Negative differential resistance of carbon nanotube electrodes with asymmetric
coupling phenomena
Woo Youn Kim,
1
S. K. Kwon,
2
and Kwang S. Kim
1,2,
*
1
Center for Superfunctional Materials, Department of Chemistry, Pohang University of Science and Technology,
Pohang 790-784, Korea
2
Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea
共Received 12 March 2007; revised manuscript received 7 May 2007; published 26 July 2007
兲
An intricate problem in molecular electronics is to control the molecule-electrodes contacts. Asymmetric
couplings between both contacts are important in driving novel nonlinear transport characteristics like negative
differential resistance 共NDR兲. We find that in the presence of an applied field, metallic carbon nanotubes
共CNTs兲 can form asymmetric couplings even if symmetric structures are employed. This origin is due to the
CNT itself, while the NDR phenomenon can be obtained by tuning the threshold voltage for the asymmetric
couplings by a proper choice of a molecule.
DOI: 10.1103/PhysRevB.76.033415 PACS number共s兲: 73.63.⫺b, 61.46.⫺w, 71.15.⫺m, 73.40.⫺c
Molecular electronics can play an important role in real-
izing electronic devices to utilize quantum interference phe-
nomena at the molecular level. One of the main problems in
current studies of molecular electronics is to find good elec-
trode materials as a part of the device without serious contact
problems at the molecular junctions.
1–3
Common metallic
electrodes might not be suitable at molecular scales because,
as an example, typical metal-molecule junctions like gold-
thiol linkage 共GTL兲 show serious reproducibility problems
which have been obstacles to overcome.
4–6
To obtain asym-
metric couplings between both contacts, which characterize
the transport phenomena, both molecule-electrode junctions
for GTL have to use different linkages or different electrode
materials, which is not practical. We note that a carbon nano-
tube 共CNT兲 can be a good electrode material to resolve such
difficulties. The CNT has the following advantages: 共i兲 it is
easy to form a robust and reproducible covalent bond with
organic molecules through well-established chemistry,
7
共ii兲 it
is possible to utilize metallic properties of either thin single-
walled or multiwalled CNTs,
8
and 共iii兲 CNTs have quasi-one-
dimensional structures useful to integrate many individual
devices.
9
Based on these facts, we have examined the intrin-
sic characteristics of a CNT as an electrode material by using
first principles methods.
Apart from strong covalent bonding with device mol-
ecules, we find that the quantum interference at the interface
inherently induces asymmetric couplings in symmetrical mo-
lecular devices linked to CNT electrodes. Metallic armchair
CNTs have two crossing
-character bands at the Fermi
level
8
共E
f
兲 and complex band structures of different charac-
ters around E
f
. These band structures can be exploited for the
design of molecular devices having unique transport phe-
nomena, which would not be observed in GTL junctions be-
cause gold has a relatively simple s-character band structure
in a wide energy range around E
f
.
10
The negative differential
resistance 共NDR兲 phenomenon in the molecular electronic
devices involving metallic electrodes has not been clearly
understood. Here, we first show that even with a symmetric
device structure, asymmetric potential drops are induced
through a molecule bridging the metallic CNTs, and this
drives NDR.
Figure 1 shows a device system composed of two CNTs
with chiral vector 共5,5兲共CNT55兲 and a molecule. Amide
endgroups were used as linkages between a molecule and
CNT55. We consider two different molecules, phenyl-
ethynyl 共PE兲 oligomers and pyrrollo pyrrole 共PP兲. In the cal-
culations, the end of each CNT is capped by H atoms and the
geometry including the H and end carbon atoms 共region A in
Fig. 1兲 was fully relaxed by using
Gaussian03 suite of pro-
grams with 3–21G basis set
11
in density functional theory
共DFT兲. The remaining part of each CNT was fixed at the
C–C bond length of graphite 共1.421 Å兲. The DFT calcula-
tions were carried out with Becke’s three parameter hybrid
functional, which employs the Lee-Yang-Parr correlation
functional. The transmission calculations were performed us-
ing nonequilibrium Green’s-function-based program
packages
12,13
with the single-zeta polarization basis set and
the local density approximation. The current 共I兲-voltage 共V兲
characteristics were obtained with the Landauer formalism
by integrating the transmission coefficient within the energy
共E兲 region restricted by the bias voltage:
I共V兲 =
冕
−⬁
⬁
T共E,V兲关f共E −
L
兲 − f共E −
R
兲兴dE,
where f represents the Fermi function,
L,R
are chemical
potentials of the left and right leads, and T共E ,V兲 is the bias-
dependent transmission coefficient.
FIG. 1. Geometry of the CNT55-molecule–CNT55 system with
amide linkage. PE and PP are the molecules used in calculations.
PHYSICAL REVIEW B 76, 033415 共2007兲
1098-0121/2007/76共3兲/033415共4兲 ©2007 The American Physical Society033415-1
Figure 2共a兲 presents the contour plots of induced potential
of the CNT55-PE-CNT55 system at various bias voltages
共V
b
兲. It is naturally expected that symmetric and asymmetric
junctions would induce symmetric and asymmetric potential
drops through the junction, respectively.
14
However, note
that even though our system was designed symmetrically in
geometric structures, its potential drop in the junctions is
asymmetric for V
b
=2.4 V, while they are almost symmetric
at lower bias. To understand this phenomenon, we draw a
diagram of the potential drop across the junction in Fig. 2共b兲.
It schematically shows how the induced charge in the mo-
lecular region determines the potential drop in the junction.
Going into detail, in Fig. 2共c兲 we analyze the change of real
charge distribution in the molecular region due to the finite
bias voltage in comparison with the zero bias case. If an
external potential is applied to an isolated molecule, the
charge in the molecule would be redistributed to screen the
potential. At low bias, the induced charges are so small that
the molecule retains electronic states almost similar to those
of the isolated molecule and the total sum of the induced
charges is almost zero. Hence, the potential drop is symmet-
ric as the case of zero electron loss “
␦
=0” in Fig. 2共b兲.
However, above the threshold voltage 共1.2 V兲, the screening
is not fully effective, leading to nonzero value of the total
induced charge in the molecule 关inset of Fig. 2共c兲兴. Such a
significant electron loss induces a local Coulomb potential
with positive charge by electron loss 共
␦
⬎0兲 in the molecular
region, resulting in asymmetric potential drop at each junc-
tion.
When an occupied molecular energy level enters into the
current window, the total charge in the molecular region de-
creases, as the discharging rate on the left side junction is
larger than the recharging rate on the right side junction. The
coupling strength in the junction is responsible for the charg-
ing rate. Therefore, the observed asymmetric potential drop
is attributed to the asymmetric coupling strength between the
molecule and the left and right electrodes during the trans-
port process.
In Fig. 3共a兲, we present the transmission curves together
with the density of states 共DOS兲 of both electrodes in the
CNT55-PE-CNT55 system. The DOS at zero bias was cal-
culated for a single CNT55 without other device parts. Those
at nonzero bias voltage V
b
are obtained by simply shifting it
up and down by ±eV
b
/2. Finally, we set up the diagrams by
aligning E
f
of the calculated transmission peaks with that of
the single CNT55. The peaks of DOSs of the electrodes are
on the same positions at zero bias on both sides. However,
the left and right chemical potentials split off by as much as
共
L
−
R
兲=eV
b
at finite bias such that the transmission peaks
are aligned with the different positions of the DOS on each
side 共i.e., different band characters or electronic states兲.
Thus, rapidly changing band characters and shapes of DOSs
of the electrodes naturally affect the coupling between mo-
lecular states and electronic states of the electrodes under a
finite bias voltage.
15–17
The change in coupling triggers drastic effects on the
transmission. First, the heights of the transmission peaks
change. In Fig. 3共a兲, the upper peak marked by an arrow at
V
b
=0 V, which corresponds to the lowest unoccupied mo-
lecular orbital 共LUMO兲 energy levels, almost diminishes at
V
b
=2.4 V, because there are few available states at the right
side electrode. In contrast, the peak corresponding to the
highest occupied molecular orbital 共HOMO兲 energy level in-
creases because of the increased available states at the right
side electrode. Second, the energies of the peaks shift as
shown in Fig. 3共b兲. At low bias, all peaks remain at the same
positions of the zero bias case. However, they begin to shift
from the threshold voltage 1.2 V, at which the peak corre-
sponding to the HOMO enters into the current window.
These changes are consistent with the change of charge in
the molecular region in the inset of Fig. 2共c兲. This reflects
that the shifts of the transmission peaks are closely related to
the discharging effect and asymmetric potential drop.
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10
-0.020
-0.015
-0.010
-0.005
0.000
0.6 V
1.5 V
2.4 V
(C)
0.0 0.6 1.2 1.8 2.4
-0.2
-0.1
0.0
ρ
e
V
b
(V)
ρ
e
Atomic indices
C2C1 C3 C4 C5 C6 C7 C8 C9 C10
FIG. 2. 共Color online兲共a兲 Contour plots of induced potential of
the CNT55-PE-CNT55 systems at V
b
=0.6, 1.5, and 2.4 V. Red to
blue color represents high to low potential. 共b兲 A schematic picture
of the potential drop through the junction. The case of zero electron
loss “
␦
=0” presents the symmetric potential drop, corresponding to
those at V
b
=0.6 and 1.5 V in 共a兲 and the case of electron loss “
␦
⬍0” presents the asymmetric potential drop due to discharging ef-
fect, corresponding to that of V
b
=2.4 V. 共c兲 Excess electron 共
e
兲
population on each atom 共C1–C10兲 for the given bias voltage. Inset
is the total excess electron population in the molecular part includ-
ing the linkages, for the given V
b
, with respect to the unbiased case.
BRIEF REPORTS PHYSICAL REVIEW B 76, 033415 共2007兲
033415-2
It is worthwhile to compare the result with that of GTL
using the same PE molecule.
14
In GTL, the energies of the
transmission peaks do not shift due to the symmetric poten-
tial drop when the same linkages are used. The reason is that
molecular states couple symmetrically to both electrode
states even at high bias voltages because bulk Au has a uni-
form DOS in a relatively wide range of energy. In contrast,
when the different linkages are used, the asymmetric poten-
tial drop is induced purely due to the asymmetric coupling of
the different linkages, and hence, the energies of the trans-
mission peaks shift. Additionally, the heights of the transmis-
sion peaks for the GTL system do not change regardless of
used linkages. This difference between CNT and GTL sys-
tems results in different behaviors in device applications be-
cause the change in transmission peaks drastically affects I-V
characteristics.
To tune the threshold voltage for the asymmetric cou-
pling, we applied the same analysis for the other molecule
共PP兲 having smaller HOMO-LUMO gap. Indeed, asymmet-
ric potential drops for PP are derived at a lower bias voltage,
as compared with that in PE.
18
The importance of the asymmetric potential drop is
clearly represented in I-V characteristics, which is the main
interest in device applications. Figure 4 shows the I-V curve
for the PE/PP system. At low bias, the current follows linear
behavior and is on the order of 1–10 nA for the PP molecule
as shown in the inset of Fig. 4. This behavior is similar to
experimental measurements.
3
At high bias, which was not
demonstrated in the experiment, however, the nonlinear ef-
fects due to the bias-dependent transmission become signifi-
cant. For example, in the PE molecule 关Fig. 3共b兲兴, the energy
level of the transmission peak corresponding to the HOMO
level, E
homo
⬃−1.0 eV, does not shift much for V
b
⬍1.2 V.
However, for V
b
⬎1.2 V, E
homo
touches the left chemical po-
tential 共
L
兲, resulting in the onset of current flow to induce
the discharging effect. This drives E
homo
to shift along with
L
at slightly lower energy. Such a shift of E
homo
suppresses
the current flow till V
b
=1.8 V, at which the LUMO begins to
enter into the current window. More interestingly, in the PP
molecule 关Fig. 3共c兲兴, E
homo
readily shifts upon the applied
voltage because E
homo
is close to E
f
. In addition, E
homo
shows
a large peak on the edge of
L
for V
b
=1.2 and 2.1 V 共due to
the strong couplings between the molecule and electrodes兲,
highly contributing to the current flow, whereas it shows a
small peak between them 共due to the weak coupling between
the molecule and electrodes兲, contributing less to the current.
The current increment from V
b
=1.5 to 2.1 V mainly comes
from the LUMO contribution. As a consequence, the PP mol-
ecule exhibits the NDR phenomenon at V
b
=1.2–1.5 and
2.1–2.4 V.
NDR has been observed from various molecular
devices.
2,15–17,19,20
The origin of the NDR observed in mo-
lecular devices with semiconducting electrodes is due to
band gaps of the electrodes. However, metallic CNTs have
no band gaps. Here, the NDR can be explained by consider-
ing two effects arising from the band structures: 共i兲 how
many states are available to couple with molecular states and
共ii兲 how well each of their state couples with the molecular
states.
17
At low bias, molecular energy levels are aligned
with the electronic states of electrodes, having equivalent
band characters on both sides. Therefore the potential drop
through the molecule is symmetric. As the bias increases
further, the energy levels are aligned with the electronic
2.4 V
µ
R
µ
L
-2
-1
0
1
2
0.0 V
E-E
f
(eV)
0.0
0.3
0.6
0.9
0.3
0.9
1.5
2.1
2.7
-1 0 1
T(E,V)
V
b
(V)
µ
µ
R
µ
µ
L
E-E
f
(eV)
0.0
0.3
0.6
0.9
0.3
0.9
1.5
2.1
2.7
-1 0 1
E-E
f
(eV)
µ
µ
R
µ
µ
L
V
b
(V)
T(E,V)
(
c) PP
(
b) PE
(a)
FIG. 3. 共a兲 Bias-dependent transmissions and DOSs of both
electrodes in the CNT55-PE-CNT55 system. A left/middle/right in-
set of each figure denotes DOS of the left electrode/transmission
curves of the device system/DOS of the right electrode. Bias-
dependent transmission curves for 共b兲 PE and 共c兲 PP.
0.0 0.6 1.2 1.8 2.4
0
1
2
-400 0 400
-10
-5
0
5
10
V
b
(mV)
I(V) (nA)
I(V) (µA)
V
b
(V)
PE
PP
FIG. 4. I-V curves at various bias voltages for each molecule of
CNT55-molecule-CNT55. Inset: I-V curves at low bias for PP
molecule.
BRIEF REPORTS PHYSICAL REVIEW B 76, 033415 共2007兲
033415-3
states of different band characters on each side, resulting in
asymmetric couplings. This makes the energies of the trans-
mission peaks shift according to the induced potential drop
关Figs. 3共b兲 and 3共c兲兴. Further, as the coupling at one side
becomes weak at a certain bias voltage, the corresponding
transmission peaks become smaller and the current is re-
duced. Thus, the bias-dependent transmission leads to an in-
triguing feature of I-V characteristics like the NDR phenom-
enon.
In summary, apart from the fact that CNTs are promising
electrodes with minimal contact problems in molecular elec-
tronics due to the strong covalent bonding with a molecule,
we find that CNTs generate asymmetric couplings with a
molecule beyond a certain threshold bias voltage, which
would not be observable in GTL junctions. These asymmetri-
cal couplings give rise to a bias-dependent transmission, and
hence, NDR as the threshold voltage for the asymmetric cou-
pling can be tuned by a proper choice of a molecule. Thus,
the CNT-molecule-CNT devices could be useful and robust
molecular electronic devices in the near future.
This work was supported by Global Research Laboratory
Project 共KOSEF兲 and BK21.
*
kim@postech.ac.kr
1
G. S. Tulevski et al., Science 309, 591 共2005兲; M. Siaj and P. H.
McBreen, ibid. 309, 588 共2005兲.
2
N. P. Guisinger et al., Nano Lett. 4,55共2004兲; P. G. Piva et al.,
Nature 共London兲 435, 658 共2005兲.
3
X. F. Guo et al., Science 311, 356 共2006兲; C. Cuniberti, G. Fagas,
and K. Richter, Chem. Phys. 281, 465 共2002兲.
4
G. K. Ramachandran et al., Science 300, 1413 共2003兲;A.H.
Flood et al., ibid. 306, 2055 共2004兲; P. Tarakeshwar et al.,J.
Phys. Chem. B 110, 7456 共2006兲; Y. H. Kim, S. S. Jang, Y. H.
Jang, and William A. Goddard III, Phys. Rev. Lett. 94, 156801
共2005兲.
5
J. J. Palacios, A. J. Perez-Jimenez, E. Louis, and J. A. Verges,
Phys. Rev. B 64, 115411 共2001兲; Y. B. Hu, Y. Zhu, H. J. Gao,
and H. Guo, Phys. Rev. Lett. 95, 156803 共2005兲; H. Basch et
al., Nano Lett. 5, 1668 共2005兲; D. Cheng, W. Y. Kim, S. K. Min,
T. Nautiyal, and K. S. Kim, Phys. Rev. Lett. 96, 096104 共2006兲.
6
L. Venkataraman et al., Nature 共London兲 442, 904 共2006兲.
7
S. Niyogi et al., Acc. Chem. Res. 35, 1105 共2002兲.
8
R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Prop-
erties of Carbon Nanotubes 共Imperial College Press, London,
1998兲;B.H.Honget al., Proc. Natl. Acad. Sci. U.S.A. 102,
14155 共2005兲.
9
H. J. Dai, Acc. Chem. Res. 35, 1035 共2002兲.
10
T. Nautiyal, S. J. Youn, and K. S. Kim, Phys. Rev. B 68, 033407
共2003兲.
11
M. Frisch et al., GAUSSIAN 03 suite of programs.
12
J. M. Soler et al., J. Phys.: Condens. Matter 14, 2745 共2002兲.
13
M. Brandbyge, J. L. Mozos, P. Ordejon, J. Taylor, and K. Stok-
bro, Phys. Rev. B 65, 165401 共2002兲; J. Taylor, H. Guo, and J.
Wang, ibid. 63, 245407 共2001兲; W. Y. Kim and K. S. Kim 共un-
published兲.
14
J. Taylor, M. Brandbyge, and K. Stokbro, Phys. Rev. Lett. 89,
138301 共2002兲.
15
W. Lu, V. Meunier, and J. Bernholc, Phys. Rev. Lett. 95, 206805
共2005兲.
16
H. Mehrez, A. Wlasenko, B. Larade, J. Taylor, P. Grutter, and H.
Guo, Phys. Rev. B 65, 195419 共2002兲.
17
B. Larade, J. Taylor, H. Mehrez, and H. Guo, Phys. Rev. B 64,
075420 共2001兲.
18
The asymmetric coupling depends on the shape of the density of
states and molecular energy gaps. The threshold voltage can be
tuned to trigger the asymmetric coupling at relatively low bias,
where the current theoretical method is clearly valid.
19
J. Chen et al., Science 286, 1550 共1999兲.
20
Y. Xue, S. Datta, S. Hong, R. Reifenberger, J. I. Henderson, and
C. P. Kubiak, Phys. Rev. B 59, R7852 共1999兲.
BRIEF REPORTS PHYSICAL REVIEW B 76, 033415 共2007兲
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