Understanding Coherent Transport through π-Stacked Systems upon

Jun 16, 2010 - We study the electron transport through π-stacked structures bound to metallic electrodes and in particular examine both the energy de...
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J. Phys. Chem. B 2010, 114, 14735–14744

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Understanding Coherent Transport through π-Stacked Systems upon Spatial Dislocation† Gemma C. Solomon,* Josh Vura-Weis, Carmen Herrmann, Michael R. Wasielewski, and Mark 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 π-stacked structures bound to metallic electrodes and in particular examine both the energy dependence and the effect of spatial dislocations on the electronic transmission. We compare these results with the predictions that can be made regarding the behavior of these systems from a model of the electronic coupling matrix elements derived from the splitting of monomer molecular orbitals in the dimer structure. We show that whereas these models agree reasonably well for predictions of nearresonant transport in a small stack of substituted benzene molecules, the relationship between these two approaches 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 governing a vast variety of biological functions. These systems have evolved to ensure that long-range electronic coupling is preserved in the face of all of the fluctuations that occur with the physical and chemical modifications of the surroundings. In these complex environments, the entire structure of the system can be used to facilitate charge transport with both throughbond and through-space interactions1 defining the paths between electron donors and electron acceptors.2,3 The robustness with which these processes occur inspires work to design artificial systems to perform the same functions, both biological mimics and synthetic systems, simply using molecular components to control electronic properties. Whereas natural systems may transfer charge with great efficiency, their complex chemical structures do not clearly reveal the important structural aspects for controlling the electronic coupling. As efforts continue to design optimal synthetic systems, basic understanding of the nature of electronic coupling and its manifestation in a wide range of structural units will be critically important. The nature of the electronic coupling through a given chemical structure will not only be determined by the functional units within the molecule but also by the environment in which it is placed. The environment will control many factors, from symmetry lowering effects, to the energetics of charge injection, to the dissipation of thermal energy and dephasing influences. All of these factors will influence the charge transfer and transport characteristics, and the dominance of any one factor may qualitatively change the behavior of a molecular 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 the conducting entity. Alternatively, molecular electronics has emerged as the study of the electrical properties of single molecules bound between (often metallic) electrodes. Whereas the observed properties do correspond to a single molecule in a conducting junction, the environmental effect of the electrodes †

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

is crucial. Here we investigate one of the fundamental consequences of binding a molecule in a conducting junction, the lowering of molecular symmetry from the presence of binding groups. In particular, we examine π-stacked systems where chemical intuition would suggest that maximal overlap would favor maximal transport. We show that this is not always the case when the specific effects of binding to electrodes are taken into account and illustrate how rich the charge injection energy dependence of the transport properties can be in these systems. We compare these results with calculations of the electronic coupling based on the splitting of monomer orbitals in the π-stacked dimer and show how and when these quantities are related. The article proceeds as follows: Section 2 gives an outline of the background to both the role of π-stacked systems in electron transport and the theory used for molecular conductance calculations; Section 3 presents the results first for a simple system of stacked benzene rings and then continues by increasing both the size of the stack and the size of the stacked molecules; finally, Section 4 discusses these results and presents some conclusions. 2. Background Electron Transfer and π-Stacked Systems. The nature of the electronic coupling in π-stacked systems has been a subject of considerable interest for many years. These structures are observed frequently, both in large biological molecules and in the crystal structures of small conjugated systems. The ease with which these structures form makes them a favorable motif for self-assembly and, consequently, a common element in designs for synthetic devices. The nature of the electronic coupling in π-stacked systems has been investigated in the context of electron transfer.6-10 The alignment between the stacked units is critically important for controlling the magnitude of the coupling. Structural changes that either translate9,10 or rotate6-8 the units with respect to each other can rapidly take either the electron or hole coupling from its maximum value to zero. Clearly then, the binding energy/ geometry of the stacked structures plays an important role in controlling whether the resulting systems will have either high or low amplitudes for electronic transport.

10.1021/jp103110h  2010 American Chemical Society Published on Web 06/16/2010

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This same sensitivity to orientation is reflected in the spin coupling properties of π-stacked biradical systems. Whether the radical centers are effectively “meta” or “para” to each other across the stack determines whether a low-spin or high-spin ground state is observed.11,12 Conversely, in molecular electronics π-stacked systems have been investigated as a possible artifact, with π-stacked structures of multiple molecules being potentially misinterpreted as the conductance signature of a single molecule.13 Careful experimental efforts have shown that π-stacked molecular systems can bridge junctions resulting in clearly measurable conductance.14 Whether π-stacked structures play an important role in molecular conductance junctions by accident or by design, understanding the nature of the electronic coupling in these systems will be important for minimizing, maximizing, or switching their contributions. Here we will focus on simple systems, controlled with the precision that theory allows, to probe the details of the electronic coupling in a transport context. In particular, we focus on the effect of spatial dislocation between the stacked units and how this change modifies transport. Whereas the electronic coupling of either electrons or holes is the relevant quantity for modulating rates of electron transfer, the charge injection energy in a transport junction can additionally be varied over a considerable range, either by changing the electrode materials or by the use of a gate electrode. Consequently, we will also examine the energy dependence of the transport through dislocated π-stacked systems. This article will proceed by examining the transport through π-stacked systems considering only coherent tunneling through the molecules. In practice, dephasing effects and incoherent hopping may play a significant role in large π-stacked structures; however, for the purposes of simply understanding the impact of symmetry breaking on the electronic coupling, we neglect these effects. In the coherent tunneling limit, the current (I) through a molecule is often calculated with Green’s function implementations of the so-called Landauer approach15

I(V) )

2e h

∫-∞∞ dE [fL(E, V) - fR(E, V)]T(E, V)

(1)

where fL(E, V) and fR(E, V) are the Fermi functions for the left and right electrodes, e is the magnitude of the charge on an electron, h is Planck’s constant, and T(E, V) is the energy- (E) and bias- (V) dependent transmission. Here we report only the transmission characteristics that underlie the zero bias differential conductance, thus fL(E, V) ) fR(E, V), and plots will give 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 is formally zero; however, the differential conductance may be nonzero. The π-stacked systems considered here are shown in Figure 1 and will be designated “para” and “meta” as a convenient shorthand to describe the manner in which the systems are bound to the electrodes. In all stacked systems, the molecules are not bound to both electrodes but rather one component of the stack binds to one electrode and another component of the stack binds to the other, with through-space interactions completing the transport path through the supermolecule. Consequently, these designations do not correspond to true para and meta substitution but rather give an indication of where the binding groups lie

Figure 1. Range of model π-stacked molecules considered here. The systems are designated “meta” or “para” as a convenient short-hand to describe the manner in which the molecules are bound to the electrodes.

on the uppermost and lowermost molecule in the stack when viewed from above. Theoretical Methods. The molecular structures were formed by optimizing the isolated component parts with density functional theory (B3LYP/6-311G**) using Q-Chem3.1.16 The stacked structures were formed by placing the component parts 3.4 Å apart in all cases. The terminal hydrogen atoms were then removed from the thiol groups, and the molecules were bound to the fcc hollow site of planar gold electrodes with the binding distance taken from the literature.17 The binding angle between the S-C bond and the surface normal varied depending on the nature of the system and was chosen to minimize interactions between the surface and any parts of the molecule other than the terminal sulfur atoms. In real systems, these interactions may play a significant role in the overall transport properties; however, for the purposes of this investigation, the coupling properties are most clear when the interactions with the electrodes are dominated by through-bond terms. Transport calculations were then performed using gDFTB.18-22 This code is an implementation of a Green’s function approach to calculating electron transport, which is used here at the Landauer (coherent tunneling) level of theory. The electronic structure is calculated for the system of two gold electrodes and a single molecule spanning the gap between periodic boundary conditions used. In the benzene-based systems, a 6 × 6 gold atom unit cell was used for the electrodes, whereas an 8 × 8 gold atom unit cell was used in the case of the pyrene-based systems. In both cases, three layers were included in each electrode. The system is then partitioned into the left and right electrodes, and an extended molecular component (including the real molecule and optionally some number of electrode atoms) in the middle. In these calculations, no gold atoms are included in the extended molecule for the molecular influences in the transmission characteristics to be highlighted. To make this study directly comparable to previous work,10 we also calculated the binding energies and the electron and hole electronic coupling matrix elements by the same method as in that work. These calculations were performed on the isolated molecules with the thiol end groups intact and no electrodes. All calculations were performed with Q-Chem3.1 using density functional theory with MO6-2X,23 an exchangecorrelation functional well suited for systems with nonbonded interactions and the 6-31G* basis set. The binding energies were

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Figure 2. Top: Transmission (right) at a series of geometries with translations (in angstroms) along the short and long axes of a “para”-substituted benzene π-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).

calculated as the difference in the total energies of the bound system and the isolated components

binding energy ) Edimer - 2Emonomer+ghost

(2)

where Emonomer+ghost is calculated by including explicitly the atoms of one component of the stack and additional ghost basis functions placed at the locations of the component in the dimer to correct for basis set superposition error.24 Electronic coupling matrix elements VHOMO and VLUMO were calculated as half the energy difference between the highest occupied molecular orbital (“HOMO”) and the HOMO-1 and the lowest unoccupied molecular orbital (“LUMO”) and the LUMO+1 dimer orbitals. This method is well-suited for symmetric dimers where the pairs of dimer orbitals are simply linear combinations of the monomers’ HOMOs and LUMOs;4,25,26 any admixture of other monomer orbitals will result in a more complicated 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 VHOMO and VLUMO as the electron transfer and transport properties depend only on the coupling squared. The requirement of a symmetric system means that only certain types of dimers can be calculated by this approach. In this instance, the groups used to bind to the electrodes decrease the symmetry of the system as a whole, and only a limited set of substitution patterns can be calculated by this approach. These systems provide a useful subset, however, to compare the results from transport calculations with prior work on electron transfer in π-stacked systems. 3. Results Basic Stacks. In practice, π-stacked structures frequently possess some sort of additional functionality to hold the

conjugated components together in the presence of thermal fluctuations, such as hydrogen bonds in self-assembled structures or saturated linkages in the example of cyclophanes. Here we neglect these components, taking some of the most simple π-stacked structures imaginable: stacked benzene rings. Whereas these structures seem unlikely to be stable under ambient conditions, they show a distinct resemblance to the structures hypothesized to dominate the measured conductance.14 Furthermore, these calculations allow us to isolate the contribution to the transport properties of the π-stacked component in larger stable structures with additional bonded or nonbonded interactions stabilizing the stack. As long as systems are designed with the idea that the π-system will be the dominant conduit for charge transfer, understanding the role of the π-stacked component in the absence of the functional groups that stabilize these structures is an important first step. Previous work on isolated, stacked benzene rings showed that the symmetry breaking influence of coupling to electrodes meant that fully eclipsed structures are not always optimal for maximizing transmission.27 In that case, two dislocated structures were chosen where transport was increased as a result of the overlap between selected sites being optimized. Here we illustrate the full range of the transmission characteristics as a function of translation along the short (x) and long (y) axes of the molecule. For a “para”-substituted benzene stack, the results are shown in Figure 2, whereas for the “meta”-substituted stack, the results are shown in Figure 3. The “para”-substituted stack shows a pronounced interference feature in the transmission near the Fermi energy at geometries around the eclipsed structure, akin to the low transmission observed frequently in meta-substituted systems. In the full spatial map at the Fermi energy, local maxima in the transmission are clearly evident for various translations. In all cases, these translated structures maximize overlap between selected

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Figure 3. Top: Transmission (right) at a series of geometries with translations along the short and long axes of a “meta”-substituted benzene π-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).

sites, either ortho or para to the binding groups, rather than maximizing the overlap between the rings as a whole. These selected positions are strongly coupled to the electrodes and consequently allow for high levels of transport through the system as a whole. The plots of the transmission at selected geometries give an indication of the migration of the interference feature across the energy range with translation. The resonant peaks in the transmission, around 2 eV below and 2.5 eV above the Fermi energy, decrease in amplitude upon translation; however, their energy location remains essentially unchanged. Conversely, the sharp interference minimum shifts by a significant amount even with a relatively small translation. For example, from the fully eclipsed structure, a translation of 0.2 Å along the short and long axis shifts the interference feature by ∼0.5 eV. The transmission through the “meta”-substituted stack behaves differently. It shows no interference features across a wide energy range in the vicinity of the Fermi energy. The full spatial map of the transmission at the Fermi energy is substantially similar to that for the “para”-substituted system; however, the seemingly minor differences in the orientation of the features with respect to translations leads to a stark difference in the properties of the fully eclipsed structures. Away from the Fermi energy, specifically at 1.5 eV above and below the Fermi energy, the similarity between the spatial maps of the transmission for the “para” and “meta” structures is striking. The features in the transmission map for the “meta” system are rotated 60°, commensurate with the rotation of the underlying benzene rings relative to the translation axes. The characteristic shapes in the maps take their form from that of the HOMO and LUMO of the component parts of the stack, as shown in Figure 4. The nodal patterns in the molecular orbitals are reflected in the regions of low transmission through the system, suggesting that electron and hole transport in these

Figure 4. Three illustrations of the properties of the molecules that comprise the stacks: the HOMO (left), the pattern of starring used to define the coupling through alternate hydrocarbons (center), and the LUMO (right). These three aspects can be used to understand the transport below, around, and above the Fermi energy, respectively.

systems are going to show different sensitivity to geometric fluctuations. In the case of hole-dominated transport, below the Fermi energy, there is a large region around the fully eclipsed structure characterized by high transmission. Conversely, the larger number of nodes in the LUMO results in regions of low transmission above the Fermi energy, indicating that electron transport is likely to be significantly more sensitive to geometric changes 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 even alternate hydrocarbon.28,29 These types of molecules have the important property of orbital pairing,30 where the occupied and virtual orbitals are symmetric about the midpoint of the HOMO-LUMO gap and each pair of orbitals differs only in the relative phase on each atom. By marking every second atom that participates in the π system with a star and continuing across the nearest-neighbor interactions in the stack, it can be determined whether the two sulfur atoms are both marked with stars (alike coupling) or only one will be (disjoint coupling). Orbital pairing and the absence of heteroatoms in the π-system means that around the midpoint of the HOMO-LUMO gap these two types of coupling correspond to very different transport properties. Disjoint coupling will yield relatively high levels of transport, whereas alike coupling results in a destructive interference feature that suppresses π-system transport. The

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Figure 5. Top: Starring assignments for the fully eclipsed “para” benzene stack (top) and a series of dislocated structures with two sites eclipsed and 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.

assignments for one component of the stacked structures are shown in Figure 4. For the fully eclipsed structures, there is alike coupling in the “para” system resulting in low levels of transport near the Fermi energy, whereas the higher levels through the “meta” system arise with the disjoint coupling. As the stacked structures are translated away from the fully eclipsed geometry, the pairs of nearest-neighbors across the stack will change, and consequently, the transmission through the system changes as well. Figure 5 illustrates how the starring assignment across the supermolecule changes with dislocation for a selected set of systems; in cases where there is alike coupling, the top molecule is colored red, and in cases where there is disjoint coupling, the top molecule is colored blue. The binding energy and electronic coupling matrix elements can also be calculated for the “para” system and are shown in Figure 6. The spatial maps of VHOMO and VLUMO show all of the same qualitative features as the transmission maps 1.5 eV above and below the Fermi energy. The transmission resonances associated with the molecular HOMO and LUMO are further from the Fermi energy, between 2 and 3 eV in both cases; however, it is clear that as soon as the transport is dominated by these resonances it qualitatively follows the patterns that can be seen in the orbital splittings of the dimers. There is no analogous feature in the dimer splitting calculations to account for the off-resonant behavior near the Fermi energy. It is clear from the transmission through both the “para” and “meta” systems that the region around the Fermi energy falls solidly in the overlapping tails of the transmission through both the HOMO and LUMO resonances. In this region, the coupling and the transport are determined by the interaction of these tails, and either constructive or destructive interference effects can dominate the transport.31 Whereas the spatial map

of the transmission at the Fermi energy clearly reflects features of the transmission near the resonant peaks, the precise mix of these features and the way in which they manifest at select locations, such as the fully eclipsed geometries, clearly cannot be deduced from the form of the resonant maps. This is yet another indication of why the nature of proximate molecular transport resonances cannot necessarily be used to predict the behavior in the off-resonant regime. The binding energy approaches a local minimum with displacement along the short axis; in fact, it is likely that the true local minimum lies below the short axis (x) in the plot because this translation allows repulsions to be reduced. The high-energy structures that result from larger displacements along the long axis (y) can be understood by appreciating that the thiol groups result in a dipole across the molecule and the long axis translation results in a transition from a favorable interaction to an increasingly unfavorable interaction between these dipoles. Extending the Stacks. Part of the motivation for investigating π-stacked structures is the potential to transport charge over long distances by utilizing self-assembled architectures of this kind. The larger the structure, the more likely that incoherent transport processes that dephase or localize charge will dominate.32 It is still interesting to examine whether the understanding of spatial characteristics of coherent transport illustrated in the previous section can be extended to longer stacks. For this purpose, we extend the stack with the addition of an unsubstituted benzene ring between the units bound to the electrodes. Now, each translation corresponds to the top part of the stack moving in the positive x and y directions and the lower part of the stack moving by the same amount in the negative x and y directions.

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Figure 7. Top: Transmission (right) at a series of geometries with translations along the short and long axes of an extended “para”-substituted benzene π-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”-substituted benzene π-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).

Unsurprisingly, the transmission through the “para”- and “meta”-substituted variants of these extended structures is significantly 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 interference feature near the Fermi energy, which it does. The interference feature in this system is now quite sensitive to geometry; there can be significant through-space interactions between carbon and gold, bypassing the thiol group and resulting in

Coherent Transport through π-Stacked Systems

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

an increase in transport with the interference feature disappearing. The features in the transmission maps, both at the Fermi energy and 1.5 eV above and below the Fermi energy, are less pronounced than in the case of the smaller stacked systems. The form of the HOMO and the LUMO in the isolated components is identical to the smaller systems for the substituted rings and largely similar for the unsubstituted system. The washed-out nature of transmission maps can be attributed to a number of weak coupling paths becoming increasingly competitive as the overall transport through the system decreases. The fact that considerable care is required to ensure that the transmission is not dominated by through-space terms between the electrode and the phenyl ring on the unsubsituted side of the top and bottom rings gives an indication of the sorts of lowlevel coupling elements that can become significant. As expected, the actual transmission maxima (ignoring interference dips) are substantially higher in the two-molecule stack of Figure 2 than in the three-molecule stack of Figure 7. There is a region in the transmission maps that exhibits a clear difference between the “para” and “meta” systems, the feature at a long axis (y) shift of between 1.5 and 2 Å. For the “para” system, this region is marked by a distinct dip in the transmission across the entire energy range, whereas a peak is evident in the “meta” system. It should be noted that this difference between the structures arises not because of some important 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 the benzene rings. In the case of the “para” system, this translation along the long axis puts the para carbon atoms in the top and bottom rings directly above and below the center of the middle ring. Conversely, in the case of the “meta” system, this same translation results in a carbon-carbon bond in one ring lying directly above a carbon-carbon bond in the adjacent ring and two sets of carbon atoms with favorable overlap. Larger-Conjugated Systems. The π-stacked structures can alternatively be extended by increasing the size of the conjugated cyclic structures that comprise the stack. This change increases the strength of the interaction between the components of the stack and the number of coupling pathways through the system. The two-component benzene systems were extended to pyrene to give an indication of the effect of translation on the transmission through larger systems. The increased system size also brings increased complexity to the molecular orbital landscape. Whereas the monomers now have significantly more π-system orbitals, not all of them couple to the thiol group used to bind the molecule to the electrodes. Four frontier orbitals for the thiol-substituted pyrene monomer are shown in Figure 9. Whereas all of these orbitals are delocalized π-system orbitals, only the HOMO and the LUMO+1 couple to the thiol group. Furthermore, the signatures of these orbitals do not all clearly appear as transmission resonances, as shown in Figure 10. The dominant transmission resonance (where the transmission approaches unity in the fully eclipsed structure) lies ∼1.8 eV below the Fermi energy, almost exactly between the HOMO

J. Phys. Chem. B, Vol. 114, No. 45, 2010 14741 and the HOMO-2 (at -7.09 eV) of the monomer. The transmission resonance above the Fermi energy lies around 2.3 eV above the Fermi energy, which is below the LUMO+1 of the monomer. In a number of the dislocated structures, there are several sharp features around the energy of the HOMO-1/ HOMO and the LUMO in systems where the interference features approach these energies. Without strong coupling to the electrodes, broad transmission resonances will never be observed 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 right circumstances. They instead appear as sharp resonances, particularly when the residual off-resonant transmission through the tails of energetically proximate orbitals is disrupted by destructive interference effects. For the “meta”-substituted pyrene system, shown in Figure 11, there are transmission peaks around the energy of the monomer HOMO and LUMO, although these peaks do not reach a maximum value of one. The interference feature present near the Fermi energy in the “para”-substituted system is also absent; this behavior is a reflection of what was seen in the smaller 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 tend to indicate that the LUMO, in particular, should not feature a transmission resonance because there is no observable density on the binding group. The same argument cannot clearly be made for the resonance around the HOMO due to the near degeneracy 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 interaction between the monomers in this meta orientation that compensates for the weak coupling between these orbitals and the electrodes, resulting in the diminished but clearly visible peaks in the transmission. This crude analysis of the transmission properties of the π-stacked dimer in terms of the molecular orbitals of the monomer should serve as a caution against interpreting molecular transmission in this fashion. Whereas this is necessary if any conclusions about transport properties are to be drawn from electronic coupling matrix elements derived from the splitting of monomer orbitals in the dimers, it can become irrelevant in many circumstances when the orbital energy spacings are small. The transmission maps 1.5 eV above and below the Fermi energy also show distinctly different patterns in the “para” and “meta” systems. Given the similarity of the transmission maps as the resonances were approached in the benzene dimer systems, this difference is perhaps surprising. Examining the electronic matrix coupling elements for the “para”-substituted pyrene dimer derived from the orbital splitting, as shown in Figure 12, provides insight into the origin of this difference. Around 1.5 eV above and below the Fermi energy, the transmission maps through the “meta” system correspond to those of the HOMO and LUMO splitting. Conversely, the transmission maps through the “para” system correspond to the splitting of the HOMO-1 and LUMO+1. In these larger structures, the different alignment of the stacked elements with changing substitution alters how favorable interactions across the stack can be and consequently changes the relative contributions of these elements. Whereas these results provide a persuasive picture of the differences between the “para” and “meta” systems, one caveat should be noted. The large energy splitting between the monomer orbitals (on the same order as the orbital energy

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Figure 10. Top: Transmission (right) at a series of geometries with translations along the short and long axes of an extended “para”-substituted pyrene π-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”-substituted pyrene π-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).

spacing in an isolated monomer) in these larger systems means that admixtures of π-system orbitals in the dimer are highly probable. In particular, the electronic coupling matrix elements determined for the HOMO and HOMO-1 monomer orbitals are likely to comprise a more complicated range of interactions and consequently cannot be rigorously interpreted in the usual fashion.

4. Conclusions These relatively simple π-stacked structures demonstrate the rich array of transport properties promised by such architectures. The relationship between the electronic coupling through the system and the relative orientation of the components cannot be summarized in a single simple

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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 various monomer orbitals makes admixtures likely in the dimer structure.

statement, yet the behavior of each system studied here can be simply understood. In the off-resonant regime, the presence of destructive interference features can have a significant role in modulating the transport properties. Whether an interference feature will be observed in this region depends on both the number of molecules in the stack and the relative orientation of the substituents used to bind the stack to the two electrodes. Both of these factors need to be considered and controlled to either maximize or minimize off-resonant transport through a π-stacked system. The off-resonant regime becomes readily accessible in molecular electronics as the typical Fermi energies of metal electrodes allow electrons to be injected into the system in this range. In the context of organic electronics, dissipation within the molecular film means that the energy range of interest will generally be much closer to the molecular resonances and consequently dominated by different behavior. In this context, it is common to consider the splitting of monomer orbitals to obtain a measure of the electronic coupling matrix elements, and it is clear that in some systems, such as the benzene systems studied here, this will provide a clear indication of the nearresonant transport properties. When the systems became larger, the electronic coupling matrix elements that could be deduced by orbital splitting became ambiguous. The stacked pyrene systems illustrated that even if the coupling deduced from orbital splitting might be clear, it is not always obvious which orbitals are dominant in the transmission spectra. Broadly, it can be said that there is clearly a relationship between the transmission calculated through a π-stacked supermolecule bound to electrodes and the coupling through the supermolecule 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 clearly predicted at the outset, which should serve as a severe caution

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