We have determined the thermal resistance for transferring heat between individual single-walled carbon nanotube devices and solid substrates. Using sapphire and comparing our results to previous results obtained from SiO2, we find that the resistance is dominated by interfacial resistance rather than the spreading resistance of heat for diffusing into the substrate. Our results are in agreement to a recent model for the thermal resistance of nanoscale constrictions. Our results suggest that relatively short contact lengths (~10–30 nm) to a typical solid should be sufficient to transfer heat efficiently into carbon nanotubes, underscoring the potential of carbon nanotubes for nanoscale thermal management

Bockrath, Marc

Chiu, Hsin-Ying

Maune, Hareem

English

Caltech Authors - Main

APPLIED PHYSICS LETTERS 89, 013109 2006Thermal resistance of the nanoscale constrictions between carbon
nanotubes and solid substrates
Hareem Maune, Hsin-Ying Chiu, and Marc Bockratha
Department of Applied Physics, California Institute of Technology, Pasadena, California 91125
Received 7 March 2006; accepted 17 May 2006; published online 6 July 2006
We have determined the thermal resistance for transferring heat between individual single-walled
carbon nanotube devices and solid substrates. Using sapphire and comparing our results to previous
results obtained from SiO2, we find that the resistance is dominated by interfacial resistance rather
than the spreading resistance of heat for diffusing into the substrate. Our results are in agreement to
a recent model for the thermal resistance of nanoscale constrictions. Our results suggest that
relatively short contact lengths 10–30 nm to a typical solid should be sufficient to transfer heat
efficiently into carbon nanotubes, underscoring the potential of carbon nanotubes for nanoscale
thermal management. © 2006 American Institute of Physics. DOI: 10.1063/1.2219095As electronic devices are further miniaturized and inte-
gration density is increased, the dissipation of heat becomes
an important issue for maintaining viability. Thermal man-
agement is thus a critical issue towards future nanoelectronic
devices. This is important also for obtaining fundamental
knowledge about how thermal energy is transported in
nanoscale systems. For example, in the nanometer scale,
thermal transport differs from the situation in bulk materials
because the mean free path for phonon scattering can be
large compared to device dimensions, leading to novel phys-
ics such as quantum thermal phenomena.1–3
Of particular interest are carbon nanotubes, which have
been proposed both as prototypical nanoscale circuit ele-
ments and by, virtue of their extraordinary current carrying
capacity, as potential materials for interconnects. Recent
studies have shown that nanotubes are remarkable thermal
conductors4–8 even at elevated temperatures9,10 and ballistic
phonon transport has been demonstrated in micron-scale
devices,11–13 with a mean free path in agreement to theoret-
ical calculations.14 Nevertheless, for thermal management it
is also of interest to determine the thermal resistance be-
tween nanotubes and different materials.
Here we show, using the electrical breakdown phenom-
enon to perform thermometry,15 that in sapphire and SiO2
substrates the dominant thermal resistance between single-
walled nanotubes and solid substrates stems from the
substrate-nanotube interface. We find a value of 3 K m/W
for sapphire substrates compared to 0.6 K m W−1 obtained
from our previously determined value for multiwalled nano-
tubes on SiO2,
11 the value 12 K m W−1 obtained from
scanned thermal microscopy for multiwalled nanotubes on
SiO2 Ref. 16, and 2 K m W−1 based on breakdown data
reported for single-walled nanotubes on SiO2.
17 Our results
are consistent with thermal transport that is limited by the
interfacial thermal impedance between the nanotube and sub-
strate, and we find quantitative agreement with a recently
reported theoretical model.18
Sample fabrication is as follows. Individual single-
walled carbon nanotubes are grown on sapphire by dipping
the chip in iron nitrate and then drying after dipping in hex-
ane. The nanotubes are grown under a mixture of CH4 and
aElectronic mail: mwb@caltech.edu
0003-6951/2006/891/013109/3/$23.00 89, 01310
Downloaded 14 Jul 2006 to 131.215.240.9. Redistribution subject to H2, and yield predominantly single-walled nanotubes with a
fraction of tubes containing multiple shells.19 On sapphire
substrates, we find aligned nanotube growth on the a and r
planes, consistent with the results of Han et al.20 Once iso-
lated individual nanotubes are located relative to predefined
alignment marks, Cr/Au electrodes are placed on the nano-
tubes using electron beam lithography.
Current voltage measurements are obtained from each
tube, and the voltage is increased until the current drops
abruptly, signaling electrically driven breakdown. Previous
studies indicate that this breakdown occurs when the nano-
tube reaches a fixed temperature TB, which has been esti-
mated to be 600 °C.15 In addition to thermalized phonons
at temperature TB, there is also a nonequilibrium population
of optical phonons present generated by emission from the
electrons. However, these phonons are expected to decay
into the thermal population before they can carry significant
thermal energy out of the nanotube.11
Figure 1 shows I-V curves obtained from individual
single-walled carbon nanotube devices for lengths ranging
from 0.25 to 8 m. The current begins to saturate at larger
FIG. 1. Current-voltage characteristics for sapphire-supported individual
carbon nanotube devices. The voltage is increased until the current drops
abruptly due to electrical breakdown. The length of each nanotube is indi-
cated. Longer nanotubes tend to require more electrical power for
breakdown.
© 2006 American Institute of Physics9-1
AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
013109-2 Maune, Chiu, and Bockrath Appl. Phys. Lett. 89, 013109 2006voltages because of the increased scattering from optical or
zone boundary phonon emission, tending towards the maxi-
mum of 4e /hEop-zb25 A/nanotube shell, where
Eop-zb0.18 eV is the characteristic zone boundary or opti-
cal phonon energy in nanotubes.21 Although the maximum
current can vary for each device, the required power to reach
breakdown increases approximately linearly with the nano-
tube length.
Figure 2 shows a plot of the power required versus
length for sapphire substrates. The data follows an approxi-
mately linear trend. This indicates that the device cooling
occurs along the entire length, and therefore longer nano-
tubes require more power to reach the same temperature.
Fitting a straight line to the data shows that P=cL, with
c=0.15 mW/m. With TB600 °C, the thermal resistance
on a per-length basis to the substrate can thus be inferred
as 3 K m/W for sapphire compared to 0.6–12 K m/W
obtained from previous work for multiwalled nanotubes on
SiO2 Refs. 11 and 16 and 2 K m/W single-walled nano-
tubes on SiO2.
17 These values are thus similar in magnitude
for both SiO2 and sapphire substrates.
However, the thermal conductivity of sapphire
is 40 W/K m, while the thermal conductivity of SiO2 is
1 W/K m. We thus expect if cooling were determined by
the substrate thermal conductivity that the required break-
down power would be approximately 40 times higher for the
case of sapphire as compared to SiO2. Instead, we find simi-
lar values. This suggests that the thermal interface resistance,
rather than the spreading resistance of thermal energy into
the substrate, is the dominant thermal resistance between the
nanotube and the substrate. Indeed, in recent work it has
been pointed out that consideration of an interfacial bound-
ary thermal resistance is necessary to understand thermal
transport across nanoscale boundaries such as the interface
between the nanotubes and the substrate.18 The author gives
the expression
Rc =
1
L
 1
w
ln2D
a
	 − 1
2w
+
1
s
ln D
a
	 + Rb
2a

 1
for the thermal resistance, where w is the nanotube thermal
conductivity, s is the substrate thermal conductivity, D is
the nanotube diameter, a is the effective contact width, L is
the nanotube length, and Rb is the interfacial resistance.
Since D is much less than the phonon mean free path in
nanotubes, the terms with w can be neglected.
Our data indicate that the term including Rb is an impor-
FIG. 2. Breakdown power vs length for sapphire-supported individual car-
bon nanotube devices. A linear fit to the data is shown with a slope of
0.15 mW/m.tant contribution to Rc. To estimate the order of magnitude of
Downloaded 14 Jul 2006 to 131.215.240.9. Redistribution subject to Rb, we use the diffuse mismatch D-M model
18,22,23 and
apply it to graphite/sapphire or graphite/SiO2 interfaces. In
this picture, phonons reaching an interface between two ma-
terials are either transmitted across the boundary or reflected
with a probability that depends only on the density of pho-
non modes in each material. This picture is most appropriate
for a disordered interface. However, the results obtained
from a model based on specular transmission or reflection
from the interface typically yield similar results,23 and thus
we use the D-M model for simplicity. In the D-M model, Rb
is given by Rb=4/ 1→2C1v1, where C1 is the specific heat
per unit volume, v1 is the characteristic speed of sound, and
1→2 is the transmission coefficient for phonons to be trans-
mitted from medium 1 to medium 2. For most solid
interfaces 1→2 is a factor of order unity, so for simplicity
we take 1→2=1. Using also v15000 m/s and
C1103 J /kg K, we find Rb410−10 m2 K/W. The pa-
rameter a can be estimated from the calculated relaxed shape
under the van der Waals forces pinning it to the substrates.24
For tubes with a characteristic diameter 2.0 nm as in our
experiment, we expect 2a0.3–0.5 nm.
Using the above value for Rb and taking D=2 nm,
SiO2 =1.0 W/m K, and sapphire=40 W/m K, we find
Rc=1.0 K m/W for a unit length for sapphire and
Rc1.4 K m/W for a unit length for SiO2. These order of
magnitude estimates are within a factor of order unity to our
measured values of 3 K m/W for sapphire substrates and
previously obtained values by our group and others
0.6–12 K m/W for SiO2 substrates. Note also that the val-
ues we have obtained are within the typical range observed
for the surface thermal resistance between most solids and
smaller than the resistance observed between surfactant-
coated nanotubes and liquids.25 Extrapolating these results
suggests that using a contact length lC of only a few tens of
nanometers between a typical solid and a nanotube should
enable approaching the minimum thermal resistance allowed
by quantum mechanics. This minimum value is 1/MQ,
where M is the number of occupied phonon branches, typi-
cally 10–20 for a single-walled nanotube at room tempera-
ture, and Q=
2kB
2T /3h is the thermal conductance
quantum.1,2 Note that this assumes ballistic phonon transport.
For sufficiently long nanotubes, phonon transport is diffusive
and the nanotube thermal resistance would be larger than the
quantum mechanical limit above. The above estimate for lC
is thus an upper bound on the required contact length so that
the intrinsic nanotube thermal resistance is dominant. Never-
theless, this prediction should be supported by more accurate
calculations, such as molecular dynamics simulations and
further experiments.
In sum, we have determined the thermal resistance be-
tween carbon nanotubes and sapphire substrates using nano-
tube electrical breakdown to determine the temperature. We
find that the thermal resistance is dominated by the interfa-
cial thermal resistance and is little affected by the substrate
thermal conductivity for both SiO2 and sapphire. Our mea-
sured values for the interfacial resistance between nanotubes
and sapphire agree with recent theory using order of magni-
tude estimates for the surface resistance based on the diffuse
mismatch model. Our results suggest that relatively short
contact lengths 10 nm to a typical solid can transfer heat
efficiently into nanotubes, further underscoring their poten-
tial for thermal management in nanoscale circuits.
AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
013109-3 Maune, Chiu, and Bockrath Appl. Phys. Lett. 89, 013109 2006This work was supported by the ONR, the Powel Foun-
dation, and the NSF. The authors thank Natalio Mingo for
helpful discussions.
1K. Schwab, E. A. Henriksen, J. M. Worlock, and M. L. Roukes, Nature
London 404, 974 2000.
2J. B. Pendry, J. Phys. A 16, 2161 1983.
3R. Maynard and E. Akkermans, Phys. Rev. B 32, 5440 1985.
4J. Hone, M. Whitney, and A. Zettl, Synth. Met. 103, 2498 1999.
5J. Hone, B. Batlogg, Z. Benes, A. T. Johnson, and J. E. Fischer, Science
289, 1730 2000.
6M. C. Llaguno, J. E. Fischer, A. T. Johnson, and J. Hone, Nano Lett. 4, 45
2004.
7J. P. Small, K. M. Perez, and P. Kim, Phys. Rev. Lett. 91, 256801 2003.
8P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, Phys. Rev. Lett. 87,
215502 2001.
9E. Pop, D. Mann, J. Cao, Q. Wang, K. Goodson, and H. J. Dai, Phys. Rev.
Lett. 95, 155505 2005.
10E. Pop, D. Mann, Q. Wang, K. Goodson, and H. J. Dai, Nano Lett. 6, 96
2006.
11H. Y. Chiu, V. V. Deshpande, H. W. C. Postma, C. N. Lau, C. Miko, L.
Forro, and M. Bockrath, Phys. Rev. Lett. 95, 226101 2005.
Downloaded 14 Jul 2006 to 131.215.240.9. Redistribution subject to 12C. H. Yu, L. Shi, Z. Yao, D. Y. Li, and A. Majumdar, Nano Lett. 5, 1842
2005.
13E. Brown, L. Hao, J. C. Gallop, and J. C. Macfarlane, Appl. Phys. Lett.
87, 023107 2005.
14N. Mingo and D. A. Broido, Phys. Rev. Lett. 95, 096105 2005.
15P. G. Collins, M. Hersam, M. Arnold, R. Martel, and P. Avouris, Phys.
Rev. Lett. 86, 3128 2001.
16P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, Physica B 323, 67
2002.
17A. Javey, J. Guo, M. Paulsson, Q. Wang, D. Mann, M. Lundstrom, and H.
J. Dai, Phys. Rev. Lett. 92, 106804 2004.
18R. Prasher, Nano Lett. 5, 2155 2005.
19J. Kong, H. T. Soh, A. M. Cassell, C. F. Quate, and H. Dai, Nature
London 395, 878 1998.
20S. Han, X. L. Liu, and C. W. Zhou, J. Am. Chem. Soc. 127, 5294 2005.
21Z. Yao, C. Kane, and C. Dekker, Phys. Rev. Lett. 84, 2941 2000.
22E. T. Swartz and R. O. Pohl, Rev. Mod. Phys. 61, 605 1989.
23D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar,
H. J. Maris, R. Merlin, and S. R. Phillpot, J. Appl. Phys. 93, 793 2003.
24T. Hertel, R. E. Walkup, and P. Avouris, Phys. Rev. B 58, 13870 1998.
25S. T. Huxtable, D. G. Cahill, S. Shenogin, L. P. Xue, R. Ozisik, P. Barone,
M. Usrey, M. S. Strano, G. Siddons, M. Shim, and P. Keblinski, Nat.
Mater. 2, 731 2003.
AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp


Thermal resistance of the nanoscale constrictions between carbon nanotubes and solid substrates

APPLIED PHYSICS LETTERS 89, 013109 2006Thermal resistance of the nanoscale constrictions between carbon
nanotubes and solid substrates
Hareem Maune, Hsin-Ying Chiu, and Marc Bockratha
Department of Applied Physics, California Institute of Technology, Pasadena, California 91125
Received 7 March 2006; accepted 17 May 2006; published online 6 July 2006
We have determined the thermal resistance for transferring heat between individual single-walled
carbon nanotube devices and solid substrates. Using sapphire and comparing our results to previous
results obtained from SiO2, we find that the resistance is dominated by interfacial resistance rather
than the spreading resistance of heat for diffusing into the substrate. Our results are in agreement to
a recent model for the thermal resistance of nanoscale constrictions. Our results suggest that
relatively short contact lengths 10–30 nm to a typical solid should be sufficient to transfer heat
efficiently into carbon nanotubes, underscoring the potential of carbon nanotubes for nanoscale
thermal management. © 2006 American Institute of Physics. DOI: 10.1063/1.2219095As electronic devices are further miniaturized and inte-
gration density is increased, the dissipation of heat becomes
an important issue for maintaining viability. Thermal man-
agement is thus a critical issue towards future nanoelectronic
devices. This is important also for obtaining fundamental
knowledge about how thermal energy is transported in
nanoscale systems. For example, in the nanometer scale,
thermal transport differs from the situation in bulk materials
because the mean free path for phonon scattering can be
large compared to device dimensions, leading to novel phys-
ics such as quantum thermal phenomena.1–3
Of particular interest are carbon nanotubes, which have
been proposed both as prototypical nanoscale circuit ele-
ments and by, virtue of their extraordinary current carrying
capacity, as potential materials for interconnects. Recent
studies have shown that nanotubes are remarkable thermal
conductors4–8 even at elevated temperatures9,10 and ballistic
phonon transport has been demonstrated in micron-scale
devices,11–13 with a mean free path in agreement to theoret-
ical calculations.14 Nevertheless, for thermal management it
is also of interest to determine the thermal resistance be-
tween nanotubes and different materials.
Here we show, using the electrical breakdown phenom-
enon to perform thermometry,15 that in sapphire and SiO2
substrates the dominant thermal resistance between single-
walled nanotubes and solid substrates stems from the
substrate-nanotube interface. We find a value of 3 K m/W
for sapphire substrates compared to 0.6 K m W−1 obtained
from our previously determined value for multiwalled nano-
tubes on SiO2,11 the value 12 K m W−1 obtained from
scanned thermal microscopy for multiwalled nanotubes on
SiO2 Ref. 16, and 2 K m W−1 based on breakdown data
reported for single-walled nanotubes on SiO2.17 Our results
are consistent with thermal transport that is limited by the
interfacial thermal impedance between the nanotube and sub-
strate, and we find quantitative agreement with a recently
reported theoretical model.18
Sample fabrication is as follows. Individual single-
walled carbon nanotubes are grown on sapphire by dipping
the chip in iron nitrate and then drying after dipping in hex-
ane. The nanotubes are grown under a mixture of CH4 and
aElectronic mail: mwb@caltech.edu
0003-6951/2006/891/013109/3/$23.00 89, 01310
Downloaded 14 Jul 2006 to 131.215.240.9. Redistribution subject to H2, and yield predominantly single-walled nanotubes with a
fraction of tubes containing multiple shells.19 On sapphire
substrates, we find aligned nanotube growth on the a and r
planes, consistent with the results of Han et al.20 Once iso-
lated individual nanotubes are located relative to predefined
alignment marks, Cr/Au electrodes are placed on the nano-
tubes using electron beam lithography.
Current voltage measurements are obtained from each
tube, and the voltage is increased until the current drops
abruptly, signaling electrically driven breakdown. Previous
studies indicate that this breakdown occurs when the nano-
tube reaches a fixed temperature TB, which has been esti-
mated to be 600 °C.15 In addition to thermalized phonons
at temperature TB, there is also a nonequilibrium population
of optical phonons present generated by emission from the
electrons. However, these phonons are expected to decay
into the thermal population before they can carry significant
thermal energy out of the nanotube.11
Figure 1 shows I-V curves obtained from individual
single-walled carbon nanotube devices for lengths ranging
from 0.25 to 8 m. The current begins to saturate at larger
FIG. 1. Current-voltage characteristics for sapphire-supported individual
carbon nanotube devices. The voltage is increased until the current drops
abruptly due to electrical breakdown. The length of each nanotube is indi-
cated. Longer nanotubes tend to require more electrical power for
breakdown.
© 2006 American Institute of Physics9-1
AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
013109-2 Maune, Chiu, and Bockrath Appl. Phys. Lett. 89, 013109 2006voltages because of the increased scattering from optical or
zone boundary phonon emission, tending towards the maxi-
mum of 4e /hEop-zb25 A/nanotube shell, where
Eop-zb0.18 eV is the characteristic zone boundary or opti-
cal phonon energy in nanotubes.21 Although the maximum
current can vary for each device, the required power to reach
breakdown increases approximately linearly with the nano-
tube length.
Figure 2 shows a plot of the power required versus
length for sapphire substrates. The data follows an approxi-
mately linear trend. This indicates that the device cooling
occurs along the entire length, and therefore longer nano-
tubes require more power to reach the same temperature.
Fitting a straight line to the data shows that P=cL, with
c=0.15 mW/m. With TB600 °C, the thermal resistance
on a per-length basis to the substrate can thus be inferred
as 3 K m/W for sapphire compared to 0.6–12 K m/W
obtained from previous work for multiwalled nanotubes on
SiO2 Refs. 11 and 16 and 2 K m/W single-walled nano-
tubes on SiO2.17 These values are thus similar in magnitude
for both SiO2 and sapphire substrates.
However, the thermal conductivity of sapphire
is 40 W/K m, while the thermal conductivity of SiO2 is
1 W/K m. We thus expect if cooling were determined by
the substrate thermal conductivity that the required break-
down power would be approximately 40 times higher for the
case of sapphire as compared to SiO2. Instead, we find simi-
lar values. This suggests that the thermal interface resistance,
rather than the spreading resistance of thermal energy into
the substrate, is the dominant thermal resistance between the
nanotube and the substrate. Indeed, in recent work it has
been pointed out that consideration of an interfacial bound-
ary thermal resistance is necessary to understand thermal
transport across nanoscale boundaries such as the interface
between the nanotubes and the substrate.18 The author gives
the expression
Rc =
1
L 1w ln2Da 	 − 12w + 1s ln Da	 + Rb2a
 1
for the thermal resistance, where w is the nanotube thermal
conductivity, s is the substrate thermal conductivity, D is
the nanotube diameter, a is the effective contact width, L is
the nanotube length, and Rb is the interfacial resistance.
Since D is much less than the phonon mean free path in
nanotubes, the terms with w can be neglected.
Our data indicate that the term including Rb is an impor-
FIG. 2. Breakdown power vs length for sapphire-supported individual car-
bon nanotube devices. A linear fit to the data is shown with a slope of
0.15 mW/m.tant contribution to Rc. To estimate the order of magnitude of
Downloaded 14 Jul 2006 to 131.215.240.9. Redistribution subject to Rb, we use the diffuse mismatch D-M model18,22,23 and
apply it to graphite/sapphire or graphite/SiO2 interfaces. In
this picture, phonons reaching an interface between two ma-
terials are either transmitted across the boundary or reflected
with a probability that depends only on the density of pho-
non modes in each material. This picture is most appropriate
for a disordered interface. However, the results obtained
from a model based on specular transmission or reflection
from the interface typically yield similar results,23 and thus
we use the D-M model for simplicity. In the D-M model, Rb
is given by Rb=4/ 1→2C1v1, where C1 is the specific heat
per unit volume, v1 is the characteristic speed of sound, and
1→2 is the transmission coefficient for phonons to be trans-
mitted from medium 1 to medium 2. For most solid
interfaces 1→2 is a factor of order unity, so for simplicity
we take 1→2=1. Using also v15000 m/s and
C1103 J /kg K, we find Rb410−10 m2 K/W. The pa-
rameter a can be estimated from the calculated relaxed shape
under the van der Waals forces pinning it to the substrates.24
For tubes with a characteristic diameter 2.0 nm as in our
experiment, we expect 2a0.3–0.5 nm.
Using the above value for Rb and taking D=2 nm,
SiO2 =1.0 W/m K, and sapphire=40 W/m K, we find
Rc=1.0 K m/W for a unit length for sapphire and
Rc1.4 K m/W for a unit length for SiO2. These order of
magnitude estimates are within a factor of order unity to our
measured values of 3 K m/W for sapphire substrates and
previously obtained values by our group and others
0.6–12 K m/W for SiO2 substrates. Note also that the val-
ues we have obtained are within the typical range observed
for the surface thermal resistance between most solids and
smaller than the resistance observed between surfactant-
coated nanotubes and liquids.25 Extrapolating these results
suggests that using a contact length lC of only a few tens of
nanometers between a typical solid and a nanotube should
enable approaching the minimum thermal resistance allowed
by quantum mechanics. This minimum value is 1/MQ,
where M is the number of occupied phonon branches, typi-
cally 10–20 for a single-walled nanotube at room tempera-
ture, and Q=2kB
2T /3h is the thermal conductance
quantum.1,2 Note that this assumes ballistic phonon transport.
For sufficiently long nanotubes, phonon transport is diffusive
and the nanotube thermal resistance would be larger than the
quantum mechanical limit above. The above estimate for lC
is thus an upper bound on the required contact length so that
the intrinsic nanotube thermal resistance is dominant. Never-
theless, this prediction should be supported by more accurate
calculations, such as molecular dynamics simulations and
further experiments.
In sum, we have determined the thermal resistance be-
tween carbon nanotubes and sapphire substrates using nano-
tube electrical breakdown to determine the temperature. We
find that the thermal resistance is dominated by the interfa-
cial thermal resistance and is little affected by the substrate
thermal conductivity for both SiO2 and sapphire. Our mea-
sured values for the interfacial resistance between nanotubes
and sapphire agree with recent theory using order of magni-
tude estimates for the surface resistance based on the diffuse
mismatch model. Our results suggest that relatively short
contact lengths 10 nm to a typical solid can transfer heat
efficiently into nanotubes, further underscoring their poten-
tial for thermal management in nanoscale circuits.
AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
013109-3 Maune, Chiu, and Bockrath Appl. Phys. Lett. 89, 013109 2006This work was supported by the ONR, the Powel Foun-
dation, and the NSF. The authors thank Natalio Mingo for
helpful discussions.
1K. Schwab, E. A. Henriksen, J. M. Worlock, and M. L. Roukes, Nature
London 404, 974 2000.
2J. B. Pendry, J. Phys. A 16, 2161 1983.
3R. Maynard and E. Akkermans, Phys. Rev. B 32, 5440 1985.
4J. Hone, M. Whitney, and A. Zettl, Synth. Met. 103, 2498 1999.
5J. Hone, B. Batlogg, Z. Benes, A. T. Johnson, and J. E. Fischer, Science
289, 1730 2000.
6M. C. Llaguno, J. E. Fischer, A. T. Johnson, and J. Hone, Nano Lett. 4, 45
2004.
7J. P. Small, K. M. Perez, and P. Kim, Phys. Rev. Lett. 91, 256801 2003.
8P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, Phys. Rev. Lett. 87,
215502 2001.
9E. Pop, D. Mann, J. Cao, Q. Wang, K. Goodson, and H. J. Dai, Phys. Rev.
Lett. 95, 155505 2005.
10E. Pop, D. Mann, Q. Wang, K. Goodson, and H. J. Dai, Nano Lett. 6, 96
2006.
11H. Y. Chiu, V. V. Deshpande, H. W. C. Postma, C. N. Lau, C. Miko, L.
Forro, and M. Bockrath, Phys. Rev. Lett. 95, 226101 2005.
Downloaded 14 Jul 2006 to 131.215.240.9. Redistribution subject to 12C. H. Yu, L. Shi, Z. Yao, D. Y. Li, and A. Majumdar, Nano Lett. 5, 1842
2005.
13E. Brown, L. Hao, J. C. Gallop, and J. C. Macfarlane, Appl. Phys. Lett.
87, 023107 2005.
14N. Mingo and D. A. Broido, Phys. Rev. Lett. 95, 096105 2005.
15P. G. Collins, M. Hersam, M. Arnold, R. Martel, and P. Avouris, Phys.
Rev. Lett. 86, 3128 2001.
16P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, Physica B 323, 67
2002.
17A. Javey, J. Guo, M. Paulsson, Q. Wang, D. Mann, M. Lundstrom, and H.
J. Dai, Phys. Rev. Lett. 92, 106804 2004.
18R. Prasher, Nano Lett. 5, 2155 2005.
19J. Kong, H. T. Soh, A. M. Cassell, C. F. Quate, and H. Dai, Nature
London 395, 878 1998.
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Thermal resistance of the nanoscale constrictions between carbon nanotubes and solid substrates

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