Molecular dynamics simulations of carbon

THE JOURNAL OF CHEMICAL PHYSICS 128, 164708 共2008兲

Molecular dynamics simulations of carbon nanotube/silicon interfacial thermal conductance Jiankuai Diao,1,a兲 Deepak Srivastava,1,b兲 and Madhu Menon2,c兲 1

University Affiliated Research Center, University of California, Santa Cruz and NASA Ames Center for Nanotechnology, Moffett Field, California 94035, USA 2 Department of Physics and Astronomy, and Center for Computational Sciences, University of Kentucky, Lexington, Kentucky 40506, USA

共Received 25 January 2008; accepted 8 March 2008; published online 22 April 2008兲 Using molecular dynamics simulations with Tersoff reactive many-body potential for Si–Si, Si–C, and C–C interactions, we have calculated the thermal conductance at the interfaces between carbon nanotube 共CNT兲 and silicon at different applied pressures. The interfaces are formed by axially compressing and indenting capped or uncapped CNTs against 2 ⫻ 1 reconstructed Si surfaces. The results show an increase in the interfacial thermal conductance with applied pressure for interfaces with both capped and uncapped CNTs. At low applied pressure, the thermal conductance at interface with uncapped CNTs is found to be much higher than that at interface with capped CNTs. Our results demonstrate that the contact area or the number of bonds formed between the CNT and Si substrate is key to the interfacial thermal conductance, which can be increased by either applying pressure or by opening the CNT caps that usually form in the synthesis process. The temperature and size dependences of interfacial thermal conductance are also simulated. These findings have important technological implications for the application of vertically aligned CNTs as thermal interface materials. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2905211兴 I. INTRODUCTION

As the size of the electronic circuits in microprocessors decreases and their density increases, there is a corresponding increase in the heat generation. Metallic heat sink materials such as copper are usually attached to electronic devices to help dissipate the heat generated in electronic devices into the environment. The thermal conductance at interfaces formed by bringing two solid surfaces that were originally separated into contact, however, is generally low due to the very low effective contact area 共less then a few percent兲 at the interface.1 This low effective contact area stems from the fact that solid surfaces are usually rough at the atomic level. To increase the effective contact area and thereby the interfacial thermal conductance, thermal interface materials such as a thermal grease are commonly used in the electronic industry, e.g., in between microprocessor chips and heat sink materials.1 The usefulness of thermal grease derives from the fact that it is an extremely flexible material that can form an efficient contact with both solid surfaces under pressure, effectively increasing the contact area between the original two solid surfaces. The thermal conductivity of the thermal grease itself is, however, low. Carbon nanotubes 共CNTs兲 have been theoretically predicted2–4 and experimentally proved5–9 to have a very high thermal conductivity along the axial direction. CNTs also have very fine structures 共a few to a few tens of nanometers in diameter兲, and are very flexible due to their high a兲

Electronic mail: [email protected]. Electronic mail: [email protected]. c兲 Electronic mail: [email protected]. b兲

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aspect ratio. Furthermore, if placed between the two solid surfaces, vertically aligned CNTs can also form good contact with the solid surfaces under reasonable pressure. These very favorable thermal, structural, and mechanical properties make vertically aligned CNTs an ideal thermal interface material. Vertically aligned CNTs are currently vigorously investigated for use as a thermal interface material both within academe and industry.10–13 In such applications, one end of CNTs is placed in contact with a metallic heat sink material such as copper, with the other end in contact with the silicon layers of microelectronic devices. The overall thermal transport performance is determined by the thermal conductivity of the CNTs themselves and the thermal conductance at the two interfaces at the two ends of the CNTs. While the thermal conductivity of CNTs has been well studied in many experiments and simulations,2–9 the thermal conductance at the interfaces has not been theoretically investigated in detail so far. An investigation of this process is, therefore, timely and necessary. In the present work, we use direct molecular dynamics 共MD兲 method to simulate thermal transport between vertically aligned CNTs and silicon materials, in an effort to understand the mechanism of the thermal transport at the CNT/silicon interface and to study its performance as a function of structural 共capped versus uncapped兲 or mechanical 共pressure兲 parameters which can be adjusted or controlled through experiments. In contrast to the response function based approaches to simulations of thermal conductivity in bulk materials, by direct MD simulations, we mean setting up hot and cold regions in the simulation sample by using appropriate thermal boundary conditions and directly computing the heat flux from the hot to the cold region as the system tries to achieve equilibrium.4

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ling processes. A few structural atomic configurations at different applied pressures are then extracted and studied for the interfacial thermal conductance using direct MD simulations. As shown in Fig. 1共a兲, by direct MD simulations of thermal transport,17,18 we mean that additional thermal energy 共kinetic energy in the amount of Q = 0.002 eV兲 is distributed among the 20 atoms at the center layer of the CNT and subtracted from the 128 atoms at the end layer in the silicon slab at each MD step by scaling their velocities as follows:

冉

␯new = ␯old 1 ⫾

FIG. 1. 共Color online兲共a兲 Setup of the simulation and 共b兲 a typical temperature profile of the system.

The setup of the simulations performed in the present work is shown in Fig. 1共a兲, where a 6.1 nm long 共10,10兲 CNT forms interfaces at its two ends with the free surfaces of two silicon slabs. Each of the Si slabs has a square cross section with 4.35 nm sides, with 6.1 nm thickness in the axial direction. Periodic boundary conditions are imposed in all three directions, and thus the two silicon slabs are connected at the ends with a total length of 6.1 nm in the axial direction. The exposed free surfaces of silicon are 2 ⫻ 1 reconstructed 共100兲 surfaces. We use both open-ended uncapped 共with dangling bonds兲 and capped CNTs in the simulations. The Tersoff bond-order potential14 is used to model the many-body reactive C–C, Si–Si, and C–Si interactions. This potential has been successfully used to simulate the thermal conductivity of CNT systems.2,15,16 Initially, the CNT ends are placed at a distance of about 5 Å from the silicon surfaces. The silicon slabs are then moved inward toward the CNT at a rate of 0.1 Å/100 000 MD steps at a constant temperature of 300 K with a time step size of t = 0.125 fs, until the CNT gradually makes contact with the silicon surface with the formation of bonds at the interface. Further inward movement causes the CNTs to indent into the silicon slabs with a stress built up in the system and eventually buckle due to the axially applied stress by the silicon slabs. The term applied pressure in the rest of the text refers to the axial normal stress acting on the silicon slab during the initial contact, indentation, and buck-

冊

1/2

,

共1兲

where ␯new and ␯old are the velocities after and before the scaling, respectively, and Ek is the total kinetic energy of all the atoms in the corresponding layer. The total energy of the system remains constant during the direct MD simulations. Due to the periodic boundary condition imposed on the system, the heat flows from the hot energy reservoir at the center in the axially opposing directions in the CNT, to the CNT/ silicon interface, and then to the cold energy sink region at the end of the silicon slab shown in Fig. 1共a兲. A steady state in the temperature distribution across the system 关Fig. 1共b兲兴 is found to be achieved after 0.4– 0.5 ns of MD simulations. The simulations were further continued for 0.5 ns more to extract the time-averaged smooth temperature profiles across the system, where the temperature at each layer of the system is calculated according to:18 N

Tl = II. SIMULATION METHOD

Q Ek

l 1 兺 mi␯2i , 3NlkB i=1

共2兲

where kB is the Boltzmann constant, Nl is the number of atoms in an atomic layer of interest, and mi and ␯i are the mass and velocity of the ith atom in the layer. A typical temperature profile for the systems is shown in Fig. 1共b兲, where the temperatures in the CNT and Si layers are linearly fitted and the temperature drops ⌬T at the two interfaces are calculated and averaged. The heat current through the system is calculated as J = Q/共2At兲,

共3兲

with A being the CNT cross-sectional area defined as A = ␲共r0 + rvdW兲2 ,

共4兲

where r0 = 0.69 nm is the distance from the center of CNT to the centers of carbon atoms and rvdW = 0.17 nm is the van der Waals radius of carbon atoms. Note that the effective contact area at the interfaces facilitating thermal transport is not necessarily the area of cross section of CNT itself but is atomistically determined by the actual number of C–Si bonds at the interface. As expected, this varies with the applied pressure during the indentation process, especially for the interfaces with the capped CNT. The above defined area is used because it is the area occupied by the CNT, no matter what the effective contact area is at the interfaces. The interfacial thermal conductance 共Kapitza conductance兲19 is then defined as


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FIG. 2. 共Color online兲 Pressure dependence of the interfacial thermal conductance for systems with both the capped and open-ended CNTs.

FIG. 3. 共Color online兲 The number of carbon atoms at the end of CNT that interact with the silicon slab as a function of pressure.

共5兲

the slabs linearly increases in the initial elastic regime, and shows discrete jumps in the value at large applied pressures as the CNT indents into the silicon slab. A closer examination of the CNT tips at the interfaces reveals that the originally hemispherical tips of the capped CNT become increasingly flattened with applied pressure, facilitating the formation of additional C–Si bonds at the interface. For the systems with the uncapped CNT, the number of C atoms bonded to Si atoms of the slabs also increases with pressure accompanied by a sudden jump in the value at a pressure of approximately 0.7 GPa. For a given pressure, it appears to be much easier for the C atoms at the ends of the uncapped CNTs to adjust their positions during the dynamics and form bonds with silicon slabs than the carbon atoms in the capped CNT tips. Thus, at low applied pressure, the uncapped CNT significantly forms more bonds with the silicon slab, and the systems with the uncapped CNT show larger interfacial thermal conductance when compared to the capped CNT case. The interfacial thermal conductance, normalized 共divided兲 by the number of C atoms bonded to silicon slabs, is shown in Fig. 4. This normalized interfacial thermal conductance can be considered as the conductance contributed by each individual bond defined in the above way. The normalized interfacial thermal conductance for systems with the

kBD = J/共⌬T兲,

and has been recently used to compute interfacial thermal conductance in direct MD simulations.17,18 III. RESULTS AND DISCUSSION

Figure 2 shows the interfacial thermal conductance for systems with both the capped and uncapped CNTs as a function of the applied pressure. We are not aware of any published experimental results for the silicon/CNT interfacial thermal conductance, but our values are of the same order of magnitude as those for the silicon/diamond20 and silicon/metal21 interfaces recently reported in the literature. As seen in the figure, the interfacial thermal conductance increases with pressure for both systems, and is higher for the system with uncapped CNT at low pressure and lower at higher pressure, with a cross-over pressure of about 1.1 GPa. This result suggests two avenues for improving the interfacial thermal conductance: 共i兲 applying pressure or 共ii兲 opening the CNT caps that usually form in the synthesis process. Since, in practice, the applied pressure cannot be too high, opening the CNT caps would be a practical way to improve thermal conductance. As the figure shows, at low pressure, the resulting enhancement in thermal conductance could be up to a factor of 3. In order to provide a mechanistic understanding of the applied pressure and structural 共capped versus uncapped兲 dependence of the interfacial thermal conductance, it is desirable to have a knowledge on the effective contact area at the interfaces. However, the effective contact area at the interfaces is not well defined at the atomic level, especially at interfaces with the capped CNTs. Since we are dealing with atomistic simulations, an effective way of taking into account the actual contact area would be to count the number of chemical bonds formed at the interfaces. Specifically, the number of carbon atoms in a CNT tip that interact with the silicon slab at the interface within the C–Si chemical bonding cutoff distance 共2.51 Å兲 of the Tersoff potential is counted and averaged over the two interfaces and also timeaveraged over many MD steps at each applied pressure. The number of C–Si bonds defined this way is shown as a function of the applied pressure in Fig. 3. For systems with capped CNT, the number of C atoms bonded to Si atoms of

FIG. 4. 共Color online兲 Interfacial thermal conductance normalized by the number of carbon atoms at the end of the tube that interact with the silicon slab.


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capped CNT fluctuates with the pressure and decreases at high pressure. This decrease is due to the indentation of the CNT tip into the substrate and the resulting local distortion of the crystal structure and the bonding geometry which scatters phonons at the interfaces. However, as the total number of carbon atoms bonded to the silicon slab continues to increase with pressure, the total interfacial thermal conductance continues to increase, as shown in Fig. 2. For the systems with the uncapped CNT, the normalized interfacial thermal conductance does not appreciably change with pressure and the increase in the total interfacial thermal conductance in Fig. 2 is mainly due to the increasing number of carbon atoms bonded to the silicon slab. Thus, in both cases, the increase in the total interfacial thermal conductance is mainly due to the increasing number of carbon atoms bonded to the silicon slab. The normalized interfacial thermal conductance, however, is higher for the systems with the capped CNTs, which implies that the bonds formed at the interface with the capped CNTs are more efficient in transferring heat, and the higher interfacial thermal conductance at low pressure for the systems with the uncapped CNT is due to the increased number of bonds formed at the interfaces. We note that classical MD schemes tend to overestimate the thermal conductivity at temperatures below the Debye temperature by about three times due to the equal excitation of phonons. For CNTs, this is the case at 300 K since the Debye temperature of CNT is close to 1000 K.22 For silicon, however, this problem is less severe since the Debye temperature 共645 K兲 is lower than that of graphitic carbon or CNTs. Therefore, we should expect a similar, but a smaller effect on the interfacial thermal conductance at the interface between silicon and CNTs. However, since we keep the same temperature of 300 K in all the simulations, the trends in the interfacial thermal conductance as a function of pressure and structure 共capped versus uncapped兲 should still be valid. We next looked at the temperature dependence of the interfacial thermal conductance. We chose the system with the uncapped CNT at a pressure of 0.2 GPa. The system is then thermally equilibrated at different temperatures with the system pressure kept roughly constant. The numbers of the bonds formed at the interfaces are also roughly constant 共around ten bonds per interface兲 at different temperatures. These thermally equilibrated systems are then used to study the interfacial thermal conductance at different temperatures using the same method as above. The temperature dependence of the interface thermal conductance is shown in Fig. 5. It is interesting to note that up to 1000 K, no peaks are observed in the CNT/Si interface thermal conductance as a function of temperature. This is in striking contrast to the thermal conductivity behavior of CNTs and silicon, where the thermal conductivity at higher temperatures is found to be lowered due to phonon-phonon scattering and exhibits peaking behavior. This monotonic increase in the interfacial thermal conductance is also seen in other simulations23 for a generic interface modeled with Lennard–Jones and Morse potentials. This suggests that the monotonic increase as a function of temperature might be a more general feature for interfacial thermal conductance. The monotonic increase might be due to the fact that as the temperature increases,

J. Chem. Phys. 128, 164708 共2008兲

FIG. 5. Temperature dependence of the interfacial thermal conductance.

more phonons are excited in both silicon and CNT, and thus, contribute more to the interfacial thermal conductance. In general, at high temperatures, the phonon-phonon scattering begins to dominate, causing a decrease in the thermal conductivity of CNT and silicon. In the present case, however, the thickness of the interfacial region is considerably smaller than the wavelengths of acoustic phonons in CNT and Si slabs, resulting in the suppression of the phonon-phonon scattering effects. This explains the lack of peaking behavior for thermal conductivity seen in the present simulations. Additionally, higher temperature also means broadening of phonon density of states in both CNT and silicon, giving rise to a better phonon-phonon coupling at the interface. We are currently conducting further investigation of this temperature dependence with the Green’s function method and intend to provide a more mechanistic understanding of the temperature dependence. The thermal conductivity of CNT has been shown to depend on length if the CNT length is smaller than the phonon mean free path.2,15,16 We have also found similar size dependence on the interfacial thermal conductance. To study this size dependence, we chose the system with the uncapped CNT at a pressure of 0.2 GPa and at 300 K, and increased the length of the CNT as well as the length of the silicon slabs by the same percentage 共150%, 200%, 250%, and 300% of the original size兲, while keeping the pressure, the interfacial structure, and the total number of bonds constant at the interfaces. The interfacial thermal conductance for these larger systems is calculated with the same direct MD method and is shown in Fig. 6. As seen in the figure, the interfacial thermal conductance shows an increase with the system size. This is probably due to the inclusion of more 共longer wavelength兲 phonons contributing to the heat transfer as the lengths of CNTs and silicon slabs are increased. With the increasing system size, the thermal conductance should increase first and then saturate to a stable value when the system size is comparable to or is larger than the phonon mean free path in the CNT and in the silicon slabs. The length of the phonon mean free path in CNT and silicon is of the order of micrometers and, therefore, beyond the reach of direct MD simulation methods using the currently available computational resources. For example, the results presented in this work typically utilized 16 CPUs on a SGI Altix 350 supercomputer. However, since we keep the system size con-


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terfacial thermal conductance also depends on size whenever the system size is smaller than the phonon mean free path. ACKNOWLEDGMENTS

This work has been supported by UC Discovery Grant Award through Nanoconduction and NASA contract 共NAS203144兲 to UARC at University of California Santa Cruz. M.M. gratefully acknowledges support from grants by DOE 共DE-FG02-00ER45817 and DE-FG02-07ER46375兲 and USARO 共W911NF-05-1-0372兲. J. P. Gwinn and R. L. Webb, Microelectron. J. 34, 215 共2003兲. S. Berber, Y. K. Kwon, and D. Tomanek, Phys. Rev. Lett. 84, 4613 共2000兲. 3 J. Che, T. Cagin, and W. A. Goddard, Nanotechnology 11, 65 共2001兲. 4 M. A. Osman and D. Srivastava, Nanotechnology 12, 21 共2001兲. 5 J. Hone, M. Whitney, C. Piskoti, and A. Zettl, Phys. Rev. B 59, R2514 共1999兲. 6 J. Hone, M. C. Llaguno, N. M. Nemes, A. T. Johnson, J. E. Fischer, D. A. Walters, M. J. Casavant, L. Schmidt, and R. E. Smalley, Appl. Phys. Lett. 77, 666 共2000兲. 7 D. J. Yang, Q. Zhang, G. Chen, S. F. Yoon, J. Ahn, S. G. Wang, Q. Zhou, Q. Wang, and J. Q. Li, Phys. Rev. B 66, 165440 共2002兲. 8 P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, Phys. Rev. Lett. 87, 215502 共2001兲. 9 M. Fujii, X. Zhang, H. Q. Xie, H. Ago, K. Takahashi, T. Ikuta, H. Abe, and T. Shimizu, Phys. Rev. Lett. 95, 065502 共2005兲. 10 J. Xu and T. S. Fisher, Int. J. Heat Mass Transfer 49, 1658 共2006兲. 11 H. Huang, C. H. Liu, Y. Wu, and S. S. Fang, Adv. Mater. 共Weinheim, Ger.兲 17, 1652 共2005兲. 12 Y. Xu, Y. Zhang, and E. Suhir, J. Appl. Phys. 100, 074302 共2006兲. 13 X. Liu, T. P. Bigioni, Y. Xu, A. M. Cassell, and B. Cruden, J. Phys. Chem. B 110, 20102 共2006兲. 14 J. Tersoff, Phys. Rev. B 39, 5566 共1989兲. 15 N. R. Raravikar, P. Keblinski, A. M. Rao, M. S. Dresselhaus, L. S. Schadler, and P. M. Ajayan, Phys. Rev. B 66, 235424 共2002兲. 16 G. Zhang and B. Li, J. Chem. Phys. 123, 114714 共2005兲. 17 P. Jund and R. Jullien, Phys. Rev. B 59, 13707 共1999兲. 18 R. J. Stevens, P. M. Norris, and L. V. Zhigilei, Proceedings of IMECE04, ASME International Mechanical Engineering Congress and Exposition, 共2004兲, 共unpublished兲, pp. 1–10. 19 P. L. Kapitza, Zh. Eksp. Teor. Fiz. 11, 1 共1941兲. 20 K. E. Goodson, O. W. Kading, M. Rosner, and R. Zachai, Appl. Phys. Lett. 66, 3134 共1995兲. 21 R. J. Stevens, A. N. Smith, and P. M. Norris, J. Heat Transfer 127, 315 共2005兲. 22 L. X. Benedict, S. G. Louie, and M. L. Cohen, Solid State Commun. 100, 177 共1996兲. 23 C. J. Twu and J. R. Ho, Phys. Rev. B 67, 205422 共2003兲. 1 2

FIG. 6. Size dependence of the interfacial thermal conductance.

stant for all other simulations, the results of structural 共capped versus uncapped兲, mechanical 共pressure兲, and temperature dependences of interfacial thermal conductance studied are still valid. To summarize, we have calculated the structural 共capped versus uncapped兲, mechanical 共pressure兲, and temperature dependences of the interfacial thermal conductance. Our results show an increase in the interfacial thermal conductance with the applied pressure, which can be explained by the increasing number of chemical bonds formed between the CNT and silicon slab due to the applied pressure and the resulting indentation process. The caps of the capped CNT become flattened at increased applied pressure, facilitating the formation of more bonds at the interface. At low applied pressure, the interfacial thermal conductance for systems with uncapped CNTs is much higher than that of the systems with capped CNTs due to the larger number of bonds formed at the interface. At high applied pressure, however, the interfacial thermal conductance for the capped CNT case is higher due to the fact that the bonds formed at the interface are more efficient in transferring heat. Our results suggest that the contact area or the number of bonds formed between the CNT and Si slab is crucial to the interfacial thermal conductance, which can be increased either by applying pressure or by opening the CNT caps that usually form in the synthesis process. The interfacial thermal conductance is found to monotonically increase with temperature up to 1000 K. In-
