1. science
2. physics

IAU PUBLISHING
JOURNAL OF THEORETICAL PHYSICS
♠ J. Theor. Phys. 2 (2013) 1-9
www.jtpc.ir
Electronic transport through a molecular junction:
Effects of dephasing, quantum interference and gate
voltage
A. Ahmadi Fouladi and Javad Vahedi
Department of Physics, Sari Branch, Islamic Azad University, Sari, Iran
E-mail: a.ahmadifouladi@iausari.ac.ir (Corresponding author) and
javahedi@iausari.ac.ir
Abstract. The role of dephasing reservoirs, quantum interference and gate voltage
on the electron transport through the single phenalenyl molecule sandwiched between
metal electrodes as a M/phenalenyl/M junction is numerically investigated. Our
calculations are performed based on a tight-binding model and a generalized Green’s
function method in the well-known Landauer-B¨
uttiker formalism, and the dephasing
effect is modeled within the B¨
uttiker-probe approach. In this study the quantum
interference effect of electron waves is related to the geometry that the device adopt
within the junction. Our results illustrate that the presence of dephasing reservoirs,
quantum interference effect and gate voltage provide a remarkable change in the
transport properties of a M/phenalenyl/M junction and the mentioned parameters
can be considered to designing molecular electronic devices.
PACS numbers: 73.23.-b
Keywords: molecular junction, Green’s function, dephasing, quantum Interference
effect, phenalenyl molecule
c 2013 IAU Publishing. All rights reserved.
°
ISSN 2251-855X (Print), 2251-9998 (Online)
A. Ahmadi Fouladi, J. Vahedi
Electronic transport through a molecular junction...2
1. Introduction
Organic molecules with low fabrication costs and high manufacturability have been
attracted investigations in the molecular electronics [1, 2, 3, 4, 5]. The concept of
electron transport through molecules was first studied in the theoretical work in 1974
[6]. Since then plenty experiments [7, 8, 9, 10, 11] have been carried out through
molecules sandwiched between two electrodes. Following experimental progresses,
theory can play key role in perception the mechanisms of electron transport through the
molecules. In coherent regime, the electron transport is rather properly explained within
the framework of the Landauer-B¨
uttiker formalism [12], which generally the inelastic
processes are neglected. The electron-phonon (e-ph) interaction is the main source of
phase-breaking in the transport through the molecular junctions. A theoretical analysis
of the e-ph interaction in mesoscopic systems is performed based on a combination
of density functional theory (DFT) and a non-equilibrium Green’s function (NEGF)
method [13, 14]. This computational procedure even for the short length molecular
(a)
(b)
(c)
(d)
Figure 1. (color online) A schematic M/phenalenyl/M junction of the different
quantum interference pattern which are labeled as (a), (b) and (c). (d) Each carbon
site of phenalenyl molecule is assumed to be coupled to a fictitious electronic reservoir,
which takes into account the effects of dephasing.
A. Ahmadi Fouladi, J. Vahedi
Electronic transport through a molecular junction...3
junctions is very time consuming. A well-known method to consider dephasing in a
mesoscopic device was originally proposed by B¨
uttiker over two decades ago [15, 16].
The basic idea consists in adding some virtual electronic reservoirs into the coherent
system which in turn causes a phase-breaking process [15, 17, 18]. When electrons move
from the left electrode to the right one get scattered into reservoirs and then re-injected
into the device with a random phase. One of the important factors to study electron
transport through the mesoscopic junctions is to control of the quantum interference
effect of electron waves related to the geometry that the device adopt within the junction.
In this study the variation of interference conditions is determined by replacement three
different configurations of phenalenyl molecule connected to the metal electrodes (Fig.1
(a, b and c)). In this paper we consider the effects of dephasing, quantum interference
and gate voltage on the electron transport of phenalenyl molecule sandwiched between
metal electrodes as a M/phenalenyl/M junction. Since electron transport through
the molecular junction is significantly affected by the gate voltage, we also study
this effect on the transport properties of M/phenalenyl/M junction. Using Green’s
function method in the framework of Landauer-B¨
uttiker formalism, we investigate
the transmission function and current-voltage characteristics of the M/phenalenyl/M
junction. The model and description of the computational methods for investigating the
electron transport properties of the model junction (Fig. 1) are introduced in section
2. The results and discussion are presented in section 3, followed by the conclusion in
section 4.
2. Computational scheme
A generalized Hamiltonian for the M/phenalenyl/M junction in the absence of dephasing
reservoirs may be expressed as
H = HL + Hp + HR + HC ,
(1)
where HL(R) represents the Hamiltonian of the left (right) metal electrode which is
described within the single-band tight binding approximation and is written as
Hβ =
X
{ε0 c†iβ ciβ − t(c†iβ ciβ +1 + c†iβ +1 ciβ )}.
(2)
iβ
Here c†iβ (ciβ ) denotes the creation (annihilation) operator of an electron at site i in the
electrode β(= L or R). ε0 and t are the on-site energy and the nearest neighbor hopping
integral, respectively. In the absence of metal electrodes, the Hamiltonian of phenalenyl
molecule may be as follows
Hp =
X
{εn d†n dn − tn,n+1 (d†n+1 dn + H.c.)}.
(3)
n
The index n runs over the π-orbitals of the carbon atom along the chain of phenalenyl
molecule. The operator d†n (dn ) creates (annihilates) an electron at site n. εn is the
on-site energy of a carbon atom. The nearest neighbor electron hopping integrals are
given as tn,n+1 = t0 , where t0 is the hopping integral of the π-electrons. Finally in Eq.
A. Ahmadi Fouladi, J. Vahedi
Electronic transport through a molecular junction...4
(1), HC denotes the coupling between the phenalenyl molecule and metal electrodes and
takes the form
HC =
XX
n
tc(n,i) (c†i di + H.c.),
(4)
i
where the matrix elements tc(n,i) represent the coupling strength between phenalenyl
molecule and metal electrodes are taken to be tc . In order to introduce the
effects of dephasing, some fictitious electronic reservoirs are connected to carbon
sites of phenalenyl molecule as shown in Fig. 1(d). The Green’s function of the
M/phenalenyl/M system in the presence of dephasing reservoirs can be written as
G(E) = [E1 − Hp − ΣL − ΣR −
X
Σj ]−1 ,
(5)
j
where ΣL(R) and Σj are the contact self-energy resulting from the coupling of phenalenyl
molecule to the left (right) electrode and fictitious electron reservoirs. Using the Dyson
equation, the self-energy may be carried out as follows [19]
τβ 2
Σβ =
.
(6)
E − εβ − φβ
q
Here, φβ = (E − εβ )/2 − i tβ 2 − (E − εβ )2 /4; and β = L, R, 1, 2, ...N. For left and
right electrodes, β = L(R), τβ = tc , εβ = ε0 and tβ = t; while considering the dephasing
probes, β = j (j = 1, 2, ...N ), τβ = γ, εβ = εd , and tβ = td . Within the D’AmatoPastawski model [20] the total effective transmission probability of an electron from the
left electrode to the right one in the presence of phase-breaking is given by
Tef f (E) = TL,R +
N
X
−1
TR,k Wk,m
Tm,L .
(7)
k,m=1
The right-hand side of the above expression contains two contributions: The first term
represents electrons that propagate coherently through the system, whereas the second
term contains the incoherent contributions owing to electron suffering the dephasing
processes. W −1 is the inverse of the matrix W defined by the following relation
Wk,m = [(1 − Rk,k )δk,m − Tk,m (1 − δk,m )].
(8)
where,
Rk,k = 1 −
X
Tk,m .
(9)
k6=m
Tl,m is the transmission probability between reservoirs and given as
Tk,m (E) = T r(Γk Gr Γm Ga ).
(10)
The broadening matrix Γk,m is defined as the imaginary part of the self-energy
Γk,m = −2Im(Σk,m ).
(11)
Transmission function tells us the rate at which electrons transmit from the left to the
right electrode by propagating through the molecule. Relying on Landauer-B¨
uttiker
formula, we evaluate the current as a function of the applied bias voltage [21]
2e Z +∞
Tef f (E)[fL − fR ]dE,
(12)
I(Va ) =
h −∞
A. Ahmadi Fouladi, J. Vahedi
Electronic transport through a molecular junction...5
0
-2
-4
(a)
-6
Log (T(E))
-5
-10
-15
(b)
-20
-25
-4
-8
(c)
-12
-6
-4
-2
0
2
4
6
Energy (eV)
Figure 2. (color online.) Logarithmic scale of transmission probability as a function
of the injecting electron energy E for the (a), (b) and (c) cases in the absence (solid
line) and in the presence of the dephasing reservoirs (dash dotted line).
where fL(R) = f (E − µL(R) ) is the Fermi distribution function on the left (right)
electrode with chemical potential µL(R) = EF ± eV2a and Fermi energy EF . For
the sake of simplicity, here we assume that the total voltage is dropped across
the molecule/electrode interface and this assumption does not eminently affect the
qualitative aspects of current-voltage characteristics. In fact the electric field inside
the molecule, particularly for short molecules, seems to have a negligible effect on the
I-V characteristics. On the contrary, for longer molecules and higher bias voltages, the
electric field inside the molecule may play a more remarkable role depending on the
structure of the molecule [22], but yet the effect is very small.
3. Results and discussion
Here, we represent the results of the numerical calculations based on the formalism
described in section 2. For a typical phenalenyl molecule, we choose the carbon on-site
energy equal to zero, i.e., εn = 0. Furthermore, the value of the hopping integrals takes
as t0 = 2.5 eV . The tight-binding parameters for metal electrodes chosen to be ε0 = 0
and t = 4 eV . As a reference energy, the Fermi energy of the metal electrodes is set
EF = 0. In addition, we set tc = 0.8 eV , γ = 0.5 eV , td = 4 eV , εd = 0 and T = 4 K.
In Fig.2, we illustrate the logarithmic scale of transmission function versus energy
A. Ahmadi Fouladi, J. Vahedi
Electronic transport through a molecular junction...6
40
(a)
20
0
-20
-40
Current (
A)
20
(b)
10
0
-10
-20
20
(c)
10
0
-10
-20
-6
-3
0
3
6
Voltage (V)
Figure 3. (color online.) The current-voltage characteristics of the (a), (b) and (c)
cases in the absence (solid line) and in the presence of the dephasing reservoirs (dash
dotted line).
of the M/phenalenyl/M junction for a, b and c cases. The solid curves represent the
alteration of logarithmic scale of coherent transmission function, while the dash dotted
curves describe the result in the presence of dephasing. As shown in Fig. 2 (a), in
the absence of dephasing, the probability of transmission function reaches its saturated
value (resonant peaks) for the specific energy values (see solid line). These resonance
peaks are related to the eigenenergies of the individual phenalenyl molecule. In the
presence of dephasing reservoirs, the magnitude of the resonant peaks decreased (dash
dotted line) due to the enhancement of rate of scattering in the virtual reservoirs,
compared to the coherent transport (solid line). Also, the widths of the resonance
peaks in the transmission function become broadened. This broadening is related to the
dominance of the phase-relaxation effects over backscattering. A considerable change in
the transmission function is observed when the phenalenyl molecule is connected to the
left and right electrodes as shown in Fig. 1 (b) and (c). In the absence of dephasing,
the amplitude of some resonant peaks decreases compared to the other resonant peaks.
When the electrons travel from left electrode to the right one through the phenalenyl
molecule, the electron waves propagating along the two branches of phenalenyl molecule
may suffer a relative phase shift between themselves. Consequently, there might
be constructive or destructive interference due to the superposition of the electronic
wave functions along the various pathways. Therefore, the transmission probability
will change. Also, we observe some anti-resonant states appear in the transmission
probability in the b and c cases. These anti-resonant states are related to the quantum
interference effect. As we see in Fig. 1 (b) and (c), in the presence of dephasing
A. Ahmadi Fouladi, J. Vahedi
Electronic transport through a molecular junction...7
0
-4
-8
(a)
Log (T(E))
-12
-10
(b)
-20
-10
(c)
-20
-6
-4
-2
0
2
4
6
Energy (eV)
Figure 4. (color online.) Logarithmic scale of transmission probability as a function
of the injecting electron energy E for the (a), (b) and (c) cases in the absence (solid
line) and in the presence of the dephasing reservoirs (dash dotted line) in the presence
of gate voltage.
all the anti-resonant states disappear. In order to provide a deep understanding of
the electron transport, we have shown the current-voltage (I − V ) characteristics in
Fig. 3. An applied voltage shifts the chemical potentials of two electrodes relative
to each other by eV , with e the electronic charge. When a molecular level (either
HOMO or LUMO) is positioned within such bias window, current will flow. We can see
I − V curves show staircase-like structure which indicates that a new channel is opened.
In Fig. 2 (a), both in the absence and presence of electron dephasing current shows
nearly identical change though a remarkable change is observed in their transmission
probabilities. This is owing to the fact that the dephasing broadens the widths of
the resonance peaks in the transmission function, while it also decreases the magnitude.
These two effects deactivate each other showing nearly identical spectrum in the currentvoltage characteristics. The effect of dephasing become much striking in the I − V
characteristic for the cases of the (b) and (c). In these configurations, in the presence of
dephasing (dash dotted line) the current amplitude increases remarkably compared to
the coherent transport (solid line). This is due to the fact that the phase-relaxation in
order to significant broadening of the widths of the resonance peaks in the transmission
function reduces destructive interference, and it can enhance transmission function and
thus increase the current amplitude. Fig. 4 shows the logarithmic scale of transmission
A. Ahmadi Fouladi, J. Vahedi
Electronic transport through a molecular junction...8
40
(a)
20
0
-20
Current (
A)
-40
(b)
20
0
-20
(c)
20
0
-20
-40
-6
-4
-2
0
2
4
6
Voltage (V)
Figure 5. (color online.) The current-voltage characteristics of the (a), (b) and (c)
cases in the absence (solid line) and in the presence of the dephasing reservoirs (dash
dotted line) in the presence of gate voltage.
probability as a function of the injecting electron energy E for the (a), (b) and (c)
cases in the absence (solid line )and in the presence of the dephasing reservoirs (dash
dotted line) with local applied gate voltage Vg = 2.5eV . Assuming the gate voltage (Vg )
affects only on one atom of the phenalenyl molecule. The application of the gate voltage
produces small changes in the energy eigenvalues of the phenalenyl molecule with respect
to the ungated molecule. consequently, we see more resonance and antiresonance states
in transmission functions than the ungated molecule. Therefore one can get on/off state
of a molecular junction by changing the external gate voltages, without modifying the
structure of phenalenyl molecule. This result is very important for the fabrication of
efficient molecular switches or gates.
4. Conclusion
In summarizing, we have investigated electron transport in the M/phenalenyl/M system
in the presence of phase breaking using the B¨
uttiker dephasing model. Using the
generalized Green’s function technique based on tight-binding model and the LandauerB¨
uttiker theory, we have calculated the transmission probability and current in the
presences of dephasing and quantum interference. Our results illustrate that the
presence of dephasing reservoirs and quantum interference effect provide a significant
A. Ahmadi Fouladi, J. Vahedi
Electronic transport through a molecular junction...9
change in the transport properties of a M/phenalenyl/M junction. We have set three
a, b and c configurations in order to check the effect of geometrical quantum interference.
Our calculations show that quantum interference effect can lead to appear anti-resonant
states in the transmission probability for the b and c cases. In the presence of dephasing
these anti-resonant states are removed. Also we have studied the effects of the gate
voltage on the transport properties of the M/phenalenyl/M junction. The application
of the gate voltage produces changes in the energy eigenvalues of the phenalenyl molecule
with respect to the ungated molecule. Therefore one can change state of a molecular
junction by varying the external gate voltages. Consequently, the gate voltage can play
control role in the transport properties of M/phenalenyl/M molecular junction. Our
presented results may be useful in understanding the dephasing effects of the electron
transport through the molecular junctions and designing the future molecular electronic
devices.
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