The limits of inelastic tunneling spectroscopy for identifying transport pathways
Inelastic Electron Tunneling Spectroscopy (IETS) is a powerful tool to study the properties of molecular junctions. In particular, it is considered useful for extracting information on electron transport pathways. We explore the limits of this approach by comparing computed interatomic transmission pathways with IETS intensities for different molecular junctions, employing a new efficient implementation for evaluating IETS intensities via the mode-tracking algorithm. We find that while a correlation between pathways and IETS intensities indeed holds when vibrations are clearly localized on atoms off the transport pathway, there is no such correlation for molecules with less localized vibrations, even if transport pathways only sample part of the molecule, and even if a statistical analysis over the vibrational modes is made. This could indicate that the significance of IETS signals for transport pathways is limited to molecules with localized vibrational modes.
<!>Methods
<p>molecules under unusual (nonequilibrium) conditions. [1][2][3][4][5][6][7][8][9][10][11][12] For understanding and controlling electron transport through molecules, it is important to know which path an electron follows through a molecule. This has been studied extensively in proteins, [13][14][15][16][17] and pathway control has been achieved via infrared excitations in donor-bridge-acceptor systems. 18 Pathway analyses are also essential for identifying whether electrons follow helical paths in helixshaped molecules 19 or in electronically helical structures, 20 which may be important for understanding chiral induced spin selectivity. [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] Circular pathways are closely linked with destructive quantum interference in molecular junctions, [39][40][41] and engineering pathways can be employed as a means of manipulating the transport properties of molecular systems. 42 While a number of theoretical approaches exist for extracting pathway information, 9,14,15,20,39,[43][44][45][46][47][48][49][50][51] access by experimental methods is limited. An experimental technique which is able to provide pathway information is inelastic tunneling spectroscopy (IETS), 1,[52][53][54][55][56] in which signatures of tunneling electrons exciting molecular vibrations are recorded. In addition to a set of propensity rules, [57][58][59][60] it has been found that molecular vibrations interact with tunneling electrons much more strongly when they are located within their path than when they are outside of it. 52,53 We explore the limits of this relation by comparing electron transport pathways and IETS intensities from first-principles simulations not only for systems with well-localized vibrations, but also with delocalized ones. We find that for the latter case (which is frequent in typical molecular junctions), IETS cannot give information on transport pathways, even if a statistical analysis over vibrational modes is made.</p><p>For short molecules off resonance and at low temperatures, the dominant transport mechanism usually is elastic coherent tunneling. 1,54,61 Aside from elastic processes, inelastic ones such as excitations into different spin or vibrational states can be measured in a scanningtunneling-microscopy (STM) or break-junction experiment at low temperatures, 1,62,63 ap-2 pearing as a peak or dip in the second derivative of the current I with respect to the voltage V . When the applied voltage provides enough energy to excite a vibration, an additional transport channel is opened as shown in Figure 1. Thus, a complete vibrational spectrum of the junction can be obtained by varying the applied voltage.</p><p>Figure 1: Elastic and inelastic tunneling through a tunneling barrier representing a molecule between two electrodes. h! corresponds to the energy of a vibrational quantum. For energies below h!, just the elastic channel is available. At energies higher or equal to h!, the inelastic channel is opened, which leads to a slight increase of the tunneling current. By calculating the second derivative of the current with respect to the voltage, the IET spectrum is obtained.</p><p>For the calculation of IET spectra, several approaches have been suggested. [64][65][66][67][68][69] We follow the approach derived by Troisi et. al., 69 who generalized the theoretical description of the elastic tunneling current through a molecular junction within the coherent tunneling regime. [70][71][72] Using a Green's function approach, 73 the transmission function T (E) is obtained as</p><p>This roughly describes the probability of an electron to be transmitted from one electron to the other at a certain energy. The elastic zero-bias conductance g is given by</p><p>L , R describe the electronic coupling between the left or right electrode and the molecular subsystem, while G C is the Green's matrix describing the molecule (or more generally the central region) in the presence of the electrodes. g 0 = 2e 2 h is the quantum of conductance (with e the unit charge and h Planck's constant).</p><p>To introduce electron-phonon coupling, the dependence of the central-region Green's function on the nuclear coordinates is taken into account. Performing a Taylor expansion of G C for a set of vibrational modes {Q ↵ } in the harmonic approximation (using mass-weighted coordinates) around the equilibrium structure, the first-order correction is nonzero only for an initial and final state, which differ by one vibrational quantum. 69 This part (G ↵ C ) describes the correction to the elastic current due to the emission (or adsorption) of one vibrational quantum and is given by 74</p><p>for each matrix element. From this matrix, the peak area for each mode ↵ can be calculated by 75</p><p>where each matrix is evaluated at the Fermi energy.</p><p>Intuitively, one would expect this IETS intensity to be larger for vibrations involving atoms in the path of the tunneling electrons. Therefore, if, for example, the intensity of a vibration localized on a side group is low, this suggests that the current bypasses that sidegroup. If vibrational modes are delocalized over substantial parts of the molecule, as is often the case for typical molecular wires studied in conductance experiments, it is less clear how to extract information on current pathways from IET spectra. Inspired by Ref., 52 we employ a statistical average in such cases.</p><p>Electron tunneling paths are not directly accessible in experiments on molecular junctions, but can be evaluated from theoretical analysis. For many different types of molecules, a decomposition of the overall transmission into atom-to-atom contributions T AB (E) has proven valuable to understand properties like switching or quantum interference 39,46,48,[76][77][78][79][80][81] (see Supporting Information for more details).</p><p>In the following, we compare this information on transmission pathways with IETS intensities of vibrations located on and off the transmission path for three molecules with increasing delocalization of vibrational modes. We treat the Fermi energy as an adjustable parameter for two reasons: (1) It is not clear how to properly choose this energy in a simulation, given that the electrode surfaces in a molecular junction may be irregular and strongly modified by adsorbed molecules, making it difficult to find a good atomistic model, and that standard DFT is not very good at describing molecule-metal surfaces, 82,83 and (2) we are interested in general relationships between IETS intensities and electron pathways, so by varying E F , we get more robust information on this relationships from each single molecule than from a single energy value. Such variations of E F can also be thought of as mimicking how chemical substituents can shift transmission curves w.r.t. the Fermi energy of the electrodes. 84,85 When varying the Fermi energy, care must be taken when approaching resonances, as the theoretical approach employed for evaluating IETS intensities is based on perturbation theory, and since the Landauer approach may fail in resonance. 61 However, changes observed when going towards the resonances from a clearly off-resonant situation may be considered reasonably safe to be interpreted in the discussion below.</p><p>First, we compare IET spectra and transmission pathways for a system with a welllocalized vibration: an iron(II) porphyrin molecular junction with a CO and an imidazole ligand in is low-spin ground state, inspired by myoglobin (see Figure 2). Porphyrin complexes are of high interest for molecular electronics, 86 for example due to their switching behavior. 87,88 For the complex chosen here, transmission was previously shown based on theoretical analysis to bypass the central iron atom (and hence the CO ligand) for all energies except close to (sharp) resonances around the iron d-orbital energies. 43 This would suggest that IETS intensities of the CO vibration should be low.</p><p>Figure 2 shows the system together with the total transmission, the local transmissions at 5.0 and 6.0 eV and the summed-up local transmissions involving the central iron atom, the carbon and oxygen atoms of the adsorbed CO, and an anchoring sulfur atom, respectively, normalized by the total transmission,</p><p>where the sum runs over all other atoms B. The contribution of the sulfur atom is flat and close to one for all energies, since current will mostly enter or leave the system through the anchoring groups. For the Fe atom and for the CO ligand, the contribution is very low except for some of the energies at which sharp dips or peaks occur, related to destructive quantum interference. In agreement with our previous study, 43 the local transmissions show that for almost all energies, the electron tunnels via the organic backbone around the metal center, while for, e.g., 6 eV ring currents occur heavily involving the Fe atom. In these regions, the sum over the atomic contributions is occasionally higher than the total transmission, since If as here only certain (e.g., localized) vibrations are of interest, first-principles calculations of vibrational modes in the harmonic approximation can be greatly accelerated by selective techniques, which only evaluate the relevant modes (without loosing accuracy com-pared with a full calculation). This is particularly important for molecular junctions, where a molecular bridge is attached to two fragments representing electrodes, resulting in a fairly large system. This gain in efficiency is achieved by the mode-tracking protocol. [89][90][91] We have therefore programmed an interface which allows for combining both full vibrational analyses and selective mode-tracking calculations with the evaluation of IETS intensities (see Supporting Information) and have applied it here to effectively evaluate the CO stretch vibration only, which takes 4 single point calculations to converge -in contrast to 324 for a full evaluation of the Hessian matrix. Since IETS calculations (and also experiments) can be quite sensitive to structural (and for the calculations, computational) parameters, we apply mode-tracking also for efficient validation of our methodology, which is also discussed in the Supporting Information.</p><p>Next, we study a nitrile-substituted biphenyl junction which was already suggested in Ref. 59 as an example where IETS shows low intensities for the vibrations located on the CN substituents due to the current bypassing them. Our local transmission analysis confirms that for most energies (Figure 3), except for those associated with sharp destructive interference features, current indeed barely passes through the substituents: The summed-up local transmission through the carbon atoms between the benzene units is more than one order of magnitude higher than through the atoms of the nitrile groups (Figure 3 b). This is in line with the IETS intensities for the two C⌘N stretching modes at 2241 cm 1 (asymmetric) and 2242 cm 1 (symmetric), which are much smaller than for the central C⌘C mode stretching mode at (2209 cm 1 ). Approaching energies where the interference features appear, the local transmissions resulting from ring currents through the nitrile groups become comparable to   In the structure just discussed, the C⌘N vibrational modes are still fairly localized, but less so than in the previous example involving the CO ligand (see Supporting Information).</p><p>Still, our analysis suggests that the correlation between IETS intensities and pathways still holds. As a final example, we will study three structures based on a meta-connected benzene wire (Figure 4, left), and two pyridine analogs differing in the location of the nitrogen atom in the ring (middle and right). Conductance in such structures is reduced by destructive quantum interferences, and inelastic contributions are larger than for molecules without interference. 60,92,93 The presence or absence of the nitrogen atom in the two different pathways around the ring should (and does) lead to differences in the preference of the tunneling electron for which pathway it takes.</p><p>Off resonance, for the benzene junction the short path is preferred over to longer one, as can be seen in the left panel of Figure 4 c), since the ratio of local contributions through the lower (green) vs the upper (orange) carbon atom is always larger than one. Inserting a nitrogen atom into the long path does not change the situation substantially (except for energies of around 6 eV, right panel). By insertion of the nitrogen in the shorter path (middle panel), a clear crossover at 4.78 eV can be observed. At higher energies (up to about 3 eV), the longer path is preferred, while at lower energies the electron takes the shorter path. Thus, when varying the Fermi level, we would hope to see a similar change of ratio of the IETS intensities.</p><p>Due to their delocalized nature (see Supporting Information), the vibrational modes of the molecule can not be easily assigned to fragments of the molecule corresponding to tunneling pathways, so we evaluate a weighted sum of the IETS peaks for each atom (inspired by Ref. 52 ),</p><p>where N Q is the number of vibrational modes, and |d A,Q↵ | is the length of the displacement vector for the x, y, z degrees of freedom of atom A in mode Q ↵ . Thus, if an atom is participating in a vibrational mode which has a high IETS intensity, this would significantly add to C A . The full IET spectra can be found in the SI.</p><p>Figure 4 e) shows the calculated atomic IETS contributions close to the crossing point discussed above. We would expect the contribution of the nitrogen atom (marked green in the Figures) to decrease with energy in comparison to the contribution through the topmost carbon atom (marked orange). This trend cannot be seen in the in the plot. The IETS intensities in general drop from E F = 5 to 4.0 eV, and rise when going to 3.5 eV. In all cases, the ratio between the contribution of the nitrogen atom and the top carbon atom stays constant, only at 3.5 eV a slight relative drop of the contributions of the nitrogen atom can be observed. This would be in line with the analysis of the local transmissions, but is to weak to draw conclusions.</p><p>The low contributions of the sulfur atoms might be puzzling at first, since more or less all of the current has to enter/leave the system through these atoms. They contribute mostly to modes with low wavenumbers (< 500 cm 1 ). These wavenumbers were left out of the analysis since the sulfur-gold interface, not being our focus, is not described very accurately by the attachment to gold clusters without further structural relaxations, and since lowwavenumber vibrations are not described well by the harmonic approximation. Also, most molecular vibrations are at higher wavenumbers (even though exceptions may exist, see e.g.</p><p>Ref. 94 ).</p><p>As a consistency check, we also perform the statistical analysis for the CN-substituted biphenyl (Figure 3 d), note that due to the larger variations in C A for this molecule, a logarithmic scale was chosen). Regardless of the choice of Fermi energy, the summed-up IETS contributions C A for the CN atoms are consistently lower than for the carbon atoms making up the biphenyl backbone (as above, the sulfur atoms have low contributions because the lower-frequency vibrations were not taken into account in the summation in Eq. ( 5)).</p><p>The only exception is in at the sharp antiresonance feature ( 7.8 eV), which involves local transmissions through the CN groups, and accordingly has larger T A on those atoms. This statistical analysis is less clear-cut than the considerations based on selected vibrational modes discussed above, but importantly, it supports the same conclusion on the ability of IETS to extract information on electron tunneling pathways in this system.</p><p>Overall, our results suggest that in systems where the vibrational degrees of freedom are strongly delocalized, it is difficult to deduce tunneling pathways using IETS. This conclusion could also be a due to approximations inherent in the theoretical methods employed here, for example there are cases in which currents do not follow bonds 51 (as they are forced to do here), which would require more flexible theoretical approaches as described in, e.g., Refs. 45,[49][50][51] Yet, these cases so far have been limited to aliphatic compounds, 51 and the past success of atom-to-atom pathway analysis for describing and elucidating IET spectra and single-molecule conductance properties of conjugated systems as studied here suggests that our conclusion is reasonable, and that if a correlation between electron transport pathways and IET spectra for molecules with delocalized vibrational modes can be made, it is at least not straightforward.</p><!><p>Molecules were structurally optimized as dithiols using Turbomole 6.6, 95-98 Ahlrichs' splitvalence triple-⇣ basis set with polarization functions on all atoms (def2-TZVP 99,100 ) and the BP86 exchange-correlation functional. 101,102 We used the resolution-of-identity (RI) approximation as implemented in Turbomole 6.6. [103][104][105] The convergence criterion of the self-consistent field (SCF) algorithm was a change of energy below 10 8 Hartree. The convergence criterion for the structure optimizations was a gradient below 10 4 atomic units.</p><p>To mimic the electrodes, after the optimization of the molecules, Au 3 clusters were attached to the terminal sulfur atoms after stripping off the terminal hydrogen atoms, mimicking hollow-site adsorption on Au (111) surfaces. The sulfur-gold distance was 2.85 Å. Vibra-tional analyses were performed in the harmonic approximation using MoViPac, 91,106 employing BP86/def2-TZVP on the cluster-molecule-cluster structures. This is reasonable even though the molecule-metal interfaces were not structurally optimized (to prevent unphysical distortions resulting from the small cluster size), as it was shown for thiolate molecules on silver clusters. 107 For both full vibrational analyses and selective calculations via modetracking, second derivatives of the total energy w.r.t. structural displacements were evaluated in a seminumerical fashion (such that each second derivative needs two single-point calculations 89,108 ). If the mode-tracking algorithm 89,90 was applied to selectively calculate a specific vibrational normal mode, the convergence criterion was set to 0.0005 for the maximum component of the residuum vector, for root homing the overlap with the last selected eigenvector was used as a selection criterion, and for preconditioning the backtransformation of the Davidson matrix was employed. Transport calculations were performed by carrying out single-point DFT calculations (B3LYP 109-111 /LanL2DZ) with Gaussian 09 112 on the cluster-molecule-cluster structures, followed by postprocessing via Artaios. 72,113 This smaller basis set was employed to prevent ghost transmission. 72 The wide-band limit was used to obtain the Green's function matrices of the isolated electrodes, with the constant LDOS set to 0.036 eV 1 as calculated by DFT for the 6s band of gold. 114 For IETS calculations, the same electronic structure settings were used as for the transport calculations. The details of the implementation using Artaios and MoViPac can be found in the Supporting Information.</p>