Communication and Network, 2010, 2, 73-78
doi:10.4236/cn.2010.21012 Published Online February 2010 (http://www.scirp.org/journal/cn)
73
A Model for Cu-Se Resonant Tunneling Diodes
Fabricated by Negative Template Assisted
Electrodeposition Technique
Meeru Chaudhri1, A. Vohra1, S. K. Chakarvarti2
1Department of Electronic Science, Kurukshetra University, Kurukshetra, India
2Department of Applied Physics, National Institute of Technology (Deemed University), Kurukshetra, India
Received November 16, 2009; accepted December 29, 2009
Abstract: In this paper, the authors present and discuss a model for Cu-Se nano resonant tunneling diodes
(RTDs) fabricated by negative template assisted electrodeposition technique and formulate the mathematical
equations for it. The model successfully explains the experimental findings.
Keywords: track-etch membrane, template synthesis, Cu-Se resonant tunneling diodes, electrodeposition
1. Introduction
For nanoelectronics to become a reality one must be able
to fabricate the devices and circuits at nanometer dimen-
sions. For this, the researchers the world over have put in
efforts in three different areas: nanofabrication, quantum
modeling and circuit innovations. Modeling of a device
is an essential part of this effort that provides a test bench
and also forms the basis for simulation tools for the de-
vice. With the help of models, one can also adjust the
structural parameters and keep at bay the undesirable
parameters through device design and optimization while
fabrication. However, the traditional device modeling is
not valid in the nanometer regime [1]. Each of these ar-
eas has their own importance. As the nano dimensioned
materials lead to new phenomenon and also possibly
novel devices based on quantum tunneling mechanisms
[2] a device theory that can properly treat quantum
transport phenomenon is, therefore called for. In our pre-
vious publications [3–6], we have discussed the fabrica-
tion and the characterization of RTDs of various diame-
ters made by utilizing different material systems. In this
paper, we have developed a model for these RTDs.
Equations have been formulated for this model and the
experimental results have been verified with the help of
these equations.
2. Experimental
Cu-Se RTDs have been fabricated by electrodepositing
Cu and Se in the pores of the polycarbonate track-etch
membranes (PC TEMs) [3,4]. (PC TEMs) with pores of
diameters 1 µm, 100 nm and 40 nm were used for this
purpose. The experimental set-up used to fabricate the
Cu-Se RTDs is shown in the Figure 1.
TEM foils with in situ Cu–Se binary structures were
used for obtaining I–V characteristics. However for SEM
characterization, membranes were dissolved in the sol-
vent dichloromethane (CH2Cl2), should be leaving be-
hind the structures. SEM view of Cu-Se RTD of diameter
1 µm in back-scattering mode is shown in the Figure 2.
This mode is used to obtain the contrast image of the
object. In the figure, dark part is indicating Se and bright
part is indicating Cu.
An ohmic contact was made by applying Ag based
paint on the top side of the Se to obtain I-V characteris-
tics. Figure 3 illustrates the schematic cross-section of
the samples in the pores of the membranes with the silver
paste.
Experimental results of I-V characteristics of Cu-Se
binary structures of diameters 2 µm, 1 µm, 100 nm and
40 nm are shown in Figures 4 and 5.
It is clear from the Figures 4 and 5 that a prominent
feature of negative differential resistance region (NDR)
appear as the diameter of the Cu-Se binary structures
reduces from 2µm to 1nm. This NDR increases with fur-
ther reduction in the diameters of the Cu-Se binary
structures. The values of peak to valley current ratios
(PVCRs) of Cu-Se RTDs of different diameters are
shown in Table 1.
3. A Model for Cu-Se RTDs
The structure consists of three different layers-Cu, Se
and Ag. As the quantum size effects in metals are nor-
mally seen at 1 nm [7], density of states (DOS) in Cu and
Ag are expected to be continuous. It, thus behaves as a
metal. Quantum size effects in semiconducting material

M. CHAUDHRI ET AL. 74
Figure 1. Experimental set-up for negative-template as-
sisted electrodeposition of nano-/micro binary structures
Figure 2. SEM micrograph of a template synthesized Cu-Se
binary structure of 1 µm diameter in back scattering mode
Figure 3. Samples with silver paste
become apparent when the size of the semiconducting
material is of the order of hundreds of nanometers [8].
Thus, Se material has quantized bands as shown in Fig-
ure 6 (a), with infinite potential on the both sides of it i.e.
Se semiconductor at small dimensions, forms a quantum
well similar to the one fabricated by exploiting the en-
ergy band discontinuities of semiconductor heterostruc-
tures. The fabricated Cu-Se-Ag structure with wire shape,
Figure 4. Experimental I-V characteristics of Cu-Se binary
structures of 2 µm and 1 µm diameters
-20
0
20
40
60
80
-4-20 2 4
Vo ltage (volts)
Current (uA)
40 nm
100 nm
Figure 5. Experimental I-V characteristics of Cu-Se RTDs
of 100 nm and 40 nm diameters
Table 1. Variation of PVCR with diameters of Cu-Se devices
Diameter (nm) PVCR
40 2.5
100 2.0
1000 1.02
(b)
Figure 6. Model utilized for explaining the I-V characteris-
tics of Cu-Se RTDs (a) equillibrium state (b) electrons flow
from Cu to Se when a suitable voltage is applied across this
system

M. CHAUDHRI ET AL. 75
hence, forms one dimensional RTD with Cu as emitter,
Ag as collector and Se as a potential well. On applying a
voltage across the device, the band diagrams can be re-
drawn as shown in Figure 6(b). The electrons from the
Cu electrode tunnel to the empty states in the conduction
band of Se. The electrons in the well stay at a particular
energy level until these electrons get enough energy to
jump to the next higher energy level. These electron
waves reflect back and forth between the two walls of
the well and interfere, causing the change in the ampli-
tude of the wave. When the energy of the electrons is
equal to the energy of the quantized level in the well, the
two waves interfere constructively and resonance of the
electron wave takes place, which results in maximum
transmission of electrons. The accumulation of electrons
in the well thus results in a decrease in the current up to
valley point current of the I-V curve.
Based on this model of quantization of energy levels,
the I-V behavior of the Cu-Se structures has been ex-
plained. The energy levels in Cu-Se-Ag structures can be
drawn as in Figure 7. The bulk behavior of the Cu-Se
binary structures of 2 µm can be explained by the energy
level diagram of Figure 7 (a) where the energy levels are
continuous.
However, as the dimensions of the device are reduced,
quantized energy levels appear in Se semiconductor and
a negative differential resistance region starts appearing.
This is shown in Figure 7 (b). On reducing the diameter
further, the negative differential resistance region in-
creases and this is illustrated in Figure 7 (c). The energy
band diagrams in Figures. 7 (b) and (c) show the increase
in spacing in energy levels in the conduction band of the
Se with decrease in the dimensions (diameter) of the fab-
ricated binary structures.
Further, the cut-in voltages of the devices increase
with decrease in diameters of the device. This indicates
an increase in the Schottky barrier height due to increase
in band gap of Se as the device dimensions are reduced.
Various workers [9,10] have reported an increase in band
gap with reduction in dimensions. A similar behavior is
expected for Se as well and has been shown in Figures
7(a), (b) and (c).
4. Theor etical Analysis of Experimental Results
In this section, the authors intend to correlate some of the
experimental observations. Figures 4 and 5 and Table 1
indicate that there is 1) an increase in cut-in voltage as
the diameters of the device is decreased 2) The PVCR
increases with decrease in diameters of the device.
4.1 Increase in Cut-In Voltage
The increase in cut in voltage as seen in Figs. 4 and 5 can
be explained due to increase in band gap. Such an in-
crease in band gap with decrease in diameter is reported
in literature [10–12].
As a metal is brought in contact with a semiconductor,
a barrier will be formed at the metal-semiconductor in-
terface. The height of the barrier is governed by metal
work function and the electron affinity of the semicon-
ductor. The voltage required to increase the energy of
electrons on the metal side to overcome the barrier is
cut-in voltage. The cut-in voltage and the band gap of the
semiconductor are related as [13].
qb=Eg - q (m -) (1)
where
Eg is band gap of the semiconductor
qm is work-function of the metal
qb is Schottky barrier height at the
metal- semiconductor
contact
q is electron affinity of the semiconductor
Figure 7. Energy band diagrams and corresponding I-V characteristics of Cu-Se resonant tunneling diodes of dif-
ferent diameters illustrating the emergence of quantum size effects (a) bulk effect (b & c) quantum size effects

M. CHAUDHRI ET AL. 76
From the Equation 1, it is clear that the cut-in voltage
is directly dependent upon the band gap of the semicon-
ductor material i.e. higher the band gap, higher will be
the cut-in voltage. Klimov while studying the absorption
spectra of CdSe material in bulk and in quantum dot
form [14], found the appearance of quantized bands and
an increase in band gap of CdSe in quantum dots. Further,
the researcher also obtained an expression for the size
dependent energy gap using the spherical “quantum box”
model which is given below.
Eg
band(d) = Eg band (bulk) + h2/8mehd2 (2)
where
meh = memh/(me+mh)
and d is the diameter of circular/cylindrical material
me is the effective mass of electron
mh is the effective mass of hole
In Equation 2, the parameter ‘d’ introduces the size
based effects. Equation 2 can be written as [15].
Eg
band(d) = Eg band (bulk) + K/d2 (3)
It is clear from Equation 3 that second term in Equa-
tion 3 tends to increase as the diameter of the device de-
creases. It implies that the value of band gap will in-
crease as the diameter of the device is reduced. As the
value of band gap increases, following Equation 1, the
cut-in voltage will also increase.
Hence, the reduction in diameter of the device leads to
an increase in the band gap of Se, which is indicated by
an increase in cut-in voltage of the device.
4.2 Tunneling Current
Tunneling current I can be expressed by Equation 4
[16,17]
I =  T(E)m(E)s(E)[Fm(E)-Fs(E)] dE, (4)
where
T(E) is tunneling probability between the occupied
level in the
Cu metal and the unoccupied level in the Se semicon-
ductor
m (E) or s(E) is Density of states (DOS) of the metal
and semiconductor, respectively
Fm and Fs is Fermi-distribution function in metal and
Semiconductor respectively
From the WKB (Wentzel-Kramers-Brillouin) ap-
proximation, the tunneling probability can be approxi-
mated as [17]
T (E)  exp (-2kt) (5)
where
k is wave vector
t is width of the barrier
Fm and Fs = 1/ (1+ exp (E-E
f
/ kT) ) [18] (6)
where, Ef is Fermi energy of metal or semiconductor
Density of states in metals can be estimated by para-
bolic approximation, resulting in an E1/2 dependency of
the density of states [18].
m = 3.14/2  volume  (8me h 2)3/2  E 1/2 (7)
where me is effective mass of an electron in the metal h
is the Planck’s constant
However, the small size of Se semiconductor implies
the presence of (Columbic) charging energy states in
addition to the density of states of the particles. Taking
into account the size distribution of the materials, we can
express the density of states of the Se material as [17].
022
0
exp[()2 / (2) / (2)
n
ss c
n
EnE  



 
 (8)
0
s

is density of states without the charging states and is
given by [19]
02
2/ 1/
se
mh E  
where s is density of states of the Se
Ec is charging energy of the Se
 is size-dependent standard deviation in energy space
As the spacing between the energy levels increases, 
will increase as the size of the semiconductor decreases.
The various parameters for Cu-Se binary structures are
given in Table 2.
The values of the various other parameters are given
below.
Free mass of the electron (m) = 9.1  10-31 kg
Planck’s constant (h) = 6.602  10-34 J-s
Ef (Cu) = 7.0 eV [23]
Ef (Se) = 5.6 eV [21]
4.3. Calculation of Charging Energies for Se
Charging energy is the energy required to put a charge q
on a conductor of capacitance C0 and is given by [24,25].
Ec =e2/2C0  e
2/40rd (7)
Table 2. Parameters of Cu, Ag and Se materials
Work
function
(eV)
Electron
affinity
(eV)
Effective
mass
(Kg)
References
Cu 4.7 1.46 m [20]
Se 5.11 3.0 0.22 m [20–22]
Ag 4.73 [20]

M. CHAUDHRI ET AL. 77
where
0 is permittivity of free space = 8.854  10-12 F/m
r is relative permittivity or dielectric constant of
Se material = 6.1 [26]
d is diameter of the semiconductor
e is charge on an electron = 1.6  10-19 C
Substituting the values of diameters of different Cu-Se
devices in Equation 7, correspondingly, charging energy
comes out to be 0.0002 eV (1µm diameter), 0.002 eV
(100 nm diameter) and 0.006 eV (40 nm diameter).
Since capacitance C0 is dependent directly on the diame-
ter of the device, clearly the charging energy will in-
crease as the diameter is reduced. Hence, an electron
will be able to enter into the nanomaterial if it has
enough charging energy and if it is in resonance with an
empty state of the well. Substituting the values of pa-
rameters of Cu, Se and Ag in Equations 5, 6, 7 and 8, m,
s, T(E) and F(m or s) are calculated for different values of
energies. Substituting the values of these terms in Equa-
tion 4, the variation of tunneling current in Cu-Se RTDs
of diameters 1 µm, 100 nm and 40 nm for various values
of voltages are calculated and plotted. The calculated I-V
curves for different diameters are shown in Figures. 8, 9
and 10.
From Figures 8, 9 and 10, we observe that the theo-
retical I-V characteristics of the Cu-Se binary structures
of diameters 1 µm, 100 nm and 40 nm show a behavior
similar to the one as seen in the experimental observa-
tions. However, the current values as calculated are very
small as compared to experimental values of the currents.
This type of behavior is expected, since the calculated
I-V characteristics is for a single Cu-Se RTD, whereas,
the experimental results are due to the collective behav-
ior of a large number of Cu-Se RTDs in parallel.
Further, the PVCR of the Cu-Se RTDs are calculated
for different diameters and are shown in tabular form in
Table 3.
Calculated values of PVCR of the Cu-Se RTDs of di-
ameters 1µm, 100 nm and 40 nm are 2.01, 2.67 and 3.14,
which show an increase in PVCR with decrease in di-
ameters. This behavior is similar to that seen in the I-V
curves obtained experimentally.
Hence, it can be inferred that the results obtained from
theoretical model of RTD show a behavior similar to that
obtained experimentally. However, the theoretical device
currents are small in values because, in experimental
set-up, several devices are working in parallel while, in
theoretical equations, the current for a single device is
calculated.
Table 3. Variation of calculated PVCR with diameter.
Diameter
(
nm
)
PVCR
40 3.14
100 2.67
1000 2.01
0
20
40
60
00.5 1 1.5 2
V o ltag e (vo lts)
C u rrent (mA)
Figure 8.Theoretical I-V characteristics of Cu-Se RTD of 1
µm diameter
0
20
40
60
00.511.52
Vo ltage (V)
Current (nA)
Figure 9. Theoretical I-V characteristics of Cu-Se RTD of
100 nm
0
0.1
0.2
0.3
00.511.52
Voltage (V)
Current (nA)
Figure 10. Theoretical I-V characteristics of Cu-Se RTD of
40 nm diameter
5. Conclusions
A suitable model for the template synthesized Cu-Se
RTDs is proposed. Experimental results have been veri-
fied with the help of equations formulated for this model.
The results obtained from theoretical model of RTD
show a behavior similar to that obtained experimentally.
However, the theoretical device currents are small in
values and PVCRs show deviation in their values from
the experimental values. This is because, in experimental
set-up, several devices are working in parallel while, in
theoretical equations, the current for a single device is
calculated.
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