VOLUME 2, NUMBER 5, MAY 2002 © Copyright 2002 by the American Chemical Society
Molecular Conductance of Dendritic Wires C. Kalyanaraman and D. G. Evans* Department of Chemistry and the Albuquerque High Performance Computing Center, UniVersity of New Mexico, Albuquerque, New Mexico 87131 Received September 27, 2001; Revised Manuscript Received March 28, 2002
ABSTRACT Steady-state rates of electron transfer through molecular bridges are used to calculate the conductance through molecular wires. The study focuses on the topology of the molecular bridge and the implications for the conductance through particular structures. Here we demonstrate an enhanced steady-state flux of electrons through dendritic wires. A calculation of the steady-state rates illustrates that branched topologies, and in particular dendritic structures, can show larger conductance than the linear chain counterparts. The position and the topology of the side branches are shown to be crucial in determining these transport properties.
Progress in the study of molecular conduction through single molecular wires over the past few years has been phenomenal.1-3 The conductive properties of single molecules have been studied experimentally, using mechanical break junction,4 scanning tunneling microscopy5 and photoinduced studies through adsorbates on metal surfaces.6 At the same time, there has also been renewed interest in the theoretical understanding of electron transport through molecules. The similar Greens function-based approaches of Datta et al.7 and Ratner et al.8 provide a method of evaluating the current through a single molecular wire attached to two metal contacts. While the effects of vibrations may be taken into account by self-energy terms in these approaches, methods based on the Redfield equations have been very useful for examining the dynamical effects of vibrations and solvent on electron transfer (ET) through molecular wires.9 The steady-state rate method, introduced recently by Nitzan, Ratner and co-workers,10,11 is based on a generalization of * Corresponding author. E-mail:
[email protected]. 10.1021/nl010074o CCC: $22.00 Published on Web 04/17/2002
© 2002 American Chemical Society
the Landauer formula12 and the ET rate, which can be calculated using the Redfield equations of the reduced density matrix. Extensive studies of the effects of bridge length of linear polymers on conduction have subsequently allowed for a calculation of molecular wire conductance from the coherent regime to the incoherent (hopping) regime. Recent work on the electrochemical properties of dendrimers [tree-like branched polymers]13 have demonstrated their promise as electronic storage devices and electroactive surface monolayers.14-16 Other applications include the experimental demonstration that ET can be controlled in branched structures to develop photon-controlled molecular switches.17 This paper uses the Redfield approach9,11 to study the conductance through dendrimer wires. Using the generalized Landauer formula to approximate the conductance through branched molecular wires, we demonstrate enhanced electron transport through branched wires, the effects of the positions of the branches, and the essential changes introduced by the branching topology. We show that the addition
of branches along a molecular wire has a cumulative effect and that the hyperbranched dendrimeric topology is consequently very efficient at enhancing the ET rate. The steady-state approach is used to compare ET processes in linear chain polymers and branched and hyperbranched macromolecules. For branched and dendritic molecules, the donor and the acceptor are both on the periphery and the ET rate is determined through the hyperbranched bridge structure. The steady-state rate through these structures is evaluated after establishing a constant flux of electron density through the molecular bridge between the donor and the acceptor site. This is achieved after a relatively short transient time if the donor state population is maintained at a constant value and there is irreversible decay off the acceptor site.10,11 The steady-state population on all bridge sites, side groups and the acceptor site is calculated by propagating the timedependent electron density matrix using the Redfield equations.9,11 As in previous studies of conductance through linear chain polymers,10,11 the steady-state rate constant, kSS, is evaluated from the steady-state populations on the acceptor site (σAA), the donor site (σDD), and the decay rate associated with damping from the acceptor site (ΓA), by using kSS ) (ΓAσAA/σDD). Accurate determination of the steady-state rate can be ensured by checking that the acceptor state population does in fact reach a steady-state value after an initial transient period. In all our calculations, the percentage population change on the acceptor site was confirmed to be less than 5% over a period of 250 fs beyond 1.5 ps time evolution. A tight-binding Hamiltonian is used to describe the electronic state of the system. Although this is a simplistic approach, it is not feasible to use much more sophisticated models and take into account the dynamical effects of the solvent. The onsite energies for each monomer unit and the nonadiabatic site-site coupling terms are chosen using chemically reasonable values. The donor and the acceptor state energies are chosen as 0 cm-1 and onsite energies of all bridge sites are 1500 cm-1. The nonadiabatic coupling strength is taken as 600 cm-1 to model a delocalized π electron system [like, for example, extended phenylacetylene structures]. In addition, coupling of these electronic states to a stochastic bath at 300 K (with coupling strength 100 cm-1 and relaxation time τ ) 8.85 fs) mimics the effects of coupling to a dissipative solvent or vibrational modes. The form of the Hamiltonian used is exactly the same as that used by Nitzan et al. to study ET in linear chain wires.11 The molecular wires considered in this study are shown in Figure 1. The structures are color coded to facilitate the mapping of the structures with the steady-state rates illustrated in Figure 2. For all the structures in Figure 1, the donor is taken as the filled circle on the left and the acceptor is the filled circle on the right. These structures have been chosen to illustrate specific features related to the length, topology and position of the side groups and their effect on conductance. In Figure 2, the steady-state rate is plotted as a function of bridge length (N), for the structures shown in Figure 1 in the corresponding color. In addition, in Figure 2, the rate for these structures is identified with a particular geometry 438
Figure 1. Geometries of the structures.
by using a simple notation introduced previously:16 for a bridge with N sites on a linear chain between the donor and acceptor, the label (n1, n2, ..., nN) indicates a side group of n1 sites attached to bridge site 1, a side group of n2 sites attached to bridge site 2, etc. For the linear chain of bridge length N with no side groups attached, the steady-state rate decreases slightly from N ) 2 through 4 and it decreases exponentially beyond N ) 4. In the adiabatic limit (large V) considered here, tunneling appears to be the dominating factor. Therefore, for relatively short chain lengths from N ) 2 to 4, the rate changes only slightly. Earlier studies on electron transfer in dendrimers16 have used a phenomenological model to compare ET rates in second and third generation dendrimers and their linear chain counterparts. This earlier study demonstrated that within this model, dendrimers showed an increased ET rate compared to the linear chain with same number of bridge sites. These observations were rationalized by the increase in the width of the band due to the additional side groups. Side groups Nano Lett., Vol. 2, No. 5, 2002
Figure 2. Steady-state electron-transfer rate as a function of bridge length for linear, branched, and hyper-branched (dendrimer) chains.
with radically different onsite energies showed strong resonances and switching behavior. This model also suggested that a systematic addition of side groups to the main bridge across the chain leads to a cumulative increase in the rate. While this previous study used phenomenological damping terms as a model for solvent and vibrational perturbations on the ET rate, Figure 2 demonstrates that these conclusions are still valid within the more sophisticated Redfield model. In addition to confirming these results of the previous simple model, we have also studied various new effects associated with the addition of side groups to linear chains. The Redfield approach has enabled us to study these phenomena and relate them directly to the conductance that can be measured in single molecule break-junction or STM experiments. The structures in blue (in Figure 1) illustrate how the length of a side group may affect the electron transport. Side groups of an increasing number of sites are placed midway along the main bridge between the donor and acceptor with three bridge units. The steady-state rate is illustrated in Figure 2. For the short bridge length, N ) 3, considered here, the branched and linear chain polymers have the same steadystate rate. Increasing the length of this side group to three and five units does not alter the rate. The rate of ET between the donor and acceptor is not affected by this linear extension. A comparison of these results with those of the structures in red in Figure 1 illustrates how the topology of the side group affects the steady-state rate. The structures shown in red in Figure 1 in which a three unit side group is varied across the chain and their corresponding steady-state rates are shown in Figure 2. The one in which the side group is in the middle results in a higher rate than the ones in which the side group is close to either the donor or the acceptor. The position of the side group alters the energy gap between the bridge eigenstates and the donor and acceptor energies and when the side group is adjacent to the donor, increased population interference along the bridge results in a smaller rate. Similar effects are also found for the structure in which the side group is added adjacent to the acceptor. This effect is not as Nano Lett., Vol. 2, No. 5, 2002
important for longer bridge structures, however. As illustrated for a main bridge with five sites along the chain (shown in magenta in Figures 1 and 2), the effects of altering the topology and structure of a side group placed midway along the main bridge is negligible. For longer chains, it might be expected a larger side group is needed to alter the rate significantly. The structures in red show that the position of a threeunit side group across the chain changes the rate by altering the energy gap to the bridge and changing interferences along the bridge. In particular, interferences result in smaller rates when the side group is attached adjacent to either the donor or the acceptor. The violet structures in Figure 1 illustrate the topological effects on the rate due to side groups connected adjacent to both donor and acceptor sites and in the middle bridge sites. These structures have four bridge sites along the main chain and despite the different positions of the side-branches along the chain, the steady-state rate is similar, and these rates are only slightly higher than the rate through the linear chain. It is important to notice the difference compared with the shorter bridge lengths and structures (300) and (003). For the (1001) structure, the addition of side groups next to the donor and the acceptor appears to increase the rate slightly. In this case, the addition of side groups reduces the interference along the bridge. This effect is also illustrated by the larger branched structures with 6 bridge sites shown in cyan in Figure 1 and their corresponding rates shown in Figure 2. The steady-state rate through (100001), where side groups are adjacent to both donor and acceptor is slightly larger than the linear chain. However, for the structure (110000), even though it has the same number of side groups (two), the steady-state rate is appreciably smaller. For the structure (100001), the addition of side groups minimizes the energy gap between the bridge eigenstates and the donor and the acceptor more than for the structure (110000). When two more side groups are added to (110000), this energy gap and consequently the interferences are minimized, resulting in a rate that is slightly increased for the structure (111100). When side groups are added to all bridge sites the rate is increased nearly an order of magnitude. Such a cumulative effect has been pointed out in earlier study.16 Figure 2 illustrates some important observations: (i) addition of branched side groups along the bridge can enhance the steady-state ET rates, (ii) this enhancement depends on the position of the side group and its proximity to the donor and the acceptor, and (iii) additional side groups have a cumulative effect. All of these features are present for end-to-end ET through a dendrimer. In Figure 1, we show structures of both second, (131), and third, (13731), generation dendrimers in green. Their corresponding steady-state rates are shown in Figure 2. Tunneling dominates in the second-generation dendrimer (with N ) 3) and there is no increase in the rate over the linear chain. However, for the third generation dendrimer, the rate is significantly increased. Figure 2 demonstrates that the higher generation dendrimer shows enhanced transport in comparison to not only the linear structure but also the other branched structures of the same 439
bridge length. This is one of the main conclusions from this study: the dendrimer topology is well-suited to enhance electron transport. It is anticipated that these structures would show higher conductances as molecular wires and this is confirmed below. The nonequilibrium steady-state formalism used in this study provides a relatively simple method to simulate conduction in molecular wires. In recent break-junction experiments, the conductance of a molecule sandwiched between two electrodes may be determined by measuring the current allowed to pass from one electrode to the other through this molecular wire. As Nitzan and co-workers11 have shown, the conductance may be obtained directly from the steady-state rate using a generalization of the Landauer formula: g)
e2 l k πp m SS
x
me 2EF
(1)
where me is the mass of electron, EF is the Fermi energy of the metal electrode and lm is the characteristic size of the donor state. To calculate the conductance as a function of applied voltage, a voltage drop across the molecular wire is included using a simple linear ramp model8 by adding terms of the form Vi ) eV
i N+1
(2)
where e is the electric charge and V is the applied voltage. This adjusts the onsite energy of the ith site in the molecular bridge between the donor and the acceptor. The energies of the side groups attached to the ith site in the main bridge are also adjusted by this same term. Using reasonably typical values of EF ) 3 eV and lm ) 5 × 10-10 m, qualitative characteristics of the conductance as a function of applied voltage can be obtained. Within the voltage range studied here, the conductance decreases as the bridge eigenstates in the presence of an external field are moved away from resonance with the metal Fermi level. The results are shown in Figure 3. As expected, the model predicts that the conductance of the linear chains decreases with an increase in the chain length. The conductance of the second generation dendrimer (with N ) 3) is similar to the linear chain with 2 bridge sites and the conductance of the third generation dendrimer (with N ) 5) is comparable to a linear chain with 4 bridge sites and an order of magnitude larger than the linear chain with 6 bridge sites. This work is a study of the conductivity properties of branched and dendrimeric wires where we have used a steady-state formalism to compare the rates of electron transport through linear chain, branched and hyperbranched (dendrimer) structures. The approach taken provides a way of predicting the measurable conductance through a molecular wire sandwiched between two metal electrodes by first calculating the steady-state rate for a variety of molecular geometries. By varying the number, position, and nature of side groups (branches) attached to the bridge sites between the donor and acceptor, the role that these structural features 440
Figure 3. Conductance versus applied voltage for linear chains and second and third generation dendrimers.
play in molecular conductance can be studied. These calculations demonstrate a number of interesting and important properties and suggest ways that experimentalists may work to increase the conductance through molecular junctions. These simulations show that proximity of side groups to the acceptor site can affect the conductance quite dramatically. Side groups alter the density of states in the molecular bridge and change the energy gap between the bridge and the donor and acceptor and, consequently, affect interferences along the bridge in long-range electron transfer. In larger chains, the addition of side groups along the molecular wire generally increases the conductance. This increase cannot be explained only by the changes in the effective donoracceptor coupling matrix element. It is related to complicated dynamics along the bridge that depends sensitively on the dissipative effects of the stochastic nuclear motion coupled with the purely electronic effects. The effects of the side groups on conductance are cumulative, ensuring that the larger branched dendrimer structure, with side groups attached to all sites in the main bridge, shows the greatest increase in conductance relative to linear chains. Dendrimeric structures are expected to show enhanced conductance as a consequence of the position and topology of the sidebranches. By varying the energies of these side branches, many other interesting properties can be harnessed: the potential exists for switching, rectification, and trapping of electrons along the bridge. This study and previous work10,11 has illustrated that molecular conductance is not only dependent on the electronic properties of the bridge, but more importantly on the mechanism of electron transport due to the coupling to vibrational modes. It is likely that altering the coupling strength to the solvent and explicit vibronic relaxation of the side groups will affect the observed properties significantly. These, and other issues, will be discussed in much more detail in a subsequent full article. Acknowledgment. This work is supported by the National Science Foundation under Grant No. 9875230, the Petroleum Research Fund administered by the American Nano Lett., Vol. 2, No. 5, 2002
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