J. Phys. Chem. C 2010, 114, 3641–3644
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Smallest Electrical Wire Based on Extended Metal-Atom Chains Te-Wei Tsai, Qian-Rui Huang, Shie-Ming Peng, and Bih-Yaw Jin* Department of Chemistry and Center for Theoretical Sciences, National Taiwan UniVersity, Taipei, Taiwan, Republic of China ReceiVed: August 14, 2009; ReVised Manuscript ReceiVed: January 3, 2010
An ideal electrical wire needs not only good conductivity for its central conductor but also a surrounding insulating layer to protect its current from leaking. We show that the extended metal-atom chain is a promising candidate to be the smallest molecular electrical wire for future practical applications. The electron can move through core metals, while the internal current is insulated from outside by the surrounding π-conjugated functional group. Moreover, we also show the existence of unavoidable hidden pathways at each site to the electrodes in a nanoscaled quantum circuit. Nevertheless, the Kirchhoff’s junction rule still holds when the current inflow and outflow arising from the additional terms of the self-energies of contacts are included. It is an important issue to find potential single molecular wires that can be functional units of nanotransistors for electronic apparatus application.1 The current understanding of the electron transfer through a single molecular wire is mainly built on organic molecules with and without π-electron conjugated systems.2-4 People have realized that in order to reach high conductivity the delocalized electrons in the bridge are needed to play the role of charge transfer. Pure organic molecules such as oligo(phenylene ethynylene) (OPE)5 and oligo(phenylene vinylene) (OPV)6 are often discussed for their long π-conjugated chains. However, in the fabrication of nanodevices, it is unavoidable to have multiple molecular wires packed in proximity. The wave function of one conductor can mix with the other through the overlap of electron clouds, which may result in the unwanted transversal hopping of charge carriers. In view of this, one-dimensional metal string complexes7 can be a promising candidate without the above problem. Extended metal-atom chain (EMAC) complexes consist of a central metal-containing backbone and four specifically designed polydentate ligands. The use of the poly(pyridylamine) ligand with a flexible 1D metal chain developed individually by Cotton and Peng has led to the isolation of metal chains with 3-9 metal atoms.7 Structurally, the EMAC is possibly the smallest version of an ordinary electrical wire that one can synthesize. In this article, we focus on the conductive properties of the trinuclear compound of the type, [M3(µ3-dpa)4(NCS)2] (M ) Cr, Co, Ni; dpa ) syn-syn-bis(R-pyridyl)amido) (see Figure 1). Experimentally, these mixed-valence stacks of organic as well as inorganic molecules exhibit unusual electrical properties.8-10 Bond orders for the symmetric and neutral complexes of nickel, cobalt, and chromium are 0, 0.5, and 1.5, which indicate the degree of the electron delocalization and thus the efficiency of the electron transfer through metal centers.8,11 To calculate the electron-transfer properties, we use the code Hu¨ckel IV 3.0,12 which is based on the nonequilibrium Green’s function (NEGF) formalism. The influence of the outside effect (contact) is incorporated into the main body of the device through self-energy matrices.13,14 The Hamiltonian of the system is constructed in the extended Hu¨ckel theory and obtained from the YAeHMOP,15 in which the orbital asymmetry parameter is * To whom correspondence should be addressed. E-mail: byjin@ ntu.edu.tw.
Figure 1. Model for the EMAC bridge connecting with Au electrodes. Central metals can be Cr, Co, and Ni. The organic functional group, dpa ) syn-syn-bis(R-pyridyl)amido), protects core metals from the outside. The internal current on the marked M-N bond (*) flows through ligands rather than the central bridge and will be discussed later.
used to take care of the counterintuitive orbital mixing.16 We assume that the molecule is connected to the surface (111) of gold atoms with the separation 1.905 Å.17 The whole system has been reduced to an “extended molecule” including the molecule itself and three connected gold atoms on each metal surface, to account for the molecular adsorption. We adopt the Landauer-Bu¨ttiker formalism13,14 which includes the effect of the phase-breaking arising from the interaction of electrons with the surrounding bath. The CNDO (complete neglect of differential overlap)18,19 method is used for the calculation of the self-consistent potential12 at nonzero bias. While most of studies focus on the total differential conductance and I-V characteristics of single molecules, we investigate the internal loop currents20,21 with an emphasis on the role of multiple pathways in EMACs. The internal current per unit energy between neighboring sites i and j is given by13,14
iij(E) )
4e 4e Im[ψ*F Im[FijGijn(E)] i ijψj] ) p h
(1)
where ψi is the wave function at the site i, F is the Fock matrix, the correlation function Gn is Gn ) G(Γ1 f1 + Γ2 f2)G†, and G is the retarded Green’s function, G ) (ES - F - Σ1 - Σ2)-1. Γ1(2) ) i(Σ1(2) - Σ†1(2)) represents the broadening of energy levels for the introduction of electrodes, in which the self-energy of the left (right) electrode Σ1(2) is numerically calculated in a recursive way.22 f1(2)(E) ) [1 + exp((E - µ1(2))/kBT)]-1 is the Fermi distribution function of the left (right) electrode. The chemical potential is µ1(2) ) Ef - 1/2eV at the applied bias, V,
10.1021/jp907893q 2010 American Chemical Society Published on Web 02/04/2010
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Figure 3. Local current distribution of [Cr3(µ3-dpa)4(NCS)2] at the bias 0.5 V which is the linear limit in experiments. The internal current is in the protection of organic ligands.
Figure 2. (a) Transmission functions of the three trinuclear EMAC molecules. The solid line refers to [Cr3(µ3-dpa)4(NCS)2], the dashed line refers to [Co3(µ3-dpa)4(NCS)2], and the dash-dotted line refers to [Ni3(µ3-dpa)4(NCS)2]. (b) The left is the molecular orbital of [Cr3(µ3dpa)4(NCS)2] with energy E ) -9.9 eV, and the right is the molecular orbital of [Co3(µ3-dpa)4(NCS)2] with energy E ) -10.18 eV.
where Ef is the Fermi level of the bulk Au and is chosen to be -9.5 eV12 to get the best fitting with experiments. The local current, Iij, is the energy integral of iij:
Iij )
∫-∞∞ iij(E) dE
(2)
To investigate the internal current further, we can expand the electron wave function of the system in terms of molecular orbitals, i.e., ψ ) ∑µaµφµ. The current from site i to j is given by iij ) 4e/p∑µ,νFµν ij |aµ||aν| sin(θν - θµ), where θµ is the phase of φµ. Thus, the coefficient aµ has a significant contribution only when the energy of the incoming electron is near the molecular orbital µ. The phase shift among molecular orbitals reflects the asymmetry of the effective coupling between the molecule and electrodes. The relative trend of conductivity, Cr > Co > Ni, observed by experiments,8 can be understood by examining the energies and spatial distributions of molecular orbitals responsible for relevant resonances in the transmission spectra as shown in Figure 2a. The trichromium and tricobalt complexes have higher transmission below the Fermi energy of Au electrodes due to the existence of two resonances at -9.9 and -10.2 eV in the transmission spectrum. In Figure 2b, the relevant molecular orbitals responsible for these two resonances mainly consist of the hybridization of d-orbitals among core metals. Unlike the previous two compounds, the delocalization on the central bridge in the trinickel complex is weak and thus leads to the low conductivity for this compound. Moreover, the spatial distribution of molecular orbital in the main channel also provides a qualitative picture of the characteristics of quantum loop currents at specified energy or bias voltage. Figure 3 shows local current distributions in the trichromium complex at the bias 0.5 V. One can see that the
electron crosses the molecular bridge mainly through the central core metals with small current contributions from the four peripheral organic ligands. That is to say, the π-conjugated functional groups in EMACs function as an insulating layer rather than a transport bridge, which is quite different from the pure organic conjugated molecules. Two other metal complexes also exhibit this trait, implying the advantage of EMACs as nanoscaled electrical wires. It is worthy to note that the local currents on S-C-N bonds as shown in Figure 3 appear to decay gradually even though there is only one bonding pathway for electrons to move. Naively, the disappearance of local currents seems to suggest that the Kirchhoff’s junction rule is violated in molecular electronic devices. But, here we would like to show that this is in fact related to the hidden outflow and inflow to or from different atoms in the molecular wire due to the long-range coupling to electrodes. To demonstrate this, we consider a twoterminal model system consisting of one orbital at each site of the molecule, H ) ε∑ic†i ci + ∑i,j(c†i cj + h.c.), where ε is the on-site energy, t is the overlap integral between neighboring sites i and j, and c†i (ci) is the creation (annihilation) operator. The retarded Green’s function expands as
G ) G0 + G0ΣG0 + G0ΣG0ΣG0 + · · ·
(3)
where Σ ) Σ1 + Σ2 and G0 ) (E - H)-1 is the zero-order Green’s function. The implicit self-energy is explicit now. In the wild band limit, the self-energy can be simplified to be Σmn ) iΛδmlδnl where l ) {l1, l2} is the molecular site directly coupled to electrodes and Λ is a real number. The internal current is
4e Im[HijGijn(E)] h 4e 2 ) Λ HijGl01l2(Gil0 1Gjl0 2 - Gil0 2Gjl0 1) × h {-1 + Λ2[(Gl01l1)2 + (Gl02l2)22(Gl01l2)2] + · · · }
iij )
(4) where we have set f1 ) 1 and f2 ) 0. Interestingly, the only term correlated with sites i and j is (Gil0 1Gjl0 2 - Gil0 2Gjl0 1) in the order of Λ2 and higher. It exhibits that the internal current is
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coupled to all molecular orbitals and cannot be distinguished into individual contribution as the transmission function.23 In this representation, it is easy to obtain iii ) 0 and iji ) -iij, which cannot be easily derived from iij(E) ) 4e/p Im[ψ*H ijψj] i in an open system. If we sum over all internal currents from sites j ) j1, j2, · · · , jn to decoupled site i, i.e. i * {l1, l2}, the value is unchanged upon the interchange of coupled sites, {l1, l2}. By doing this, we obtain
∑ iij ) ∑ (Gil0 Gjl0 j
1
j
)
∑ (Gil0 Gjl0 2
j
2
- Gil0 2Gjl0 1)( · · · )
1
- Gil0 1Gjl0 2)( · · · )
) 0
(5)
and the Kirchhoff’s junction rule is conserved even in quantum systems. However, if site i does couple to electrodes, e.g., i ) l1, the summation over j is
∑ iij ) ∑ (Gl0 l Gjl0 j
j
11
2
- Gl01l2Gjl0 1)( · · · ) * 0 in general
(6) We see the contribution from the long-range coupling to electrodes, or in other words, “hidden” inflow and outflow pathways. The Kirchhoff’s junction rule still holds with an additional term from contacts included. Thus, in a molecular circuit each junction (atom) in it should show the conservation of current because of the continuity of wave function across junctions. However, the sum of “values of internal currents” across a particular atom may not satisfy the conservation law of current as shown in Figure 3. Only atoms far away from contacts or weakly coupled to electrodes display the Kirchhoff’s junction rule as general electronic circuits without the correction from the self-energies of contacts. In Figure 4a, we examine the magnitude of current leaking from the central bridge of EMACs to the peripheral ligands by looking at the local current on the marked M-N bond in Figure 1 relative to the absolute value of the local maximum in the molecular circuit. It is noted that the local absolute maximum is on the central M-M bond in the trichromium and trinickel complexes, but on the side M-N-C bond in the tricobalt complex. Each subfigure in Figure 4a has four different ratios of leaking corresponding to four M-N bonds, and the positive value means that the local current flows from atom M to N. Nonequal values of four local currents in metal complexes are from the configuration asymmetry. The magnitude of current leaking for trimetal complexes is in the following order, Ni > Co > Cr, and remains small compared with the current through the central bridge. The local current leaking in the molecular wires composed of the trichromium and trinickel complexes is nearly symmetric at both sides of zero bias but is asymmetric in the tricobalt complex. For this complex, the local current leaking shows an abrupt change around bias 0.2 V because of a particular resonance coming from the molecular orbital at E ) -9.4 eV, as shown in Figure 4b. The asymmetric spatial distribution of this molecular orbital directly reflects the characteristics of local currents. It implies the strong correlation between the spatial distribution of the electron density and the current distribution in a molecular network. Besides, this also indicates the possibility that the distribution of local currents in a multipathway molecular circuit can be controlled by a gate
Figure 4. (a) Ratio of the marked local current to the local absolute maximum in the molecule. The four different markers correspond to the ratio for the four M-N bonds. The positive value means that the local current flows from central metal atom to nitrogen atoms on the ligands. (b) Molecular orbital of [Co3(µ3-dpa)4(NCS) 2] at E ) -9.4 eV.
potential through changing the electrostatic energy in the molecular bridge. The new circuit reflects the spatial density of state in the main channel and the phase shift with neighboring molecular orbitals. Overall, we have showed the advantage of EMACs to be molecular electrical wires. In a nanoscaled conductor, the molecular bridge is unavoidably under the long-range coupling from electrodes, and this results in the hidden pathway at each site. The Kirchhoff’s junction rule is conserved in the correction of the self-energies of contacts, which is significant in nanoscale systems. The local currents of EMACs among core metals remain maximum and leak to surrounding organic ligands little. The π-conjugated functional groups shield the internal current from the outside rather than transfer charges in a multipathway system. This assures the excellent insulating property in EMACs and thus the benefit for the fabrication of nanodevices. Acknowledgment. We acknowledge the financial support of NSC, Taiwan, ROC. B.-Y.J. acknowledges the support from the Center of Quantum Science and Engineering, NTU, Taiwan. References and Notes (1) Joachim, C.; Gimzewski, J. K.; Aviram, A. Electronics using hybridmolecular and mono-molecular devices. Nature 2000, 408, 541–548. (2) Kushmerick, J. G.; Holt, D. B.; Pollack, S. K.; Ratner, M. A.; Yang, J. C.; Schull, T. L.; Naciri, J.; Moore, M. H.; Shashidhar, R. Effect of bondlength alternation in molecular wires. J. Am. Chem. Soc. 2002, 124, 10654– 10655. (3) Choi, S. H.; Kim, B.; Frisbie, C. D. Electrical resistance of long conjugated molecular wires. Science 2008, 320, 1482–1486. (4) Bumm, L. A.; Arnold, J. J.; Cygan, M. T.; Dunbar, T. D.; Burgin, T. P.; Jones, L., II; Allara, D. L.; Tour, J. M.; Weiss, P. S. Are single molecular wires conducting? Science 1996, 271, 1705–1707. (5) Sachs, S. B.; Dudek, S. P.; Hsung, R. P.; Sita, L. R.; Smalley, J. F.; Newton, M. D.; Feldberg, S. W.; Chidsey, C. E. D. Rates of interfacial
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