Quantum Interference: the Structural Dependence of Electron Transmission Through Model Systems and Cross-Conjugated Molecules

J. Phys. Chem. C 2008, 112, 16991–16998 16991

Quantum Interference: The Structural Dependence of Electron Transmission through Model Systems and Cross-Conjugated Molecules

David Q. Andrews,* Gemma C. Solomon,* Randall H. Goldsmith, Thorsten Hansen, Michael R. Wasielewski, Richard P. Van Duyne, and Mark A. Ratner Department of Chemistry, Northwestern UniVersity, EVanston, Illinois 60208 ReceiVed: June 24, 2008; ReVised Manuscript ReceiVed: August 25, 2008

We report on a class of molecules that exhibit nonlinear current/voltage behavior in the low bias tunneling regime. This interesting behavior is attributed to quantum interference. Using site models, we show that interference features, while common, do not necessarily occur at experimentally relevant energies, hindering realization in transport measurements. Calculations made using a nonequilibrium Green’s function code show that quantum interference can be experimentally relevant in cross-conjugated molecules. A detailed bond length analysis of cross-conjugated molecules gives insight into why these molecules have interference at energetically accessible regions. The interference features are shown to be stable to both an electronic dephasing analysis and geometric ﬂuctuations provided by molecular dynamics.

Introduction The potential for molecular electronics is rooted in the unique chemical properties of molecules. The suggestion that single molecules could be used as discrete electronic components was made over 30 years ago.1 Since then, experimental studies of charge transport through molecules have largely been completed in an electrochemical transport junction or optical donor- bridge-acceptor context. These measurements have been completed on a large range of molecular systems as well as cross-conjugated molecules. In junction measurements, the Figure 1. The transmission probability is a sum of the couplings between source/donor-molecular orbitals-drain/acceptor for a single molecular conductance is studied, while in optical measurements channel. The electron “sees” all available molecular orbital pathways the transport through the molecules is measured as a charge in energy space. This can be compared to Young’s double slit separation or recombination rate. Single molecule two probe experiment; in electron transport, destructive interference can occur transport measurements have opened up a new area of research when summing the coupling terms across the system.18 where the average transmission through a molecule can be studied without charging the molecule, allowing the ability to tune the incident electron energy.2-4 If the charge transport lecular frontier energy states sandwiching the electrode Fermi behavior of a molecule is extremely sensitive to incident electron level. Within this picture, explaining conductance relies on four energy near the Fermi level, this control of voltage bias should main factors, the electrode/molecule coupling strength, the yield new information on the transport behavior of molecules. associated molecular state broadening, the alignment and Measuring transport through single molecules has largely distance of the Fermi energy to the frontier molecular orbitals, focused on molecular wires. These generally consist of linear and the energy level shift of the molecular states upon contact molecules (both saturated and unsaturated) or fully conjugated to the electrode. aryl systems. Making such measurements is critical in establish- While this picture has proven extremely useful, it is based ing new experimental techniques and allows comparison among on an independent electron model that assumes the molecular different methods. There has also been strong theoretical energy states do not interact in the transport process. We show development of models to explain the transport behavior in a here that interactions between molecular orbital energy states, metal-molecule-metal junction.5-9 This field of research is sometimes including those far from the Fermi level, can have relatively young, and the transport through less common organic a very large effect on the transport properties of a molecule in structures should open up new areas where the electron the tunneling regime at low bias voltage. Figure 1 shows a transmission as a function of energy behaves in unique ways. diagram illustrating schematically how the transmission probability for an electron to tunnel through a molecule is calculated In analyzing the charge transport through molecules in the 18 coherent tunneling limit, the Simmons equation, which pictures as the sum of the coupling terms from donor to acceptor. The a molecular junction as a tunneling barrier where the height of coupling terms between donor and acceptor vary with the the barrier is essentially the distance from the Fermi level to geometry, orientation, and electrode linking sites of the mol- the nearest molecular resonance, is often used.10-17 This picture ecule, and a single molecule can have drastically different incorporates a common illustration showing the discrete mo- conductance behavior for different linking geometries even though the frontier molecular orbitals are similar. * Corresponding author. E-mail: [email protected] (D.Q.A.), The largest deviation from the Simmons model of charge [email protected] (G.C.S.). transport occurs when the coupling terms across the molecule 10.1021/jp805588m CCC: $40.75  2008 American Chemical Society Published on Web 10/07/2008 16992 J. Phys. Chem. C, Vol. 112, No. 43, 2008 Andrews et al. destructively interfere in the energy region between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). In the context of electron transport, interference can occur among the terms that are summed to calculate the transmission probability. At the energy level of a molecular resonance, the transmission probability f 1; con- versely, when there is quantum interference, an antiresonance can occur where the transmission probability f 0. Interference features have been studied extensively in quantum dots.19-22 Quantum dots provide a discrete energy spectrum similar to molecules and, due to the increased size scale, enable greater experimental control. In the quantum dot literature, measurements have been made on the transmission phase in a quantum dot using interferometers.19,23 In this work, the transmission phase change going through a resonance was π, Figure 2. Transmission through a two-site model. The electrode but unexpected were the measured abrupt phase changes of π attachments determine the presence of an antiresonance feature. The 23 plot on the left shows transmission as a function of energy, and the plot between sequential resonances. This abrupt phase change was on the right shows phase as a function of energy. The antiresonance later attributed to a nonspanning node in the specific energy feature in the perpendicular model is seen at E ) 0 as a zero in level of the quantum dot.24 A nonspanning node is defined as transmission and a corresponding phase jump of π. an in-phase resonance where the change in wave function density leads to a node that touches only one boundary or zero boundaries of a quantum dot.24 In the limit where a quantum dephasing calculations, we show that the interference feature dot can be treated as a large molecule, we show how the phase calculated in cross-conjugated molecules is stable. change in molecular resonances directly correlates to the nonspanning node explanation. A Two-Site Model In visual similarity to cross-conjugated molecules, T-shaped To understand how quantum interference can occur in electron quantum dots have been proposed as quantum interference transport through molecules, it is illustrative first to analyze a devices.22,25 Experimentally observing interference effects in donor-bridge-acceptor model of a two-site system. As shown nanostructured materials is hindered by the need to maintain in Figure 2, there are two possible electrode/molecule site electron phase coherence over long distances.20,21 Here, we linkages (hereafter simply referred to as binding site geometries) discuss systems that are similar to a T-shaped quantum dot for a two-site bridge between a donor and an acceptor. We refer system but do not require extremely long electron coherence to these two binding site geometries as the linear model and lengths; specifically, we consider cross-conjugated molecules. the perpendicular model. In the linear model shown in Figure In focusing our research on single molecule effects, it is 2a, the donor is connected to one site and the acceptor to the important to realize that experimental measurements of single other site. In the perpendicular model shown in Figure 2b, the molecule junctions are quite tricky. Controlling the formation donor and acceptor are both connected to the same site. Setting of an angstrom-nanometer scale gap as well as maintaining the site energy to 0 and the intersite coupling, ,to-0.5, we contact to both electrodes is extremely difficult. Once these calculate the Hu¨ckel molecular orbitals, for the isolated two- measurements are made, care has to be taken in analyzing the site system as shown. data. It has become apparent through many calculations and The energy-dependent electronic coupling between the elec- experimental measurements that molecules can sample many trodes determines the transport properties of a molecular different geometries and possible mechanistic regimes with junction. To calculate the coupling between the donor and the respect to time. These fluctuations in current can be of the same acceptor, we solve the one-electron Hu¨ckel Hamiltonian. This order of magnitude as the observable.26 Hu¨ckel coupling matrix is then transformed into a molecular Because of the difficulties in controlling small molecular orbital basis. junctions, many early measurements have been reanalyzed.27-29 The transmission is calculated using the Landauer equation:30,31 For useful device applications, a molecule with novel transport ) L r R a behavior should be stable to geometric fluctuations while still T(E) Tr[Γ (E)G (E)Γ (E)G (E)] (1) providing interesting transport properties that can be experi- with ΓL/R as the left and right spectral densities and Gr/a as the mentally realized. retarded and advanced Green’s functions. Using a transformation To investigate interference features, we ask the questions: to a molecular conductance orbital basis,32 we have shown that How can interference occur in model systems? Why is the the transmission can be calculated using the sum of the squared 18 interference present in cross-conjugated molecules? How stable couplings tR through each molecular orbital (labeled by i), is this interference feature to electronic dephasing and molecular given by vibrations? ) L † r R Two- and four-site model systems are studied to give an tR(E) ∑ VRi (E)Gii (E)Vi (E) (2) indication of how interference features can occur in a transport i picture and how interference features appear to be quite where R and are the basis dimensions of the electrode, VL(R) common. A detailed analysis of cross-conjugated bonds is are the bridge-electrode couplings, and Gr′ is the retarded presented to give an understanding of why this molecular unit Green’s function18 (calculation of the transmission from the has such a large effect on electron transport. Using nonequi- coupling is shown in the Supporting Information). librium Green’s function codes, we calculated the transmission To calculate the coupling as a function of energy, we leave and current-voltage characteristics for a series of cross- the site energies at 0 and sweep the electrode energy. In both the conjugated molecules. Using molecular dynamics and electronic linear model and the perpendicular model, the coupling through Quantum Interference J. Phys. Chem. C, Vol. 112, No. 43, 2008 16993

Figure 3. Four-site model system with six possible donor-acceptor couplings. In all model systems with a side group there is an antiresonance feature, although not necessarily at E ) 0. the highest occupied molecular orbital (HOMO) level is In this two-site model, the antiresonance is correlated with either identical. The coupling through the lowest unoccupied molecular no change or a decrease in the number of spanning nodes orbital (LUMO) differs only in a sign change. The overall between the donor and the acceptor. A spanning node is taken coupling through the system can be determined by taking the here to be a node in the shortest path between the donor and absolute value of the sum of the individual coupling terms. acceptor. At E ) 0, the sum of the coupling terms in the perpendicular two-site model is zero. The coupling terms through the HOMO A Four-Site Model and LUMO molecular orbitals are equal and opposite, leading to destructive interference. This destructive quantum interference We now look at the coupling through a four-site model shown gives the perpendicular two-site model the sharp antiresonance in Figure 3. The Hu¨ckel molecular orbitals for the isolated four- at E ) 0. To compare directly with two probe measurements, site system are shown in the upper left. As with the two-site we calculate the transmission as a function of energy18 as shown model, the isolated molecular orbitals are the same, independent in Figure 2c. of how we connect the donor and acceptor. In the four-site In Figure 2d, we show the phase of the transmission as a model, there are 10 different ways to connect the donor and function of energy. The transmission phase is deﬁned as the the acceptor. We limit our discussion to the transmission plots arctangent of the imaginary/real parts of the transmission.18 As for six selected systems. Figure 3b compares the transmission the energy passes a molecular resonance, the transmission phase through D1A4 and D1A2. There are two antiresonances, both always changes by π. In the perpendicular two-site model, there occurring in the D1A2 model. The antiresonance occurs when is an abrupt phase change of -π at E ) 0, the location of the there is a molecular orbital that introduces a nonspanning node, antiresonance. This antiresonance occurs whenever there is a taken here to be a node that is not in the shortest path between zero in both the real and the imaginary plane of the transmission. the donor and acceptor. A zero in both the imaginary and the real components of the In Figure 3c, D1A3 and D2A3 both have an antiresonance transmission need not cause a phase jump of π, and for this at E ) 0 in the transmission. In both models, a nonspanning reason we have included the more complex parametric plots node is introduced in the LUMO. The HOMO (E ) 0.618)in (shown in the Supporting Information). Sweeping the energy Figure 3a has one node in the electron density between site from very low to very high, the transmission phase changes by pairs 1 and 3, and 2 and 3. The LUMO (E )-0.618) has nπ, where n is equal to the smallest number of sites between only 1 node between sites 1 and 3 and 0 nodes between sites 2 the donor and acceptor or electrode binding sites. In the and 3. In the D2A3 model, there are two nonspanning nodes perpendicular two-site model, there is only one site between introduced, causing the width of the antiresonance to increase. donor and acceptor, indicating that there will be one abrupt phase Figure 3d shows two models, D1A1 and D2A2, where the change of -π. donor and acceptor are both connected to the same site. In this Using a two-site model, we can see that an antiresonance example, each subsequent molecular orbital introduces a non- occurs at E ) 0 for the perpendicular model. This zero in the spanning node in the Hu¨ckel orbitals. Correspondingly, in the transmission is a direct result of the destructive interference transmission plot for both of these four-site models, there is caused by the coupling to the HOMO and the LUMO orbitals. antiresonance between each subsequent higher energy molecular 16994 J. Phys. Chem. C, Vol. 112, No. 43, 2008 Andrews et al.

Cross-conjugation is a common motif in organic chemistry, found in a wide range of naturally occurring and synthetic molecules; yet detailed analysis of the electron delocalization and transport properties is sparse. Recent synthetic work has detailed a large number of cross-conjugated molecules.46,47 A number of theoretical studies of HOMO’s and LUMO’s of cross- conjugated molecules have shown that the electron delocalization is split at the cross-conjugated unit.44,48 This difference between linear and cross-conjugation has been borne out in absorbance measurements, as the cross-conjugated unit results in a measured blue shift in the absorbance peak.49 In donor-bridge-acceptor systems, the cross-conjugated con- nectivity leads to measured electron transfer rates lower than Figure 4. Resonance structures for a cross-conjugated molecule. The those of linearly conjugated systems but generally higher than electron delocalization shown in (b) and (d) illustrates how in a three- those of nonconjugated systems.50 way junction the electron can only delocalize in two directions; there is no direct electron delocalization possible between atoms 1 and 5. Bond Length Analysis The molecules shown in Figure 5 are used to illustrate the resonance. The antiresonances occur at the energies of the fundamental bond length differences between cross conjugation, molecular orbitals of the isolated side chain. linear conjugation, and saturated structures. It has been previ- The two- and four-site models show that quantum interference ously established that bond length and bond length alternation and corresponding antiresonances are quite common in toy are correlated to π electron delocalization in both linear and systems. Within these controlled systems, an antiresonance cross conjugation molecules.44,45,51,52 In this analysis, we occurs whenever there is a zero in both the real and the compare the bond lengths for 15 molecules with three different imaginary portion of the transmission. These transmission zeros central bonding motifs; cross conjugation, linear conjugation, appear whenever there is a molecule having sites not in the and a saturated carbon atom. To allow direct comparison with shortest geometric path between donor and acceptor. In the metal-molecule-metal transport calculations presented later, molecular orbitals, this zero is seen as the introduction of a node all of the molecules are thiol terminated, and the geometry in the molecular orbitals that does not span the path between optimized gas-phase structures are calculated using density donor and acceptor. While this feature may be common in toy functional theory with B3LYP53,54 and 6-311G** in Q-Chem systems, it has been noticeably absent in experimental measure- 3.0.55 Our analysis focuses on the delocalization of the charge ments. This absence is due mainly to the occurrence of between the sulfur atoms because this will be the direction of interference features not energetically positioned between the charge transport in a molecular junction. The cross-conjugated 33 HOMO and LUMO. If the interference is not near the Fermi molecules labeled 6-10 all have a double bond perpendicular energy, experimental verification is difficult and the appearance to the direction of charge transport. Within this subset of of the antiresonance may have little effect on charge transport. molecules, 6, 7, and 9 meet the specific definition of cross One commonly studied molecule that has an interference feature conjugation. Molecules 11-15 have a double bond connected near the Fermi energy is benzene connected to the electrodes in the trans orientation between the thiol terminations. Molecules through the meta positions. Experimental verification of these 11, 12, and 14 are completely linearly conjugated between thiol differences has been noted in I/V measurements in a mechanical groups. Molecules 1-5 are the same as molecules 6-10, but 34 35 break junction, in absorbance measurements, as well as in with hydrogen atoms replacing the cross-conjugated double 35-41 numerous theoretical calculations. We will now focus on bond. the acyclic cross-conjugated bond, a common molecular bonding To simplify the comparison among so many molecules, we motif that has been calculated to have interference between the focus on the bonds labeled a, b, and c in Figure 5. Bonds a and 42 HOMO and LUMO. c are both formally single bonds, and bond b is a double bond. The carbon single bonds are systematically longest in the Cross-Conjugated Molecules molecules 1-5 and shortest in the molecules with linear “A cross-conjugated compound may be defined as a com- conjugation, 11-15. The corresponding bond lengths in the pound possessing three unsaturated groups, two which although cross-conjugated molecules 6-10 are between the length of the conjugated to a third unsaturated center are not conjugated to saturated and unsaturated linear molecules. The changes in bond each other. The word “conjugated” is defined here in the lengths agree well with previous studies completed on molecules classical sense of denoting a system of alternating single and 1, 2, 6, 7, and 9.45 double bonds.”43 By definition and as illustrated in the resonance To compare the carbon single bond lengths, we have picture shown in Figure 4, cross-conjugation in transport calculated the percentage of double bond character. This is done junctions depends on the position linking to the electrodes, by comparing all of the bond lengths with those calculated for reminiscent of our observations with the toy models. In a ethene and ethane. Ethene by definition has 100% double bond molecule that is fully π-conjugated, this binding dependence character, and ethane has 0% double bond character. This bond will affect the low bias delocalization when there are an odd length comparison shows that in a cross-conjugated molecule number of carbon atoms between the donor and acceptor. the π electron delocalization is less than that in a linearly Electron delocalization and electron transport are correlated,44,45 conjugated molecule but greater than that in a fully saturated and from Figure 4 it seems evident that electron transport should carbon atom. behave differently if the electrodes are connected to atoms 1 Using the bond length characterization to analyze these data, and 6 versus connection to atoms 1 and 5, as there is no we come to three important conclusions. First, a cross- delocalization between atoms 1 and 5. conjugated molecule and a linearly conjugated molecule show Quantum Interference J. Phys. Chem. C, Vol. 112, No. 43, 2008 16995

Figure 5. A bond length analysis for a series of molecules with sections of conjugated, nonconjugated, and cross-conjugated carbon atoms. Red indicates bond a or c is bonded to an alkyne group, blue to an alkene group, and green to an alkane group. By deﬁnition, the ethane bond length has 0% double bond character, and the ethene bond length has 100% double bond character. A negative double bond character indicates a carbon-carbon bond longer than that in ethane.

Figure 6. Transmission and current-voltage calculated in Hu¨ckel IV 3.0 for molecules 4, 5, 9, and 14. In the transmission plot shown on the left, the cross-conjugated molecule 9 shows an interference feature at E ) 0. The corresponding current-voltage plot on the right shows that current through molecule 9 has a greater voltage dependence than molecules 4, 5, and 14 (the deviation from linearity in molecule 9 is discussed in the Supporting Information). different behavior with respect to electron delocalization. Second, in cross-conjugated molecules, the electron delocalization is reduced between cross-conjugated regions of the molecules, as is evident by the increase in both C-C bond lengths. The saturated C-C bond length only changes due to direct coupling to an unsaturated carbon. This effect is local, and the bond length changes are largely a nearest neighbor effect. Within the three sets of molecules, 1-5, 6-10, and 11-15, the C-C bond lengths a and c remain weakly correlated to each other (except when required to match by symmetry in molecules 1, 4, 6, 9, 11, and 14), implying that the addition of Figure 7. Electronic dephasing effects calculated for meta- and para- the cross-conjugated bond induces very little electron delocal- benzene as compared to linear and cross-conjugated site representations. ization across the cross-conjugated carbon atom. The decay time is deﬁned as the time necessary for the total population to decrease 95%. Transport Calculations To calculate the transport properties of cross-conjugated the Laundeur-Imry low bias tunneling regime.30,31,56,57 Figure molecules, we look at a representative series of molecules from 6 shows a series of transmission spectra and I/V plots for this Figure 5, speciﬁcally molecules 4, 5, 9, and 14. These molecules family of molecules. These calculations were completed in represent a complete series of double bonded molecules with Hu¨ckel IV 3.0.7,58 The qualitatively similar plots calculated using cross-conjugation, linear conjugation, and replacing the cross- ATK 2.0.459-62 are shown in the Supporting Information. conjugated unit with a saturated carbon atom or saturation of The transmission plot for the cross-conjugated molecule, 9, one double bond. The transport calculations are completed in has HOMO and LUMO molecular orbitals at an energy and 16996 J. Phys. Chem. C, Vol. 112, No. 43, 2008 Andrews et al.

Figure 8. Molecular dynamics results for molecules 5, 9, and 14. Plots (a) and (b) show 100 transmission traces calculated for molecules 9 and 14. Plots (c) and (d) show Gaussian fits to the distribution of conductance values calculated at 10 mV and 2 V. with transmission similar to those of the fully linearly conjugated Measuring or estimating an absolute electronic dephasing molecular 14. Yet at E ) 0 the transmission through molecule strength can be difficult with estimates varying from 10 s to 9 is over 3 orders of magnitude lower. The transmission through thousands of cm-1.70-74 However, the rate of dephasing between the cross-conjugated molecule 9 is also lower than that through two sites will depend on the correlation function between site molecules 4 and 5. This remarkable result occurs because of energies.63 Indeed, the true dephasing strength for a cross- quantum interference: the coupling terms in the cross-conjugated conjugated unit will likely be quite weak, as the stochastic unit cancel the π contribution to transport near the Fermi energy fluctuations in site energy and intersite communication will almost completely. For an antiresonance to occur, a zero in the likely be strongly correlated within the small cross-conjugated transmission is required. In this example, there is complete unit as the unit is of the same dimension as the individual solvent destructive interference in the dominant π transport channel, molecules. The physical processes behind the estimates of but there is still transport through the σ channel. In the I/V plot dephasing strengths are largely from energy fluctuations from shown on the right in Figure 6, the current through molecule 9 uncorrelated solvent motions between distal molecules in the is the lowest at low voltage but increases at a higher nonlinear case of photon echo experiments75,76 or the fluctuating energy rate than any of the other molecules. The molecules 4, 5, and gap between ground and excited states in resonance Raman 14 behave more like resistors with approximately linear voltage experiments.73 Contrasting these uncorrelated energies with the dependence (shown in the Supporting Information). largely correlated site energies in the small cross-conjugated unit suggests these above estimates should be regarded as an Electronic Dephasing upper limit for dephasing strength. For comparison, the level of dephasing required for erasing Interference stability is of primary importance for experi- the effect of interference on transport in benzene, a similarly mental measurement of the unique properties of cross-conjugated sized molecule to the cross-conjugated unit, is significantly - molecules. Electronic dephasing,63 65 frequently caused by smaller; yet these interference effects persist, and have been fluctuations in molecular environment or geometry, might lead experimentally measured,34 and dominate the substitution to decoherence of the transmitted electron and destruction of chemistry of phenyl compounds. Consequently, we believe that the interference feature.66,67 In earlier studies, we looked at how the transport interference calculated in cross-conjugated mol- the traditional reactivity series in ortho-, meta-, and para- ecules will survive electronic dephasing. benzene could be recast as an interference effect67 and how this effect could be erased by purely local dephasing.66,67 The Molecular Dynamics calculation of transport dynamics is done using the quantum Liouville equation with dephasing included by reducing the The transport through a molecule would ideally be constant magnitude of the off-diagonal elements of the density matrix despite room temperature molecular vibrations and binding site (coherence).68,69 At t ) 0, all population is placed on the donor/ fluctuations. Experimental measurements on single molecule source site, while an absorbing boundary condition on the conductance have primarily been made using Au elec- acceptor/drain site is used to simulate irreversible electron trodes.26,27,77-80 The Au surface is quite mobile, and the transfer. We have calculated the decay time necessary for the distribution of conductance values through a sulfur-terminated total system population to decrease to 5% of the initial value, molecule can be quite large. This variation has been well studied shown in Figure 7. Our results indicate that a dephasing strength both experimentally and theoretically, with typical distributions of γ > 100 cm-1 is necessary for coalescence of the transport of (60% in conjugated molecules on Au surfaces.77,78,81-84 The time through both two-site models. unique conductance characteristics of cross-conjugated mol- Quantum Interference J. Phys. Chem. 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