Phonon Engineering in Carbon Nanotubes by Controlling Defect

Oct 3, 2011 - Division of IT Convergence Engineering and National Center for Nanomaterials Technology, POSTECH, Pohang 790-784,. Republic of Korea...
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Phonon Engineering in Carbon Nanotubes by Controlling Defect Concentration Cem Sevik,*,†,§ H^aldun Sevinc- li,*,‡,§ Gianaurelio Cuniberti,*,‡,^ and Tahir C- agın*,† †

Artie McFerrin Department of Chemical Engineering and Material Science and Engineering, Texas A&M University, College Station, Texas 77845-3122, United States ‡ Institute for Materials Science and Max Bergmann Center of Biomaterials, Dresden University of Technology, 01062 Dresden, Germany ^ Division of IT Convergence Engineering and National Center for Nanomaterials Technology, POSTECH, Pohang 790-784, Republic of Korea ABSTRACT: Outstanding thermal transport properties of carbon nanotubes (CNTs) qualify them as possible candidates to be used as thermal management units in electronic devices. However, significant variations in the thermal conductivity (k) measurements of individual CNTs restrict their utilizations for this purpose. In order to address the possible sources of this large deviation and to propose a route to solve this discrepancy, we systematically investigate the effects of varying concentrations of randomly distributed multiple defects (single and double vacancies, Stone Wales defects) on the phonon transport properties of armchair and zigzag CNTs with lengths ranging between a few hundred nanometers to several micrometers, using both nonequilibrium molecular dynamics and atomistic Green’s function methods. Our results show that, for both armchair and zigzag CNTs, k converges nearly to the same values with different types of defects, at all lengths considered in this study. On the basis of the detailed mean free path analysis, this behavior is explained with the fact that intermediate and high frequency phonons are filtered out by defect scattering, while low frequency phonons are transmitted quasi-ballistically even for several micrometer long CNTs. Furthermore, an analysis of variances in k for different defect concentrations indicates that defect scattering at low defect concentrations could be the source of large experimental variances, and by taking advantage of the possibility to create a controlled concentration of defects by electron or ion irradiation, it is possible to standardize k with minimizing the variance. Our results imply the possibility of phonon engineering in nanostructured graphene based materials by controlling the defect concentration. KEYWORDS: Thermal transport, carbon nanotubes, phonon engineering, thermal management, defects, disorder

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ontinuous miniaturization of electronic devices gives rise to an exponential growth of power densities in integrated circuits.1 The recent advances in processing/manufacturing nanostructured materials provide wider possibilities of controlling the thermal transport and novel designs for integrating them within the device. In particular, carbon-based nanomaterials such as CNTs in both individual2 4 and composite forms5 11 have attracted significant attention as candidates for thermal cooling applications due to their excellent thermal transport properties in addition to their remarkable thermal stability and mechanical strength.12,13 The experimental values for k of individual single-wall and multiwall CNTs range between 400 and 7000 W m 1 K 1,14 19 and between 300 and 3000 W m 1 K 1,18 24 respectively. The lack of a control mechanism and large variance in k measurements hinder the utilization of CNTs in practical thermal management applications and complicate the device design. In literature, the source of this large discrepancy in experiments has been described as problems resulting from the measurement techniques, such as uncertainty in contact resistance and difficulty in generating and accurately determining temperature gradients over nanoscale lengths.18,25 Some experiments particularly designed to alleviate the influence of these r 2011 American Chemical Society

problems have shown that the morphology and quality of CNTs have conspicuous effects on k as well.18 21 This led to several theoretical studies focusing on the influence of morphological properties such as length, chirality, diameter, number of walls, bending, and defects.26 33 In these studies, minor effects of diameter and chirality on k, and a significant increase with increasing length have been reported.34 36 Although the influence of individual defects has been studied,37,31 the potential effects of several types of defects have not been examined in detail so far. In this paper, the effects of different defect types (single and double vacancies, Stone Wales defects) (Figure 1) on thermal properties of single wall CNTs of various lengths are studied with both nonequilibrium molecular dynamics (NEMD) simulations,38 and atomistic Green’s functions (AGF)39 technique. We show that small changes in defect concentration give rise to a large variance in lattice thermal conductivity when the defect concentration is low, and more interestingly, when the defect concentration is above a certain value, thermal Received: August 23, 2011 Revised: September 22, 2011 Published: October 03, 2011 4971

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Figure 1. Schematic description of nonequilibrium molecular dynamics simulations for CNTs and the representation of single vacancy, double vacancy, and Stone Wales type defects. Figure 3. Lattice thermal conductivities of pristine semiconductor (10,0) and metallic (10,10) CNTs obtained from NEMD simulations as functions of length at different temperatures. Thermal conductivity increases monotonically for both CNTs with increasing length, where k of CNT(10,10) is larger than that of CNT(10,0), in general. The symbols, black circle, red square, and blue diamond represent different temperatures, 200, 300, and 400 K, respectively. Here, the insets show the room temperature thermal conductivities up to 3 μm of CNT lengths. Figure 2. Room temperature lattice thermal conductivity of pristine CNT(10,0) evaluated with NEMD for different CNT lengths and temperature gradients (a) and also with varying the lengths of constant temperature regions (b). Thermal conductivity increases with increasing tube length, where different implementations of temperature gradient do not alter the thermal transport significantly, as shown in (a). The constant temperature regions should be long enough in order to get rid of the spurious interference effects (b).

conductivity is no more sensitive to the changes in defect concentration. On the basis of these observations, we propose a route to minimize the variance due to intrinsic scattering processes by increasing the defect concentration intentionally. Our proposal for standardizing the thermal conductivity of CNTs makes use of the persistent quasi-ballistic behavior of low-frequency phonons, and strong suppression of higher frequency modes under intentional disorder. The range of defect concentration that is necessary for standardizing thermal conductivity of CNTs does not detoriate the thermal transport performance, still well above other good conductors, and we believe that this is a feasible proposal since recent electron or ion beam experiments indicate positive results in tailoring structural and electronic properties of CNTs.40 42 The paper is organized as follows. We first discuss thermal properties of pristine CNTs and address the issues of length and temperature dependencies. Second, we investigate the effects of various defect types and their ensembles. Employing AGF technique, we analyze the frequency resolved phonon transport regimes of defected CNTs as a function of CNT length. Then we concentrate on the variance of k as the defect concentration is altered, and propose a route for the standardization of k. Theoretical methods used in this work are summarized after the conclusion part. Results and Discussions. Pristine Nanotubes. In order to get an accurate estimate of lattice thermal conductivity with NEMD simulations, the number of the atoms in the constant temperature regions, shown in Figure 1, and the temperature difference between these parts (ΔT = Thot Tcold) are crucially important. Therefore, different sets of test simulations are performed for both pristine and defected CNTs with intention to determine optimum values for these parameters. Eventually, considering the results, such as those shown in Figure 2, 50 nm for the length of constant temperature regions and 20% of T for ΔT are adopted at all the simulations performed in this study.

The length dependence of lattice thermal conductivity of zigzag (10,0) and armchair (10,10) types of CNTs for three different temperatures, namely, 200, 300, and 400 K, is demonstrated in Figure 3. As the length increases, k increases monotonically without converging even for L ≈ 1 μm, which is attributed to the predicted very long phonon mean free paths of these materials.17,43 Lattice thermal conductivity of zigzag CNT (ZCNT) is prominently higher than that of armchair CNT (ACNT), especially for lower temperatures. However, the calculated room temperature k values for L = 3 μm are 1565 ( 34 W m 1 K 1 for ACNT and 1405 ( 69 W m 1 K 1 for ZCNT. As seen in the insets of Figure 3, the length dependence of k diminishes after L = 1 μm, and chirality dependence is less pronounced. These results are consistent with other previously reported calculations.31 33,36,44 46 Additionally, it is worth mentioning that the agreement between the periodic boundary condition Green Kubo calculations, using the same Tersoff type interatomic potential parameter (IIP) sets,47 and nonequilibrium simulations is established when the CNT length is longer than 1 μm. The room temperature k values we predict are smaller than those reported experimentally,16,23 which, we believe, is due to differences in experimental and simulation conditions. In addition, the temperature-dependent lattice thermal conductivity of CNTs for different lengths, shown in Figure 4, clearly represents that lattice thermal conductivity of longer nanotubes decreases considerably with the increase in temperature, while that of shorter nanotubes changes slightly. This prediction might be explained with the explicit dependence of Umklapp scattering rates on nanotube length, and due to the apparent competition between the scattering time and the time required a phonon to pass through the nanotube. Defected Nanotubes. Now, we systematically study the effects of varying densities of Stone Wales defects and single and double vacancies on the lattice thermal transport properties of CNTs; see Figure 1. We treat each defect as an individual scattering center (independent of the defect type or the number of atoms that forms it), by defining the defect concentration as NDefect/Natom  100, where NDefect and Natom are the number of defects and number of atoms in pristine CNTs, respectively. First, the effects of each defect type on the lattice thermal conductivity are investigated separately by performing NEMD simulations for armchair and zigzag CNTs of various lengths, 4972

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Figure 4. Lattice thermal conductivities of pristine semiconductor (10,0) and metallic (10,10) CNTs as functions of temperature with different lengths as obtained from NEMD. The symbols, black circle, red square, blue diamond, green star, and magenta triangle represent different lengths: 5, 20, 80, 240, and 900 nm, respectively. k increases with increasing length and decreases with increasing temperature in the temperature region of interest. The slopes of temperature dependent k decrease with decreasing tube length. This behavior is due to the fact that longer CNT increases the probability of a phonon to be scattered by other phonons.

including randomly distributed defects. As seen in Figure 5, a rapid drop in thermal conductivity is observed for all defect types and chiralities. When the defect concentration is increased to 0.625%, thermal conductivity is suppressed by up to 90% with respect to its pristine value. Interestingly, after 0.5% defect concentration, conductivities of all defect types for both tubes converge nearly to the same value of k ∼ 160 W m 1 K 1 at all lengths considered in this study. These results show that extremely long convergence lengths for thermal conductivities of CNTs (see the insets in Figure 3) decrease down to a few hundred nanometers, even for low defect concentrations. This is due to strong reduction of phonon mean free paths by defect scattering. This significant effect manifests itself in the temperature dependence of lattice thermal conductivity, as well. Weak dependence of k on temperature, when the defect concentration is 0.625%, as shown in the inset of Figure 5b, suggests that the 1 1 1 = L elastic +L anharmonic ) is dominated total mean free path (L mfp by defect scattering, masking the temperature dependence. Furthermore, in order to investigate a more realistic case, we perform NEMD simulations for zigzag CNTs, including randomly distributed defects of all types in each sample. As shown in the inset in Figure 5a, very similar values for lattice thermal conductivity are predicted for all the random structures, which, in fact, is expected because of thermal conductivity being almost independent of the defect type. In order to shed more light on the behavior of phonons in defected CNTs, we employ the AGF method (see Methods Section for details). We consider the case of CNT(10,0) with randomly distributed single vacancies and calculate phonon transmission functions with varying defect concentration and length, for different ensembles of CNTs consisting of 10 samples. After performing ensemble averages, we obtain elastic mean free paths L elastic by fitting the numerical data. In Figure 6(upper panel), L elastic is plotted for various defect concentrations. We observe that L elastic drops by an order of magnitude at ω = 150 cm 1 for all considered concentrations. Such a difference in frequency dependence of L elastic points to the possibility to filter out the higher frequency phonons, at least partially. For a 1 μm long CNT with d = 1%, for example, low frequency modes (ω < 150 cm 1) are quasi-ballistic, while higher frequency modes are suppressed very strongly. The crossover at ω > 150 cm 1 is observable in the length dependent transmission spectrum as well (Figure 6(lower panel)). Since L elastic is generally shorter than 10 7 m for ω > 150 cm 1 with d = 1%, the transmission of

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Figure 5. Evaluation of NEMD lattice thermal conductivity with defect concentration for (a) CNT(10, 0) and (b) CNT(10, 10) containing randomly distributed single vacancy, double vacancy, and Stone Wales type defects. The symbols, black circle, red square, and blue diamond represent different lengths, 200, 400, and 600 nm, respectively. The insets in (a) and (b) show the k values for three different randomly defected CNT (10,0) structures and the effect of temperature on 0.625% defected CNTs, respectively. At low defect concentrations (d < 0.1%), k decreases very fast with increasing defect concentrations. At high concentrations (d > 0.25%), k decreases quite slowly with increasing d, approaching very similar values for both armchair and zigzag CNTs and for all types of defects.

higher frequency modes is already suppressed within the first 1 μm, while lower frequency modes are conducting almost perfectly. Owing to the large step in L elastic around 150 cm 1, lower frequency modes keep transmitting almost ballistically for a long distance up to 7.5 μm as tested with AGF calculations. Therefore, k remains almost unchanged for a wide range of CNT lengths. A comparison of thermal conductivities of defected CNTs with graphene is in order here. Graphene is the system with the highest thermal conductivity ever reported48 with striking features13,39,42,43,49,50 and defects can be expected to affect its thermal transport properties in a similar way to CNTs.13,51,52 The available experimental data for graphene displays a large discrepancy, k ranging from 600 to 5300 W m 1 K 1.48,53 Apart from the differences in experimental techniques, such a large difference might also stem from the difference in defect concentrations of the samples as it is observed in CNTs (see Figure 5). In wide graphene nanoribbons, for example, k is reduced from 2300 to 1700, 800, and 500 W m 1 K 1 when defects are introduced with concentrations d = 0.01%, 0.05%, and 0.1%, respectively.13 That is, the reduction in thermal conductivity is fast at small d and tends to diminish as d is increased, as is the case in CNTs. Second, anharmonic processes are detrimental in thermal transport through pristine structures of both graphene and CNTs,13,51,52 but their role is minimized at elevated defect concentrations.13 For defect-free graphene, the effect of anharmonic processes is amplified with increasing temperature and k is reduced by up to a factor of 2 when temperature is increased from 200 to 400 K.51 Likewise, in 900 nm long pristine CNTs k is reduced from 1300 (1750) W m 1 K 1 to 750 (900) W m 1 K 1 for CNT(10,0) (CNT(10,10)) (see Figure 4). Also, weakened temperature dependence of k is predicted for graphene when defects are introduced.51 In parallel to this, k is found to change 4973

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Figure 7. Variance analysis of thermal conductivity of defected CNTs as obtained from NEMD simulations. Thermal conductivity of 1 μm long zigzag CNTs are plotted as a function of defect concentration (b), where random distributions of different defect types are implemented. We assume ensembles of CNTs with Gaussian distributions of defect concentration (a), and plot the k distributions (c), keeping the variance in concentration (σ2d) constant, and shifting the mean concentration, d. A small mean concentration d = 0.05% results in a large deviation in thermal conductivity, σk = 92 W m 1 K 1 (distributions indicated in blue), where larger d values give rise to narrower distributions of k (red and green). When d = 0.5% (∼1 defect per 20 Å along the tube axis) the deviation becomes as low as 4 W m 1 K 1.

Figure 6. Elastic phonon mean free paths (upper panel) and transmission spectrum (lower panel) calculated with atomistic Green’s functions for CNT(10,0) with randomly distributed single vacancies. In the upper panel, elastic mean free path, L elastic, is plotted as a function of frequency for different defect concentrations. L elastic scales as ω 2, and includes details of phonon density of states. This ends in approximately 3 orders of magnitude difference between the L elastic of high and low frequency modes. Sharp dips correspond to van Hove singularities in the phonon density of states, where scattering rates are maximized. The transmission spectrum as a function of frequency and CNT length is plotted for d = 1% in the lower panel. High frequency phonons are suppressed almost completely when the CNT is longer than 1 μm. Low frequency phonons preserve their quasi-ballistic nature for several micrometers.

only slightly in wide graphene nanoribbons at defect concentration d = 0.1%, which suggests that L elastic dominates over L anharmonic in all graphene-based structures at high enough defect concentrations. Variance and Standardization. Besides the advances in the synthesis of CNTs, small concentrations of defects are still unavoidable and variations in defect concentrations, at low densities, result in large changes in thermal conductivity, as discussed above. On the other extreme, higher defect concentration gives rise to k values which are only slightly sensitive to changes in defect concentration, sample length, and temperature. In order to demonstrate this behavior, we analyze the variance in thermal conductivity for ensembles of CNTs having Gaussian distribution of defect concentrations with different mean values, d. In Figure 7b, we plot the thermal conductivity of zigzag CNT of 1 μm length, as a function of defect concentration

obtained from NEMD simulations. Normalized distributions of concentration of three ensembles are plotted in different colors in Figure 7a, and the corresponding k distributions are given in Figure 7c. We choose the deviation in concentrations to be σd = 0.02%, and the mean concentrations are set to d = 0.05, 0.25, and 0.5%. The corresponding mean thermal conductivities are k ̅ = 547, 235, and 160 W m 1 K 1, and the deviations in k are σk = 92, 11, and 4 W m 1 K 1, respectively. These results suggest that defect scattering is a major factor of large σk in the experiments with low defect samples. More interestingly, σk can be reduced by more than a factor of 20 with increasing the defect concentration. Furthermore, taking advantage of the possibility to create a controlled concentration of defects by electron or ion irradiation,40,41 one can set L mfp of CNTs. Here, the resulting L mfp becomes almost independent of the initial state of the CNT, due to the fact that defect induced L elastic is considerably shorter than the mean free paths due to other sources of scattering like isotopic disorder and anharmonic interactions. Therefore, it is possible to obtain CNTs of different radii, chiralities, and initial qualities, but all having similar k, which is tunable by the exposure time of the electron or ion beam. Summary and Concluding Remarks. In summary, the effects of different types of defect structures on lattice transport properties of zigzag and armchair CNTs of several lengths are elaborately investigated with both nonequilibrium molecular dynamics and atomistic Green’s function methods. It is predicted that thermal conductivities of CNTs of different structural aspects converge to very similar values by increasing the defect concentration, due to the significant reduction of phonon mean free paths. These results not only explain the significant variations in experimentally measured thermal conductivities of CNTs but also point to the possibility of standardizing the thermal transport properties of CNTs of different radii, 4974

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Nano Letters chiralities, and initial qualities, with intentional disorder. We note that electrons enter the Anderson localization regime when the double vacancy concentration is around 0.03%, and the resistance increases by 3 orders of magnitude.40,54 Therefore, we do not expect electrons to have a significant contribution to heat transport of disordered CNTs. Although the standardized value of k is small compared to its pristine value, it is still much higher than the thermal conductivities of semiconductor nanowires, e.g., for silicon nanowires k < 50 W m 1 K 1 at room temperature.55 62 Methods. In order to estimate the thermal transport properties of pristine and defected (10,0) and (10,10) CNTs, we perform NEMD simulations with a recently optimized Tersoff44,63 type IPP set using LAMMPS64 code. In the course of the simulations, different temperatures, Thot (T + 10%T) and Tcold (T 10%T), are assigned to the left and the right regions (∼50 nm) of the simulated CNTs, to generate a thermal current from the left to the right and fixed boundary conditions are imposed at the ends; see Figure 1. A Nose Hoover thermostat65,66 (NVT) is used to maintain the temperature of the reservoirs, and a microcanonical ensemble (NVE) is imposed to the central CNT region. Before performing MD simulations, the structures are relaxed, and they were observed to be stable during the simulations within the temperature range of this work. Namely, no bond-breaking or defect migration is observed during data collection. Thermal conductivity, k, calculations are carried out with the same method published by Hu et al.38 With a 0.5 fs time step, 4 ns MD simulations are performed and the Nose Hoover thermostat relaxation time τ is set to be 1 ps for all temperatures. Time averaging of the temperature and heat current is performed from the last 1.5 ns. During the heat current conversion to k, π((r + Δr)2 (r Δr)2) is used as the cross sectional area of the CNTs; here r is the CNT radius and Δr is the half of the interlayer (van der Waals) distance of graphite, 0.335 nm. Atomistic Green’s function (AGF) method is employed within the recursive scheme for calculating the mean free paths and the transmission spectra. Free Green’s function for the central region is defined as gC(ω2) = ((ω + i0+)2 kCC) 1, where kCC is mass normalized dynamical matrix for the central region, kCC = MC 1/2 DCC MC 1/2 with DCC being the force constant matrix and MC being the diagonal matrix of atomic masses. Force constant matrices are obtained from the same parametrization of the Tersoff potential as used in NEMD simulations. A single vacancy is implemented by setting the corresponding atomic mass as mvacancy f 0+. This way of introducing single-vacancies neglects the reconstruction effects in the vicinity of the vacancy but includes the major contributions correctly. The central region is contacted to left (L) and right (R) reservoirs, which are semiinfinite pristine CNTs of the same kind as the central region. Selfenergies due to coupling to the reservoirs are defined as ∑L(R) = kCL(CR) kL(R) gLC(RC). Here, gL(R) is the free Green’s function of the semi-infinite reservoir and kCL(CR) and kLC(RC) are the parts of the mass normalized dynamical matrix which stand for the coupling between the central region and the reservoirs. Then Green’s function for the central region including coupling to the reservoirs is Gc (ω2) = ((ω +i0+)2 kcc ∑L(ω2) ∑R(ω2)) 1, and the transmission spectra are obtained by using T = Tr[GC ΓLGC†ΓR], where ΓL(R) = 2Im[∑L(R)] stand for broadenings due to coupling to the reservoirs. The mean free paths are obtained by fitting T (ω,L) 1 = T o(ω) 1 (1 + L/L mfp (ω)), where T (ω,L) is the ensemble averaged transmission value of

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samples of length L and T o(ω) is the transmission value for the pristine system.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]; [email protected]. de; [email protected]; [email protected]. Author Contributions §

The first two authors contributed equally and share the first authorship of this work.

’ ACKNOWLEDGMENT C.S. and T.C. acknowledge the support from NSF (DMR 0844082) to International Institute of Materials for Energy Conversion at Texas A&M University as well as AFRL. Part of the computations are carried out at the facilities of Laboratory of Computational Engineering of Nanomaterials also supported by ARO, ONR, and DOE grants. We are grateful for the generous time allocation made for this project by the Supercomputing Center of Texas A&M University. H.S. and G.C. acknowledge support by the priority program Nanostructured Thermoelectrics (SPP-1386) of the German Research Foundation (DFG) (Contract No. CU 44/11-1), the cluster of excellence of the Free State of Saxony ECEMP—European Center for Emerging Materials and Processes Dresden (Project A2), and the European Social Funds (ESF) in Saxony (research group InnovaSens). G.C. further acknowledges support from the WCU (World Class University) program sponsored by the South Korean Ministry of Education, Science, and Technology Program, Project no. R31-2008000-10100-0. The Center for Information Services and High Performance Computing (ZIH) at the TU-Dresden is also acknowledged. ’ REFERENCES (1) Pop, E.; Goodson, K. E. Thermal Phenomena in Nanoscale Transistors. J. Electron. Packag. 2006, 128, 102. (2) Tong, T.; Zhao, Y.; Delzeit, L.; Kashani, A.; Meyyappan, M.; Majumdar, A. Dense Vertically Aligned Multiwalled Carbon Nanotube Arrays as Thermal Interface Materials. IEEE. Trans. Compon. Packag. Technol. 2007, 30, 92. (3) Panzer, M. A.; Zhang, G.; Mann, D.; Hu, X.; Pop, E.; Dai, H.; Goodson, K. E. Thermal Properties of Metal-Coated Vertically Aligned Single-Wall Nanotube Arrays. ASME J. Heat Transfer 2008, 130, 052401. (4) Pop, E. Energy Dissipation and Transport in Nanoscale Devices. Nano Res. 2010, 3, 147–169. (5) Kordas, K.; Toth, G.; Moilanen, P.; Kumpumaki, M.; Vahakangas, J.; Uusimaki, A.; Vajtai, R.; Ajayan, P. M. Chip Cooling with Integrated Carbon Nanotube Microfin Architectures. Appl. Phys. Lett. 2007, 90, 123105. (6) Mohammed, O. F.; Samartzis, P. C.; Zewail, A. H. Heating and Cooling Dynamics of Carbon Nanotubes Observed by TemperatureJump Spectroscopy and Electron Microscopy. J. Am. Chem. Soc. 2009, 131, 16010. (7) Cola, B. A.; Xu, X.; Fisher, T. S. Increased Real Contact in Thermal Interfaces: A Carbon Nanotube/Foil Material. Appl. Phys. Lett. 2007, 90, 093513. (8) Zhang, K.; Chai, Y.; Yuen,M.M. F.; Xiao, D. G.W.; Chan, P. C. H. Carbon Nanotube Thermal Interface Material for High-Brightness Light-Emitting-Diode Cooling. Nanotechnology 19, 215706. (9) Biercuk, M.; Llaguno, M.; Radosavljevic, M.; Hyun, J.; Johnson, A.; Fischer, J. Carbon Nanotube Composites for Thermal Management. Appl. Phys. Lett. 2002, 80, 2767–2769. 4975

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