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Vibrational Excitations in Weakly Coupled Single-Molecule Junctions: AComputational AnalysisJohannes S. Seldenthuis, † Herre S. J. van der Zant, †, * Mark A. Ratner, ‡ and Joseph M. Thijssen †

†Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands, and‡Department of Chemistry, Northwestern University,Evanston, Illinois 60208

I n recent years, vibrational excitationsof single molecules have been mea-sured in the inelastic tunneling regime

with scanning tunneling microscopes (ST-Ms),1 in mechanical breakjunctions(MBJs),2,3 and in the sequential tunneling

regime (SET) with electromigrated breakjunctions (EBJs).4–6 Measurements in EBJson an oligophenylenevinylene derivative byOsorio et al.7 reveal a vibrational spectrumof 17 excitations in the sequential tunnelingregime (see Figure 5a). It has been shownthat these modes are consistent with Ra-man (above 15 meV) and infrared (above50 meV) spectroscopy data. However, theRaman and IR data show more peaks thanare observed in the transport measurement.Moreover, the Raman and IR measure-

ments were performed on polycrystallinesamples and KBr pellets, respectively, whichdo not reect the conditions of the mol-ecule in the junction.

Theoretical investigations on vibrationalexcitations in the SET regime have so farmainly concentrated on small systems withonly one vibrational mode. 8–13 Chang et al.14 use density functional theory (DFT) cal-culations on the C 72 fullerene dimer to cal-culate all vibrational modes, restrictingthemselves to transitions from the ground

state in one charge state to the vibrationalexcited state of the other charge state. Wehave developed a computationally efcientmethod to calculate the vibrational spec-trum of a sizable molecule in the sequen-tial tunneling regime, based on rst prin-ciples DFT calculations to obtain thevibrational modes in a three-terminal setup. This method takes the charge state andcontact geometry of the molecule into ac-count and predicts the relative intensities of vibrational excitations. In addition, transi-tions from excited to excited vibrational

state are accounted for by evaluating theFranck Condon factors involving severalvibrational quanta. Our method can there-fore predict qualitatively different behaviorcompared to calculations that only includetransitions from ground state to excited vi-brational state. 13

A schematic picture of a gateable elec-tromigrated breakjunction containing a

single molecule is shown in Figure 1a. Thecouplings to the leads are given by L andR. In the weak coupling limit, L, R, kT E , E C , the level spacing ( E ) and charging

energy (E C) of the molecule allow only oneelectron to tunnel onto the molecule at atime (sequential tunneling). The transportmechanism for this case is shown in Figure1b. While tunneling on or off a molecule, anelectron can excite a vibrational mode,which may show up as a line running paral-lel to the diamond edges in the stability dia-gram (a plot of the conductance as a

*Address correspondence [email protected].

Received for review March 20, 2008and accepted June 03, 2008.

Published online June 21, 2008.

10.1021/nn800170h CCC: $40.75

© 2008 American Chemical Society

ABSTRACT In bulk systems, molecules are routinely identied by their vibrational spectrum using Ra

infrared spectroscopy. In recent years, vibrational excitation lines have been observed in low-temperatu

conductance measurements on single-molecule junctions, and they can provide a similar means of ident

We present a method to efciently calculate these excitation lines in weakly coupled, gateable single-m

junctions, using a combination of ab initio

density functional theory and rate equations. Our method takestransitions from excited to excited vibrational state into account by evaluating the FranckCondon factors for

an arbitrary number of vibrational quanta and is therefore able to predict qualitatively different behavio

calculations limited to transitions from ground state to excited vibrational state. We nd that the vibratio

spectrum is sensitive to the molecular contact geometry and the charge state, and that it is generally nece

to take more than one vibrational quantum into account. Quantitative comparison to previously reported

measurements on -conjugated molecules reveals that our method is able to characterize the vibrational

excitations and can be used to identify single molecules in a junction. The method is computationally fe

commodity hardware.

KEYWORDS: single-molecule junction · three-terminal transport · Coulomb

blockade · vibrational modes · rate equations · Franck Condon factors · densityfunctional theory

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function of bias and gate voltage). 15 In this paper, wecalculate such stability diagrams with a rate equationapproach and present a comparison with experimentalresults.

RESULTS AND DISCUSSION

We have applied our rate equationmethod to three molecules with increas-ing length: benzenedithiol (see Figures 2and 3), the oligophenylenevinylene de-rivative OPV-3 (see Figure 4), and OPV-5(see Figure 5). All vibrational mode calcu-lations are carried out using the Amster-dam density functional code 16,17 with theAnalytical Second Derivatives module. 18

As a rst example, we present our re-sults for benzenedithiol adsorbed on gold. This example is used in order to demon-strate our method. In experiments, thissystem is generally not weakly coupled.We test the method by studying the inu-ence of the number of vibrational quanta,the charge state, and the presence of gold contacts on the stability diagrams. The stability diagrams are calculated with

a symmetric coupling to the leads of 1meV, a bias ( ) and gate ( ) coupling of

0.5, and a temperature of 1.6 K. The resulting stabilitydiagram for the 1 ¡ 0 transition in bare benzene-dithiol with one vibrational quantum is shown in Fig-ure 2a. Of the 25 vibrational modes with energies

Figure 1. (a) Schematic picture of a gateable junction containing a single molecule capaci-tively coupled to theleft ( C L ),right ( C R ),and gate( C G ) electrode. (b) Transportmechanisminthe sequential tunneling regime. The distribution of the bias voltage over theleads is givenby (C R

1 /2 C G )/(C L C R C G ) and the gate coupling by C G /(C L C R C G ).21

Figure 2. Calculated stability diagrams of the 1 ¡ 0 (a c) and 0 ¡ 1 (d) transitions in benzenedithiol with increasingnumber of vibrational quanta. The arrows point to the main differences between the diagrams (see text). Since the calcula-tion is symmetric in the bias voltage, they are only shown for positive bias.

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below 200 meV, 8 excitation lines are visible belonging

to the 1 state and 7 belonging to the neutral state. Forboth states, there are three excitation lines that do notcontinue all the way to the diamond edge but stop atthe strong line at 43 meV.

Taking two vibrational quanta into account (see Fig-ure 2b) reveals two new excitation lines for the neutralstate, a strong line at 86 meV and a weak line at 180meV (see the white arrows). Taking one more quan-tum into account (see Figure 2c) adds a weak excita-tion line for the neutral charge state at 129 meV, againindicated by a white arrow. For the 0 ¡ 1 transition(Figure 2d), small changes are found. The arrows point

to excitation lines that are absent in Figure 2c. Com-pared to that diagram, the higher energy excitations(above 100 meV) have shifted by several millielectron-volts.

In a junction, the molecule is bonded to the goldcontacts. We have modeled this by adding two gold at-oms on either side of the molecule. The resulting stabil-ity diagrams (with l 2) are shown in Figure 3. Thesediagrams are quite different from those of the samecharge state transitions in Figure 2c,d. Depending onthe charge state, ve to eight excitations are visible be-low 75 meV, but no higher modes are observed. Theelectron phonon couplings for the neutral chargestate in the transition of Figure 3a are shown in Figure

7a. Two modes have a large electron phonon cou-

pling (with coupling strengths larger than 1), showingthat it is necessary for this system to take more than onevibrational quantum into account. 13

The calculations show that only a few of the 30 vi-brational modes of benzenedithiol are expected to bevisible in transport measurements, and that they are de-pendent on the charge state and sensitive to the con-tact geometry. For some modes in this molecule, it isnecessary to take more than one vibrational quantuminto account. For example, the modes at 86 meV (for l 2) and 129 meV (for l 3) are probably higher harmon-ics of the strong excitation at 43 meV. The fact that sev-

eral other lines stop at this excitation shows that it isalso necessary to take the Franck Condon factors forexcited vibrational state to excited vibrational state intoaccount.

The second molecule for which we have calculatedthe vibrational spectrum is OPV-3. As with benzene-dithiol, the gold contacts are simulated by adding twogold atoms on either side of the molecule. The results fortwo charge state transitions with gold and one withoutare shown in Figure 4. The calculations take two vibra-tional quanta into account. Comparing the calculationsto those on benzenedithiol indicates that OPV-3 is lesssensitive to the contact geometry. OPV-3 without goldhas more modes at lower energies than benzenedithiol,

Figure 3. Calculated stability diagrams of the 1 ¡ 0 (a) and 0 ¡ 1 (b) transition in benzenedithiol with two gold atomson either side to simulate the leads. Two vibrational quanta are taken into account.

Figure 4. Calculated stability diagrams for OPV-3 with (a and b) and without (c) gold for two charge state transitions. Thecalculations take two vibrational quanta into account.

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and the modes at higher energies are much less sup-pressed when the two gold atoms are added. Also, theelectron phonon couplings for OPV-3 are smaller thanfor benzenedithiol (see Figure 7b). These trends are notunexpected since OPV-3 is a larger molecule and the at-oms will on average be further away from the leads, lead-ing to a smaller sensitivity to the contact geometry. Also,since OPV-3 is conjugated, an extra electron will be delo-calized over the entiremolecule,and theatomicdisplace-ments will be smaller, resulting in a smallerelectron phonon coupling. In the case of OPV-3,wehave performed several calculations with different con-tact geometries. We nd that adding up to 19 gold atoms

on either side of the molecule has no signicant effecton the vibrational modes above 20 meV. 31

The calculated stability diagram of the third mol-ecule, OPV-5, is shown in Figure 5b. The temperatureand coupling parameters of the calculation are chosento be the same as in the experiment (Figure 5a). Al-though we are unable to determine the charge statesin the measurement, the fact that the degeneracy pointis the rst at a negative gate voltage suggests a 1 ¡

0 transition (see also Figure 3 in ref 7). In the calculation,the nonconjugated dodecane side arms of the mea-sured molecule are omitted. These arms are not ex-

pected to inuence the electronic transport and willmost likely only affect the low-energy vibrationalmodes. As with benzenedithiol, the contacts are mod-eled by adding two gold atoms on either side of themolecule. The calculation takes one vibrational quan-tum into account.

Figure 6 shows a d I /dV trace along the diamondedge of the neutral charge state in Figure 5b. How-ever, since this is a smaller calculation, three vibrationalquanta can be taken into account. The peaks in this g-ure correspond to the excitation lines in the calculatedstability diagram. In the experimental stability diagram,a background conductance makes it difcult to resolve

all excitation lines at the same color scale, but close in-spection reveals 17 modes in the energy range below125 meV (see Figure 6 and Table 1 in ref 7). The ener-gies of the excitations in the measurement are deter-mined from the bias voltage at which they cross the dia-mond edge. Broadening due to the temperature andthe leads introduces an uncertainty, indicated by thehorizontal bars in Figure 6.

Figure 6 reveals a close match between the experi-ment and the calculation for the modes between 10and 80 meV. The calculation shows several small peaksin this range not observed in the measurement. Itshould be noted that, in the rate equation approach,

broadening of excitation lines is solely due to tempera-ture. Broadening due to the couplings to the leads isnot accounted for. Calculations which do take thisbroadening into account show that these small peaksare smeared out, and the calculation and measurementshow the same number of peaks in the aforemen-tioned range.

Figure 5. (a) Measured stability diagram of a junction containing OPV-5. 7 (b) Calculated stability diagram of an OPV-5 mol -ecule with onevibrational quantum. (c) The conguration of theOPV-5 molecule in thecalculations. The dodecane side armsof the measured molecule (see Figure 1a in ref 7) are omitted, and two gold atoms are added on either side to simulate theleads.

Figure 6. Calculated d I /d V trace of the diamond edge of the neutral charge state in Figure 5b, taking three vibra-tional quanta into account. The inset shows the same cal-culations, but with the gold atoms omitted. All mea-sured excitations in this energy range (see Figure 5a)are shown. The uncertainties in the measured energiesare indicated by the horizontal bars.

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The inset in Figure 6 shows the same calculation,but with the gold atoms omitted. Comparison with themeasurement shows a large discrepancy for excitationsbelow 50 meV. It is clear from this gure that the addi-tion of two gold atoms on either side of the moleculecan already account for most of the inuence of the

contact geometry on modes above 10 meV. The charg-ing energy of an OPV-5 molecule in a junction is an or-der of magnitude smaller than the difference betweenthe ionization energy and electron afnity of the mol-ecule in the gas phase, 19 probably due to screening inthe leads. While this effect changes the energies of theorbitals, theshape of theorbitals and therefore the elec-tron density will remain relatively unaffected. Since theFranck Condon factors primarily depend on the differ-ence in electron density between different chargestates, we have chosen not to take image charges intoaccount in the calculations.

While this effect changes the energies of the orbit-als, the shape of the orbitals, and therefore the elec-tron density, on which the Franck Condon factors pri-marily depend, will remain relatively unaffected, and wehave chosen not to take image charges into accountin the calculations.

The omission of the side arms in the calculation low-ers the mass of the molecule, which might explain thediscrepancy between the calculation and the measure-ment for the modes below 10 meV, which involve mo-tions of the whole molecule. Also, the contact geometryin the measurement is unknown, so any mode involv-

ing a signicant distortion of the gold sulfur bond isexpected to be inaccurate. Like OPV-3, the vibrationalspectrum of OPV-5 is less sensitive to the charge stateand contact geometry than benzenedithiol and theelectron phonon couplings are smaller (see Figure7c). As in the case of benzenedithiol and OPV-3, the cal-culation of OPV-5 predicts the intensity of the excita-tion lines to be much weaker above 30 meV than be-low. This is also observed in the measurement. Theintensities gradually increase for energies up to 30 meV,after which they suddenly drop, a trend also visible inthe electron phonon couplings. For excitations above80 meV, the low intensities make a quantative compari-

son between the measurement and the calculationdifcult.

Most of the vibrational modes haveelectron phonon couplings below 0.1 and are not ex-pected to give rise to extra excitation lines when an-other vibrational quantum is taken into account. Themodes at 17 and 27 meV, with coupling strengths of 0.6and 0.7, respectively, are expected to give rise to excita-tion lines at 34, 51 54, and possiby 81 meV. Theselines are indeed observed in the measurement and thecalculation (see Figure 6).

It should be emphasized that, in Figure 6, all visiblevibrational excitations for both the calculation and themeasurement are shown. Comparing the spectrum toRaman and IR spectroscopy data reveals a close match, 7

but the optical spectra predict many more modes notobserved in the measurement and calculation. The cal-culation predicts only a handful of visible excitationsout of a total of a 129 vibrational modes under 150 meV.Our method is thus able to provide what we might call“selection rules” for vibrational excitations in single-molecule junctions.

CONCLUSIONS

Our results show that the vibrational spectrum of asingle molecule in a junction is sensitive to the contactgeometry and charge state, although this inuence be-comes smaller for larger molecules. Contrary to Ramanand IR spectroscopy, calculations can take these inu-ences into account, provide selection rules, and predictthe relative intensity of excitation lines in transportmeasurements. Our calculations also show that it is nec-essary to take more than one vibrational quantum intoaccount for small molecules, but that due to decreasingelectron phonon couplings, this becomes less impor-tant for larger molecules. Finally, our method is compu-tationally efcient. All Franck Condon and transportcalculations have been performed on an HP xw9300workstation, with the largest calculation (over 54 mil-lion Franck Condon factors in OPV-3 for 250.000 biasand gate points) taking just over 4 h.

Figure 7. Dimensionless electron phonon couplings ( ) for the vibrational modes of the neutral charge state at the 1 ¡ 0 transitionfor (a) benzenedithiol, (b) OPV-3, and (c) OPV-5. All calculations include two gold atoms on either side of the molecule to simulate theleads. Analysis of the atomic displacements shows that primarily modes that distort the overlap give rise to a nonzeroelectron phonon coupling.

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METHODS

Rate Equations.Our method to calculate stability diagrams isbased on the rate equation approach, which is generally used inthe sequential tunneling regime. 20,21 The central quantity in thisapproach is the vector of occupation probabilities P n, , for stateswith charge quantum number n, spin quantum number , and vi-brationalquantum number . These probabilitieschange by transi-tions from one state to the other with rates Rn, , ¡ n = , = , = . Thetime evolution of the occupation probabilities as a function of these rates is described by the master equation:

dP n,σ,ν

dt ) ∑

n ,σ ,ν * n,σ,ν

(P n ,σ ,ν Rn ,σ ,ν f n,σ,ν - P n,σ,νRn,σ,νf n ,σ ,ν ) (1)

The rates are given by Fermi’s golden rule:

Rn,σ,νf n ,σ ,ν )1h

|(n σ ν |H ˆ |nσν)|2Fn ,σ ,ν (2)

These rates contain contributions from the electronic andnuclear wave functions. The electronic contributions aredescribed by the couplings to the leads and the Fermi function,and the nuclear contributions are described by theFranck Condon factors, which we will discuss below.

At low bias, only two charge states are relevant: the initialstate (n, = , = ), where the level in the bias window (see Figure1b) is unoccupied, and the nal state (n 1, , ), where it is oc-cupied. The rates for these states are 13,22

Rn,σ ,ν f n+ 1,σ,νL ) F ν ν

ΓLp f (E n+ 1,σ,ν - E n,σ ,ν - eβV g - eRV b)

Rn,σ ,ν f n+ 1,σ,νR ) F ν ν

ΓRp f (E n+ 1,σ,ν - E n,σ ,ν - eβV g - e(R - 1)V b)

Rn+ 1,σ,νf n,σ ,νL ) F ν ν

ΓLp [1 - f (E n+ 1,σ,ν - E n,σ ,ν - eβV g - eRV b)]

Rn+ 1,σ,νf n,σ ,νR ) F ν ν

ΓRp [1 - f (E n+ 1,σ,ν - E n,σ ,ν - eβV g - e(1 - R )V b)]

(3)

where F = are the Franck Condon factors and f is the Fermifunction. E n 1, , E n, = , = is the energy difference between theinitial and nal state. This difference is composed of the levelspacing, the charging energy, the vibrational reorganizationenergy, and the vibrational energy of the states = and . For asingle degeneracy point, all but the latter of these terms can beneglected by choosing a suitable reference point for V g .

With the rate equations in place, we write eq 1 in matrix-vector form:

dP

dt ) MP (4)

where P haselements Pn, = , = and P

n 1, , . M isthe2 N 2N ratematrix, where N numbers the vibrational states. Its elements are

given by

Mij ) {- ∑ ν) 1

N Rn,σ ,i f n+ 1,σ,ν if i ) j and i , j e N

- ∑ ν ) 1

N Rn+ 1,σ,i - N f n,σ ,ν if i ) j and i , j > N

Rn+ 1,σ, j - N f n,σ ,i if i e N and j > N Rn,σ , j f n+ 1,σ,i - N if i > N and j e N 0 otherwise

(5)

To calculate the current, we need the stationary occupationprobabilities, i.e., dP /dt 0, which can be obtained bycalculating the null space of M, with the condition that allelements of P are non-negative and n, , P n, , 1. Once thestationary occupation probabilities and the rates are known, thecurrent can be calculated by summing over the total rate

through one of the leads. For the left rate this becomes 8,21

I ) e∑ν ) 1

N

∑ν) 1

N

(P n,σ ,ν Rn,σ ,ν f n+ 1,σ,νL - P n+ 1,σ,νRn+ 1,σ,νf n,σ ,ν

L ) (6)

Franck Condon Factors.The Franck Condon factors ( F = ) ineq 3) are a measure of the probability that a tunneling eventwill be accompanied by a vibrational excitation. When an elec-tron tunnels on or off a molecule, the change in electron den-sity will shift the equilibrium position of the nuclei and possiblycause a transition to a different vibrational excited state. Theprobability for this to happen is equal to the square of the over-lap integral of the vibrational wave functions in both chargestates. 23 To calculate the overlap integral, the normal coordi-nates of one charge state have to be expressed in those of theothers. This procedure is known as the Duschinskytransformation. 24,25 This transformation yields the Duschinskyrotation matrix and a mass-weighted displacement vector k . Thelatter can then be used to calculate the dimensionlesselectron phonon couplings i k i i /2 .22,26

Two methods are mainly used for calculating theFranck Condon factors from the Duschinsky transformationand the frequencies. One is the generating function method of Sharp and Rosenstock; 27 the other is the recursion relationmethod of Doktorov et al. 28 We have used the latter as imple-mented in the two-dimensional array method of Ruhoff andRatner. 29,30 This method returns an N N array of Franck Condon factors, where N (l

l ) is the number of per-mutations of up to l vibrational quanta over vibrational modes.

Relaxation Rates.Since Franck Condon factors representprobabilities, the elements of each row and each column of thismatrix add up to 1 for l ¡ . Typically, for a large system withmany Franck Condon factors, there are many factors for whichF = 1. This can present numerical difculties when calculat-ing the stationary occupation probabilities. The rates are propor-tional to the Franck Condon factors, and if all the rates intoand out of a certain level are (very nearly) 0, any occupation of that level will be stationary, resulting in an innite number of so-lutions for the stationary occupation probabilities. There aretwo ways to prevent this from happening. The rst is to take in-tramolecular vibrational relaxation into account. We have imple-

mented a simple relaxation model in which a single relaxationtime is used for all states: 20,12

dP n,σ,ν

dt ) ∑

n ,σ ,ν * n,σ,ν

P n ,σ ,ν Rn ,σ ,ν f n,σ,ν -

P n,σ,νRn,σ,νf n ,σ ,ν -1τ(P n,σ,ν - P n,σ,ν

eq ∑ν ''

P n,σ,ν '')(7)

where

P n,σ,νeq )

e -En, σ, ν

kT

∑ ν 'e-E n,σ,ν '

kT

(8)

is the equilibrium population according to the Boltzmanndistribution. This term can be included in the rate matrix byadding the relaxation matrix:

Mij R )

1τ

×{P n,σ ,i

eq - 1 if i ) j and i , j e N

P n,σ ,i eq if i * j and i , j e N

P n+ 1,σ,i - N eq - 1 if i ) j and i , j > N

P n+ 1,σ,i - N eq if i * j and i , j > N

0 otherwise

(9)

For sufciently small relaxation times, the previously mentionedstationary states will decay to the ground state and there willbe only one solution.

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Iterative Solution.The second approach is to calculate the nullspace iteratively by starting from the equilibrium population.Since both the equilibrium population of higher energy statesand the rates to those states are 0, they will never become popu-lated and the method will converge to the physically correct so-lution. This approach has several additional benets. The ratematrix for most realistic systems will be sparse, and an iterativemethod can make use of this and scale better than a directmethod. Also, the stationary population at neighboring biasand gate points will be the same unless a new state becomes

available, so by using the previous population as a starting point,most points will only need a single iteration to converge. Wehave implemented a direct method using singular value decom-position and an iterative method using a Jacobi preconditionedbiconjugate gradient method. The second implementation isgenerally several orders of magnitude faster. It turns out thatimplementing a combination of both approaches yields a one-dimensional null space of the rate matrix, even for larger mol-ecules with several vibrational quanta.

Acknowledgment. We thank M. Galperin and S. Yeganeh fordiscussions. Financial support was obtained from Stichting FOM(project 86), from the EU FP7 programme under the grant agree-ment “SINGLE”, and from the Division of Chemistry and the Of-ce of International Science and Engineering of the NSF in theU.S. This work was also sponsored by the Stichting Nationale

Computerfaciliteiten (National Computing Facilities Foundation,NCF, project mp-06-111) for the use of supercomputer facilities,with nancial support from the Nederlandse Organisatie voorWetenschappelijk Onderzoek (Netherlands Organization for Sci-entic Research, NWO).

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31. We have also performed measurements on vibrational

excitations in OPV-3. However, broadening of the linesdue to large couplings to the leads prevents us fromobtaining measurements with sufcient resolution tomake a quantitative comparison to the calculationspossible. The measurements do show the same trends asthe calculations. None of the samples show anyexcitations above 30 meV, and only a few below.

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