Cotunneling Model for Current-InducedEvents in Molecular WiresThorsten Hansen,*,† Vladimiro Mujica,†,‡ and Mark A. Ratner†

Department of Chemistry and International Institute for Nanotechnology, NorthwesternUniVersity, 2145 Sheridan Road, EVanston, Illinois 60208-3113, and Argonne NationalLaboratory, Center for Nanoscale Materials, Argonne, Illinois 60439-4831

Received April 8, 2008; Revised Manuscript Received July 8, 2008

ABSTRACT

Many phenomena in molecular transport junctions involve transitions between electronic states of the molecular wire, and, therefore, cannotbe described adequately using the Landauer picture. We present a model for cotunneling processes in molecular wires. These are coherentsecond-order processes that can be observed at low temperatures. As an example, we consider the STM-induced dissociation of acetylene.The large voltage threshold for dissociation observed experimentally is naturally accounted for within the cotunneling model.

Most theoretical descriptions of electrons tunneling throughmolecules, such as the generalization of the Landauerequation used in molecular electronics,1 share a commonfeature. They are effective one-electron descriptions that treatthe electronic structure either uncorrelated, at the Hartree-Fockor tight-binding level, or correlated, using DFT. Transportin the elastic regime is then assumed to occur throughmolecular orbitals on a single electronic potential energysurface.

Regardless of the inclusion of correlation, effective one-electron descriptions have one major drawback: Theyconsider one electronic state at the time, usually the groundstate of either the neutral molecule or an ionic species. Thisis in sharp contrast to numerous experiments in recent yearsthat demonstrate various effects, for example, switching inmolecular wires, that clearly involve several molecularstates.2,3 We present a model for the description of coherentelectron current in molecular junctions that include both anelastic tunneling and an inelastic cotunneling component. Themany-electron states of the molecule form the basis for ourdescription; thus, multiple states are naturally accounted.

Electronic cotunneling events are taken here to meancoherent second-order processes that are electronicallyinelastic; that is, the molecule makes a transition from onepotential energy surface to another. This is in contrast tovibrationally inelastic events that transfer energy to themolecular vibrations yet leave the molecule on the samepotential energy surface. Cotunneling processes are signifi-cant when incoherent first-order hopping processes aresuppressed. The ionic states are assumed to be energetically

out of reach and will only appear as virtual intermediate statesin the cotunneling processes. The concept of cotunneling wasoriginally invoked to describe leakage currents throughquantum dots in the Coulomb blockade regime.4-6

The theory is a scattering theory, where the current, I )-ewfi, is described in terms of a transition rate, wfi, obtainedfrom a generalized Fermi’s golden rule. The initial and finalstates, i and f, involve both the electrodes and the molecularbridge. We show that the description of cotunneling processesamounts to approximating the transition matrix by the self-energy, Σfi

ret, introduced by the molecule-electrode coupling.The self-energy therefore plays the role of an effectiveperturbation:

wfi )2πp

|Σfiret|2δ(Ei -Ef) (1)

This is the key element of the our theoretical work, sinceit represents the link between the electron transport and themixing of electronic states in the molecular bridge that canlead to chemical reactions and other processes involvingenergy transfer between the tunneling electron and themolecule. This is an important generalization of simpler one-electron models that consider only elastic tunneling, wherethe role of the interaction with the continua of levels in theelectrodes is limited to broaden and shift the molecular levels.

A preliminary discussion of our model was presented inref 7, where we speculated that the cotunneling mechanismprovides a qualitative explanation for the most conspicuousfeatures of STM-induced unimolecular reactions as reportedfor the dissociation of acetylene in ref 8. In this contribution,we formulate our model in a more general way and analyzein detail the experimental information to provide a quantita-tive estimate of the parameters of the theory. In this waythe voltage threshold, the different time scales, and the

* Corresponding author. E-mail: thorsten@chem.northwestern.edu.† Northwestern University.‡ Argonne National Laboratory.

NANOLETTERS

2008Vol. 8, No. 10

3525-3531

10.1021/nl801001q CCC: $40.75 2008 American Chemical SocietyPublished on Web 09/23/2008

isotope effect observed in the experiment are interpreted ina consistent way.

Using the STM-induced dissociation of acetylene on acopper surface, as reported in ref 8, as a case study, we showhow the large voltage threshold of several electronvolts isnaturally related to the excitation energy between the groundstate of the molecule and a doorway state. The energy forthis transition is provided by the cotunneling process.Following the excitation, the dissociation proceeds via thecoupling between the doorway state and a radical electronicstate, which is strongly stabilized by the metal surface. Asimple kinetic model is then used to discuss the timescalesof the dissociation process. The unusually large isotope effectobserved in the experiment is also explained as primarilydue to the difference in geometries and dipole moments ofthe molecular states of the H- and D-substituted species onthe surface.

Current-induced events in molecular transport junctionsis an actively studied topic both experimentally8-20 andtheoretically.21-30 For example, Seideman and co-workersdiscuss how the current can trigger nuclear dynamics viathe electron-phonon coupling, thus inducing rotations,conformational changes, or unimolecular chemical reactionsin the molecular junction.21-25

Cotunneling provides a complementary mechanism fordynamic events that, so far, has not been discussed in thecontext of molecular transport junctions. The transfer ofkinetic energy of the tunneling electron to the molecule isanalogous to the vertical excitation invoked in photon-induced gas phase reactions, but it is mediated by theelectrode via the self-energy. Physically, our model combineselectron transport and energy transfer in a way that is suitablefor the description of surface chemical reactions. This kindof description may have important implications for nano-electrochemistry and nanocatalysis.

Cotunneling Model. We shall consider a molecular wirethat in the relevant regimes of voltage, gate voltage,temperature, and so forth can be found in a number of neutralquantum states. In the following, we shall work with twomolecular states, |a⟩ and |b⟩, for simplicity, but generalizationto more states is straightforward.

If the charged states of the molecule are energeticallyaccessible, redox processes will occur and the junction canconduct by incoherent hopping processes. This is the regimeof nanoelectrochemistry. At sufficiently low bias, or if thecharged states are higher in energy, the incoherent hoppingof electrons will be suppressed. This is analogous to the low-bias situation in the Coulomb blockade regime of metallicquantum dots.31

Any current in this regime is the result of coherentprocesses that are higher order in the, presumably weak,metal-molecule coupling. We shall consider only the lowestorder processes, being second order in the metal-moleculecoupling. First, there are elastic processes that leave themolecule in the same state, also known as superexchangeprocesses. Second, there are inelastic processes that inducetransitions between the molecular states. These are thecotunneling processes. This choice of words implicitly

assumes that the states |a⟩ and |b⟩ differ in energy. This, ofcourse, need not be the case; they could be degenerate anddiffer in some other quantum number. This case is alsonaturally accounted for in our model.

We think of the Hamiltonian for the entire system as asum of three parts,

H)Hlead +Hmol +V (2)

corresponding to the leads, the molecule, and theelectrode-molecule coupling, respectively.

The electrons in the leads are described in the conventionalway, using operators ck

† and ck that create or annihilateelectrons in single-electron states indexed by k. The leadHamiltonian reads

Hlead )∑k∈ L

(εk - µL)ck†ck +∑

k∈ R

(εk - µR)ck†ck (3)

where µL and µR denote the chemical potentials in the leftand right leads, respectively.

In the molecular Hamiltonian we include not only the twomolecular states of interest, |a⟩ and |b⟩ , but also the twocharged states, a cationic state |n - 1⟩ and an anionic state|n + 1⟩ , where n is the number of electrons in states |a⟩ and|b⟩. The two charged states will appear as virtual intermediatestates during superexchange and cotunneling processes. Themolecular part of the Hamiltonian is expressed as

Hmol ) ∑j)a,b,n+1,n-1

Ej|j⟩ ⟨ j| (4)

This Hamiltonian is defined in Fock space, in that it includesstates that differ in the number of electrons. In general, wecould write each many-particle state as a superposition ofSlater determinants, |�i⟩ ) ci1

† ci2† ...cin

† |0⟩ , where cij† creates an

electron in the jth molecular orbital used in the constructionof the ith Slater determinant, for example,

|a⟩ )∑i

ai|�i⟩ )∑i

aici1† ci2

† ...cin† |0⟩ (5)

where ai is a coefficient, but we use the projection, orHubbard, operators because the results derived below do notrely on the details of the electronic structure and are thereforevalid for the exact many-particle states or approximate statesat any given level of electronic-structure theory.

Because of the differing choices of representation above,the electrode-molecule coupling Hamiltonian is an uncon-ventional mixture of two types of operators. We write it asa sum of two terms, V ) Vcat + Van, which are given by

Vcat ) ∑j)a,b

∑k∈ L,R

{Vkn-1,jck†|n- 1⟩ ⟨ j|+V kn-1,j

/ |j⟩ ⟨ n- 1|ck} (6)

Van ) ∑j)a,b

∑k∈ L,R

{Vkj,n+1ck†|j⟩ ⟨ n+ 1|+V kj,n+1

/ |n+ 1⟩ ⟨ j|ck} (7)

The two terms, eq 6 and eq 7, correspond to the couplingof the neutral molecular states, |a⟩ and |b⟩ , to the cationic,|n - 1⟩ , and anionic state, |n + 1⟩ , respectively. Eachindividual term describes a virtual charge transfer processto or from the molecule. The term Vkn-1, ack

†|n - 1⟩⟨ a|, forinstance, describes the virtual oxidation of state |a⟩ to thecation |n - 1⟩ accompanied by the creation of an electronin the electrode. The Hamiltonian, eq 2, with couplingterms as in eq 6 and eq 7 is also known as the ionic model.32

3526 Nano Lett., Vol. 8, No. 10, 2008

Current. In a molecular transport junction, we can thinkof the current as the net result of individual scatteringevents that take electrons across the junction. We shallconstruct system wide scattering states and calculate thetransistion rates among them using a generalized Fermi’sgolden rule,

wfi )2πp

|Tfi|2δ(Ei -Ef) (8)

Here, T is the transition matrix and the indices i and f indicatethe initial and final state of a particular scattering event. Thecontribution of this transition to the current is then simplygiven by

I)-ewfi (9)

The scattering states are constructed as direct products ofmany-particle states describing each part of the junction. Asan example, we can write

|i⟩ ) |L⟩ |a⟩ |R⟩ (10)

|f ⟩ ) cl|L⟩ |b⟩ cr†|R⟩ (11)

where |L⟩ and |R⟩ represent the Fermi sea of electrons inthe respective electrode. This choice of initial and finalstates corresponds to a situation where an electron istransferred from a state l in the left electrode to a state rin the right. If the bias is reversed or at higher tempera-tures, different or additional types of initial and final statesmust be considered.

The transition rate, wfi ≡ wbalr , from |i⟩ to |f⟩ depends on

the electrode indices l, r. To obtain the full transition rate ofthe molecule from state |a⟩ to |b⟩ , we must sum over allavailable electron states,

wba )∑lr

wbalr (12)

Transition Matrix. The transition matrix, T(E), for ascattering process can be defined as

T(E))V+VGret(E)V (13)

Here, Gret is the retarded Green’s function with respect tothe full Hamiltonian,

Gret(E)) 1E-H+ iη

(14)

where η ) 0+ is a positive infinitesimal. Likewise, G0ret(E)

is the retarded Green’s function with respect to the uncoupledHamiltonian, H0 ) Hlead + Hmol.

For a coupling V as in eq 6 and eq 7, the retarded Green’sfunction with respect to the full Hamiltonian, Gret(E), canbe expanded as

G)G0 +G0VG0 +G0VG0VG0 + ... (15)

Here, however, we consider tunneling through a mol-ecule in a situation where the charging energy is large.Thus, both the initial and the final states are neutral, andthe charged states appear only virtually. And since theperturbation V changes the charge on the molecule, onlyterms of even order in V will appear the expansion of theGreen’s function.

G)G0 +G0VG0VG0 +G0VG0VG0VG0VG0 + ... (16)

Defining the retarded self-energy, Σret(E) ) VG0ret(E)V, we

can rewrite eq 16 as a Dyson’s equation,

G)G0 +G0ΣG (17)

Introducing the expansion 16 into expression 13 for thetransition matrix and disregarding the first term, V, leadto

T)VGV)VG0V+VG0VG0VG0V+ ... (18)

When we consider cotunneling processes, meaning includ-ing only the first term of the expansion 18, we have T(E)) Σret(E). And the generalized Fermi’s golden rule nowreads

wfi )2πp

|Σfiret|2δ(Ei -Ef) (19)

Equation 19 is an important result because it represents thelink between electron transport and energy transfer that cangenerate the driving force for a chemical reaction.

Transition Rates. To calculate the cotunneling rate, wefollow the strategy laid out in refs 4-6. Since both the initialand the final states are neutral, the matrix element of theself-energy operator, to second order in the coupling, canbe written as a sum of two terms.

Σfiret ) ⟨ f |VG0V|i ⟩ ) ⟨f|VcatG0V

cat|i ⟩ + ⟨f|VanG0Van|i⟩ (20)

For now, we use specific electrode states l and r and writea bra with the final state as ⟨f | ) ⟨i|(|a⟩⟨ b|cl

†cr). The first termin the self-energy is then calculated to be

⟨ f |VcatG0Vcat|i ⟩ )∑

l′r′

⟨i|(|a ⟩ ⟨ b|cl†cr)(Vl′n-1,b

/ |b ⟩ ⟨ n- 1|cl′) ×

1Ei -H0

(Vr′n-1,acr′†|n- 1⟩ ⟨ a|)|i ⟩

)Vln -1,b/ Vrn-1,a

Ea -En-1 - εr⟨L|cl

†cl|L ⟩ ⟨ R|crcr†|R⟩ (21)

Calculation of the second term, describing the couplingto the anion, is very similar and the self-energy amountsto

Σfiret ) ⟨L|cl

†cl|L⟩ ⟨ R|crcr†|R⟩ ( Vln -1,b

/ Vrn-1,a

Ea -En-1 - εr+

Vla,n+1/ Vrb,n+1

Ea -En+1 + εl)

(22)

We can now take the square and sum over all states in theelectrodes, with indices l and r. Assume, for simplicity, thatthe coupling elements are constant at each electrode; that is,VL ≡ Vln -1, b ) Vla, n+1 and VR ≡ Vrn-1, a ) Vrb, n+1. We nowhave

∑lr

|Σfiret|2 )∑

lr

⟨L|cl†cl|L⟩ ⟨ R|crcr

†|R⟩ | VLVR

Ea -En-1 - εr+

VLVR

Ea -En+1 + εl|2(23)

Define the spectral density as ΓL ) 2π|VL|2ρ(εl), which isthe same expression used in the single-particle formulation,1

and assume a flat density of states. The summations can thenbe rewritten as

Nano Lett., Vol. 8, No. 10, 2008 3527

∑l

|VL|2f∫ dεl ρ (εl)|VL|2=ΓL

2π∫ dεl (24)

The matrix element ⟨L|cl†cl|L⟩ corresponds to the zero

temperature limit of the Fermi distribution for the left leadand should be replaced with a thermal average

fL(εl)) f(εl - µL))∑i

Wi⟨Li|cl†cl|Li⟩ (25)

where Wi represents the Boltzmann weight of the configu-ration |Li⟩ .

Including thermal averaging over electrode configurations,we arrive at a general result for the inelastic scattering rate:

wba )ΓLΓR

2πp∫ dεl∫ dεr( 1Ea -En-1 - εr

+ 1Ea -En+1 + εl

)2×

fL(εl)(1- fR(εr))δ(Ea -Eb - εr + εl) (26)

The elastic tunneling rate, waa, corresponding to electrontransport involving only the ground state |a⟩ can now beobtained simply by choosing Eb ) Ea,

waa )ΓLΓR

2πp∫ dεl∫ dεr( 1Ea -En-1 - εr

+ 1Ea -En+1 + εl

)2×

fL(εl)(1- fR(εr))δ(εl - εr) (27)

In comparing the expressions for the inelastic and elasticscattering rates, the only difference is the energy conservationcondition appearing in the argument of the delta function.This condition is harder to satisfy in terms of availability ofstates in the inelastic case, eq 26, than in the elastic case, eq27, and this difference will eventually translate into theappearance of a voltage offset of the inelastic contribution.

Expressions such as eq 26 and eq 27 could be mappedonto a single-particle theory. If we associate the electronaffinity with a LUMO energy, Ea - En+1 ∼ εLUMO, and theionization potential with a HOMO energy, En-1 - Ea ∼εHOMO, the excitation energy, ∆E ) Eb - Ea, appearing ineq 26 would be the only reference to the many-particle states.At the Hartree-Fock level of theory, this mapping is ensuredby Koopmans’ theorem, but it is also often used, without aformal proof, within the Kohn-Sham approach to densityfunctional theory.

A Simple Case. Consider the inelastic cotunneling processin a simple case where one intermediate state, say the virtualcation, |n - 1⟩ , is dominant. In this case, eq 24 reduces to

wba )ΓLΓR

2πp∫ dεl∫ dεr

fL(εl)(1- fR(εr))

(Ea - (En-1 + εr))2δ(Ea -Eb - εr + εl)

) ΓLΓR

2πp∫ dεl

fL(εl)(1- fR(Ea -Eb + εl))

(Eb -En-1 - εl)2

(28)

In the zero temperature limit, the rate can be expressedanalytically. Define the ionization potentials for the states|a⟩ and |b⟩ as A ) En-1 - Ea and B ) En-1 - Eb, respectively.Now we substitute the integration variable by u ) B + εl,and at zero temperature, we obtain

wba )ΓLΓR

2πp∫ du

u2(1- ϑ(u-B- µL))ϑ(u-A- µR)

) ΓLΓR

2πp∫A+µR

B+µL du

u2

) ΓLΓR

2πp [ 1A+ µR

- 1B+ µL] (29)

For identical electrodes, the chemical potentials are givenby µL ) -W + eV/2 and µR ) -W - eV/2, where W is thework function of the metal, and the expression for theinelastic tunneling becomes

wba )ΓLΓR

2πpeV-∆E

(A-W)(B-W)+ eV2

∆E- e2V2

4

(30)

where ∆E )Eb - Ea.47 To obtain the corresponding elasticrate, we set b ) a to obtain

waa )ΓLΓR

2πpeV

(A-W)2 - e2V2

4

(31)

Consider the numerator in eqs 30 and 31. In eq 30 for thecotunneling rate, an additional term ∆E appears. This is thevoltage threshold.

We have considered cotunneling as an excitation mech-anism, using a positive ∆E ) Eb - Ea. If we want to invokecotunneling as a deexcitation mechanism, we must redo thecalculations above and arrive at a deexcitation rate of

wab )ΓLΓR

2πpeV+∆E

(A-W)(B-W)- eV2

∆E- e2V2

4

(32)

Due to a larger availability of states that can participate indeexcitation, one sees that wab > wba.

Surface Reaction. Recent experiments have shown thatrelatively weak tunneling currents passing through a molec-ular junction may induce different processes ranging fromheating to desorption and even chemical reactions.8-20 Weare particularly interested in the remarkable experimentdescribed in ref 8 that shows the successive breaking of thetwo CH bonds in an acetylene molecule adsorbed on the(001) surface of copper, under the influence of an STM tip.

Generally speaking, we can consider two different mech-anisms for surface processes associated with tunnelingcurrents: first, current-induced processes that involve excita-tion of vibrations via the electron-phonon coupling. Theseinclude desorption, junction heating, and mechanical vibra-tions of the type studied by Seideman and co-workers.21-25

One can think of these as a sort of molecular Joule effectwhere the emission of phonons gradually heats up thejunction.

Second are the processes that involve directly electronicexcitations of the molecule. Such processes, like STM-inducedchemical reactions, will show a large voltage threshold. Anytheoretical description of such processes must involve at leasttwo electronic states of the substrate-adsorbate system, withthe coupling between them mediated by the contacts. We are

3528 Nano Lett., Vol. 8, No. 10, 2008

proposing a mechanism involving cotunneling, and in thefollowing, we shall apply it to the dissociation of acetylene.

A different but related theoretical model was proposed byHasegawa et al.26 It implies, however, an unlikely energyreconcentration effect and assumptions about the energyparameters that we think are not adequately justified.

Energy Diagram. We focus on the dissociation of acety-lene, as reported in ref 8,as a specific example. Considerthe energy diagram in Figure 1. We treat the ground state,|a⟩ , of acetylene, and an excited state, |b⟩ , at the bottom ofan excited potential energy surface. From |b⟩, the system cantunnel through a potential energy barrier to a dissociativechannel |c⟩ . To map out the diagram, we rely on energyvalues from the literature.

For treatments of the excited states of acetylene see refs33-36. Theoretical calculations show that the lowest excitedstates of acetylene are triplet states. In optical spectroscopysuch transitions would be forbidden, but the selection rulesdo not apply to excitation by cotunneling. The lowest excitedstate is the cis-bent state 13B2. It lies 3.81 eV above theground state.36 This is a viable candidate for the state |b⟩ inFigure 1, especially since the ground-state geometry ofacetylene bound on a copper surface is cis-bent.8,37-40

An excitation energy of 3.81 eV is, of course, larger thanthe threshold of 2.8 eV. We mention two possible explana-tions for this. First, the reported excitation energies arevacuum numbers. The ground-state geometry changes uponadsorption. We assume that the main role of the surface isto stabilize the adsorbed species, but such a stabilization canbe very different for polarized or charged species comparedwith the ground state. This could lower the excitation energy.

Second, the excitation could occur from a vibrationallyexcited state of the electronic ground state. This is in linewith optical dissociation experiments where vibrations areexcited before dissociation. Note that these are not necessarilythermal excitations. In the STM-junction, this energy wouldbe provided via the mechanism described by Seideman andco-workers.21-25

The vibrational energies of acetylene are discussed in refs8, 10. An average value of the CH stretch modes is 377 meVon the Cu surface. Likewise, a value of 101 meV is reportedfor CCH bending mode.

An increase in temperature, from 9 to 45 K, caused anincrease of the voltage threshold by 0.8 V.8 We speculatethat the heating of the substrate could partly quench the

selective predissociation excitations of the CH stretch andCCH bending modes.

To gauge the position of state |c⟩ in Figure 1, we need toknow the dissociation energy for acetylene into ethynyl(C2H•) and hydrogen. Values for dissociation in vacuum are5.712 eV from experiment41 and 5.795 eV from theory.42

This value is lowered dramatically at the Cu surface, sincethe stabilization of the radical is much larger than that ofthe neutral species. The binding energies for acetylene andethynyl to a Cu(001) surface are reported to be 1.31 and4.22 eV, respectively.10 A similar calculation gives 1.09 and4.36 eV.43 Both references used DFT with the generalizedgradient approximation (GGA). By correcting the experi-mental vacuum value using the calculated stabilizationenergies, this gives values of 2.802 and 2.442 eV for thedissociation energy.

Kinetic Model. To estimate the impact of the cotunnelingscheme on the rate of dissociation, we invoke a simple kineticmodel. The overall dissociation process is described by threekinetic equations,

dPa

dt) -wbaPa +wabPb (33)

dPb

dt)wbaPa - (wab +wdis)Pb (34)

dPc

dt)wdisPb (35)

where Pa and Pb describe the populations of states |a⟩ and|b⟩ of acetylene, respectively, and Pc ) 1 - Pa - Pb describesthe effective population of the product state. The rates wba

and wab are cotunneling rates, and wdis is the rate for thesystem to tunnel from |b⟩ to |c⟩ .

The set of kinetic equations can be solved,44 for example,by Laplace transformation, and the solution is governed bythe two roots, u1 e u2, of the equation

u2 - (wab +wba +wdis)u+wbawdis ) 0 (36)

For the initial conditions, Pa ) 1 and Pb ) Pc ) 0, the exactsolution for Pc(t) can be expressed as

Pc(t)) 1-u2

u2 - u1e-u1t +

u1

u2 - u1e-u2t (37)

As we discuss below, we assume that we are in a regimewhere the cotunneling processes occur at a much faster ratethan the effective dissociation, thus wab, wba . wdis. This inturn implies that u1 , u2, and the overall process will, afterestablishing a kinetic equilibrium between states |a⟩ and |b⟩ ,be dominated by a single exponential. We have

Pc(t)= 1- e-u1t (38)

where u1 to a good approximation is given by

u1=wbaw

dis

wab +wba +wdis=

wba

wab +wbawdis (39)

To estimate values for the transition rates wab and wba forour case study of acetylene dissociating on copper, weassume that the overall current is well-described by I )-ewaa, where waa is the elastic current from eq 31.Experimentally, a current of ∼0.85 nA is reported at a bias

Figure 1. Potential energy surfaces

Nano Lett., Vol. 8, No. 10, 2008 3529

of 3.0 V. This corresponds to electrons passing through thejunction at a rate of 0.53 × 1010 s-1. We use a theoreticalvalue of A ) 11.5 eV for the ionization potential ofacetylene.45 An average value for the electrode work functionof W ) 4.6 eV is used since WCu ) 4.7 eV and WW ) 4.5eV. With these numbers, we are able to estimate an averagevalue for the spectral density; �ΓLΓR ) 32 meV.

Having a value for the spectral densities, we can nowcalculate the rate of cotunneling excitations, wba, using eq32. We use ∆E ) A - B ) 2.8 eV and get wba ) 1.68 ×109 s-1. Likewise, the deexcitation rate, calculated using eq30, is given by wab ) 6.73 × 1010 s-1. Other mechanismsfor deexcitation, such as exciton transfer or phonon emission,are disregarded in this crude estimate of rates.

The ratio between the deexcitation and the excitation rates,wab and wba, respectively, is given by

wab

wba) eV+∆E

eV-∆E×

(A-W)(B-W)+ eV2

∆E- e2V2

4

(A-W)(B-W)- eV2

∆E- e2V2

4

) 40.1 ≈ 40 (40)

Now we can estimate the impact of the kinetic equilibriumbetween |a⟩ and |b⟩ on the overall dissociation rate. From eq39, we now have u1 ) (1/41)wdis, and since the overallexperimental rate is of the order of u1 ∼ 1 s-1,8 we havewdis ) 41 s-1. This is much slower than the excitation anddeexcitation rates, as assumed above. So, even though theexcitation kinetics does constitute a bottleneck and does slowdown the overall rate, in this case, by a factor of 41, themagnitude of the rate is by far dominated by the dissociatingprocess, represented by the rate wdis.

Time Scale. One intriguing aspect of the experiment isthe relatively long time scale of the overall dissociationprocess. Our simple kinetic model suggests that once thethreshold has been overcome the excitation kinetics has onlylimited impact on the overall rate of dissociation.

Such activated processes have been discussed extensivelyin the literature on photochemical processes.46 And also inthe context of current-induced processes.21-23 The mainconclusion of these works is that a dissociation time τ ∼p/wdis dominates the bond breaking process.

The excited state |b⟩ is playing the role of what is knownas a doorway state in the literature on photochemicalprocesses.46 The coupling between the excited state and adissociation channel of state |c⟩ is physically responsible forthe breaking of the bond.

Isotope Effect. Another important experimental finding thatcan be understood within our model is the massive isotopeeffect in the observed dissociation threshold. The thresholdfor HCCH is 2.8 V, compared with 3.8 V for DCCD.8

It is well-known that isotopic substitution in a moleculeleads to changes in the vibrational and rotational frequenciesthat are responsible for changes in the IR spectra, and inour problem this can have some bearing on the voltagethreshold. For acetylene on copper, the change from C-Hto C-D stretch frequency is on the order of 100 meV.8,10 If

the excitation from |a⟩ to |b⟩ is indeed occurring from avibrationally excited state of |a⟩ , this will explain some, butnot all, of the change in threshold.

The electron-phonon coupling constant is another mag-nitude that depends on the isotope mass. However, this wouldbe of importance mostly in modifying the efficiency of thevibrational-electronic energy transfer between the surfaceand the molecule, and since no significant change in thedissociation time scale was reported for the different isotopes,we can disregard this effect.

These considerations leave us with the possibility ofisotope-induced geometry changes as the main explanationfor the very large isotope effect observed in the experiment.In gas phase, the ground-state geometry of both HCCH andDCCD is linear, but experimental studies of normal anddeuterated acetylene on a Pt surface suggests that the cis-bending of the molecule is much larger for HCCH and thatDCCD is nearly linear.37 This implies that the dipole momentof normal acetylene on a surface will be larger than that ofthe deuterated species. We expect these differences to bepreserved in the excited states, hence leading to a higherstabilization of the doorway state for HCCH compared withthat of DCCD. Electronic structure calculations of theexcited-state potential energy surfaces of acetylene invacuum33,36 indicate that the energy is very sensitive to thebending angle. In fact, the 130° cis-bend geometry is morethan 1 eV lower in energy than the linear geometry,33,36 anenergy difference that is well within the right order ofmagnitude to explain the variation in the threshold voltageassociated with the isotopic substitution.

Conclusion. The description of charge and energy transferprocesses occurring under the nonequilibrium conditionspresent in electrode-molecule-electrode junctions underbias is a formidable theoretical challenge. In principle, thenonequilibrium Green’s functions (NEGF) formalism affordsa complete physical picture, but its complexity demands anumber of approximations to extract the relevant information.

Cotunneling is an inelastic process that can be importantfor the description of chemical reactions because it can playan important role in providing a route for energy transferbetween otherwise uncoupled electronic states. Our modelidentifies the self-energy with the effective coupling betweenthe electrodes mediated by the bridge. The resulting expres-sion for the current includes an elastic contribution, arisingfrom superexchange processes, and an inelastic contribution,involving the transfer of energy that triggers the chemicalreaction. The description of this explicit pathway connectingelectron transport and the onset of a chemical reaction in amolecular junction is the main result of our work.

Using our model, we were able to rationalize in terms ofexcitation processes and the relative stability of the differentchemical species on the surface, the main features of theSTM-induced dissociation reaction of acetylene on a coppersurface, a system for which detailed experimental observa-tions are available. Although further confrontation withempirical information is required to validate our model, it isremarkable that the three main features of this system, thevoltage threshold for unimolecular dissociation, the long time

3530 Nano Lett., Vol. 8, No. 10, 2008

scale for current variation associated with the chemicalreaction, and the unusually large isotope effect, are naturallyaccounted within the framework of our theory.

The inclusion of several electronic states is a key elementof our model. This is achieved through a generalization ofthe Landauer model to a Fock space description whereseveral states with a varying number of electrons aresimultaneously considered. The potential of this approachfor the description of current-induced nonadiabatic processesand other effects due to electronic correlation, like Coulombblockade and Kondo effect, is currently investigated.

Acknowledgment. We thank L. J. Lauhon for very helpfuldiscussions. We thank the Danish Research Council forNature and Universe (FNU), The NSF-MRSEC program, TheNASA URETI program, and CNM at Argonne NationalLaboratory for support for the research reported in this work.

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