First-principles calculation on the transport properties of molecular wires between Au clusters under equilibrium

Abstract

Based on the matrix Green's function method combined with hybrid tight-binding / density functional theory, we calculate the conductances of a series of gold-dithiol molecule-gold junctions including benzenedithiol (BDT), benzenedimethanethiol (BDMT), hexanedithiol (HDT), octanedithiol (ODT) and decanedithiol (DDT). An atomically-contacted extended molecule model is used in our calculation. As an important procedure, we determine the position of the Fermi level by the energy reference according to the results from ultraviolet photoelectron spectroscopy (UPS) experiments. After considering the experimental uncertainty in UPS measurement, the calculated results of molecular conductances near the Fermi level qualitatively agree with the experimental values measured by Tao et. al. [{\it Science} 301, 1221 (2003); {\it J. Am. Chem. Soc.} 125, 16164 (2003); {\it Nano. Lett.} 4, 267 (2004).] Comment: 12 pages,8 figures

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arXiv:cond-mat/0504594v2 [cond-mat.mes-hall] 13 Jul 2005
First-principles calculation on the transport properties of molecular
wires between Au clusters under equilibrium
Zhanyu Ning∗,1, Jingzhe Chen1, Shimin Hou2, Jiaxing Zhang2, Zhenyu Liang1,
Jin Zhang1, and Rushan Han1
1School of Physics, Peking University, Beijing 100871, People’s Republic of China
2Department of Electronics, Peking University, Beijing 100871, People’s Republic of China
Abstract
Based on the matrix Green’s function method combined with hybrid tight-binding / density func-
tional theory, we calculate the conductances of a series of gold-dithiol molecule-gold junctions including
benzenedithiol (BDT), benzenedimethanethiol (BDMT), hexanedithiol (HDT), octanedithiol (ODT) and
decanedithiol (DDT). An atomically-contacted extended molecule model is used in our calculation. As
an important procedure, we determine the position of the Fermi level by the energy reference according
to the results from ultraviolet photoelectron spectroscopy (UPS) experiments. After considering the
experimental uncertainty in UPS measurement, the calculated results of molecular conductances near
the Fermi level qualitatively agree with the experimental values measured by Tao et. al. [Science 301,
1221 (2003); J. Am. Chem. Soc. 125, 16164 (2003); Nano. Lett. 4, 267 (2004).]
1Introduction
Molecular devices are now playing an important role in nanoelectronic researches. The construction, mea-
surement and understanding of the transport properties in molecular electronic circuits have drawn more and
more attention. At present, a lot of theoretical work is concentrating on the sandwichtype structure of an in-
dividual molecule between metallic electrodes [1–4]. Although current simulation methods can qualitatively
describe some electronic properties of such simple systems, there are still a lot of issues waiting to be made
understandable for the purpose of predicting, designing and controlling new molecular devices. For example,
the methods used to calculate the coupling strength of electrodes-molecule interaction remain unconvincing.
It’s difficult to model the metal-molecule-metal junction accurately because of the uncontrollable interface
∗Corresponding author. Email:zyning@pku.edu.cn
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structures between molecules and leads. In addition, the position of the Fermi energy is another focal point.
The common simple viewpoint is that the Fermi level of electrodes lies within the gap of molecular orbitals
between HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) [5].
Further calculation and experimental results are essentially needed to clarify the doubt of precise location
of the Fermi energy. On the other hand, the importance of the modification of molecules due to the non-
equilibrium situations and the determination on the spatial profile of the electrochemical potential across
the interface also need more discussions.
For device under equilibrium, the above problems reduce to calculating electronic structures of the
molecules sandwiched between two metal electrodes, which is the first step to understand the transport be-
haviors of molecular devices under any circumstances. Recently, the conductances of a series of gold-dithiol
molecule-gold systems were measured near low bias by the tip of scan tunneling microscope (STM) [6–8].
Though the molecular junctions based on Au-S bond have been investigated extensively [9–14], our calcu-
lation and further discussions according to the experimental results still can show more enlightenments for
understanding the interaction between molecules and metals. Ultraviolet photoelectron spectroscopy (UPS)
was referenced for the purpose of determining “line up” about the Fermi level [15–19]. UPS experiments
can provide direct spectroscopic information about the gap between HOMO and EF, which is one of the key
points in the transport calculation of molecule-metal sandwichtype structures. In this paper, we present a
first-principles calculation, which is based on the methods of Xue-Datta-Ratner [2,3,9], on the conductance
of molecular wires including benzenedithiol (BDT), benzenedimethanethiol (BDMT), hexanedithiol (HDT),
octanedithiol (ODT) and decanedithiol (DDT). An atomically-terminated extended molecule model is used
and our calculations are in qualitative agreement with the experimental results [6–8]. We also try to analyze
the influence from the delocalization of molecular orbitals on the transport properties of molecular junctions.
2 Methods
We use the matrix Green’s function method combined with hybrid tight-binding / density functional theory
(DFT) for modeling molecular devices, which was firstly introduced by Datta et. al. [20,21]. Since what
we now consider is the transport properties under equilibrium, the most important thing is to calculate
electronic structures of metal-molecule-metal junctions. This problem is the generalization of the familiar
chemisorption where a single isolated molecule is considered to be adsorbed onto one metal surface. The
molecular device can be divided into two different parts: “extended molecule” region (the molecule and
the neighbour metallic atoms on the electrodes perturbed by the adsorption of the molecule) and “contact
region”(the unperturbed part of the electrodes) [21,22]. This separation provides a convenience to treat the
two parts by well-established techniques independently. The solution of the total device system is based on
Green’s Function method, which supplements the coupling of the molecule and electrodes via the concept of
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self-energy. The retarded Green’s function GRof the extended molecule is given by
GR= (ES − F − ΣR
1− ΣR
2)−1
(1)
where E is energy, S is the overlap matrix, F is the Fock matrix of the isolated extended molecule. F and S
can be obtained through DFT calculation, which is performed using GAUSSIAN 03 program package [24].
The self-energy matrices ΣR
1and ΣR
2can be written as
ΣR
1= τ†
1gR
1τ1 and ΣR
2= τ†
2gR
2τ2
(2)
where τ1,2 is the coupling between the extended molecule and electrodes, and g1,2 is the surface Green’s
function of the electrodes.
The computation of the surface Green’s function of gold electrodes and the coupling matrices is described
using tight-binding scheme with the parameters given by Papaconstantopoulos [23]. Here we suppose that
the gold electrode is semi-infinite Au(111) surface and calculate the Green’s function layer by layer with
an iteration procedure [21]. Then we can obtain the transmission function through the molecule by the
following equation
T(E) = Tr(Γ1GRΓ2GA) (3)
where
Γ1,2= i(ΣR
1,2− ΣA
1,2) (4)
Since what we concern is the conductance under equilibrium condition, the relation can be given by
G =2e2
hT(Ef) (5)
The density of state of the extended molecule D(E) can also be calculated by Green’s function
D(E) = −1
πTr[Im(GRS)](6)
3 Results and discussions
3.1Optimization and calculation models
In the following DFT calculations we have used the Becke’s three-parameter hybrid functional using the
Lee, Yang, and Parr correlation functional [25,26] (B3LYP). The basis set chosen for C, H, S atoms is the
triply split valence with polarization functions, 6-311G** [27,28]. When the Au atoms are included in the
calculations, two Los Alamos National Laboratorybasis sets [29,30] are used with effective core potentials, the
LanL2DZ basis set for the geometry optimization and the LanL1MB basis set for the transport calculations.
The use of the pseudopotentials with relativistic corrections has been widely demonstrated to be a good
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compromise with the alternative use of full-electron procedures, especially when we deal with relatively large
numbers of heavy atoms. It requires less computational effort without loss of accuracy [31].
Under equilibrium condition, we have considered the electronic transport properties in molecular junc-
tions consisting of benzenedithiol (BDT), benzenedimethanethiol (BDMT), hexanedithiol (HDT), octanedithiol
(ODT) and decanedithiol (DDT). As the first step in our calculation, we optimize the molecules by a gold
trimer-molecule-gold trimer structure, which is shown in Figure 1. The optimization scheme is similar to our
recent work on calculating 4,4-bipyridine connnected to gold electrodes [32]. The Au-S bond lengths of five
optimized metal-molecule-metal structures are 2.32 ∼ 2.33˚ A. All the optimized bond lengths are consistent
with other theoretical data [33]. On the other hand, the calculation of Kru¨ ger et. al. also indicates that
the gold trimer-molecule structure is stable after leaving from the gold surface [34], which supports our
optimization scheme.
Assuming the molecular structure remains unchanged during the pulling process by STM tip, we can
use the above optimized structures in the following transport calculations. The contact geometry is always a
debating problem in the construction of molecular junctions. Some previous work have already done careful
discussions on this issue [34, 35]. Because different adsorption geometries may correspond to dissimilar
local minimal energy points [36], we can not determine the accurate contact by only a simple junction
model. The work of Emberly’s group shows that the conductance of gold-BDT-gold junction changes little
when the geometries of contact vary near zero bias [35]. Their results indicate that an atomically-contact
extended molecule model could be one of the reasonable qualitative descriptions for molecular junctions under
equilibrium. Thus we model the junctions by inserting a single gold atom between the end sulfur atom and
the gold surface at each contact. The inserted gold atom is settled above the center of 3-fold hollow site on
Au(111) surface and maintains the Au-S bond length calculated in the above optimized structures. Three
nearest-neighbor atoms on the surface are included in the extended molecule, which is showed as an example
in Figure 2. Within the range of nearest-neighbour coupling, each side of the outmost 3-fold gold atoms in
the extended molecule is coupled to the gold electrode through nine gold atoms in the first layer and seven
gold atoms in the second layer. All the gold-gold distance is fixed on 2.88˚ A, which is the bond length of
body gold. With the above procedure, we construct all the five gold-molecule-gold systems and show them
in Figure 3. For simplicity, all the structures are treated symmetrically.
3.2The alignment of the Fermi level
Another key point for calculating the conductance in molecular junctions is the position of the Fermi level EF.
As what Patrone has pointed out [16], UPS experiments can give direct information about EF− EHOMO,
which is an important energy reference in the molecular transport calculation. Because we calculate the
electrodes and extended molecule separately, we have to face the difficulty that how to unify the zero
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energy points of two different methods. Fortunately, introducing the gap of EF−EHOMOinformed by UPS
data, we surpass this problem. With energy translation on the Papaconstantopoulpos diagonal Hamiltonian
matrix elements [23], we can adjust the Fermi level in gold electrodes and locate it in the suitable place
within molecular levels obtained by DFT calculations. Thus, the adjustment of the relative energy scale
with respect to the experimental data of UPS [15,17–19] becomes one of the meaningful procedures in our
calculations. In Table 1 we show the calculation results of all the five different molecular junctions compared
with the experimental values [6–8]. To our satisfaction, we found in our calculation that the conductance
measured in STM experiments qualitatively coincide well with the UPS data via the bridge of EF−EHOMO.
As what we have shown in Table 1, after considering the UPS data of EF− EHOMO 0.6eV [18], the
calculation of BDT’s transmission near the Fermi level (T(Ef) = 0.0086) agrees well with the experimental
result 0.011. UPS experiments also show that the position of HOMO is larger than 5eV from the Fermi
level in octandecanethiol monolayers on Au [15,19]. For the similar electronic structures of three saturated
alkanedithiol molecules (HDT, ODT, DDT), here we assume 5.5eV as an median value for the calculation of
all the three molecules. The results of 0.85× 10−3(HDT), 3.17 × 10−4(ODT), 2.62× 10−5(DDT) could be
qualitatively in accordance with the experimental results 1.2×10−3(HDT), 2.5×10−4(ODT), 2×10−5(DDT)
[6–8]. Although we do not find direct UPS experimental results about BDMT, the measurement of 4,4-
(ethynylphenyl)-1-benzenethiol [17], which has a comparable electronic structure with the incorporation
of phenyl and carbon-chain group like BDMT, can give us useful enlightenments. Unlike the small value
(0.5 ∼ 0.6eV) in pure π-bonded molecules [16,18] and the large value (> 5eV ) in saturated carbon-chain
molecules [15,19], the above mixed electronic structure has a 1.9eV gap of EF−EHOMO[17]. It provides the
prompt on estimating the EF− EHOMO value of BDMT. Here we use 2.1eV and find that the calculation
result 6.79 × 10−4is in good agreement with the experimental data 6 × 10−4. In addition, we consider
the influence on the calculation results due to the uncertainty in UPS experiments. In Table 1, we change
EF−EHOMOinformed by UPS data with a value of ±0.2eV. Our calculation shows that the results are not
affected qualitatively. It indicates that introducing UPS experimental information might provide a relatively
reliable energy reference for qualitative calculations on molecular conductance under equilibrium.
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