Gemma C. Solomon et al- Understanding Coherent Transport through π-Stacked Systems upon Spatial...

8/3/2019 Gemma C. Solomon et al- Understanding Coherent Transport through -Stacked Systems upon Spatial Dislocation

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Understanding Coherent Transport through -Stacked Systems upon Spatial Dislocation

Gemma C. Solomon,* Josh Vura-Weis, Carmen Herrmann, Michael R. Wasielewski, andMark A. Ratner

Department of Chemistry and Argonne-Northwestern Solar Energy Research (ANSER) Center,Northwestern UniVersity, EVanston, Illinois 60308-3113

ReceiVed: April 6, 2010; ReVised Manuscript ReceiVed: May 30, 2010

We study the electron transport through -stacked structures bound to metallic electrodes and in particularexamine both the energy dependence and the effect of spatial dislocations on the electronic transmission. Wecompare these results with the predictions that can be made regarding the behavior of these systems from amodel of the electronic coupling matrix elements derived from the splitting of monomer molecular orbitalsin the dimer structure. We show that whereas these models agree reasonably well for predictions of near-resonant transport in a small stack of substituted benzene molecules, the relationship between these twoapproaches is less clear in larger structures, where the mechanisms and pathways can become more complex.

1. Introduction

Electron transfer is a ubiquitous process in nature governinga vast variety of biological functions. These systems haveevolved to ensure that long-range electronic coupling ispreserved in the face of all of the fluctuations that occur withthe physical and chemical modifications of the surroundings.In these complex environments, the entire structure of the systemcan be used to facilitate charge transport with both through-bond and through-space interactions1 defining the paths betweenelectron donors and electron acceptors.2,3 The robustness withwhich these processes occur inspires work to design artificialsystems to perform the same functions, both biological mimicsand synthetic systems, simply using molecular components tocontrol electronic properties.

Whereas natural systems may transfer charge with great

efficiency, their complex chemical structures do not clearlyreveal the important structural aspects for controlling theelectronic coupling. As efforts continue to design optimalsynthetic systems, basic understanding of the nature of electroniccoupling and its manifestation in a wide range of structural unitswill be critically important. The nature of the electronic couplingthrough a given chemical structure will not only be determinedby the functional units within the molecule but also by theenvironment in which it is placed. The environment will controlmany factors, from symmetry lowering effects, to the energeticsof charge injection, to the dissipation of thermal energy anddephasing influences. All of these factors will influence thecharge transfer and transport characteristics, and the dominance

of any one factor may qualitatively change the behavior of amolecular unit in different systems.The charge transport properties of synthetic systems have

been studied extensively in the area of organic electronics4,5

where films or crystals of often small molecules constitute theconducting entity. Alternatively, molecular electronics hasemerged as the study of the electrical properties of singlemolecules bound between (often metallic) electrodes. Whereasthe observed properties do correspond to a single molecule ina conducting junction, the environmental effect of the electrodes

is crucial. Here we investigate one of the fundamental conse-quences of binding a molecule in a conducting junction, the

lowering of molecular symmetry from the presence of bindinggroups. In particular, we examine -stacked systems wherechemical intuition would suggest that maximal overlap wouldfavor maximal transport. We show that this is not always thecase when the specific effects of binding to electrodes are takeninto account and illustrate how rich the charge injection energydependence of the transport properties can be in these systems.We compare these results with calculations of the electroniccoupling based on the splitting of monomer orbitals in the-stacked dimer and show how and when these quantities arerelated. The article proceeds as follows: Section 2 gives anoutline of the background to both the role of-stacked systemsin electron transport and the theory used for molecular con-

ductance calculations; Section 3 presents the results first for asimple system of stacked benzene rings and then continues byincreasing both the size of the stack and the size of the stackedmolecules; finally, Section 4 discusses these results and presentssome conclusions.

2. Background

Electron Transfer and -Stacked Systems. The nature ofthe electronic coupling in -stacked systems has been a subjectof considerable interest for many years. These structures areobserved frequently, both in large biological molecules and inthe crystal structures of small conjugated systems. The ease with

which these structures form makes them a favorable motif forself-assembly and, consequently, a common element in designsfor synthetic devices.

The nature of the electronic coupling in -stacked systemshas been investigated in the context of electron transfer.6-10 Thealignment between the stacked units is critically important forcontrolling the magnitude of the coupling. Structural changesthat either translate9,10 or rotate6-8 the units with respect to eachother can rapidly take either the electron or hole coupling fromits maximum value to zero. Clearly then, the binding energy/geometry of the stacked structures plays an important role incontrolling whether the resulting systems will have either highor low amplitudes for electronic transport.

Part of the Michael R. Wasielewski Festschrift.* To whom correspondence should be addressed. E-mail: g-solomon@

northwestern.edu.

J. Phys. Chem. B 2010, 114, 1473514744 14735

10.1021/jp103110h 2010 American Chemical SocietyPublished on Web 06/16/2010


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This same sensitivity to orientation is reflected in the spincoupling properties of-stacked biradical systems. Whether theradical centers are effectively meta or para to each otheracross the stack determines whether a low-spin or high-spinground state is observed.11,12

Conversely, in molecular electronics -stacked systems havebeen investigated as a possible artifact, with -stacked structuresof multiple molecules being potentially misinterpreted as theconductance signature of a single molecule.13 Careful experi-

mental efforts have shown that -stacked molecular systemscan bridge junctions resulting in clearly measurable conduc-tance.14 Whether -stacked structures play an important role inmolecular conductance junctions by accident or by design,understanding the nature of the electronic coupling in thesesystems will be important for minimizing, maximizing, orswitching their contributions.

Here we will focus on simple systems, controlled with theprecision that theory allows, to probe the details of the electroniccoupling in a transport context. In particular, we focus on theeffect of spatial dislocation between the stacked units and howthis change modifies transport. Whereas the electronic couplingof either electrons or holes is the relevant quantity for modulat-

ing rates of electron transfer, the charge injection energy in atransport junction can additionally be varied over a considerablerange, either by changing the electrode materials or by the useof a gate electrode. Consequently, we will also examine theenergy dependence of the transport through dislocated -stackedsystems.

This article will proceed by examining the transport through-stacked systems considering only coherent tunneling throughthe molecules. In practice, dephasing effects and incoherenthopping may play a significant role in large -stacked structures;however, for the purposes of simply understanding the impactof symmetry breaking on the electronic coupling, we neglectthese effects.

In the coherent tunneling limit, the current (I) through a

molecule is often calculated with Greens function implementa-tions of the so-called Landauer approach15

where fL(E, V) and fR(E, V) are the Fermi functions for the leftand right electrodes, e is the magnitude of the charge on anelectron, h is Plancks constant, and T(E, V) is the energy- (E)and bias- (V) dependent transmission. Here we report only thetransmission characteristics that underlie the zero bias dif-

ferential conductance, thus fL(E, V))

fR(E, V), and plots willgive the injection energy (E) relative to a common Fermi energy(Ef) for the two electrodes, which is set to -5 eV throughout.Under these conditions, the current through the system isformally zero; however, the differential conductance may benonzero.

The -stacked systems considered here are shown in Figure1 and will be designated para and meta as a convenientshorthand to describe the manner in which the systems are boundto the electrodes. In all stacked systems, the molecules are notbound to both electrodes but rather one component of the stackbinds to one electrode and another component of the stack bindsto the other, with through-space interactions completing thetransport path through the supermolecule. Consequently, these

designations do not correspond to true para and meta substitutionbut rather give an indication of where the binding groups lie

on the uppermost and lowermost molecule in the stack whenviewed from above.

Theoretical Methods. The molecular structures were formedby optimizing the isolated component parts with densityfunctional theory (B3LYP/6-311G**) using Q-Chem3.1.16 Thestacked structures were formed by placing the component parts3.4 apart in all cases. The terminal hydrogen atoms werethen removed from the thiol groups, and the molecules werebound to the fcc hollow site of planar gold electrodes with thebinding distance taken from the literature.17 The binding anglebetween the S-C bond and the surface normal varied dependingon the nature of the system and was chosen to minimizeinteractions between the surface and any parts of the moleculeother than the terminal sulfur atoms. In real systems, theseinteractions may play a significant role in the overall transport

properties; however, for the purposes of this investigation, thecoupling properties are most clear when the interactions withthe electrodes are dominated by through-bond terms.

TransportcalculationswerethenperformedusinggDFTB. 18-22

This code is an implementation of a Greens functionapproach to calculating electron transport, which is used hereat the Landauer (coherent tunneling) level of theory. Theelectronic structure is calculated for the system of two goldelectrodes and a single molecule spanning the gap betweenperiodic boundary conditions used. In the benzene-basedsystems, a 6 6 gold atom unit cell was used for theelectrodes, whereas an 8 8 gold atom unit cell was usedin the case of the pyrene-based systems. In both cases, three

layers were included in each electrode. The system is thenpartitioned into the left and right electrodes, and an extendedmolecular component (including the real molecule andoptionally some number of electrode atoms) in the middle.In these calculations, no gold atoms are included in theextended molecule for the molecular influences in thetransmission characteristics to be highlighted.

To make this study directly comparable to previous work,10

we also calculated the binding energies and the electron andhole electronic coupling matrix elements by the same methodas in that work. These calculations were performed on theisolated molecules with the thiol end groups intact and noelectrodes. All calculations were performed with Q-Chem3.1using density functional theory with MO6-2X,23 an exchange-

correlation functional well suited for systems with nonbondedinteractions and the 6-31G* basis set. The binding energies were

I(V) )2eh-

dE[fL(E, V) - fR(E, V)]T(E, V) (1)

Figure 1. Range of model -stacked molecules considered here. Thesystems are designated meta or para as a convenient short-hand todescribe the manner in which the molecules are bound to the electrodes.

14736 J. Phys. Chem. B, Vol. 114, No. 45, 2010 Solomon et al.


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calculated as the difference in the total energies of the boundsystem and the isolated components

where Emonomer+ghost is calculated by including explicitly theatoms of one component of the stack and additional ghost basisfunctions placed at the locations of the component in the dimerto correct for basis set superposition error.24

Electronic coupling matrix elements VHOMO and VLUMO werecalculated as half the energy difference between the highestoccupied molecular orbital (HOMO) and the HOMO-1 andthe lowest unoccupied molecular orbital (LUMO) and theLUMO+1 dimer orbitals. This method is well-suited forsymmetric dimers where the pairs of dimer orbitals are simplylinear combinations of the monomers HOMOs and LUMOs;4,25,26

any admixture of other monomer orbitals will result in a morecomplicated molecular orbital spectrum for the stacked system,

and clear coupling matrix elements cannot be deduced so simply.We note that we have reported the absolute values of VHOMOand VLUMO as the electron transfer and transport propertiesdepend only on the coupling squared. The requirement of asymmetric system means that only certain types of dimers canbe calculated by this approach. In this instance, the groups usedto bind to the electrodes decrease the symmetry of the systemas a whole, and only a limited set of substitution patterns canbe calculated by this approach. These systems provide a usefulsubset, however, to compare the results from transport calcula-tions with prior work on electron transfer in -stacked systems.

3. Results

Basic Stacks. In practice, -stacked structures frequentlypossess some sort of additional functionality to hold the

conjugated components together in the presence of thermalfluctuations, such as hydrogen bonds in self-assembled structuresor saturated linkages in the example of cyclophanes. Here weneglect these components, taking some of the most simple-stacked structures imaginable: stacked benzene rings. Whereas

these structures seem unlikely to be stable under ambientconditions, they show a distinct resemblance to the structureshypothesized to dominate the measured conductance.14 Further-more, these calculations allow us to isolate the contribution tothe transport properties of the -stacked component in largerstable structures with additional bonded or nonbonded interac-tions stabilizing the stack. As long as systems are designed withthe idea that the -system will be the dominant conduit forcharge transfer, understanding the role of the -stacked com-ponent in the absence of the functional groups that stabilizethese structures is an important first step.

Previous work on isolated, stacked benzene rings showed thatthe symmetry breaking influence of coupling to electrodes meantthat fully eclipsed structures are not always optimal formaximizing transmission.27 In that case, two dislocated structureswere chosen where transport was increased as a result of theoverlap between selected sites being optimized. Here weillustrate the full range of the transmission characteristics as afunction of translation along the short (x) and long (y) axes ofthe molecule. For a para-substituted benzene stack, the resultsare shown in Figure 2, whereas for the meta-substituted stack,the results are shown in Figure 3.

The para-substituted stack shows a pronounced interferencefeature in the transmission near the Fermi energy at geometriesaround the eclipsed structure, akin to the low transmissionobserved frequently in meta-substituted systems. In the fullspatial map at the Fermi energy, local maxima in the transmis-

sion are clearly evident for various translations. In all cases,these translated structures maximize overlap between selected

Figure 2. Top: Transmission (right) at a series of geometries with translations (in angstroms) along the short and long axes of a para-substitutedbenzene -stacked system (left). Bottom: Full spatial maps of the transmission as a function of translation at 1.5 eV below the Fermi energy (left),at the Fermi energy of-5.0 eV (center), and 1.5 eV above the Fermi energy (right).

binding energy ) Edimer - 2Emonomer+ghost (2)

Coherent Transport through -Stacked Systems J. Phys. Chem. B, Vol. 114, No. 45, 2010 14737


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sites, either ortho or para to the binding groups, rather thanmaximizing the overlap between the rings as a whole. Theseselected positions are strongly coupled to the electrodes andconsequently allow for high levels of transport through thesystem as a whole.

The plots of the transmission at selected geometries give anindication of the migration of the interference feature acrossthe energy range with translation. The resonant peaks in thetransmission, around 2 eV below and 2.5 eV above the Fermienergy, decrease in amplitude upon translation; however, theirenergy location remains essentially unchanged. Conversely, thesharp interference minimum shifts by a significant amount evenwith a relatively small translation. For example, from the fullyeclipsed structure, a translation of 0.2 along the short andlong axis shifts the interference feature by 0.5 eV.

The transmission through the meta-substituted stack be-haves differently. It shows no interference features across a wideenergy range in the vicinity of the Fermi energy. The full spatialmap of the transmission at the Fermi energy is substantiallysimilar to that for the para-substituted system; however, theseemingly minor differences in the orientation of the featureswith respect to translations leads to a stark difference in theproperties of the fully eclipsed structures.

Away from the Fermi energy, specifically at 1.5 eV aboveand below the Fermi energy, the similarity between the spatialmaps of the transmission for the para and meta structuresis striking. The features in the transmission map for the metasystem are rotated 60, commensurate with the rotation of theunderlying benzene rings relative to the translation axes. Thecharacteristic shapes in the maps take their form from that ofthe HOMO and LUMO of the component parts of the stack, asshown in Figure 4. The nodal patterns in the molecular orbitals

are reflected in the regions of low transmission through thesystem, suggesting that electron and hole transport in these

systems are going to show different sensitivity to geometricfluctuations. In the case of hole-dominated transport, below theFermi energy, there is a large region around the fully eclipsedstructure characterized by high transmission. Conversely, thelarger number of nodes in the LUMO results in regions of lowtransmission above the Fermi energy, indicating that electrontransport is likely to be significantly more sensitive to geometricchanges in the stacking geometry of the molecule.

Near the Fermi energy, the spatial map of the transmission

is most clearly understood by considering the system as an evenalternate hydrocarbon.28,29 These types of molecules have theimportant property of orbital pairing,30 where the occupied andvirtual orbitals are symmetric about the midpoint of theHOMO-LUMO gap and each pair of orbitals differs only inthe relative phase on each atom. By marking every second atomthat participates in the system with a star and continuing acrossthe nearest-neighbor interactions in the stack, it can bedetermined whether the two sulfur atoms are both marked withstars (alike coupling) or only one will be (disjoint coupling).Orbital pairing and the absence of heteroatoms in the -systemmeans that around the midpoint of the HOMO-LUMO gapthese two types of coupling correspond to very differenttransport properties. Disjoint coupling will yield relatively high

levels of transport, whereas alike coupling results in a destructiveinterference feature that suppresses -system transport. The

Figure 3. Top: Transmission (right) at a series of geometries with translations along the short and long axes of a meta-substituted benzene-stacked system (left). Bottom: Full spatial maps of the transmission as a function of translation at 1.5 eV below the Fermi energy (left), at theFermi energy of-5.0 eV (center), and 1.5 eV above the Fermi energy (right).

Figure 4. Three illustrations of the properties of the molecules thatcomprise the stacks: the HOMO (left), the pattern of starring used to

define the coupling through alternate hydrocarbons (center), and theLUMO (right). These three aspects can be used to understand thetransport below, around, and above the Fermi energy, respectively.



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assignments for one component of the stacked structures areshown in Figure 4. For the fully eclipsed structures, there isalike coupling in the para system resulting in low levels of

transport near the Fermi energy, whereas the higher levelsthrough the meta system arise with the disjoint coupling. Asthe stacked structures are translated away from the fully eclipsedgeometry, the pairs of nearest-neighbors across the stack willchange, and consequently, the transmission through the systemchanges as well. Figure 5 illustrates how the starring assignmentacross the supermolecule changes with dislocation for a selectedset of systems; in cases where there is alike coupling, the topmolecule is colored red, and in cases where there is disjointcoupling, the top molecule is colored blue.

The binding energy and electronic coupling matrix elementscan also be calculated for the para system and are shown inFigure 6. The spatial maps ofVHOMO and VLUMO show all of thesame qualitative features as the transmission maps 1.5 eV aboveand below the Fermi energy. The transmission resonancesassociated with the molecular HOMO and LUMO are furtherfrom the Fermi energy, between 2 and 3 eV in both cases;however, it is clear that as soon as the transport is dominatedby these resonances it qualitatively follows the patterns that canbe seen in the orbital splittings of the dimers.

There is no analogous feature in the dimer splitting calcula-tions to account for the off-resonant behavior near the Fermienergy. It is clear from the transmission through both the paraand meta systems that the region around the Fermi energyfalls solidly in the overlapping tails of the transmission throughboth the HOMO and LUMO resonances. In this region, thecoupling and the transport are determined by the interaction of

these tails, and either constructive or destructive interferenceeffects can dominate the transport.31 Whereas the spatial map

of the transmission at the Fermi energy clearly reflects featuresof the transmission near the resonant peaks, the precise mix ofthese features and the way in which they manifest at select

locations, such as the fully eclipsed geometries, clearly cannotbe deduced from the form of the resonant maps. This is yetanother indication of why the nature of proximate moleculartransport resonances cannot necessarily be used to predict thebehavior in the off-resonant regime.

The binding energy approaches a local minimum withdisplacement along the short axis; in fact, it is likely that thetrue local minimum lies below the short axis (x) in the plotbecause this translation allows repulsions to be reduced. Thehigh-energy structures that result from larger displacementsalong the long axis (y) can be understood by appreciating thatthe thiol groups result in a dipole across the molecule and thelong axis translation results in a transition from a favorable

interaction to an increasingly unfavorable interaction betweenthese dipoles.Extending the Stacks. Part of the motivation for investigating

-stacked structures is the potential to transport charge overlong distances by utilizing self-assembled architectures of thiskind. The larger the structure, the more likely that incoherenttransport processes that dephase or localize charge will domi-nate.32 It is still interesting to examine whether the understandingof spatial characteristics of coherent transport illustrated in theprevious section can be extended to longer stacks. For thispurpose, we extend the stack with the addition of an unsubsti-tuted benzene ring between the units bound to the electrodes.Now, each translation corresponds to the top part of the stackmoving in the positive x and y directions and the lower part of

the stack moving by the same amount in the negative x and ydirections.

Figure 5. Top: Starring assignments for the fully eclipsed para benzene stack (top) and a series of dislocated structures with two sites eclipsedand one site eclipsed. In each case, the top molecule is colored red when there is alike (low) coupling and blue when there is disjoint (high)coupling.

Figure 6. Binding energy (left), VHOMO (center), and VLUMO (right) for the simple para stacked benzene system for comparison with Figure 2.



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Unsurprisingly, the transmission through the para- andmeta-substituted variants of these extended structures issignificantly attenuated, as shown in Figures 7 and 8. The

transport is now through two nonbonded layers. Additionally,for the meta system, the fully eclipsed structure is again

predicted by starring arguments to exhibit an interferencefeature near the Fermi energy, which it does. The interferencefeature in this system is now quite sensitive to geometry;

there can be significant through-space interactions betweencarbon and gold, bypassing the thiol group and resulting in

Figure 7. Top: Transmission (right) at a series of geometries with translations along the short and long axes of an extended para-substitutedbenzene -stacked system (left). Bottom: Full spatial maps of the transmission as a function of translation at 1.5 eV below the Fermi energy (left),at the Fermi energy of-5.0 eV (center), and 1.5 eV above the Fermi energy (right).

Figure 8. Top: Transmission (right) at a series of geometries with translations along the short and long axes of an extended meta-substitutedbenzene -stacked system (left). Bottom: Full spatial maps of the transmission as a function of translation at 1.5 eV below the Fermi energy (left),at the Fermi energy of-5.0 eV (center), and 1.5 eV above the Fermi energy (right).



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an increase in transport with the interference feature

disappearing.The features in the transmission maps, both at the Fermienergy and 1.5 eV above and below the Fermi energy, are lesspronounced than in the case of the smaller stacked systems.The form of the HOMO and the LUMO in the isolatedcomponents is identical to the smaller systems for the substitutedrings and largely similar for the unsubstituted system. Thewashed-out nature of transmission maps can be attributed to anumber of weak coupling paths becoming increasingly competi-tive as the overall transport through the system decreases. Thefact that considerable care is required to ensure that thetransmission is not dominated by through-space terms betweenthe electrode and the phenyl ring on the unsubsituted side ofthe top and bottom rings gives an indication of the sorts of low-

level coupling elements that can become significant. Asexpected, the actual transmission maxima (ignoring interferencedips) are substantially higher in the two-molecule stack of Figure2 than in the three-molecule stack of Figure 7.

There is a region in the transmission maps that exhibits aclear difference between the para and meta systems, thefeature at a long axis (y) shift of between 1.5 and 2 . For thepara system, this region is marked by a distinct dip inthe transmission across the entire energy range, whereas a peakis evident in the meta system. It should be noted that thisdifference between the structures arises not because of someimportant difference in the coupling through para- and meta-substituted systems but rather because of the disparity of the

definition of the long axis with relation to the orientation of thebenzene rings. In the case of the para system, this translationalong the long axis puts the para carbon atoms in the top andbottom rings directly above and below the center of the middlering. Conversely, in the case of the meta system, this sametranslation results in a carbon-carbon bond in one ring lyingdirectly above a carbon-carbon bond in the adjacent ring andtwo sets of carbon atoms with favorable overlap.

Larger-Conjugated Systems. The -stacked structures canalternatively be extended by increasing the size of the conjugatedcyclic structures that comprise the stack. This change increasesthe strength of the interaction between the components of thestack and the number of coupling pathways through the system.The two-component benzene systems were extended to pyreneto give an indication of the effect of translation on thetransmission through larger systems.

The increased system size also brings increased complexityto the molecular orbital landscape. Whereas the monomers nowhave significantly more -system orbitals, not all of them coupleto the thiol group used to bind the molecule to the electrodes.Four frontier orbitals for the thiol-substituted pyrene monomerare shown in Figure 9. Whereas all of these orbitals aredelocalized -system orbitals, only the HOMO and the LU-MO+1 couple to the thiol group. Furthermore, the signaturesof these orbitals do not all clearly appear as transmissionresonances, as shown in Figure 10.

The dominant transmission resonance (where the transmission

approaches unity in the fully eclipsed structure) lies 1.8 eVbelow the Fermi energy, almost exactly between the HOMO

and the HOMO-2 (at -7.09 eV) of the monomer. Thetransmission resonance above the Fermi energy lies around 2.3eV above the Fermi energy, which is below the LUMO+1 ofthe monomer. In a number of the dislocated structures, thereare several sharp features around the energy of the HOMO-1/HOMO and the LUMO in systems where the interferencefeatures approach these energies. Without strong coupling tothe electrodes, broad transmission resonances will never beobserved at these orbital energies; however, these orbitals

constitute an integral part of the molecular electronic structure,and their signatures can nonetheless be evident under the rightcircumstances. They instead appear as sharp resonances, par-ticularly when the residual off-resonant transmission throughthe tails of energetically proximate orbitals is disrupted bydestructive interference effects.

For the meta-substituted pyrene system, shown in Figure11, there are transmission peaks around the energy of themonomer HOMO and LUMO, although these peaks do notreach a maximum value of one. The interference feature presentnear the Fermi energy in the para-substituted system is alsoabsent; this behavior is a reflection of what was seen in thesmaller benzene systems. The different peak structure is a

distinctly new feature that emerges only in the larger system.The form of the molecular orbitals of the monomer would tendto indicate that the LUMO, in particular, should not feature atransmission resonance because there is no observable densityon the binding group. The same argument cannot clearly bemade for the resonance around the HOMO due to the neardegeneracy of the HOMO (with density on the binding group)and the HOMO-1 (with no density on the binding group).Evidently however, there is some particularly favorable interac-tion between the monomers in this meta orientation thatcompensates for the weak coupling between these orbitals andthe electrodes, resulting in the diminished but clearly visiblepeaks in the transmission.

This crude analysis of the transmission properties of the-stacked dimer in terms of the molecular orbitals of themonomer should serve as a caution against interpreting molec-ular transmission in this fashion. Whereas this is necessary ifany conclusions about transport properties are to be drawn fromelectronic coupling matrix elements derived from the splittingof monomer orbitals in the dimers, it can become irrelevant inmany circumstances when the orbital energy spacings are small.

The transmission maps 1.5 eV above and below the Fermienergy also show distinctly different patterns in the para andmeta systems. Given the similarity of the transmission mapsas the resonances were approached in the benzene dimersystems, this difference is perhaps surprising. Examining theelectronic matrix coupling elements for the para-substitutedpyrene dimer derived from the orbital splitting, as shown inFigure 12, provides insight into the origin of this difference.Around 1.5 eV above and below the Fermi energy, thetransmission maps through the meta system correspond tothose of the HOMO and LUMO splitting. Conversely, thetransmission maps through the para system correspond to thesplitting of the HOMO-1 and LUMO+1. In these largerstructures, the different alignment of the stacked elements withchanging substitution alters how favorable interactions acrossthe stack can be and consequently changes the relative contribu-tions of these elements.

Whereas these results provide a persuasive picture of thedifferences between the para and meta systems, one caveat

should be noted. The large energy splitting between themonomer orbitals (on the same order as the orbital energy

Figure 9. HOMO-1, HOMO, LUMO, and LUMO+1 of the pyrenemonomer; note that the gold Fermi energy is set to -5.0 eV.



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spacing in an isolated monomer) in these larger systems meansthat admixtures of -system orbitals in the dimer are highlyprobable. In particular, the electronic coupling matrix elementsdetermined for the HOMO and HOMO-1 monomer orbitals arelikely to comprise a more complicated range of interactions and

consequently cannot be rigorously interpreted in the usualfashion.

4. Conclusions

These relatively simple -stacked structures demonstratethe rich array of transport properties promised by sucharchitectures. The relationship between the electronic cou-

pling through the system and the relative orientation of thecomponents cannot be summarized in a single simple

Figure 10. Top: Transmission (right) at a series of geometries with translations along the short and long axes of an extended para-substitutedpyrene -stacked system (left). Bottom: Full spatial maps of the transmission as a function of translation at 1.5 eV below the Fermi energy (left),at the Fermi energy of-5.0 eV (center), and 1.5 eV above the Fermi energy (right).

Figure 11. Top: Transmission (right) at a series of geometries with translations along the short and long axes of an extended meta-substitutedpyrene -stacked system (left). Bottom: Full spatial maps of the transmission as a function of translation at 1.5 eV below the Fermi energy (left),at the Fermi energy of-5.0 eV (center), and 1.5 eV above the Fermi energy (right).



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statement, yet the behavior of each system studied here canbe simply understood.

In the off-resonant regime, the presence of destructiveinterference features can have a significant role in modulatingthe transport properties. Whether an interference feature willbe observed in this region depends on both the number of

molecules in the stack and the relative orientation of thesubstituents used to bind the stack to the two electrodes. Bothof these factors need to be considered and controlled to eithermaximize or minimize off-resonant transport through a -stackedsystem.

The off-resonant regime becomes readily accessible inmolecular electronics as the typical Fermi energies of metalelectrodes allow electrons to be injected into the system in thisrange. In the context of organic electronics, dissipation withinthe molecular film means that the energy range of interest willgenerally be much closer to the molecular resonances andconsequently dominated by different behavior. In this context,it is common to consider the splitting of monomer orbitals toobtain a measure of the electronic coupling matrix elements,and it is clear that in some systems, such as the benzene systemsstudied here, this will provide a clear indication of the near-resonant transport properties. When the systems became larger,the electronic coupling matrix elements that could be deducedby orbital splitting became ambiguous. The stacked pyrenesystems illustrated that even if the coupling deduced from orbitalsplitting might be clear, it is not always obvious which orbitalsare dominant in the transmission spectra.

Broadly, it can be said that there is clearly a relationshipbetween the transmission calculated through a -stacked su-permolecule bound to electrodes and the coupling through thesupermolecule calculated by the splitting of monomer orbitals.The precise nature of this relationship, however, is less clear.

The nature of these relationships could not have been clearlypredicted at the outset, which should serve as a severe caution

against predicting the transmission properties of -stackedsystems in terms of the properties of the isolated components.The transmission through a molecule is a complex, energy-dependent quantity that can involve both constructive anddestructive interference terms arising from different aspects ofthe electronic structure. Whereas a desire for models that are

both chemically intuitive and predictive encourages the develop-ment of minimal approaches to understand electronic transport,caution must be taken to ensure that the richness these systemsoffer is fully incorporated.

Acknowledgment. We would like to thank David Q.Andrews for helpful discussions. This work was partiallysupported by the ANSER Center, an Energy FrontierResearch Center funded by the U.S. Department of Energy,Office of Science, Office of Basic Energy Sciences, underaward number DE-SC0001059. C.H. would like to thank theGerman Research Foundation (DFG) for generous supportthrough a postdoctoral research fellowship.

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Figure 12. Calculations on the isolated para-substituted pyrene dimer. Top: Binding energy (left), VHOMO-1 (bottom), and VHOMO (right). Bottom:VLUMO (left) and VLUMO+1 (right). Note that VHOMO-1 and VHOMO cannot be considered to be reliable because the energetic proximity of variousmonomer orbitals makes admixtures likely in the dimer structure.



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