PHYSICS APPLIED LETTERS 91, 123119 2007
Length-dependent thermal conductivity of an individual single-wall carbon nanotube
Zhao Liang Wang, Da Wei Tang,a Xiao Bo Li, and Xing Hua Zheng
Institute of Engineering Thermophysics, Chinese Academy of Science, Beijing 100080, China
Wei Gang Zhang
Institute of Process Engineering, Chinese Academy of Science, Beijing 100080, China
Li Xin Zheng and Yuntian T. Zhu
Los Alamos National Laboratory, Los Alamos, New Mexico 87545
Ai Zi Jin, Hai Fang Yang, and Chang Zhi Gu
Institute of Physics, Chinese Academy of Science, Beijing 100080, China
Received 11 May 2007; accepted 15 August 2007; published online 21 September 2007 The thermal conductivity of single-wall carbon nanotubes SWCNTs is predicted to increase with length, but this has never been proved experimentally because of limitations in previous measurement methods. Here, the authors report the measurement of the length-dependent thermal conductivities of individual SWCNTs on a Si substrate using a four-pad 3 method. An increase in thermal conductivity with length was observed at room temperature, which is consistent with a theoretical prediction that considers higher order three-phonon processes. When SWCNTs are longer than the phonon mean path, they showed dissipative thermal transport. The observed increase of thermal conductivity with length makes SWCNTs ideal for thermal management. 2007 American Institute of Physics. DOI: 10.1063/1.2779850 Theoretical and experimental studies have shown that single-wall carbon nanotubes SWCNTs have outstanding thermal and electrical transport characteristics, which makes them good candidates for applications in integrated circuits as microtransistors, probes of atomic force microscope AFM , and thermal interface materials in thermal management. Although the thermal properties of millimeter-sized carbon nanotube mats and packed carbon nanofibers have been measured, the measurement on an individual nanotube has been a challenge. A "thermal conductance" technology,14 a T-type nanosensor,5 and a 3 method68 have been utilized to measure the thermal conductivities of multiwall carbon nanotubes MWCNTs and the temperature-dependent thermal conductance of individual SWCNTs. However, it is not easy for these technologies to measure the length-dependent thermal conductivity of individual MWCNT or SWCNT because of difficulties in sample preparation, including synthesis and suspension of MWCNTs or SWCNTs. The four-pad 3 method8 has an advantage over other methods because it has no contact resistance problem and much reduced spurious signals, but the suspension of nanotubes is still a big challenge. For these reasons, experimental study on length-dependent thermal conductivity of individual SWCNTs has been rare. Probably due to the limitation of the approximation made on phonon relaxation rate, both an analytical model9 and the linearized Boltzmann-Peierls phonon transport equation that includes three-phonon processes to first order10 predict a nearly linear increase of thermal conductivity of SWCNTs with length size over some range and this is not consistent with the molecular dynamics1113 MD and the couple mode theory14 results. Therefore, it is of great interest to experimentally measure the length-dependent thermal
a
conductivity of SWCNTs in order to evaluate the accuracy of these theoretical predictions. The recent synthesis of ultralong individual SWCNTs on a Si substrate15 has overcome some of the experimental difficulties. However, the SWCNT interacts with the Si substrate via van der Waals energy,16 which makes it difficult to measure the thermal conductivity of the SWCNT. We recently presented a relationship among the third harmonics, electrical current, and line contact thermal resistance, which takes into account the heat loss to the substrate along the line contact. This relationship, coupled with the four-pad 3 method, overcomes all the obstacles in measuring the thermal conductivities of individual SWCNTs on a substrate at different lengths at room temperature. It is the objective of this letter to measure the length-dependent thermal conductivity of SWCNTs on a Si substrate. The thermal resistances of suspended SWCNTs have been calculated using a heat conduction equation,4 which reveals their longitudinal thermal dissipation nature. For an individual SWCNT on a substrate in vacuum, assuming that the contact is realized by van der Waals interaction energy, neglecting convection and radiation losses, a thermal contact resistance term can be introduced into the one-dimensional 1D heat conduction equation,6 which yields
2
cp
t
=
x
2
+
I2 sin2 t R0 + R 2LS
-
LSRc
,
1
Author to whom correspondence should be addressed; electronic mail: [e-mail]
where , c p, and are the density, specific heat, and thermal conductivity of the SWCNT, = T - T0 is the local temperature change at temperature T, t is the time, x is the coordinate along the length direction, I and are the effective amplitude and angular frequency of ac, R0 is the initial electrical resistance of the SWCNT at initial temperature T0, L and S are the length and cross-sectional area, respectively because of the difficulty in defining S of SWCNT, two kinds of definitions, S = d2 / 4 and S = d ,5 are used here to fit experi 2007 American Institute of Physics
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FIG. 2. Third harmonics at different frequencies for L = 4.919 FIG. 1. Image of SWCNT. a samples Image of SWCNT. b SEM image of SWCNT samples with four 1 m wide electrodes.
m SWCNT.
mental data, where d is the outer diameter of the SWCNT and = 0.335 nm is the layer separation distance in graphite , Rc is the line contact thermal resistance between the SWCNT and the substrate,17 and R is temperature coefficient of electrical resistance. By retaining only the first term of a series expansion, the mean temperature change responding to heating is found to be from Eq. 1 as 8I2RL = . 2 4 S + L/ 2Rc According to the relationship between the third harmonic response U3 and temperature change fluctuating at 2 , the relationship between U3 and can be described as U3 = 4I3RR L , 4 S + L/ 2Rc 3
where R = R0 + R is the electrical resistance of SWCNT at temperature rise . In the case of a wire on substrate in vacuum, Eq. 3 shows a linear relationship between U3 and I 3. The synthesis of SWCNTs on a Si substrate has been described previously.15 In this study, ultralong SWCNTs of the semiconducting type were grown on a Si substrate with a SiO2 film on its surface. Figure 1 a is the scanning electron micrograph SEM image of SWCNTs on the Si substrate. The focused ion beam technique is used to deposit a 2 3 nm thick Ni layer over SWCNT segments for contact pads first, and then a 1 m wide, 200 nm thick, and 1 mm long Au pad was deposited on top of the Ni layer. Following this procedure, several groups of four-pad structures with different lengths were fabricated, as shown in Fig. 1 b . The diameter of the SWCNT was measured by AFM as 1.9 nm. The four-pad 3 setup is similar to that by Choi et al.8 except for the current source. The main components of apparatus include a lock-in amplifier 7280 and a well controlled constant amplitude current source based on an amplifier AMP01 unit with minimum overshoot current. In the experiment, I must be less than several microamperes so as to validate the expected relationship U3 I3 and maintain the sample integrity. The SWCNT sample was tested in an antistatic, high vacuum 1.5 10-3 mbar chamber. To minimize the effects of parasitic inductances and capacitances, low frequencies in the range of 110 550 Hz were used to measure the first and third harmonics. The phase of the lock-in at each frequency was zeroed to improve the phase accuracy for small amplitude 1 current and all were line
filters turned off to keep them from distorting 3 signals. The measured third harmonics on a 4.919 m SWCNT segment as a function of frequency is shown in Fig. 2. It is clear that spurious signals were brought into U3 at two resonance points of 50 and 100 Hz probably due to the heat loss to the substrate. The electrical resistances of three segments from a single SWCNT with lengths of 0.509, 4.919, and 6.941 m in the temperature range of 28.5 31.5 C are shown in Fig. 3. It is clearly seen that there exists a linear relationship between the electrical resistance and temperature over this temperature range and the electrical resistance decreased with temperature. Also, the electrical resistance of SWCNT with different lengths increased nearly linearly with length and no anomalous change of electrical resistance with length were observed. Although it is extremely difficult to measure the junction electrical contact resistance Rcont, simultaneous measurements of Rcont and electrical resistance of test section of the SWCNT can be attempted through changing the length of the SWCNT with manipulation under SEM. Neglecting the electrical resistance of the leads 70 , Rcont turned out to be less than 10% of electrical resistance of the test section. At 400 Hz, the 3 harmonics measured on three SWCNTs with different lengths as a function of current are shown in Fig. 4. The three curves showed the expected relationship of U3 I3. By fitting the results according to Eq. 3 , the exponent of the current was found to be 2.723, 2.89, and 2.908 for the three different lengths, 0.509, 4.919, and 6.941 m, respectively. The thermal conductivity fitted using two definitions of the cross-sectional area multiplied by the tube diameter, d, is presented in Fig. 5. The corresponding values of the thermal conductivities are comparable with and higher than the thermal conductivity of diamond, indicating that these SWCNTs are outstanding interface materials for thermal management. The measured thermal conduc-
FIG. 3. Electrical resistance measured at different temperatures.
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FIG. 4. Third harmonics as a function of current.
tance of individual SWCNTs by the 3 method at room temperature is smaller than the result by thermal conductance technique,4 but close to the result in Ref. 3, likely depending on the measurement method, the SWCNT sample size, and structure. The measured is of the same order of magnitude as the measured values of individual SWCNTs.3,4 According to Eq. 2 and Fourier's law for heat conduction, the heat loss to substrate could be found to be about 30% 40% of the total heating power. The uncertainty in measured thermal conductivity of individual SWCNTs is mainly related to uncertainty in diameter and length of SWCNT, temperature coefficient of resistance, 3 voltage, and the estimation of line contact thermal resistance. The total uncertainty is estimated to be within 10.5%. Now we can estimate the mean free path MFP of phonons in the SWCNT at room temperature. Assuming the phonon group velocity in SWCNT as v 2 104 m / s,8 the thermal capacity was deduced inversely as c 1.33 103 kJ/ m3 K, according to thermal conductivity and MFP values of individual SWCNTs, measured with thermal conductance technology at room temperature.5 Using the measured on 6.941 m long SWCNT and the thermal conductivity formula = cvl / 3 the factor of 1 / 3 should be dropped from the formula for a 1D SWCNT, and l is the MFP , MFP was calculated as 180 nm, which is shorter than the reported values of 0.5 Ref. 3 and 0.375 Ref. 4 at room temperature. The discrepancy of the values of MFP is caused by the errors in the v and c values. As the SWCNT samples are semiconducting type, the thermal transport is mainly due to phonons. The length dependence of thermal conductivity can be interpreted based on the phonon MFP at room temperature. If the sample length is shorter than or comparable to MFP, the phonon transportation is dominated by the ballistic regime, in which is limited by phonon-boundary scattering rather by phonon-phonon scattering processes. If the
sample length is longer than MFP, the effect of the short wavelength phonons reaches a stable level while the long wavelength phonons still have some positive effect on the heat transport process. Furthermore, the higher order threephonon processes yield a finite thermal conductivity for carbon nanotubes.10 From the slopes of the three curves corresponding to different lengths in Fig. 4, it was found that the longer the SWCNT length, the closer the exponent of the curve is to 3. The two longer tubes may be claimed to agree with the expected power of 3.0 for the third harmonic voltage, but the shortest one deviates substantially. Since the MFP is calculated from the measured data as 180 nm and the MD model and analytical model by Mingo and Broido10 cannot be used to fit the experimental data for the determination of length-dependent thermal conductivities of an individual SWCNT, the 3 method should be approximately applicable. Therefore, there exists size effect of thermal conductivity for individual SWCNTs over the measured length range at room temperature. These observations are consistent with the Boltzmann-Peierls phonon transport equation including three-phonon processes to second order.10 In summary, the four-pad 3 method was employed to measure the thermal conductivities of SWCNTs on a Si substrate. Over the length range of 0.5 7 m, an increase in thermal conductivity with length at room temperature was observed. The mean free path of phonons was estimated to be 180 nm. It is necessary to take into account heat loss to the substrate in the heat conduction equation as it accounts for about 30%40% of the total heat power. The measured thermal conductivities of SWCNTs are comparable and higher than that of diamond, making them a good thermal conductor. Our experimental observations are consistent with the Boltzmann-Peierls phonon transport equation including three-phonon processes to second order.10 The authors would like to thank L. H. Zhang of the Institute of Physics, Chinese Academy of Sciences, for solving the noise cancellation and dynamic reserve problems of lock-in amplifier. This work was supported by the National Natural Science Foundation of China Grant No. 50376066.
P. Kim, L. Shi, A. Majumdar, and P. McEuen, Phys. Rev. Lett. 87, 215502 2001 . L. Shi, D. Li, C. Yu, W. Jang, Z. Yao, P. Kim, and A. Majumdar, J. Heat Transfer 125, 881 2003 . 3 E. Pop, D. Mann, Q. Wang, K. Goodson, and H. Dai, Nano Lett. 6, 96 2006 . 4 C. Yu, L. Shi, Z. Yao, D. Li, and A. Majumdar, Nano Lett. 5, 1842 2005 . 5 M. Fujii, X. Zhang, H. Xie, H. Ago, K. Takahashi, T. Ikuta, H. Abe, and T. Shimizu, Phys. Rev. Lett. 95, 065502 2005 . 6 L. Lu, W. Yi, and D. L. Zhang, Rev. Sci. Instrum. 72, 2996 2001 . 7 T. Y. Choi, D. Poulikakos, J. Tharian, and U. Sennhauer, Appl. Phys. Lett. 87, 013108 2005 . 8 T. Y. Choi, D. Poulikakos, J. Tharian, and U. Sennhauer, Nano Lett. 6, 1589 2006 . 9 P. Chantrenne and J. L. Barrat, Superlattices Microstruct. 35, 173 2004 . 10 N. Mingo and D. A. Broido, Nano Lett. 5, 1221 2005 . 11 S. Berbert, Y. K. Kwon, and D. Tomanek, Phys. Rev. Lett. 84, 4613 2000 . 12 J. Che, T. Cagin, and W. A. Goddard, Nanotechnology 11, 65 2000 . 13 S. Maruyama, Physica B 323, 193 2002 . 14 S. Lepri, R. Livi, and A. Politi, Phys. Rep. 377, 1 2003 . 15 L. X. Zheng, M. J. O'Connell, S. K. Doorn, X. Z. Liao, Y. H. Zhao, E. A. Akhadov, M. A. Hoffbauer, B. J. Roop, Q. X. Jia, R. C. Dye, D. E. Peterson, S. M. Huang, J. Liu, and Y. T. Zhu, Nat. Mater. 3, 673 2004 . 16 V. Bahadur, J. Xu, and T. S. Fisher, J. Heat Transfer 127, 664 2005 . 17 C. Yu, S. Saha, J. Zhou, and L. Shi, J. Heat Transfer 128, 234 2006 .
2 1
FIG. 5. Thermal conductivities multiplied by tube diameter of SWCNT as a function of length at room temperature.