NANO LETTERS
Current versus Temperature-Induced Switching in a Single-Molecule Tunnel Junction: 1,5 Cyclooctadiene on Si(001)
2009 Vol. 9, No. 8 2996-3000
Christophe Nacci,† Stefan Fo¨lsch,†,§ Karl Zenichowski,‡ Jadranka Dokic´,‡ Tillmann Klamroth,‡ and Peter Saalfrank*,‡ Paul-Drude-Institut fu¨r Festko¨rperelektronik, HausVogteiplatz 5-7, D-10117 Berlin, Germany, and Institut fu¨r Chemie, UniVersita¨t Potsdam, Karl-Liebknecht-Strasse 24-25, D-14476 Potsdam-Golm, Germany Received May 4, 2009; Revised Manuscript Received June 11, 2009
ABSTRACT The biconformational switching of single cyclooctadiene molecules chemisorbed on a Si(001) surface was explored by quantum chemical and quantum dynamical calculations and low-temperature scanning tunneling microscopy experiments. The calculations rationalize the experimentally observed switching driven by inelastic electron tunneling (IET) at 5 K. At higher temperatures, they predict a controllable crossover behavior between IET-driven and thermally activated switching, which is fully confirmed by experiment.
The capability to address and to manipulate single atoms and molecules at surfaces by cryogenic scanning tunneling microscopy (STM)1-3 represents a powerful tool to study fundamental processes in surface physics and chemistry. In addition, it makes possible to explore prototypical functionalities based on single atomic and molecular building blocks supported on a surface. For example, the reversible transfer of adsorbed atoms between two stable configurations, or the conversion of molecules from one conformation to another, have gained increasing attention as model systems with switching behavior at the atomic scale. In both cases, initial and final arrangements are bound and can be described by two wells of a double-minimum potential. This locally excited switching is of interest because it is often accompanied by an abrupt change of macroscopic properties, such as the tunneling current in the junction of an STM. By far the most cases of reversible switching between two (or more) potential minima have been realized on transition metal surfaces. Established examples include the vertical transfer of atoms4 and molecules5 between metal surface and STM tip, the lateral back-and-forth transfer of atoms between adjacent adsorption sites6,7 or rotations of single molecules8 excited by the STM tip. The related STM-driven conformational switching of single adsorbed molecules has been realized on metal surfaces9-12 as well as on ultrathin insulating spacer layers on metals.13 * To whom correspondence should be addressed. E-mail: petsaal@ uni-postdam.de. † Paul-Drude-Institut fu¨r Festko¨rperelektronik. ‡ Universita¨t Potsdam. § E-mail address:
[email protected]. 10.1021/nl901419g CCC: $40.75 Published on Web 07/07/2009
2009 American Chemical Society
Theoretical investigations suggest that in many cases the switching is due to inelastic electron tunneling (IET), which causes vibrational ladder climbing in the double-well potential from one side to the other.14-18 Much less work has been reported so far for semiconductor surfaces such as silicon, despite their utmost potential for molecular electronics. A comparatively simple example is the reversible switching of single hydrogen atoms on a partially H-covered Si(001) 2 × 1 surface from one Si atom of a Si dimer to the neighboring, dangling bond site.19 Also here, IET-driven vibrational ladder climbing in a doublewell potential is assumed to take place.20 To date, only few examples exist in which STM-driven conformational switching of single molecules has been achieved on Si surfaces.21-23 Most of the above studies have been carried out at fixed, cryogenic temperatures. On the other hand, chemical transformations at surfaces in general are often characterized by a nontrivial and interesting temperature dependence. This is also true for bound-bound transitions of the double-well type. For example, in ref 24 it was found, by scanning tunneling microscopy, that above a “crossover temperature”, Tc, the diffusive hopping of H atoms on Cu(001) from a 4-fold hollow site to a neighboring one is a thermally activated, over-the-barrier process. In this regime the hopping rate R follows an Arrhenius law R ) Ae-Ea/(kbT)
(1)
where A is a frequency factor, and Ea an effective activation energy for atom transfer. Below Tc ≈ 60 K for H, a nearly
Figure 1. Left: Stick-and-ball models of the COD/Si(001) molecule in the double dimer-bound bridge structure obtained by geometry optimization using a cluster model. The molecule is shown in orange and the underlying Si cluster in blue. Two degenerate conformers (a,b) are given in side view (top row) and in top view (bottom row). Right panel, (c,d) Constant-current STM topographs (105 Å × 105 Å, 0.1 nA, 5 K) with nine COD molecules on Si(001) imaged at -2.00 V [occupied states, (c)] and at +1.85 V sample bias [empty states, (d)]; the Si dimer rows are aligned vertically and exhibit missing dimers as intrinsic defects. (e) Empty-state image of a single molecule (28 Å × 14 Å, 0.7 nA, +1.50 V, 5 K); crosses mark equivalent positions at which time spectra of the tunnel current were taken.
temperature independent rate R is observed instead, which is due to atom tunneling through the barrier. For the heavier D, no tunneling was seen. Thus, the low-T behavior depends sensitively and exclusively on intrinsic properties of the system, for example, barrier height and masses of reactants. Note that in ref 24, the STM was only used to monitor the hopping process, rather than driving it. In the present work, a related phenomenon is described, namely the appearance of a crossover temperature in STMinduced biconformational switching of single cyclooctadiene (COD) molecules chemisorbed on a Si(001) surface. Here the STM is used to induce and also to monitor the reaction. For this system, switching at very low temperatures, T ) 7 K, was recently realized experimentally in refs 23 and 25. In this work, we apply a quantum mechanical model based on electronic structure calculations and open-system density matrix theory to disclose the quantum dynamics of the experimentally observed switching process. The model rationalizes the experimental finding of a reaction driven by IET, at low T. Almost no T-dependence of the isomerization rate is predicted in this regime. Above a certain crossover temperature, however, the reaction is thermally activated and the rate grows exponentially with T. By extending the experiments to higher temperatures, the crossover is confirmed. According to the model, Tc and the low-T IET rate depend on, and can be controlled by, external parameters such as the tunneling current. Thus, the electron tunneling current can be used to control the onset of tunneling of atoms through a reaction barrier. Biconformational Switching of COD/Si(001) at Low Temperature. On the experimental side, STM measurements were carried out in ultrahigh vacuum (UHV) and at cryogenic temperatures to characterize the biconformational switching dynamics of the COD/Si(001) molecule in the double dimerbound bridge structure.26 Details on the n-type Si(001) substrate preparation and the COD deposition procedure in UHV are described elsewhere.23,25 During the tunneling experiment, the sample temperature is first kept at 5 K, while later higher temperatures will be considered. Nano Lett., Vol. 9, No. 8, 2009
The left panel of Figure 1 shows stick-and-ball models of two equivalent ground state conformers (a,b) in side view and in top view (bottom). The structures were obtained from geometry optimization on the B3LYP/6-31G(d) level of theory27,28 of a COD molecule on a Si15 cluster that was saturated with 16 H atoms at the cluster edges as shown. The calculations were carried out with the GAUSSIAN03 program suite.29 Consistent with previous work,26 our calculations reveal buckling patterns of the C atoms resembling the doubly degenerate twisted-boat structure30 of free COD. The experimental constant-current STM images Figure 1c,d show a surface area with nine discrete COD molecules adsorbed in the bridge structure appearing as uniform protrusions (height ∼ 0.6 Å) on top of the Si dimer rows.23 Note that the apparent height contrast of the Si dimer rows is inverted in occupied (c) compared to empty-state imaging (d) due to tunneling out of (into) π surface states (π* surface and Si backbond states). Figure 1e shows a close-up view of a single COD molecule at a set point current and sample bias of 0.7 nA and +1.50 V. It is found that (i) at the present set point parameters the apparent molecular height is reduced to ∼0.2 Å and that (ii) frequent discontinuities in the height contour are observed when scanning the STM tip across the molecule. As shown earlier,23,25 these discontinuities are due to IET-induced switching of the molecule between its two degenerate conformers depicted in Figure 1a,b. The conformational switching gives rise to a binary telegraph signal in the tunnel current when placing the tip over the molecule at constant height. We find that a maximum high-to-low current level ratio of 1.2:1 is obtained at four equivalent off-center tip positions as indicated by the white crosses in Figure 1e. This observation is consistent with the presence of a lefthanded and a right-handed double dimer-bound COD/Si(001) conformer (cf. Figure 1a,b) each having C2 symmetry. To demonstrate the single-molecule switching experimentally, the STM tip was placed over the molecule at a set point of 0.7 nA and + 1.50 V, the feedback loop was turned off and current versus time spectra were measured. Figure 2, lowest panel, shows the current trace recorded at T ) 5 K (blue 2997
Figure 2. Tunnel current vs time recorded at constant tip height with the STM tip placed over a single COD molecule at an initial set point of 0.7 nA and +1.50 V prior to opening the feedback loop, tip position according to the locations marked in Figure 1e; the current traces detected at different temperatures are offset by 0.3 nA along the y axis. Inset: Logarithmic plot of the switching rate Rsw ) 1/τ versus the reciprocal temperature indicating a crossover behavior at Tc ) 45 K. Different symbols correspond to different experimental runs probing different molecules.
Figure 3. Model: (a) Sketch of coordinates along a C2-preserving path with two Si dimers and C atoms indicated. (b) Doubleminimum potential V(φ) (with φ ) (1/2)(φl + φr)) and doublets n ) 0...6. (c) Switching rate Rsw as a function of current, for various librational lifetimes τvib. (d) Arrhenius plots Rsw vs 1/T for three different tunneling currents at τvib ) 10 ps. For I ) 0.7 nA, experimental data (crosses) from Figure 2 are shown for direct comparison.
curve). The switching rate Rsw ) 1/τ (τ: mean residence time in a specific current state) of 3.7 Hz obtained at 5 K is consistent with our previous experiments in this lowtemperature regime indicating a single-electron IET excitation process with a quantum yield of (Iτ/e)-1 ≈ 1 × 10-9.23,25 Quantum Dynamics of the Switching Process. To set up a theoretical model for switching, the same method and cluster as before were used to calculate a one-dimensional potential curve for a simultaneous change of the dihedral angles defined by C atoms a, b, c, and d in Figure 3a. Along the curve, all atoms except the lowest two layers were relaxed. For the resulting path, an approximately linear relationship to an isomerization angle φ ) (φl + φr)/2 was 2998
found, which describes the assumed simultaneous but opposite, rotation of two adjacent C2H4 units of adsorbed COD, around the indicated axes. During the rotations, the C2 symmetry with respect to the molecular center is preserved. We obtain a double-minimum potential V(φ) as shown in Figure 3b with a barrier of 0.179 eV separating the two conformers at φ0 ) (0.321 rad. This barrier is lower than the B3LYP/6-31G(d) value of 0.311 eV found for the free molecule. The potential supports vibrational (better: librational, i.e., rotation) bound states |n(〉 with energies E(n , which are of gerade (+) or ungerade (-) parity. The eigenpairs ˆ are obtained by diagonalizing the switching Hamiltonian H 2 2 2 ) -(p /2I)(d /dφ ) + V(φ) where I is the moment of inertia for the rotating C2H4 units. This has been achieved by using the Fourier Grid Hamiltonian (FGH) method31 on a dense grid, with V(φ) represented as a cubic spline and employing a potential cutoff of 1.27 eV. Here, I ) 109476.09 mea20 is the moment of inertia for the rotating C2 H 4 units, calculated from the known masses of the moving atoms and their average distances from the rotation axes. The lowest 40 eigenstates, that is, doublets n ) 0 to n ) 19 extending up to 0.81 eV were included in the dynamics. A fundamental + librational frequency ω10 ) (E+ 1 - E0 )/p of 36 meV is obtained. As shown in Figure 3b, the lowest five doublets are below the barrier, all higher doublets are either around (n ) 5) or above it (n > 5). Each doublet is characterized by + a “tunneling time”, τtun n ) pπ/(En - En ), which is 26 µs for n ) 0 and ∼1 ps at the barrier top (n ) 4,5). The STMdriven isomerization is enforced by interlevel transitions, treated by Markovian open-system density matrix theory and using a model analogous to the one developed for IET-driven H transfer on Si(001).20 Accordingly, we solve generalized Master equations for the diagonal elements of the reduced density matrix (populations), and for the off-diagonal elements (coherences) between librational levels of the doubleminimum potential. In practice, due to lattice imperfections and/or the presence of the nearby STM tip, librational states will be localized to one well. Left (L) and right (R) localized states can be constructed from delocalized levels according to |nL,R〉 ) (1/2)(|n+〉 ( |n-〉). The degree of localization achieved was above 99% for the first five doublets, and higher than 82% for states above the barrier. The initial state and all transition rates Wnfm will be defined in this localized basis. In the localized basis, denoted as {|n〉} for short, the equations of motion are dFnn i )dt p
∑ (H
npFpn
- FnpHpn) +
p
∑ (W
pfnFpp
- WnfpFnn)
(2)
p
for the diagonal elements of the density matrix, and dFnm i )dt p
∑ (H
npFpm
- FnpHpm) - γnmFnm
(3)
p
for the off-diagonal elements. Here, γnm ) 1/2 Σf (Wnff + Wmff) is a dephasing rate. In the localized basis, the system Nano Lett., Vol. 9, No. 8, 2009
ˆ is blockdiagonal with 2 × 2-blocks. The Hamiltonian H En Vn nth 2 × 2 -block is given as H _n ) where En ) Vn En + (1/2)(En + En ) is the average energy of doublet n, and Vn ) (1/2)(E+ n - En ) half of its tunnel splitting. The Vn allow for the possibility of nuclear tunneling through the potential barrier. Other than in ref 20, basis set transformations between localized and unlocalized state basis are avoided, since eqs 2 and 3 are solved in the localized state basis. The transition rates Wnfm account for vibrational relaxation due to adsorbate vibration-phonon coupling, and for STMinduced IET
(
)
rel IET Wnfm ) Wnfm + Wnfm
(4)
rel IET The transition rates Wnfm and Wnfm between localized states are calculated as follows. The localized states define sets of harmonic-oscillator-like states for left and right wells. It is thus reasonable to adopt a model for vibrational relaxation in which two harmonic oscillators are bilinearly coupled to a “bath” of surface oscillators. Perturbation theory then predicts (T ) 0 K)20
rel ) Wnfm
1 nδ τvib m,n-1
(5)
τvib is the relaxation time of librational level n ) 1. Equation 5 implies downward transitions with selection rules ∆n ) -1 between localized states within one well, left (|n〉 ) |nL〉) or right (|n〉 ) |nR〉). At finite temperature, upward transitions are enforced by rel nonzero upward rates given by detailed balance as Wmfn ) rel -(En - Em)/(kBT) Wnfme . Typical lifetimes τvib of adsorbate vibrations at Si surfaces range from below 1 ps to several nanoseconds.32-34 Below, τvib will be treated as a model parameter. In ref 20, two contributions to the STM-driven IET rates dip WIET nfm were considered: a dipole term Wnfm, and a resonance res rate Wnfm. The former accounts for charge carriers that tunnel from the STM tip to the surface and couple via their electric field to the dipole moment of the adsorbate. In a perturbative model, the dipole rates are20,35 dip Wnfm )
(|| )
I 1 〈n µz m〉 e ea0
2
dip ) Wmfn
(6)
where I is the STM tunneling current, and µz(φ) the dipole moment component perpendicular to the surface (z, the direction of the tunneling current), as a function of φ. µz(φ) was determined from cluster calculations as above, and fitted to a Gaussian shape, µz(φ) ) A0 exp(-(φ/σ)2) + dip A1.36 For a current of 1 nA, the rate W0T1 connecting the lowest two localized states of either well is 1.2 × 105 s-1. Since the system is anharmonic, overtone excitations are also possible, and important, despite being small. res Other than in ref 20, resonance contributions Wnfm due to electron or hole attachment were not included in the following, because experimentally the switching rate of Nano Lett., Vol. 9, No. 8, 2009
COD/Si(001) is independent of bias voltage U in the range U ∈[- 2.5, +2.75] V.23,25 In contrast, a perturbative rate res 20,37,38 expression for Wnfm predicts a strong dependence on U. Test calculations for the present system and model showed that Rsw is approximately independent of U only if (i) the resonance(s) is (are) energetically well separated from the Fermi level of the tip (by ∆ > 8 eV) and/or (ii) if the resonance width Γ is extremely broad (1/Γ e 0.1 fs). Under these extreme conditions, resonance contributions are small. On the other hand, the single-electron mechanism and the experimental quantum yield per electron (≈ 1.0 × 10-9 at low T) is also compatible with a resonance mechanism, when a slightly less restrictive parameter choice ∆ ) 8 eV, 1/Γ ) 1 fs, and τvib ) 1 ps is made. Under these conditions the resonance contribution can dominate over the dipole mechanism,37 albeit with a predicted but unobserved, U-dependence. In lack of detailed knowledge of the resonance and lifetime parameters for COD/Si(001), we therefore cannot exclude resonance contributions to IET at the moment. Fortunately, it is found numerically that the latter have a quantitative, but no qualitiative effect on the results to be presented below. Equations 2 and 3 are solved by a fourth order RungeKutta scheme with an initial state localized in the left well, Fˆ (0) ) |0L〉〈0L|. The switching rate was defined as time derivative of the probability PR to be on the right, Rsw ) dPR/dt where PR is calculated by projection operator techniques.20 After an onset, a steady state is reached and Rsw is a constant. The overall propagation time was ≈10 × τvib. Within this model, STM-induced isomerization is possible. Figure 3c shows isomerization rates Rsw as a function of tunneling current for various assumed librational lifetimes, τvib, at T ) 0 K. Efficient libron-phonon coupling and short lifetimes well below nanoseconds were assumed because the librational mode of 36 meV lies in the phonon band of Si(001).39 Double-logarithmic plots Rsw versus I in Figure 3c have slope 1, indicating a linear increase of Rsw with I in agreement with experiment.23 If one-by-one stepladder climbing was at work, a scaling I ∝ In would be expected instead, where n ≈ 5 is the number of levels below the barrier. Inspecting the IET excitation pathways we find that in our anharmonic system isomerization is dominated by “multilibron” transitions. Thus, a single electron brings the wavepacket close to the barrier top from where it easily tunnels to the other side. The switching rate depends on the vibrational relaxation time τvib according to Figure 3c. Short lifetimes (τvib ) 0.1 ps) cause low rates, because quenching by libron-phonon coupling competes with librational excitation by IET. For the probably more realistic, longer lifetime τvib ) 10 ps one finds Rsw ≈ 7 Hz at I ) 1 nA. This corresponds to a quantum yield (switches per electron) of 1.1 × 10-9, independent of I, which agrees well with the experimental value of 1 × 10-9 at T ) 5 K. Switching at Higher Temperatures. Increasing the temperature has a clear effect according to theory. This is demonstrated in Figure 3d, where Rsw is logarithmically plotted versus 1/T. A lifetime τvib ) 10 ps was assumed and 2999
three different currents, I ) 0.1, 0.7, and 10 nA were chosen. We observe a plateau region with a largely T-independent switching rate up to a crossover temperature Tc, which is about 55 K for I ) 0.7 nA. In this regime, isomerization is caused by IET. Above Tc, Rsw is dominated by thermal overthe-barrier switching and follows an Arrhenius law. The effective activation barrier Ea according to eq 1 is 0.14 eV and thus smaller than the computed barrier of 0.179 eV due to quantum corrections. The theoretical prediction of Tc is fully supported by experiment. In experiment, the sample temperature was varied between 5 and 100 K by slowly heating the entire STM scanner at a temperature rate of 5 K per hour. The switching rate at 5 K of 3.5 Hz reported in the lowest curve of Figure 2 is essentially unchanged at 17 K (purple curve), whereas it appears to be clearly increased at 53 K (red curve) and at higher temperatures. The inset in Figure 2 summarizes the measured temperature dependence of Rsw in an Arrhenius plot. As evident, a clear-cut crossover behavior is found between a temperature-independent plateau rate of Rsw ) (3.4 ( 0.9) Hz below 45 K, and an Arrheniuslike rate increase above the transition temperature. The experimental data are shown for direct comparison also as crosses in Figure 3d. A straightforward analysis of the measured two-level current traces was attainable at temperatures up to 100 K. This experimental limitation makes it difficult to determine the slope of the experimental curve for temperatures T > Tc accurately and may also explain why, from simple inspection at least, the theoretical slope in the Arrhenius region seems to be higher than in experiment. Theory predicts in Figure 3d as another effect, the dependence of the low-temperature tunneling rate and crossover temperature on the current, I. At higher currents, the low-T rate is shifted upward, in a linear fashion according to Figure 3c. In parallel the crossover temperature increases with increasing I from Tc ∼ 50 K (I ) 0.1 nA) over Tc ∼ 55 K (I ) 0.7 nA) to Tc ∼ 65 K (I ) 10 nA) and Tc ∼ 75 K (I ) 100 nA, not shown). This suggests that the transition from a “thermal”, classical to a “tunneling” quantum regime can be controlled by an external parameter, I. The experimental verification of this expectation is underway. Conclusions. A generalized Master equation approach has been developed for the STM-driven biconformational switching of COD on Si(001). At temperatures below a critical temperature, Tc, switching is by one-electron IET, with a T-independent switching rate. Above Tc, isomerization proceeds by a thermal mechanism with exponential T-dependence. The predicted temperature variation of the rate is fully supported by experiment. Theory further suggests that the transition temperature and, therefore, “classical” versus “quantum” behavior of the moving atoms can be controlled by external parameters. This is different from the diffusion experiments cited above,24 where the transition temperature depends on intrinsic system properties only.
3000
Acknowledgment. Support by the Deutsche Forschungsgemeinschaft through SFB 658, Projects A2 and C2, is gratefully acknowledged. References (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37)
(38) (39)
Stroscio, J. A.; Eigler, D. M. Science 1991, 254, 1319. Ho, W. J. Chem. Phys. 2002, 117, 11033. Bartels, L.; Meyer, G.; Rieder, K. H. Phys. ReV. Lett. 1997, 79, 697. Eigler, D. M.; Lutz, C. P.; Rudge, W. E. Nature 1991, 352, 600. Bartels, L.; Meyer, G.; Rieder, K. H.; Velic, D.; Knoesel, E.; Hotzel, A.; Wolf, M.; Ertl, G. Phys. ReV. Lett. 1998, 80, 2004. Stroscio, J. A.; Celotta, R. J. Science 2004, 306, 242. Stroscio, J. A.; Tavazza, F.; Crain, J. N.; Celotta, R. J.; Chaka, A. M. Science 2004, 313, 948. Stipe, B. C.; Rezaei, M. A.; Ho, W. Science 1998, 279, 1907. Gaudioso, J.; Ho, W. Angew. Chem., Int. Ed. 2001, 40, 4080. Qiu, X. H.; Nazin, G. V.; Ho, W. Phys. ReV. Lett. 2004, 93, 196806. Iancu, V.; Hla, S. W. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 13718. Silien, C.; Liu, N.; Ho, W.; Maddox, J. B.; Mukamel, S.; Liu, B.; Bazan, G. C. Nano Lett. 2008, 8, 208. Liljeroth, P.; Repp, J.; Meyer, G. Science 2007, 317, 1203. Gao, S.; Persson, M.; Lundqvist, B. I. Solid State Commun. 1992, 84, 271. Walkup, R. E.; Newns, D. M.; Avouris, P. Phys. ReV. B 1993, 48, 1858. Salam, G. P.; Persson, M.; Palmer, R. E. Phys. ReV. B 1994, 49, 10655. Teillet-Billy, D.; Gauyacq, J. P.; Persson, M. Phys. ReV. B 2000, 62, R13306. Liu, K.; Gao, S. Phys. ReV. Lett. 2005, 95, 226102–1. Quaade, U. J.; Stokbro, K.; Thirstrup, C.; Grey, F. Surf. Sci. 1998, 415, L1037. Zenichowski, K.; Klamroth, T.; Saalfrank, P. Appl. Phys. A 2008, 93, 319. Riedel, D.; Cranney, M.; Martin, M.; Guillory, R.; Dujardin, G.; Dubois, M.; Sonnet, P. J. Am. Chem. Soc. 2009, 131, 5414. Pitters, J. L.; Wolkow, R. A. Nano Lett. 2006, 6, 390. Nacci, C.; Lagoute, J.; Liu, X.; Fo¨lsch, S. Phys. ReV. B 2008, 77, 121405(R). Kua, J.; Lauhon, L. J.; Ho, W.; Goddard, W. A., III. J. Chem. Phys. 2001, 115, 5620. Nacci, C.; Lagoute, J.; Liu, X.; Fo¨lsch, S. Appl. Phys. A 2008, 93, 313. Cho, J. H.; Oh, D. H.; Kleinman, L. Phys. ReV. B 2001, 64, 241306– 1(R). Becke, A. D. J. Chem. Phys. 1993, 98, 5648. Ditchfield, R.; Hehre, W. J.; Pople, J. A. J. Chem. Phys. 1971, 54, 724. Frisch, M. J. et al. Gaussian 03, rev. B.05; Gaussian, Inc.: Wallingford, CT, 2004. Rocha, W. R.; De Almeida, W. B. J. Comput. Chem. 1997, 18, 254. Marston, C. C.; Balint-Kurti, G. G. J. Chem. Phys. 1989, 91, 3571. Guyot-Sionnest, P.; Lin, P.; Miller, E. J. Chem. Phys. 1995, 102, 4269. Andrianov, I.; Saalfrank, P. J. Chem. Phys. 2006, 124, 034710–1. Sakong, S.; Kratzer, P.; Han, X.; Lasse, K.; Weingart, O.; Hasselbrink, E. J. Chem. Phys. 2008, 129, 174702–1. Persson, B. N. J.; Demuth, J. E. Solid State Commun. 1986, 57, 769. The fitting parameters are A0 ) 0.0575 ea0, σ ) 0.2223 rad, and A1 ) 1.4056 ea0. The perturbative expression for the resonance rates according to ref res 20 is Wnfm )(I/e)[|〈n|Ve(φ)- Vg(φ)|m〉|2]/[∆2 + (pΓ/2)2], where Vg is the ground state potential along the reaction coordinate, Ve is the resonance state potential, Γ is the resonance width, and ∆ the biasshifted excitation energy ∆E, that is, ∆ ) ∆E + eU. The Ve are obtained from Koopmans’ theorem by attaching an electron to the LUMO (“electron attachment”), or removing an electron from the HOMO (“hole attachment”) of the cluster used above.20 With the choice ∆ ) 8 eV, 1/Γ ) 1 fs and τvib ) 1 ps, inclusion of the resonance rates leads at T ) 0 K to a switching rate that is by about 5 (electron attachment) or 8 (hole attachment) times larger than in the pure dipole case. Persson, B. N. J.; Baratoff, A. Phys. ReV. Lett. 1987, 59, 339. Tu¨tujncuj, H. M.; Jenkins, S. J.; Srivastava, G. P. Surf. Sci. 1998, 402404, 42.
NL901419G
Nano Lett., Vol. 9, No. 8, 2009
