First-Principles Study for Detection of Inelastic Electron Transport in

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First-Principles Study for Detection of Inelastic Electron Transport in Molecular Junctions by Internal Substitution Hisao Nakamura* Research InitiatiVe for Computational Sciences (RICS), Nanosystem Research Institute (NRI), National Institute of AdVanced Industrial Science and Technology (AIST), Central 2, Umezono 1-1-1, Tsukuba, Ibaraki 305-8568, Japan ReceiVed: December 2, 2009; ReVised Manuscript ReceiVed: May 20, 2010

The idea of doping and chemical substitutions for bridge molecules is one of promising techniques to control the transport properties and engineering of molecular devices. In the present article, we extend this concept and propose the use of internal substitutions to detect the mechanism of ballistic and inelastic transport through molecular junctions. By performing first-principles transport calculations for several internally substituted systems, we show systematic pathway analysis of tunneling electrons and vibronic states as well as electron-phonon couplings on bridge molecules. The correlation of inelastic electron tunneling spectroscopy (IETS) and Raman spectroscopy is also discussed. 1. Introduction There has been progress in the conductance of molecular junctions toward applications for molecular electronics.1-3 However, a more detailed understanding and advanced characterization for atomic structures in junctions are required to form suitable contacts for practical devices.4 Furthermore, finding correlations between the chemical properties of adopted molecules and transport characters in molecular junctions will be useful for designing functional devices.5,6 Inelastic tunneling electron spectroscopy (IETS) is one of the most promising approaches for probing molecular-scale junctions.7,8 The IETS technique can identify chemical signatures of the bridge molecule in the junction by using the interaction between electron and molecular vibrations (phonons). According to recent progress in experimental studies,9-11 it is possible to measure the conductance and IETS of each single molecule in the junction simultaneously, where the bridge molecule is connected to both electrodes. In addition, the high resolution of the resulting IET spectra provides information potentially about the local Joule heating mechanism12 and tunneling pathway of electrons (current profile) in the junctions.13,14 The assignment and interpretation of IET spectra is generally difficult without theoretical computation. Two basic theories of IETS have been known for many years; one is based on the one-dimensional potential model,15 and the other adopts the sum of the Coulomb interactions between a transport electron and the partial site charges of the atoms in the molecule.16 Both theories lead to the correlation between infrared (IR) and IETS activities, though the latter also predicts similar correlation strength with Raman-active modes. However, IETS experiments of semifluorinated alkanethiol junctions by Beebe et al. show that IETS-active C-H modes have a Raman preference and are not related to IR intensities.17 Theoretically, several groups have proposed various new symmetry propensities combined with computational simulation of IETS, where correlation with IR/Raman spectroscopy is only partial and not rigorous.18-21 To address these issues, we perform a systematic analysis of IETS intensity with correlations of mode symmetry, electron* E-mail: [email protected].

phonon interactions, and local current vectors for both ballistic and inelastic currents using first-principles calculations. The central idea in Beebe et al.’s study is substitutions from H to F atoms in alkanethiol.17 In the present study, we adopt this concept of “chemical substitutions for a bridge molecule” to detect the inelastic transport mechanism. Since an alkanethiol molecule has a low-symmetry conformation, we adopt benzenedithiol (BDT), which has a high-symmetry structure (D2h) as a template. Then, we substitute one C atom for one B atom and the other C atoms for N atoms, known as internal substitutions, by means of an analogue to internal p(n)-type doping.22-24 In the present study, we employ three substituted molecules, n1, n2, and n3, where the C atom at the β position is substituted by the N atom and then the other C atom is replaced by a B atom for ortho (n1), para (n2), and meta (n3) positions, as shown in Figure 1. The advantages of adopting BDT and relating n1-n3 are as follows: (i) The molecular conformations are almost invariant due to the rigid benzene ring structure. Therefore it is easy to assign vibrational modes of n1, n2, and n3 to the modes of symmetric BDT although the electronic structures (dipole, polarizability, etc.) are changed for each others. (ii) The influence of symmetry breaking, which is particularly important for analyzing the influence of electron-phonon interactions caused by multilevel molecular orbitals (MOs), can be easily analyzed. (iii) The transmission coefficient close to the Fermi level is much smaller than 1 for all of the cases, as shown later; hence, these substitutions are free from the complicated IETS shape/dip problem, such as the 0.5 rule.25,26 2. Theoretical Framework In order to perform first-principles transport calculations, we adopt the nonequilibrium Green’s function (NEGF) formalism combined with density functional theory (DFT).27-30 The current for the given bias V is defined as a flux on the left (right) lead, L (R), connected to each bulk. When electron-phonon interactions are omitted, the total current consists of only the ballistic current, Ibal, and it can be written using the Green’s function, G(E), as follows

10.1021/jp9114223  2010 American Chemical Society Published on Web 06/24/2010

Inelastic Electron Transport in Molecular Junctions

Ibal(V) ) g0

∫ dE T(E)(fL(E) - fR(E))

T(E) ) Tr[ΓL(E)G(E)ΓR(E)G†(E)]

J. Phys. Chem. C, Vol. 114, No. 28, 2010 12281

(1)

where fL/R is the Fermi distribution function with chemical potential µL (µR) of the left (right) leads and is related to V by µL - µR ) V. The transmission coefficient, T(E), is a function of one-electron energy, E. The term, ΓL/R is i(ΣL/R - ΣL/R†), where ΣL/R is the lead self-energy. In the present study, we adopt the unit as p ) e ) 1; hence, the conductance unit, g0 is 1/π.

In the inelastic transport, electron-phonon interactions should be included in the NEGF calculations. When electron-phonon couplings are weak, the lowest-order expansion (LOE) of the electron-phonon interaction is sufficient.31 Recently, we proposed the conventional LOE formulation by using the rigorous LOE form.29,31,32 The details of the derivation can be found in the Appendix of ref 29. In the present report, we summarize only the results. The total current in the conventional LOE is expressed as the sum of the ballistic (Ibal), elastic correction (δIec), and

Figure 1. The atomic structure of the molecular junctions on the Au(001) electrodes, where only the bridge molecule and nearest four Au toms are shown. Each inset represents the constitutional formula of the relating bridge molecule, that is, (a) BDT, (b) n1, (c) n2, and (d) n3.

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inelastic (Iinel) current terms. For low bias, the elastic correction and inelastic currents can be represented as

∑ δI ) g V ∑ {(2N + 1)Tj - Tj - Tj j )F(V, Ω )} j g ∑ {(T +T j (2N V + F(V, Ω ))} ) ∑ I ) g ∑ {T

δIec )

ec R

0

R

0

Iinel

ecL R

ec R

R

R ecR R

ecL R

ecR R }

+

R

R

inel R

0

R

in R

R

R

R

(2) where ΩR is the frequency of the mode QR (simply labeled as R). The function F(V,ΩR) has the following form

F(V, ΩR, Tm) ) (ΩR - V)NBE(ΩR - V) - (ΩR + V)NBE(ΩR + V) (3) where, NBE(ω) is the Bose-Einstein distribution. The term NR is the current-induced nonequilibrium phonon distribution j ecL/R, and T j in can be obtained from j ec, T function. The terms T ec the energy-dependent functions T (E), TecL/R(E), and Tin(E) with averaging over the range, E ⊂ [EF - V/2, EF + V/2]. The concrete expressions for these functions are given in ref 29. The normalized IET intensity for the mode R is written as

IIETS ) R j ec 2(T R

+

inel d2(δIec R + IR ) 2

dV j Rin) T

d2NR dV2

+

≈

{

dN 1 j Rec + T j Rin) R + 4(T j dV g0T

j RecL (T

+

j RecR T

+

j Rin) T

d2F(V, ΩR) dV2

}

(4)

bal Jµfν ) g0V Im{H*µν(GΓLG†)µν - H*µν(GΓRG†)µν}

(5) where µ f ν represents that the this bond current is current density from the AO φµ to φν.34,35 Now, we can define the local current vector on the atom A as follows

∑ ∑

bal Jµfν b u AB

inel Jµfν ) g0V(2NR + 1) Im{H*µν(GMRGΓLG†MRG†)µν -

H*µν(GMRGΓRG†MRG†)µν} (7) where MR is the electron-phonon coupling. The local inelastic bal current vector on the atom A can be obtained by replacing Jµfν inel by Jµfν in eq 6, and we refer to it as the vibronic pathway. Using the vibronic pathway, one can trace the flow of the exchanged energy inside of the molecule. This will be a useful quantity to analyze the relations between molecular properties (substituted functional groups, atoms, and the vibrational heating) directly. In the present study, the transmission coefficients of all of the systems are much smaller than 1 (i.e., low conductance), as subsequently reported. Usually, the IETS intensity is dominated 2 inel ec by (d2Iinel R /dV ) in eq 4 and |I | . |δI | when the conductance 25,26 Hence, the analysis of the vibronic is sufficiently low. pathway will shed light on the IETS propensities as well as the mechanism of vibrational and/or local heating in the molecule. 3. Electronic Structure

Note that only the inelastic current Iinel is the term caused by exchanging the energy with molecular vibrations by electronphonon interactions.26 In this sense, term Iinel can be regard as the current of vibronic states rather than that of electrons.33 Since the current is defined as the flux observed on the lead region L or R, the current density and/or local current vector in the bridge molecule is not given in eqs 1 and 2. The local current vector is very useful to analyze the relation between a single molecular property and its role for the transport because this “current profile” represents the pathway of electrons. In the NEGF method expanded by an atomic orbital (AO) basis, the local current vector on each atomic site can be easily obtained by the bond current.34 The bond current of the ballistic current term, which relates to the pathway of an electron, is written as follows

b J Abal ) g0V

A bond current for each mode-specific inelastic current Iinel R can be also derived by using the NEGF framework, just as the case of the ballistic current term. Since Iinel R is the result of energy exchange between electrons and phonons, the local current vector for Iinel R can be regarded as the pathway of a vibronic state for the mode R. Applying the LOE, the resulting from of the inelastic bond current is35

(6)

B*A µ∈A,ν∈B

where B is the label of other atoms and b uAB is the unit vector connected to the positions of A and B. We refer to this local ballistic current vector as the electron pathway.

We set a 42 × 42 unit cell of Au(001) as the electrodes, and each lead part (L and R) contains two gold (Au) layers by fixing the gap at 14.24 Å. The central region consisted of the molecule (BDT, n1, n2, or n3) and two Au layers on each side. Further, we added additional Au layers as buffer for part of the DFT calculations. In the DFT calculations, we adopted the PBE exchange-correlation functional and took 3 × 3 × 1 points for k-sampling. The basis set was a pseudoatomic-type orbital at the single-ζ polarized (SZP) level. The SIESTA program was applied for standard DFT calculations.36 The hollow site was adopted as the adsorption position, and then, the geometry optimization was performed by relaxing the bridge molecule and four Au atoms on each side. We refer to this relaxed region as the vibrational box. The relating atomic configurations are given in Figure 1a-d for BDT, n1, n2, and n3, respectively. The bridge BDT keeps the structure quite close to D2h symmetry. The n1 molecule adopts a similar configuration, that is, the molecule is in the plane and the line of the hollow-S-C-CS-hollow, which corresponds to the C2 axis of the BDT and is almost equal to180°. Due to the polarization, the most deviated angle from the benzene ring is ∠CBN (113.9°), but the backbone of the ring is sufficiently close to the benzene ring. On the contrary, the adsorption site is slightly shifted (about 0.15 Å) from the center of the hollow position for both n2 and n3. The C2 axis in the BDT is now nonlinear. For instance, the angle created by the two S-C bond axes is 160.0° for n3, though the benzene ring backbone is still maintained. We applied the normal-mode analysis by relaxing only the atoms in the same vibrational box and obtained the molecular internal vibrations, whose atomic displacements are localized on each molecule. In order to find the correspondence between the mode Rn1 (n1 molecule) and the mode RBDT (BDT molecule), we calculated the inner product b e(Rn1) · b e(RBDT), where b e(Rn1) and b e(RBDT) are the normalized displacement vectors of Rn1 and



Figure 2. The three PMOs lower than the Fermi level, ψσ, ψπ(1), and ψπ(2), for each molecular junction. The plots in the first, second, third, and fourth row represent the PMOs of the BDT, n1, n2, and n3 systems, respectively.

RBDT, respectively. When the value of b e(Rn1) · b e(RBDT) is sufficiently large (close to 1), we can assign the mode Rn1 by the relating RBDT. We applied the same procedure and made the list of one-to-one correspondence with BDT for n2 and n3. Thanks to the rigid benzene ring backbone, we could find the clear one-to-one correspondence, that is, most of the modes of n1-n3 can be labeled by using BDT. (Typically, the value e(RBDT) is larger than 0.7 when the correspondence b e(Rn13˜) · b exists.) For convenience, we use the D2h irreducible representation as the label of these vibrational modes. For the subsequent analysis, we calculated projected molecular orbitals (PMOs), where the PMOs are defined as the eigenstates of the Hamiltonian projected onto the molecule. In Figure 2, the three PMOs closest to EF at the Γ point are shown for each molecule. Although one PMO of the BDT has no amplitude on the terminal S atoms, which have been already predicted by the previous works,37 the relating PMOs for the other molecules have sufficient states on the S atom. Clearly, one can identify each PMO as one σ-bond-type PMO (ψσ) and two π-type PMOs (ψπ(1) and ψπ(2)) from Figure 2. The PMO energy “dressed” by the coupling with the electrodes, which is denoted as EK, can be defined by using the calculated Green’s function as follows

EK ) EK0 + Re(EK0 - 〈ψK |G-1(EK0)|ψK〉) (K )

ψσ, ψπ(1), or

ψπ(2))

(8)

where EK0 is the eigenvalue of the molecular projected Hamiltonian and the second term is the dressed energy part. Since this dressed PMO energy relates to the resonant energy level directly, the relative position of EK and the Fermi level EF

TABLE 1: PMO Energies for the BDT, n1, n2, and n3, Where the Connected Bulk Au Fermi Level Is Set to 0a EK0 (eV)

EK (eV)

γK (eV)

(BDT) ψσ ψπ(1) ψπ(2)

-2.63 -2.32 -1.40

-1.65 -2.31 -0.75

0.73 0.00 0.27

(n1) ψσ ψπ(1) ψπ(2)

-2.67 -2.64 -1.12

-1.72 -2.12 -0.61

0.73 0.40 0.18

(n2) ψσ ψπ(1) ψπ(2)

-2.58 -1.77 -1.55

-1.75 -1.36 -0.98

0.63 0.21 0.22

(n3) ψσ ψπ(1) ψπ(2)

-2.57 -2.04 -1.07

-1.74 -1.46 -0.74

0.32 0.13 0.13

a Each orbital is labeled as ψσ and so forth, as given in the text. The energy without molecule-electrode interactions (bare energy: EK0) is in the first column, while the energy including the couplings (PMO energy dressed by the interaction with electrodes: EK) is given in the second column. The third column is the molecule-electrode coupling. The units are eV.

determines the tunneling is either resonant tunneling or offresonant for the transport. Each energy value is listed in Table 1. Here, we set EF to 0. When a molecule is completely isolated, one of the three PMOs for each molecule belongs to the lowest unoccupied molecular orbital. On the contrary, any PMOs lower than the Fermi level are due to the molecule-electrode couplings. The energy level of ψπ(2) is closest to EF, and the energy differences between ψπ(2) and ψπ(1) are changed by the

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substitutions. However, the gap of ψπ(2) and EF is large, that is, 0.75, 0.61, 0.98, and 0.74 eV for BDT, n1, n2, and n3, respectively. Therefore, the present “internal substitution” for the BDT does not tune the energy gap with the Fermi level drastically, and the off-resonant tunneling mechanism is maintained for all cases. 4. Conductance and IETS NEGF-DFT was performed by the same parameters (basis set, b k| points, etc.) as those used in the previous DFT calculations.38 Since the coverage rate of the molecule is sufficiently low, the phonons at the Γ point are sufficient to treat molecular internal vibration. However, in order to be consistent with the calculation of the ballistic current, we calculated electron-phonon couplings for each b k| point to calculate the inelastic transport and IET signals. Figure 3a shows the calculated transmission coefficients for each system. The conductance values of ballistic transport are 0.14g0, 0.23g0, 0.06g0, and 0.10g0 for BDT, n1, n2, and n3, respectively. Using the calculated Green’s function, we also calculated the projected density of states (PDOS) of the PMOs (ψσ, ψπ(1), ψπ(2)) for the BDT and n1-n3, as shown in Figure 3b. Since ψ(1) π of the BDT has no amplitude on the terminal S atoms, the couplings with the leads are almost equal to 0. The coupling strength of each PMO with the rest of the (outside) region, denoted as γK, can be estimated at the Fermi level

γK ) Im(EF - 〈ψK |G-1(EF)|ψK〉) (K ) ψσ, ψπ(1), or ψπ(2))

(9)

and Table 1 shows the results. Since the amplitudes of the ψπ(1)/(2) orbitals of the BDT are localized on the one-side (γψ(2) ) comes terminal S atom, one can expect that most of γψ(1) π π from only the left (right) lead side in the n3 molecule. The γK values in the other molecules are determined by contributions from both the left and right sides. According to the peak position of the PDOS and γψπ(2), the ballistic transport is mainly determined by tunneling through the PMO ψπ(2) for each BDT, n1, and n2 system. Hence, the transmission coefficient can be expressed by simple Breit-Wigner form.39 Since the present tunneling is off-resonant tunneling dominated by the ψ(2) π channel, we name this transport mechanism a Breit-Wigner-type off-resonant tunneling. On the contrary, the transport mechanism in the n3 system is slightly different because most of the term γψπ(2)(γψπ(1)) comes from only the coupling with the one-side electrode. Recall that an electron must transverse from left (right) to right (left) leads to observe the current, and the molecules have to couple with the two leads. For instance, most of γψπ(2) is the coupling with the right lead (see Figure 2), and hence, the element of ΓL corresponding to ψπ(2) should be close to 0. From eq 1, one can find that the component of T(E) determined by the diagonal term of is also close to 0. Thus, the off-diagonal coupling, γψ(1)ψ(2) ) Im(EF -〈ψψπ(1)|G-1(EF)|ψψπ(2)〉), should contribute to π π the transport for n3. In other words, the interference of the two PMOs should be considered for the transport mechanism. When the dominant channel interferes with other channels (background channels) in scattering processes, it is called Fano resonance.39 In the present n3 tunneling, ψπ(2) is closest to EF, but the interference with ψπ(1) is required for the transport. Therefore, as an analogue, the transport mechanism of n3 can be termed as Fano-type off-resonant tunneling. Figure 4 gives the local current vectors (electron pathway) for all of the systems. For

the BDT, the electron pathways are symmetric against the C2 axis. This is consistent with the delocalized π character for ψ(2) π . However, it is deformed to quasi-symmetric by the internal substitution. The substitution of the bonded B+dN- reduces the bond current, and separated substitutions of N (B) enhance (reduce) the current vectors. Next, we calculated the IETS of the BDT and n1-n3 by including all of the molecular internal modes (in the present case, 30 modes for each case). In the conventional LOE, we adopt 200 meV as the damping factor for the nonequilibrium phonon, NR, which is almost close to external damped limit.29 The temperature is set to 5.0 K, and bias is applied from 0.0 to 0.25 V. Figure 5 shows the resulting IETS, where the intensity is normalized by the derivative conductance to eliminate the conductivity dependence of the spectrum intensity (see eq 4). The normal-mode vector of the visible totally symmetric modes, whose label is expressed as Ag in the D2h irreducible representation, is shown in Figure 5a. Here, we insert the envelope functions to identify distinct moieties in the spectra, where each function is fixed with the Gaussian form. The height and width parameters for the Gaussian function are determined by a simple fitting procedure, where the minimum of the width parameter is set to 15.0 meV. The four peak moieties can be found in the IET signal of the BDT junction (Figure 5b), and each peak envelope includes one IETS-active totally symmetric mode. In the present calculation, we could not find a clear one-to-one correspondence for the Ag(5) mode of n2 with BDT. Thus, we omit this Ag(5) of the n2 molecule in the present analysis. The four IETS-active Ag modes have frequencies of 327, 710, 1030, and 1548 cm-1, which agree well with the recent IETS experimental data in refs 10 and 11. In addition, our results reproduce the relative intensities of the moieties in the IETS reported by Tsutsui et al.,11 although they differ slightly from the results by “mechanically controlled break junction” IETS (MCBJ-IETS) experimental data.10 The two moieties in the lowest bias region consist of not only Ag modes but also nonsymmetric modes (B3u, B2 g, and Au), and this is consistent with previous theoretical work.18,21,29 We note that there is another Ag mode (1123 cm-1), which is the fourth lowfrequency mode in Ag, but its IET intensity is much lower than 0.1; thus, we conclude that this Ag mode is IETS-inactive. While the previous studies by Gagliardi et al.18 and us29 give a weak signal for this inactive Ag mode, the present result has a much weaker intensity than that seen in the above studies. This will be the result of the washing-out of a small numerical error at the Γ point approximation by sufficient b k| sampling. In Figure 5c-e, the IETS of n1-n3 labeled by D2h irreducible representation is presented. Most of the IETS-active modes in the BDT are also visible in the case of n1, n2, and n3, while the relative intensities are different. From now on, we adopt the notation (Label)(n) to specify the mode for convenience, where (Label) is the label of an irreducible representation in D2h and (n) is nth low frequency on the same (Label) modes in the BDT junction system. We then checked the relation of symmetry propensities of IR/Raman and IETS activity. The calculated result of BDT shows that all of IETS-active Ag modes have a Raman preference but are inactive for IR. The recent experiments also observed the Raman preference of the IETS-active Ag modes.10,11 However, nonsymmetrical modes, whose IETS intensities are significantly large (e.g., B3u(1), B2 g(1), Au(1)), are inactive for both IR and Raman. To detect details of the above IETS-Raman correlation in the Ag modes, we performed further analysis for Ag modes by using n1-n3. For instance, let us consider the



Figure 3. (a) The transmission coefficient T(E) for each molecular junction and (b) the PDOS of the PMOs, ψσ, ψπ(1), and ψπ(2), where the top is the BDT (left) and n1 (right) and the bottom is n2 (left) and n3 (right), as shown in the insets. The plots PDOS of ψσ, ψπ(1), and ψπ(2) are represented by the solid, dashed, dashed-dotted, and dotted lines, respectively, Note that the PDOS of ψπ(1) for the BDT is almost equal to 0 in the present energy range and has a quite narrow peak in the lower-energy region.

Ag(3) modes. The Raman intensity for n2 and n3 is reduced to about 30% compared with that of BDT, and the IETS intensity is reduced to about 50%. On the contrary, the IETS intensities of Ag(2) are enhanced more than 120% for n1-n3, while the

Raman intensity is reduced to about 70%. Furthermore, any kinds of the present internal substitutions reduce the Raman intensity of Ag(1), but the enhancement of the IETS intensity for Ag(1) is found in n2 and n3. Thus, even if we focus only on

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Figure 4. The ballistic local current vector (electron pathway) on each bridge molecule. The left panel of the top row is the BDT, and the right is the n1. The left and right panels of the bottom row are n2 and n3, respectively. The large current on each atom is represented by the brightness of the arrow, that is, the brighter arrows mean a larger current intensity.

the Ag symmetric modes, clear evidence of the correlation between Raman and IETS activities cannot be confirmed. 5. Electron-Phonon Couplings and Vibronic Pathway To analyze electron-phonon scattering processes inside of the bridge molecule directly, we calculated the projection of R electron-phonon couplings on the PMOs, MKK′ ) 〈ψK|(∂H/ ∂QR)|ψK′〉, and estimated the vibronic pathway by using eq 7. First, let us focus on the symmetric IETS-active modes, Ag(3),(5), R are given in and compare BDT with n1. The elements of MKK′ Table 2, where the listed values were calculated at the Γ point and scaled by setting MψR (2)ψ(2),R ) A(3) g to 1. The off-diagonal π π elements of the BDT are 0 for both Ag(3) and Ag(5) due to the R group theory. MψR (2)ψ(2) is the largest element in MKK′ for the π π (3) (2) Ag ; thus, the single PMO ψπ should dominate the inelastic current of Ag(3) in the BDT. This is same with the case of the ballistic current, where the single ψ(2) π orbital plays the dominant role for the Breit-Wigner-type off-resonant tunneling. This indicates that the tunneling of the Ag(3) vibronic state can be classified as the Breit-Wigner-type off-resonant. Next, we checked the n3 case. According to our analysis of the ballistic current for n3, the sufficiently large values for (i) both MψR (2)ψ(2) π π and MψR (1)ψ(1) or (ii) MψR (2)ψ(1) are required. In Table 2, one finds π π π π that condition (i) is actually satisfied, and this provides that the tunneling of the Ag(3) vibronic state in n3 can be classified as the Fano-type off-resonant as an analogue to the ballistic current. The analysis for Ag(5) is same as that for Ag(3) for the BDT and n1. On the contrary, the current of Ag(5) in n3 is slightly different. The largest coupling is now MψR (2)ψ(1) which is about twice as π π large as MψR (1)ψ(1) and four times larger than MψR (2)ψ(2) thus, π π π π condition (ii) is satisfied. Hence, off-diagonal elements of both molecule-electrode coupling and electron-phonon coupling

play an important role in tunneling of the Ag(5) vibronic state, where more complicated interference than the tunneling of an electron and/or Ag(3) vibronic states should exist. To see the above analysis of inelastic current more directly, the local current vectors (vibronic pathway) of the Ag(5) terms of the BDT, n1, and n3 were calculated and are shown in Figure 6. Although the vibronic pathway of the BDT and n1 agrees well with the ballistic ones (see Figure 4), one can see the clear difference for n3. In addition, we give a brief analysis for the IETSinactive symmetric mode Ag(4). The largest electron-phonon (4) Ag(4) g coupling of Ag(4) for the BDT is MψA(1) (1), and the value of Mψ(2)ψ(2) π ψπ π π is almost equal to 0. Furthermore, the coupling elements related to ψπ(2), that is, Mψ(2) (K ) ψπ(1)/(2) or ψσ), are close to 0 for the π other systems. Hence, the vibronic pathway of Ag(4) through the ψπ(2) is blocked, and this leads to the inactivity for the IETS. Finally, we discuss the IETS-active nonsymmetric mode. Although many of theoretical calculations have predicted sufficiently strong IET signals of nonsymmetric modes, the details of the relative spectrum intensities and assignments of modes strongly depend on each theoretical model. Furthermore, there is no assignment of such nonsymmetric modes in the experimental studies.9-11 In the present analysis, we focus on the large IETS peak of the lowest-frequency mode, B3u(1) (see B3u(1) expanded by the Figure 5). We calculated the coupling MKK′ PMOs just as the case of the symmetric modes. We found that B3u(1) , where (KK′) is the first column in all of the elements MKK′ Table 2, were smaller than 6.0 × 10-3, although the IETS intensity was as large as Ag(3),(5). This is caused by the delocalization of the electron-phonon couplings on the Au layers rather than that on the bridge molecules. This is very interesting because the B3u(1) mode belongs to the molecular internal mode and is localized on the molecule. In other words,



Figure 5. (a) The displacement vectors of the resulting normal modes for the BDT junction corresponding to the IETS-active totally symmetric modes, Ag(1), Ag(2), Ag(3), and Ag(5). (b-e) The IET signal for the BDT, n1, n2, and n3 junctions, respectively, where the dashed line is the envelope line to specify the four moieties relating to visible Ag modes.

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TABLE 2: Electron-Phonon Coupling Elements in the PMO Basis for the BDT, n1, and n3 Bridge Moleculesa Ag(3) (1) (1)

π π π(2)π(2) σπ(2) π(1)π(2)

Ag(5)

BDT

n1

n3

BDT

n1

n3

0.55 1.00 0.00 0.00

0.12 0.67 0.00 0.39

0.48 0.68 0.00 0.14

1.45 0.94 0.00 0.00

0.57 0.59 0.57 0.00

0.17 0.34 0.00 0.62

a The couplings are for the two symmetric modes Ag(3) and Ag(5). R The indexes of matrix MKK′ are denoted as KK′; for example, the symbol π(1)π(1) in the first column represents the element MψR π(1)ψπ(1). The values are scaled to set MψR (2)ψ(2), R ) Ag(3) to 1. π

π

the inelastic current term of B3u(1) corresponds to the nonlocal nature of the Green’s function, and this Fano-type off-resonant tunneling is supported by many virtual states just like the “superexchange”.40 The vibronic pathway related to B3u(1) of the BDT is also shown in Figure 6. The resulting pathway in the BDT seems somewhat of a loop current and is very different from the electron pathway. From the above analysis, the lowfrequency internal modes can be strongly linked to the modes of the contact and/or electrodes due to the long tail of the electron-phonon couplings, even if it is molecular and internal and localized on the molecule. This will be one reason why the IETS of low-frequency (nonsymmetric) modes is sensitive for theoreticalmodels.Furthermore,thelongtailoftheelectron-phonon couplings can trigger the energy dissipation between molecular vibrations and contact/electrodes modes, which are important for softening of the modes and heat dissipation within the fully self-consistent Born approximation.41,42 6. Summary We have presented the detection of ballistic and inelastic transport processes of molecular junctions by tuning “internal”

substitutions based on the first-principles NEGF calculations. First, due to the high symmetric structure, the BDT junction was adopted as a template, and the Raman preference in IETSactive modes was confirmed only for the symmetric modes of the BDT, which were also assigned by the recent experiments. Then, in order to eliminate the structure factors such as an adsorbed orientation in the analysis of the correlation between Raman and IETS activity, we estimated the relation of enhancement/reduction of Raman and IETS intensities for the internal substituted molecular junctions, n1-n3. The results did not find any additive or proportional enhancement of IETS to Raman. Furthermore, since the relations between IETS and IR activity are completely lost in all cases, the simple potential barrier and/ or on-site charge Coulomb potential models are insufficient. A description based on the molecular orbital is mandatory. To trace the change of IETS signals and the transport mechanism caused by the substitutions, both the moleculeelectrode and the electron-phonon couplings were estimated in the PMO basis, and the rigorous pathway analysis was carried out by using the local current vectors. By the substitutions, the inelastic transport mechanism can be changed from single-level (i.e., Breit-Wigner-type) off-resonant tunneling to multilevel (i.e., Fano-type) off-resonant tunneling, and this leads to the importance of nonlocal couplings in electron-phonon interactions on the molecule. In addition, the calculations of the inelastic transport relating to low-frequency nonsymmetrical modes are quite sensitive to the model due to its nonlocal tunneling nature. Acknowledgment. The author thanks Koichi Yamashita for fruitful discussion. The author would also like to thank Giacomo George for useful comments. This research was supported by a Grant-in-Aid for Scientific Research on Priority Area #20027002 and #2003801 from MEXT of Japan and by aGrant-in-Aid for Scientific Research (C) #20613002 from JSPS.

Figure 6. The inelastic local current vector (vibronic pathway) on each bridge molecule. The top two panels are for the BDT; the left is the inelastic current relating to Ag(5), while the right is that for B3u(1). The left and right panels in the bottom row are the inelastic current of Ag(5) in the n1 and n3, respectively.

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