Ab initio analysis of electron currents in thioalkanes

Abstract

A combined density functional theory and Green's function procedure is used to calculate the electrical characteristics of a group of alkanethiols representing possible experimental settings. It is found that the current running through the molecule is the sum of the contributions from all molecular orbitals each presenting a barrier to electron transport equal to their energy difference from the Fermi level of the contacts. For the alkanethiols the location of these intrinsic barriers are at 0.78 eV (highest occupied molecular orbital [HOMO] and HOMO-1), 1.99 eV (HOMO-2 and HOMO-3), 2.07 eV (LUMO and LUMO+1), and 2.67 eV (HOMO-4). However, barriers obtained through fittings to known models do not bear any physical meaning at the molecular level, as they are sort of exponential average of the intrinsic barriers. We have also found that the exponential dependence of the current with the length of the alkane is practically independent of the contact nature, perhaps due to the large resistance of the alkanes. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

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Ab Initio Analysis of Electron Currents
in Thioalkanes
JORGE M. SEMINARIO, LIUMING YAN
Chemical Engineering Department and Electrical Engineering Department, Texas A & M
University, College Station, Texas 77843, USA
Received 1 August 2004; accepted 9 August 2004
Published online 23 November 2004 in Wiley InterScience (www.interscience.wiley.com).
DOI 10.1002/qua.20384
ABSTRACT: A combined density functional theory and Green’s function procedure is
used to calculate the electrical characteristics of a group of alkanethiols representing
possible experimental settings. It is found that the current running through the molecule is
the sum of the contributions from all molecular orbitals each presenting a barrier to electron
transport equal to their energy difference from the Fermi level of the contacts. For the
alkanethiols the location of these intrinsic barriers are at 0.78 eV (highest occupied
molecular orbital [HOMO] and HOMO-1), 1.99 eV (HOMO-2 and HOMO-3), 2.07 eV
(LUMO and LUMO?1), and 2.67 eV (HOMO-4). However, barriers obtained through
fittings to known models do not bear any physical meaning at the molecular level, as they
are sort of exponential average of the intrinsic barriers. We have also found that the
exponential dependence of the current with the length of the alkane is practically
independent of the contact nature, perhaps due to the large resistance of the alkanes.
© 2004 Wiley Periodicals, Inc. Int J Quantum Chem 102: 711–723, 2005
Key words: alkanes; DFT; electron currents; Green’s function; molecular electronics;
electron transport
Introduction
W
Pople. A direct product of his extraordinary work is
e are honored to participate in this special
issue of the journal in memory of John A.
now used for the analysis, design, and simulation
of molecular electronic devices. His methods have
reached the extraordinary degree of precision that
applications to real and interesting systems are
helping the development of the field of molecular
electronics, which is of paramount importance to
perform quantum chemistry calculations to obtain
electrical characteristics needed for the design of
molecular electronic circuits. This work shows how
important Pople’s methods are in complementing
and in most cases clarifying results obtained exper-
imentally. As electronic systems become smaller,
Pople’s methods yield more precise results as ex-
perimental techniques become more difficult to the
Correspondence to: J. M. Seminario; e-mail: seminario@
tamu.edu
Contract grant sponsor: DARPA/ONR.
Contract grant number: N00014-01-1-0657.
Contract grant sponsor: ARO.
Contract grant numbers: DAAD19-00-1-0592; DAAD19-00-1-
0634.
Contract grant sponsor: NRL.
Contract grant number: N00173-01-1-G017.
International Journal of Quantum Chemistry, Vol 102, 711–723 (2005)
© 2004 Wiley Periodicals, Inc.

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point that obtaining electrical characteristics for sin-
gle molecules is almost impossible.
Extensive theoretical and experimental effort has
been invested in the understanding of electron
transfer mechanisms in single organic molecules
due to their potential applications in nano-scale
electronic systems [1–15]. One of the most studied
types of molecules is alkanethiol [8, 12, 16–20],
because they can easily be chemisorbed (covalently
bonded) to a gold surface through the S–Au bond in
liquid phase, making a robust and mechanically
stable attachment of a self-assembled molecular
monolayer (SAM) [21–24]. This process allows ac-
cessing electrically one of the ends of the molecules;
the other end is much more difficult and several
alternatives have been performed. For a review on
these techniques, the reader is referred to the exten-
sive literature (e.g., [16, 21] and references therein).
For example, the other end of the molecule can be
addressed by a scanning tunneling microscope
(STM) [18, 25], or AFM tips [19, 20, 26–28]. These
techniques may address one or several of the self-
assembled molecules and the nature of this second
contact may be very different than the former one.
An alternative is to evaporate a gold layer on a
small region of the SAM, for instance, in the nano-
pore experiments [17]. Another alternative is to
have two SAMs at atomistic proximity with the
hope that one or very few molecules get attached
between the two regions (for instance, the break
junction experiment and the approaching wires [29]
are variants of this alternative). As can be imagined,
the nature of this second contact is very random in
nature and there is no consensus on whether this
contact may be covalently bonded (chemisorbed) or
nonbonded (physisorbed) in several experiments.
Thus, there is a lack, so far, of a well-characterized
experimental setting that can be used for the vali-
dation of theoretical procedures regarding to the
electrical characteristics of single molecules. Several
attempts have been performed in order to get the
two terminals of the molecule chemisorbed to test
electrodes (e.g., [20, 30]). Although alkanes have no
major interest for the development of molecular
devices due to their high resistance and their lack of
switching behavior at moderate bias potentials,
they are still attractive due to the straightforward
conditions to prepare good SAMs and therefore to
perform electron transport experiments. For other
molecules, like the oligo-phenylene-ethynylenes
(OPE) and others, the situation is not as straightfor-
ward as for the alkanes [31].
Therefore, our goal in the current work is to
understand better the reasons for the disagree-
ments and to establish the rules governing these
experiments from the basic but exact laws of nature.
There are certainly plenty of reasons for the dis-
agreements from measurements in several experi-
ments [19, 27, 32]; for example, the electronic decay
constant ?Nranges from 0.57 to 1.2 [12, 17, 19, 26,
32–34] perhaps, as explained much earlier by Smith
[35] because the atomic force microscopic (AFM) or
STM tips were contaminated by the adsorbed mol-
ecules in the experiments. In the experiments with
two chemisorbed contacts [20, 26, 32], this problem
is circumvented because both terminals are chemi-
cally bonded to the molecule; therefore, reproduc-
ible results are obtained. The electronic decay con-
stant ?Nis defined empirically from
R ? R0exp??Nn?,(1)
where n is the number of methylene groups in the
alkane, R is the resistance of the molecule, and R0is
the resistance extrapolated to a molecule with no
methylene groups or the resistance contributed by
the contacts. It was found that ?Nis independent of
the contact force exerted on the AFM tip and with a
value of 0.57 per methylene group [26, 32]. How-
ever, in the measurement with physisorbed contact,
the electronic decay constant is still dependent on
the contact force and yields different values. For
example, Lindsay reported values from 0.6 to 1.2
per methylene group by AFM measurements in
toluene [26], Cui et al. [32] reported a value of 0.8
per methylene group by AFM measurements in
toluene, Wold et al. [19] reported a value of 0.75 by
AFM measurements in vacuum, Holmlin et al. [12]
and Rampi et al. [33] reported a value of 0.70 by
mercury-drop experiments, and Wang et al. [17]
reported a value from 0.58 to 0.83 by nanopore
experiments.
Phenomenological electron transport mecha-
nisms have been developed for single-band insula-
tors [36] following the extensive experience in solid-
state physics experiments. Unfortunately, much of
the current work for the conduction on molecules
tries to follow such empirical approach ignoring to
some extent the availability of quantum chemistry
procedures that can be easily adapted to study the
electron conduction through molecules using ab
initio procedures where no a priori empirical input
is needed. Direct tunneling, Fowler–Nordheim tun-
neling, thermionic emission, and hopping conduc-
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tion are the most common mechanism in semicon-
ductors [36]. It is of interest to analyze the atomistic
features associated with these typical mechanisms.
Direct tunneling is a mechanism in Metal Oxide
Semiconductor (MOS) structures with ultrathin ox-
ide barrier (?4 nm), where electrons from the con-
duction band of the semiconductor are transferred
across the barrier into the conduction band of metal
without changing energy; the transmission proba-
bility of direct tunneling is a very strong function of
the width of the barrier. Fowler–Nordheim tunnel-
ing is the mechanism of conduction in oxides when
a high electric field is applied, electrons tunnel from
the semiconductor conduction band into the oxide
conduction band through part of the potential bar-
rier at the semiconductor–oxide interface (most
likely to dominate current with oxide 5–10 nm
thick). Thermionic emission is transport of electrons
excited by heat over a barrier of oxide insulator.
Hopping conduction is the mechanism in which
charge carriers “hop” from defect to defect in a
nonconductor. As can be noticed, these mecha-
nisms are not orthogonal to each other, their defi-
nitions are not unambiguous, and most of the times
their assignment is confusing when they are trans-
lated into conduction in single molecules. The
study of electron conduction using ab initio proce-
dures for the calculation of currents should be able
to perform under any of the above empirical mech-
anisms; and indicators may or may not be available
to correlate the results with those from phenome-
nological methods. There is experimental evidence
that temperature does not affect the I-V character-
istics of metal–alkane–metal junctions; therefore,
this excludes the thermionic emission and the hop-
ping conduction mechanisms because both trans-
port mechanisms depend on the reciprocal of tem-
perature exponentially [17]. Thus, considering the
experimental feedback, direct tunneling or Fowler–
Nordheim tunneling is supposed to be the trans-
port mechanism in metal–alkane–metal junctions.
Both direct tunneling and Fowler–Nordheim tun-
neling yield an exponential decay of probabilities as
the barrier width increases; thus, the resistance R
increases exponentially [36]. For metal–alkane–
metal systems, it has been proved that the resis-
tance increases exponentially with the length of the
alkane molecules [8, 17, 27, 32]. It was found that
the nanopore [17] setting obeys the direct tunneling
mechanism instead of the Fowler–Nordheim. A
barrier height of 1.42 eV and a shape constant of
0.65 were found. The shape parameter accounts for
the deviation of the barrier shape from a square
barrier. In a quantum transport calculation, it was
found that the current also obeys the direct tunnel-
ing mechanism [8, 37, 38]. Quantum transport cal-
culations and experiments agree at low electric bias
(?0.5 V) [8], but at high bias surface electron charg-
ing is possible [26] and a large disagreement is
obtained.
We calculate the density of states (DOS), trans-
mission probabilities, and I-V characteristics of a
series of alkanethiols addressed on one terminal by
chemisorbed Au and on the other terminal, with
chemisorbed and physisorbed contacts. The calcu-
lations use our combined density functional theory
and Green’s function theory procedure for the de-
termination of electron currents in bulk–molecule–
bulk junctions.
Methodology
The Green’s function and density functional
theory (DFT) are used to study the electron trans-
port characteristics of the alkanethiol junctions,
[AubulkAu]–[SxH1?xCnH2nS]–[AuAubulk], n ? 8, 12,
16, x ? 1, 0 (x ? 0, physisorbed contact; x ? 1,
chemisorbed contact). This nomenclature indicates
that the molecule under test, [SxH1?xCnH2nS], is
extended to Au–SxH1?xCnH2nS–Au and interfaced
to a semi-infinite bulk of gold through the atoms
extending the molecule (Fig. 1).
Our procedure follows these steps: (a) the geom-
etry of the extended molecule is optimized using
Gaussian 03 program [39] until a local minimum is
obtained. Usually, several initial geometries are
tried at this stage, and thus several local minima are
obtained; (b) a second-derivative calculation is
done (usually known as “frequency calculation”) to
guarantee the stability of the extended molecule,
those with negative force constants are reoptimized
using modified initial geometries until a global
minimum is obtained; (c) a series of single-point
calculations of the extended molecule including the
applied bias electric field are run for each point for
which we want to get the I-V, thus each point of the
I-V curve correspond to a full ab initio calculation;
the Hamiltonian and overlap matrices in electric
field are obtained. Some molecules require that a
geometry optimization with an applied electric
field is run because their geometry may be modi-
fied, but this is not the case for the alkanes: (d) the
DOS for the bulk material is obtained using the
Crystal 98 program [40]; (e) a Green’s function
transport procedure using the Hamiltonian and
ELECTRON CURRENTS IN THIOALKANES
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overlap matrices of the extended molecule as well
as the DOS of gold contact as input is used to
calculate the DOS, the electron transmission prob-
abilities, and the I-V characteristics of the molecule.
Care is taken that the electron transport procedure
does not ignore the chemistry of the molecule, as it
occurs in most of the nonequilibrium Green’s func-
tion procedures. Using the applied electric field in a
self-consistent manner on the extended molecule
assures that the chemistry of the molecule is not lost
in the calculation. We must make sure that the
extended molecule is enough to create an accept-
able interface to the bulk. For alkanes or very-high-
impedance molecules, this is not a problem, as we
will explain.
The quantum mechanics calculations are run at
the B3PW91/LANL2DZ/6-31G(d) level of theory
as implemented in Gaussian 2003 program [39]
and Crystal 98 programs [40]. The B3PW91/
LANL2DZ/6-31G(d) level of theory uses the Kohn–
Sham (KS) Hamiltonian [41, 42] with the Becke-3
hybrid exchange functional [43] and the general-
ized-gradient approximation (GGA) Perdew–Wang
91 correlation functional [44, 45]. In geometry opti-
mization and frequencies calculation, the basis sets
used are the LANL2DZ for Au and 6-31(d) basis
sets for S, C, and H; the effective core potential for
Au isthe LANL2. In
LANL2DZ basis sets are used for all elements, and
the LANL2 effective core potentials are used for Au
and S [46–49].
In Green’s function approach, the Hamiltonian
matrix HKSused is a combination of the extended
molecular Hamiltonian matrix and the contact self-
energy matrices, which are obtained from bulk con-
tact material (for detail information readers are re-
ferred to Ref. [7]). The Hamiltonian matrix HKSis
field calculation,the
HKS? S?1?
H11
HM1
H21
H1M
H12
HM2
HMM? ¥1? ¥2
H2M
H22?,(2)
where S is the overlap matrix and Hijis the Ham-
iltonian of the extended molecule (including the
alkane, the sulfur, and the gold atoms) obtained
from Gaussian 2003 program [39] with an electric
field applied to the extended molecule. Further
transformations are performed to the overlap ma-
trix and Hamiltonian matrix: they are first projected
to atomic orbital basis, and the Hamiltonian is then
rearranged into submatrices, HMMfor the molecule,
H11and H22for the contacts, HM1and H1Mfor the
coupling between cluster and left contact, HM2and
H2Mfor the coupling between cluster and right
contact, and H12and H21for the coupling between
left and right contact, which is assumed zero. The
coupling between the contact i and the cluster (the
molecule without gold atoms) yields the self-energy
term ¥i, which contributes to the shifting and
broadening of the molecular electronic levels and
depends on the Green’s function, gidescribing the
contacts [7]:
¥i? HMigi?E?HiM.(3)
The contact Green’s function provides the infor-
mation from the contacts to the transport formal-
ism. But the self-energy of the contact is in general
not Hermitian [50] and does neither the Hamilto-
nian HKS[7], resulting complex eigenvalues
E ? ?0? ? ? i?/2.(4)
The complex part ?/2 is responsible for the broad-
ening of the electronic levels; ?0is the eigenvalue of
the alkanethiol molecule, the ? counts for the shift
of the eigenvalue of the infinite system from the
alkanethiol molecule. The transmission probabili-
ties, the I-V characteristics, and the DOS formalism
used is the same as that in Ref. [7].
In summary, the Hamiltonian describing the ex-
tended molecule is obtained from first-principle
quantum chemistry calculations where an Au atom
is added at each side of the alkanethiol molecule,
and the coupling between the alkanethiol molecule
and the contacts are well described. The effect of the
applied electric field is directly expressed in the
Hamiltonian matrix. The semi-infinite leads are ap-
proximated by the contact atoms, which are located
between the lead and the alkanethiol molecule, al-
lowing a first-principle description of the molecu-
lar-lead system.
Results and Comments
In this section we present and discuss the results
of the quantum chemistry calculations on the ex-
tended molecules, emphasizing the description and
use of the molecular orbitals. In the following sub-
section we study the DOS on the molecule, which is
due to the original discrete molecular orbitals
(MOs) widened and shifted due to the effect of the
metallic contacts; this is followed by the calculation
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FIGURE 1. Systems studied with chemisorbed contact: 1, octanedithiol [AubulkAu]–[SC8H16S]–[AuAubulk]; 2, dode-
canedithiol [AubulkAu]–[SC12H24S]–[AuAubulk]; 3, hexadecanedithiol [AubulkAu]–[Sc16H32S]–[AuAubulk] and with phy-
sisorbed contact; 4, octanethiol [AubulkAu]–[HC8H16S]–[AuAubulk]; 5, dodecanethiol [AubulkAu]–[HC12H24S]–[AuAubulk];
6, hexadecanethiol [AubulkAu]–[HC16H32S]–[AuAubulk]. The green parts at the two ends represents the infinite gold
tips, the yellow ball is sulfur atom, the dark gray ball is carbon atom, and the light gray ball is hydrogen atom. The
distances between gold–contact–atom and sulfur are optimized by running Gaussian 2003 calculations. The distance
between gold–contact–atom and carbon, d, is set to three different values of 3.8 Å, 4.0 Å, and 4.4 Å for each mono-
thiol system.
FIGURE 2. (a) The HOMO energy (solid curve) and LUMO energy (dotted curve) merge at about 3 V for alkanedi-
thiolates; however the two energies are practically unchanged for the pure alkane. (b) Relative total energies for the
alkanes with respect to the energy with no field. The Au–C distance for the monothiolates is 3.8 Å (level of theory,
B3PW91/LANL2DZ).
ELECTRON CURRENTS IN THIOALKANES
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of the transmission probabilities for electron trans-
port through the junction. The next subsection cov-
ers the calculation of the current-voltage character-
istics covering aspects related to the conduction
channels, and the last two sections cover the elec-
tronic decay constant as well as the contact resis-
tance and equivalent circuits.
QUANTUM MECHANICS CALCULATION OF
GOLD ALKANETHIOLATES
All systems studied are chemisorbed at one end;
however, the other end may be chemisorbed or
physisorbed as shown in Figure 1. 1, 2, and 3 have
S–Au bonds at both contacts resembling the exper-
iments with two chemisorbed contacts [32]. 4, 5,
and 6 have one S–Au bond at right and a van der
Waals contact –CH3- - - Au at the left contact, re-
sembling the experiment with one chemisorbed
and one physisorbed contacts [19, 26]. Contact dis-
tances of 3.8 Å, 4.0 Å, and 4.4 Å between the Au - - -
C are used. These distances may reflect different
contact forces in AFM experiments (forces of 6 nN,
8 nN, and 14 nN were used recently [26]).
Total Energy and HOMO–LUMO Gap Under
Electric Field
The alkanes are saturated organic molecules
with large HOMO–LUMO gaps. The Fermi energy
of gold (?5.31 eV) is located between their highest
occupied molecular orbital (HOMO) (?8.0 eV) and
lowest unoccupied molecular orbital (LUMO) (?2.7
eV, i.e., unstable cations) energies. An applied bias
voltage results in polarization of the molecule,
causing a slight decrease in the total energy, as well
as in the energy of the HOMO and LUMO (Fig. 2).
This so-called total energy includes the effect of the
electrical field but the field-molecule interaction en-
ergy is not added. If polarization causes a large
change in total energy, special considerations have
to be taken if a geometry reoptimization is required.
The polarization effect in alkanes is weak; however,
the HOMO–LUMO gap and polarizability are
greatly affected by the Au and S. Both the S–Au
bonding and antibonding orbitals lie above the
HOMO of the alkane; as a result, the HOMO–
LUMO gap of the corresponding alkanethiolate de-
creases to 2.5 eV; and the HOMO energy increases,
whereas LUMO energy decreases under the ap-
plied electric field (Fig. 2). The polarization effect is
large as the HOMO–LUMO gap decreases to almost
0 at a bias of 3 volts.
Orbital Energy and Orbital Shape
Although the HOMO–LUMO gap and polariz-
ability strongly changes when the Au and S atoms
are added to the alkanes, the energy and shape of
the original MOs in alkanes are slightly affected
when they show up in the alkanethiols. The HOMO
in the alkane is highly delocalized, but it is far from
the Fermi energy, and it transforms into HOMO-4
in alkanedithiolates. Four occupied MOs localized
in Au and S are added above the original HOMO of
the alkanes, keeping them practically in high im-
pedance. Thus the alligator-clip S does not affect
the conduction characteristics of the alkane but al-
lows a robust attachment to the Au. From Figure 3
it can be inferred that adding additional Au atoms
to represent the contacts only increases the amount
of highly localized states that do not contribute to
the conduction process. This figure also shows that
having alkanes in the junction is equivalent to hav-
ing vacuum if bias voltages are larger than ?7 eV,
which is needed to reach the conducting MOs of the
alkanes. However, at this bias voltage, the electric
field applied is so intensive that it is able to break
the chemical bonds in the molecule. Therefore,
these systems are so far limited in their use as
electronic devices. Under this picture, the conduc-
tion in alkanes is very straightforward: electrons
from any of occupied MOs tunnel to one contact
with a tunneling probability that depends on the
separation between the orbital energy and the
Fermi level and the shape of the MO. Once the
electron has tunneled, another electron from the
other contact can take its place, and the process
repeats. The conduction through the unoccupied
MOs follows a different sequence. First, an electron
from one terminal tunnels to the unoccupied MOs,
and then this electron transfers to the other contact.
The tunneling probability also depends on the sep-
aration between its orbital energy and the Fermi
energy and the shape of the MO. Notice that for the
sake of clarity, the above description seems to dis-
tinguish the electrons during the transfer; in a real
situation and during the calculations, the electrons
are indistinguishable.
DENSITY OF STATES (DOS) AND
TRANSMISSION PROBABILITIES
Figure 4 shows the DOS in the molecule attached
to the contacts. Each DOS is the result of the shift,
broadening, and overlap of the discrete MOs of the
isolated molecule due to its interaction with the
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continuous DOS of the bulk metal. For the energies
within the HOMO–LUMO gap (from ?6.06 eV to
?3.24 eV), the DOS is usually small (Fig. 4). The
HOMO and HOMO-1 are practically degenerate at
?6.12 eV in the isolated molecule and transform
into a broad but small peak in [AubulkAu]–
[SxH1?xCnH2nS]–[AuAubulk].
closer to the Fermi level—the HOMO-2 and
HOMO-3 at ?7.29 and ?7.32 eV, the LUMO and
LUMO?1 at ?3.25 and ?3.22 eV, and the
LUMO?2 and LUMO?3 at ?0.36 eV—are totally
localized MOs yielding poor contributions to con-
duction. Actually, similar conduction could be ob-
tained if the molecule is substituted by vacuum,
and the electrons at the HOMO and lower energies.
However, the fully delocalized HOMO-4 and
HOMO-5 at ?7.98 and ?8.35 eV, respectively,
which are directly related to the HOMO of the pure
alkane, become a well-defined peak shifted to
higher energies due to their interaction with the
other MOs. The peaks below ?8.5 eV overlapping
with the DOS of the gold bulk are unreachable
Theotherorbitals
experimentally, because energies at such range are
large enough to break the molecule and the con-
tacts. For example, the strong C–C and C–H bonds
FIGURE 3. Energies and shapes of the MOs for oc-
tane and gold–alkanedithiolates. The MOs to the left of
the Fermi level (vertical red line) are occupied MOs and
to right are unoccupied MOs. There are four sets of
vertical bars starting from the bottom for the pure al-
kane C8H18(blue) followed by those for the gold–al-
kanedithiols. The HOMO and LUMO for the pure alkane
is plotted at the bottom of the figure. Note that the
LUMO is out of the scale of the plot. Both HOMO and
LUMO are conducting MOs; however, they are too far
from the Fermi energy. The MOs near the Fermi energy
are fully localized in the Au and S; thus, all MOs yield
poor conduction (functional, B3PW91; basis sets,
6-31G(d) for C, H, and S; LANL2DZ basis and effective
core potential for Au).
FIGURE 4. The DOS for the [AubulkAu]–
[SxH1 ? xCnH2nS]–[AuAubulk], (a) for n ? 8, (b) n ? 12,
and (c) n ? 16. Each plot shows the curves for the
chemisorbed (actually doubly chemisorbed) case and
the three physisorbed (or singly chemisorbed) ones.
The insets show the DOS in a large range.
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correspond to an energy of 3.47 and 4.26 eV, respec-
tively, and the weak C–S, S–Au, and Au–Au bonds
correspond to an energy of 2.68, 1.16, and 0.36 eV,
respectively, whereas the nonbonded Au–C inter-
action can be as low as 3 meV.
The DOS for physisorbed and chemisorbed con-
tacts differ because the additional states of the S
atom are larger than the H. Note that this DOS is a
“global” quantity for the whole extent of the mol-
ecule under study; therefore, the DOS in the unoc-
cupied region increases with the number of atomic
basis set. On the other hand, the DOS in the occu-
pied region is independent of the atomic basis set
used but increases with the number of electrons in
the system. However, the DOS does not provide an
idea of the quality of those states for electron con-
duction; this has to wait for the calculation of the
transmission probabilities. Nevertheless, excellent
estimates can be obtained from the shapes and en-
ergies of the molecular orbitals of the extended
molecule [2, 7, 51]. Note also that the attachment of
the Au to the molecule is stronger in the doubly
chemisorbed case than in the singly chemisorbed
one. For the physisorbed system (or singly chemi-
sorbed), the distance between contact gold atom
and the carbon does not affect the DOS greatly and
it should be equivalent to the DOS of the molecule
attached to just one contact.
The transmission probabilities for all the systems
are shown in Figure 5. The integral on these prob-
abilities as the bias field is applied yields the cur-
rent through the molecule. Thus, as expected, trans-
mission probabilities for short molecules are much
higher than the corresponding ones for long mole-
cules. The doubly chemisorbed contact allows bet-
ter electron transport, whereas the physisorbed
contact blocks the electron transport, and the length
of the physisorbed attachment certainly reduces the
transmission probability. In the energy range for
the incoming electrons from 0 to ?5.5 eV, the trans-
mission probabilities slightly decrease, whereas be-
low ?6.5 eV, the transmission probabilities increase
sharply. The increase in transmission probabilities
right below ?2.0 eV of the Fermi level yields a
sharp increase in conductance at a bias of 3.8 V.
Note that the junctions with d ? 4.0 and 4.4 Å show
similar transmission, yet the one with d ? 4.0 Å is
close to the chemisorbed one for C8 and C12 but
similar to the other two (d ? 4.0, 4.4 Å) for C12. This
more erratic behavior is typical in smaller wires [8]
where the application of standard conduction mod-
els is not possible.
CURRENT-VOLTAGE CHARACTERISTICS,
CONDUCTION BARRIERS, AND CHANNELS
Figure 6 shows the I-V characteristics calculated
using our combined DFT-GF approach for the
chemisorbed and the three physisorbed cases. In
each of these four cases, the currents for C8, C12,
and C16 are calculated. Thus, a total of 12 full
optimizations on the extended molecules are per-
formed followed by full self-consistent DFT calcu-
lations for each point of the applied bias electric
field, which allows us to get the currents for each
point of the bias using the Green’s function ap-
proach. The ab initio calculated currents are fitted
(Fig. 6) to a direct tunneling mechanism considered
satisfactory with experimental results at low bias
[17], which is given by
I ?
e
4?2?d2???B?eV
???B?eV
2?e??2?2m/?????B??eV/2? d
2?e??2?2m/?????B??eV/2? d?,(5)
where ?Bis the barrier height, ? is the shape con-
stant of the barrier, d is the barrier width (approx-
imately the length of the molecule), V is the applied
bias, and ?, e, and m are the Plank’s constant, elec-
tron charge, and effective mass of the electron, re-
spectively. Comparisons with SAM-based measure-
ments would require knowing the number of
molecules participating in the conduction process.
Wang et al. [17] reported experimental I-Vs com-
patible with the direct tunneling mechanism with a
barrier height of 1.42 eV obtained after a fitting of
the experimental measurements to Eq. 5. However,
no physical meaning can be assigned to this “bar-
rier,” except perhaps for some particular average of
a group of intrinsic barriers of the molecule. The
main reason for the lack of even qualitative agree-
ment with the precise theoretical calculations is that
no frontier MO with an energy difference from the
Fermi level of 1.42 eV is found. The intrinsic barri-
ers corresponding to the energies of the molecular
orbitals in the neighborhood of the Fermi level, for
the alkanedithiolates, are approximately 0.78 eV for
the HOMO and HOMO-1; 1.99 eV for HOMO-2,
and HOMO-3; 2.07 eV for the LUMO and
LUMO?1; and 2.67 eV for HOMO-4. Needless to
say, calculations of this type for alkanes are precise
enough that the discrepancy in energies cannot be
attributed to the level of theory used in this calcu-
lation. The most reasonable explanation is that sev-
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eral MOs with different energies participate in the
conduction process and all of them can be averaged
to one barrier, but this individual barrier is not able
to explain the electron transport through the mole-
cules.
The contributing MOs orbitals are shown in Fig-
ure 7 for C8 at zero bias and at 0.7 V bias. Note that
at zero bias, the closest MOs to the Fermi level are
the degenerate HOMO and HOMO-1, which are
highly delocalized (poorly conducting), at 0.78 eV
from the Fermi level. Electron conduction can take
place through the occupied and unoccupied MOs.
In the particular case of the alkanethiols, most of the
conduction is done through the occupied MOs, also
referred to loosely as hole conduction, with the
intent to resemble the conduction in doped semi-
conductors, thus calling conduction through the
unoccupied MOs, “electron” conduction; however,
this is not a precise description because the conduc-
tion channels in semiconductors are fully delocal-
ized but the unoccupied MOs close to the Fermi
level are still localized for the case of the alkanethi-
FIGURE 5. Transmission probability of the [AubulkAu]–
[SxH1 ? xCnH2nS]–[AuAubulk] for the physisorbed with
C–Au lengths, d, of 4.4 (green), 4.0 (black), and 3.8
(blue) and the chemisorbed (red) junctions are shown.
Vertical lines (red) represent the location of the Fermi
level, and the insets show the transmission probabilities
from ?0 to ?10 eV; (a) C8, (b) C12, and (c) C16.
FIGURE 6. The I-V characteristics for (a) the alkane-
thiols with chemisorbed contact; and (b–d) the three
physisorbed cases with Au–C distances of 3.8 Å, 4.0 Å,
and 4.4 Å, respectively. The dots correspond to the ab
initio currents obtained directly with our combined Den-
sity Functional Theory-Green Function (DFT-GF) proce-
dure. The solid curves are from fittings to a direct tun-
neling fitting corresponding to four barriers from the
nearest MOs to the Fermi level (green for C8, purple for
C12, and blue for C16).
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ols. The next group of MOs correspond to the oc-
cupied degenerate and poorly conducting HO-
MO-2 and HOMO-3 at 1.99 eV and on to the
unoccupied, degenerate, poorly conducting LUMO
and LUMO?1 at 2.07 eV. The first conducting (fully
delocalized) MOs for the thioalkane is the occupied
HOMO-3 at 2.67 eV from the Fermi level; this MO
is a direct consequence of the HOMO for a pure
alkane, which in this case combines additively with
the MOs of the attached Au and S atoms. All these
channels combine such that an equivalent “experi-
mental” barrier can be determined. In the biased
case, there is a very small variation in the location
of the MOs; however, the applied bias allows elec-
trons to get closer to the Fermi energy, increasing
theirtransmission probabilities
through either the occupied or unoccupied MOs.
The biased case in Figure 7 shows the first step in
the conduction process: electrons from the negative
(left) terminal jump into the unoccupied MOs and
electron from the occupied MOs jump into the pos-
itive terminal (right).
exponentially
ELECTRONIC DECAY CONSTANT
The resistance measured in experiments is also
affected by a contact resistance; therefore, a direct
comparison between theory and experiment is im-
possible unless the contact resistance can be evalu-
ated [19, 24, 52]; however, for the alkanes, the elec-
tronic decay constant ?N, (decay constant per
methylene group) should not depend on the contact
resistance, at least if the resistance of the alkenes
units is larger. Figure 8(a,b) shows how the total or
equivalent resistance of the junction depends on the
number of methylene group n and the Au–C dis-
tance d. For the same Au–C distance, the logarithm
of resistance changes linearly as the number of
methylene group increases, corresponding to decay
constants of 0.67, 0.63, 0.63, and 0.66 with respect to
chemisorbed contact and physisorbed contact at
d ? 3.8 Å, 4.0 Å, and 4.4 Å (Fig. 9), respectively.
Clearly, the decay constant does not change as the
Au–C distance increases, which is equivalent to a
change in the nature of the contact.
The transmission probabilities at the Fermi level
for the thioalkanes are very low due to the com-
bined effect of the shape and energies of the molec-
ular orbitals. The MOs closest to the energy level of
bulk Au are totally localized (poorly conducting,
[51]), and the fully delocalized (the best for conduc-
tion, [51]) MOs are too far in energy from the Fermi
level. As mentioned in the last section, the alkane
would be similar to a vacuum between the contacts
and with electrons at the MO’s energies. This is
verified from the Wentzel–Kramer–Brillouin (WKB)
approximation, the transmission probability T(E) of
an electron with energy E tunneling a one-dimen-
sional barrier of height ?(x) and width w is,
T?E? ? exp???
0
w
dx?
8m
?2???x? ? E??
? exp????w?, (6)
where the meaning of m, ?, and ? are the same as in
Eq. (5), and ? is the electronic decay constant. This
model yields an electronic decay constant of
1.02?? per Å, with the barrier in eV. Assuming the
projected length of each methylene group of 1.22 Å
on their longitudinal axis, we get an electronic de-
cay constant of 0.84?? per methylene group.
CONTACT RESISTANCE CONTRIBUTION
FACTOR AND EQUIVALENT CIRCUIT OF
ALKANES
In order to evaluate how the contact contributes
to the resistance of the molecular device, the loga-
rithm of the total resistance is plotted with the
Au–C distance in Figure 8(c,d) for the three phy-
sisorbed cases; we also included the gold–alkanedi-
thiolates (two chemisorbed contacts) assuming an
equivalent Au–C distance of 3.7 Å, because the
chemisorbed contact resistance is equivalent to a
physisorbed one at Au–C distance of 3.7 Å. The
effective contact resistance of the two contacts is the
intercept on vertical axis in Figure 8(a,b) extrapo-
lating to n ? 0 (see, e.g., Ref. [19]). We make use of
the symmetric doubly chemisorbed system to de-
termine the contact resistance.
The metal–molecule junction is too small with
respect to the electron coherence length (i.e., Ohm’s
Law is not obeyed [50]) to allow us to provide
lumped resistances to each of the three parts of the
junction. Electrons at atomistic dimensions “know”
they are in different parts of the small nano-sized
junction, therefore, according to Landauer’s [53,
54]:
R ?
h
2e2M?1T?1?12.9k?
M
Tl
?1Tm
?1Tr
?1, (7)
where R is resistance of the molecular device, M is
the number of conduction modes, T is the overall
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720
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transmission probability, and T1, Tm, and Trare
transmission probability of the left contact, the mol-
ecule, and the right contact, respectively. The e and
h are charge of electron and Plank’s constant. Re-
writing Eq. (7) in practical terms:
R ? 12.9k? ?
Tl
?MTm
?1
?1Tr
?1
?M? 12.9k? ? rlrmrr,
(8)
where rl, rr, and rmare the resistance contribution
factors, unitless parameters accounting for the re-
sistance contribution from the left and right con-
tacts and the molecule, respectively.
The resistance increases exponentially with the
number of methylene groups in alkane, rm? e?Nn
and for the (doubly) chemisorbed contact, rl? rr;
thus, the resistance contribution factor of a chemi-
sorbed contact as well as the resistance contribution
factor of a physisorbed contact can be calculated.
At low bias, ?1.5 V, the resistance contribution
factor of a chemisorbed contact is 18.7, the resis-
tance contribution factor of physisorbed contact at
Au–C distances 3.8 Å, 4.0 Å, and 4.4 Å are 28.4, 66.5,
FIGURE 7. Multichannel conduction for octanedithio-
late (left, zero bias; right, biased). For conduction of
electrons from left to right terminal, electrons from oc-
cupied MOs must tunnel a barrier from their level to the
Fermi level (red dashed horizontal lines) of the right
contact, and electrons from the left terminal must tun-
nel a barrier from the Fermi level to the energy of the
unoccupied MOs. Green curves are the DOS for the
gold contacts obtained also by a DFT procedure, and
the blue bars correspond to the eigenvalues of oc-
tanedithiolate on the vertical scale in eV. In the biased
case, the Fermi level of the right contact gets closer to
the LUMO energy, and simultaneously the HOMO en-
ergy gets closer to the Fermi level of the right contact.
FIGURE 8. Semilog plots of the resistance versus the
number of methylene groups n, (a) at low bias and (b)
at high bias and of the resistance versus the Au–C dis-
tance d, (c) at low bias and (d) at high bias. Also in (c)
and (d), the resistance for the chemisorbed contact is
shown at d ? 3.7 Å (see text for explanation).
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and 364, respectively. The molecule resistance con-
tribution factor is e0.65n; thus, the resistance at low
bias (Fig. 10),
RL? 12.9k? ? e4.25?d?3.01??0.65n?2.93. (9)
At high bias (from 1.5 V to 3.0 V), the resistance
contribution factor of a chemisorbed contact is 12.0
and the resistance contribution factor of phy-
sisorbed contact at contact–carbon distances 3.8 Å,
4.0 Å, and 4.4 Å are 15.8, 28.1, and 89.3, respec-
tively. And the molecule resistance contribution
factor is e0.58n. Therefore, the resistance at high bias
is
RH? 12.9k? ? e2.89?d?2.85??0.58n?2.48.(10)
Conclusion
Several channels participate in the electron trans-
port; the main ones from the HOMO-4 through the
LUMO?1 grouped by pairs with barriers of 0.78,
1.99, 2.67, and 2.07 eV. These conduction channels
are difficult to distinguish experimentally, and their
combined effect is equivalent to a single barrier of
1.42 eV.
The resistance of a molecular device can be ex-
pressed as a product of contact and molecular fac-
tors. At bias voltages smaller than 1.5 V, the resis-
tance contribution factor by a chemisorbed contact
is 18.7, and the resistance contribution factor by a
physisorbed contact is dependent on the Au–C dis-
tance, ranging from 28.4 to 364 for Au–C distances
from 3.8 Å to 4.4 Å. At bias higher than 1.5 V, the
resistance contribution factor by a chemisorbed
contact is 12.0, and the resistance contribution fac-
tor by a physisorbed contact is dependent on the
contact–carbon distance, ranging from 15.8 to 89.3
for contact–carbon distance from 3.8 Å to 4.4 Å.
The electronic decay constant (?0.65 at biases of
lower than 1.5 V and ?0.58 at biases greater than
1.5 V) depends on the number of methylene groups,
but is independent on the contacts.
ACKNOWLEDGMENTS
This work has been supported by DARPA/ONR
grant N00014-01-1-0657, ARO grants DAAD19-00-
1-0592 and DAAD19-00-1-0634, and NRL grant
N00173-01-1-G017.
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