alessandro troisi and mark a. ratner- propensity rules for inelastic electron tunneling spectroscopy...

Propensity rules for inelastic electron tunneling spectroscopyof single-molecule transport junctions

Alessandro Troisia�

Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdomand Centre of Scientific Computing, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL,United Kingdom

Mark A. Ratnerb�

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113and Institute for Nanotechnology, Northwestern University, 2145 Sheridan Road, Evanston,Illinois 60208-3113

�Received 27 April 2006; accepted 16 October 2006; published online 7 December 2006�

Using a perturbative approach to simple model systems, we derive useful propensity rules forinelastic electron tunneling spectroscopy �IETS� of molecular wire junctions. We examine thecircumstances under which this spectroscopy �that has no rigorous selection rules� obeys welldefined propensity rules based on the molecular symmetry and on the topology of the molecule inthe junction. Focusing on conjugated molecules of C2h symmetry, semiquantitative argumentssuggest that the IETS is dominated by ag vibrations in the high energy region and by out of planemodes �au and bg� in the low energy region. Realistic computations verify that the proposedpropensity rules are strictly obeyed by medium to large-sized conjugated molecules but are subjectto some exceptions when small molecules are considered. The propensity rules facilitate the use ofIETS to help characterize the molecular geometry within the junction. © 2006 American Institute ofPhysics. �DOI: 10.1063/1.2390698�

I. INTRODUCTION

The interaction of the nuclear motions with electrontransport across a molecular junction gives rise to a varietyof phenomena whose understanding is crucial for the correctinterpretation of the conductance properties.1 In a recentoverview an attempt was made to classify the different mani-festations of the nuclear degrees of freedom in transportmeasurements at the single-molecule level.2 The most com-plex cases occur when the energy of the electron injectedfrom the electrodes is close to an electronic resonance withone molecular level. In these cases such phenomena asdouble-barrier tunneling into vibronic states3,4 or polaronictransport5,6 can be observed, depending on the strength of theinteraction �spectral density� between the molecule and theelectrodes. In the low voltage regime, inelastic electron tun-neling spectroscopy �IETS� spectra can be recorded at lowtemperature.7–16 Here, molecular vibrations modulate thetunneling matrix element between states localized on the leftand right electrodes, and this modulation is related to thesmall probability that the electron tunnels inelasticallythough the molecule, losing �or gaining� a quantum of vibra-tional energy. IETS are displayed as plots of d2I /dV2 versusV, and inelastic peaks appear in such plots, reflecting whichmolecular vibrations are most effective in modulating thetunneling probability. The simpler interpretation14 is that

IETS corresponds to the opening of a new, inelastic channel,so that there is an increase in the current as voltage is in-creased and a peak in d2I /dV2. IETS is an extremely prom-ising technique because it allows an unmistakable proof ofthe presence of the molecule inside the junction and canprovide information about the structure of the junctions.Such measurements at the single-molecule level are rela-tively recent, and the richness of information they provide�structure/bonding� requires an appropriate theoretical ap-proach.

Interpretation of IETS is not trivial, since there are nostrict selection rules derived by the theory and propensityrules have not been clearly identified. Numerical approachesfor the interpretation of the IETS spectra have been proposedby several groups.17–26 In one recent paper27 we showed thata numerical approach based on a perturbative treatment ofthe tunneling is able to reproduce correctly the spectra ofthree molecules measured by Kushmerick et al.15 Numericaltreatments are valuable for understanding transport experi-ments in molecular electronics, but it is also very useful toderive simple principles more closely related to intuition,which can guide interpretation of the results. For example, ininfrared �IR� spectroscopy, the most intense modes modulatemost strongly the molecular electric dipole and, withoutmuch calculation, it is relatively easy to predict which vibra-tions are more likely to give strong infrared absorption. Se-lection rules are strictly obeyed in both Raman and IR spec-troscopies and greatly simplify the assignment of thevibrational modes of symmetric molecules. Lorente et al.18

a�Electronic mail: [email protected]�Electronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 125, 214709 �2006�

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reported elegant work on IETS of deuterated acetylenes ab-sorbed on Cu�111�. The experiments were based on scanningtunneling microscopy �STM� measurements, for which themolecular energies are pinned to the metal, the transport in-volves tunneling through vacuum to the STM tip, and theresulting currents are in the nanoampere range. Then theanalysis can be based on the Tersoff-Hamann limit, depend-ing only on state densities. We are concerned here with themolecular junction case, where the molecule bridges the in-terelectrode space, the spectral density is comparable at bothmolecular termini, currents are in the microampere range,and the Tersoff-Hamann approach28 is not appropriate.

Herein we use the Landauer-Imry approach29,30 to tun-neling transport to derive a series of propensity rules forIETS that can be used to predict the activity of a given vi-brational mode. The derivation is perturbative, assuming thatthe conductance is much smaller than the quantum conduc-tance and the electrodes are at equilibrium during the mea-surement. The set of propensity rules will be deduced apply-ing the general formalism �outlined in Sec. II� to modelsystems �Sec. III� and applying standard point group theory.Some calculations on realistic systems will be used in Sec.IV to verify the validity of the propensity rules.

II. BACKGROUND

A. Elastic conductance

We will recall here the general formalism for the pertur-bative simulation of IETS �for a wider discussion see Refs.19 and 23�. The elastic component of the conductance can bewritten as31,32

gel�E� = gc Tr��L�E�G�E��R�E�G�E�+� , �1�

where gc is the quantum conductance and �L and �R aretwice the imaginary part of the self-energy matrices associ-ated with the interaction of the molecular subsystem with theleft and right electrodes. G is the matrix representation of theretarded Green function operator, related to the molecularHamiltonian H as

G�E� = �E − H − i��→0−1 . �2�

The elastic contribution to the current is

Iel = �−�

+� gel�E�e

�fL�E� − fR�E��dE ,

where fL= f�E+e�L� and fR= f�E+e�R� are the Fermi func-tions of the left and right electrodes, modified by the externalpotential �L or �R applied to the respective electrode.

B. Inelastic channel

The interaction of the tunneling electron with the vibra-tional levels can be treated perturbatively, allowing exchangeof a single vibrational quantum per tunneling electron andassuming that the molecule is in its ground vibrational state�low temperature limit�. These assumptions are discussed inRef. 19 where a detailed derivation of the following equa-tions is given. Green’s function matrix elements are para-metrically dependent on the set of normal modes �Q��; this

dependence can be expressed in a Taylor series expansionaround the equilibrium position �Q��=0 �we use dimension-less coordinates�,

Gij�E,�Q�� Gij�E,0� + � �Gij�E,�Q��

�Q��Q�. �3�

The matrix G�, with Gij� =�2/2��Gij�E , �Q�� /�Q��Q��=0,

quantifies the importance of the inelastic channel involvingthe vibrational mode �. The inelastic contribution to the con-ductance due to mode �, separated in left to right �L→R�and in right to left �R→L� tunneling, is

g�inel�L→R��E� = gc Tr��L�E�G��E��R�E − ��G��E�+� ,

�4a�

g�inel�R→L��E� = gc Tr��L�E − ��G��E��R�E�G��E�+� ,

�4b�

and the inelastic current from mode � is

I�inel = I�

inel�L→R� − I�inel�R→L�, �5a�

I�inel�L→R� = �

−�

+� ginel�L→R��E�e

fL�E��1 − fR�E − ��dE ,

�5b�

I�inel�R→L� = �

−�

+� ginel�R→L��E�e

fR�E��1 − fL�E − ��dE .

�5c�

The elastic and inelastic components are additive in this lim-ited model; therefore,

I = Iel + �

I�inel. �6�

Equations �5� imply that the inelastic channel � opens whenthe applied bias is equal to �� and it stays open for largerbias. A plot of d2I /dV2 versus V therefore shows a peak forV=�� /e �or −�� /e� and the integral below the peak isg�

inel�L→R� �or −g�inel�R→L�� the proof is given in Appendix B�.

At low bias, the energy dependence in Eqs. �1� and �4�can be neglected and E set to EF. The integrated intensity W�

of the peak due to mode � in an IETS spectrum �d2I /dV2

versus V� is

W� = gc Tr��L�EF�G��EF��R�EF�G��EF�+� . �7�

C. Localized representation

Any basis set appropriate for the one-electron wavefunction can be used to build the matrices in Eqs. �1� and �7�.It is convenient to consider localized atomic orbitals. Sincewe use a linear combination of atomic orbitals/molecular or-bital descriptor overall, this choice of basis can still describedelocalization within the molecule.

We assume geometry for the junction as illustrated inFigs. 1 and 2. The coupling matrix element between anatomic orbital far from the electrode and the electrode states

214709-2 A. Troisi and M. A. Ratner J. Chem. Phys. 125, 214709 �2006�


is negligible �coupling falls exponentially with interatomicdistance�. Mathematically, recalling the definition of �L �for�R is similar�:

�ijL�E� = 2�

ViVj* �E − E� �8�

� is an index running over the left electrode states and theVj are the molecular or atom/electrode couplings�, it is clearthat �ij

L is nonzero only for i and j close to the electrode. Wecan partition the space of the atomic orbitals in three sub-spaces L, C, and R where L and R contain the orbitalsinteracting only with the left and right electrodes, respec-tively, and C is the space of the “central” atomic orbitals notinteracting with either electrode. Of the nine blocks that formthe matrices �L and �R only one is non-null:

�L = �LLL �LC

L �LRL

�CLL �CC

L �CRL

�RLL �RC

L �RRL � = �LL

L 0 0

0 0 0

0 0 0� , �9a�

�R = �LLR �LC

R �LRR

�CLR �CC

R �CRR

�RLR �RC

R �RRR � = 0 0 0

0 0 0

0 0 �RRR � . �9b�

This form of the two matrices permits a basis set transforma-tion that makes both �LL

L and �RRR diagonal. Then the trace

in the expression of the elastic conductance �Eq. �1�� can bewritten as a summation over two indices

g�E� = gc m�Ln�R

�mmL �E��nn

R �E��Gmn�E��2. �10�

Equation �10� is essentially a sum over the tunneling chan-nels m-n: the electron is coupled to the left electrode throughorbital m and to the right electrode through orbital n. Notethat all the elements of the summation are positive �there isno interference between tunneling channels, all the interfer-ence is incorporated into the term Gmn�E��. In this represen-tation, the intensity W� of the peak due to mode � is

W� = gc m�Ln�R

�mmL �E��nn

R �E�1

2� �Gmn�E�

�Q� �2

. �11�

As we will see at the end of the next section �and in Appen-dix A� the localized representation allows simple inclusionof symmetry considerations in the evaluation of W�.

III. MODEL SYSTEMS

A. Point groups in molecular junctions

It is probably very common that a realistic molecularjunction has no “rigorous” symmetry element, because of thepoorly defined structure of the electrodes in many experi-ments �break junctions, cross-bar junction�, the interactionwith other molecules �e.g., in self-assembled monolayers�,and the coordinative molecule-electrode interaction thatcauses frequent fluctuation of the interfacial geometry.33

Nevertheless, it is convenient to describe the symmetry pro-pensity rules in the idealized case where both electrodes pre-serve the maximum allowed degree of symmetry of the mol-ecule in the junction.

Two types of molecular symmetry element can be foundin a molecular junction: �i� symmetry elements that inter-change the left and right electrode interfaces �like the inver-sion center, a symmetry plane, or C2 rotation axis parallel tothe electrodes� and �ii� symmetry elements that transform theleft and right electrode interfaces into themselves �e.g., sym-metry planes or rotational axis perpendicular to the elec-trodes�. When one of the elements of the first group ispresent the junction is electrically symmetric since the leftand right electrodes are equivalent. We will first discuss thesystem where there is one symmetry element of the first type,but the conjugated planar molecule may possess the secondset of symmetry elements, and, at the end of Sec. IV, we willprovide some guidelines for the general case. These symme-try elements are illustrated in Fig. 3.

FIG. 1. Schematic of a molecular junction and of the three portions that aretreated with different computational approaches: �i� accurate density func-tional theory �DFT� calculations of the isolated molecule �M� are used toevaluate the molecular vibrations and the intramolecular Hamiltonian matrixelements, �ii� a tight binding Hamiltonian is used to evaluate the LDOS onthe surface of the bare electrodes �E�, and �iii� a cluster calculation includingfew gold atoms is used to compute the metal-molecule Hamiltonian matrixelement at the interface �I�.

FIG. 2. Model system used to compute the coupling matrix element be-tween the molecule and the electrode. The benzenethiolate molecule is onthe fcc adsorption site of the Au9 cluster and the C–S bond is perpendicularto the surface. The geometry of the adsorbed molecule was optimized at theB3LYP/6-31G* level, keeping frozen the coordinates of the gold atoms.

214709-3 Inelastic electron tunneling spectroscopy J. Chem. Phys. 125, 214709 �2006�


B. Linear symmetric tight binding chain

We consider a linear chain of atoms with one orbital �j�per site j, each interacting with its neighbor. Vij = �i�H�j� isthe coupling matrix element between adjacent sites i and j.We assume that there is one molecular symmetry element�for example a C2 rotation axis parallel to the electrodes� thatexchanges left and right electrode-molecule interfaces. Onlythe first ��1�� and last ��N�� atoms of the chain interact, re-spectively, with the left and right electrodes, so that the elas-tic conductance for this system is

gel�E� = gc�11L �E��NN

R �E��G1N�E��2, �12�

and the intensity of the inelastic peak due to mode � is

W� = gc�11L �E��NN

R �E�1

2� �G1N�E�

�Q��2

. �13�

If the system has a symmetry element S that interchanges �1�and �N�, i.e., S�1�= �N�, it is easy to verify that only totallysymmetric vibrational modes lead to non-null �G1N�E� /�Q�

matrix elements. In fact, building the symmetrized combina-tions ��+�= �1�+ �N� and ��−�= �1�− �N� the matrix element ofinterest can be written as34

�1��G�E�/�Q��N� = 14 ��+��G�E�/�Q��+�

− ��−��G�E�/�Q��−�� 14�

that is non-null only if Q� transforms as the totally symmet-ric representation, because G transforms as the totally sym-metric representation �the general case with more than onesymmetry element relating �1� and �N� is discussed below�.

It is convenient to confirm this finding in a more intui-tive manner. Using the Dyson equation35 we express the ma-trix element G1N�E� and �G1N�E� /�Q� as

G1N�E� =�E − E1�

�E − E1 − �11L ��E − EN − �NN

R � �j=1,N−1

Vj,j+1�Q��E − Ej�

,

�15�

�G1N�E��Q�

= j=1,N−1

�Vj,j+1�Q��Q�

G1N�E�Vj,j+1�Q�

. �16�

The elements of the summation in Eq. �16� corresponding tosymmetry related j , j+1 couples are identical if the mode Q�

transforms as the totally symmetric representation while theyare opposite in sign �and cancel each other� if the mode is nottotally symmetric. For example, the following chart consid-ers an idealized symmetric molecule described by fouratomic orbitals. The arrows exemplify a symmetric �Qs, �a��and antisymmetric �Qa, �b�� mode obtained by combining themotion of atoms 2 and 3 along their bond.

In the evaluation of Eq. �14� for the symmetric mode Qs,the symmetry related terms containing �V12/�Qs and�V34/�Qs are identical and add up. The evaluation of Eq.�14� for the antisymmetric mode gives zero because the sym-metry related terms containing �V12/�Qa and �V34/�Qa areopposite in sign and cancel �the term containing �V23/�Qa iszero in the antisymmetric case�.

C. Branched symmetric tight binding chain with onesite in contact with the electrodes

The previous observation holds also if the chain of at-oms contains branches or loops, as long as there is a singleatomic orbital interacting with each electrode as in the four

FIG. 3. Using the HS–CHvCH2–CH2vCH–SH molecule as an example, the symmetry operations that help define the propensity rules are illustrated.



examples below �the electrode in contact with atoms 1 and 6is omitted�:

Each node of the graphs above represents an atomic or-bital and connected nodes correspond to interacting orbitals.If there is more than one path connecting sites 1 and N �con-ventionally in contact with the electrode� the matrix elementG1N�E�, evaluated via the Dyson equation, is similar to thatof Eq. �15� except that a summation over all the possiblepaths from �1� to �N� needs to be included:

G1N�E� = paths P

� �E − E1��E − E1 − �11

L ��E − EN − �NNR �

�j=1,M

VP�j�,P�j+1��Q�

�E − EP�j�� =

paths P

�G1NP �E�� . �17�

P�j� is the jth element of a vector containing the ordered listof neighboring atoms from 1 to N, i.e., it describes a stepalong a specific tunneling path. We have indicated withG1N

P �E� the contribution to Green’s function matrix elementgiven by the path P. Considering for example graph �e�above, the most important paths are 1-2-5-6 and 1-2-3-4-5-6.Assuming a similar coupling between neighbors, the formerhas a much larger �G1N

p �E�� and dominates the transport. Thederivative with respect to the nuclear coordinates is

�G1N�E��Q�

= paths P

� j=1,M

�VP�j�,P�j+1��Q�

�Q�

G1NP �E�

VP�j�,P�j+1��Q�� 18a�

= paths P

� j=1,M

G1P�j�P �E�

�VP�j�,P�j+1��Q�

�Q�

GP�j+1�NP �E�� .

�18b�

This example demonstrates the expected result that if anelectron path for the tunneling is more favorable, the vibra-tions that modulate it are also more effective in mediating theinelastic tunneling. In fact, a vibration Q� that mostly modu-lates a given coupling Vkl, i.e., with a large �Vkl�Q� /�Q�, willaffect the matrix element �G1N�E� /�Q� only if there is a pathP with large G1N

P �E� that contains the coupling Vkl. For thepractical point of view, side groups attached to a molecularwire give weaker signals in IETS while the most importantsignals derive from the vibrations that influence the mostfavorable transmission path. Considering chart 2, the vibra-

tional modes sketched in �d� and �e� are not likely to givestrong signals in the IETS measurements because they do notinfluence the main tunneling path. Experimental evidence insupport of this conclusion comes from the comparison of theIETS measurement of dithiolates,14 where the C–H groupsbehave as “side group” and give a negligible C–H stretchingsignal at �3000 cm−1, with the IETS measurement ofmonothiolates,15 where the −CH3 �terminal� group is alongthe most favorable transmission path and the C–H stretchingsignal is very intense. Further experiments are certainly nec-essary for a more quantitative assessment of the role of sidegroups in IETS measurements.

D. Symmetric tight binding chain with more than onesite in contact with the electrodes

In this case more than one Green’s function matrix ele-ment should be computed following Eq. �10�:

g�E� = gc m�Ln�R

�mmL �E��nn

R �E�� paths P

�GmnP �E��2

. �19�

The conductance is the sum of the �always positive� contri-butions of each tunneling channel m-n, each resulting fromthe interference of the terms Gmn

P �E� connecting the elec-tronic states m and n, on the left and right electrodes throughthe coupling path P. The inelastic tunneling peak has inten-sity

W� = gc m�Ln�R

�mmL �E��nn

R �E�1

2� paths P

� �GmnP �E�

�Q� ��2

.

�20�

It is easy to verify that the existence of more than one tun-neling channel also makes the contribution of antisymmetricmodes non-null. In chart 3, symmetric and antisymmetriccombinations are shown for a system with two sites in con-tact with each electrode. While the coupling through the�symmetric� path 1-2-3-4 is modulated only by the symmet-ric modes �for the same reason valid for the �a� and �b�examples�, the coupling through the asymmetric path 1-2-3-5is modulated by both symmetric and antisymmetric modes.

More formally, in a system with only one molecularsymmetry element that interchanges the orbitals interactingwith the right and left electrodes, if the orbitals m and n incontact with the electrodes are not related by a symmetryoperation, the matrix element �Gmn�E� /�Q� is in generalnon-null regardless of the symmetry property of Q�. Al-though this lack of selection rules in the general case mightseem distressing, it is very often the case that one tunnelingchannel dominates the transport because one m-n orbitalcouple is related to a much larger matrix element Gmn�E�. As



seen in Eq. �18� the larger inelastic tunneling matrix ele-ments �Gmn�E� /�Q� are associated with larger Gmn�E� sothat the inelastic intensity Wa can be considered as originat-ing from a single tunneling channel. When this happens andif the orbitals m and n are related by a symmetry element thetotally symmetric modes will dominate strongly the IETSsignal. Next, we will see why this situation is relatively com-mon for molecular wires.

E. Planar conjugated molecules

An important subset of molecules studied in molecularelectronics36–38 consists of planar conjugated systems in con-tact with the electrodes through a S–Au link. We assumethat, together with the symmetry plane that contains the mol-ecule, there is a symmetry element that interchanges the twoelectrode-molecule interfaces, so that the point group of theoverall molecular system is C2h or C2v �see Fig. 3�. We willdiscuss only the C2h point group in the following �because itis more common� but the results can be easily transferred tothe �isomorphic� C2v group. To an excellent degree of ap-proximation the only important matrix elements for the elas-tic and inelastic channels, Gmn�E� and �Gmn�E� /�Q�, are theones where m and n are orbitals localized on the sulfur atomin contact, respectively, with the left and right electrodes.The m and n orbitals may be either symmetric ��-type� orantisymmetric ��-type� with respect to the molecular sym-metry plane.39 Group theory can be used to determine thesymmetry of the vibrational modes that give non-null�Gmn�E� /�Q� matrix elements �details are given in AppendixA�. It is convenient to distinguish three cases:

�i� When both m and n are �-type orbitals, the vibrationsthat transform as the ag or bu representation give non-nullmatrix elements. However, if m and n are symmetry relatedorbitals interchanged by a C2 axis only the totally symmetricmodes give a non-null Green function derivative. If a mol-ecule interacts with the electrodes by a S–Au contact, onlytwo symmetry related � orbitals interact with the electrode,corresponding to the pz valence orbital of the sulfur atom.40

Consequently only ag modes modulate the �Gmn�E� /�Q� ma-trix elements between equivalent orbitals. Since the channelthrough the � system is the dominating channel, the ag vi-brations �that more effectively modulate the tunneling matrixelements� are expected to be the most intense in an IETSmeasurement ag vibrations in planar conjugated moleculesare in plane motions involving C–C and C–H stretchings andC–C–C and C–C–H bendings. It is expected that C–Cstretching and C–C–C bending combinations would give thestrongest contribution to the IETS spectra.

�ii� When m and n are both �-type orbitals, the modestransforming as the ag or bu representations lead to non-null�Gmn�E� /�Q� matrix elements. However, the contribution ofthe � channel to the total current is small �and decreases asthe size of the conjugated molecule increases� so that theinelastic peaks related to the � tunneling channel are ex-pected to be very weak.

�iii� When m is of � type and n of � type �or vice versa�the vibrational modes that modulate the matrix elements arethose responsible for the �-� mixing, i.e., out of plane

modes that transform as the au or bg representation. To esti-mate the intensity of these modes we can consider again thecase of a linear chain of atoms, but this time we assume thateach site has one � orbital and several other � orbitals, as fora generic conjugated molecule. The IETS matrix elementbetween a � orbital on one end ��1�� and a � orbital on theother end ��N�� of the molecule is composed of contribu-tions of the form

G1�,P��j�P �E�

�VP��j�,P��j+1��Q�

�Q�

GP��j+1�,N�P �E� , �21�

where P��j� is a � orbital and P��j+1� a � orbital alongone particular coupling path �see Eq. �18b��. Each term likethe one in Eq. �21� corresponds to a path of the tunnelingelectron divided in two portions, one through the � orbitalsfrom 1� to P��j� and another through the � orbitals fromP��j� to N� �while the term �VP��j�,P��j+1��Q� /�Q� is re-sponsible for the �-� mixing�. It can be argued that thesecontributions are larger for conjugated systems when thelongest part of the path is through the � system, i.e., whenthe P�j� orbital is very close to the electrode. This argumentsuggests that, among the out of plane vibrations, the mostactive in IETS are the ones with the largest component onatoms close to the electrode.

F. Summary

The considerations of this section can be summarized asgiving the following propensity rules for IETS:

�i� If there is a single tunneling channel, only the totallysymmetric modes are active in the IET spectroscopy,or equivalently, the greater is the dominance of asingle tunneling channel the stricter is the propensityrule that favors totally symmetric modes.

�ii� Vibrations that do not modify the interelectrode cou-pling along the main tunneling paths are less active inIETS. For example, the modes that involve mostlyside-group motions of a molecular wire are less activein IETS. Vibrations localized on atoms that do notcontribute to the important tunneling paths are alsoless active in IETS.

�iii� If a � system is present, the largest contribution to theIETS is given by the totally symmetric vibrations thatmodulate the interaction between � orbitals �ag

modes in C2h systems�. The contribution of the out ofplane modes �au or bg in C2h� can be important whenthe tunneling path involves orbitals of � type at oneend and � at the other. Then it is largest for modeswith a stronger component close to the electrodes.The bu modes are the least intense, visible only if theconjugation is not too extended. Table I presents asummary view of the propensity rules.

IV. REALISTIC EXAMPLES

We present a small set of accurate calculations of theIETS spectra to verify the validity of the propensity rulessuggested in the previous section. We will consider two



groups of molecules belonging to the C2h group and depictedin Figs. 4 and 5. A series of conjugated polyenes, terminatedwith thiol groups, is studied to investigate systematically theeffect of conjugation length on the IET spectra. We com-

puted also the IET spectra of benzene dithiolate and twomolecular wires of the type prepared by Tour’s group, con-taining the phenyl ring parasubstituted with ethynyl groups.The second set of molecules is more similar to that investi-

TABLE I. Summary of the principal propensity rules for IETS in a metal/molecule/metal junction.

Symmetry of the m-ncouple with respect tothe molecular plane

Are the m and norbitals relatedby the C2 axis?

Active modesaccording togroup theory Other considerations

Modes seenexperimentally

�-� Yes Ag Very weak because ofenergy denominator

NoneNo Ag, Bu

�-� Yes Ag There is only one �orbital per sulfur �pz�,so m and n must besymmetry related

Ag

No Ag, Bu

�-� No �cannot be� Au, Bg Au, Bg

FIG. 4. Simulated spectra of three all-trans polyenedithiolates �showed at the left of each spectrum� in the60–1900 cm−1 range. Peaks in red, blue, and cyan cor-respond to vibrational modes of symmetry ag, au, andbg, respectively. The peaks deriving from the bu vibra-tions are not observed for any of these molecules. Theline length is proportional to the calculated intensity.



gated experimentally in Ref. 41. Detailed �and satisfactory�comparisons between some experimental measurement andcalculations performed with the method described abovehave been presented elsewhere.27 Here we will use this set ofmolecules to discuss the general patterns arising from thecomputed IET spectra and to validate through accurate nu-merical calculations the propensity rules derived from thequalitative arguments of the previous section.

A. Computational details

Most of the computational details are similar to thosepresented in Ref. 27. We improved the computation of theself-energy and the metal-molecule interaction using an ap-proach close to that presented in Ref. 42 or 43 Three differ-ent kinds of calculation have been performed to computeseparately �i� the isolated molecule properties, �ii� themolecule-electrode interaction, and �iii� the electrode localdensity of states �see also Fig. 1�:

�i� Isolated molecule computations are employed to com-pute the vibrational frequencies of the molecule andthe matrix elements Gnm�E� and �Gnm�E� /�Q� fromfirst principles at the 6-31G*/B3LYP level. We havetherefore not included the effect of the gold substrateon the molecular vibrations. Calculations have beenperformed on the indicated compounds with–S–H ter-minations instead of the–S–Au �electrode�.44

�ii� To compute the electrode-molecule interaction, weperformed a cluster calculation �again at the B3LYPlevel� on a model system made by nine gold atomsand a benzene thiolate. We assumed that the S atomsare in a fcc-like adsorption site, and that the S–C bondis perpendicular to the surface �see Fig. 2�. We used aS–Au distance of 2.85 Å. We used the 6-31G* basisset for the organic part of the cluster and the“Lanl2mb” basis set45,46 and pseudopotential for thegold atoms. This cluster was used to compute theHamiltonian matrix elements between the S and theAu atomic orbitals. These S–Au matrix elements areassumed to be similar for all investigated systems.This approximation is plausible since we consideronly molecules where the sulfur is connected to ansp2 or sp carbon of a conjugated system.

�iii� The local density of states �LDOS� at the Fermi en-ergy on the gold surface is needed, together with thematrix element computed in �ii� to obtain the � ma-trices of Eqs. �1� and �7� according to

�ijL = 2�

k

VkiVkj* LDOS�EF�k. �22�

Here k is an index running on the molecular orbitalson the left electode, i and j correspond to orbitals on

FIG. 5. Simulated IET spectra of three conjugated com-pounds �shown on the left� in the 60–2500 cm−1 range.The same color code as in Fig. 4 is used here to labelthe symmetry of the vibrational modes associated witheach peak. bu vibrations �in green� are observed onlyfor the shortest molecule of the series �see text�.



the molecule, and the Vik are the couplings betweenthese two groups of orbitals computed in step �ii�. TheLDOS was computed using a tight binding �TB�model for metallic gold and a very large cluster ofatoms to simulate the LDOS on the Au�111� gold sur-face. The TB parameters were obtained from firstprinciples �B3LYP� using small cluster calculationswith the same pseudopotential and basis set used forgold in the calculation of the molecule-metalinteraction.42

The approach outlined above is quite general, and itsaccuracy can be systematically improved by increasing theportion of the system considered in �i� as the “molecule” toinclude a few atoms from the electrode. It is, however, verydifficult to perform a frequency analysis on a cluster contain-ing a molecule and few metal atoms because the optimizedstructures are very far from the actual structure at the inter-face. Here we took the simplest route, neglecting the effectof the electrode on the molecular vibrations and excludingfrom the calculation the vibrations of the Au–S interface. Thesymmetry considerations we are focusing on are not affectedby this approximation.

We report here the computed spectra, with each moderepresented as a vertical bar on the energy axis with heightproportional to its intensity. The effect ofbroadening5,16,24–27,41 will not be considered. All modes re-lated to the CH stretching are not very active �in agreementwith propensity rule �ii��, and the spectra are only displayedin the low frequency ��2500 cm−1� range. The position ofthe electrode Fermi level with respect to the molecular levelswas set to −3.8 eV, a value intermediate between the highestoccupied molecular orbital �HOMO� and lowest unoccupiedmolecular orbital of all considered systems. We observedpreviously23 that the exact value of EF does not alter thestructure of the computed spectra if changed within a reason-able range �the IETSs are essentially independent of thechoice of EF when its distance from the HOMO and LUMOis larger than �0.4 eV�.

B. Results

The simulated spectra of the all-trans polyene dithiolateseries are collected Fig. 4. We used different colors to indi-cate signals arising from vibrations belonging to differentirreducible representations, as described in the figure caption.In Sec. II, qualitative arguments led to the propensity rulethat the totally symmetric �ag� vibrations would be the mostimportant in the IETS spectra, followed by the vibrationsbelonging to the au or bg representation. We predicted thatthe weakest contribution would be given by the bu vibrations,noting that the validity of these propensity rules increaseswith the length of the conjugated portion of the molecule�increased dominance of the � channel�. The simulated spec-tra of octatetraene and hexatriene dithiolates reported in Figs.4�a� and 4�b� confirm the validity of the predictions. The bu

transitions are effectively forbidden for all the polyenes con-sidered here �more than 30% of all modes are of bu type�.This trend confirms the reasoning in the previous section:there are no strict selection rules, but some definite propen-sity rules appear when one channel dominates the transport.Since the au or bg vibrations correspond to out of planemodes that are usually at a lower frequency than the ag �inplane� combination of stretching and bending, the spectraappear separable into two regions: the low frequency region�0–800 cm−1� where the au and bg transitions are seen andthe high frequency region �800–2000 cm−1� dominated bythe ag modes.

Figure 5 collects the simulated spectra of the aromaticmolecules studied in the C2h group.47 The spectra of thesemolecules obey the same trend seen for the polyenes, with ahigh frequency region of ag transitions and a low frequencyregion where the au and bg are seen. Transitions of bu sym-metry appear in the simulated spectra of the benzene dithi-olate, the shortest molecule considered here, evidently be-cause the transport �and the inelastic channel� through the �

orbitals of the systems is not completely negligible for veryshort molecules. Several complications make the low fre-quency region of the spectra more difficult to assign: theregion has a higher density of modes and their energy can beseriously affected by the approximation that the electrodes

FIG. 6. Comparison of the calculated and observed�Ref. 15� IETSs for the phenylene-ethynylene trimerdithiol. The calculation included Gaussian broadeningof 100 cm−1, but the intensities and frequencies are di-rectly calculated in the isolated molecule using DFT�the frequencies here have been scaled by the recom-mended factor of 0.961 for B3LYP calculations�. Thebar lines show the unbroadened calculation results: thepeak heights are proportional to the calculated intensi-ties, and their color indicates the symmetry with thesame code as in Fig. 4 and 5.



do not affect the molecular vibrations. Moreover at low biasthe recorded spectra suffer from a still unexplained feature�the so-called zero-bias feature48 �ZBF�� that covers the in-elastic tunneling signal at the lowest frequencies. On theother hand the high frequency region seems to be dominatedby very few modes so that many features of the spectra canbe assigned very confidently and rationalized well by thepropensity rules.

Extensive comparisons with experimental IETS will begiven elsewhere. Figure 6 demonstrates, by direct compari-son, that the current perturbative formalism indeed accu-rately calculates IETS for single-molecule junctions and thatthe propensity rules developed here work. In this case thefrequencies have been scaled by the recommended factor�0.961� for this level of computational theory49 and an em-pirical Gaussian broadening of 100 cm−1 was applied to eachpeak. The possibility of predicting from first principles thebroadening of IETS signals is discussed in Ref. 50.

C. Generalization and conclusion

The discussion above covers most situations that can befound in practical metal/molecule/metal junctions. The gen-eralization to different symmetry groups is straightforward.One has to consider the orbitals localized close to both elec-trodes and use group theory to evaluate which pairs of orbit-als on the left and right electrodes are coupled by whichnormal modes, using reasoning similar to that in AppendixA. Computation or intuition can suggest which orbital pairsmight give larger contributions to the conductance and to theinelastic channel. In fact, the most intense modes in IETS areassociated with the vibrations that most effectively modulatethe overall coupling among orbitals in contact with the elec-trodes and are connected by an efficient tunneling path at theFermi energy.

The IETS spectrum can yield structural informationabout the molecule in the junction. The propensity rules de-scribe modifications that may be observed if the moleculewere to isomerize, as well as differences that will occur fordiffering interfacial geometry or binding �� or �, single ormultiple site�.

We have shown that is possible to define propensity rulesfor IET spectroscopy that could guide the assignment of ex-perimental spectra. Such rules are more strictly obeyed forlarger conjugated molecules while it is more probable thatthe interpretation of the results for very small junctions willrequire accurate computational modeling. These simple rulescould contribute to an intuitive understanding of the experi-mental IETS measurements, one of the prerequisites for its

broader experimental use. Further joint experimental andtheoretical work may be necessary for the clarification ofseveral aspects of IETS.

ACKNOWLEDGMENTS

We are grateful to the Research Council UK, to theMOLEAPPS program of DARPA, to the NASA URETI pro-gram, to the Northwestern MRSEC, to the NSF InternationalDivision, and to the DoD MURI/DURINT program for sup-port of this research.

APPENDIX A: NON-NULL MATRIX ELEMENTS FORJUNCTIONS BELONGING TO THE C2h POINTGROUP

We consider the generic atomic orbitals �siL� and �si

R� of �type closer, respectively, to the left and right electrodes andthe atomic orbitals �pi

L� and �piR� of � type also closer to the

left and right electrodes, respectively. We assume that the

L-R pairs are related by the C2 symmetry element, i.e.,

C2�siL�= �si

R� and C2�piL�= �pi

R�, and we define the symmetrizedcombination:

�si+� = �si

L� + �siR� �Ag� , �A1a�

�si−� = �si

L� − �siR� �Bu� , �A1b�

�pi+� = �pi

L� + �piR� �Au� , �A1c�

�pi−� = �pi

L� − �piR� �Bg� �A1d�

�in parentheses we have indicated the irreducible representa-tion of the symmetrized combination�. The matrix elements

of the operator O between two orbitals localized on oppositeends of the molecule are

�siL�O�sj

R� = 14 ��si

+�O�sj+� − �si

+�O�sj−� + �si

−�O�sj+�

− �si−�O�sj

−�� , �A2a�

�piL�O�pj

R� = 14 ��pi

+�O�pj+� − �pi

+�O�pj−� + �pi

−�O�pj+�

− �pi−�O�pj

−�� , �A2b�

�siL�O�pj

R� = 14 ��si

+�O�pj+� − �si

+�O�pj−� + �si

−�O�pj+�

− �si−�O�pj

−�� . �A2c�

The symmetry properties of Eqs. �A1� imply that

�siL�O�sj

R� � 0 if O transforms as the Ag or Bu representation,

�piL�O�pj

R� � 0 if O transforms as the Ag or Bu representation,

�siL�O�pj

R� � 0 if O transforms as the Au or Bg representation.



However, in the case of S–Au termination there is not morethan one contributing �-type orbital on the left and rightsulfur atoms, and the only relevant matrix elements are of

type �piL�O�pi

R�, that are non-null only if O transforms as thetotally symmetric representation.

For the applications in this paper it is sufficient to recallthat Green’s function operator transforms as the totally sym-metric mode, and the operator �G�E� /�Q� transforms as themode � does.

APPENDIX B: THE INTENSITIES OF IETS PEAKS

The integral below one peak of an IET spectrum occur-ring at V=�� /e is simply the difference between the con-ductance measured for frequencies above and below the peak��V is an arbitrary small voltage interval�. In fact,

W� = ��/e−�V

��/e+�V d2I

dV2dV

= dI

dV�

V=��/e+�V− dI

dV�

V=��/e−�V

= dI�inel

dV�

V=��/e+�V− dI�

inel

dV�

V=��/e−�V. �B1�

The last equality derives from the observation that only theinelastic conductance due to mode � changes around V=�� /e �if no other vibrational modes are active in the in-terval �� /e−�V¯�� /e+�V��.

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alessandro troisi and mark a. ratner- propensity rules for inelastic electron tunneling spectroscopy...

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