experimental workseparates electronic effects from structural relaxationswell as detailed calculations that include the effects ofmolecule, leads, and their coupling are required. In this stion we discuss and compare our findings with previoupublished theoretical concepts and experimental results.

Experimental work reported by Costa-Kra¨mer et al.54,55

have shown clear nonlinearI -V characteristics in Au nanocontacts starting at biasVb50.1 V, and its origin was attrib-uted to stronge-e interactions. To rule out the impurity effect, the authors54 have used scanning electron microscopyanalyze contacts of diameter;300 nm. They also used energy dispersive x-ray analysis and determined a contamtion concentration below detection sensitivity. Experimencleanliness checks performed for the large nanocont(;300 nm), however, cannot be extrapolated to junctionsfew atoms in size due to the exquisite chemical sensitivitydemonstrated by our model. In other experiments56 with Aurelays, it was observed that the conductance quantizationtogram survived for even larger bias (13Go peak persistsfor Vb;1.8 V). These data show clearly that in suchexperiment nonlinearI -V behavior cannot occur at low bia(Vb.0.1 V). We believe that these junctions were formbetween atomically clean gold contacts. These experimwere performed by forming and breaking the contacts vquickly ~approximately in microseconds!, thus removing theimpurity atoms from the Au junction even if they existed.72

Recently, elegant experimental work by Hansenet al.57

found that contaminated Au nanostructures show nonlinI -V characteristics, whereas experiments done with a cltip sample in UHV show perfect linearity forVb,0.7 V.The nature of the contamination was not determined.

A fundamental question that is to be addressed insection is transport through an impurity~or a tunneling bar-rier at the contact!, the most likely physical picture that caexplain the observed nonlinearI -V characteristics. In the fol-lowing we compare and discuss some of the conceptsscribed in the literature relevant to this issue.

Free-electron models have been used to describe thehavior of nanostructures under external bias.31,39 In thesesystems, depending on the potential profile across the deand the external bias voltage drop, variousI -V characteris-tics can be extracted. In passing, we note that these calctions neglected the voltage dependent couplingT(E,Vb)

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~discussed in Fig. 7!. This is obviously a major drawback focalculations at the molecular level, as shown by our analyHowever, a voltage drop at the contacts is still a reasonaapproximation. In Fig. 8, we plot the Hartree potential acrothe four-Au-atom chain. It is clearly seen that the potendrops mostly at the interfaces. However, assuming a unifpotential across the constriction is not adequate due toatomic structure and the small variation of charge transfea function of bias voltage.

A self-consistent tight-binding~TB! model has also beeimplemented to find the conduction dependence of eeigenchannel in the Au nanocontact as a function of thevoltage.73 In these calculations, the TB parameters are calated from a bulk system and the charge neutrality of eatom of the nanocontact is enforced in order to carry outself-consistent calculations. Although it is not clear if the Tparameters determined from bulk structures are diretransferable to nanocontacts with atoms of low coordinatnumber, especially when put under a bias potential, ourabinitio results show that charge neutrality is a valid appromation for Au devices without impurities since charge trafer is quite small. In self-consistent tight-binding modecharge neutrality is accomplished by locally adjustingchemical potential. For a zero-bias calculation and nanoctacts with three atoms, Ref. 73 shows that a local poten;3 eV needs to be added to the central atom to achcharge neutrality. This seems to be a large value for Hamtonian correction. Therefore, we suggest that for pure

FIG. 8. ~a! Average Hartree potential (VH) across the centracross section (;9 Å2) of the ideal contact at the top site: dotteline at zero bias and dashed line atVb50.5 V and~b! their differ-ence showing that most of the potential drops symmetrically atinterfaces. Black circles correspond to atom positions alongconstriction for guidance.

19541

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nanostructures, zero-biasab initio calculations should bedone to extract the TB parameters~rather than from bulk!.These can then be used more safely with charge neutrconstraints to deduceI -V characteristics of nanocontactSince we have seen only small effects of the bias voltagethe atomic charge transfer for a S doped Au structure in ouab initio calculations, we suggest that for this particulstructure it is possible to deduce the charge distributionthe TB parameters from zero-bias calculations. Using thTB interaction parameters and the zero-bias charge at eatom as a constraint, theI -V characteristics of these structures can then be solved self-consistently. The results areaccurate compared to a fullab initio calculation, but theyshould give better results than using the conventionalparameters derived from bulk systems.

There are other important theoretical calculations gobeyond the single-particle picture. Since the device consttion is narrow,e-e interaction can be strong and non-Fermliquid behavior might have to be taken into account. Hoever, it was shown by Maslov and Stone74 that for thesesystems the resistance is due to contact and thus the reare independent from the Luttinger-liquid behavior in tconstriction. Therefore, in these calculations74 transmissionis found to be identical to noninteracting particles. An impotant, single-particle transmission assumption is incorporain this model.74 However, at nonzero temperature or/and bvoltage, a finite number of particles are injected into tnanostructure resulting in backscattering effects. Theseto charge accumulation at the interface, creating an epotential in addition to the original constriction potentiaThis additional potential is both Hartree (VH) and exchangecorrelation (Vxc) in nature, and it is the reason for the scalled resistance dipole.75 Due to this additional potentiacontribution, it was shown76 that for one-dimensional~1D!systems the transmission coefficient is renormalized. It wfurther proposed that this charging effect can even closconducting channel that is 90% transmissive.55 This channelis gradually opened asVb is increased for a complete tranmission atVb;0.35 V, thereby inducing a nonlinearI -Vcurve. From a theory point of view, the picture of chargiinduced nonlinearI -V characteristics should overcome twfurther difficulties: that the renormalized transmission dpends on an interaction parametera, which cannot yet bedetermined for atomic wires; and that previous calculatiosolved a 1D case with no effect ofVb on the transmissionT(E,Vb). In the rest of this section, we follow the interestinidea of the charging effect and analyze it in greater detaiunderstand if this effect, which leads to channel closing54,55

can give rise to nonlinearI -V curves of atomic devices. Tostart, we follow the work of Yueet al.76 by assuming a stronginteraction and write the renormalized transmission coecient as55,76

TR~E!5T0~E/D0!2a

R01T0~E/D0!2a, ~4!

where,T0 andR0 are the transmission and reflection coefcients of the noninteracting model such thatT01R051, anda is a parameter to describee-e interaction in the constric-

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tion. The parameterD0 is the energy range nearEF , whichcontributes to renormalizingT0 and is determined76 by D05\vF /W, where vF is the Fermi velocity of the system(108 cm/s for Au! and W the width of thenanoconstriction,55 W;10–20 Å. Therefore, we find thafor these devicesD0;1.0 eV. From Eq.~4!, we computethe currentI (Vb) at zero temperature by integratingTR(E)from zero toeVb , assuming no bias dependence ofTR(E).The conductance is then deduced byG5I (Vb)/Vb . We notethat due to the finite length effect of the constriction,55 L;100 Å, the renormalization effect is cut off for bias volagesVb,Vs , whereVs52p\vF /L;0.1D0. Within this ap-proach and using the free parameters as specified, weplotted in Fig. 9~a! the conductance of a single channel afunction of bias voltageVb . Qualitatively, the results showthat G increases withVb due to the channel opening. However, quantitatively they do not give a complete chanopening at the experimental value ofVb;0.35 V ~as sug-gested in Ref. 55!, if the channel is less than 10% transmsive atVb50. In fact, we found that a bias of 2 V is needeto overcome the charging potential barrier, thereby the nlinearity in I -V curve can only set in at much larger voltageWe also note that since the extra charge due to backscattis accumulated at the interface, which has a larger crosstion, we expect that DFT and the local-density approximtion to Vxc should work well. Our self-consistent calculations, with all the charge transfer and rearrangemeaccounted for, have already partially included the ba

FIG. 9. Conductance calculated from renormalized transmisdue to backscattering as a function of bias voltage~a! for T0

50.99, 0.9, and 0.8 corresponding to solid, dashed, and dolines, respectively, with interaction parametera51.0 and~b! forT050.9 and interaction parametera50.1, 0.5, and 1.0 corresponding to solid, dashed, and dotted lines, respectively.

19541

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scattering effects. Our results show that it is possible to ptially close a channel;60%, but not completely. Our expermental data, which shows zero-bias conductance;2.2Go , indicate that at least one channel is 20% transmsive and that channels areonly partially closed, consistentwith our theoretical work. Therefore, we can conclude at tpoint that the physical picture of electron interactionscompletely close and open a channel as a function ofVb ,while interesting, cannot explain our experimental data.

To further address the importance of backscattereffects,76 especially for our devices with realistic atomleads, we have investigated the dependence on the intetion parametera in Eq. ~4!. Tuning this parameter can dramatically change the renormalized transmission coefficias a function ofVb as shown in Fig. 9~b!. Therefore we havea wide range of this parameter for fitting our experimendata. However, if we change the Au nanocontact by onlsingle sulfur atom, we should not expect a very differente-einteraction parametera, thereby this would predict anI -Vcurve not too different from that without the sulfur. Thsuggests that strong interaction alone would not prestrong sensitivity of theI -V curve dependence on smaamount of impurities, in contradiction with experimentresults.56,57 Transmission renormalization is important in 1scattering problems where there is no charge in the leadsrealistic atomic leads, the backscattered charge is a sfraction of the original one, and its effect should be wscreened. With all these considerations, we believe it islikely that the dynamically established charging effect alois large enough to cause the observedI -V nonlinearity for Aunanocontacts. Finally, we note that any tunneling barrie71

could be responsible for the experimental nonlinearI -V datareported here. Although we explored the possibility, frotheory, that the barrier is induced by a single S atom~or otherimpurity atoms, with S being the more likely one due toaffinity to Au!, more comprehensive experiments are neeto firmly establish the origin of the tunneling barrier.

VI. CONCLUSION

In this paper, we discuss the electronic transport propties of Au nanocontacts from both experimental and theoical studies. Our experimental data analysis enables useparate electronic effects from structural relaxations, alloing a better comparison to theoretical modeling of these nodevices. Our theoretical work shows that transport propties at the molecular scale need to be analyzed at the syslevel: leads, the molecule, and their interactions have tostudied simultaneously. The self-consistently determintransmission coefficientT(E,Vb) is shown to vary as a function of E andVb . This gives rise to differential conductancfluctuations of the order of 1% at temperature 300 K takinto account the experimental averaging process. Thesetuations are attributed mainly to the effects of the lead bastructure. Most striking, however, is the possibility thatsingle impurity atom at the contact region can alterI -Vcurves qualitatively in these devices: pure and perfect

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nanocontacts do not give the observed nonlinearity, whisulfur doped device does. Importantly, we have shownthe measured nonlinearI -V characteristics of Au nanowirecan be quantitatively modeled by impurity effects that crea tunneling barrier at the nanostructure junction. Howevother effects such as heating and charging can stillpresent, but we believe that their role is secondary. Tanalysis clearly points to the vital importance for understa

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ing contacts in nanoelectronic devices, and, perhaps, toploit it for the benefit of device operation.

ACKNOWLEDGMENTS

We gratefully acknowledge financial support froNSERC of Canada and FCAR of Quebec. H.M. thanksN. Agraıt for a useful discussion on their experimental wor

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