Letter pubs.acs.org/NanoLett
Achieving Predictive Description of Molecular Conductance by Using a Range-Separated Hybrid Functional Atsushi Yamada,† Qingguo Feng,† Austin Hoskins,† Kevin D. Fenk,† and Barry D. Dunietz*,† †
Department of Chemistry and Biochemistry, Kent State University, Kent, Ohio 44242, United States S Supporting Information *
ABSTRACT: The conductance of molecular bridges tends to be overestimated by computational studies in comparison to measured values. While this well-established trend may be related to difficulties for achieving robust bridges, the employed computational scheme can also contribute to this tendency. In particular, caveats of the traditional functionals employed in first-principles-based calculations can lead to discrepancies reflected in exaggerated conductance. Here, we show that by employing a range-separated hybrid functional the calculated values are within the same order as the measured conductance for all four considered cases. On the other hand, with B3LYP, which is a widely used functional, the calculated values greatly overestimate the conductance (by about 1−2 orders of magnitude). The improved description of the conductance with a RSH functional builds on achieving a physically meaningful treatment of the quasi particles associated with the frontier orbitals. KEYWORDS: Molecular junctions, electron transport, generalized Kohn−Sham, Green’s function, Landauer equation, range-separated hybrid functional
M
constant fraction of the local density approximation (LDA) in the exchange part. However, these functionals typically suffer from a lack of the derivative discontinuity in the exchangecorrelation part13−15 and therefore tend to underestimate the fundamental orbital gap.13,15−20 In particular, such functionals have been associated with the tendency to overestimate the conductance.5,21−23 These limitations have been addressed through various correction schemes that improve the agreement with benchmark measurements.11,24−28 Nevertheless the salient trend of the formative GF-DFT scheme to overestimate the transport persists.25,28,29 Sublimely, a predictive quality can be achieved by an ab initio approach that reproduces benchmark conductance measurements without resorting to ad hoc corrective measures. High-quality GW-based calculations that address the electrode self-energies have achieved improved agreement with measured conductances.30,31 However, the high computational cost of such methods limits their wide applicability. Therefore, it remains desirable to design a DFT-based approach to achieve predictive quality in calculating molecular conductance using relatively large molecular bridge models. Indeed, a promising route to achieve this goal is presented by a generalized Kohn−Sham (KS) DFT framework of long-range corrected (LRC)32,33 or optimally tuned range separated hybrid (RSH)13,34,35 functionals. These relatively novel and of high quality functionals address well-documented DFT caveats by
olecular scale bridges attract a wide research attention both through experimental efforts of their fabrication and characterization and through computational modeling of their structure−function relationships.1−3 Ostensibly, advances in both fronts are needed to gain understanding of related fundamental physical phenomena and ultimately to enhance the use of such bridges in technological applications. More recently molecular scale bridges have been promoted as a viable route to improve energy conversion based on thermoelectric functionality.4 Alas, the progress in the field has been obstructed by the discrepancies between measured conductances and calculated values. A notorious trend of the computational studies is to overestimate the conductance.1,5 The challenges in realizing well-characterized bridges, which are required for establishing robust molecular models,6−8 are likely contributing to this trend. Indeed physical processes as due to thermal fluctuations and surface reorganization are expected to strongly affect the conductance.1,6,8−12 Nonetheless, to acquire understanding of the molecular bridges activity a reliable electronic structure perspective to address the coupling and nonequilibrium aspects of the conducting molecular bridges is required. Unequivocally, achieving such a physically meaningful electronic structure treatment of the bridges is challenging as both the solid state properties of the electrodes and the molecular properties of the bridges have to be addressed at a well balanced quality. The workhorse for calculating molecular scale electron transport has become the combination of Green’s function (GF) formalism with density functional theory (DFT). Most GF-DFT treatments are based on functionals that bear a © XXXX American Chemical Society
Received: June 3, 2016 Revised: September 5, 2016
A
DOI: 10.1021/acs.nanolett.6b02241 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters invoking a distance related switching between expressions in the exchange-correlation part of the functional. According to this framework, essentially the exact exchange is turned on for long enough interactions.15,36−38 Of formidable importance is the established success upon tuning the parameter to achieve physically relevant orbital energies,13,15,19,34,37,39−42 where the fundamental orbital gap increases properly. The importance of proper energy alignment of the orbital energies in conductance calculations has been shown earlier through a GW-based approach.30,31 We emphasize that the RSH functional achieves the correct frontier orbital gap alignment through a firstprinciples based scheme with no arbitrary shift of orbital energies. An underestimated orbital gap, that is associated with traditional DFT, is expected to increase the transmission at the electrodes Fermi energy (FE) and therefore to overestimate the conductance at low bias.5,25,28 We demonstrate this trend in Figure 1 with the B3LYP transmission that is shown to be
Figure 2. The molecular gold bridges of (a) BDT, (b) HDT, (c) BDMT, and (d) BPDT. Inset: The gold clusters (Au931) included in the bridge models, where the broken line indicates the region included in the GF calculations.
Figure 1. Characteristic transmission spectra calculated using B3LYP and BNL functionals for a 1,4-dithiol phenyl bridge. The BNL transmission peaks around the FE are shifted away in energy relative to those of B3LYP.
addition to the two outmost tip layers.) We find the following key structural features of the thiol-gold bonding: • BDT with AuS bond length of 2.490 Å and CSAu angle of 106.5°. • HDT with an optimized AuS bond length of 2.476 Å and a CSAu angle of 102.1°. • BDMT with AuS bond length of 2.417 Å and CS Au angle of 102.8°. • BPDT with AuS bond length of 2.530 Å and CSAu angle of 102.1°. (The atomic coordinates of the optimized four systems are provided in the SI.) The molecular bridges are then constructed by adding to the optimized molecules electrode models of hexagonal close packed (hcp) Au crystal shown in Figure 2. In these tip models, the outermost layer is of a single gold adatom forming the first layer that sits on the hollow position of three gold atoms in the second layer and completed with seven layers of nine atoms denoted as Au931. The AuAu bond distance is 2.884 Å with a corresponding interlayer Au distance (in the 111 crystal) of 2.355 Å. In calculating the transmission, the central region includes a total of six layer with four of the larger gold atom layers (see the inset in Figure 2). Consistent with previous studies, we find that including a total of six gold layers provides a wellconverged result.28,45 In SI Figure S1, we compare the transmission using a different number of gold layers included in the region used for the GF calculation. Interestingly the BNL transmission seems to be less sensitive to the partitioning choices than in the case of B3LYP. Computational Details. In molecular bridges, upon contacting the electrodes their electronic band structure is projected on the otherwise discrete molecular-based electronic
about 2 orders of magnitude higher than that of the Baer− Neuhauser−Livshits (BNL) functional,13,34 which is an optimally tuned RSH functional. The BNL frontier orbitals exhibit a larger energy gap and therefore are shifted further away from the FE. Recently the BNL functional was used to study electron transmission in a donor−acceptor based molecular bridge.29,43,44 Below we quantify the effect of using such an advanced functional on the calculated conductance of several benchmark systems. Molecular Bridge Models. We consider four bridges that are all relatively widely studied both computationally and experimentally. The four molecular bridges that are shown in Figure 2 are contacted to the electrodes through thiol groups and are as follows: • benzene ring with thiols at para positions (BDT; 1,4benzeneditiol); • hexane chain with thiols at 1,6 positions (HDT; 1,6hexanedithiol); • benzene ring with methylene-thiolated bridges (BDMT; 1,4-benzenedimethanethiol); • biphenyl dimer with thiols groups at para positions of each ring (BPDT; 4,4′-biphenyldithiol). In establishing the bridge models, the dithiolated molecules are first optimized with a single gold atom attached to each sulfur. The S−Au bond length is then adjusted to the bond length of a reoptimized adsorbed thiolated molecule on a surface of a gold cluster of 22 gold atoms with their positions kept fixed. (The gold atoms form three layers of six atoms in B
DOI: 10.1021/acs.nanolett.6b02241 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters states. The celebrated Landauer picture46 relates the conductance of molecular bridges to the electronic transmission function (T(E)) resulting from such broadened energy levels. The current is calculated by integrating the transmission over the potential bias induced energy window of nonequilibrium state population: I (V ) =
e h
∫ν
ν+
Table 1. HOMO/LUMO Energies of the Various Bridge Models, where the Range Switching Parameter γ Is Tuned for the Free Molecules, Molecules with Single Au atoms, and the Bridge Modelsa B3LYP HOMO
LUMO
HOMO
LUMO
γ
H2−BDT Au2−BDT Au9312−BDT
−6.88 −6.46 −5.30
−0.90 −3.92 −4.84
−8.56 −7.95 −6.23
1.18 −2.12 −3.84
0.29 0.26 0.23
H2−HDT Au2−HDT Au9312−HDT
−6.48 −6.23 −5.29
0.22 −3.81 −4.82
−9.28 −8.14 −6.23
2.63 −1.74 −3.81
0.39 0.31 0.27
H2−BDMT Au2−BDMT Au9312−BDMT
−6.48 −6.23 −5.33
−0.93 −3.63 −4.84
−8.28 −8.12 −6.26
1.54 −1.71 −3.89
0.30 0.31 0.23
H2−BPDT Au2−BPDT Au9312−BPDT
−6.47 −6.23 −5.28
−1.30 −3.72 −4.84
−7.78 −7.57 −6.18
0.94 −2.16 −3.86
0.24 0.24 0.22
system
Δv f (E)T (E)dE
−
(1)
Here ν−/ν+ are the electrodes chemical potentials shifted below/above the Fermi energy (FE) by the potential bias (V): ν± = FE ± V/2, and Δν f(E) is the difference between positively and negatively bias-shifted Fermi distributions. The transmission is obtained by tracing the bridge electronic structure with the broadening functions of the electrodes. The electronic structure calculations and geometry optimizations are calculated using the Q-Chem47 software package. All DFT calculations (unless otherwise stated) were implemented with basis sets augmented with the Los Alamos double-ζ effective core potential (LANL2DZ).48−50 The computational cost addressing BDT bridge using BNL is about 60 CPU-hours/core using a single node with 16 core processors. The bridge model includes a total of 3026 basis functions. The semilinear structure of the bridge as gold atoms are added along the transport axis allows for relative effective scaling with respect to molecular model size. The quantum transport T-Chem utility within Q-Chem47 is used to calculate the transmission functions.45,51−57 The transmission is calculated with wide band limit (WBL) selfenergies (SEs)58 using a tight binding (TB) procedure at the Fermi energy of gold (−5.1 eV). Comparison to a full TB treatment of the SEs is provided in SI Figure S3. As expected the full TB calculation is more sensitive to the choice of electrode models and partitioning. Nevertheless, the two approaches for evaluating the SEs provide conductance values that are only slightly different (within a factor of 2), where the TB values are smaller than those obtained by WBL. For the BDT system, the conductance calculated using BNL is 0.012G0 with WBL and 0.0061G0 with TB. (The B3LYP values are 0.14G0 and 0.085G0.) Because the WBL approach is easier to converge the SEs we continue our analysis of the conductance using WBL-based SEs. We have performed unrestricted spin calculations to eliminate artifacts that may be introduced by an enforced diradical electronic structure nature.7,59 The transmission is calculated using BNL and B3LYP functionals. In the BNL approach, the switching parameter is tuned such that the highest occupied molecular orbital (HOMO) energies of the neutral (with N electrons) and the anionic (with N + 1 electrons) systems are as close as possible to their ionization potential (IP).17,60,61 Namely, the following J2(γ) error measure
a
The Au9312 sytem is the actual bridge model used for the transmission analysis (includes nine layers of gold atoms in each side; see Figure 2).
LUMO energies correspond well to the IP/EA for these molecules, whereas the B3LYP HOMO/LUMO absolute energies are underestimated. Also, for the gold-contacted molecular systems the BNL HOMO−LUMO energy gap is larger in comparison to the B3LYP gap. As expected, including more gold atoms reduces the orbital gap (for both RSH-BNL and B3LYP functionals). Clearly, further reduction is expected upon full treatment of the electrodes. Recently the image charge effect representing long-range polarization of the metallic electrodes was evaluated.16,28−31 Below, we estimate such effects of the metallic bulk by recalculating the conductance after retuning the γ parameter to impose a decrease of the orbital gap. We next consider the transmission functions calculated using both functionals. In Figure 3, we highlight the bridge frontier orbitals energies that contribute to the transmission near the FE. As the BNL frontier orbital energies are shifted away from the FE more than the B3LYP ones, so are the corresponding transmission peaks. The actual frontier orbital energies of the bridge models are listed in Table 1 and a reproduced version of the transmission figure, where these energies are also indicated, is provided in the SI Figure S2. An illustration of the frontier orbitals and the key transporting orbitals is provided in SI Figure S4. Importantly, the increase in the gap energy between the transmission peaks around the FE decreases the conductance at the low voltage. In addition, the BNL transmission peaks heights tend to be lower than those of the B3LYP peaks. Therefore, we find a common theme where B3LYP transmission at the FE is substantially elevated when compared to those of the BNL. In addition to shifting the frontier orbitals away from the FE, the transmission peak heights obtained with BNL are either lower or equal to those of B3LYP (note the lower LUMO band transmission in BDT as an example). This trend of lower BNL transmission peaks than those of B3LYP reflects the increased electronic localization of the relevant orbitals. To demonstrate and understand this relation, we follow the transmission spectra
J 2 (γ ) = [IPγ (N ) + εHγ (N )]2 + [IPγ (N + 1) + εHγ (N + 1)]2
BNL
(2)
is minimized. Unless otherwise mentioned, the BNL range separation parameter is determined using bridge model systems with the outmot two gold layers attached to each sulfur. Results and Discussion. We begin by comparing the frontier MO energies to IP and EA using the B3LYP and BNL functionals for the molecules with and without the gold contacts. These energies (and the tuned BNL γ values) are provided in Table 1. We confirm that the BNL HOMO/ C
DOI: 10.1021/acs.nanolett.6b02241 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 4. (a) Transmission spectra for BDT with B3LYP and BNL of varied γ-parameter. (b) Charge localization of the LUMO to LUMO +20 band of orbitals. The parameter follows the average localization of electrons residing in these orbitals, where zero value corresponds to a fully symmetrically distribution (no localization). The LDA limit of the BNL functional (low γ paramter) is found to be associated with close to 0.3 localization. A formal definition of this measure is provide in the SI. Figure 3. Transmission obtained with B3LYP (red/dashed) and BNL (green/solid) functionals. Vertical dotted black lines indicate the Fermi energy. The key transporting orbitals around the FE are indicated by “H*” and “L*” arrows, respectively. (The actual HOMO/ LUMO of the bridge model are localized on the gold with their energies indicated in SI Figure S2 with H/L labeled arrows).
the other hand, is within the same order of magnitude of the measured values. A similar trend of the calculated conductances by GF-DFT in comparison to measured values is established for the three other systems: • For the HDT bridge, several measured values have been reported: The two lower values of 2.6 × 10−4G067 and 2.1 × 10−5G066 are likely to be characteristic of the fully extended hexane chain (all trans conformation). In particular the 2.6 × 10−4G0 value was assigned to an electrode model of an atop contact site.67 Our calculated values are 1.1 × 10−3G0 and 6.5 × 10−5G0 with B3LYP and BNL, respectively. The BNL value is in reasonable agreement with the two measured values. • For the BDMT bridge, the measured value 6 × 10−4G065,69,70 is reproduced well by the BNL value of 4.2 × 10−4G0. The B3LYP value is overestimating the measurements by two orders. • For the BPDT bridge, the experimental values are 1.5 × 10−4G073 and more recently 1 × 10−3G0.71,72 Our calculated values are 6.3 × 10−2G0 with B3LYP and 1.9 × 10−3G0 with BNL. Therefore, also here we find that the BNL value offers better agreement with the measured values than the B3LYP value. We stress that our molecular bridge models focus on a single representative structure, whereas the measured values result from analyzing the distribution of large number of measurements typically represented by histograms. Therefore, our aim cannot (and should not) be to reproduce numerically the measured values. Instead our goal is to demonstrate that using RSH scheme provides a bridge model that is associated with conductance that is close to statistically significant experimental values. The alternative approach based on B3LYP that
as the BNL γ-parameter is varied in Figure 4a. We stress that with a vanishing switching parameter value we effectively reduce the weight of the exact exchange to achieve the limit of the LDA functional. The properly tuned BNL functional involves an increased weight of the exact exchange (γ = 0.23) and is associated with an order of magnitude increased localization than that of LDA. On the other hand, following this series for this system we find B3LYP to relate to a BNL functional of a smaller γ = 0.10. The average localization of the orbitals contributing to the transmission peak above the FE for the different functionals is provided in Figure 4b. Functionals with increased weight of the exact exchange tend to localize more the electronic density. Here the B3LYP is associated with lower localization of 0.64, whereas the proper BNL functional (γ = 0.23) is associated with over 0.89 localization. (The procedure to calculate the localization measure is outlined in detail in the SI.) Therefore, the tendency of the BNL functional to localize the electronic density of the transporting orbitals sems to be reflected by the relatively lower transmission peaks. We now turn to compare the calculated conductance to measured values in Table 2. We start by considering the BDT junction with measured conductances of 0.005G063 and 0.01G0.62,64,65 The B3LYP calculated value of 0.14G0 is substantially higher (and in good agreement with earlier calculated values9,74−76). Our BNL-calculated value 0.012G0, on D
DOI: 10.1021/acs.nanolett.6b02241 Nano Lett. XXXX, XXX, XXX−XXX
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Table 2. Calculated Conductance Using BNL and B3LYP (under Low Bias) and Earlier Reported Calculated and Measured Values (in Electrical Conductance Units: G0 ≡
2e 2 h
= 7.748 × 10−5Ω−1)
previous research system BDT
HDT
BDMT
BPDT
measured
this work calculated
B3LYP
BNL
10−2a 10−3b 10−2c 10−2d
7.5 × 10−1l (2.1−7.9) × 10−1m 6.5 × 10−1n 4.5 × 10−1o
1.4 × 10−1
1.2 × 10−2
2.1 × 10−5e 2.6 × 10−4f 1.2 × 10−3g
2 × 10−3p 9.6 × 10−2q
1.1 × 10−3
6.5 × 10−5
6 × 10−4h 6 × 10−4i
8.3 × 10−2r
2.5 × 10−3
4.2 × 10−4
1 × 10−3j 1.5 × 10−4k
5 × 10−2s 1.5 × 10−2t 1 × 10−1u
6.3 × 10−2
1.9 × 10−3
1.32 5 1 1.1
× × × ×
a
At 0−0.1 V.62 bAt 0.020 V.63 cAt 0.2 V.64 dAt 0.3 V.65 eAt 0.1 V.66 fAt 0.1 V.67 gAt 0.3 V.68 hAt 0.025 V.69 iAt 0.3 V65 and at 0.4 V.70 jAt 0.09 V71 and at 0.1 V.72 kAt 0.18 V.73 lPBE at FE.74 mB3LYP at 0.3 V.9 nB3LYP at 0.4 V.75 oLDA at 0.4 V.76 pBP86 at FE.67 qPBE at FE.74 rPBE at FE.74 s BP86 at FE.73 tBPW91 at 0.2 V.77 uB3LYP at 0.4 V.75
We also capitulate that computational details including the molecular structure of the contact, electrode representation,7,9,78 basis set, self-energy calculations, and possibly other aspects may affect significantly the numerical values. Furthermore, the Landauer picture simplifies the system by imposing a static description. Clearly, only a dynamical prespective can address the nonequilibrium nature of the transport systems,79−81 as well as treating important effects as due to thermal fluctuations and vibrations. We also refer to the difficulty in converging the GF-DFT calculations with respect to the size of the electrode region included in the bridge models. Clearly, addressing more completely the long-range electrostatic effects due to the metallic electrodes environment should further enhance the quality of the treatments. Indeed, recently, the image charge effect due to metallic electrode has been shown to improve the agreement of calculated conductances with the measured values.28,29,31 These treatments that considered similar electrode model as used in the current study found decrease of the orbital gap by up to 1.0 eV.28−31 We therefore estimate the image charge effect on the conductance by the RSH procedure discussed above. We find that the BDT conductance with such bulk affected γ = 0.15 is 0.03G0, which is only 2.5 times larger than the one obtained without the correction. To obtain this estimate, we imposed a 0.66 eV decreased gap to evaluate the γ-parameter affected by the image charge of the electrode. We stress that our molecular bridge model is already including a substantial gold cluster and therefore is expected to involve an overall smaller effect than would be the case for the free molecule. Conclusions. To conclude, we show that using optimally tuned range-separated hybrid functional has substantial impact on the calculated conductance of molecular bridges in comparison to using traditional functionals. As a trend, the increased fundamental orbital gap in BNL calculations relative to the B3LYP case leads to reduced transmission around the FE and therefore to lower the current at low bias. Indeed by benchmarking against experimental measurements we find that
represents widely implemented GF-DFT calculations is shown to be associated with unacceptable overestimated values. To follow the overall trend, we observe the calculated I−V relationships shown in Figure 5 and where available measured values (at various low voltages) are explicitly noted by circles. In all cases, we find the currents calculated using BNL to be smaller by 1−2 orders of magnitude than those obtained using B3LYP. In particular, the calculated currents based on an optimally tuned RSH functional afford a much improved agreement with measured currents (within the same order of the reproducible measurements (up to a factor of 4 difference)). For completeness, we compare to conductances based on RSH tuning parameter evaluated for the gold-free molecule in SI Table S2, where we show that by using a goldaffected parameter the resulting conductances are only mildly affected. We also emphasize that no empirical tuning is employed in any of the calculations. Importantly, here we use exactly the same calculation framework where only the functional choice is changed. In this way, we can gauge unequivocally the role of the functional on the calculated current. In particular, we focus on the importance of utilizing an electronic structure protocol that in principle provides proper description of fundamental aspects. Therefore, comparing the BNL values to B3LYP ones in such framework is of high interest. Indeed, we find that in all cases the transmission around the FE drops substantially when calculated with the BNL functional in comparison to the B3LYP calculated transmission. Overall, our B3LYP-calculated conductances (see Table 2) are similar to earlier reported values using a GF-DFT approach (these treatments may employ B3LYP or other functionals with a preset weight of LDA in the exchange part). Namely, we reproduce the expected trend upon using a DFT approach that improperly lacks the derivative discontinuity in the exchangecorrelation part. In such cases the calculated conductance is overstimated in comparison to the measured values. On the other hand, calculated values based on BNL are in much better agreement with the measured values. E
DOI: 10.1021/acs.nanolett.6b02241 Nano Lett. XXXX, XXX, XXX−XXX
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A.Y. and Q.F. contributed equally to this work. A.H. contributed through a dual enrollment program as a student at Green high school, OH. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors would like to acknowledge the financial support by a DOE-BES award through the Chemical Sciences Geosciences and Biosciences Division (Grants DE-SC0004924 and DEFG02-10ER16174) as well as the support from the Ohio supercomputer and the Kent State University for access to computing resources. B.D.D and K.D.F thank the National Science Foundation for funding support for NSF-REU program (CHE-1263087).
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(1) Nitzan, A.; Ratner, M. A. Science 2003, 300, 1384−1389. (2) Ratner, M. Nat. Nanotechnol. 2013, 8, 378−381. (3) Nichols, R. J.; Higgins, S. J. Annu. Rev. Anal. Chem. 2015, 8, 389− 417. (4) Aradhya, S. V.; Venkataraman, L. Nat. Nanotechnol. 2013, 8, 399−410. (5) Di Ventra, M.; Pantelides, S. T.; Lang, N. D. Phys. Rev. Lett. 2000, 84, 979−982. (6) Pontes, R. B.; Novaes, F. D.; Fazzio, A.; da Silva, A. J. R. J. Am. Chem. Soc. 2006, 128, 8996−8997. (7) Pal, P.; Dunietz, B. D. J. Chem. Phys. 2012, 137, 194104. (8) French, W. R.; Iacovella, C. R.; Rungger, I.; Souza, A. M.; Sanvito, S.; Cummings, P. T. Nanoscale 2013, 5, 3654−3659. (9) Basch, H.; Cohen, R.; Ratner, M. A. Nano Lett. 2005, 5, 1668− 1675. (10) Li, Z.; Kosov, D. S. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 035415. (11) de Melo Souza, A.; Rungger, I.; Pontes, R. B.; Rocha, A. R.; da Silva, A. J. R.; Schwingenschlöegl, U.; Sanvito, S. Nanoscale 2014, 6, 14495−14507. (12) Wang, H.; Leng, Y. J. Phys. Chem. C 2015, 119, 15216−15223. (13) Baer, R.; Neuhauser, D. Phys. Rev. Lett. 2005, 94, 043002. (14) Mori-Sanchez, P.; Cohen, A. J.; Yang, W. Phys. Rev. Lett. 2009, 102, 066403. (15) Kronik, L.; Stein, T.; Refaely-Abramson, S.; Baer, R. J. Chem. Theory Comput. 2012, 8, 1515−1531. (16) Neaton, J. B.; Hybertsen, M. S.; Louie, S. G. Phys. Rev. Lett. 2006, 97, 216405. (17) Stein, T.; Eisenberg, H.; Kronik, L.; Baer, R. Phys. Rev. Lett. 2010, 105, 266802. (18) Baer, R.; Livshits, E.; Salzner, U. Annu. Rev. Phys. Chem. 2010, 61, 85−109. (19) Phillips, H.; Zheng, Z.; Geva, E.; Dunietz, B. D. Org. Electron. 2014, 15, 1509−1520. (20) Mosquera, M. A.; Borca, C. H.; Ratner, M. A.; Schatz, G. C. J. Phys. Chem. A 2016, 120, 1605. (21) Koentopp, M.; Burke, K.; Evers, F. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 121403. (22) Koentopp, M.; Chang, C.; Burke, K.; Car, R. J. Phys.: Condens. Matter 2008, 20, 083203. (23) Kurth, S.; Stefanucci, G.; Khosravi, E.; Verdozzi, C.; Gross, E. K. U. Phys. Rev. Lett. 2010, 104, 236801. (24) Tada, T.; Kondo, M.; Yoshizawa, K. J. Chem. Phys. 2004, 121, 8050−8057. (25) Toher, C.; Sanvito, S. Phys. Rev. Lett. 2007, 99, 056801. (26) Toher, C.; Sanvito, S. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77, 155402. (27) Pontes, R. B.; Rocha, A. R.; Sanvito, S.; Fazzio, A.; da Silva, A. J. R. ACS Nano 2011, 5, 795−804.
Figure 5. Current-bias relationships obtained with B3LYP (red/ dashed) and BNL (green/solid) functionals. Measured currents (evaluated using the conductances as listed in Table 2) are noted by black circles and with the same letter as used in the table notes.
the BNL-based values are within 1 order of magnitude of the measured values, whereas the B3LYP calculated values overestimate the conductance by up to 2 orders of magnitude. We therefore demonstrate that using a RSH functional in calculating electronic transmission provides an important avenue to achieve predictive description of molecular bridges functionality.
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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.nanolett.6b02241. Description of the localization estimate, transmission and conductance plots, MO plots, list of atomic coordinates, and complete authorship of ref 47(PDF)
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail: bdunietzatkent.edu. F
DOI: 10.1021/acs.nanolett.6b02241 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters (28) Liu, Z.-F.; Wei, S.; Yoon, H.; Adak, O.; Ponce, I.; Jiang, Y.; Jang, W.-D.; Campos, L. M.; Venkataraman, L.; Neaton, J. B. Nano Lett. 2014, 14, 5365−5370. (29) Darancet, P.; Widawsky, J. R.; Choi, H. J.; Venkataraman, L. Nano Lett. 2012, 12, 6250−6254. (30) Quek, S. Y.; Venkataraman, L.; Choi, H. J.; Louie, S. G.; Hybertsen, M. S.; Neaton, J. B. Nano Lett. 2007, 7, 3477−3482. (31) Quek, S. Y.; Choi, H. J.; Louie, S. G.; Neaton, J. B. Nano Lett. 2009, 9, 3949−3953. (32) Yanai, T.; Tew, D. P.; Handy, N. C. Chem. Phys. Lett. 2004, 393, 51−57. (33) Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. J. Chem. Phys. 2004, 120, 8425−8433. (34) Livshits, E.; Baer, R. Phys. Chem. Chem. Phys. 2007, 9, 2932−41. (35) Kümmel, S.; Kronik, L. Rev. Mod. Phys. 2008, 80, 3−60. (36) Tsuneda, T.; Song, J.-W.; Suzuki, S.; Hirao, K. J. Chem. Phys. 2010, 133, 174101. (37) Stein, T.; Autschbach, J.; Govind, N.; Kronik, L.; Baer, R. J. Phys. Chem. Lett. 2012, 3, 3740−3744. (38) Manna, A. K.; Lee, M. H.; McMahon, K. L.; Dunietz, B. D. J. Chem. Theory Comput. 2015, 11, 1110−1117. (39) Zheng, S.; Phillips, H.; Geva, E.; Dunietz, B. D. J. Am. Chem. Soc. 2012, 134, 6944−6947. (40) Lee, M. H.; Dunietz, B. D.; Geva, E. J. Phys. Chem. C 2013, 117, 23391−23401. (41) Zheng, Z.; Manna, A. K.; Hendrickson, H. P.; Hammer, M.; Song, C.; Geva, E.; Dunietz, B. D. J. Am. Chem. Soc. 2014, 136, 16954−16957. (42) Wilcox, D. E.; Lee, M. H.; Sykes, M. E.; Niedringhaus, A.; Geva, E.; Dunietz, B. D.; Shtein, M.; Ogilvie, J. P. J. Phys. Chem. Lett. 2015, 6, 569−575. (43) Baratz, A.; Baer, R. J. Phys. Chem. Lett. 2012, 3, 498−502. (44) Baratz, A.; Galperin, M.; Baer, R. J. Phys. Chem. C 2013, 117, 10257−10263. (45) Prociuk, A.; Van Kuiken, B.; Dunietz, B. D. J. Chem. Phys. 2006, 125, 204717. (46) Landauer, R. IBM J. Res. Dev. 1957, 1, 223. (47) Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X.; et al. Mol. Phys. 2015, 113, 184−215. (48) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 270−283. (49) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 299−310. (50) Wadt, W. R.; Hay, P. J. J. Chem. Phys. 1985, 82, 284−298. (51) Chen, Y.; Prociuk, A.; Perrine, T. M.; Dunietz, B. D. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 245320. (52) Das, M.; Dunietz, B. D. J. Phys. Chem. C 2007, 111, 1535−1540. (53) Perrine, T. M.; Dunietz, B. D. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 75, 195319. (54) Perrine, T. M.; Dunietz, B. D. Nanotechnology 2007, 18, 424003. (55) Perrine, T. M.; Dunietz, B. D. J. Phys. Chem. A 2008, 112, 2043. (56) Perrine, T. M.; Smith, R. G.; Marsh, C.; Dunietz, B. D. J. Chem. Phys. 2008, 128, 154706. (57) Perrine, T. M.; Berto, T.; Dunietz, B. D. J. Phys. Chem. B 2008, 112, 16070−16075. (58) López Sancho, M. P.; López Sancho, J. M.; Rubio, J. J. Phys. F: Met. Phys. 1985, 15, 851−858. (59) Feng, Q.; Yamada, A.; Baer, R.; Dunietz, B. D. J. Chem. Theory Comput. 2016, 12, 3431−3435. (60) Stein, T.; Kronik, L.; Baer, R. J. Am. Chem. Soc. 2009, 131, 2818. (61) Stein, T.; Kronik, L.; Baer, R. J. Chem. Phys. 2009, 131, 244119. (62) Song, H.; Kim, Y.; Jang, Y. H.; Jeong, H.; Reed, M. A.; Lee, T. Nature 2009, 462, 1039−1043. (63) Kiguchi, M.; Nakamura, H.; Takahashi, Y.; Takahashi, T.; Ohto, T. J. Phys. Chem. C 2010, 114, 22254−22261. (64) Tsutsui, M.; Taniguchi, M.; Kawai, T. Nano Lett. 2009, 9, 2433− 2439. (65) Xiao, X.; Xu, B. Q.; Tao, N. Nano Lett. 2004, 4, 267−271. (66) Nishikawa, A.; Tobita, J.; Kato, Y.; Fujii, S.; Suzuki, M.; Fujihira, M. Nanotechnology 2007, 18, 424005.
(67) Li, C.; Pobelov, I.; Wandlowski, T.; Bagrets, A.; Arnold, A.; Evers, F. J. Am. Chem. Soc. 2008, 130, 318−326. (68) Xu, B.; Tao, N. J. Science 2003, 301, 1221−1223. (69) Ulrich, J.; Esrail, D.; Pontius, W.; Venkataraman, L.; Millar, D.; Doerrer, L. H. J. Phys. Chem. B 2006, 110, 2462−2466. (70) Ghosh, S.; Halimun, H.; Mahapatro, A. K.; Choi, J.; Lodha, S.; Janes, D. Appl. Phys. Lett. 2005, 87, 233509. (71) Dell, E. J.; Capozzi, B.; DuBay, K. H.; Berkelbach, T. C.; Moreno, J. R.; Reichman, D. R.; Venkataraman, L.; Campos, L. M. J. Am. Chem. Soc. 2013, 135, 11724−11727. (72) Song, H.; Kim, Y.; Jeong, H.; Reed, M. A.; Lee, T. J. Appl. Phys. 2011, 109, 102419. (73) Mishchenko, A.; Vonlanthen, D.; Meded, V.; Burkle, M.; Li, C.; Pobelov, I. V.; Bagrets, A.; Viljas, J. K.; Pauly, F.; Evers, F.; Mayor, M.; Wandlowski, T. Nano Lett. 2010, 10, 156−163. (74) Hoshiba, Y.; Maeda, Y.; Hamada, K.; Fukuoka, S.; Ishikawa, Y.; Kurita, N. Chem. Phys. Lett. 2012, 521, 39−44. (75) Cohen, R.; Stokbro, K.; Martin, J. M. L.; Ratner, M. A. J. Phys. Chem. C 2007, 111, 14893−14902. (76) Stokbro, K.; Taylor, J.; Brandbyge, M.; Mozos, J.-L.; Ordejon, P. Comput. Mater. Sci. 2003, 27, 151−160. (77) Xue, Y.; Ratner, M. A. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 68, 115406. (78) Solomon, G. C.; Herrmann, C.; Ratner, M. A. Top. Curr. Chem. 2011, 313, 1−38. (79) Stefanucci, G.; Almbladh, C. O. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 69, 195318−195334. (80) Kurth, S.; Stefanucci, G.; Almbladh, C. O.; Rubio, A.; Gross, E. K. U. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 72, 035308. (81) Prociuk, A.; Dunietz, B. D. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 165112.
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DOI: 10.1021/acs.nanolett.6b02241 Nano Lett. XXXX, XXX, XXX−XXX
