CHIN. PHYS. LETT. Vol. 30, No. 6 (2013) 067302
Electron Transport through a Silicon Atomic Chain
*
LIU Fu-Ti(柳福提)1,2** , CHENG Yan(程艳)2** , CHENG Xiao-Hong(程晓洪)1 ,
YANG Fu-Bin(羊富彬)2 , CHEN Xiang-Rong(陈向荣)2
1
2
College of Physics and Electronic Engineering, Yibin university, Yibin 644000
College of Physical Science and Technology, Sichuan University, Chengdu 610064
(Received 22 February 2013)
The electron transport properties of a silicon atomic chain sandwiched between Au (100) leads are investigated by
using the density functional theory combined with the non-equilibrium Green’s function method. The breaking
process of Au-Si4 -Au nanoscale junctions is simulated. The conductance and the corresponding cohesion energy
as a function of distance 𝑑z are obtained. With the increase of distance, the conductance decreases. When
𝑑z = 18.098 Å, there is a minimum value of cohesion energy. The nanoscale structure of junctions is most stable,
and the equilibrium conductance is 1.71𝐺0 (𝐺0 = 2𝑒2 /ℎ) at this time. The 𝐼–𝑉 curves of junctions at equilibrium
position show linear characteristics.
PACS: 73.63.Rt, 61.46.+w, 81.16.Ta
DOI: 10.1088/0256-307X/30/6/067302
When a system’s dimensions reach the order of
the electron mean free path, the electron transport
becomes ballistic and its conductance is quantized
in units of 𝐺0 (2𝑒2 /ℎ), contrary to diffusive electron
transport in a macroscopic system, where 𝑒 is the electron charge and ℎ is Planck’s constant.[1] With the recent progress in micro-fabrications and self-assembly
techniques, downsizing of electronic devices, the electron transport properties through atomic scale contacts, which have been investigated for more than a
decade,[2] have attracted great interest. Chains of
metal atoms can be considered as the ultimate conductors of future nano-electronics.[3] Apart from possible technological applications such as interconnects
in molecular electronics, the simple structure of these
monatomic wires makes them an ideal test ground for
developing and validating our understanding of electron transport on the nanometer length scale.[4] Recently, it has been reported that the conductance of
chains of Au, Pt, and Ir atoms oscillates as a function
of the number of atoms in the chain.[5] Apart from
chains of metal atoms, the chains of semiconductor
or insulator atoms have also been investigated. The
conductance of monatomic chains of C, Si, Ge, Sn elements, and of binary compounds such as InP, GaAs,
AlSb, BN, SiC, GaN, AlN have been reported.[6−9]
In mechanically controlled break junction (MCBJ) experiments it is possible to measure conductance. However, the detailed structure of the chain, including
the number of atoms in the chain, the distance and
the electrode environment, remains unknown in the
MCBJ measurements. This clearly calls for support
from first-principles calculations.[3−10]
Roland et al. developed an ab initio non-
equilibrium Green’s-function formalism and investigated the transport behavior of small silicon clusters
between Al and Au electrodes.[11] Dai et al. investigated the effect of the contact geometry on the electronic transport properties of Si4 cluster.[12] Their results show that small Si nanoclusters act primarily as
metals, with typical conductance ranging between 2
and 4 in units of 𝐺0 . To realize the application of
silicon clusters in nanodevices, we are interested in
investigating their transport properties based on our
previous research.[13] In this Letter, we explore the
conductance and the 𝐼–𝑉 characteristics of Si atomic
chains sandwiched between Au electrodes. In order to
address the quantum transport properties in Si chains,
we focus on atomic contacts in the geometry of top-totop arrangement, such as those of a scanning tunneling
microscope (STM). Thus, we will simulate the junctions breaking process, and calculate the 𝐼–𝑉 characteristics of junctions at equilibrium position. We expect that our theoretical predictions might be realized
in experiment.
The device system, as shown in Fig. 1, is divided
into three regions: left leads, right leads, and a central
extended molecule (scattering region), which includes
some lead atomic layers respectively at each side of the
junction to screen the perturbation effect. The unit
cell of the extended molecule comprises the chains,
which is made up of 4 silicon atoms, and 13 (100)oriented Au atomic lays with a (3 × 3) super cell. The
left and the right leads are considered to be perfect
crystals, and the potential is well approximated by
that of a perfect bulk electrode.
The electron transport through the scattering region has been calculated with the ab initio transport
* Support by the National Natural Science Foundation of China under Grant Nos 11174214 and 11204192, and the Research
Project of Education Department in Sichuan Province under Grant No 13ZB0207.
** Corresponding author. Email: [email protected]; [email protected].
© 2013 Chinese Physical Society and IOP Publishing Ltd
067302-1
CHIN. PHYS. LETT. Vol. 30, No. 6 (2013) 067302
-
-
-
-
Right
lead
Fig. 1. The two-probe geometry of the Si chains attached
to Au (100) leads.
The conductance 𝐺 associated to the two-probe
device can be calculated by using Fisher–Lee’s
relation[16]
𝐺=
2𝑒2
R
Tr[ΓL 𝐺R+
M ΓR 𝐺M ],
ℎ
(1)
where the ΓL/R is the anti-hermitian part of the self
energy, 𝐺R
M , which contains all the information about
the electronic structure of the extended molecule attached to the leads, is the retarded Green’s function of the scattering region. It is easy to see that
R
Tr[ΓL 𝐺R+
M ΓR 𝐺M ] is simply the energy-dependent total transmission coefficient of standard scattering theory. The current 𝐼 can be calculated by using the
formula[17]
∫︁
𝑒
R
𝑑𝐸Tr[ΓL 𝐺R+
𝐼=
M ΓR 𝐺M ]
ℎ
· [𝑓 (𝐸 − 𝜇L ) − 𝑓 (𝐸 − 𝜇R )],
(2)
where 𝑓 is the electron distribution function of the
two leads, and 𝜇L/R is the chemical potential for the
left/right lead. Note that now the transmission coefficient depends both on the energy and bias. More
calculation details on how this procedure is performed
in SMEAGOL can be found in the literature.[18]
In this work, we apply the Perdew–Zunger[19] form
of the local-density approximation to the exchangecorrelation functional. Valence electrons are expanded in single-zeta basis sets for Au atoms and
the double-zeta basis sets for Si atoms. Scalarrelativistic Troullier–Martins[20] pseudo potential in
nonlocal form is generated. A periodic boundary condition is applied in the basal plane (orthogonal to the
transport direction) with four irreducible 𝑘-points in
the two-dimensional Brillouin zone. A 𝑘-grid sampling
of 2 × 2 × 100 for the gold electrodes is employed. The
cut-off energy and iterated convergence criterion for
total energy are set to 200 Rydberg and 10−4 , respectively. Furthermore, the charge density is integrated
over 50 energy points along the semi-circle, 20 energy
points along the line in the complex plane and 20 poles
are used for the Fermi distribution.
2.5
-3
Conductance
D
2.0
-4
1.5
1.0
-5
0.5
0.0 17
(eV)
-
D
Au Si Si Si Si Au
0)
Left
lead
In order to calculate the equilibrium structure under different distances, we perform geometry relaxation by keeping all gold atoms in the leads fixed and
relaxing the apexes of the point contact in the center
until the force on each atom is smaller than 0.1 eV/Å
in the optimization.[21] The ground state energy is
therefore calculated as a function of the distance 𝑑z
between the outer slices, i.e., we simulate a junction
breaking process.
During the simulations of the junction breaking
process, we calculate the junctions’ conductance (circles and left-hand side axis) as a function of distance
𝑑z (shown in Fig. 2). we find that the conductance
varies with 𝑑z . This demonstrates the sensitivity of
the conductance to the local atomic re-arrangement of
the contact region.[22] For more details, as the junctions are stretched, the conductance changes from
2.28𝐺0 at 𝑑z ∼16.898 Å to 0.75𝐺0 at 𝑑z ∼19.698 Å,
then it increases a little to 1.03𝐺0 at 𝑑z ∼20.498 Å,
and jumps abruptly to 0.08𝐺0 at 𝑑z ∼20.898 Å, where
the Si–Si bonds breaks point. On the whole, the
conductance decreases with the stretched junctions in
nano-contacts.
Conductance (
code SMEAGOL,[14] which calculates the density matrix and the transmission coefficients of a two-probe
device using the non-equilibrium Green’s function formalism. The scattering potential is calculated selfconsistently by using the SIESTA[15] implementation
of density functional theory.
18
19
(
A)
20
21-6
Fig. 2. Conductance (circles and left-hand side axis) and
the cohesion energy (squares and right-hand side axis) as
a function of distance 𝑑z .
In order to obtain the conductance of the junctions
at the most stable structure in different distances, we
calculate the cohesion energy as a function of 𝑑z during the simulation process. The cohesion energy is
defined as follows: 𝐸 = 𝐸Au leads+Si chain − 𝐸Si chain −
𝐸Au leads . The cohesion energy 𝐸 as a function of 𝑑z is
shown in Fig. 2 (squares and right-hand side axis). In
the range of 16.898 Å to 20.898 Å, there is a minimum
value of the cohesion energy. The minimum is located
at the equilibrium distances, 𝑑z,eq = 18.098 Å. The
distances 𝑑z,eq corresponding to those minima describe
the optimal position, where the system will naturally
form if the leads are free to relax. When the junctions
are in the optimal equilibrium positions, the Au-Si
bond length 𝑑Si−Au is 2.28 Å, and 𝑑Si−Si = 2.14 Å. It
is consistent with the results in Ref. [23]. The equilibrium conductance of Si chains at the optimal posi-
067302-2
CHIN. PHYS. LETT. Vol. 30, No. 6 (2013) 067302
-1
0
1
2
3
(eV)
Fig. 3. Transmission coefficient as a function of energy at
the equilibrium position.
0.3
2.0
4
1.6
1.2
PDOS (arb. units)
Si-
0.1
-1
0
(eV)
1
2
0.4
0.8
1.2-8
Fig. 5. The conductance (squares and left-hand side axis)
and the current (circles and right-hand side axis) as a function of the bias.
0.2
-2
0.0
(V)
Si-
0.0
-3
-4
0.8
-1.2 -0.8 -0.4
Si-
0
A)
8
-5
-2
2.4
(10
0V
0.4 V
0.8 V
1.2 V
0)
3.0
2.5
2.0
1.5
1.0
0.5
0.0-3
dow and therefore the conductance becomes larger.
When the external bias 𝑉 > 0.4 V, the HOMO drifts
away from the Fermi energy, thus the conductance becomes smaller. The conductance as a function of bias
is shown in Fig. 5 (squares and left-hand side axis).
With the increase of the positive or negative voltage,
it increases firstly, and when |𝑉 | > 0.4 V, the conductance then becomes smaller, which is consistent with
the transmission coefficient as a function of the bias.
The results of the relationship between current and
bias are also shown in Fig. 5 (circles and right-hand
side axis). When |𝑉 | > 0.4 V, the conductance decreases, however the current is still increasing. The
reason is that with the increase of the voltage, the
bias window increases, and the scope of the energy
integral is expanded. In short, although the conductance varies with the voltage, the 𝐼–𝑉 curves display
a linear relationship.
Conductance (
Transmission
tion is 1.71𝐺0 , which indicates that the junctions have
good conductivity.
In order to understand the transport properties
of the junctions in the optimal position, we can analyze the transmission spectrum 𝑇 (𝐸, 𝑉 = 0) (shown
in Fig. 3) and the projected density of states (PDOS)
(shown in Fig. 4) of the Si chains at the optimal distance. 𝑇 (𝐸, 𝑉 = 0) is dominated by a resonance
corresponding to the energy of the lowest unoccupied
molecular orbital (LUMO). It provides a large conductance thus the junctions have large transmission.
Such a transport channel is almost formed by the 𝑝x
and 𝑝y orbital electrons of Si atoms.
3
Fig. 4. The PDOS for Si chains at the equilibrium position.
We now look at the dependence of 𝑇 (𝐸, 𝑉 ) on bias.
Because the two electrodes maintain different chemical potentials due to the external bias, the devices are
in the non-equilibrium states. In order to investigate
non-equilibrium properties of this two-probe system,
we calculate the transmission of the junctions for biases in the range from −1.2 to 1.2 V. When the bias is
0 V, 0.4 V, 0.8 V, and 1.2 V, respectively (as shown in
Fig. 3), we observe a significant drift of the LUMO resonance to lower energies as the bias increases. Such
a drift moves the LUMO near to the Au Fermi energy. The transmission peak moves into the bias win-
In summary, we have examined the transport properties of a silicon chain attached to Au (100) leads
using the density functional theory with the nonequilibrium Green’s function method. We simulate
the silicon atomic chain nanoscale junction breaking
process and calculate the corresponding cohesion energy at different distances. The equilibrium conductance of the Si atomic chain at the optimal position
is 1.71𝐺0 . It shows large conductance and has good
transport properties. The transport channel is mainly
formed by the 𝑝x and 𝑝y electron orbitals of Si atoms.
With the increase of the bias voltage, the conductance
of the silicon chain first increases, then decreases. We
investigate the 𝐼–𝑉 characteristics of junctions at the
optimal position, which shows a linear 𝐼–𝑉 relationship. We hope that these results are useful in designing silicon nanoscale devices and engineering contacts.
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