JOURNAL OF APPLIED PHYSICS 107, 104907 !2010"
Thermal conductance and phonon transmissivity of metal–graphite
interfaces
Aaron J. Schmidt, Kimberlee C. Collins, Austin J. Minnich, and Gang Chena!
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA
!Received 20 January 2010; accepted 17 April 2010; published online 26 May 2010"
The thermal boundary conductances between c-axis oriented highly ordered pyrolytic graphite and
several metals have been measured in the temperature range 87–300 K and are found to be similar
to those of metal–diamond interfaces. The values obtained are indicative of the thermal interface
conductance between metals and the sidewalls of multiwall carbon nanotubes !CNTs" and,
therefore, have relevance for the accurate characterization of the thermal properties of CNTs,
graphene, and the design and performance of composite materials and electronic devices based on
these structures. A modified diffuse mismatch model is used to interpret the data and extract the
phonon transmissivity at the interface. The results indicate that metal–graphite adhesion forces and
interfacial mixing effects play important roles in determining the boundary conductance. © 2010
American Institute of Physics. #doi:10.1063/1.3428464$
I. INTRODUCTION
The thermal boundary conductance between metals and
carbon nanotubes !CNTs", graphene, and other graphitebased materials is a subject of both fundamental and practical interest. An accurate picture of thermal transport in these
situations is essential for the design of next generation electronic devices based on CNTs !Refs. 1 and 2" and graphene3,4
as well as composites and high-performance graphite–metal
thermal interface materials.5,6 In addition, accurate measurements of the thermal properties of CNTs and graphene, a
matter of theoretical importance, requires knowledge of the
thermal interface conductance between these structures and
the metal contacts necessary to perform the measurements.7–10
The interface conductance between metal and the c-axis
direction of highly ordered pyrolytic graphite !HOPG" is an
excellent approximation to the thermal interface between
metal and the sidewall of a multiwall CNT !MWCNT".8 Experimental data for thermal interface conductance between
metal and graphite is scarce.11 Chen et al.12 measured the
boundary conductance between SiO2 and single-layer and
multilayer graphene sheets. They found values between 8.3
! 107 – 1.8! 108 W / !m2 K" and no clear dependence on the
number of layers. A few studies have considered metal–
diamond interfaces,13,14 and estimates of the thermal conductance of single-walled CNTs with platinum contacts have
also been reported. Pop et al.15 estimated the thermal conductance per unit length between a 2 nm CNT and a SiO2
substrate as %0.17 W / m K, which translates into a thermal
interface conductance of %1 ! 108 W / !m2 K", depending
on contact area. Similar results were obtained by Maune et
al.16
Here we present measurements of the thermal interface
conductance between HOPG in the c-axis direction and Al,
Au, Cr, and Ti, and Al with a 5 nm Ti adhesion layer !rea"
Electronic mail: gchen2@mit.edu.
0021-8979/2010/107"10!/104907/5/$30.00
ferred to as Al/Ti" in the temperature range 87–300 K. HOPG
is characterized by large, single-crystal regions with hexagonal covalent bonding in the plane and weaker aromatic bonding between planes, resulting in large mechanical and thermal anisotropy.17 Room-temperature values for the crossplane thermal conductivity are on the order of 5–10 W/m K,
while the in-plane thermal conductivity is near 2000
W/m K.17,18
II. EXPERIMENT
Our HOPG samples were obtained from SPI corporation
and the Dresselhaus group at the Massachusetts Institute of
Technology. The samples were prepared using double-sided
carbon tape to peel off flakes of crystalline graphite several
millimeters in size. The freshly exposed interfaces were
coated with approximately 90 nm of metal via electron-beam
deposition. Figure 1 shows representative scanning electron
microscope !SEM" images of HOPG samples coated with Al,
Au, Cr, and Ti. The Al and Ti films were smooth and continuous; the large scale cracks and defects result from the
underlying HOPG structure and some dust particles. The Cr
(a)
(b)
10 µm
10 µm
(c)
(d)
10 µm
10 µm
1 µm
FIG. 1. SEM images of HOPG samples: !a" Al, !b" Au, !c" Cr, and !d" Ti.
107, 104907-1
© 2010 American Institute of Physics
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104907-2
J. Appl. Phys. 107, 104907 "2010!
Schmidt et al.
140
120
100
3
Ti
Al/Ti
Al
Cr
Au
1
Ge = &
4 j
'
$max,j
0
%1−2&$v1D1
#f
d$ ,
# Te
!1"
G (MW/m2K)
80
60
40
20
100
150
200
250
300
Temperature (K)
FIG. 2. !Color online" The thermal interface conductance, G, between
HOPG in the c-axis direction and Ti, Al with a 5 nm Ti adhesion layer, Al,
Cr, and Au.
film formed grains several tens of microns in size, an indication of weaker adhesion, while the Au film formed pockets, also indicative of the poor adhesion between Au and
HOPG.19 The pocketing was observed on several samples
coated using e-beam and sputter deposition.
Measurements were performed using time-domain thermoreflectance, a technique widely used to study thermal interface conductance.13,14 Our experimental system is built
around a Ti-sapphire laser which emits a train of 150 fs long
pulses at a repetition rate of 80 MHz. We use a pump modulation frequency on the order of 10 MHz and a focused pump
spot 80 "m in diameter to eliminate radial conduction effects and isolate the transport in the cross-plane direction.11
The values obtained for the thermal interface conductance are shown in Fig. 2. The interface conductance increases with temperature, a general trend predicted by models based on the phonon spectra of the two materials.20 The
interface conductance is highest for Ti and Al/Ti, followed
by Al and Cr, and finally lowest for Au. For Ti at higher
temperatures the signal became extremely weak, possibly
due to a decreasing thermoreflectance coefficient at 800 nm.
Error bars for the Ti data, the most uncertain set, indicate the
standard deviation in multiple measurements at different locations on the sample. For the other data sets we estimate the
uncertainty to be % # 20%. The temperature at the sample
was determined using a Si diode with an uncertainty of less
than 1 K, including steady-state heating effects from the laser.
where the sum is taken over the three acoustic polarizations
!one longitudinal, two transverse", %1−2 is the phonon transmissivity from material 1 !in this case the metal" to material
2 !HOPG", v1 is the phonon group velocity in the direction
of transport, D1 is the density of states !DOS", and f is the
Bose–Einstein occupation function. Under the condition of
elastic scattering, $max is taken as the lower of the maximum
frequencies in either material. The subscript e following temperature and conductance is to emphasize that this definition
is based on the temperature of the emitted phonons, not the
equivalent local equilibrium temperature.21,22
Under the DMM, the transmissivity is given by
%1−2 =
(0$max&$v2D2 fd$
(0$max&$v1D1 fd$ + (0$max&$v2D2 fd$
We use a modified diffuse mismatch model !DMM"
!Ref. 20" to explain our results and obtain a clearer picture of
the transport at the HOPG–metal interface. In the DMM it is
assumed all phonon scattering is diffuse, which is reasonable
at high temperatures where the average phonon wavelength
approaches the length scale of atomic roughness. The boundary conductance is given by
!2"
where now we have lumped together the three polarizations
using an average phonon velocity. In our implementation, we
use a sine-type dispersion relation for the metal because high
frequency roll-off of the group velocity provides a more realistic picture than the linear dispersion of the Debye
model.23 The dispersion for each polarization is given by
) *
$ = $0 sin
'(
,
2 (0
!3"
where (0 is the Debye cutoff wave-vector and $0 is defined
as 2 / 'vs(0 to match the low-frequency speed of sound, vs.
We obtain the cut-off wave-vector from the experimentally
determined Debye temperatures for the metals24 using (0
= kB)D / &vs where )D is the Debye temperature and kB is
the Boltzmann constant. On the graphite side, we follow the
approach of Duda et al.25 and use an effective twodimensional !2D" DOS, which is a better approximation to
the true graphite DOS than a three-dimensional !3D" DOS
based on either the c-axis or a-axis sound speeds.25
Finally, we convert our result for conductance based on
emitted temperature to one based on an equivalent local
equilibrium temperature, since an equilibrium temperature is
assumed when using a diffusion-based model to extract
boundary conductance from our measurement. Under the
gray medium assumption,22
+ )'
T1 − T2 = 1 −
III. MODELING AND DISCUSSION
,
1
2
1
0
%1−2d"1 +
'
1
0
%2−1d"2
!!Te,1 − Te,2",
*,
!4"
where " is the directional cosine and again the subscript e
signifies the temperature of emitted phonons while the absence of a subscript indicates an equivalent equilibrium temperature. Under the DMM, we can further assume that scattering is independent of angle and that %1−2 = 1 − %2−1, which
leads to
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104907-3
J. Appl. Phys. 107, 104907 "2010!
Schmidt et al.
1000
Cr
G (MW/m2K)
(a) c-axis HOPG
Al
(c) Diamond
Ti
100
Al
Al
Au
Cr
Ti
Ti
Al
Ti
10
Cr
Cr
Au
Au
Au
(b) a-axis HOPG
(d) Average HOPG
1
100
150
200
250
300
100
150
200
250
300
100
150
200
250
300
100
150
200
250
300
Temperature (K)
FIG. 3. !Color online" Diffuse mismatch calculations for the thermal interface conductance between metals and !a" c-axis HOPG, !b" a-axis HOPG, !c"
diamond, and !d" HOPG with the average of the a-axis and c-axis sound speed. In all cases the curves for Al and Ti are nearly coincident.
!5"
From the definition of thermal boundary conductance, q
= Ge*Te, where q is the heat flux. Substituting Eq. !5", we
see that q = G*T where G = 2Ge and *T is now the equilibrium temperature. Thus, accounting for the equilibrium temperature has added a factor of two to Eq. !1".
Calculations for thermal interface conductance for c-axis
oriented HOPG are shown in Fig. 3!a". Also shown are the
calculations for a-axis oriented HOPG in Fig. 3!b", diamond
in Fig. 3!c", and HOPG with the mean of the a-axis and
c-axis sound speeds, va and vc, in Fig. 3!d". The mean sound
2
speed is calculated using 3 / vavg
= 1 / v2c + 2 / v2a,23 accounting
for two transverse polarizations in the basal plane. The sound
speeds for metal and diamond were taken from Refs. 20 and
13 while the graphite values are from.17 For c-axis oriented
HOPG, the model captures the general trend of the data with
temperature and the order of magnitude. However, the relative values of G for different metals do not match the data
such as: the Ti–HOPG and Al/Ti–HOPG conductances are
significantly larger than Al–HOPG or Cr–HOPG, which are
of similar magnitude. Consistent with the model, the Au–
HOPG interface conductance is the smallest.
A possible explanation for the difference between model
and data is that the DMM does not account for the adhesion
force between metal and substrate. Using lattice dynamics,
Stoner and Maris13 and Young and Maris26 showed that the
strength of the interfacial bond can strongly affect the predicted interface conductance. The wettability of carbonbased materials, including graphite and diamond, by metals
has been extensively studied.19 Gold does not wet carbon
materials while aluminum does, forming covalent bonds and
carbide compounds. Transition metals such as Ti and Cr
bond strongly to carbon because carbon donates valence
electrons to help fill the d-bands of transition metals, and the
resulting bond strength is related to the number of available
empty valence locations in d-band. Consequently, Ti has a
stronger bond with carbon than Cr.19
The importance of the interfacial bond is further emphasized by the fact that the data for Al with a 5 nm Ti adhesion
layer are very similar to the pure Ti data. As shown in Fig. 3,
based on sound speed and DOS considerations Al and Ti
should give similar results. In addition, the thermal conductance of metal–metal interfaces is many times higher than
metal–nonmetal conductances,27 so the interface conductance of the Al/Ti sample is dominated by the 5 nm Ti layer
bonded to the HOPG.
In Fig. 4, we plot our results for Al, Au, and Ti together
with data taken by Stoner and Maris for the same metals on
isotopically enriched diamond substrate.13 Surprisingly, the
two sets of data are of similar magnitude, while the DMM
predicts that the metal–diamond values should be significantly larger. The metal–HOPG data also have a slightly
weaker temperature dependence than the metal-diamond
data, which is predicted by the DMM and arises from the fact
that HOPG is described by an effective 2D DOS while diamond has a 3D DOS.
The similarity between metal–HOPG and metal–
diamond interface conductances is not expected from the
DMM. However, there are several factors not included in the
model that may contribute to the discrepancy. One is that
interfacial defects and impurities at the atomic level could
disrupt the graphite lattice, effectively creating atomic-level
mixing close to the interface. The region of interfacial roughness may effectively take a weighted-average crystallographic direction for energy propagation across the interface.
120
100
80
Graphite
Diamond
60
40
Titanium (a)
20
50
2
G (MW/m K)
1
T1 − T2 = !Te,1 − Te,2".
2
40
30
20
Aluminum (b)
10
40
30
20
Gold (c)
10
100
150
200
250
300
Temperature (K)
FIG. 4. !Color online" Comparison of the thermal interface conductance, G,
of metal–c-axis HOPG and metal–diamond for !a" Ti !circles" and Al/Ti
!squares", !b" Al, and !c" Au. Contrary to the prediction of the DMM, the
measured conductances are of similar magnitude.
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104907-4
J. Appl. Phys. 107, 104907 "2010!
Schmidt et al.
32
50
(a)
45
G (MW/m2K)
30
TABLE I. Phonon transmissivities from metal to HOPG obtained from fitting Eq. !1".
(b)
40
35
28
30
26
50
40
(c)
(d)
100
30
80
20
Ti
Al/Ti
60
100
200
300
100
200
300
Temperature (K)
FIG. 5. !Color online" Predictions obtained for the phonon transmissivity
from metal to HOPG obtained by fitting Eq. !1" using a sine dispersion
model with an effective constant phonon transmissivity for !a" Au, !b" Al,
!c" Cr, and !d" Ti and Al/Ti.
This would mean replacing the c-axis group velocity in Eq.
!2" with a weighted-average group velocity that accounts for
some fraction of phonons traveling in the a-axis direction.
While a realistic weighting for the average velocity requires
detailed knowledge of the interface morphology, the simple
weighting used in Fig. 3!d" shows that the averaging effect
does bring the value for c-axis HOPG closer to that of diamond. Also, the phonon dispersion in the vicinity of the interface would be modified, further complicating the situation.
Another possibility, as we discussed in the context of the
different metals, is that the interfacial bond may play an
important part and act as the limiting factor in the interfacial
heat flow to a high thermal conductivity material such as
diamond.
Ultimately, none of these explanations is entirely satisfactory and fully explains our results. The modeling of
boundary conductance for imperfect interfaces is a complex
problem and it may not be possible to predict fully with the
DMM, which assumes a perfect interface and diffuse scattering. The specific mechanical and chemical details of each
interfacial region need to be characterized in more depth, and
more complex models are needed that could take these additional details into account.
Nonetheless, the DMM is appealing due to its simplicity
and ease of application. For this reason, we use our data to
extract the effective phonon transmissivity under the DMM.
We treat the transmissivity from the metal into HOPG in Eq.
!1" as an adjustable, temperature-independent parameter, and
determine which value best matches the experimental data.
This effective phonon transmissivity depends on the form of
the dispersion relation used in Eq. !1". In Fig. 5, we show the
best fit curves for our data obtained using the sine dispersion
relation. The effective phonon transmissivities from metal
into HOPG are listed in Table I. We also list the transmissivities obtained using Debye !linear" dispersion for the
metal. In this case the transmissivities are all two to three
times smaller than for sine dispersion.
Due to the reasons discussed in the preceding paragraphs, we do not have a simple explanation of the values
obtained from fitting the phonon transmissivity. However,
when the fitted values are used, Fig. 5 shows that the DMM
does a reasonable job capturing the temperature dependence
Dispersion
Sine
Debye
Gold
Aluminum
Chromium
Titanium
0.061
0.032
0.017
0.067
0.023
0.013
0.008
0.027
of the boundary conductance and could be used in calculations if the effective transmissivity is known. Additionally,
the comparison between the data from c-axis HOPG and diamond shown in Fig. 4 suggests that, contrary to the predictions of the DMM, the boundary conductance between metal
and a-axis HOPG may not be much different from c-axis
HOPG or diamond, and by extension, the thermal conductance into the sidewall of a MWCNT may be quite similar to
the conductance into the end of a MWCNT.
IV. SUMMARY
In conclusion, we have presented measurements of the
thermal interface conductance between metal and c-axis
HOPG. Previous work indicates that these results are an excellent approximation to the thermal interface conductance
between metal and the sidewalls of MWCNTs, and they
should also serve as a rough approximation for other graphite
structures such as single-walled CNTs and graphene sheets.
Our experimental results show that the interfacial conductances between metal and c-axis HOPG are similar to those
between the same metals and diamond. A DMM model fails
to adequately explain our results. Interfacial roughness,
modified phonon dispersion near the interface and metal–
carbon adhesion forces are all factors that complicate the
picture, and more detailed studies of the interfaces are necessary to fully understand the problem. Regardless, the
DMM model is used to extract effective phonon transmissivities at the interfaces. This information is valuable to those
seeking to design new devices and materials using carbonbased nanostructures.
ACKNOWLEDGMENTS
The authors would like to thank Professor Mildred
Dresselhaus, Xiaoting Jia, Alfonso R. Reina Cecco, and
Daniel Nezich for their assistance providing and preparing
HOPG samples. We are also grateful to Dr. Patrick Hopkins
for his helpful discussions on the application of the DMM to
graphite. This work was supported in part by DARPA NTI
program under Grant No. G9U535369 and MIT Energy Initiative Seed funding. The views, opinions, and/or findings
contained in this article/presentation are those of the author/
presenter and should not be interpreted as representing the
official views or policies, either expressed or implied, of the
Defense Advanced Research Projects Agency or the Department of Defense.
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