Huge negative differential conductance in Au-H2 molecular nanojunctions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Huge negative differential conductance in Au-H molecular nanojunctions A. Halbritter, P. Makk, Sz. Csonka, and G. Mih´aly
Department of Physics, Budapest University of Technology and Economics and Condensed MatterResearch Group of the Hungarian Academy of Sciences, 1111 Budapest, Budafoki ut 8., Hungary (Dated: November 5, 2018)Experimental results showing huge negative differential conductance in gold-hydrogen molecularnanojunctions are presented. The results are analyzed in terms of two-level system (TLS) models: itis shown that a simple TLS model cannot produce peaklike structures in the differential conductancecurves, whereas an asymmetrically coupled TLS model gives perfect fit to the data. Our analysisimplies that the excitation of a bound molecule to a large number of energetically similar looselybound states is responsible for the peaklike structures. Recent experimental studies showing relatedfeatures are discussed within the framework of our model.
PACS numbers: 73.63.Rt, 73.23.-b, 81.07.Nb, 85.65.+h
I. INTRODUCTION
The study of molecular nanojunctions built fromsimple molecules has attracted wide interest in recentyears. Contrary to more complex molecular electron-ics structures, the behaviour of some simple moleculesbridging atomic-sized metallic junctions can already beunderstood in great detail, including the number ofconductance channel analysis with conductance fluctu-ation and shot-noise measurements, the identifica-tion of various vibrational modes with point-contactspectroscopy and the quantitative predictive power ofcomputer simulations. The above methods were success-fully used to describe platinum-hydrogen junctions: itwas shown that a molecular hydrogen bridge with a sin-gle, perfectly transmitting channel is formed between theplatinum electrodes. Point-contact spectroscopy turned out to be an espe-cially useful tool in the study of molecular nanojunctions,as a fingerprint of the molecular vibrational modes can begiven by simply identifying the small steplike signals inthe dI/dV ( V ) curves of the junction. However, the de-tection of the small vibrational signals is difficult for junc-tions with partially transmitting conductance channels,where quantum interference (QI) fluctuations may givean order of a magnitude larger signal. Surprisingly, in-stead of observing small steplike vibrational signals uponthe background of QI fluctuations, molecular nanojunc-tions frequently show peaklike structures in the differen-tial conductance curves with amplitudes comparable toor even much larger than the QI fluctuations. The peakscan be either positive or negative and their amplitude canbe as small as a few percents, but – as we demonstrate inthis manuscript – the peak-height can be several ordersof magnitude larger showing a huge negative differentialconductance (NDC) phenomenon (Fig. 1).As a general feature of the phenomenon it can be statedthat at low bias the conductance starts from a constantvalue, at a certain threshold voltage the peaklike struc-ture is observed, and at higher bias the conductance sat-urates at a constant value again. The high-bias conduc-tance plateau can be either higher or lower than the low- -2.0-1.5-1.0-0.50.00.5-150 -100 -50 0 50 100 150-2-1012
I [ A ] d I/ d V [ / h ] V [mV]
FIG. 1:
Huge negative differential conductance (upper panel)and the corresponding I − V curve (lower panel) in gold-hydrogen nanojunctions. The solid black lines show the theo-retical fits with the asymmetrically coupled TLS model. Theparametrs are: E = 54 meV, T = 5 . K, σ = 0 . G , σ ∞ = 0 . G , N = 60 and W = 0 . bias plateau. In the first case the peak at the transitionenergy is positive, whereas in the second case it is neg-ative. In other words, the junction shows linear I − V characteristics both at low and high biases but with adifferent slope, and at the threshold energy a sharp tran-sition occurs between the two slopes. The switching ofthe conductance between two discrete levels implies thedescription of the phenomenon with a two-level system(TLS) model, but as we demonstrate in this paper a sim-ple TLS model only gives steps in the dI/dV at the exci-tation energy of the TLS, and it cannot account for sharppeaks.The above phenomenon is not a unique feature of aspecial atomic configuration, but it frequently occurs ina wide conductance range and it appears almost in all thestudied molecules and contact materials. In our exper-iments on gold-hydrogen molecular junctions we haverecognized huge negative differential conductance curvesin the conductance regime of 0 . − . showing a char-acteristic threshold energy of 30 −
100 meV. We have alsoobserved similar phenomenon in niobium-hydrogen nano-junctions. In parallel the group of Jan van Ruitenbeek– focusing on the conductance range close to the quan-tum conductance unit – has demonstrated the occurrenceof smaller peaklike structures. In their study the peaksappeared with H , D , O , C H , CO, H O and ben-zene as molecules, and Pt, Au, Ag and Ni as contactelectrodes. An scanning tunneling microscopy (STM)study of a hydrogen-covered Cu surface revealed bothsharp gaplike positive peaks and large negative differen-tial conductance peaks in the tunneling regime. The lat-ter study has also demonstrated a clear telegraph fluctu-ation between the two levels on the millisecond timescale.An earlier STM study demonstrated smaller negative dif-ferential conductance peaks due to the conformationalchange of pyrrolidine molecule on copper surface. The above results imply the presence of a general phys-ical phenomenon of molecular nanojunctions resulting ina similar feature under a wide range of experimental con-ditions. In Ref. 8 it was shown that in certain systems thepeaklike structures are related to the vibrational modesof the molecular junctions and the observed features weresuccessfully described in terms of a vibrationally medi-ated two-level transition model. This model, however,cannot describe the large negative differential conduc-tance curves in our studies, and the large peaklike struc-tures in the STM experiments. In the following, we present a detailed analysis of theobservations in terms of two-level system models. Wedemonstrate the failure of a simple TLS model and thenecessary ingredients for producing peaklike structuresinstead of simple conductance steps. We propose a modelbased on an asymmetrically coupled two-level system,which can describe all the above observations. The com-parison of the model with experimental data implies ahuge asymmetry in the coupling strength, which canbe explained by exciting a strongly bound molecule toa large number of energetically similar loosely boundstates. In our opinion, the proposed model is appropri-ate for describing the appearance of peaklike structures(or even negative differential conductance) in the dI/dV curves of molecular nanojunctions under a wide range ofexperimental conditions.
II. THE FAILURE OF A SIMPLE TWO-LEVELSYSTEM MODEL
The scattering on two-level systems in mesoscopicpoint-contacts has been widely studied after the discov-ery of point-contact spectroscopy (see Ref. 11 and refer- ences therein). In the following, based on the results fora general point-contact geometry, we shortly give anoverview of the scattering process on a two-level systemlocated near the center of an atomic-sized nanojunction.The TLS is considered as a double well potential, wherethe two states in the two potential wells have an energydifference ∆ (see Fig. 2). The two states are coupled bytunneling across the barrier with a coupling energy Γ.The coupling between the wells causes a hybridization ofthe two states, resulting in two energy eigenvalues witha splitting of E = √ ∆ + Γ . It is considered that in thelower state of the TLS the contact has a conductance σ ,whereas in the upper state the conductance is σ . Theoccupation number of the two states are denoted by n and n = 1 − n . These are time-averaged occupationnumbers, the system is jumping between the two statesshowing a telegraph fluctuation. On a time-scale muchlonger than that of the TLS the fluctuation is averagedout, and the I − V characteristic is determined by thevoltage dependent occupation numbers: I ( V ) = ( σ n + σ n ) V, (1) dIdV = σ + ( σ − σ ) (cid:20) n + V dn dV (cid:21) . (2)The occupation of the upper state can be calculated fromthe rate equation: dn dt = P → − P → , (3)where the transition probabilities are determined byFermi’s golden rule: P → = ρ F n γ Z d ǫf ( ǫ, eV )(1 − f ( ǫ − E, eV )) , (4) P → = ρ F n γ Z d ǫf ( ǫ, eV )(1 − f ( ǫ + E, eV )) . (5)Here an electron from an occupied state with energy ǫ scatters on the TLS to an unoccupied state with energy ǫ ± E . The nonequilibrium distribution function of theelectrons is denoted by f ( ǫ, eV ), whereas ρ F stands forthe density of states at the Fermi energy. The transi-tion matrix element from an initial electron state and theground state of the TLS to a final electron state and theexcited state of the TLS is considered as a constant cou-pling strength: γ = 2 π/ ¯ h · |h i, | H e − T LS | f, i| . Assum-ing that the TLS is situated at the middle of the contact,where the half of the electrons is coming from the leftand the half from the right electrode, the nonequilibriumdistribution function can be approximated as: f ( ǫ, eV ) = f L ( ǫ ) + f R ( ǫ )2 , (6)where f L ( ǫ ) and f R ( ǫ ) = f L ( ǫ − eV ) are the equilibriumFermi functions of the left and right electrodes, for whichthe chemical potential is shifted by the applied voltage.By inserting the distribution function to Eq. 4-5 and us-ing the formula: Z f ( ǫ )(1 − f ( ǫ − a ))d ǫ = a a kT −
1) (7)the rate equation can be written as: dn dt = n ν − n ν , (8)where ν and ν are the inverse relaxation times of thelower and upper state of the TLS, ν = ρ F γ (cid:20) eV + E eV + E kT − − eV + E − eV + E kT −
1) + E (coth E kT − (cid:21) ν = ρ F γ (cid:20) eV − E eV − E kT − − eV − E − eV − E kT − − E (coth E kT − (cid:21) . The steady state solution for the occupation number ofthe upper state is: n = 1 − n = ν ν + ν . (11)In the zero temperature limit the inverse relaxation timesand the occupation number take the following simpleforms: ν = ρ F γ · (cid:26) e | V | < Ee | V | − E for e | V | ≥ E (12) ν = ρ F γ · (cid:26) E for e | V | < Ee | V | + 3 E for e | V | ≥ E (13) n = (cid:26) e | V | < E − Ee | V | + E for e | V | ≥ E (14)Fig. 2 shows the evolution of the occupation numbers,the I − V curve and the differential conductance curvein the zero temperature limit. At eV ≫ E both statesare equally occupied with n = 1 /
2, but the transi-tion towards this is very slow. A characteristic energyscale, δE for the variation of the occupation numberscan be defined by extrapolating the linear growth of n at eV = E + to the saturation value (see the upper panelin the figure). Due to the small slope of n ( V ) this char-acteristic energy scale is larger than the excitation energy,precisely: δE = 2 E . This slow change is reflected by asmooth variation of the I − V curve. The differentialconductance curve shows a steplike change at eV = E ,and then saturates to σ ∞ = ( σ + σ ) /
2. The size ofthe step is smaller than the overall change of the conduc-tance (∆ σ E = ∆ σ ∞ / dI/dV curve. d I/ d V eV/E I E n n n E FIG. 2:
Results in the zero temperature limit of a symmetri-cally coupled TLS model. The upper panel shows the voltagedependence of the occupation numbers. The characteristic en-ergy describing the growth of n above the excitation energy isdenoted by δE . The middle panel shows the I − V curve, thelow and high bias slopes ( σ · V , σ ∞ · V ) are illustrated by dot-ted lines. The lower panel shows the differential conductanceexhibiting a jump of ∆ σ E at the excitation energy and anoverall change of ∆ σ ∞ between the zero and high bias limits.On the right side the two-level system model is demonstrated. In the above considerations the calculation of the cur-rent (Eq. 1) includes only elastic scattering of the elec-trons on the two states of the TLS with different corre-sponding conductances, and the inelastic scattering pro-cesses are just setting the voltage dependence of the oc-cupation numbers. However, the inelastic scattering ofthe electrons in principle gives direct contribution to thecurrent: above the excitation energy the backscatteringon the contact is enhanced due to the possibility for in-elastic scattering. More precisely the inelastic currentcorrection can be written as: δI in = e · [( − P L + , → L − , + P L + , → R + , (15) − P R − , → L − , + P R − , → R + , ) + ( ↔ )] . Here L + and R − are incoming states to the contact on theleft and right side, while R + and L − are outgoing states,and e.g. P L + , → L − , is the probability that an electronfrom an incoming state on the left side excites the TLSand scatters to an outgoing state on the left side. Theincoming states have the distribution function of the cor-responding reservoir, while the distribution functions ofthe outgoing states are a mixture of the left and rightFermi functions with the transmission probability of thecontact, τ :The sign of the different current terms are determined bythe direction of the outgoing states. After evaluating theenergy integrals with the appropriate distribution func- (cid:87) RL )1( ff (cid:87)(cid:87) (cid:14)(cid:16) f L f R RL )1( ff (cid:87)(cid:87) (cid:16)(cid:14) L f R f tions one obtains: δI in = n δI in + n δI in , (16)where the first and second term correspond to the exci-tation and relaxation of the TLS, and: δI in = − eρ F γ (1 − τ )4 · (17) (cid:2) eV + E (coth eV + E kT − − − eV + E (coth − eV + E kT − (cid:3) δI in = − eρ F γ (1 − τ )4 · (18) (cid:2) eV − E (coth eV − E kT − − − eV − E (coth − eV − E kT − (cid:3) . The inelastic correction also causes a steplike changeof the conductance, which has a magnitude of δG in ≈ e ρ F γ (1 − τ ) /
4. For a contact with large transmissionthe conductance decreases at the excitation energy, whilefor a tunnel junction it increases, with a transition be-tween the two cases at τ = 1 /
2. We note that our simplemodel gives the same result for the transition betweenpoint-contact spectroscopy and inelastic electron tunnel-ing spectroscopy at τ = 1 / The contribution of the inelastic process can easily beestimated. According to Eq. 13 the relaxation time ofthe TLS is τ TLS ≈ ( ρ F γE ) − , thus the inelastic correc-tion to the conductance is | δG in | < e / ( τ TLS · E ). Itmeans that a relaxation time of 1 µ sec corresponds to aninelastic correction smaller than 10 − G . In other wordsa correction of 0 . would correspond to a TLS witha sub-picosecond relaxation time. These are unphysicalnumbers, especially if the telegraph fluctuation can beresolved, thus in the following the inelastic correction isneglected.As a conclusion a simple TLS model cannot producepeaklike structures in the differential conductance curve.Regardless of the fine details of the model, for any TLSa characteristic energy δE > ∼ E is required for the tran-sition, whereas for the observation of sharp peaks in thedifferential conductance δE ≪ E is desired. As an otherconsequence of a simple TLS model, the high voltageconductance is limited by σ ∞ > σ / σ ∞ ≈ σ /
5. For suchlarge change of the conductance a population inversion, n ( ∞ ) ≫ n ( ∞ ) is required. III. TWO-LEVEL SYSTEM MODEL WITHASYMMETRIC COUPLING
Both a population inversion and a sharp transition ofthe occupation numbers can be introduced by insertingan asymmetric coupling constant to the model, that is the d I/ d V eV/E E I n E n n N energetically equivalent states ( / =N) FIG. 3:
Results in the zero temperature limit of an asymmet-rically coupled TLS model. The upper panel shows the volt-age dependence of the occupation numbers, the middle panelshows the I − V curve, and the lower panel shows the differen-tial conductance. On the right side the two-level system witha degenerated upper level is demonstrated. coupling of the lower state of the TLS to the electrons inEq. 4 is γ , whereas the coupling to the upper state inEq. 5 is γ ≪ γ . The asymmetry parameter is definedas N = γ /γ . With this modification, once the TLScan be excited, it cannot relax back easily, thus a sharptransition occurs.The original definition of the coupling constantdoes not allow any asymmetry, as γ = 2 π/ ¯ h ·|h f, | H e − T LS | i, i| = 2 π/ ¯ h · |h i, | H e − T LS | f, i| = γ due to the hermicity of the Hamilton operator. An asym-metry arises, however, if the phase spaces of the twolevels are different. For instance if the ground state iswell-defined, but the upper state is not a single level, butN energetically equivalent states (Fig. 3), then the effec-tive coupling constant contains a summation for the finalstates resulting in γ = N γ .With this modification the occupation numbers in the T = 0 limit are defined by: n = ( eV < E γ ( e | V |− E ) γ ( e | V |− E )+ γ ( e | V | +3 E ) for eV ≥ E . (19)The corresponding occupation numbers, I − V and dI/dV curves are plotted in Fig. 3. The population inversionis obvious with n ( ∞ ) /n ( ∞ ) = N . The characteris-tic energy of the variation of the occupation numbers is δE = 4 E/ ( N + 1). The I − V curve already shows asharp transition from the initial to the final slope, andthe differential conductance shows a sharp peak if theasymmetry parameter is high enough. The appearanceof the dI/dV peak is determined more precisely by cal-culating the ratio of ∆ σ E and ∆ σ ∞ :∆ σ E = N σ − σ ); ∆ σ ∞ = NN + 1 ( σ − σ ) (20)∆ σ E ∆ σ ∞ = N + 14 . (21)With an asymmetry parameter N > σ − σ ) both the width andthe height of the conductance peak are determined bythe asymmetry parameter, i.e. the peak width and thepeak hight are not independent parameters. Depend-ing on the sign of ( σ − σ ), the peak can be eitherpositive or negative. The conductance at large bias is σ ∞ = ( N σ + σ ) / ( N + 1), thus it can be arbitrarilysmall compared to the zero-bias conductance.At finite temperature the dI/dV curve is calculatedby inserting Eqs. 9, 10, 11 into Eq. 2 using asymmetriccoupling constants, γ , γ . The finite temperature causesa smearing of the curves, but the transition from stepliketo peaklike structure is similarly observed at N = 3.A more realistic generalization of the model can begiven by assuming a distribution of the energy levels atthe excited state with a density of states ρ ( E ), for whichthe standard deviation ( W ) is kept much smaller than themean value ( E ). The asymmetry parameter is definedby the normalization: N = R ρ ( E )d E . With this mod-ification the occupation of the excited states is energydependent: n ( E ), but the probability that any of theexcited states is occupied is given by a single occupationnumber, n = R n ( E )d E . The calculation of the tran-sition probability P → includes the integration of theexcitation rate in Eq. 9 with the energy distribution: ν ( eV, E , W ) = Z ρ ( E ) ν ( eV, E ) dE. (22)The precise calculation of the relaxation probability( P → ) would require the knowledge of the energy depen-dent occupation number, n ( E ). However, in the narrowneighborhood of the excitation energy, where the peak isobserved the voltage dependence of the relaxation ratecan be neglected beside the constant value of ≈ E (seeEq. 13), thus the original formula for ν (Eq. 10) is a goodapproximation for the finite distribution of the levels aswell. Our analysis shows, that the relaxation rate caneven be replaced by a voltage independent spontaneousrelaxation rate without causing significant changes in theresults.As demonstrated in the following a TLS model withan asymmetry parameter, N , and a narrow width of theupper states, W , shows perfect agreement with the ex-perimental observations. b d I/ d V [ / h ] Displacement [nm] a C oun t s [ a . u .] G [2e /h] V=120 mV
V=20 mV
FIG. 4:
Panel (a) shows a conductance trace on which NDCcurves were observed when the electrodes were approachingeach other after rupture. In this region the differential con-ductance measured at low DC bias ( σ , V DC = 0 mV) andhigh bias ( σ ∞ , V DC = 120 mV) exhibit a large splitting. Thedifferential conductance was measured with an AC modulationof µ V. Panel (b) shows conductance histograms of Au-H junctions measured at a DC bias of mV and mV, re-spectively. IV. EXPERIMENTAL RESULTS
In the following we present our experimental resultson the negative differential conductance phenomenon ingold-hydrogen nanojunctions. The high stability atomicsized Au junctions were created by low-temperature me-chanically controllable break junction technique. Thehydrogen molecules were directed to the junction froma high purity source through a capillary in the sampleholder. The molecules were dosed by opening a solenoidneedle valve with short voltage pulses, adding a typicalamount of ∼ . µ mol of hydrogen molecules.The appearance of the NDC phenomenon cannot begenerally attributed to definite parts of the conductancetraces, but in Au-H junctions the NDC curves werequite frequently observed when the junction was closedafter complete disconnection, as demonstrated by thetrace in Fig. 4a. The conductance curve was measuredby recording a large number of dI/dV ( V ) curves dur-ing a single opening – closing cycle, and extracting thedifferential conductance values both at zero bias andat V DC = 120 mV. During the opening of the junc-tions no difference is observed, but during the closingof the junction the two curves show large deviation. Thehigh bias trace resembles the traditional traces of puregold junctions: during the approach of the electrodesan exponential-like growth is observed, but already at asmall conductance value ( < . ) the junctions jumpsto a direct contact with G = 1 G . At low bias voltagethe conductance grows to a much higher value ( ≈ . )before the jump to direct contact. The behavior of theconductance trace in Fig. 4a agrees with the general trendshown by the conductance histograms (Fig. 4b): at lowbias a large variety of configurations is observed at any -150 -100 -50 0 50 100 150-0.3-0.2-0.10.00.1 -2.0-1.5-1.0-0.50.00.5-100 -50 0 50 100-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0 b d I/ d V [ / h ] V [mV] a V [mV]
FIG. 5:
Panel (a) shows an experimental NDC curve (thickorange curve in the background) and the fit with a Dirac-delta distribution (open circles) and a finite width uniformdistribution (thin black line). The fitting parameters are: E = 66 . meV, N = 42 , σ = 0 . G , σ ∞ = 0 . G ;and T = 25 / . K, W = 0 / . meV for the Dirac-delta andthe uniform distribution, respectively. Panel (b) shows exper-imental curves exhibiting smaller features between the NDCpeaks. conductance value, whereas at high bias the weight inthe region G = 0 . − . is very small. Note thatonly a part of the configurations (a few percent of alltraces) show NDC features, for other configurations irre-versible jumps are observed in the I − V curve at a certainthreshold voltage. We have not found any evidence thatthe gold-hydrogen chains reported in our previous work would show NDC phenomenon or any peaklike structuresin the differential conductance curve.Figure 1 has already presented an example of the hugenegative differential conductance phenomenon. Usingthe finite temperature results of the asymmetrically cou-pled TLS model with a Dirac-delta distribution of thelevels ( W = 0) we have fitted our experimental data inFig. 1. The fitting parameters are the energy, the tem-perature and the asymmetry parameter, whereas σ and σ ∞ are directly read from the experimental curve. Asshown by the black solid lines in the figure, the modelprovides perfect fit to the data. The striking result ofthe fitting is the extremely large asymmetry parameter, N = 60.Figure 5a shows an other example for the NDC phe-nomenon. For this curve the fitting with a Dirac-deltadistribution provides a nonrealistic temperature of T =25 K. By fitting with a finite width uniform distribution of the excitation levels the temperature can be kept at theexperimental value (4 . W = 4 . E = 37 . . -100 -50 0 50 100-2.0-1.5-1.0-0.50.00.5 d I/ d V [ / h ] V [mV] G [ /h ] E [meV] W [meV] G [ ] N G [ /h ] C oun t s E [meV]
FIG. 6:
The lower panel shows the variation on a NDCcurve as a function of electrode separation. The curves wererecorded during the closing of a disconnected junction. Theupper panels show the variation of the fitting parameters as afunction of the zero-bias conductance. The curves with zerobias conductances of . , . and . G correspond to thepushing of the electrodes by ≈ . ˚ A , ≈ . ˚ A and ≈ . ˚ A with respect to the initial curve with zero bias conductance of . G . The inset in the lower panel shows the distributionof the peak positions for 60 independent NDC curves. introducing further energy levels with different degener-acy, but such a model would be too much complicatedwithout aiding the understanding of the general phe-nomenon. The upper curve shows a steplike decrease of ∼
10% at E = ±
39 meV resembling the vibrational signalof molecular junctions, although its amplitude is ratherlarge for a vibrational spectrum. (The typical conduc-tance change due to vibrational excitations is 1 −
3% atthe conductance quantum, and it should vanish towards0 . , where the negative point-contact spectroscopysignal turns to positive inelastic electron tunneling spec-troscopy signal. ) The observed steps could also beexplained by the scattering on a symmetrically coupledTLS, for which the step-size can be as large as 50% ofthe zero bias conductance.The bottom panel in Fig. 6 shows the variation of thenegative differential conductance curves by changing theelectrode separation. The curves were recorded duringthe closing of a disconnected junction, similarly to thetrace in Fig. 4a. The upper panels present the change ofthe fitting parameters as a function of the zero bias con-ductance. Both the excitation energy, E and the widthof the excited levels, W decrease with increasing con-ductance, whereas the asymmetry parameter, N variesaround a constant value.We have repeated our measurements for a largeamount of curves showing NDC feature. The observed FIG. 7:
An illustration for the proposed model: a moleculestrongly bound between the electrodes is excited to a looselybound configuration, where different configurations have havesimilar binding energy. excitation energies have shown a broad distribution inthe range 30 −
100 meV, as demonstrated by the inset intho bottom panel of Fig. 6. Below 30 meV we have notobserved any peaks, whereas above 100 meV the junc-tions frequently become unstable, and the recording ofreproducible I − V curves is not possible. In every casewhen the electrode separation dependence was studied E was decreasing by increasing zero-bias conductance show-ing a shift of the excitation energy by even a few tens ofmillivolts. The standard deviation of the excited levelswas typically in the range 0 . −
10 meV, also showing adecreasing tendency with increasing conductance. Theasymmetry parameter was in the range ≈ − V. DISCUSSION
Our analysis of the two-level system models shows thatin the case of two single levels the population of the up-per level remains very small in the narrow neighborhoodof the excitation energy, which inhibits the appearanceof sharp peaklike structures in the differential conduc-tance curve. By introducing a large asymmetry in thecoupling constants, already a small voltage bias abovethe excitation energy causes a sudden flip of the occupa-tion numbers, resulting in peaks or even huge negativedifferential conductance in the dI/dV curves.The analysis of our experimental curves shows that thenegative differential conductance phenomenon is success-fully fitted by the asymmetrically coupled TLS model,yielding extremely large asymmetry parameters ( N ≈ − dI/dV curves. From the fitting of the ex-perimental curves only the large asymmetry parametercan be deduced, but the precise microscopic origin ofthe asymmetry cannot be determined without detailedmicroscopic calculations. Possible candidates are the dif-ferent arrangements and rotational states of the moleculewith respect to the contact, but it is also very much prob-able, that the bound molecule is desorbed to differentpositions on the side of the contact, and then it can evendiffuse away on the contact surface. In the latter case notnecessarily the same molecule relaxes back to the initialbound state. All the above processes cause an effectiveasymmetry in the coupling.Similar results were obtained on tunnel junctions withhigher resistance in Ref. 9, where a hydrogen-covered Cusurface is studied with an STM tip. The authors presenta phenomenological two-state model, which gives perfectfit to the data, but the microscopic parameters are hid-den in the model, and for the more detailed understand-ing of the results further analysis is required. The fittingwith our asymmetrically coupled TLS model shows thatthe experimental curves in Ref. 9 correspond to similarlylarge asymmetry parameters. In the case of an STM ge-ometry a direct picture can be associated with the asym-metry: a molecule bound between the surface and the tipcan be excited to several equivalent states on other sitesof the surface, away from the tip. The observation of thevoltage dependent telegraph noise in Ref. 9 provides adirect measure of the occupation numbers, proving thepopulation inversion in the system.The group of Jan van Ruitenbeek has demonstratedpeaklike structures concentrating on contacts close to theconductance unit. In this conductance range the rel-ative amplitudes of the peaks are much smaller than ourNDC curves in the lower conductance regime, but theoverall shape of the curves is similar. In some cases adirect transition from a steplike vibrational signal to a -100 -50 0 50 1000.71.01.31.61.92.22.5 d I/ d V [ / h ] V [mV] h FIG. 8:
In the inset the vibrationally mediated TLS model isillustrated. The thick curve in the background is a fit to oneof the experimental curves in Ref. 8 using the vibrationallymediated TLS model (the parameters are: σ = 0 . G , σ =1 . G , ¯ hω = 40 meV, ∆ = 3 meV, T = 7 K). The thin blackline shows a fit with asymmetrically coupled TLS model with E = 41 meV, T = 4 . K, N = 120 and W = 2 . meV. peaklike structure was observed, which implies the cou-pling of the phenomenon to molecular vibrations. Thecoupling to the vibrational modes was also indicated bythe isotope shift of the peak positions. A TLS model il-lustrated in the inset of Fig. 8 was proposed, in which thetwo base levels of the double well potential (with a split-ting of ∆ ) are separated by a wide potential well, whichcannot be crossed. In both wells, however, a vibrationalmode of the molecule can be excited (with E = ¯ hω ), andat the vibrational level the potential barrier can alreadybe crossed, causing a hybridization of the excited lev-els between the two wells. In this model the upper baselevel is unoccupied for eV < ¯ hω , but above the excitationenergy the transition between the two wells is possible,and for ¯ hω ≫ ∆ the upper base level suddenly becomesalmost half occupied above the excitation energy. Simi-larly to the asymmetrically coupled TLS model, this vi-brationally mediated TLS model produces sharp peaklikestructures in the dI/dV due to the sudden change of theoccupation numbers. The authors claim that the cou-pling of a vibrational mode to a TLS magnifies the oth-erwise tiny vibrational signal, and thus the sharp peaksdirectly show the vibrational modes of the junction.Our analysis shows that the asymmetrically coupledTLS model proposed by us gives almost identical fits tothe experimental curves in Ref. 8 as the vibrationally me-diated TLS model, which means that the automatic iden-tification of the peaklike structures with the vibrationalmodes of the junction is only possible if their growth froma vibrational spectrum is detected, and so the vibrationalsignal is anyhow resolved. In contrast, the negative dif-ferential curves demonstrated in our manuscript cannotbe fitted with the model in Ref. 8 due to the large ratio of σ /σ ∞ ( ≈ − σ /σ ∞ < eV > ∼ ¯ hω ), and σ /σ ∞ < eV ≫ ¯ hω ),where the third level starts to be populated as well. Themodel could be generalized by inserting asymmetric cou-pling, however in this case the inclusion of the vibrationallevel is not necessary any more for fitting the curves. Thebroad distribution of the NDC peak positions also indi- cates that the phenomenon is not related to vibrationalenergies.In our view the vibrationally mediated TLS model re-quires unique circumstances: the molecule needs to bereally incorporated in the junction with well-defined vi-brational energies. Furthermore, two similar configura-tions are required, which have the same vibrational en-ergy, and for which the transition is only possible in theexcited state. This phenomenon may be general in cer-tain well-defined molecular junctions, for which the tran-sition of a vibrational-like signal to a peaklike structureand the isotope shift of the peaks provides support. Incontrast, the asymmetrically coupled TLS model can beimagined under a much broader range of experimentalconditions, just a bound molecular configuration and alarge number of energetically similar loosely bound (e.g.desorbed) states are required with some difference in theconductance. The fine details of the underlying physicalphenomena may differ from system to system, howeverboth models provide an illustration of a physical processthat may lead to the appearance of peaklike structuresin dI/dV curves of molecular junctions. VI. CONCLUSIONS
In conclusion, we have shown that gold-hydrogen nano-junctions exhibit huge negative differential conductancein the conductance range of 0 . − . . The position ofthe peaks shows a broad distribution in the energy range30 −
100 meV. Similar features were observed in tunneljunctions with higher resistance using an STM setup, whereas in break-junctions with G ∼ peaklike struc-tures with smaller amplitude were detected. These re-sults show that sharp peaklike structures are generallyobserved in the differential conductance curves of molec-ular nanojunctions under a wide range of experimentalconditions: using different molecules, different electrodematerial and different contact sizes. All these experimen-tal results imply the explanation of the observations interms of two-level system models.We have shown that a simple two-level system modelcannot produce peaklike structures, but a TLS modelwith asymmetric coupling successfully fits all the experi-mental data. Our analysis shows that the peaklike curvescorrespond to large asymmetry parameters, which im-plies that a molecule from a bound state is excited toa large number of energetically similar loosely boundstates. This picture provides a physical phenomenon thatcan appear under a wide range of experimental condi-tions and may explain the frequent occurrence of peaklikestructures in dI/dV curves of various molecular systems.Our analysis also shows that a recently proposed vi-brationally mediated two-level system model may beapplicable for certain well-defined molecular configura-tions, but the general relation of the conductance peaksto vibrational modes is not possible without further ex-perimental evidences.
ACKNOWLEDGEMENTS
This work has been supported by the Hungarian re-search funds OTKA F049330, TS049881. A. Halbritter is a grantee of the Bolyai J´anos Scholarship. N. Agrait, A.L. Yeyati, J.M. van Ruitenbeek, Physics Re-ports , 81-279 (2003). R.H.M. Smit, Y. Noat, C. Untiedt, N.D. Lang, M.C. vanHemert, J.M. van Ruitenbeek, Nature
906 (2002). D. Djukic and J.M. van Ruitenbeek, Nano Letters, D. Djukic, K.S. Thygesen, C. Untiedt, R.H.M. Smit,K.W. Jacobsen, J.M. van Ruitenbeek, Phys. Rev. B. ,161402(R) (2005). K.S. Thygesen, K.W. Jacobsen, Phys. Rev. Lett. B. Ludoph and J.M. van Ruitenbeek, Phys. Rev. B. Sz. Csonka, A. Halbritter, and G. Mih´aly, Phys. Rev. B , 075405 (2006). W.H.A. Thijssen, D. Djukic, A.F. Otte, R.H. Bremmer, J.M. van Ruitenbeek, Phys. Rev. Lett. , 226806 (2006) J.A. Gupta, C.P. Lutz, A.J. Heinrich, D.M. Eigler, Phys.Rev. B. , 115416 (2005). J. Gaudioso, L. J. Lauhon, and W. Ho Phys. Rev. Lett. , 1918 (2000). A. Halbritter, L. Borda, A. Zawadowski, Advances inPhysics , 939-1010 (2004). M. Paulsson, T. Frederiksen, M. Brandbyge, Phys. Rev. B. , 201101(R) (2005). L. de la Vega, A. Martin-Rodero, N. Agrait, A.L. Yeyati,Phys. Rev. B. , 075428 (2006). We have found that the model is not sensitive to the choiceof the distribution, all the distributions with a well-definedwidth (e.g. Gaussian) provide similar results. Note, that ac-cording to the general definition W is always the standarddeviation and not the width of the distribution.15