Int. 1. Nanoelectronics and Materials 2 No.1 (2009) 23-39
~
U· MAP
Estimation of thermal contact resistance in metal-plastic interface of
semiconducting electronic devices
S. A. Oke\", A. A. Oyekunle', T. A. O. Salau', A. A. Adegbernile", K. O. Lawal3
1.1) Department of Mechanical Engineering, University of Lagos, Nigeria.
1) Department of Mechanical Engineering, University of Ibadan, Nigeria.
3) Faculty of Engineering, University of Ado-Ekiti, Nigeria ..
Abstract
For decade, thermal contact resistance (TCR) has been measured experimentally.
Unfortunately, the database, which should regularly support decision-making on TCR
coefficients, seems not to exist. Thus, companies result to using outdated or irrelevant data
that limits lifespan of electronics devices, their performance and reliability. This paper
mathematically models the problem of TCR between two media (plastic-metal interface) in
semi conductors with reference to resistance and the flow of heat across or interface of two
surfaces that are in contact, particularly in engineering applications. In this paper, a semi-
conductor/heat sink assembly is used to model the behaviour of thermal contact resistance.
A cylindrical shaped semi-conductor was conceptualised, with the governing differential
equations derived and the boundary conditions for the problem stated. The effect of
parameters such as surface roughness, contact pressure, density of interstitial gas, heat
capacity, thermal and mechanical properties on the temperature at the center of the semi-
conductor was studied. From the analysis, it can be inferred that by effectively lowering
thermal contact resistance, efficient heat transfer results, which helps to prolong the life and
reliability of the semi-conductor. The current work is motivated to fill an important gap that
may be beneficial to practitioners in the semi-conductor industry.
Keywords: Thermal, Resistance, Plastics, Performance, Electronic devices.
1. Introduction
Effective description of therrrial energy is a necessary and important aspect in the
electronic industry. In order to increase the performance, reliability and life span of an
electronic device, the efficient removal of heat from the device to the atmospheric
environment is very important. This concept is referred to as the thermal energy transfer
(TET). The thermal energy transfer (removal) involves the following stages: (a) The
volumetric heat generation within the device; (b) The transfer of heat from the semi-
conducting device to the heat sink; (c) The heat transfer from the heat sink to the
.) For correspondence, E-mail: saoke@yahoo.com
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S.A. Oke et al./ Estimation of thermal contact resistance in metal-plastic interface ...
, environment (Rajput, 1998; Yang, 2007; MIT, 2007). We are most concerned with (b), that
is, the efficient transfer of heat from the semi-conducting device to the heat sink..
Now, consider the semi-conducting device/heat sink assembly, in place of air, which
.fills up the interstitial spaces between the two non-conforming surfaces, there are some
applications where materials of higher thermal conductivity known as thermal interface
material, are used. With the use of thermal interface materials, there are two possibilities: (1)
the thermal interface material fills up the gaps within the interface such that there are still
contacts between the two surfaces at some points. Thus, in the circuit, the resistance is
replaced with the estimated thermal resistance of the thermal interface material; (2) the
thermal interface material could be such that it completely separates the two surfaces, filling
the gaps between them (Rajput, 1998; MIT, 2007).
From the foregoing, it is important to define the problem solved in this paper. This is
stated as follows:
"In a semiconductor device (i.e. field effect transistor), the heat is generated in the
channel and conducted to the chip on the silicon substrate. The heat is removed
from the chip to the environment by appropriate packaging techniques. Typically,
semiconductor electronics and packaging problems involve planar structures (chip,
package, boards, etc). This problem is of importance to researchers in thermal
contact resistance. "
Thus, the problem is approached by considering a semi-conductor/heat sink
assembly and modelling the behaviour of the thermal contact' resistance. A cylindrical-
shaped conductor is used for the modelling. This is formulated as a problem and an
appropriate solution methodology sought for.
The following is a review of related studies on thermal contact resistance: Barber
(1986) studied the effect of mechanism on the axial temperature variation in parallel or
counter flow heat exchangers. A method was developed for integrating the controlling
equations for an arbitrary, non-linear, pressure-dependent contact resistance without
iteration and the effect of prestress and other parameters on the occurrence of multiple
solutions is discussed. Cong et al. (2006) carried out numerical simulation of both the three-
dimensional heat conduction and two-dimensional elastic wave propagation at the interface
of contact solids. Numerical results of heat conduction simulations show that both the true
contact area and thermal contact conductance increase linearly with an increase in the
contact pressure.
Yang (2007) applied a conjugate gradient method based on an inverse algorithm to
estimate the unknown time-dependent 'thermal contact resistance' in a single-coated optical
fiber, which is subjected to transient 'thermal' loading. Vermeersch and DeMercy (2007)
numerically calculated the 'thermal' impedance/ ZthGro) for a silicon chip glued on a ceramic
substrate by modelling the chip-substrate interface as a 'thermal contact resistance' (Ye). In
the investigation by McDonald, et al. (2007) plasma-sprayed molybdenum and ythia-
stabilized zirconia particles (38-63fl diameters) were sprayed onto glass and Inconnel 625
held at either room temperature and 400°C samples of Inconnel 625 were preheated for 3h,
and then air-cooled to room temperature before spraying. Photographs of the splats were
captured using a fast charge-coupled device (CCD) camera. A rapid two-color pyrometer
was used to collect "thermal radiation from the particles during flight and spreading to
follow the evolution of their temperature. The temperature evolution was used to determine
the cooling rate of spreading particles. An analytical heat conduction model was developed
to calculate the 'thermal contact resistance' at the interface of the plasma-sprayed particles
and the surfaces from splat cooling rates.
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lnt. 1. Nanoelectronicsand Materials 2 No.1 (2009) 23-39
Shinji et al. (2005) proposed the quantitative measuring method of the thermal
contact resistance in consideration of the real contact area of the joint and make clear its
validity and the possibility of formulation of the resistance. Bahrami et al. (2004) developed
a compact analytical model for predicting thermal contact. resistance of non conforming
rough contacts of bare solids in a vacuum by employing the 'Scale analysis method'. It is
demonstrated that the geometry of heat sources on a' half-space for micro contacts is
justifiable for applicable range of contact pressure. Bahrarni et al. (2005) developed a new
model which is for low pressures. The effect of elastications beneath the plastically
deformed micro contacts is determined by superimposing normal deformations due to self
and neighboring contact spots in an elastic half space. Muzuhara and Ozawa (1999)
examined an estimation method of the titled resistance value in the contact surface between
objects. The paper showed the composition of a measuring device and a measurement
method of both resistance, and measurements were conducted for lapped surface, ground
surface and scraper surface of cast iron. Cook et al. (1984) presented a novel concept for
reducing thermal contact resistance. Gavarella et al. (2003) considered an idealized problem,
which a thermo elastic rod slides against a rigid plane with both frictional heating and a
contact resistance. From the foregoing review, it seems obvious that investigations relating
to the inclusion of diverse parameters in thermal contact resistance studies appear lacking in
the literature. The need to close this important gap has therefore motivated the current work.
The structure of the paper is as follows: introduction, mathematical framework,
results and discussion, and conclusion. The introduction provides the motivation for the
study and a justification for the approach utilised in the current work by pointing out the gap
in the literature. The problem definition is also included ill this section. The problem is
defined in order to have a clear focus of the challenge faced in concise form. The
mathematical framework is commenced by stating the nomenclature and then the problem.
It models the problem in a practically acceptable form. The section on mathematical
framework also considers the problem analysis and the application of dimensional analysis
in solving the problem. Results are displayed in the next section. The last section contains
the concluding remarks.
2. Mathematical framework
2.1 Definition of terms
qg uniform volumetric heat generation per unit volume per unit time
Qg volumetric heat source in device. We assume a value of IOW/m3
A surface area of contact. We assume a cylinder of length 3cm and radius 0.5cm and
we have chosen 1I1ih of the whole body for our analysis since there is symmetry.
T surface temperature
T, ambient temperature (Temperature. of the environment' chosen as 23°C)
H convective heat transfer coefficient of air. We assume natural convection and a value
ofh = 30W/m2k
Tmax the temperature at the center of the semi-conductor, the maximum temperatures
T, the temperature at the outer surface at the heat sink
RJ thermal resistance of the semi-conductor
R, thermal resistance of the interstitial gas
R2 thermal resistance of the heat sink
Rm thermal resistance of interface material
P pressure
cr surface roughness
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S.A. Oke et al.! Estimation of thermal contact resistance in metal-plastic interface ...
p density of interstitial gas
~l viscosity
cp heat capacity
k thermal conductivity
T temperature
V velocity (of interstitial gas)
kl thermal conductivity of semi-conducting device. We assume a value of 60WmJk
k2 thermal conductivity of heat sink. We assume a value of250WmIk
x the thickness of the interface between semi-conducting device/heat, assumed as
10~m
k, thermal conductivity of air = 0.13 W/mk
Ag gap heat transfer area
D a dimensionless parameter which should be a property of the fluid (air). The Prandtl
number is one of the dimensionless parameter we expect to have effect on D.
2.2 Problems analysis
At the interface of the semi-conducting device and heat sink, there exists a
phenomenon known as thermal contact resistance (TCR). Thermal contact resistance is a
resistance to the flow of heat across an interface of two surfaces that are in contact. An
illustration is shown in Figure 1. The contacting surfaces of the semi-conducting device and
heat sink are not perfectly smooth. Therefore, the real contact area is less than the nominal
contact area. Below are some of the factors considered which may influence the value of
thermal contact resistance. '
(i) Contact/Bearing Pressure: It is reasoned that as the contact pressure between the
two surfaces increases, heat transfer becomes more efficient and thermal contact
resistance is reduced.
(ii) Deformation: Preceding from (i) above, that if it is possible to deform the two
materials up to an extent where the thermal properties are not too affected, actual
surface area of contact can be increased thereby reducing thermal contact resistance.
(iii) Nature of gap materials: Since we have established that the two surfaces in contact
are not perfectly smooth, the gaps or voids between the mating surfaces will be filled
with some sort of gases or fluids (since we are not expecting there to be a vacuums).
For the purpose of this paper, we will take air as the material filling up these gaps.
The pressure of air within the mating surfaces and its thermal properties will have
effect on the TCR.
RMS value a: We shall also take a look at the quantity rms which is used as a
(iv) measure of the roughness of a surface.
(v) Temperature of the interstitial material: There is possibility of higher temperatures
of interstitial material increasing the rate of heat transfer. Since with higher
temperature, the gas particles will collide more often.
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Int. J. Nanoelectronics and Materials 2 o. 1 (2009) 23-39
C11 p I" Iq -,( ) ( )
LJ L2
T
TJ
Effect ofTCR, discontinuity
at the interface
Fig. I' Illustrating the phenomenon of thermal contact resistance.
Without the effect of TCR, we would expect the temperature gradient to be like that
illustrated in Figure 2.
T
x
Fig. 2: Temperature gradient in thermal contact resistance system.
For the sake of simplicity, the semi-conducting device is modeled as a cylinder
shape, which generates uniform volumetric heat from within. The semi-conducting device
is modeled as shown in Figure 3 while the heat sink as shown in Figure 4.
Fig. 3: Cylinder oflength L and radius R.
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S.A. Oke et aLl Estim'ation ofthennal contact resistance in metal-plastic interface ...
On a 2-D plane, the semi-conducting device and the heat sink together can be
imagined to understand its structure. Since the whole assembly is symmetrical, we can pick
a part of it, write the governing equations and apply them to the whole of the body.
Fig. 4: Elemental part of semi-conducting device and heat sink.
Derivation of governing equation
From first principle, we have
Y'. (kY'T) + qg = pcp dT , where Y' is defined as ~
dt ax; (1)
For a constant k, (i) becomes:
(2)
For the steady state situation, we are considering, the right hand side value becomes zero in
equation (2),
(3)
2 .
For a one-dimensional heat transfer, equation (iii) is reduced to: aax: + ~ = 0k
Heat transfer from heat generated within volume to the interface (Figure 5) from Fourier's
law,
Q=-KA-dTdx (4)
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lnt. 1. Nanoelectronics and Materials 2 No.1 (2009) 23-39
Considering an elemental strip of the cylinder with radius r and thickness dr,
dT
Qr=-K2mL - (5)
dr
L
Fig. 5: Heat generation profile.
But A = 2rcrL, therefore heat generated in the elementQ, = qg x volume
Qg = qg x (Area) x L = qg x (2rcr.dr) x L = qg . (2rcrdr).L (6)
Heat out at radius r + dr is derived from Q(r + dr) = Qr + ~ (Qr) dr
dr
---.
Q(r + dr)
r r + dr
For energy balance, Qr + Qg = Q(r + dr). Therefore
d
Qg = - (Qr) dr (7)
dr
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S.A. Oke et al J Estimation of thermal contact resistance in metal-plastic interface ...
By substituting (5) and (6) into (7), we have
qg. 2m.dr.L = ~ [- k. 2mL dT] dr or ~ [;. dT] = _qgt:-
dr dr dr dr k
B.y integratin.g, r -dT = - qgr2 Thi-2- + a . IS means
dr k
dT qg r a
-=---+- (8)
dr k 2 r
q r2
Integrating (8): T=--.!.-+a lnr+bk 4
(8a)
The constants a and b can be evaluated from the boundary conditions as follows:
(1) at r = R, T = Tin!>and (2) qg (nR 2L) = -K . 2nRL . ILdT-] (9)
dr r=R
At r = 0, dT = O. From (8), (dT) = _ qg R + ~. Also, From (2), (dT) = _ qg R
dr . dr re R k 2 R dr r=R k 2
Then, equating both terms, it is clearly seen that a =O.Therefore, Tint = _ ~ ~2 + b, and
• 2
b = Tint+ - qg ~. By substituting values of a and b in (8a):
k 4
T-_ --q-g R2 qg _ qg 2 2 _ qg 2 rk 4 +T t+ ---k'
R2
4--T t+ -4-k (R -r )-T t+ -R
( 1--2 J
m m III 4k R 2
In terms of dimensionless parameters,
(10)
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Int. 1. Nanoelectronics and Materials 2 No. I (2009) 23-39
If we plot equation (10) on a graph, we get a parabolic curve as shown below:
~-'
Tmax
r/R
Fig. 6: Parabolic shape of curve of parameters.
Tmax occurs at r = 0 (Figure 6). Our aim is now to limit Tmax to the. level that will not cause
damage to the semi-conducting device. .
Tmax --'T'mt+-, qg R2 (11 )
4k
At the interface, there is a jump or slip in time, which is due to thermal contact
resistance. Let us represent this jump or slip by f:.Tint' Using the electrical circuit analogy,
we can ~raw our circuit as shown in Figure 7:
Ra
,,,
,,
,
, ,
,,
,,, ,,
,
, heat sink ~,'" ,
"', inter face..
r= 0
r=R
Fig, 7: Circuit diagram for the system under study.
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S.A. Oke et aU ESli~alion of thermal contact resistance in metal-plastic interface ...
We have
_ Tmax - Ts
q- IR (12)
T
But (13)
,0.Tint
Therefore (14)
Note that in this work, interstitial gas is assumed to be air.
2.2.1 Dimensional analysis
With the use of Buckingham 1t theorem, we want to try to derive dimensionless
parameters that may affect the thermal contact resistance.
Supposing the thermal contact resistance depends on the following parameters:
Parameters Dimensions
Pressure, P ML-1r2
Surface roughness, o L
Density of interstitial gas, p ML·3
Viscosity, ~ , ML-lrl
Heat capacity, cp L2r2e'l
Thermal conductivity, k MLT-3e·1
Temperature, T e
Velocity, Y (of interstitial gas) M·IL-2T3e
Note that Rer = f(P, rr, p, ~,cp, k, T, v), Then, we could state that f(Rer, P, c, p, u, cp, k, T, v)
= O. Then f(11I~112,1t3, 1t4, 1t5) = 0
1t1 _- Tal- . crbl . pc,l v" , P
1t2 _- Ta2 • crb2 . p.e2 yd2 . P
1t3 = TaJ • cr,b3 pc3 . v" . P
114= Ta4 , crb4 d4. pe4 , y • P
1t5 -_ Ta5 ,cr.b5, p.c5 yd5 , P
T, c, p, v have been chosen as our repeating variable.
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lnt. 1. Nanoelectronics and Materials 2 No.1 (2009) 23-39
p _ pvo' kT _ pv3O'2
By solving the rt-term, we obtain: 7tJ= --2' 7t2- --, 7t3- --3-' 7t4- ~r---, and
pv f-l pv 0' T
7ts= cvpT. Note that 7t2gives us the Reynold's coefficient, and'cOl~bination of7t2, 7t3and 7ts2
gives ~, which is Prandth number, which is a property of the fluid.
=.
By referring to our problem model, we have the following (Figure 8)
Heat sink
Heat dissipation
through convec- (No heat generated)
Semi-conducting device
(volumetric heat source)
An (kYT) = 0
Fig. 8: Detailed diagram of semi-conducting device and heat sink.
Using the series analogy of heat transfer, we model k, (thermal conductivity at points
of contact in the interface) as:
(15)
We assume effective surface roughness a as 151lm, we can now write
(16)
where c would be a dimensionless parameter which would be a constant for any
particular semi-conducting device.
3. Results and discussion
To bring in the effect of the contact/bearing pressure mentioned in the earlier part of
L
2
this work, we are more inclined to pick c as Pv . Density of air p = 1.3kg/m3 If we assume
.,.,
.J.J
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S.A. Oke et al./ Estimation ofthennal contact resistance in metal-plastic interface ...
P 2 60x 250 15000
2
P = 2Pa and v = 2m1s , c = pv l.3x22 = 0.385. From (15), kc = 60+250= 310
48.39Wmlk.
We model thermal resistance of air R. as: (_X_)D
k.Ag
A = -11tRL 1 x -22= - x 0.5 x 3 x 10-4 = 0.79 x 10-4mJ
6 6 7
6
Let D = 0.007, then n,= ( 10 X 10. 4 )0.007 = 6~82x 1O·3kJwa,nd
0.13xO.79xl0'
n,= ( 1.5X 610. )0.385 = 2.39 10·4kJw
48.39 x 0.5 x 10.4
Going back to Figure 8, the values could be indicated on the assembly as follows:
340k (assu- T, = (23 + 273)
med) = 300k
Fig. 9: Part of the semi-conducting device with specification.
With the combination of equations (xii), (xiii) and (xiv), a lower value of Tmax is
obtained when the value of Rer is not considered. The following results are then obtained
from the simulated data used for the work (Tables 1 and 2).
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Int. 1. Nanoelectronics and Materials 2 No. I (2009) 23-39
cr(llm S(MNI
) m FCN) m) AIl(m2) x(llm) Y(,lm) b(llm) A,(m2) PCN/m2) HCN/m2) k(W/mk) Rc(k/W)
5 0.2 200 500000 0.0025 0.0004 4.9996 0.002 1.257E-17 80000 2000000 230 3.452536
5 0.2 250 500000 0.0025 0.0005 4.9995 0.0025 1.964E-17 100000 2000000 230 2.760651
5 0.2 300 500000 0.0025 0.0006 4.9994 0.003 2.828E-17 120000 2000000 230 2.299395
5 0.2 350 500000 0.0025 0.0007 4.9993 0.0035 3.849E-17 140000 2000000 230 1.969928
5 0.2 400 500000 0.0025 0.0008 4.9992 0.004 5.027E-17 160000 2000000 230 1.722828
5 0.2 450 500000 0.0025 0.0009 4.9991 0.0045 6363E-17 180000 2000000 230 1.53064
5 0.2 500 500000 0.0025 0.001 4.999 0.005 7.855E-17 200000 2000000 230 1.376891
5 0.2 550 500000 0.0025 0.0011 4.9989 0.0055 9.505E-17 220000 2000000 230 1,.251096
5 0.2 600 500000 0.0025 0.0012 4.9988 0.006 1.131E-16 240000 2000000 230 1.146268
5 0.2 650 500000 0.0025 0.0013 4.9987 0.0065 1.327E-16 260000 2000000 230 1.057568
5 0.2 700 500000 0.0025 0.0014 4.9986 0.007 l.54E-16 280000 2000000 230 0.981539
5 0.2 750 500000 0.0025 0.0015 .4.9985 0.0075 1.767E-16 300000 2000000 230 0.915648
5 0.2 800 500000 0.0025 0.0016 4.9984 0.008 2.011E-16 320000 2000000 230 0.857994
5 0.2 850 500000 0.0025 0.0017 4.9983 0.0085 2.27E-16 340000 2000000 230 0.807123 '
5 0.2 900 500000 0.0025 0.0018 4.9982 0.009 2.545E-16 360000 2000000 230 0.761905
5 0.2 950 500000 0.0025 0.0019 4.9981 0.0095 2.836E-16 380000 2000000 230 0.721447
5 0.2 1000 500000 0.0025 0.002 4.998 001 3.142E-16 400000 2000000 230 0.685035
Table I: Simulated data of various parameters of the model.
F(N) n nA,(m2)
200 7.95672E+12 0.0001
250 6J6537E+12 0.000125
300 5.30448E+ 12 0.00015
350 4.54669E+12 0.000175
400 3.9783GE+12 0.0002
450 3.53632E+ 12 0.000225
500 3.18269E+12 0.00025
550 2.89335E+12 0.000275
600 2.65224E+12 0.0003
650 2.44822E+12 0.000325
700 2.27335E+ 12 0.00035
750 2.12179E+12 0.000375
800 1.98918E+12 0.0004
850 1.87217E+ 12 0.000425
900 1.76816E+ 12 0.00045
950 1.6751E+12 0.000475
1000 1.59134E+ 12 0.0005
Table 2: Simulated values ofF(N) against nA~(n/).
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S.A. Oke et al./ Estimation of thermal contact resistance in metal-plastic interface ..
F against b
1XOr---------~------------------------------------------------------_,
1000
BOO
~ 600
-co
200
0
0 10 12 " 16 teb{m)
Fig. 10: Plotted values ofF and b.
1200r-----------------------------------------------------------------------,
1000
800
SE-17 lE-la 1.SE-'6 2E-1S 2.5E-16 3E-,a 3.SE-'6
Fig. 11: Plotted values ofF and AT.
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ITY OF IBADAN LIBRARY
lnt. J. Nanoelectronics and Materials 2 No.1 (2009) 23-39
1200
1000
800
~ 600
'l ,.
"'"
200
0
0 lE+12 2E+12 3E+12 4E+12 5E+12 6E+1Z 7E1'12 8E+12 9E+12
Fig. 12: Plotted values ofF and n.
I" agalnstR"
1200
1000
"'"
~ 600
400
200
0
0 0.5 1.5 2.5 3.5
Fig. 13: Plotted values ofF and Re.
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S.A. Oke et al.! Estimation of thermal contact resistance in metal-plastic interface ..';
1~,------------------------------------
1000
'00
400
o~------~--------~------__----------------~------~
o 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006
nA,(m'J
Fig. 14: Plotted values ofF and nAT.
4. Conclusion
In the present paper, we have investigated the interaction among the parameters of
thermal contact resistance that is suspected to have effects on it. The results of the
experimentation shows that the important parameters of surface roughness, pressure,
mechanical and heat properties have effect on thermal contact resistance. The investigation
presented here also states that a more efficient heat transfer is obtainable; hence thermal
contact resistance can be effectively lowered. The results of the paper may provide some
information for those who deal with the systems in which thermal resistance is concerned,
particularly thermal management of electronics.
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Int. J. Nanoelectronics and Materials 2 No.1 (2009) 23-39
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