Article pubs.acs.org/JPCC
When Conductance Is Less than the Sum of Its Parts: Exploring Interference in Multiconnected Molecules Tim Hansen and Gemma C. Solomon* Nano-Science Center and Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark S Supporting Information *
ABSTRACT: We investigate the electron transmission through molecules with multiple connections to the leads and compare this with the transmission through the same molecules where only select connections have been made. This enables us to probe the transmission through the individual pathways through the molecules and investigate their interaction. Generally, we see that the transmission of the multiconnected molecules differs from those obtained from a sum of their parts because of coherence effects between the paths through the molecules. The only exception to this trend is a case where the molecule can be considered as two separate parts, isolated electronically from each other via meta connections. We also explore the local currents though these molecules and separate these into channels, which reveals how this coherence comes into play.
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INTRODUCTION Molecular electronics is a field that has been studied intensely for the past few decades and one that holds great promise, some of which has been realized. A challenge for the field is to embrace the complexity of molecular conductance and build new physical understandings, without simply falling back on old intuition. On one hand, it is a well-known fact that the molecules that are investigated are small enough for quantum effects to play a significant role in how a current flows through a molecule, while on the other hand, there is a great desire to use our intuition from traditional electronics to better understand the observed phenomena. The pervasiveness of this semiclassical intuition means that experiments demonstrating significant deviations hold special interest. Recently Vazquez et al.1 demonstrated that Kirchhoff’s law for conventional electronic circuits does not hold for molecules with two equivalent pathways. They showed that such molecules might exhibit constructive interference to have a greater total conductance than the sum of the individual conductances from each pathway. Another recent approach has been to study molecules with multiple connections to the electrodes.2,3 These molecules are interesting in that they allow for higher conductance, not being limited to a single atom at any point between the electrodes. We have chosen to investigate the later category and start from the same system as used in refs 2 and 3, namely benzene. In addition to the structures considered in the prior work, we also consider the situation when the molecule is turned 90°, and we compare the transmission through the fully connected molecule (or “spider”) with the two possible connections schemes with only one connection to each electrode (a “direct” and a “crossed”). The different connection schemes are obtained by rotations of the vinyl groups, which enable us to © XXXX American Chemical Society
have the sulfurs pointing toward or away from the gold electrodes. Based on the work by Marsda et al.,4 we also include fenestrane and annunenoannulene in our collection of test molecules, to have a reference to large molecules where the transmission cannot be said to be limited by a small central region. See Figure 1 for drawings of all of the molecules investigated and a diagrammatic representation of the different connections considered in this work. In this paper we compare the transmission of the spider molecules with that given by “the sum of its parts”. In this context we mean two times the transmission of the directly connected molecule plus two times the transmission of the ̈ first crossed connected molecule. This is a somewhat naive guess inspired by a classical frame of mind, but we will later see that if this rule is obeyed this is a sign of incoherence between the connections to the same electrodes. This paper proceeds by first considering a set of model calculations to see the qualitative differences between the two types of transport a multiconnected molecule may exhibit: coherent transport or incoherent transport. Next we perform a set of atomistic calculations on the molecules considered in this work as mentioned above and compare these results with the qualitative results from the model calculations.
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MODEL CALCULATIONS A Qualitative Model. As a first step toward understanding the mechanisms at work in a multipath molecule, we consider simple model calculations. These calculations were performed numerically for a simple six-site tight-binding model of Received: November 16, 2015 Revised: February 4, 2016
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DOI: 10.1021/acs.jpcc.5b11211 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Figure 1. (a−f) Three molecules in the two binding conformations used for our calculations. (g−i) Schematics for the connections. (a, b) Benzene based on (a) a meta configuration or (b) an ortho configuration. (c, d) Fenestrane based on (c) a meta configuration or (d) an ortho configuration. (e, f) Annulenoannulene based an (e) a parallel configuration or (f) an orthogonal configuration. (g) “Direct” connection, (h) “crossed” connection, and (i) “spider” connection.
benzene. We chose the on-site energies to be 0.0 eV relative to the Fermi energy, and the hopping elements were set to −2.7 eV. The connectivity of the molecule to the electrode then entered as a self-energy. To get an idea of how to visualize the difference between coherence and incoherence between the connections for the spider molecules, one can imagine the incoherent case as a molecule connected to different wide band leads to each binding site, while the coherent case is equivalent to a molecule where the connection sites on the same side of the molecule bind to only a single wide band lead. The two cases are shown diagrammatically in Figure 2. Effectively, this coherence entered as off-diagonal elements in the self-energy for the coherent case, while these elements are strictly zero for the incoherent (see Supporting Information for further details on the derivation of the matrix representation of the Green’s
functions used for these calculations.) Here, we had chosen the coupling element between the connecting atoms and the electrodes to be −0.3 eV. This rather crude model enables us to make fast calculations that show the role of coherence in molecules with multiple connections to the electrodes. Model System Results. In Figure 3 we plot the transmission through benzene obtained from our qualitative model. For the incoherent case, the transmission in the middle of the HOMO−LUMO gap is exactly given by the sum of its pathways. But for the coherent case, it is more difficult to predict, as the resonances of the multiconnected molecule might have their shape altered (a) or interference might appear (b). We also observe a clear difference in the transmission at the resonances: for the incoherent case, it goes to 2, while for the coherent it only reaches 1. Both of these cases are interesting, each in their own way. The incoherent case makes it possible to observe transmission resonances above one. If this situation dominates, new methods of fitting experimental data will need to be employed, as many existing methods model the transmission as a Gaussian with the level position and the level-broadening as the only fitting parameters, assuming a transmission of 1 at the resonance. For example, see refs 5−9. The coherent case, however, holds interest in that new features might be introduced into the transmission, such as the asymmetry in the resonance line shapes for the meta-based benzene or the interference appearing for the ortho-based benzene.
Figure 2. Diagrammatic view of (a) a coherently connected molecule and (b) an incoherently connected molecule. B
DOI: 10.1021/acs.jpcc.5b11211 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Results and Discussion. The transmission for the benzene-based junctions can be seen in Figure 4 as calculated
Figure 3. Transmission of benzene from model calculations. (a) Transmissions based on the meta-connected benzene and (b) the ortho-connected benzene. The blue line shows the transmission for the direct-connected molecule, the green for the crossed-connected, the yellow and red for the spider-connected molecules, and the dashed black line is the transmission predicted by the sum rule (i.e., 2 times the direct plus 2 times the crossed).
Figure 4. gDFTB transmission of benzene based on the (a) meta configuration and (b) ortho configuration. The blue line shows the transmission for the direct-connected molecule, the green it for the crossed-connected, the yellow for the spider-connected molecule, and the dashed black line is the transmission predicted by the sum rule.
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in gDFTB, while exact transmission values reported at an energy at the center of the HOMO−LUMO gap of the molecule in the crossed connection are reproduced in Tables 1
ATOMISTIC CALCULATIONS Computational Trifles. To study the effect of multiple connections on the transmission beyond the Hückel models, we made a series of calculations with GPAW10,11 and gDFTB.12−16 All of these calculations were based on isolated molecules first optimized in GPAW using PBE17 with a doubleζ plus polarization (DZp) basis-set.18 Then, the terminal hydrogen atoms at the binding sites were removed and two (111) gold slabs added with fcc hollow sites chosen as the binding sites. To ensure an optimal binding geometry as a starting point, the arms of the multiconnected molecules (or “spiders”) were twisted the minimum angle so that both arms were at a hollow site, though not restricted to both being fcc sites. After addition of the electrodes to the molecules, the junctions were then reoptimized using a single-ζ (SZ) basis set for the gold and locking the geometry of the gold electrodes. Finally, the transmission was calculated using the respective methods. For the GPAW calculations, a DZp basis set for the molecule and SZ for the gold atoms were used, and three layers of gold atoms were included in the extended molecule. The gDFTB calculations did not include gold atoms in the extended molecule, the Fermi energy was set to be −5.0 eV, and the standard parameters are used. The three molecules used are depicted in Figure 1 in the two binding configurations. Just as two sets of calculations were performed on benzene, one based on a meta configuration and one based on an ortho configuration, as per the model system above, two sets were performed for each macrocycle, each related to the other by a 90° turn.
Table 1. Transmission for Benzene Based on the Meta Configuration at the Energy of the Center of the HOMO− LUMO Gap of the Crossed Connected Molecule (at 0.340 eV Relative to the Fermi Energy for gDFTB and 0.550 eV for GPAW)a transmission (10−3) benzene direct crossed spider 2D+2C
gDFTB 0.0015 1.0837 1.2505 2.1703
(0%) (50%) (58%) (100%)
GPAW 0.32283 (3%) 4.8741 (47%) 8.1704 (79%) 10.394 (100%)
a
The numbers in brackets are percentages of the transmission predicted by the sum rule, as given by the last row.
and 2. This energy has been chosen so that we can compare the transmissions at the same energy for the different connections, and having this energy defined in a uniform way that can be applied to all of the molecules and both calculation methods. It can be seen that these transmissions most closely resemble the incoherent case, with the exception of the resonances only going to 1. Looking closer at the exact values, it becomes clear that there are small deviations from the “sum rule”. These deviations become even more apparent in the GPAW calculations, leading to the conclusion that coherence must be at play. C
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between the two atoms the arrow connects, but rescaled in each figure so that the largest arrow has a fixed size. The linker groups to the leads are truncated in these images so only a single carbon atom represents the connected groups, while those linkers that are disconnected from the leads have been removed from the images. This allows us to zoom in on the central parts of the molecule and have an unobscured view of the differences between the molecules. Figures 5c and 6c show that the local current though a multiconnected benzene appears as two separate currents going through the molecule. It is, however, possible to construct such pictures by adding the components for the two constituent direct connections or the two crossed connections. In this case, the central ring current from each component cancels out, leaving the picture for the spiders where it would appear as if the two pathways are disconnected, due to a cancelation of terms. Considering meta benzene in the crossed connection shown in Figure 6b, for example, we see that the local current comes from one lead, splits into two contributions of half the magnitude across the ring going to the right and down, and recombines before heading out the other lead. If we were to make the other crossed connection, that would be equivalent to flipping the image vertically meaning that the ring current would go to the right and up. Adding these two cases would result in the arrows to the right across the ring adding up so they would have full magnitude, while those going up or down would cancel out due to the fact that the direction of an arrow indicates the sign of the corresponding term. In an effort to delve further into this, the local current is split into its channels20 (see Supporting Information for derivation). For the spider based on the ortho benzene, there is only one dominant channel, meaning that the two separate pathways that we see in Figure 6 are in fact only a single pathway, stressing the point that the molecule displays coherent transport. For the spider based on the meta benzene, however, two channels of approximately the same magnitude appear, and the local currents for each of those channels are plotted in Figure 7. Each of these channels look like the local current thought the crossed connected benzene. This is what would have been expected if we applied our semiclassical intuition of the current being predicted by summing up the individual parts. The transmissions for each of these crossed channels, however, are only roughly half of that for the crossed connected benzene, meaning that while the local current pictures echoes our semiclassical intuition, the fact that the magnitude of each channel is only half of what is expected is a clear consequence of the coherent nature of the transport. The resulting image would look exactly like Figure 6c. The small remnant arrows in Figure 7a, or to a higher degree Figure S6e, are ascribed to an incomplete deconvolution into spatially separate channels, but we suspect that complete
Table 2. Transmission for Benzene Based on the Ortho Configuration at the Energy of the Center of the HOMO− LUMO Gap of the Crossed Connected Molecule (at 0.335 eV Relative to the Fermi Energy for gDFTB and 0.560 eV for GPAW)a transmission (10−3) benzene direct crossed spider 2D+2C
gDFTB 0.7878 1.0070 3.3752 3.5896
(22%) (28%) (94%) (100%)
GPAW 2.4382 (17%) 4.6618 (33%) 11.467 (81%) 14.200 (100%)
a
The numbers in brackets are percentages of the transmission predicted by the sum rule, as given by the last row.
If the coherent picture is the correct way to describe these systems, then the missing interference of Figure 4b compared with Figure 3b raises some concerns. This could occur for a number of reasons, one being that transmissions in Figure 4 include the full electron structure of the molecules which among other things include the σ electrons, and these are unlikely to exhibit interference in this region. Another possibility is that the models used in Figure 3 are too simple, in that the couplings to the same electrode are set to be the same. It is unlikely that the size of the couplings should differ, but if we allow their argument in the complex plane to differ, we see that it is possible to reproduce the result of Figure 4b by setting the argument difference to π (see Supporting Information for more details). Note that the high reported transmission for the directly connected meta-based benzene in Table 1 compared to that of the interference in Figure 4a arises because the energy of the interference is slightly different from the energy at which the transmission is reported in the table. The qualitative results for the macrocycles as calculated with GPAW are summarized in Table 3, while the plots of the transmissions and the tables of absolute transmission values in the middle of the HOMO−LUMO gap can be found in the Supporting Information. These calculations repeat what we already have seen for benzene; they mainly exhibit the signs of coherence between the two leads. Only annulenoannulene in the parallel configuration shows the signs of incoherence. This might have been predicted considering the two constituent annulenes are only connected through meta connections, turning off any current that might cross from one unit into the other, and thus isolating the two into incoherent pathways. Local Channels. In an effort to see if it is possible to “see” the coherence in the current, we look at the local current through the molecule19 as calculated by gDFTB. These are shown in Figure 5 and Figure 6. The arrows in the figures have a thickness proportional to the size of the local element
Table 3. Qualitative Results from the GPAW Calculations for Fenestrane and Annulenoannulenea
Red cells indicate signs of coherence, while green cells indicate signs of incoherence. The ≥1 is used to indicate that the HOMO resonance has a transmission well above 1 (1.68) while the LUMO resonance has a transmission of ≈1. The ≤1 is used to indicate that the HOMO resonance has a transmission below 1 (0.63) while the LUMO resonance has a transmission of ≈1. a
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DOI: 10.1021/acs.jpcc.5b11211 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Figure 6. Local transmission elements of meta benzene plotted at the Fermi energy: (a) direct connection, (b) crossed connection, and (c) spider connection. The binding linkers have been truncated, and the nonbinding linkers have been removed altogether for clarity.
the connections on the opposite side of the molecule from a given entry point (in the sense that pathways to the other connection cannot occur without crossing a meta connection), the transport is dominated by two channels. On the other hand, where no such restriction is imposed on the current, there is only one dominant channel. Alternatively, the two channels can be ascribed to molecules where there is a direct pathway between the two connections to the same lead (again in the sense that the current does not cross a meta connection), while the single channel cases are molecules without this direct connection. Our sample of molecules, unfortunately, does not allow us to differentiate between the two explanations.
Figure 5. Local transmission elements of ortho benzene plotted at the Fermi energy: (a) direct connection, (b) crossed connection, and (c) spider connection. The binding linkers have been truncated, and the nonbinding linkers have been removed altogether for clarity.
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deconvolution occurs at the energy at which the two channels have the same transmission. When we consider all of the molecules examined in this paper, we see that when the current can only exit from one of
CONCLUSIONS We have seen that the transmission through a multiconnected molecule differs from that which would have been expected E
DOI: 10.1021/acs.jpcc.5b11211 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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depicting the local transmission elements for the two macrocycles (PDF)
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP7/2007- 2013)/ERC Grant Agreement No. 258806.
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Figure 7. Local transmission elements of the two dominant channels of the meta benzene in the spider connection plotted at the Fermi energy. The binding linkers have been truncated for clarity.
from the sum of its parts. This is attributed to coherence between the connections to the leads. Only in the case with the annulenoannulene in a parallel configuration did transmission behave as the sum of its parts, indicating signs of incoherence. This is attributed to the molecule being made up of two separate parts that are electronically isolated from each other via meta connections. When the current enters from one connection and can exit from any of the opposite connections (in the sense that there are pathways without meta connections to both binding groups that link to the opposite lead), we see that only a single dominant channel emerges, fortifying the idea of the current though the molecule being coherent. But when the current is only allowed to exit via one connection, the current splits into two channels, and the observed coherence must materialize as interplay between the two.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b11211. Matrix representation of the Green’s functions used for the model calculations, derivations for the factorization of the local current into the channels, plots of the transmission across benzene, fenestrane, and annulenoannulene, tables for absolute values in the middle of the HOMO−LUMO gap of the transmission, and figures F
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The Journal of Physical Chemistry C Materials Simulations in Physics, Chemistry, and Biology. Phys. Status Solidi B 2000, 217, 41−62. (15) Frauenheim, T.; Seifert, G.; Elstner, M.; Niehaus, T.; Köhler, C.; Amkreutz, M.; Sternberg, M.; Hajnal, Z.; Di Carlo, A.; Suhai, S. Atomistic Simulations of Complex Materials: Ground-State and Excited-State Properties. J. Phys.: Condens. Matter 2002, 14, 3015− 3047. (16) Pecchia, A.; Di Carlo, A. Atomistic Theory of Transport in Organic and Inorganic Nanostructures. Rep. Prog. Phys. 2004, 67, 1497−1561. (17) Perdew, J. P.; Burke, K.; Wang, Y. Generalized Gradient Approximation for the Exchange-Correlation Hole of a Many-Electron System. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 16533− 16539. (18) Larsen, A. H.; Vanin, M.; Mortensen, J. J.; Thygesen, K. S.; Jacobsen, K. W. Localized Atomic Basis Set in the Projector Augmented Wave Method. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 195112. (19) Solomon, G. C.; Herrmann, C.; Hansen, T.; Mujica, V.; Ratner, M. A. Exploring Local Currents in Molecular Junctions. Nat. Chem. 2010, 2, 223−228. (20) Gagliardi, A.; Solomon, G. C.; Pecchia, A.; Frauenheim, T.; Di Carlo, A.; Hush, N. S.; Reimers, J. R. A Priori Method for Propensity Rules for Inelastic Electron Tunneling Spectroscopy of SingleMolecule Conduction. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 75, 174306.
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