Length Dependence of Conductance in Aromatic Single-Molecule

Sep 14, 2009 - Veronika Obersteiner , Georg Huhs , Nick Papior , and Egbert Zojer. Nano Letters 2017 17 (12), 7350-7357. Abstract | Full Text HTML | P...
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NANO LETTERS

Length Dependence of Conductance in Aromatic Single-Molecule Junctions

2009 Vol. 9, No. 11 3949-3953

Su Ying Quek,† Hyoung Joon Choi,‡ Steven G. Louie,†,§ and J. B. Neaton*,† Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, Department of Physics and IPAP, Yonsei UniVersity, Seoul, Korea, and Department of Physics, UniVersity of California, Berkeley, California 94720 Received July 4, 2009; Revised Manuscript Received August 27, 2009

ABSTRACT Using a scattering-state approach incorporating self-energy corrections to the junction level alignment, the conductance G of oligophenyldiamine-Au junctions is calculated and elucidated. In agreement with experiment, we find G decays exponentially with the number of phenyls with decay constant β ) 1.7. A straightforward, parameter-free self-energy correction, including electronic exchange and correlations beyond density functional theory (DFT), is found to be essential for understanding the measured values of both G and β. Importantly, our results confirm quantitatively the picture of off-resonant tunneling in these systems and show that exchange and correlation effects absent from standard DFT calculations contribute significantly to β.

Individual molecular wires constitute the basic circuitry for the realization of single molecule electronic devices, and a quantitative understanding of their electronic conductance as a function of length is of fundamental importance in nanoscience.1 Whereas the conductance of macroscopic systems is inversely proportional to their length, the lowbias conductance G of single-molecule junctionssshort closed-shell molecules bonded to metal electrodes by thiol, amine and other end groupsshas been reported to decay exponentially with the number of molecular units N, G ∼ exp(-βN),1-7 a behavior that has been explained conceptually as off-resonant tunneling through evanescent states at the Fermi level EF.8-11 However, to date, quantitative comparisons between theoretically predicted and experimentally measured β are lacking. As a result, central questions about the length dependence of molecular junction conductance and off-resonance tunnelingsand the quantitative connection between β and the electronic structure of specific molecules and metal contactssremain open. Most recent first-principles conductance calculations rely upon density functional theory (DFT) within the local density approximation (LDA) or generalized gradient approximation (GGA). Within these common approximations and in general, DFT underestimates quasi-particle energy gaps between the highest occupied (HOMO) and lowest unoccupied (LUMO) orbital energies, leading to molecular resonances too close to EF. While the use of DFT orbital energies can result in a significant overestimate of the conductance (e.g. ref 12 and * Corresponding author: [email protected]. † Molecular Foundry, Lawrence Berkeley National Laboratory. ‡ Department of Physics and IPAP, Yonsei University. § Department of Physics, University of California. 10.1021/nl9021336 CCC: $40.75 Published on Web 09/14/2009

 2009 American Chemical Society

references therein), trends such as the decay constant β for wide-gap molecules (such as alkanes) have been reported to be in better agreement with experiment.13-15 However on physical grounds, shallow molecular resonances relative to EF should result in reduced tunneling barriers and, hence, more penetrating evanescent states at EF and a smaller β value. Indeed, for alkanedithiol-Au junctions, configurationinteraction (CI) conductance calculations,16 including nonlocal exchange and correlation, result in a predicted β value smaller by nearly a factor of 2 relative to DFT.14,17,18 A similar factor of 2 variation in the experimental β reported for alkanedithiol-Au junctions19-21 obscures a clear comparison between theory and experiment and, thus, has limited a clear understanding of the conductance mechanism in these junctions. Recent experiments have shown that the conductance of oligophenyldiamine-Au single-molecule junctions can be reliably and reproducibly measured.7 Conductance histograms constructed from thousands of measurements are peaked around well-defined values, and the peak conductance values from histograms taken from different data sets for the same molecule exhibit a standard deviation of only ∼10%, resulting in a β of 1.7 ( 0.1/ring.7 These robust experimental results provide an excellent benchmark system for quantitative comparison with theory. In this Letter, we develop a physically motivated parameter-free self-energy correction to DFT frontier orbital energy alignment and show it to be essential to elucidate the nature of tunneling in junctions formed with aromatic molecules. Applying our self-energy corrected approach to likely oligophenyl-Au junction geometries, we compute conductance values that are in close

agreement with experiment. Our calculated conductance values decay exponentially with the number of phenyl rings with an β of 1.7, giving quantitative confirmation of an offresonance tunneling mechanism for conductance of these junctions. In our calculations, junction geometries are constructed from the knowledge that the amine group binds selectively to undercoordinated atop Au sites and that the variation of conductance with contact geometry is relatively small for BDA.12 We relax all junctions using DFT within GGA (PBE)22 as implemented in SIESTA.23 (Details of our DFT calculations follow those given in ref 12.) The conductance is obtained from the Landauer formula using a scatteringstate approach.24,25 To correct for inaccuracies in using DFT Kohn-Sham eigenvalues for quasi-particle energy level alignment in the junction, we employ a physically motivated electron self-energy correction to the molecular orbital energies in the junction, consisting of two parts, initially introduced in a more approximate form in ref 12: first, a “bare or molecular” term correcting for the difference between DFT HOMO and LUMO energies and the ionization potential (IP) and electron affinity (EA) of the gas-phase molecule computed from total energy differences;26 and second, an “image-charge” term accounting for the effect of electrode polarization on the energy of the added electron (LUMO) or hole (HOMO)27 which is calculated from the charge distribution of the LUMO/HOMO and including polarization effects from both electrodes. This junctiondependent self-energy has no adjustable parameters and is found to give an accurate description of quasi-particle level alignment for weakly coupled molecule-metal substrate systems where the frontier orbitals are not significantly altered by coupling to the surface.27 In this work, the self-energy correction is added explicitly to the scattering-state Hamiltonian24 by an orbital-dependent term of the form Σˆ ) ∑n∆n|ψnmol〉〈ψnmol|, where |ψnmol〉 denotes an eigenstate of the isolated molecule and ∆n is the self-energy corrections for the nth molecular level. In the present study, for simplicity, we compute ∆n for only the HOMO and LUMO, as described above, and apply ∆HOMO to all occupied states, and ∆LUMO to all unoccupied states. (Since the HOMO dominates the zero-bias conductance for the oligophenyldiamine system (see below), this approximation is reasonable.) The modified Hamiltonian is then solved for the corrected scattering states and transmission function in a “one-shot” calculation using the DFT GGA (PBE) charge density, an approach we refer to as DFT+Σ. Our calculations indicate that the gas-phase correction to the HOMO is ∼-2.9, -2.4, and -2.1 eV for one, two, and three phenyl rings, and inclusion of electrode polarization effects changes the self-energy correction significantly to ∼-1.9, -1.6, and -1.4 eV, respectively. Since these corrections are large compared to those for the metallic bulk and surface Au states, especially for states near EF, we neglect self-energy corrections to the Au states. Figure 1a shows relaxed geometries of oligophenyldiamineAu junctions with one, two, and three phenyl rings (denoted by BDA, DBDA, and TBDA, respectively) bonded to a Au 3950

Figure 1. (a) Optimized structures of BDA, DBDA, and TBDA trimer junctions. Gold, gray, blue, and pale blue circles denote Au, C, N, and H, respectively. Transmission spectra for trimer (red) and adatom (black dashed) junctions calculated within (b) DFT and (c) DFT+Σ. Gray arrows span the HOMO-LUMO gaps (labeled). (d) Red and blue denote positive and negative isocontours taken at 10% of the maximum value for the HOMO of the isolated molecules (left) and at 2.5% (BDA and DBDA) and 1.25% (TBDA) of the maximum value for the eigenchannel wavefunctions in the junctions (right). The arrow denotes the direction of incident states in the junctions.

trimer motif, consisting of an adatom binding site coordinated by two adatoms, on Au(111). The optimized dihedral angles between phenyl rings are ∼35° for all junction geometries considered.28 Parts b and c of Figure 1 show the transmission computed using DFT and DFT + Σ for trimer and adatom contact motifs. In all cases, transmission at EF occurs through a single eigenchannel that is predominantly derived from the HOMO, as determined by examination of the eigenchannel wavefunction symmetries (Figure 1d). The additional features in the transmission spectra between -1.5 and -2.0 eV in Nano Lett., Vol. 9, No. 11, 2009

Figure 3. Right (left) panels: Real (imaginary) band structures of periodic chain with different band gaps Eg. Right: Highest valence and lowest conduction bands. Left: Complex band corresponding to the smallest imaginary k within the gap. The energy E is plotted against β(E) ) 2aIm(kmin(E)). All other bands are explicitly omitted. Black solid curve: DFT result (Eg ) 2.4 eV). Red dashed (blue dotted) curves: Results after a scissors shift is applied, to give band gaps matching the average DFT (DFT+Σ) molecular gaps in the junctions. The energy is referenced relative to the valence band maximum of the DFT band structure. Inset: Atomic geometry of a repeat unit of the periodic polyphenyl chain. The dihedral angles between adjacent rings alternate between (35°; the transmission spectra for TBDA junctions are very similar whether the dihedral angles are (35°, or both 35°.

Figure 2. (a) Calculated DFT (black filled symbols) and DFT+Σ (red hollow symbols) conductances plotted against the experimentally measured conductance peak positions on a log scale. Each cluster of data points, labeled by the number of phenyl rings N, represents conductance values calculated for oligophenyldiamine-Au junctions with different geometries. (b) Plot of ln(G/G0) against the number of phenyl rings N. The linear fit to ln G vs N is excellent for the full range of N from 1 to 7, for both DFT+Σ and DFT (inset). The fits give β ) 1.7 (1.3) for DFT+Σ (DFT).

Figure 1c arise from the HOMO hybridizing with the d states of the undercoordinated Au contact atoms. As evident from Figure 2, DFT (GGA-PBE) leads to significant differences with experiment for both the absolute conductance and its trend with length. However, DFT+Σ brings both G and β into much better agreement with measurements7 for all junctions considered.29 Remarkably, the conductances are computed to be within about a factor of 1.5 of experiment with DFT+Σ, as opposed to 6-20 times larger for DFT (Figure 2a). (Similar errors in DFT GGA were obtained in an independent calculation on BDA-Au junctions.30) Moreover, DFT+Σ results in a β of 1.7, in excellent agreement with the measured value of 1.7 ( 0.1,7 whereas the DFT β (within GGA) is predicted to be only 1.3. The quantitative agreement with experiment achieved with self-energy corrections provides verification that the conductance mechanism for oligophenyldiamine-Au junctions is in fact off-resonant tunneling, and shows that manyelectron effects absent in standard DFT calculations contribute significantly to β: The larger value of β predicted with DFT+Σ is a direct result of the junction EF being further from the Σ-corrected molecular resonances and a more abrupt Nano Lett., Vol. 9, No. 11, 2009

decay of evanescent junction quasi-particle states in the larger DFT+Σ molecular energy gap. In the limit of long oligomers,31 the decay constant β can be related to the complex band structure of the corresponding extended (periodic) polymer17,18,31-35 via β(E) ) 2aIm(kmin(E)), where kmin is the wavevector with the smallest imaginary component (representing the most penetrating evanescent state) and where a ) L/N, L is the molecular length, and β ) β(EF). In this analysis, the metal contacts influence β only through placement of the junction EF in the complex band structure.17,18,31-35 In Figure 3, we plot the DFT-GGA complex band structure of a periodic polyphenyl chain. Notably, the DFT polyphenyl gap (2.4 eV) is significantly smaller than those determined for short oligomers in the junctions, all of which also change with N. (Averaged over N ) 1-3, these gaps are 3.4 (DFT) and 6.0 eV (DFT+Σ).) The difference in these gaps leads to ambiguities in positioning EF in the DFT complex band structure. To illustrate, we add a “scissors-shift” operator to the Hamiltonian (obtained within our DFT+Σ approach, varying Σ by hand) and recompute the complex band structure with gaps of 3.4 and 6.0 eV (Figure 3). The placement of EF in the complex band structure is then estimated by setting the average HOMO position in the junctions (averaged over N ) 1-3) to the valence band maximum in the complex band structure. This gives β ) 2.0 for DFT+Σ, much too large relative to experiment. To avoid the ambiguities associated with the energy gap and Fermi level position in the above analysis, we introduce an “effective complex band structure”, βN,N′(E), obtained as the negative of the natural logarithm of the ratio of the junction transmission functions for molecules of different 3951

Figure 4. (a) DFT and DFT+Σ β1,2(E) (solid) and β1,3(E) (dashed) curves for trimer junctions, where β1,2(E) ) -ln(TDBDA(E)/TBDA(E)) and β1,3(E) ) -ln(TTBDA(E)/TBDA(E))/2. (b) DFT+Σ β1,3(E) curves for Au and Ag trimer junctions: red, Au; green, Au replaced with Ag in Au junction geometry; blue, Ag (relaxed).

lengths N and N′, divided by (N - N′). Figure 4a reports β(E) calculated from the DFT and DFT + Σ transmission for junctions with trimer bonding motifs. Three features stand out. First, the differences between DFT and DFT+Σ curves are significant, reflecting the importance of self-energy corrections to DFT for interpreting conductance for the oligophenyldiamine system. Second, while these semielliptical curves are qualitatively similar to the complex band structure shown in Figure 3, they also exhibit prominent, contact-dependent attributes near the HOMO and LUMO resonances that stem from changes in level alignment with N. Finally, for both DFT and DFT+Σ, β1,2(E) and β1,3(E) are nearly identical near EF and for energies away from the resonances, indicating that T(E) ∼ exp(-β(E)N) over a broad energy range. The energy dependence of the β(E) curves in Figure 4a reveals a strong contact dependence and suggests that if the Fermi energy position were controllably variedsthrough a different link chemistry or electrode work functionsthe magnitude of conductance with length could be significantly altered. For example, the smaller work function of Ag (∼4.7 eV) relative to Au (∼5.2 eV) implies a positive shift of EF of about 0.5 eV and a concomitant increase in β. To investigate this in more detail, we replace Au with Ag in the trimer junctions and repeat all calculations. With Ag contacts, the DFT+Σ conductance drops to 4.6, 0.6, and 0.080 × 10-3 G0 for relaxed BDA, DBDA, and TBDA junctions, respectively, and we predict a β of ∼2.0, significantly larger than that found for Au junctions. However, noticeable differences in the form of β(E) are observed: β(E) for Au and Ag are in fact not related by a simple energy shift equal to the difference in their work functions, especially after the molecule-Ag junction geometries are fully optimized (Figure 4b). This comparison indicates that for finite oligomers, the β(E) computed from transmission calculations is itself highly contact-dependent, in contrast to the complex band structure of the periodic polymer. We expect similar 3952

differences in β(E) for oligophenyl-Au junctions with different link groups, such as thiols instead of amines. It is interesting that experiments on oligophenylthiol-Au selfassembled monolayers36 report a β of 1.8 per phenyl ring, very close to that measured for oligophenyldiamine-Au single molecule junctions (1.7 ( 0.17). However, the fact that the measured numbers are almost identical may be fortuitous because the contact coupling is expected to be much larger in the thiol case, and the level alignment changes with the extent of charge transfer, which would in general differ for different link groups. In conclusion, we have confirmed quantitatively the picture of off-resonant coherent tunneling in oligophenyldiamine-Au junctions and have shown that a straightforward, parameterfree self-energy correction to DFT molecular orbital energies in the junction is necessary for agreement with the experimentally measured β. For relatively short oligophenyl molecules in the tunneling regime, the β(E) computed from transmission calculations is extremely sensitive to the junction level alignment, and quantitatively different from that obtained from the complex band structure of the extended polymer. For sufficiently large N, β(E) should be given by the polymer complex band structure,31 provided that electron self-energy effects are properly taken into account. However, a crossover from coherent to incoherent tunneling with length for longer molecules2,3 may limit the utility of the complex band structure in this case. Acknowledgment. We thank L. Venkataraman for providing us with numbers for the standard deviation in the conductance peak positions. Portions of this work were performed at the Molecular Foundry, Lawrence Berkeley National Laboratory, and were supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy. This work was supported in part by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering Division, U.S. Department of Energy under Contract No. DE- AC02-05CH11231, and by computational resources from NERSC and the KISTI Supercomputing Center (KSC2008-S02-0004). H.J.C. acknowledges support from the KRF (KRF-2007-314-C00075) and the KOSEF (Grant No. R012007-000-20922-0). References (1) Lafferentz, L.; Ample, F.; Yu, H.; Hecht, S.; Joachim, C.; Grill, L. Science 2009, 323, 1193–1197. (2) Choi, S. H.; Kim, B.; Frisbie, C. D. Science 2008, 320 (5882), 1482– 1486. (3) Davis, W. B.; Svec, W. A.; Ratner, M. A.; Wasielewski, M. R. Nature 1998, 396 (6706), 60–63. (4) He, J.; Chen, F.; Lindsay, S.; Nuckolls, C. Appl. Phys. Lett. 2007, 90 (7), 3. (5) Engelkes, V. B.; Beebe, J. M.; Frisbie, C. D. J. Am. Chem. Soc. 2004, 126 (43), 14287–14296. (6) Wang, W. Y.; Lee, T.; Reed, M. A. Phys. ReV. B 2003, 68 (3), 7. (7) Venkataraman, L.; Klare, J. E.; Nuckolls, C.; Hybertsen, M. S.; Steigerwald, M. L. Nature 2006, 442 (7105), 904–907. (8) Nitzan, A. Annu. ReV. Phys. Chem. 2001, 52, 681–750. (9) McConnell, H. J. Chem. Phys. 1961, 35 (2), 508–&. (10) Magoga, M.; Joachim, C. Phys. ReV. B 1997, 56 (8), 4722–4729. (11) Mujica, V.; Nitzan, A.; Mao, Y.; Davis, W.; Kemp, M.; Roitberg, A.; Ratner, M. A. AdV. Chem. Phys. 1999, 107, 403–429. Nano Lett., Vol. 9, No. 11, 2009

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