PHYSICAL REVIEW B 15 OCTOBER 2000-IIVOLUME 62, NUMBER 16

Nonlinear effects in conductance histograms of atomic-scale metallic contacts

A. Garcıa-Martın,1 M. del Valle,1 J. J. Sa´enz,1 J. L. Costa-Kra¨mer,2 and P. A. Serena31Departamento de Fı´sica de la Materia Condensada, Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain

2Instituto de Microelectro´nica de Madrid, Consejo Superior de Investigaciones Cientı´ficas, Isaac Newton 8, PTM,28760 Tres Cantos, Madrid, Spain

3Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cientı´ficas, Cantoblanco, 28049 Madrid, Spain~Received 25 February 2000!

General properties of conductance histograms of atomic-scale metallic contacts are discussed. Nonlineareffects are included assuming a quasiadiabatic representation for electron transport. Histograms of conductanceand histograms of differential conductance markedly differ when increasing the bias voltage. Averaging overdifferent nanowire breakage paths at high bias rapidly smears out the peaks inI /V histograms, while it leadsto new peaks at half integer values indI/dV histograms. Recent experimental results on the histogramsevolution with the applied bias are discussed.

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I. INTRODUCTION

During the last decade, nanotechnology has becomeessential topic where multiple scientific disciplines meet,ing a key for future technological developments. A fundmental branch of nanotechnology concerns the study of etronic transport through mesoscopic and low dimensiosystems.1,2 The practical realization of such systems has bbased on devices working at low temperatures, whereelectronic current flows in a two-dimensional electron g~2DEG! under a bias voltage. With this background, it hbeen possible to develop two-dimensional~2D! constrictions,quantum dots, cavities presenting chaotic features, etcparticular, the study of electronic propagation through smconstrictions has received attention in order to test baphysical ideas and propose new electronic devices.

Although 2DEG based devices have been for a long tsuitable to analyze basic aspects of mesoscopic systemgreat effort has been made to use metallic contacts of nmetric size, metallic nanowires, to study the basic featurequantum transport, opening a path to use them as internects among active elements in integrated systems.3 Thecharacteristic small dimensions of such contacts~only a fewtimes the Fermi lengthlF) make them appropriate to studphenomena such as ballistic transport or conductance qtization ~CQ!. CQ is an appealing phenomenon which rveals the quantum nature of electron transport when onfew transversal modes~due to the electron spatial confinement! are populated. In this case, the measured conductG5I /V ~where I is the current intensity andV the appliedbias voltage across the constriction! is given, in the smallbias limit, by the expressionG5N2e2/h5NG0, whereG0 isthe conductance quantum,N reflects the number of allowepropagating modes,e is the electron charge, andh is thePlanck constant.4 Although CQ was confirmed in 2DEGdevices,5,6 the study of such phenomenon in metallic nancontacts has attracted special attention because it shounoticeable even at room temperature due to the relativlarge energy separation between transversal modes.

Nowadays, it is possible to create metallic nanocontawith minimal sections formed by a few atoms, using tec

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niques based on either the scanning tunneling microsc~STM!,7–10 or the mechanically controllable break junctio~MCBJ! method.11,12 Using both experimental approachebased on piezo-controlled motions, it is feasible to stabilthe contact structure at atomic level during a long time, keing a constant electron current. The presence of conductasteps of the order ofG0 when the nanocontact sectiochanges has been interpreted as a CQ signature.

The electrical characterization of these low dimensiosystems is a topic of great interest and several experimhave measured current-voltage (IV) characteristics at differ-ent conductance values.8,10,13–15In general, the measuredIVcurves show a characteristic nonlinear behavior mstrongly marked in situations where only a few conductanchannels are involved.8,10 These nonlinear features can bnegligible when the nanocontact is formed over a substra15

A remarkable feature observed in the free-standing metananocontactIV curves is that the nonlinear component icreases with a power lawI;Vq where the exponentq rangesbetween 2 and 3.10 This behavior differs from that observein microfabricated 2DEG constrictions where the strong nlinear behavior is seen as a change of slope in theIVcurve.16,17 These changes in the slope also give rise toG0/2steps in the differential conductanceg5dI /dV when thenanocontact sectionA varies.18 These new plateaus ing(A)are explained in terms of a model that assumes the appbias voltage drops symmetrically with respect to the nacontact middle point.19,20 A similar behavior is found inthree-dimensional constrictions21,22although, in this case, theIV curves show an additional dependence on the 3D indudegeneracy of the transversal modes. The predicted behafor the curveg(V) in 3D nanocontacts21 has not been experimentally observed. Several explanations have been propin order to explain this disagreement, ranging from the extence of Coulomb blockade23 to the presence of a nonnegligible tunnel current contribution which depends onnanocontact geometry22 or to the presence of impurities ithe contact region.24

In a different context, theIV traces appearing in metallinanocontacts in the superconductor regime have been usestimate the number of channels involved in the elect

11 139 ©2000 The American Physical Society

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11 140 PRB 62A. GARCIA-MARTI N et al.

transport when conductances close toG0 are measured.25

The obtained experimental results seem to support thethat the chemical valence determines the number of alloconducting channels in single-atom contacts,26 although thepresence of geometrical effects should be accounted for27

Not only the electrical characterization of metallic nancontacts and nanowires has been the subject of study, buCQ phenomenon itself in such systems has been extensanalyzed. A tool to get insight on CQ lies in the elaboratiof histograms of conductance. Due to the inherent nonreducibility of similar atomic structures during nanocontabreakage experiments, some statistical tool is needed. Coing conductance occurrences for different retraction expments and building a histogram of conductance valueone, perhaps the simplest, of such tools.12,28 Although thereis a vivid debate about the origin and meaning of such cductance histograms,25,27,29 the evaluation and constructioof histograms has become a standard tool in the study ofconductance quantization phenomenon under different stions. Conductance histograms obtained for nanocontformed between two macroscopic metal wires demonstrathe strength of the CQ phenomenon30 and its independencon the experimental setup. Different kinds of experimeusing conductance histograms have been presented into analyze the CQ dependence on different external pareters such as chemical environment,31 temperature,32 mag-netic field,33–35 retraction speed in STM-like nanowirbreaking experiments,33 etc.

Following the same experimental methodology, a recwork36,37 shows a strong dependence of the histogram peon the applied voltage. With increasing bias voltage the pheight decreases and finally disappears at some criticalageVc . In particular the peak corresponding toG5G0 dis-appears atVc'1.9 V at room temperature whereasVc'2.2V at liquid-nitrogen temperature. This experimental reswas interpreted in terms of electromigration inducedhuge current densities in the nanocontact. Although thisplanation presents a reasonable physical framework itglects the nonlinearity in theIV curve which can play animportant role at such high voltages. A very interesting qution arises when observing the conductance histograms uapplied voltage. In principle, the nonlinear term appearingIV measurements should shift the histogram peaks towhigher G values but experimental findings do not revealnoticeable shift. There are even results indicating thatshift is towards lower G values for increasing applivoltage.38 The aim of the present work is to explain the oserved evolution of conductance histograms as consequof the nonlinear behavior of theIV characteristic curve, thuoffering a contribution which coexists with the diffusion anelectromigration ones.

The structure of the paper is as follows: in Sec. II wpresent the model for a 3D metallic contact and the metto determine the conductance histograms including finite bvoltages; in Sec. III we present calculated histogramsanalyze the voltage influence and its dependence on theage drop across the nanowire; finally, in Sec. IV the mconclusions are summarized.

II. THEORETICAL MODEL

A. Nonlinear conductances

A deep insight in the physics of quantum electronic traport through small contacts is obtained from Landaue

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scattering approach to electrical conductance.4 An atomic-scale metallic contact can be considered as a small conning nanowire between two three-dimensional electron gaThe quantization of the transversal momentum of the etrons defines different subbands or channels, each havinassociated transmission probabilityTn to go from the sub-bandn in one reservoir to the other.

In a metallic 3D constriction, the electrons are confinedthe transversal directions whereas transport occurs oalong the other one, let us call it thez direction. Thereforeeach channel has a momentumkz in the transport direction,and the dispersion relation for each channel in the reservresults ine5e(n,kz)5Ez(kz)1en , whereen is the charac-teristic transversal energy of the channeln. Thus the numberof channelsN in each reservoir is given by the maximuvaluen satisfyingen,EF . The contribution to the current oa given channel in the left reservoir for a differential enerinterval de is given by the product of the electric chargee,the group velocity (1/\)de/dkz , the density of states alonthe wire rn(e)de, and the transmission probability for aelectron to go from the channeln in the left reservoir to theright reservoir,

dInl→r5e

de

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A similar expression is obtained for electrons in the rigreservoirarriving to the channeln in the left one,

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de

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where applied bias voltage shifts the energy at whichtransmission probability is evaluated. After including thspin degeneracy, the density of states is given byrn(e)5(1/p)dkz /de. The total current is therefore given by thsummation over all the channels and integration overenergy spectrum

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whereN8(e) gives the number of channels satisfyingen,e.Here we have used the fact that the Fermi-Dirac distribut~implicit in the density of states accounting for temperatueffects! at room temperature is well approximated by the sfunction.

For finite biasV, we must distinguish between the coductanceG and the differential conductanceg. The conduc-tanceG is defined as the inverse of the resistanceG[I /Vwhereas the differential conductanceg is defined asg[dI/dV,

g5]

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Following previous approaches21,39,40 we assume that, ingeneral, the voltage drops a fractionbV between the left

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PRB 62 11 141NONLINEAR EFFECTS IN CONDUCTANCE HISTOGRAMS . . .

reservoir and the contact and (12b)V between the constriction and the right reservoir, while the potential remains costant inside the constriction itself41 (b accounts for anyasymmetry of the potential drop!. b51/2 implies a symmet-ric drop while b51 implies a perfect drop betweeen thnegative electrode and the contact~see the scheme in Fig. 1!.The actual potential drop along atomic-size constrictionsdifficult problem that has been recently addressed witself-consistent tight-binding42 and ab initio local-density43

calculations. For gold contacts and small bias (V& 0.5 V!,the potential drop was found to be approximatelly symme(b'1/2).43 Interestingly, as a consequence of asymmetin the local-density of states, the potential drop at higher bcan be highly asymmetric (b'1) even for geometricallysymmetric contacts.

Under the assumption of quasiadiabatic transport,18,21,39,40

g(e,V) only depends on the energy difference betweenincoming electrons and the bottom of the electrostatic potial, i.e., g(e,V)'g(e1b eV!. Thus we obtain for the conductance

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Under the same approach the differential conductanceg isthen given by the weighted average of two zero-voltage cductances at different effective Fermi energies:44

g~V!'bg@EF1b eV#1~12b!g~EF2~12b! eV!.~7!

As expected, in the limit of zero bias, Eqs.~6! and ~7! re-cover Landauer’s formula

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FIG. 1. Schematic representation of the contact model. Thetential profile at a biasV is also sketched.

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We should emphasize that Eqs.~6! and ~7! provide asimple approach to an otherwise extremelly complex prlem. The energy dependence of transmission probabiliTn(e,V) in actual contacts depend on the local chemistry ageometry around the contact and may change with theplied biasV. This bias dependence, which for gold contais important only for high bias (V*2 V!,43 should then betaken into account in any detailed quantitative analysis ofexperimental results.

B. Saddle-point contact model

In order to calculate transmission coefficients, we cosider a simple saddle-point contact~SPC! model, originallyintroduced by Bu¨ttiker,45 and used to describe electrontransport in two-dimensional systems.39 The SPC model wasshown to be very useful in the analysis of different transpproperties in three-dimensional constrictions.27,46 In thismodel the constriction geometry connecting the two resvoirs is given by the equipotential surfaceV(x,y,z)5e,wheree is the electron kinetic energy at the narrowest partthe constriction.V(x,y,z) can be written as

V~x,y,z!5ep

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Electronic transport takes place along thez direction,whereas electrons are confined in the transversal directby parabolic potentials. The constriction geometrysketched in Fig. 1 looks like a parabolic geometry with ‘‘eliptical’’ cross section.A is the area of the narrowest sectioRc is the radius of curvature andh describes the degree oanisotropy of the elliptical cross section. We may also defiin this case an opening angleu,47 given by

tan2~u![~AA/p!/Rc . ~10!

The transmission probability of then,m channel can becalculated exactly in closed form and takes the simple fo

Tnm~e!5H 11 expS 22p@Ap~A/lF2 !~e/EF!2enm#

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~11!

with enm5(n11/2)/Ah1Ah(m11/2). We take a constanvalue h50.668 in order to break the channel degenera~this is known to be a good approximation for gocontacts27!. The qualitative results will not depend on thexacth value, however.

III. RESULTS AND DISCUSSION

A. Conductance versus differential conductance at finite bias

Although for very small biasG andg are exactly the samequantities, there are strong differences at higher bias. Inder to emphasize these differences in Fig. 2 we have plothe currentI and bothg andG versus the applied biasV fora contact with fixed opening angleu510° at different crosssections. Figure 2~b! shows the characteristic evolution ofgversusV for b51/2. The evolution from quantum plateausinteger multiples ofG0 towards half integer values can bclearly seen. Nonlinear effects change in a noticeable w

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11 142 PRB 62A. GARCIA-MARTI N et al.

g(V) for voltages larger than 0.15EF /e ~i.e., 0.8V for gold!.In contrast, the behavior of the conductanceG @see Fig. 2~c!#does not show this evolution to noninteger multiples ofG0and the quantum plateaus gradually disappear.

The effect of a possible asymmetry in the potential dracross the constriction is illustrated in Fig. 3 where we haplotted bothg andG versusV for b51/3. At very small bias,the asymmetry induces a linear dependence of the contance withV ~except very close to integer multiples of thconductance quantum!. At higher bias, the quantum plateauin g evolve gradually towards new plateaus at (n11/3)G0(V.0) and (n12/3)G0 (V,0) following Eq.~7!. As it wasshown forb51/2 the conductanceG does not present anplateau at nonintegerG/G0 values in the high-voltage regime.

B. Conductance histograms

In metallic contacts, the behavior of the conductance welongation is a stepwise curve. In general, the position ofsteps and the conductance plateau value change fromexperiment to another. However, and notably for noble mals such as gold, copper, and silver, plateaus tend to apclose but slightly below integer values ofG0. This is evi-denced in the histograms as peaks near, but slightly beinteger values ofG0. This has been explained in termsstructural disorder making transmission coefficients devfrom perfect transmission.48,49

A detailed theoretical description of the mechanical aelectrical properties of the contact during elongation is a v

FIG. 2. Current intensityI ~a!, differential conductanceg5dI/dV ~b!, and conductanceG5I /V ~c! ~both in units ofG0),versus voltageV for different sectionsA of the three-dimensionaconstriction depicted in Fig. 1 withu510° andb51/2.

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difficult task.50 A qualitative picture, however, can be obtained from the model system discussed above, togetheran effective elongation path:27 Let us assume that, for smacontacts, the cross sectionA follows an almost exponentiadependence with the displacementd. The actual evolution ofthe contact geometry with elongation does not correspona constant angleu but, in general, would be a nontriviafunction u(A) which will change from one contact toanother.51 In a typical experiment the conductance verselongation curves present conductance plateaus whichgeneral, do not correspond to exact integer multiples ofG0.Repetitive cycles of elongation-retraction processes usugive similar stepped curves but with plateaus at differconductance values, i.e., the stepped conductance cchange from one experiment to another. This is schemcally illustrated in Fig. 4 where dashed line and thin continous line would represent two different single experimenrealizations in a typical elongation-contraction process!. Theeffective elongation path would correspond to the expemental average of many individualG vs d curves~thick linein Fig. 4!. This effective path can be estimated as followwithin the SPC model, the radius of curvature of the contRC /lF should be of the order of or larger than 1 in orderhave no unphysical tunneling contributions to the condtance~actually, in metallic contactsRC should be larger thanan atomic radius'lF). For simplicity, we consider an effective path at constantRC52lF , being the effective pathu(A) given by Eq. ~10! ~the results will not depend—qualitatively—on the exact dependence ofA on d). Assum-ing that the contact can take any section along this pathobtain the conductance histogram shown in Fig. 5~a! ~notice

FIG. 3. Differential conductanceg and conductanceG in unitsof G0 versusV for different sectionsA of the three-dimensionaconstriction shown in Fig. 1 withu510° andb51/3.

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PRB 62 11 143NONLINEAR EFFECTS IN CONDUCTANCE HISTOGRAMS . . .

that at zero bias the histograms ofG and g are exactly thesame!. Despite the simplicity of the model, the obtained htogram at zero bias is very similar to the experimental hisgrams for Au contacts. Although experimental conductahistograms present a rich structure~peaks asymmetry, peakdisplacements towards lower values, etc.! the important pointin this study will concern the height of the histogram peaThe relative peaks height obtained with our effective elontion path approximation is rather similar to that found expe

FIG. 5. Differential conductanceg and conductanceG ~in unitsof G0) histograms@H(g) and H(G)] obtained from the effectiveelongation path sketched in Fig. 4 at different bias,~a! eV50, ~b!0.2EF , ~c! 0.4EF . The potential drop is assumed to be symmet(b51/2).

FIG. 4. Conductance in units ofG0 as a function of the contacelongationDd for an effective elongation path with constantRC

52lF and zero bias~see text!. The dashed line and thin continuouline would represent two different single experimental realizatioin a typical elongation-contraction process. The thick line repsents the average conductance along the effective path.

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mentally and thus represents a good starting point to inclvoltage effects. Most of the experimental studies are ccerned with histograms at low biases. Recently, YasudaSakai36,37 have obtained a strong dependence of the condtance~G! histogram peaks with the applied voltage. As ware going to show, our simple model can qualitatively eplain their experimental observations.

The behavior ofg and G histograms@H(g) and H(G)]changes completely for finite bias@Figs. 5~b! and 5~c!#. As Vincreases, the peaks at integer multiples ofG0 gradually dis-appear, except the first one which remains at high voltagIn H(g) new peaks at half integer values can clearly be sfor V.'0.3EF . In contrast,H(G) do not show traces oconductance quantization above'0.4EF ~for gold contacts,EF'5.5 V!. Except for the behavior of the first peak at 1G0,the agreement with the experimental results ofG histogramsof Yasuda and Sakai is remarkable. In order to makecomparison more quantitative, following Ref. 37, in Fig.we have plotted both the experimental~a! and theoretical~b!heights of the first and second peak inH(G) versus the ap-plied biasV @the peak height is measured with respect tobackground inH(G)]. The qualitative agreement betweetheory and experiment suggests that the dissapearancpeaks atG larger than 1G0 may be partially explained interms of nonlinear effects in the ballistic conductance. Tfaster decrease in the experimental peak heights togewith the absence of the first peak at high bias could indicthat effects like atomic diffusion and electromigration bcurrent-induced forces52,53 may also play an important roleThese effects would lead to fluctuations of the conducta~contact cross section! with time. If the experimental timewindow used to measure a conductance value averagbetween different cross sections the peak structure ofhistogram would disappear.54 As a matter of fact, the experi

s- FIG. 6. Experimental~a! and theoretical~b! bias dependence othe normalized peak height for the first 1G0 peak~closed circles!and for the second 2G0 peak ~open circles! of the conductancehistogramsH(G). The peak heights are normalized at their valuat V50.1EF .

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mental behavior of the height of the first peak changes ab'1.5 V ~see Fig. 4 in Ref. 37!, which could indicate a biasthreshold for voltage induced electromigration. Therefohistogram evolution under increasing applied bias hasdifferent regimes: a first one governed by nonlinear effeand a second one where electromigration plays a moreportant role. Each regime should show its own rate of pheight decrease as might be extracted from the experimedata. It would be interesting to study the peak height behior as a function of temperature~the only available resultscorrespond to room temperature!. Since electromigration isalso a thermal activated process, we expect a shift ofthreshold bias towards higher values. This trend with teperature in experimental data can be seen when inspeFig. 4 of Ref. 36 where the rate at which conductance hisgram peaks disappear strongly depends on the temperaIn particular, we expect that the experimental curve showFig. 6~a! will change at lower temperatures towards the thoretical predictions shown in Fig. 6~b!.

We would like to point out that there is another factor thmay prevent the observation of peaks at high bias, eveH(g). Let us suppose that for some reason~particular geo-metrical arrangements, chemical environment, etc.! the po-tential drop along the nanocontact is different for eachperimental realization. In that case, the resultiexperimental histogram would be an average of histograhaving differentb factors. In Fig. 7 we have plotted thahistogram assuming a uniform distribution ofb values be-tween 0 and 1. As it can be seen, the final histogram is vsimilar to H(G), i.e., it does not show any peak other ththat corresponding to the first quantum.

IV. CONCLUSIONS

In this work we have constructed theoretical histogramof both the total (G) and differential~g! conductances, fornanometric-sized nanocontacts involving only a few pro

FIG. 7. Differential conductanceg and conductanceG ~in unitsof G0) histograms@H(g) and H(G)] obtained from the effectiveelongation path sketched in Fig. 4 atV50.4EF , after averagingbetweenb50 andb51.

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gating modes. Nonlinear effects induced by the bias voltV applied between the two electrodes~electronic reservoirs!have been included assuming a quasiadiabatic representfor electron transport, obtaining explicit dependences foGand g on V. Histograms calculation involves basically oningredient: the knowledge of the nanocontact geometry tevolution. We have used two approximations to describeevolution during the metallic nanowire breakage expements. On one hand, we have described the geometry ocontact using a saddle-point contact~SPC! model where themain geometrical features~minimum cross section and opening angle! are easily included. On the other hand, the nanoire dynamics has been averaged giving rise to an effecelongation path.

The main result is that histograms of conductanceH(G)and differential conductanceH(g) markedly differ for in-creasing applied voltages. The histogramH(G) losses itspeaked structure at voltages;0.2EF whereasH(g) keeps itsstructure up to higher voltages. This difference has toessentially with the richest structure noticed byg(V) in com-parison to that found inG(V). In this last case, the averagover many different nanowire breakage paths rapidly smeout the peaked histogram. This evolution withV agrees withexperimental observations, although some interpretationsuch results were given in terms of electromigration ehanced diffusive processes in the nanocontact region. Hedifferent explanation is provided, based on nonlinear cductance effects that appear for increasing voltages. Ourplanation does not exclude electromigration contributionshigh voltages. In fact, experimental histograms reflectexistence of two voltage dependence regions with differpeak disappearance rate. The histogram structure in thevoltage interval should be governed by the appearancenonlinear effects onG(V). Above a threshold voltage, thelectromigration contribution washes out the peaked strture.

In order to clarify the actual origin for the histogram deradation at high voltages different kinds of experiments cbe proposed. For instance, accumulate histograms for boGand g in order to see if the corresponding degradation rwith voltage differs or not. In the same fashion, interestiinformation could be gathered from controlled environmeexperiments, where surrounding substances~oil, glycerin,etc.! will hinder diffusive contributions.

ACKNOWLEDGMENTS

We acknowledge stimulating discussions with T. Lo´pez-Ciudad and J. I. Pascual. This work has been partially sported by the Comunidad Auto´noma de Madrid~Spain!through Project No. 07N/0024/1998, the Spanish DGES~MEC! through Project No. APC1998-0174, and the DGCyT ~MEC! with Project No. PB98-0464.

H.A.

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PRB 62 11 145NONLINEAR EFFECTS IN CONDUCTANCE HISTOGRAMS . . .

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