LETTER pubs.acs.org/NanoLett
Molecular Conduction through Adlayers: Cooperative Effects Can Help or Hamper Electron Transport Matthew G. Reuter,*,†,‡ Tamar Seideman,† and Mark A. Ratner† †
Department of Chemistry, Northwestern University, Evanston, Illinois 60208-3113, United States
bS Supporting Information ABSTRACT: We use a one-electron, tight-binding model of a molecular adlayer sandwiched between two metal electrodes to explore how cooperative effects between molecular wires influence electron transport through the adlayer. When compared to an isolated molecular wire, an adlayer exhibits cooperative effects that generally enhance conduction away from an isolated wire’s resonance and diminish conductance near such a resonance. We also find that the interwire distance (related to the adlayer density) is a key quantity. Substrate-mediated coupling induces most of the cooperative effects in dense adlayers, whereas direct, interwire coupling (if present) dominates in sparser adlayers. In this manner, cooperative effects through dense adlayers cannot be removed, suggesting an optimal adlayer density for maximizing conduction. KEYWORDS: Electron transport, cooperative effects, adlayers, tight-binding models
he flow of electric current through molecules (electron transport) and the transfer of charge between molecules and substrates are interesting problems for both their inherent conceptual issues and their applications to solar cells, thermoelectrics, scanning probe microscopies, and molecular electronics.14 Even though electron transport through a single molecular wire is reasonably well understood,3,4 cooperative effects5 between molecular wires complicate the scaling of these results to multiple molecular wires in parallel.622 Using Ohm’s law,23 conventional electronics predicts that the conductance through N wires is simply N times an isolated wire’s conductance
T
GN ¼ NG1
ð1Þ
however, previous studies have found that molecular wires obey a pseudo-Ohm’s law7 GN ¼ NGeff 1
ð2Þ
where Geff 1 is an effective single wire conductance that should be independent of N. 68,12,14,18,21,22 In many cases, cooperative effects cause Geff 1 > G1, 8,1113,18,22 although cases where Geff < G have been reported. 1 1 Besides our fundamental interest in how cooperative effects change G1 into Geff 1 , this phenomenon has practical implications. For instance, analyses of certain experimental setups, including several mechanical break junctions and scanning probe microscope-based junctions, often assume that only a single molecule connects the electrodes when electron transport is measured.4 In actuality, there could be multiple molecules, and understanding when cooperative effects increase (decrease) the conductance per wire may aid in the interpretation of these experiments. Moreover, cooperative effects should be visible in conductance histograms (methods for r 2011 American Chemical Society
statistically analyzing experimental data),24 and the ability to infer more about the underlying physics of conduction from existing data would be an exciting advance. Computational studies using ab initio methods (usually within the density functional theory) have been used to investigate such changes in per-wire conductances,7,8,14,15,17,21 finding that cooperative effects (i) arise from direct interwire coupling and substrate-mediated coupling8,14,15 and (ii) can cause significant electrostatic effects at the molecule-electrode interfaces.8,17 Despite this effort, ab initio methods exhibit several problems that have confounded the development of a set of guidelines for interpreting the underlying physics of cooperative effects in electron transport. First, ab initio methods convolute many effects together (including any cooperative or electrostatic effects), making it difficult to attribute changes in conductance to a specific source. Second, numerous approximations are required to calculate electron transport with ab initio methods.2 Alternatively, simple model systems (e.g., tight-binding models) can provide qualitative, mechanistic insights and, in some cases, exact, analytical results at the expense of quantitative accuracy. We recently investigated this change in conductance per wire using tight-binding model systems with two molecular wires,22 generally finding that Geff 1 > G1 when the Fermi level is far from an isolated wire’s resonance and Geff 1 < G1 when the Fermi level is near such a resonance. The combination of direct, interwire coupling (e.g., ππ interactions) and substrate-mediated coupling, each a source of cooperative effects, causes this behavior by energetically shifting the system’s conduction channels away Received: July 10, 2011 Revised: September 19, 2011 Published: October 18, 2011 4693
dx.doi.org/10.1021/nl202342a | Nano Lett. 2011, 11, 4693–4696
Nano Letters
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Figure 1. (a) Cartoon depiction of an example adlayer. (b) System schematic of the adlayers modeled in this work. Each molecular wire (maroon circle) sits atop an electrode site (light orange circle), and is separated from its four nearest neighbors by (m2 + n2)1/2 electrode sites.
from those of an isolated wire. Separating the molecular wires reduces the impact of cooperative effects on conduction; however, an interference effect between the two couplings allows unconventional methods to control cooperative effects. Specifically, balancing the direct and substrate-mediated couplings removes cooperative effects in systems with two wires. In this Letter, we extend our analysis to systems of uniformly dense adlayers, seeking to establish the relationship between G1, Geff 1 , and the Fermi level as the number of parallel molecular wires increases from two to infinity. This generalization is important for several reasons. First, eq 2 is approximate and is more accurate when N is large.18 Second, adlayers are generally easier to characterize and have applications in sensing, logic, and computation.2427 As before,22 we use a one-electron, tight-binding model within the LandauerImry (coherent scattering) formalism,28 yielding exact, semianalytical (i.e., analytical up to numerical quadrature) results. After introducing our model, we calculate electron transport properties through adlayers of various densities. In addition to confirming that Geff 1 is usually larger than G1 far off resonance (and less than G1 near resonance), we find that in certain cases there is an optimal adlayer density for maximizing electron transport through the adlayer. We model each electrode by a semi-infinite cubic lattice of single-state sites, where each site has energy ε and couples to its nearest neighbors with element Ve.18,22,29,30 Likewise, each molecular wire is represented by a single state with energy α; this state is conceivably the highest occupied or lowest unoccupied molecular orbital. We construct a uniformly dense adlayer by placing wires in a regular grid, as depicted in Figure 1. Each wire is m electrode sites in one direction and n in the other from its four nearest neighbors; such an adlayer is denoted (m,n) and has density (m2 + n2)1. For simplicity, we limit direct, interwire coupling to nearest neighbors, and assume that this direct coupling, β, is real. The wire Hamiltonian, Hwire, is explicitly stated in Section 1 of the Supporting Information. Next, each wire sits atop a single electrode site (as shown in Figure 1), and couples to this site with element V. The electrode-wire coupling operator, V, is also discussed in Section 1 of the Supporting Information. Finally, we appeal to the detailed discussion of these parameters in Section 5 of ref 18 for choosing values representative of realistic systems. Specifically, Ve = 0.82 eV, ε = 0 eV (centering the metal band around E = 0), and V = 0.45 eV. Due to our focused interest in electron transport through the adlayer of molecular wires, we write an effective Hamiltonian for the wires, H f Hwire + 2Σ(E), where Σ(E) is the self-energy of coupling the wires to an electrode1 and the factor of 2 reflects
Figure 2. Transmission (per wire) spectra for molecular wire adlayers of various densities. The density of adlayer (m,n) is (m2 + n2)1. In the absence of direct, interwire coupling (β = 0), cooperative effects dramatically change the transmission spectra of dense adlayers when compared to the transmission through an isolated wire (top rows). Since the substrate-mediated coupling decays as the density decreases, the transmission spectra approach that of an isolated wire as m and n increase, as expected. While the addition of direct coupling causes small changes in the transmission through dense adlayers, it seems unable to remove cooperative effects, unlike systems with two wires.22 Direct coupling tends to amplify cooperative effects in sparser adlayers.
identical coupling of the wires to both electrodes (for simplicity). Formally,1 Σ(E) = VGelec(E)V†, where Gelec(E) is the (retarded) Green function of an isolated electrode, which is semianalytical for these electrodes.30 Within a coherent scattering formalism, the zero-bias conductance per wire at zero temperature is Gð0Þ ¼
2e2 TðEF Þ h
ð3Þ
where EF is the Fermi level, and after exploiting identical coupling to both electrodes the transmission (per wire) through the adlayer is TðEÞ ¼ Tr½ΓðEÞGðEÞΓðEÞG† ðEÞ
ð4Þ
For use in eq 4, Γ(E) is the spectral density, Γ(E) = i[Σ(E) Σ†(E)], and G(E) is the Green function of the adlayer connected to both electrodes, GðEÞ ¼ ½EI Hwire 2ΣðEÞ1
ð5Þ
where we assume the wire states are orthonormalized. While omitted for clarity, our model produces analytical or semianalytical forms for all of these quantities, which are derived and discussed in Sections 2 and 3 of the Supporting Information. We begin by examining the transmission through an isolated wire, that is, β = 0 and m, n f ∞, which provides a basis for comparison. This transmission, denoted Tiso(E), is depicted in Figure 2. As is typical of conduction through a single molecule,3 Tiso(E) is quasi-Lorentzian, where the sole maximum corresponds to the wire’s resonance. This resonance signifies that the 4694
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Figure 3. The conductance (per wire) through an adlayer relative to that through an isolated wire as a function of EF, each wire’s site energy (α), and the direct, interwire coupling (β). The solid black curves trace a ratio of one; cooperative effects do not affect conductance in these cases. At the isolated wire’s resonance (E1R, dashed black lines), cooperative effects never increase conductance over the isolated wire. Moreover, only sufficiently sparse adlayers attain the isolated wire conductance near resonance. Off resonance, cooperative effects tend to enhance conductance by increasing the density of wire states at these Fermi levels.18 Note that E1R 6¼ α, in general, due to molecular level shifting induced by the electrodes; see Section 2 of the Supporting Information for details.
conduction channel is completely open [Tiso(E1R) = 1], and occurs at the energy E1R, which is shifted from α by the coupling to the electrodes [Σ(E1R), see eq 5]. Figure 2 also shows the transmission spectra through adlayers of various densities. Dense adlayers (those with smaller m and n, see Figure 1) exhibit substantially different transmission spectra from that of the isolated wire. Transmission decreases near the isolated wire’s resonance and increases away from this resonance. Moreover, substrate-mediated coupling induces most of the cooperative effects in dense adlayers, as evidenced by the fact that variations in the direct coupling produce only modest changes in the transmission. Sparser adlayers [here (0,2) and less dense] largely mimic isolated wires in the absence of direct coupling, confirming that substrate-mediated coupling decays as the distance increases.7,14,22,30 In these adlayers, direct coupling is the primary source of cooperative effects, when present (interwire coupling is also expected to decay with the interwire distance). In this manner, cooperative effects diminish as the interwire distance increases, yielding a true Ohm’s law relationship [eq 1]. Similar to their role in systems with two wires,22 cooperative effects in adlayers tend to decrease transmission near the isolated wire’s resonance (E1R), while increasing it away from E1R. The isolated wire’s conduction channel is completely open at resonance, and cooperative effects cannot improve the transmission. Conversely, cooperative effects lead to a band of wire states (generalizing the energetic splitting seen in two wires22), and the availability of more wire states in the adlayer increases transmission off resonance.18 Translating to conductance with eq 3, these results confirm the generalization of our findings from systems with two wires to adlayers. As displayed in Figure 3, Geff 1 > G1 off resonance and Geff 1 < G1 near resonance, in most cases. Unlike for systems with two wires, the adlayer density, which fixes the interwire distance, becomes an important factor in electron transport through adlayers. As we previously showed,22
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Figure 4. The effects of interwire distance, (m2 + n2)1/2, on transmission/conductance (per wire) for various energies. (a) The amount of direct coupling that maximizes transmission, βmax, for systems with two wires.22 (b) βmax for adlayers. (c) The corresponding maximum transmission (per wire) for adlayers. In both cases, βmax f 0 as the distance increases, as expected. Unlike systems with two wires, which always have maximum transmissions (per wire) of one22 (indicating the net cancellation of cooperative effects), cooperative effects cannot be negated in dense adlayers. This suggests an optimal adlayer density for maximizing conduction through adlayers.
cooperative effects between two wires could always be negated at an arbitrary energy, regardless of interwire distance, by balancing the direct and substrate-mediated couplings. This effect is depicted in Figure 4a, where we plot the value of direct coupling required to remove cooperative effects, βmax, as a function of the interwire distance for systems with two wires. Since substratemediated coupling decays with distance, βmax approaches 0 with increasing distance, and in this sense the convergence of βmax toward 0 estimates the persistence of cooperative effects. At the isolated wire’s resonance (E1R), βmax can also be interpreted as the amount of direct coupling that maximizes transmission/ conductance (per wire) through the wires. In systems with two wires, this maximum transmission (per wire) is always one.22 A close examination of Figures 2 and 3, though, suggests that dense adlayers do not realize transmissions (per wire) of one at E1R, partly resulting in our general conclusion that Geff 1 < G1 near resonance. Investigating further, Figure 4b plots βmax for adlayers and Figure 4c shows each adlayer’s corresponding maximum transmission. Akin to systems with two wires, adlayers exhibit larger |βmax| when the wires are close; |βmax| decays to zero as the wires separate. On the other hand, the maximum transmission (per wire) through an adlayer is only one after the wires are sufficiently separated. This result seems unintuitive at first but is easily explained. In two-wire systems, the direct coupling (βmax) balances the substrate-mediated coupling, a quantity that strongly depends on the interwire distance. With only one pair of wires, the choice of βmax is optimal, and high transmissions are realized. In adlayers, conversely, we cannot find a βmax that simultaneously balances the substrate-mediated couplings to all other wires unless the substrate-mediated couplings are uniform. This only occurs when the adlayer is sufficiently sparse, that is, all substratemediated couplings are zero. In the context of maximizing conductance through an adlayer, this suggests an optimal adlayer density. When EF ≈ E1R, we put as many molecules as possible in 4695
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Nano Letters the adlayer (maximizing the number of conduction channels), but require them to be sufficiently separated to minimize cooperative effects. Away from resonance (EF 6¼ E1R), cooperative effects tend to increase conductance, and the optimal adlayer density is instead limited by the adsorption chemistry (the number of molecules in the adlayer). In summary, we have presented a semianalytical formulation of electron transport through parallel molecular wires in uniformly dense adlayers, exploring the role of cooperative effects (i.e., direct, interwire coupling and substrate-mediated coupling) on transmission and conductance. Reminiscent of inelastic effects,31,32 cooperative effects generally increase conductance (Geff 1 > G1) when the Fermi level is far from an isolated wire’s resonance and decrease it (Geff 1 < G1) near such a resonance. These results help explain previous observations regarding changes in the conductance per wire as the number of wires varies.616,1821 Although our model is of single-electron type and thus neglects interelectronic interactions, it has the significant advantage of providing exact answers, much like the Newns self-energy or the Ising model. The effects of electronic repulsion can, to some extent, be introduced into one-electron models by reparameterization; however, true multielectron effects, such as correlation, will be absent from our results. For several reasons, the interwire distance, or equivalently, the adlayer density, is a key property. First, substrate-mediated coupling dominates cooperative effects in dense adlayers, whereas direct coupling (if present) induces most cooperative effects in sparser adlayers. Second, only in the absence of substratemediated coupling (sparser adlayers) can wires in an adlayer behave like isolated wires. Such net negation of cooperative effects requires the simultaneous balance of direct and substratemediated couplings between all wire pairs, which is only possible when the substrate-mediated coupling is uniform (and thus zero). Taken collectively, these results suggest optimal adlayer densities for maximizing current through adlayers. Near an isolated wire’s resonance the wires need to be sufficiently separated to eliminate substrate-mediated coupling, but away from such a resonance cooperative effects usually increase conductance, favoring denser adlayers.
’ ASSOCIATED CONTENT
bS
Supporting Information. A thorough description of our model, including derivations for all of the analytical and semianalytical results, is presented in the Supporting Information. This material is available free of charge via the Internet at http:// pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Present Addresses ‡
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, U.S.A.
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for a fellowship. We thank the NSF (Grant CHE-1012207) and the MRSEC program of the NSF (DMR-0520513) for support.
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’ ACKNOWLEDGMENT We are grateful to Abraham Nitzan, Gemma C. Solomon, Thorsten Hansen, and Lisa A. Fredin for helpful conversations. M.G.R. thanks the DoE CSGF (Grant DE-FG02-97ER25308) 4696
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