NANO LETTERS
First-Principles Simulations of Inelastic Electron Tunneling Spectroscopy of Molecular Electronic Devices
2005 Vol. 5, No. 8 1551-1555
Jun Jiang,†,‡ Mathias Kula,† Wei Lu,‡ and Yi Luo*,† Theoretical Chemistry, Royal Institute of Technology, AlbaNoVa, S-106 91 Stockholm, Sweden, and National Lab for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, China Received April 27, 2005
ABSTRACT Inelastic electron tunneling spectroscopy (IETS) is a powerful experimental tool for studying the molecular and metal contact geometries in molecular electronic devices. A first-principles computational method based on the hybrid density functional theory is developed to simulate the IETS of realistic molecular electronic devices. The calculated spectra of a real device with an octanedithiolate embedded between two gold contacts are in excellent agreement with recent experimental results. Strong temperature dependence of the experimental IETS spectra is also reproduced. It is shown that the IETS is extremely sensitive to the intramolecular conformation and the molecule-metal contact geometry changes. With the help of theoretical calculations, it has finally become possible to fully understand and assign the complicated experimental IETS and, more importantly, provide the structural information of the molecular electronic devices.
Molecular electronics has attracted great attention in the past decade. One of the most exciting recent developments in this field is the application of inelastic electron tunneling spectroscopy (IETS) to molecular junctions.1-3 The measured spectra show well-resolved vibronic features corresponding to certain vibrational normal modes of the molecule. The IETS not only helps to understand the vibronic coupling between the charge carriers and the nuclear motion of molecule but also provides a powerful tool to detect the geometrical structures of molecular electronic devices. It has been shown that the spectra of two alkanethiol molecular junctions with different geometries1,2 can have very different spectral profiles. In fact, the importance of the IETS for the molecular electronic devices cannot be overstated because the lack of suitable tools to identify the molecular and contact structures has hampered the progress of the field in recent years.4 The basic theory of IETS has been known for many years.5 Its extension to molecular wires,6,7 atomic metal wires,8 and molecular junctions,6,9-12 however, has just appeared in the literature very recently, partly motived by the success of the experiments. The theoretical modelings are generally based on Green’s function theory at different levels and have contributed to many fundamental under* Corresponding author. E-mail:
[email protected]. † Royal Institute of Technology. ‡ Chinese Academy of Sciences. 10.1021/nl050789h CCC: $30.25 Published on Web 07/01/2005
© 2005 American Chemical Society
standings of the inelastic electron tunneling processes inside a molecular/atomic junction. However, most of the theoretical studies have not been able to make realistic comparisons to the experimental spectra. The usefulness of the theoretical models has not been fully exploited. In this letter we present a new computational scheme based on our early developed quantum chemical approach for electron transport in molecular junctions.13 The vibronic coupling is introduced by expanding the electronic wave function along different vibrational normal modes. We have applied this new method to the gold-octanedithiolate-gold junction. The electronic structures and vibrational modes are calculated using the hybrid density functional theory (DFT). The simulated spectra are in excellent agreement with the experimental ones. Our computational scheme can be applied easily to other systems. Theoretical simulations are extremely useful for assigning the experimental spectra and revealing detailed information that is not accessible in the experiments. Our approach is also based on Green’s function formalism. We divide the molecular device into three parts, source, drain, and extended molecule, as shown in Figure 1. The source and drain are described by an effective mass approximation (EMA), whereas the extended molecule is treated with the hybrid density functional theory. The extended molecule is in equilibrium with the source and drain through the line up of their effective Fermi level. This approach has proven to
Figure 1. Structures of the gold-octanedithiol-gold junctions with triangle (A) and chain (B) local contacts.
lead to very good descriptions for the experimental results.13 The current density for the 3D electrodes along molecular axis z is given as13
*
iSD )
em kBT p3
∫0∞ ln
(
)
Ef + eVD - Ez kBT × Ef - E 1 + exp kBT
1 + exp
( )
S 2 2 ΓJK η (Q) ) πn VJS (Q)|〈J(Q)|η(Q)〉| +
|T(VD, Q)|2nSnD dEz (1) where nS and nD are the density of states (DOS) of the source and drain, respectively. Ef is the Fermi energy, and T is the device working temperature. VD is the external voltage. The transition matrix element from the source to the drain, T(VD, Q)T(VD, Q), is dependent on the vibrational motion, Q, and can be written as13 T(VD, Q) )
∑J ∑K VJS(Q)VDK(Q)∑η
〈J(Q)|η(Q)〉〈η(Q)|K(Q)〉 zη(Q) - η(Q)
(2)
where J and K run over all atomic sites of the molecule, which are denoted as 1, 2, ..., N, site 1 and N are two end sites of molecule that connect with two electron reservoirs. VJS (VDK) represents the coupling between atomic site J (K) and reservoir S (D). Orbital |η〉 is the eigenstate of the Hamiltonian (Hf) of a finite system that consists of the molecule sandwiched between two clusters of metal atoms: 1552
Hf|η〉 ) η|η〉. The product of two overlap matrix elements 〈J|η〉〈η|K〉 represents the delocalization of orbital |η〉. Here parameter zη is a complex variable, zη ) Eη + iΓJK η , where Eη is the energy related to the external bias and Fermi level, at which the scattering process is observed. ΓJK η is the escape rate which is determined by the Fermi Golden rule
πnDVDK2(Q)|〈η(Q)|K(Q)〉|2 (3) All of the key parameters are obtained from the calculations of the finite metal-molecule cluster. The use of the finite system allows us to easily include the vibronic coupling because the description of vibronic coupling in a molecular system has long been established. The effect of the infinite leads has been taken into account through their density of states (DOS), nS,D, which is calculated using the EMA. The line width of the IETS spectral profile is determined by the escape rate ΓJK η (Q). The nuclear motion-dependent wave function can be expanded along the vibrational normal mode using a Taylor expansion. The IETS experiment is often done at electronic off-resonant region. The adiabatic harmonic approximation can thus be applied, therefore, only the first derivative, ∂Ψ(Q)/∂Qa, needs to be considered, where Qa is the vibrational normal mode, a, of the extended molecule. The working formula we have used for IETS calculations follows the same principle as the one given by Troisi, Ratner, and Nitzan.6 We have calculated an octanedithiolate, SC8H16S, embedded between two gold electrodes through S-Au bonds. The Nano Lett., Vol. 5, No. 8, 2005
extended molecule consists of two gold trimers bonded with an octanedithiolate molecule. The gold trimers were tested in two basic conformations, a triangle and a linear chain, see Figure 1. For the triangle configuration, the sulfur atom is placed above the middle of the triangle, resembling the hollow site of a Au(111) surface. Geometry optimization and electronic structure calculations have been carried out for the extended molecules of different configurations at the hybrid DFT B3LYP level14 using the Gaussian03 program package with the LanL2DZ basis set.15 In the case of the triangle configuration, two different schemes have been used for the geometry optimization. The first geometry is obtained by optimizing octanedithiol in the gas phase and then replacing the terminal hydrogens with the gold contacts without further optimization (Tr1). The second geometry is obtained by optimizing the first geometry with the same fixed gold and sulfur distance (Tr2), which is also the approach for the chain configuration (Ch1). The S-Au distance in all of the calculations is fixed to 2.853 Å. The QCME program16 has been employed for all of the IETS calculations. The calculated IETS of the octanedithiolate junction with the triangle gold trimers are shown in Figure 2, together with the experimental spectrum of Wang et al. at temperature 4.2 K.2 One can see clearly that the calculated results for both configurations are in very good agreement with the experiment. The calculations reproduce all of the major spectral features observed in the experiment. The Tr1 and Tr2 configurations have the same Au-S bonding, but slightly different intramolecular conformations, which result in quite different spectral profiles. It thus indicates that the sensitivity of the IETS with respect to the molecular geometry is really high. Both theory and experiment show that the intensity of the vibronic feature follows the order: ν(C-C) (131 mV) > γ(CH2) (155 mV) > δ(CH2) (185 mV). In comparison with the experiment, the Tr1 configuration results in an intensity that is too large for the ν(C-S) (79 mV) and δ(CH2) (124 mV) modes, whereas the Tr2 configuration gives spectral features that are too weak in the ν(Au-S) modes region. One could conclude that the molecular geometry and contact configuration of the device in the experiment of Wang et al.2 might be the mixture of Tr1 and Tr2 configurations. It should be mentioned that theory and experiment also agree very well for the current-voltage characteristics (IV), which are determined by both elastic and inelastic scatterings, see the insets of Figure 2A and B. The calculated absolute current is actually in very good agreement with the experimental result of a single octanedithiolate molecular junction.17,18 The calculated temperature dependence of the IETS of the Tr1 and the Tr2 configurations is shown in Figure 3 together with the experimental results of Wang et al.2 The agreement between the theory and the experiment is more than satisfactory. It shows again that the geometry of the experimental device is between the Tr1 and Tr2 configurations. The evolution of the spectral bands upon the increase of the temperature is the same for both the experiment and the calculations. As an example, in both cases, the peak for Nano Lett., Vol. 5, No. 8, 2005
Figure 2. Inelastic electron tunneling spectrum of the octanedithiol junction from (A) experiment in which the background due to the encasing Si3N4 is marked with stars,2 (B) calculation for the Tr1 configuration, and (C) calculation for the Tr2 configuration. The I-V curves are given in the insets of (A) and (B). The working temperature is 4.2 K.
mode δ(CH2) at 185 mV disappears at 35 K, and the peak for mode γ(CH2) at 155 mV becomes invisible at 50 K. Our simulations indicate that the observed temperature dependence is due mainly to the changes of the Fermi distribution. It should be noted that the experimental spectra contain the background due to the encasing Si3N4.2 The discrepancy between theory and experiment could be due partially to the different temperature dependence of those background profiles. In Table 1, the assignments of the vibronic bands observed in theoretical and experimental spectra are shown. It can be seen that the vibrational frequencies given by the B3LYP calculations are in very good agreement with the experiments. Our computational scheme also allows one to calculate the spectral line width directly, which is determined by the orbital characters and molecule-metal bonding, see eq 3. The 1553
Figure 3. Temperature-dependent IETS of the octanedithiolate junction from (A) experiment2 and calculations for the Tr1 (B) and Tr2 (C) configurations. The intensity is in arbitrary units. Table 1. Assignments of the Vibrational Modes Observed in the IETS of the Octanedithiolate Junctiona mode
peak(meV)
band
exptlb
cal
fwhm
begin
end
∆
ν(Au-S)
34
ν(C-S) δ(CH2)
79
28 44 57 79 95 104 114 124 132 155 185 379
5.0 4.7 2.1 2.0 2.1 2.2 3.0 3.8 6.1 9.0 2.0 9.3
228 335 453 639 742 841 917 991 1012 1225 1484 3012
250 353 474 652 783 841 917 998 1100 1402 1536 3146
2.7 2.3 2.6 1.6 1.7 0.0 0.0 1.0 11.0 22.0 6.4 17.0
104
ν(C-C) γ(CH2) δ(CH2) ν(CH2)
132 159 180 354
a The calculated peak positions (in meV), the full width at half-maximum (fwhm) of a single spectral line (in meV), the beginning and the ending of a spectral band (in cm-1), and the total width of the band ∆ (in meV) are given. The experimental results of Wang et al.2 are also listed for comparison. b Reference 2.
calculated full width at half-maximum (fwhm) for the spectral profile of mode ν(C-C) at 132 mV is found to be around 6.1 meV for the Tr1 configuration and 2.0 meV for the Tr2 configuration, in good agreement with the experimental result of 3.73 ( 0.98 meV.2 However, it is also noticed that in the spectrum of the Tr1 configuration such a band is attributed from several modes of the same character with vibration frequencies 1012, 1038, 1061, 1069, 1084, 1090, 1097, and 1100 cm-1, respectively, covering a range of 11.0 meV. The overlapping between different vibration modes makes it impossible to determine the actual intrinsic line width of the spectral profile from a single vibration mode. We have calculated the IETS of the gold chain configuration (Ch1), shown in Figure 4b, to examine the dependence of the IETS on the molecule-metal bonding structure. Indeed, the IETS of Ch1 shows a distinct difference in spectral intensity distribution from those of the Tr1 and the Tr2 configurations. The spectral peak of mode δ(CH2) at 382 mV has become the absolute dominant feature in the spectrum. Furthermore, the intensities of different spectral 1554
Figure 4. Inelastic electron tunneling spectroscopy from (A) experiment for C11 monothiol1 and (B) calculation for the Ch1 configuration. The working temperature is 4.2 K.
features follow the order of γ(CH2) (172 mV) < ν(C-C) (132 mV) < δ(CH2) (382 mV), completely different from the results of the Tr1 and Tr2 configurations. The Au-S bonding structure of the Ch1 configuration differs from those of the Tr1 and Tr2, resulting a noticeable difference in their spectral profiles for the mode ν(Au-S) around 40 mV. The changes in molecular conformations seem to be the major cause for the large difference in the spectral intensity distributions of the two devices. It is found that the molecule Nano Lett., Vol. 5, No. 8, 2005
in the Ch1 configuration is twisted around the otherwise linearly oriented molecular backbone in the Tr1 configuration. It is interesting to note that the spectrum of the Ch1 configuration resembles the experimental IETS of an alkanemonothiol molecule, HS(CH2)8H (C11),1 quite well, as clearly demonstrated in Figure 4. The molecule-metal bonding structure of the C11 is very different from that of octanedithiol. Such a difference should be reflected by the spectral profiles of the Au-S modes at the low energy region. The large difference in the experimental spectral intensity distribution1,2 related to the molecular vibration modes implies that the molecular conformations in the two experimental setups are very different. The molecular backbone of the octanedithiolate junction in the device of Wang et al.2 should be linear, whereas it is slightly twisted for the C11 in the device of Kushmerick et al.1 In conclusion, we have proposed a new computational scheme that is capable of describing the IETS of molecular junctions with unprecedented accuracy. Our first-principles calculations provide reliable assignments for the experimental spectra and reveal important details that are not accessible in the experiment. With the help of the theoretical simulations, it can be expected that the IETS will play an essential role in the field of molecular electronics. Acknowledgment. This work was supported by the Swedish Research Council (VR), the Carl Trygger Foundation (CTS), and Chinese state key program for basic research (2004CB619004). References (1) Kushmerick, J. G.; Lazorcik, J.; Patterson, C. H.; Shashidhar, R.; Seferos, D. S.; Bazan, G. C. Nano Lett. 2004, 4, 639. (2) Wang, W.; Lee, T.; Kretzschmar, I.; Reed, M. A. Nano Lett. 2004, 4, 643. (3) Yu, L. H.; Keane, Z. K.; Ciszek, J. W.; Cheng, L.; Stewart, M. P.; Tour, J. M.; Natelson, D. Phys. ReV. Lett. 2004, 93, 266802.
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(4) Flood, A. H.; Stoddart, J. F.; Steuerman, D. W.; Heath, J. R. Science 2004, 306, 2055. (5) Wolf, E. L. Principles of Electron Tunneling Spectroscopy; Oxford University Press: New York, 1985; Vol. 71. (6) Troisi, A.; Ratner, M. A.; Nitzan, A. J. Chem. Phys. 2003, 118, 6072. (7) Asai, Y. Phys. ReV. Lett. 2004, 93, 246102. (8) Frederiksen, T.; Brandbyge, M.; Lorente, N.; Jauho, A.-P. Phys. ReV. Lett. 2004, 93, 256601. (9) Galperin, M.; Ratner, M. A.; Nitzan, A. Nano Lett. 2004, 4, 1605. (10) Chen, Y.-C.; Zwolak, M.; Di Ventra, M. Nano Lett. 2004, 4, 1709. (11) Troisi, A.; Ratner, M. A.; privite communication. (12) Chen, Y.-C.; Zwolak, M.; Di Ventra, M. Nano Lett. 2005, 5, 621. (13) (a) Jiang, J.; Lu, W.; Luo, Y. Chem. Phys. Lett. 2004, 400, 336. (b) Wang, C.-K.; Fu, Y.; Luo, Y. Phys. Chem. Chem. Phys. 2001, 3, 5017. (c) Wang, C.-K.; Luo, Y. J. Chem. Phys. 2003, 119, 4923. (14) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (15) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, Jr., J. A.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; AlLaham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, Revision C.02; Gaussian, Inc., Wallingford CT, 2004. (16) Jiang, J.; Luo, Y. QCME-V1.0: Quantum Chemistry for Molecular Electronics; Royal Institute of Technology: Stockholm, Sweden; thanks to Chuan-Kai Wang and Ying Fu. (17) Cui, X. D.; Primak, A.; Zarate, X.; Tomfohr, J.; Sankey, O. F.; Moore, A. L.; Moore, T. A.; Gust, D.; Nagahara, L. A.; Lindsay, S. M. J. Phys. Chem. B. 2002, 106, 8609. (18) Cui, X. D.; Primak, A.; Zarate, X.; Tomfohr, J.; Sankey, O. F.; Moore, A. L.; Moore, T. A.; Gust, D.; Harris, G.; Lindsay, S. M. Science 2001, 294, 571.
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