11-15-14
Supplementary Material
for
Quantum Interference in Polyenes
Yuta Tsuji,1 Roald Hoffmann,1,* Ramis Movassagh,2 and Supriyo Datta3
1
Department of Chemistry and Chemical Biology, Cornell University, Baker Laboratory,
Ithaca NY 14853
2
Department of Mathematics, Northeastern University, Boston, MA 02115 and Department of
Mathematics, Massachusetts Institute of Technology, Building E18, 77 Massachusetts Avenue
Cambridge, MA 02139-4307
3
Purdue University, School of Electrical and Computer Engineering, Electrical Engineering
Building, 465 Northwestern Ave., West Lafayette, Indiana 47907-2035
*To whom correspondence should be addressed.
E-mail: rh34@cornell.edu
Table of Contents
S1. Computational methodology for conductance through a two-probe system
page S2
S2. Zeroth order Green’s function
S5
S3. Computational methodology for conductance through a multi-probe system
S7
S4. Computational methodology for current computations
S10
S5. Effects of bond alternation
S11
S6. Effects of non-nearest neighbor interactions
S15
S7. Including the overlap in the calculations
S19
S8. ON/OFF ratios
S20
S9. Non-nearest-neighbor effects on multiprobe attachment
S22
S10. Comparison between transmissions calculated in this work and the ones computed
by Kiguchi et al.
S24
S11. Supplementary References
S26
S1
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S1. Computational methodology for conductance through a two-probe system
According to the Landauer model, the conductance of a molecular junction
composed of a molecule and two gold electrodes, in the limit of zero temperature and zero
bias voltage, is expressed as1,2
g
2e 2
T E F  ,
h
(S1)
where 2e2/h is the quantum of conductance, T is the transmission probability, and EF is the

Fermi
energy. To calculate the transmission probability, we employ the nonequilibrium
Green’s function method combined with the Hückel molecular orbital method (NEGF-HMO).
The transmission probability can be calculated from eq S2,2
T(E) = Trace[ΓLGΓRG†],
(S2)
where G is the Green’s function matrix and ΓL(R) is the broadening function matrix for the left
(right) electrode. The Green’s function is written as2
G(E) = [(E + iη)I – H – ΣL – ΣR]-1,
(S3)
where η is an infinitesimal positive number, I is the unit matrix, H is the Hamiltonian matrix
for the molecule, and ΣL(R) is the self-energy matrix describing the connection to the
macroscopic contact on the left (right) side. In the NEGF-HMO method, the Hückel
Hamiltonian matrix is employed for H. The relation between the broadening function matrix
and the self-energy matrix is written as2


Γ LR   i Σ LR   Σ †LR  .
(S4)
The self-energy matrix elements for the left and right electrodes may be expressed as3
Σ L mn   m g L   †n
(S5)
Σ R mn   m  g R    † n ,
(S6)
 ,  L
and
 ,  R
S2
11-15-14
respectively. τmα is the hopping integral between the site m in the molecule and site α in the
electrode. Here the Latin alphabet label is used for the molecule site, while the Greek alphabet
one labels the electrode site. gL(R) is the Green’s function matrix for the left (right) electrode.
In the case that only the terminal atoms of the left and right electrodes, i.e. site α of the left
electrode and site α’ of the right electrode, are attached to the sites r and s of the molecule,
respectively (see Figure S1), the nonzero elements of the self-energy matrix are
ΣL rr   r g L   †r
(S7)
ΣR ss   s  g R    † s ,
(S8)
and
because the hopping integrals between the electrode and the molecule are 0 except for τrα and
τsα’. τrα and τsα’ were set to be 0.2β,4 where β is the resonance integral between adjacent
carbon 2pπ orbitals. A standard value of the resonance integral β is -2.4 eV.5 As the electrodes
we used an ideal one-dimensional (1D) linear gold chain, and the Green’s function of the left
(right) electrode can be obtained analytically as follows:2,6
g L   g R    
eika
 elec
,
(S9)
where ka satisfies the condition E = εelec + 2τelec coska. εelec and τelec are the site energy and the
nearest-neighbor hopping integral of the electrode, respectively. This equation is valid for the
ideal 1D electrode. The site energy of the electrode was assumed to be the same as the Fermi
energy. τelec was set to be 0.6β.4
S3
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Figure S1. Schematic description of the two-probe molecular junction modeled in this study.
The blue and black circles indicate atoms of the molecule and electrode, respectively. The
solid lines indicate chemical bonds and the dotted lines indicate interaction between the
molecule and electrode.
S4
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S2. Zeroth order Green’s function
All the transmission spectra in this paper have been calculated by using eq S2,
which includes the nonequilibrium Green’s function, but the qualitative discussion is based on
eq 1, i.e. zeroth order Green’s function. Here is how we would connect the two.
In this paper we specifically consider molecules weakly coupled to the contacts
such that the conduction properties can be understood from the approximate Green’s function
at the Fermi energy:
G(EF) ≈ G(0)(EF) = [(EF + iη)I – H]-1.
(S10)
Furthermore we restrict our attention to the half-filled π-bands of linear chains and cyclic
molecules with an even number of atoms, such that the Fermi energy does not coincide with
any of the eigenenergies of H, allowing us to neglect the infinitesimal η. Using the Fermi
energy EF as our energy reference by setting EF = 0, we have what we call the zeroth order
Green’s function:
G(r, s) ≡ [G(0)(EF = 0)]rs ≈ [– H-1]rs,
(S11)
whose properties (which we discuss in this paper) should be reflected in the measured
conductance.
Note, however, that the measured conductance also depends on the broadening
function matrix ΓL(R) describing the connection of the molecule to the contacts. In this paper
we start with the simplest possibility, namely one in which two specific atoms denoted by r
and s in the molecular chain are connected through electrodes to the macroscopic terminals as
shown in Figure S1. In this case the matrices ΓL(R) each have only one nonzero element:
[ΓL]rr = γL,
(S12)
and
[ΓR]ss = γR,
(S13)
so that from eq S2 and eq S11 we can write
T(EF = 0) = γLγR|G(r, s)|2,
(S14)
S5
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making the transmission and hence the conductance, directly proportional to the zeroth order
Green’s function. The transmission also depends on γL and γR, but at the Fermi energy γL and
γR take the constant value of 0.13β (≈ -0.31 eV). Note, therefore that the conductance depends
only on the magnitude of the Green’s function.
S6
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S3. Computational methodology for conductance through a multi-probe system
We will show that a slightly different contacting scheme, the multiprobe geometry
shown in Figure S2, should allow us to probe the sign of the Green’s function as well. Let us
consider the case that the left electrode is attached to site r of the molecule and the right
electrode is attached to sites s and s’. There are two possible models for multiprobe
attachment; these are shown schematically in Figure S2a and S2b.
Figure S2. Two possible models for multiprobe attachment. (a) The model we adopted for the
detailed calculations in this study. In this model the right electrode is a 1D chain of Au atoms,
whose end atom interacts with two sites of the molecule at the end of the electrode. (b) The
right electrode is bifurcated and has two ends, each of which interacts with the molecule. The
blue and black circles indicate atoms of the molecule and electrode, respectively. The solid
lines indicate chemical bonds and the dotted lines indicate interaction between the molecule
and electrode.
Geometrically, the model shown in Figure S2a would be difficult to arrange, but
computationally it is easy to model. On the other hand, the microscopic geometrical situation
of Figure S2b, an electrode that splits into two strands, seems to be more realistic than Figure
S2a. However, it is difficult to calculate the situation shown in Figure S2b because it is
unclear how many electrode atoms should be there between the ends and the “branch point”
(perhaps this is something that can be studied, both experimentally and theoretically) and we
S7
11-15-14
find it difficult to obtain the Green’s function for the electrode. Therefore we adopted the
model shown in Figure S2a.
To calculate the conductance through the three-probe system shown in Figure S2a,
all we have to do is to modify the self-energy matrix for the right electrode. The nonzero
element of the self-energy matrix for the right electrode, i.e. eq S8, can be modified as
follows:
 Σ R ss

 Σ R s s
ΣR ss    s  g R    † s

ΣR ss   s  g R    † s
 s  g R    † s  
.
 s   g R    † s  
(S15)
Since these four nonzero elements take the same values in our model, the corresponding
broadening function matrix ΓR having four nonzero elements reflecting the coherent
connection to two atoms denoted s and s’, can be written as follows:
 Γ R ss

 Γ R ss
ΓR ss    R

ΓR ss    R
R 
.
 R 
(S16)
From eq S2 and eq S11 we now obtain
T(EF = 0) = γLγR|G(r, s) + G(r, s’)|2.
(S17)
Note that to observe this interference effect it is important to extract/insert the electron
coherently from/into the two atoms s and s’. If the atoms s and s’ are separately connected to
the external terminal and there is little phase coherence between the injected/ejected electrons
on the atoms s and s’, the appropriate broadening function matrix will look like
 Γ R ss

 Γ R ss
ΓR ss    R

ΓR ss   0
0
,
 R 
(S18)
so that eq S2 can give a transmission that is insensitive to the sign of the Green’s functions:
T(EF = 0) = γLγR(|G(r, s)|2 + |G(r, s’)|2).
(S19)
In short, the properties of the zeroth order Green’s function may be reflected in the
transmission and hence the conductance in different ways, by eqs S14, S17, and S19
depending on whether the connection is described by eq S13, S16, or S18. The conductance
calculation for a system with further multiprobe attachment can be achieved by modifying the
S8
11-15-14
self-energy matrix in a similar way. A description of the actual physical electrode connection
corresponding to eq S13, S16, or S18 could be very interesting and instructive; we propose its
study as a challenge to our experimentalist friends.
S9
11-15-14
S4. Computational methodology for current calculations
Using the nonequilibrium Green’s-function formalism, the current I through the
molecular junction can be described as a function of the applied bias V:1,2
I V  
2e 
nE T E , V  dE ,
h  
(S20)
where n(E) is the distribution function. n(E) can be written as
n ( E ) = f ( E - mL ) - f ( E - mR ) ,
(S21)
where f is the Fermi function, and μL and μR are the electrochemical potentials of the left and
right electrodes, respectively. From the expression for n(E) we can expect that only electrons
with energies within a small range around the Fermi energy contribute to the total current. We
can rewrite eq S20 in the form7
I V  
2e  R
T E , V  dE .
h  L
(S22)
Here, we can assume that the transmission function under low bias should be nearly the same
as that under zero bias.8 Hence, the current under low bias can be obtained simply by
integrating the zero-bias transmission function from μL to μR. The electrochemical potentials
in the left and right electrodes can be written as7
μL = EF – eV/2
(S23)
μR = EF + eV/2,
(S24)
and
respectively.
S10
11-15-14
S5. Effects of bond alternation
Our transmission calculations are for a polyene model with one β, effectively a
polyene with all equal bond lengths. In any real polyene, one has bond alternation, and it
persists as one approaches the infinite chain of polyacetylene (Peierls distortion9). We have
investigated the effect of the expected bond alternation in butadiene.
First we optimized with density functional theory (DFT) calculations the structure
of butadiene, in both s-trans and s-cis conformations. The DFT calculations have been carried
out with the B3LYP 10 functional and 6-311++G(d,p) basis set, as implemented in the
Gaussian 09 software.11 To properly account for dispersion interactions, Grimme’s D312
dispersion correction with the Becke and Johnson damping function13 has been added. The
optimized geometries are shown in Figure S3; the bond lengths match the experimental
structure of the molecule.14
Figure S3. Optimized geometries of (a) s-trans and (b) s-cis butadienes at the
B3LYP-D3BJ/6-311++G(d,p) level of theory.
How does the resonance integral, the coupling element, in fact depend on the
distance? This integral between atoms separated by an internuclear distance R may be
approximated by Mulliken’s formula15 as follows:
S11
11-15-14
 R  
S R 
,
S 1.40 
(S25)
where S is the overlap integral and β is the resonance integral between adjacent 2pπ orbitals in
a benzene ring, in which the CC bond length is 1.40 Å. This is also roughly the equilibrium
CC distance in a long-chain polyacetylene (polyene) with bond lengths equalized along the
chain.
Mulliken et al.16 provided explicit formulas for the overlap integral S between
atomic orbitals (AOs) of two overlapping atoms a and b. These formulas are based on
approximate AOs of the Slater type, each containing a parameter μ, which equals to Z/n*,
where Z and n* are Slater’s effective nuclear charge and effective principal quantum number
respectively for the corresponding AO. The S formulas are given as functions of two
parameters p and t defined as follows:
p
1
 a   b R ,
2a H
(S26)
and
t
 a  b
,
 a  b
(S27)
where aH (= 0.529 Å) is the Bohr radius. In the case of carbon-carbon 2pπ-bonds, where μ =
1.625, p = 3.072R, and t = 0, the overlap integral is given by
2
1 3

S  e  p 1  p  p 2 
p .
5
15 

(S28)
Using eqs S25 and S28, we can construct the “extended Hückel” Hamiltonian
matrices for s-trans and s-cis butadienes (in units of β) as follows:
H trans

0
1.11
1.26  101 7.77  103 


1.11
0
9.11  101 1.26  101 


,
1.26  101 9.11  101
0
1.11 


3
1.26  101
1.11
0
7.77  10

and
S12
(S29)
11-15-14
H cis

0
1.11
1.20  101 3.25  102 


1.11
0
8.91  101 1.20  101 


.
1.20  101 8.91  101
0
1.11 


2
1.20  101
1.11
0
3.25  10

(S30)
In this section we want to examine first the effect of bond alternation; the next
section will take up the consequence of including non-nearest-neighbor interactions. Bond
alternation was explored with the simplified Hamiltonian of eq. S31:
0
 0 1.1 0
1.1 0 0.9 0 
.
H
 0 0.9 0 1.1


0 1.1 0 
0
(S31)
Figure S4 shows the comparison between the butadienes with and without the bond
alternation. The 1-4 connection of electrodes is the one that does not show quantum
interference (QI), while the 2-3 connection shows QI. As expected from the consequence of
bond alternation, the gap between the peaks due to the HOMO and LUMO is opened, but the
bond alternation does not significantly affect the general feature of the transmission in our
model.
As the discussion in the main text of the paper indicates, the QI antiresonances
remain if bond alternation is introduced.
S13
11-15-14
Figure S4. Computed transmission spectra for (a) 1-4 and (b) 2-3 connections in butadiene.
Blue lines indicate the transmission spectra calculated with the same resonance integrals for
all bonds while red lines indicate the transmission spectra calculated with different resonance
integrals for different bonds.
S14
11-15-14
S6. Effects of non-nearest neighbor interactions
To estimate the effect of non-nearest neighbor interactions on transmission
described by the “hard” zeroes of the matrix representation of the zeroth Green’s function, we
carry out “extended Hückel” calculations of the transmission. The relevant Hamiltonians,
deduced from the optimized structures by using the Mulliken approximation, are given above
in eqs. S29 and S30. Note that there are no significant difference in the elements of the
Hamiltonian matrix between the s-trans and s-cis butadienes except for the (1, 4) and (4, 1)
elements. The (1, 4) and (4, 1) elements for s-cis butadiene are 1 order of magnitude larger
than those for s-trans butadiene.
The Hamiltonians of eqs. S29 and S30 include both the effects of bond alternation
and of non-nearest-neighbor interaction. We wanted to take apart the effect of the two
perturbations, so we began by keeping all the bond lengths along the polyene chain equal, and
just added the nonnearest-neighbor interactions. This leads to an “extended Hückel”
Hamiltonian matrix for the non-bond-alternated butadiene that can be parameterized as
follows:
 0 1  '  ' '
 1 0 1 '
,
H
' 1 0 1 


 ' '  ' 1 0 
(S32)
where β’ and β’’ are the second- and third-nearest-neighbor-coupling parameters, respectively.
Let us first investigate the effect of the second-nearest neighbor coupling, setting β’’ is to be
0.
Figure S5 shows the transmission spectra for the 2-3 connection in butadiene, where β’
is small. As β’ increases, the splitting of the antiresonance peak becomes larger. For the
smallest β’, we cannot observe the splitting on this scale. As β’ increases, the transmission
probability at the Fermi level becomes larger. Interestingly, the position of one of the split
antiresonance peaks remains at the Fermi level.
S15
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Figure S5. Computed transmission spectra for the 2-3 connection in butadiene, where β’ is
very small and β’’ is set to be 0.
Figure S6 shows the transmission spectra for the 2-3 connection in butadiene, if β’ is
larger than those in Figure S5 and β’’ is still set to be 0. As β’ increases, the splitting of the
antiresonance peak gets larger and larger; one of the split peaks remains at the Fermi level. As
β’ increases, the energy centroid of the HOMO- and LUMO-resonance peaks shift to higher
energy.
S16
11-15-14
Figure S6. Computed transmission spectra for the 2-3 connection in butadiene, where β’ is
larger than those in Figure S5 and β’’ is still set to be 0.
Let us next investigate the effect of the third-nearest neighbor coupling. Here the
second-nearest-neighbor interaction β’ is set to be 0.1β, which is a realistic value, consistent
with the distances shown in Fig. S3. Figure S7 shows the computed transmission spectra for
the 2-3 connection in butadiene as β’’ is then varied. As β’’ increases, the splitting becomes
smaller and eventually returns to the single antiresonance peak when β’’ = 0.01β. However,
the peak position is not located at the Fermi level but at the mid-point of the formerly split
peaks. The variation of β’’ in this range does not affect the position of the HOMO- and
LUMO-resonance peaks. In the case of s-trans butadiene, β’’ is close to 0.005β, so the
splitting can be observed.
Figure S7. Computed transmission spectra for the 2-3 connection in butadiene, where β’’ is
very small and β’ is set to be 0.1β.
Figure S8 shows the transmission spectra for the 2-3 connection in butadiene for
values of β’’ larger than those in Figure S7 (β’ is still set to be 0.1β). As β’’ increases, the
S17
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antiresonance peak becomes “blunter” and shallower. Since β’’ is close to 0.03β for the case
of s-cis butadiene, neither the splitting nor the single sharp antiresonance peak can be
observed.
Figure S8. Computed transmission spectra for the 2-3 connection in butadiene, where β’’ is
larger than Figure S7 and β’ is set to be 0.1β.
S18
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S7. Including the overlap in the calculations
In the above calculations we employed the modified Hamiltonian matrix H’ instead
of the simple Hückel Hamiltonian, so as to examine the effect of more distant through-space
overlapping. The transmission spectra have been calculated by using the Green’s function of
eq. S3. However, eq. S3 is actually valid for an orthogonal basis set of atomic wave functions.
For a real, nonorthogonal basis, the overlap matrix has to be taken into account; we should
replace the unit matrix I in eq. S3 with the overlap matrix S as follows:17
G '  E  i S  H' Σ L  Σ R  .
1
(S33)
Here we can construct the overlap matrix S based on eq. S28.
Figure S9 shows the transmission spectra for the 2-3 connection in s-trans butadiene.
The spectra calculated with eqs S3 and S33 are indicated by red and blue, respectively.
Inclusion of the overlap characteristically destabilizes (relative to the Fermi level) the LUMO
compared with the HOMO. However, both spectra show nearly the same splitting pattern in
the vicinity of the Fermi level; hence the conductance and low-bias current are not
significantly affected by including the overlap matrix.
Figure S9. Computed transmission spectra for the 2-3 connection in s-trans butadiene with
(indicated by blue) and without (indicated by red) the overlap matrix.
S19
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S8. ON/OFF ratios
Figure S10 shows the computed ON/OFF ratios of the CB-BD and CH-HT systems
as a function of bias voltage. Here the simple HMO method is used and the ON/OFF ratio is
defined as ION(V)/IOFF(V), where ION(V) and IOFF(V) are bias-dependent currents through ON
and OFF states, respectively. We can see that both CB-BD and CH-HT systems show
extremely high ON/OFF ratio in the low bias region.
Figure S10. The ON/OFF ratio of the (a) CB-BD and (b) CH-HT systems as a function of
bias voltage. The red and blue curves at right (1-4 CH, 2-3 CH) are superimposed.
Figure S11 shows the ON/OFF ratios of the CB-BD and CH-HT systems calculated
from the transmission spectra, where the effects of both bond alternation and
non-nearest-neighbor interactions are taken into account. All switches show smaller ON/OFF
ratios than those in Figure S10. The seeming constancy of the ON/OFF ratios for some of the
curves would be modified if a larger bias voltage were examined.
S20
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Figure S11. The ON/OFF ratio as a function of bias voltage for the (a) CB-BD and (b)
CH-HT systems calculated from the transmission spectra, where the effects of both bond
alternation and non-nearest-neighbor interactions are taken into account.
S21
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S9. Non-nearest-neighbor effects on multiprobe attachment
We have investigated the effect of non-nearest-neighbor interactions in the multiprobe
attachment scenario. Figure S12a shows the calculated transmission spectra for the 1 to 4 and
8 connection and 1 to 4 and 6 connection in all-trans-1,3,5,7-octatetraene, which is the
topologically same as the case that is shown in Scheme 6.
Figure S12. (a) Computed transmission spectra for the three-probe molecular junctions
composed of all-trans-1,3,5,7-octatetraene, where both the non-nearest neighbor interaction
and the effect of bond alternation are included. (b) Computed transmission spectra for the
three-probe
molecular
junctions
of
all-trans-1,3,5,7-octatetraene,
where
the
non-nearest-neighbor interactions are included but all nearest-neighbor elements in the
Hamiltonian are fixed to 1 to eliminate the effect of bond alternation. The left electrode is
connected to the site 1 and the right electrode is split between the sites 4 and 6 (indicated by
the dotted line), and 4 and 8 (indicated by the solid line).
Since the effect of the bond alternation is also included in this calculation, the shape of
the transmission spectra is similar to that of octatetraene shown in Figure 5a, where the
non-nearest neighbor interaction is not included but the effect of the bond alternation is. To
S22
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separate the effect of the bond alternation and that of the non-nearest-neighbor interaction, we
have recalculated the transmission spectra fixing all nearest-neighbor elements in the
Hamiltonian to 1 (in units of β); the transmission spectra are shown in Figure S12b. Although
there is a negligible splitting of the antiresonance in the 1 to 4 and 6 connection, the shape of
the spectra is similar to that of octatetraene shown in Figure 4a, where neither the non-nearest
neighbor interaction nor the effect of the bond alternation is included. Hence, the
non-nearest-neighbor interaction does not significantly change the shape of the transmission
spectra for the multiprobe attachment, but just shifts the peaks to higher energy.
S23
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S10. Comparison between transmissions calculated in this work and the ones computed
by Kiguchi et al.
Kiguchi et al.18 calculated transmission spectra of 4-TEB and 2-TEB by using the
NEGF
method
combined
with
the
Perdew-Burke-Ernzerhof
generalized
gradient
approximation (GGA-PBE). In their calculation, the whole system comprised of the left and
right gold electrodes, which were modeled by Au(111) surfaces, and the 4-TEB / 2-TEB
molecule, where the butyl groups were replaced with methyl groups, was taken into accounts.
Figure S13 shows the transmission spectra of Figure 7 on the scale of eV for ease of
the comparison. While in the transmission spectra of Kiguchi et al.18 the peaks due to the
HOMO resonance are located around -0.5 eV, in our transmission spectra the peaks due to the
HOMO resonance are located around -2.5 eV. The difference arises from the lining or
anchoring groups. In our case only the benzene backbone was taken into account, within the
framework of the HMO, so the HOMO resonance peaks coincide with the HOMO energy of
benzene itself, i.e. the value of β. On the other hand, in the case of Kiguchi et al.
thienylethynyl linkers, which make the π electrons more delocalized and the HOMO-LUMO
gap smaller, are included; hence the HOMO energy is located closer to the Fermi level than
benzene.
Figure S13. Computed transmission spectra for the coherent four-probe (indicated by blue
line), incoherent four-probe (indicated by red line), and two-probe (indicated by black line)
molecular junctions composed of cyclic 6 carbon atoms on the energy scale of eV.
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Figure S14 shows the molecules and transmissions calculated by Kiguchi et al. As
mentioned above, a direct comparison is difficult. Both calculations show greater
transmission by a factor of 4-5 at the Fermi level for the multiple junction arrangement.
Figure S14. Molecules and calculated transmissions reproduced from Kiguchi et al.18 paper,
and Supporting Information for that paper.
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11-15-14
S11. Supplementary References
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