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
Hückel 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 Hückel 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-Büttiker 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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J. Phys. Chem. C, Vol. 114, No. 8, 2010
Tsai et al.
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
Extended Metal-Atom Chain
J. Phys. Chem. C, Vol. 114, No. 8, 2010 3643
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.
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JP907893Q