Proceedings of HT2005
2005 ASME Summer Heat Transfer Conference
July 17-22, 2005, San Francisco, California, USA
HT2005-72172
Thermal Transport in Nanotube Composites for Large-Area Macroelectronics
Satish Kumar
Mohammad A. Alam
Department of Mechanical Engineering
Department of Electrical and Computer Engineering
Purdue University
Purdue University
585 Purdue Mall
465 Northwestern Avenue
W. Lafayette IN 47907
W. Lafayette IN 47907
Jayathi Y. Murthy
Department of Mechanical Engineering
Purdue University
585 Purdue Mall
W. Lafayette IN 47907
emerging as an alternative in many non-display applications.
For high performance applications, however, the choices are
limited: single crystal silicon or poly-silicon based TFTs [8,9]
cannot be manufactured at low temperature (<200C) and are
therefore not suitable for plastic substrates. As a result,
researchers are exploring a new class of nano-composite TFTs
based on bundles of silicon nanowires (Si-NWs) or carbon
nanotubes (CNTs) [10-12]. Here, high-quality, nearlycrystalline NWs and CNTs are grown at high temperature on a
temporary substrate, then released from the temporary substrate
into a carrier fluid, and finally, the wire-saturated carrier fluid is
spin-coated onto arbitrary (flexible) substrates at room
temperature to form a thin film of randomly-oriented NWs or
CNTs. Once the source/drain contacts are defined, this thin film
of nearly crystalline nanowires or nanotubes constitutes the high
performance channel of a TFT (see Fig. 1).
The unique physical properties of carbon nanotube
composites make them a promising novel material for use in
TFTs. Individual nanotubes exhibit excellent electrical and
thermal conductivities [13-15] and high Young’s modulus and
strength-to-weight ratio [16]. However, it is yet unclear to what
extent these properties will be manifested in polymer
composites, where scattering at polymer/tube boundaries and at
tube/tube contacts may substantially decrease electrical mobility
and thermal conductivity. For silicon nanowires, a significant
decrease in axial thermal conductivity has been noted in freestanding nanowires either due to phonon boundary scattering or
due to phonon confinement effects [15]. Previous experiments
and numerical simulations report that the interfacial contact
conductance between nanotubes and substrate [18] may be
extremely low. Similarly, the contact resistance between
ABSTRACT
Thermal transport in a new class of nanocomposites
composed of isotropic 2D ensembles of nanotubes or nanowires
in a substrate is considered for use as the channel region of
thin film transistors. The random ensemble is generated
numerically and simulated using a finite volume scheme. The
effective thermal conductivity of a nanotube network embedded
in a thin substrate is computed. Percolating conduction in the
composite is studied as a function of wire/tube densities and
channel lengths. The conductance exponents are validated
against available experimental data for long channels devices.
The effect of tube-tube contact conductance, tube-substrate
contact conductance and substrate-tube conductivity ratio is
analyzed for various channel lengths. It is found that beyond a
certain limiting value, contact parameters do not result in any
significant change in the effective thermal conductivity of the
composite. It is also observed that the effective thermal
conductivity of the composite saturates beyond a limiting
channel-length/tube length ratio for the range of contact
parameters under consideration.
INTRODUCTION
In recent years, there has been growing interest in lowcost large-area manufacture of thin film transistors (TFTs) on
flexible substrates for use in applications such as displays, epaper, e-clothing, biological and chemical sensing, conformal
radar, and others. TFTs based on amorphous silicon (a-Si) now
dominate the market for large-area flat-panel displays [1-2].
When transistor performance is not critical, low-cost organic
TFTs on flexible, lightweight, plastic substrates [3-7] are
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Copyright © 2005 by ASME
nanotubes has a major influence on the transport properties of
the composite.
There is little in the literature that provides a
framework for understanding, controlling and designing
nanowire and nanotube composites suitable for TFTs. The
network of nanotubes used for the channel region is embedded
in a substrate such as plastic, and is covered with glass on one
side and polymer on the other. Source and drain contacts are
deposited on this 2D matrix, and typically, the device is backgated. Thus the network layer is very thin, and may essentially
be considered two-dimensional. Moreover, unlike bulk
composites, source and drain may only be a few microns apart
in typical macroelectronics, and the channel length may be
comparable to the length of the tube or wire. Thus analyses
employing classical percolation theory for periodic ensembles
are not suitable. The finite length of the sample must be
accounted for in the analysis.
There are few experimental, theoretical and
computational models which attempt to predict the thermal
conductivity of nanotube composites. Biercuk et al. has
reported 125 % increase in thermal conductivity of epoxySWNT composites for 1 wt % SWNT loading at room
temperature [16]. Liu et al. used a silicon elastomer as the
matrix and CNT as the filler in their experiments and reported
65 % enhancement in thermal conductivity with 3.8 wt % CNT
loading [19]. Nan et al. employed an effective medium theory
and predicted that the interfacial resistance can significantly
degrade the thermal conductivity of the CNT composite
[17,20]. Lusti and Gusev predicted the thermoelastic properties
of nanotube-reinforced polymers using the finite element
method [21]. Yang and Chen has used the phonon Boltzmann
equation to study the phonon thermal conductivity of
nanocomposites using a periodic two-dimensional model [22].
To our knowledge, there have been no published analyses of
thermal transport in thin film composites based on
nanotube/nanowire bundles for TFT applications.
Our ultimate interest is in the coupled electro-thermal
analysis of nanotube bundle transistors. Heat dissipation by the
current carrying nanotubes can degrade the performance of the
transistor. In this paper, the effective lateral thermal
conductivity of finite planar nanotube/nanowire composites is
investigated. The present work is a first step towards building a
self-consistent electro-thermal model to compute the
temperature distribution and I-V characteristics of network
transistors. We analyze the geometry-dependent conductance
properties of the composite for tube densities above the
percolation threshold using a finite volume method. The general
framework may be applied to both tubes and wires. The analysis
is used to predict the conductance of the composite as a
function of wire/tube density and channel length. Simulations
are performed to study the effect of contact resistance between
nanotubes as well as nanotube and substrate. The effect of
substrate to nanotube conductivity ratio on transport properties
is also analyzed.
NOMENCLATURE
A
tube cross-sectional area
Bic
contact-conductance parameter between tubes
Bis
contact-conductance parameter between tube and
substrate
d
diameter of tube
h c, h s
heat transfer coefficient characterizing tube-tube
contact and tube-substrate contact
H
height of the channel
keff
effective lateral thermal conductivity
kt
ks
thermal conductivity of tube
LC
Lt
n
P c, P s
t

channel length
tube length
conductance exponent
contact perimeters
substrate thickness
temperature
drain temperature
non-dimensional temperature
temperature difference across the channel
displacement vector from tube segment centroid to
substrate cell centroid
conductance
0
scaling parameter for conductance

th
*
v
tube density 
tube density at percolation threshold
T
Td

T

thermal conductivity of substrate
th
contact geometry parameter
ROD GENERATION
In the present analysis, we consider a percolating
random network of nanotubes or nanowires of length Lt and
diameter d randomly dispersed in the mid-plane of a matrix of
thickness t. Thus the nanotube network is essentially 2D, while
the matrix containing it is 3D. The geometry in the mid-plane is
shown in Fig. 1. The domain is assumed to be of height H, and
the top and bottom boundaries are assumed periodic. The
channel length is LC. For LC<Lt some tubes would span the
source-to-drain distance LC, while others may not, depending on
orientation.
The source, drain and channel regions in Fig. 1 are
divided into finite rectangular control volumes. A fixed
probability p of a control volume originating a nanotube is
chosen a priori. A random number is picked from a uniform
distribution and compared with p. If it is less than p, a nanotube
is originated from the control volume. The length of source and
drain for tube generation is Lt, which ensures that any tube that
could penetrate the channel region either from the left or the
right is included in the simulations. The orientation of the tube
is also chosen from a uniform random number generator. Since
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contact parameter v. In Eq. 1(a), the summation term denotes
heat exchange between tube i and all tubes j which have an
intersection with it; the term is non-zero only at the point of
intersection. Similarly, in Eq.1(b), the summation terms denotes
the volumetric source due to tubes intersecting the substrate.
The dimensionless parameters are defined as:
the tube length is fixed at Lt, all tubes may not span the channel
region even for shorter channel lengths LC. Tubes crossing the
top and bottom boundaries are treated assuming translational
periodicity; that part of the tube crossing one of these
boundaries reappears on the other side. Tube-tube intersections
are computed from this numerically generated random network
and stored for future use. The analysis is conducted only on the
tubes that lie in the channel region.
Bic 
Here, hc and hs are the heat transfer coefficients characterizing
tube-to-tube and tube-to-substrate contact, Pc and Ps are the
corresponding contact perimeters, kt is the thermal conductivity
of the tube and A its cross-sectional area. The parameter v
characterizes contact geometry and v is the contact area per
unit volume of substrate. For low volume fractions, a starting
estimate for Bis may be computed assuming cylindrical wires in
an infinite medium [23]. Bic is more difficult to characterize and
requires detailed contact modeling. Because of the uncertainty
in these values, both are treated as parameters in the present
study. Additional dimensionless parameters governing the
problem are the conductivity ratio ks /kt, tube aspect ratio d/Lt,
as well as the geometric ratios LC /Lt, t /Lt and H/ Lt.. The
probability p of nanotube origination, described previously, is
related to the tube density  ; the corresponding dimensionless
parameter is *, which is obtained by normalizing with the
percolation threshold th. The percolation threshold for the
network is estimated as the density at which the average
distance between nanotubes equals the average length of the
H
S
Lt
D
LC
Lt
Figure 1: Computational domain showing nanotubes, source
(S), drain (D) and channel region.
tubes, so that th ~ 1 LS
GOVERNING EQUATION
Though microscale heat transfer effects such as
phonon ballistic transport and confinement may be important, in
some regimes, boundary scattering effects are expected to
dominate in long tubes, and Fourier conduction may be
assumed as a starting point, albeit with a thermal conductivity
that may differ significantly from bulk values. Using the
dimensionless
variable
=(T-Td)/( T )
and
nondimensionalizing all lengths by the tube diameter d, the
governing equations in the tubes and substrate may be written
as:
d 2 i
ds *2


 Ak
hc Pc d 2
h Pd2
; Bis  s s ;  v   v   t
kt A
kt A
 Ps  ks
2
. All computations done in this
study are restricted to d/Lt = 1000.0 and t/d=2.0.
The equations for charge transport in a resistor are
analogous to the thermal transport equations in the Fourier
conduction limit. In this limit, drift-diffusion theory based on
Kirchoff’s law may be used to compute the potential
distribution in the network, analogous to the temperature
distribution. For electrical transport, the i term in Eq. 1 (a) now
represents the dimensionless potential distribution along the
tube. The quantity Bic is the dimensionless charge-transfer
coefficient between tubes i and j at their intersection point and
is specified a priori. Only those segments with intersections
with other tubes include the Bic term in their discretization. The
Bis term is zero for the charge transport since the substrate may
be considered insulating.
Boundary conditions s =1.0 (source) and s =0
(drain) are applied at the two ends of the channel region for the
substrate. Similarly, for the tube tips embedded in the source, i
=1.0 is imposed, while for the tube-tips embedded in the drain
i =0 is imposed. All the tube tips terminating inside the
substrate are assumed adiabatic.
The top and bottom
boundaries of channel are assumed as periodic boundaries for
both substrate and tubes. The remaining two boundaries
(parallel to the network plane) are assumed adiabatic in order to
assess the lateral thermal conductivity of the composite.
Bic  j   i   Bis  s   i   0
intersecting tubes j
1(a)
N tubes
*2 s   Bis  v  i   s   0
i 1
1(b)
Here, i (s ) is the non-dimensional temperature of the ith tube at
the axial location s*, and s(x*,y*,z*) is the substrate temperature.
Thermal contact between tubes i and j is characterized by the
contact Biot number Bic. Heat exchange between each tube and
the substrate is governed by substrate Biot number Bis and the
*
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Copyright © 2005 by ASME
corresponding coupling terms in tube and substrate equations
(Eq. 1(a) and 1(b)):
NUMERICAL METHOD
The finite volume method [24] is used to solve the
temperature field in the tubes and the substrate. Each tube is
divided into 1-D segments, and a control volume balance is
performed on each tube segment. Possible contact with other
tubes is checked for each segment. Bic is zero for the segment if
there is no contact. Similarly, the substrate is divided into 3-D
control volumes, and a control volume balance is performed.
Segments of tubes located in each substrate control volume are
identified and their heat exchange is ascribed to the control
volume to ensure conservation. Second-order accurate central
differencing is used for discretization. The discrete equations
for all the tubes and the substrate are solved sequentially and
iteratively until convergence. For the long channel lengths and
large numbers of tubes, the sequential solver eventually stalls
due to the high degree of coupling between tubes and also
between the tubes and substrate. For these situations, a direct
sparse solver developed by Kundert [25] is used to solve the
resulting system of equations.
Most of the results reported here are computed by
taking an average over 100 random realizations of the network,
though more realizations are used for low densities and short
channel lengths.
 tube centroid 
substrate
 substrate centroid
(2)
where  is the position vector of tube segment centroid
relative to the cell centroid of substrate, as shown in Fig. 2.
COMPUTATUTION
OF
EFFECTIVE
LATERAL
THERMAL CONDUCTIVITY
The lateral thermal conductivity is computed using the
expression:
K t A d 
K A d
ds substrate s s
dx
tubes
K eff 
Ht 
LC


where the first term in the numerator is heat flow through the
tubes in the lateral direction, while the second term represents
heat flow in the substrate in lateral direction. The heat flow in
both tubes and substrate is computed at the source-channel
junction, where the s =1.0 boundary condition is imposed.
Here A , As and  represents cross-sectional area of the tube,
face-area of a control volume of substrate perpendicular to
lateral (x) direction and the non-dimensional temperature drop
across the channel.
GRADIENT CALCULATION FOR COUPLING TERMS
While solving Fourier conduction equations for the
tubes and the substrate, the tube temperature (i) and substrate
temperature (s) in the coupling terms of the equation are
calculated at the centroid of the tube segments and the centroid
RESULTS
GENERAL BEHAVIOR
The temperature distribution in the substrate and in the
tube network is shown in Fig. 3 (a) and (b) for the case LC/Lt =
2.0, H/Lt = 2, Bic = 10.0, Bis = 1.0e-5, ks/kt = 0.001 and * =
18.0, corresponding to LC =4m, Lt = 2m, H = 4m, and  =
4.5 m-2. Contours of constant temperature in the substrate
would be one-dimensional in x for Bis = 0, but due to the
interaction with the tubes, distortion in the contours is observed,
consistent with the temperature plots in the tube in Fig. 3 (b).
For ks/kt <<1, the tubes convey the boundary temperature much
further into the interior that does the substrate, leading to a local
energy source in the substrate. The distortion is related to the
local density of the tubes.
Tube centroid

Cell centroid
Fig. 2. A substrate control volume with two segments of a tube
lying inside it. The displacement vector from substrate cell
centroid to tube segment centroid is shown.
of the substrate cell under consideration. The centroid of the
substrate cell does not coincide with the centroid of tubesegments lying inside the cell, as shown in Fig. 2. This can
introduce significant error in the temperature distribution since
the substrate discretization is usually coarser than that of the
tube. To account for this effect, the substrate temperature
gradient in the plane of the nanotube network is computed and
is used to interpolate the substrate temperature to the tube
centroid location. This interpolated temperature is used in the
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Copyright © 2005 by ASME
using a mesh of size 100 segments per tube and a mesh of
10x20x1 cells in the substrate.
0.08 0.26 0.44 0.62 0.80
Y/L
(a)
(b)
X/L
Fig. 4. Increment in composite effective thermal conductivity
(keff) for two different grid sizes in the substrate and rod. LC/Lt =
2.0 H/Lt = 2, Bic = 10.0, Bis = 1.0e-5, ks/kt = 0.001 and * = 10.0.
NETWORK CONDUCTANCE
In the absence of the substrate, the temperature and electrical
potential distributions are analogous, and electrical conductance
measurements [11] may be used to establish the validity of the
network model. The conductance of the network is computed by
taking an average over 200 random realizations of different
orientations. The length of the tubes in [11] ranges from 1 to 3
m. The length distribution of nanotubes has not been reported
in [11]. Therefore, we have taken a simple approach and
assumed that all tubes are 2 m long in this simulation.. The
percolation threshold for the network is estimated to be 0.25
m-2. Simulations are performed for densities in the range 1-10
m-2, for channel lengths varying from 1 to 25 m and with a
width H of 90 m, corresponding to the dimensionless
parameters LC/Lt ~ 0.5-12.5, H/Lt = 45 . The device dimensions
and tube lengths are chosen to match those of the experiments
of Snow et al. [11].
In Fig. 5 (a), the network conductance  is shown as a
function of LC/Lt for several tube densities above the
percolation threshold for nearly perfect tube-tube contact (i.e.
Bic ~ 5.0). For long channels (LC < Lt) there are no tubes
directly bridging the source and drain and heat (current) can
only flow because of the presence of the network. If the tube
density is sufficiently high (greater than the percolation
threshold), a continuous thermal (electrical) path exists from
source to drain, and  is seen to be non-zero even for LC/Lt >1.
Fig. 5 (a) shows that the conductance exponent, n, is close to 1.0, which, in the electrical analogue, corresponds to ohmic
conduction for the high densities (=10m-2; *=40). The
exponent increases to -1.80 at lower densities (1.35m-2; *=5),
Fig. 3. Non-dimensional temperature distribution in (a)
substrate (b) tube network. LC/Lt = 2.0, H/Lt = 2, Bic = 10.0, Bis
= 1.0e-5, ks/kt = 0.001 and * = 18.0.
GRID INDEPENDENCE
Grid independence tests were conducted for the case of
LC/Lt = 2.0, H/Lt = 2, Bic = 10.0, Bis = 1.0e-5, ks/kt = 0.001 and
* = 10.0 (LC =4m, H = 4m,  = 2.5 m-2) for nondimensional channel length LC/Lt varying from 0.5-9 (LC = 1-18
m) Fig. 4 shows the percentage change in the composite
thermal conductivity for two different grid sizes. For the first
case there are 100 cells/unit tube length and 10 cells/unit tube
length for the substrate i.e. a channel having a length equal to
the tube length is divided into 10 cells. The second case
corresponds to 200 segments per tube, with 20 substrate cells
per unit length of tube. The grid spacing is the same in the y
direction in the substrate. Only one cell is used in the direction
normal to the network layer. The results are seen to differ by
less than 0.5% between the two cases. The simulations
presented in the rest of this paper were therefore performed
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Copyright © 2005 by ASME
indicating a non-linear dependence of conductance on channel
length. Note that the asymptotic limit of the conductance
Some evidence of this is visible in the scatter in the
experimental data at low densities.
The dependence of conductance exponent on channel
length is explored in Fig. 5(b) for Bic ~ 5.0 and for densities in
the range 1.35-10m-2, corresponding to * values of 5-40. For
densities >3.0 m-2 (* >12), the exponent approaches -1.0 with
increasing channel length (the ohmic limit). Larger exponents,
corresponding to non-ohmic transport, are observed for the
shorter channel lengths. This is consistent with experimental
observations, where conductance is seen to scale more rapidly
with channel length for small LC [11].
SUBSTRATE-TUBE
CONTACT
CONDUCTANCE
EFFECT
Recently, experimental studies conducted in [18]
suggest that heat transport in nanotube composites may be
limited by exceptionally small interfacial thermal conductance
values. The value of the interfacial resistance between the
carbon nanotube and the substrate was reported to be 8.3x10-8
m2 K/W [17,18] for carbon nanotubes in D2O. The nondimensional contact parameter Bis evaluated using this contact
conductance is 10-5. Values of Bis in the range 10-1-10-5 were
considered in this study. There are no reliable guidelines to
determine the value of the tube-tube contact parameter Bic, and
a range 10-5 – 1.0 is chosen. For the thermal conductivity ratio,
again, a wide variation is possible. Free-standing MWCNTs
have exhibited thermal conductivity values of 3000 W/mK or
more [13,14], but these values are expected to fall substantially
when the CNTs are encased in plastic. Free-standing Si-NWs
show an order of magnitude reduction with respect to bulk
thermal conductivity [15]. The thermal conductivity of typical
plastic substrates is in the range of 0.1-1 W/mK. In this study,
we explore a range of ks/kt of 10-1 – 10-3.
Fig. 5 (a) Computed conductance dependence on channel length
for different densities () in the strong coupling limit (Bic = 5)
is compared with experimental results from [11].  0 = 1.0
(simulation),  0 = 1.0 (experimental) for =10.0 m-2).
 0 =1
(simulation) and  0 =1.40 (experimental) for =1.0 m . The
-2
number after each curve corresponds to  and the number in [ ]
corresponds to the  value used in the experiments from [11].
(b) Dependence of the conductance exponent n on channel
length for different densities based on Fig. 6(a)
(i.e.  /  0 ~ LC ). Other parameters, chosen to reflect the
n
experimental conditions in [11], are: H=90m, Lt = 2 m and
LC=1 - 25m.
exponent for infinite samples with perfect tube/tube contact is 1.97 [26,27]. The linear scaling of conductance with LC for high
densities agrees very well with experimental observations of
Snow et al. [11]. The observed non-linear behavior for low
density is expected because the density value is close to the
percolation threshold. Snow et al. reported a conductanceexponent of -1.80 for a density of 1.0 m-2 and channel length >
5m. For the same device dimensions, this value of the
exponent is close to that obtained from our simulations for a
density of 1.35m-2. Small variations in experimental
parameters such as tube diameter, nanotube contact strength,
tube electronic properties as well as the presence of a
distribution of tube lengths (1-3m), which is not included in
the simulation, may explain the difference. The contact
resistance between the nanotubes and the source and drain
electrodes as well as insufficiently large samples for ensemble
averaging in the experimental setup may also be responsible.
Fig. 6. Effect of substrate-tube contact conductance on keff for
varying channel length. LC/Lt = 2.0, H/Lt = 2, Bic = 10.0, ks/kt =
0.001 and * = 10.0.
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The percentage increase in keff of the composite is
plotted against LC/Lt ratio for different Bis in Fig. 6. The tubetube contact parameter is held at Bic =10, denoting nearly
perfect contact. The conductivity ratio is 10-3, denoting highly
conducting tubes in a relatively insulating substrate. The overall
shape of the curve reflects the percolation behavior of the
network for this low value of ks/kt. A sharp increase in keff is
observed for shorter channel lengths as a result of highly
conducting tubes directly bridging source and drain. As the
channel length increases, thermal conduction is dominated by
the network conductance, which achieves invariance beyond
LC/Lt >6 or so. The tube-substrate contact parameter Bis ceases
to be limiting for Bis>10-3, and the keff variation with Lc/Lt
becomes independent of Bis beyond this value. Since the tubes
are much more conducting than the substrate, large increases in
thermal conductivity are observed. For LC/Lt ~ 0.5, a 280 %
increase in thermal conductivity is observed even for low values
of Bis (~ 1.0e-5 ), while increasing Bis to 1.0e-1 leads to a 375%
increase in the thermal conductivity. For larger channel lengths,
increases of 175 % and 140 % in thermal conductivity are
observed for Bis ~ 1.0e-3 and Bis ~1.0e-5. Since the ks/kt value
used here is expected to be typical of many composites for TFT
applications, the results in this section demonstrate that if the
percolation properties of the network could be maintained by
high tube-tube contact, the network itself could provide a
pathway for heat removal.
the overall shape of the curves is similar to that in Fig. 6.
Decreasing Bic from 1.0 to 10-5 decreases keff by about 50% for
long channels. Beyond Bic >1.0e-2, tube-tube contact is
sufficiently perfect that it ceases to be limiting; consequently
the keff curves become coincident in Fig. 8. Previous theoretical
models for computing keff of nanotube composites ignore
contact conductance between the nanotubes [17,20]. The
present analysis reveals that for tube densities higher than the
percolation threshold, Bic emerges as an important parameter
controlling keff for low values of ks/kt.
CONDUCTIVITY RATIO EFFECT
We now consider the effect of thermal conductivity
ratio on keff. Simulations are performed for ks/kt ratio varying
from 10-1 to 10-3 keeping other parameters constant LC/Lt = 2.0,
H/Lt = 2, Bic = 10.0, Bis = 1.0e-5 and * = 10.0 (LC =4m, H =
4m,  = 2.5 m-2). For the high value of tube-tube contact
considered here, Fig. 8 shows that percolating conduction in the
network dominates for
ks/kt=10-3, as expected; this is
evidenced by the steep rise in keff for low Lc/Lt. As ks/kt is
increased, some evidence of percolation may still be detected
for ks/kt <5x10-3. For higher values, though, network
conductance ceases to be a dominant contributor and the
increase in keff over ks drops to zero, signifying that the
substrate now dominates conduction through the composite.
These computations point to the necessity of accurately
characterizing the thermal conductivity of CNTs and Si NWs
embedded in substrates. If the presence of the substrate
substantially reduces kt, it is possible that ks/kt would be
relatively high and the network would no longer provide a
significant pathway for heat transfer even if effective tube-tube
contact could be maintained. In this limit, the large surface area
of contact between the tubes and the substrate allows heat to
leak from the network to the substrate, and the primary
mechanism for heat removal would be the substrate. However,
TUBE-TUBE CONTACT CONDUCTANCE EFFECT
In this section, the effect of tube-tube contact is
investigated by varying Bic in the range 100-10-5.
Fig. 7. Effect of tube-tube contact conductance on keff for
varying channel length. LC/Lt = 2.0, H/Lt = 2, Bis = 1.0e-5, ks/kt =
0.001 and * = 10.0.
Fig. 7 show the plots for different Bic, while other
parameters are kept constant at LC/Lt = 2.0, H/Lt = 2,Bis = 1.0e-5,
ks/kt = 0.001 and * = 10.0 ( corresponding to LC =4m, H =
4m,  = 2.5 m-2). As expected for this low value of ks/kt, the
overall behavior is dominated by the network conductance and
Fig. 8. Effect of substrate-tube conductivity ratio on keff for
varying channel length. LC/Lt = 2.0, H/Lt = 2, Bic = 10.0, Bis =
1.0e-5 and * = 10.0.
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Copyright © 2005 by ASME
to vary linearly with the density of the nanotubes. The range of
tube-densities considered here is much higher than the
percolation threshold (~ 0.25 m-2), but the volume fraction of
the tubes is still relatively low, and may be considered to be in
the dilute limit. This linear behavior suggests that the increase
in thermal conductivity is directly proportional to the volume
fraction of nanotubes, which is in agreement with the model
proposed by Nan et al. [17] in the dilute limit.
since both substrate and network conductance would be low in
this limit, high channel temperatures and degradation of
electrical performance would be expected.
DENSITY EFFECT
Nanotube density plays an important role in both the
electrical and thermal performance of the composite. On the
one hand, high densities tend to increase transport in the
composite by increasing the number of available highconducting pathways between source and drain. On the other
hand, high densities also imply many more contact locations
between tubes, and a concomitant increase in the number of
scattering locations for both electrons and phonons, leading to a
drop in tube electrical and thermal conductivity. Furthermore,
higher current in the network could lead to high local heat
generation which could burn nanotubes and eventually cause
transistor breakdown. The present analysis assumes kt to be
known a priori, and does not consider coupled electro-thermal
transport. However, the direct effect of tube density on keff may
be deduced from our analysis. Fig. 9 shows the effect of tube
density on thermal conductivity enhancement for LC/Lt = 2.0
(LC =4m), H/Lt = 2 (H = 4m), Bic = 10.0, Bis = 1.0e-5 and
ks/kt=0.001.
CONCLUSIONS
A computational diffusive transport model has been
developed to explore the dependence of thermal conductance
and conductivity on the channel length for thin films made of
random percolating nanotubes. In the limit of an insulating
substrate, model predictions are compared with measurements
of electrical conductance and found to be in good agreement.
Transport in finite networks with tube-to-tube contact for
medium channel lengths (LC/Lt ~1-10) is studied and provides
insight into the dominant mechanisms for thermal transport in
these composites.
For low values of ks/kt, typical of highly conducting
CNTs in plastic substrates, percolating conduction in the
network is seen to dominate over a wide range of tube-tube and
tube-substrate contact parameters. On the other hand, as ks/kt
increases, the resistance to heat flow offered by the substrate is
no longer limiting. The high surface area of contact between the
substrate and the network implies that heat will leak from the
network to the substrate, and be transported primarily by the
substrate; in this limit, percolating conduction in the network
ceases to be relevant. This switchover signals a significant
decrease in effective thermal conductivity and is critical to
characterize accurately. Accurate measurements of kt as well as
tube-tube and tube-substrate contact resistance are necessary.
The present analysis employs Fourier theory as a basis.
It is likely that the large aspect ratio of tubes implies the
predominance of boundary scattering, and diffusive transport is
likely to dominate. However, the value of kt is expected to
depart substantially from bulk, and would not be known a
priori. Effects such as the influence of tube-substrate and tubetube scattering events in decreasing kt may be determined
computationally by using molecular dynamics of tube and
substrate. Ultimately, the self-consistent computation of device
I-V performance requires a similar characterization of electrical
transport in the network under drift-diffusion assumptions, and
the coupling of electrical and thermal computations.
Fig. 9. Effect of tube density on keff for varying channel length.
LC/Lt = 2.0, H/Lt = 2, Bic = 10.0, Bis = 1.0e-5 and ks/kt = 0.001
Since all other parameters are held constant with
respect to Figs. 6 and 7, and all tube densities are above the
percolation threshold, the overall shape of the curves in Fig. 9
indicate the dominance of the network in determining keff.. As
expected, higher tube densities result in an increase in keff as the
number of available pathways increases; the present model is
not capable of incorporating the effect on increased scattering in
decreasing kt. Substantial increases in keff are possible for
higher densities, with values of over 500% for long channels.
For higher densities, the increase in thermal conductivity is seen
ACKNOWLEDGMENTS
Support of S. Kumar and J. Murthy under NSF Grant
EEC-0228390 and the Purdue Research Foundation is
gratefully acknowledged.
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