A theoretical analysis of the thermal conductivity of hydrogenated graphene

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A theoretical analysis of the thermal conductivity of hydrogenated graphene Qing-Xiang Pei *, Zhen-Dong Sha, Yong-Wei Zhang Institute of High Performance Computing, Singapore 138632, Singapore

A R T I C L E I N F O

A B S T R A C T

Article history:

We investigate the thermal conductivity of hydrogenated graphene using non-equilibrium

Received 28 March 2011

molecular dynamics simulations. It is found that the thermal conductivity greatly depends

Accepted 22 June 2011

on the hydrogen distribution and coverage. For random hydrogenation, the thermal con-

Available online 29 June 2011

ductivity decreases rapidly with increasing coverage up to about 30%. Beyond this limit, however, the thermal conductivity is almost insensitive to the coverage. For patterned hydrogenation with stripes parallel to the heat flux, the thermal conductivity decreases gradually with increasing coverage from 0% to 100%. In contrast, when the stripe direction is perpendicular to the heat flux, a small (5%) coverage causes a sharp (60%) drop of thermal conductivity. The deterioration of thermal conductivity is due to the sp2-to-sp3 bonding transition upon hydrogenation, which softens the G-band phonon modes. Percolation theory can be used to explain the variation of thermal conductivity at different hydrogenation distributions and coverages. The applicability of the rule of mixtures in predicting the thermal conductivity is also discussed. The work suggests that hydrogenation is a possible route to tune graphene thermal conductivity and manage heat dissipation in graphenebased nanoelectronic devices. Ó 2011 Elsevier Ltd. All rights reserved.

1.

Introduction

Graphene, a single layer of carbon atoms in a honeycomb lattice, has attracted a great deal of attention since it was discovered in 2004 [1]. Owing to its novel electronic, magnetic, mechanical and thermal properties, researchers are actively exploiting this material for various applications [2–4]. Because graphene is a zero bandgap semiconductor and exhibits much higher electron mobility than silicon, it has great potential to be used in nanoelectronics. As Si-based semiconductor technology is approaching its fundamental limits in miniaturizing electronic devices, graphene could replace silicon as the next-generation semiconductor material for electronic devices [3,4]. Since pristine graphene has no bandgap, how to controllably and reliably open its bandgap becomes an

important research topic. Chemical functionalization has been used to modify graphene electronic properties and engineer its bandgap, enabling its conduction to be tuned among insulating, semiconducting, and conducting [5–8]. Among various functional groups, hydrogen has attracted considerable interest [9–15]. Recently, Singh and Yakobson [16] theoretically demonstrated that through patterned hydrogenation, a graphene sheet can be divided into different stripe domains, called nanoroads, which can exhibit interesting electronic properties. Experimentally, bandgap opening in graphene sheets by patterned absorption of hydrogen was also observed [17]. Using multi-scale simulations, Fiori et al. [18] showed that a patterned hydrogenated graphene could be used to make transistors with excellent properties. In addition to its fascinating electronic properties, the thermal properties of graphene are also of fundamental and

* Corresponding author: Fax: +65 64674350. E-mail address: [email protected] (Q.-X. Pei). 0008-6223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2011.06.083

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practical significance. As electronic devices undergo drastic miniaturization, there is a strong demand for efficient heat dissipation to ensure device performance and lifetime. Hence, it is both important and necessary to have in-depth understanding of the thermal transport properties of graphene in order to develop graphene-based electronic devices. Recent experimental measurements showed that graphene has very high thermal conductivity of 2500–5000 W/m K [19–21], which are close to those measured for carbon nanotubes [22]. In addition to experiments, theoretical analyses, especially molecular dynamics (MD) simulations, have been widely used to study the thermal properties of graphene and carbon nanotubes. Different from carbon nanotubes, it was found that graphene nanoribbons showed strong anisotropic thermal conduction due to the edge effects [23–25]. The thermal conductivity of zigzag nanoribbons is 20–50% larger than that of armchair nanoribbons [23,25]. With the increase of ribbon width, however, the edge effect decreases and the anisotropic effect is weakened. For an infinitely large graphene sheet, recent theoretical analysis [26] showed that the anisotropic effect still exists, though it is very weak. Beside the edge and chiral direction effects, the thermal conductivity of graphene was found to be length-dependent [23,27,28], which is similar to that of carbon nanotubes [29–31]. The thermal conductivity k was observed to increase with simulation cell length L, following a power law relationship k  Lb, with b ranging from 0.3 to 0.47 for graphene [23]. In contrast to the intensive research performed on the thermal conductivity of pristine graphene, the thermal conductivity of chemically functionalized graphene has not been well studied. To the best of our knowledge, so far only one work [32] was performed to study the thermal conductivity of an armchair nanoribbon for random hydrogenation with coverage from 0% to 50%. It was found that the random hydrogenation can significantly affect the thermal conductivity of graphene. However, several important issues remain unsolved: (1) How does the thermal conductivity change in the full range of hydrogen coverage? (2) How is the thermal conductivity affected by patterned hydrogenation? (3) What is the underlying mechanism for the reduction in thermal conductivity? (4) Does the hydrogenation effect on thermal conductivity depend on the chiral direction of graphene? and (5) In a graphene sheet with patterned hydrogenation, does the rule of mixtures for thermal conduction hold? Studies on these issues are not only important in understanding the basic thermal properties of graphene, but also important for graphene application in nanoelectronics. In this paper, we present a detailed study on the thermal conductivity of hydrogenated graphene using non-equilibrium molecular dynamics (NEMD) simulations, with the aim to answer the questions raised above. Our study covers both random and patterned distributions of hydrogenation with coverage ranging from 0% to 100%. Our simulation results show that the distribution and coverage of hydrogenation can significantly influence the thermal conductivity of graphene. Our work suggests that hydrogenation could be an effective approach for tuning the thermal conductivity of graphene, which could be useful in nanoscale engineering of thermal transport and heat management.

2.

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Simulation methods

In MD simulations, the thermal conductivity can be computed using either NEMD [33] or equilibrium molecular dynamics (EMD) [34]. The NEMD approach is a ‘‘direct method’’, in which a temperature gradient across the simulation cell is imposed and the thermal conductivity is calculated from the Fourier’s law. In contrast, the EMD approach uses the Green–Kubo relation to extract the thermal conductivity from the heat current fluctuations. Therefore, the EMD method generally needs a long simulation time to converge the heat current autocorrelation function. We employ NEMD simulations with a time step of 0.5 fs for integration on the equations of atomic motion. The adaptive intermolecular reactive empirical bond order (AIREBO) potential [35] as implemented in the large-scale atomic/molecular massively parallel simulator (LAMMPS) [36] is used to describe the carbon–carbon interactions, carbon-hydrogen interactions and non-bonding atomic interactions. This potential is an extension of Brenner’s reactive empirical bond order (REBO) potential [37] and has been successfully used in many carbon-based systems, such as carbon nanotubes and graphene for studying thermal as well as mechanical properties [27,38–40]. Other potentials, such as the first-principles-based reactive force field (ReaxFF) [41] that has recently been used for modeling graphene [42], can also be used for the present study. However, the ReaxFF is more computationally intensive, thus it is more suitable for simulations with either small model size or short time scale. To calculate the thermal conductivity of graphene using NEMD, we impose heat fluxes on the system as illustrated in Fig. 1, and extract the temperature gradient along the X direction. The sample size is about 20 nm long and 5 nm wide. Periodic boundary conditions are applied in both the X (length) and Y (width) directions. The thermal conductivity k is calculated based on Fourier law of heat conduction ~ J ¼ krT

ð1Þ

where rT is the temperature gradient and ~ J is the heat flux, which is defined as the amount of energy transferred per unit time through per unit cross sectional area of the graphene. The initial configuration is well equilibrated at room temperature (300 K) to ensure that the system is in equilibrium state. The equilibrium is realized by the constant volume and constant temperature (NVT) simulations for 105 time steps, followed by the constant pressure and constant temperature (NPT) simulations for another 105 steps. Then, the heat flux is applied and the system is switched to the constant volume and constant energy (NVE) ensemble to keep the energy conserved. Under the NVE ensemble, we run 106 time steps to allow the system to reach steady-state. At each time step, a small amount of heat De is added into a thin slab of thickness 2d at the middle region (hot region) and removed from the thin slabs of thickness d at the two ends (cold region) of the graphene. Heat addition or removal is ensured by modifying the kinetic energies through a velocity rescaling procedure in both hot and cold regions. When the steadystate is achieved, the heat current can be calculated by

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Fig. 1 – Schematic of the simulation model for the non-equilibrium molecular dynamics. A small amount of heat is added into the hot region and removed from the cold regions to create the heat fluxes. Periodic boundary conditions are applied in both the X and Y directions.

J¼

De 2ADt

ð2Þ

where A is the cross-sectional area and Dt is the time step. To calculate the temperature gradient, the graphene is divided into 80 slabs with thickness d. The temperature in each slab is calculated based on the velocities of all the particles in the slab as follows Ti ¼

Ni 1 X mj v2j 3Ni kB j¼1

ð3Þ

where Ni is the number of atoms in the slab i, mj and vj represent the mass and velocity of the atom j, respectively, and kB is the Boltzmann constant. We run another 106 time steps to obtain the time-averaged temperature profile, which is used to calculate the temperature gradient @T=@x. The thermal conductivity can then be calculated as k¼

3.

J De ¼ @T=@x 2ADtð@T=@xÞ

ð4Þ

Results and discussion

We first perform NEMD simulations for pristine graphene sheets and compare our results with published ones to validate our approach. For the graphene sheet of length 20 nm, the calculated thermal conductivities of armchair and zigzag graphene are 89.6 and 92.3 W/m K, respectively. Those values are in good agreement with previously simulated value of 102 W/m K [38] using AIREBO potential. Our calculated thermal conductivities are also close to the recent simulation results in [28] using Tersoff potential. It is noted that the experimentally measured thermal conductivities for micrometer-sized graphene are in the range of 2500–5000 W/m K [19– 21], which are much higher than the MD simulation results. This discrepancy is due to the length-dependence of thermal conductivity at nanoscale. In fact, the length-dependent thermal conductivities of graphene and carbon nanotubes have been well-studied, and now it is well-established that the thermal conductivities increase with increasing the sample size at the nanoscale [23,27–31]. This length-dependence has been accepted as the major cause for the lower computational values obtained in previous works [23,27–31,43] and the present work. Another possible cause is the electron thermal transport, which is not considered in the MD simulations. However, the previous work [19] showed that the thermal

transport in graphene is dominated by phonons, thus the electron contribution to the thermal conductivity should be minor. It is noted that the simulated thermal conductivity along zigzag direction is slightly (3%) higher than that along armchair direction. Recently it was shown by Jiang et al. [26] that the thermal conductivity of a graphene sheet depends weakly on chirality and that the thermal conductivity along the zigzag direction is the highest. Hence the present simulation results are consistent with the reported ones in [26]. In the following sections, we proceed to study the effect of hydrogenation on the thermal conductivity of graphene.

3.1.

Random hydrogenation

We generate the hydrogenated graphene sheets by random adsorption of hydrogen atoms on graphene to obtain the desired coverage. The hydrogen coverage is varied from 0% (pristine graphene) to 100% (graphane). It should be mentioned that graphane is a graphene derivative first predicted by Sofo et al. [11] based on first-principles total-energy calculations and later synthesized by Elias et al. [14]. Fig. 2a–c show hydrogenated armchair graphene sheets with the hydrogen coverage of 5%, 10% and 30%, respectively. The calculated thermal conductivities of randomly hydrogenated graphene at different coverages are shown in Fig. 2d. In the plot, the calculated thermal conductivities of armchair and zigzag graphene sheets are normalized by the values of pristine armchair and zigzag graphene sheets, respectively. It can be seen that hydrogenation greatly reduces the thermal conductivity of graphene. The thermal conductivity of graphane is only about 32% of that of pristine graphene. With increasing hydrogen coverage from 0% to 30%, the thermal conductivity decreases rapidly. At 5% coverage, the thermal conductivity is about 65% of that of pristine case, while at 30% coverage, the thermal conductivity drops to only about 27% of that of pristine case. With further increasing hydrogen coverage from 30% to 100%, the thermal conductivity shows little change and maintains at a low value, and then undergoes a small increase near 100% coverage. It can also be observed from Fig. 2d that the two curves, corresponding to the armchair graphene and zigzag graphene, respectively, are nearly the same, indicating that the hydrogenation exhibits the same extent of effect on the thermal conductivity along both armchair and zigzag directions.

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Fig. 3 – Phonon spectra of graphene with hydrogen coverage ranging from 0% to 100%. The high frequency peaks are the G-bands.

Fig. 2 – Atomistic configurations of an armchair graphene with random hydrogenation at (a) 5% coverage, (b) 10% coverage, and (c) 30% coverage. (d) Variation of thermal conductivity with hydrogen coverage for both the armchair and zigzag graphene sheets.

Similar to pristine graphene, the functionalized graphene sheets also show weak anisotropy in thermal conductivity. It is well known that hydrogenation of graphene leads to the conversion of local carbon bonds from in-plane sp2 bonding to out-of-plane sp3 bonding [9,11,14,44]. For hydrogenated graphene with coverage between 0% and 100%, the sheet can be viewed as a mixed structure of two domains, one with high thermal conduction (sp2 C–C bonds) and the other with low thermal conduction (sp3 C–H bonds). Hence, the degradation in thermal conductivity of hydrogenated graphene can be attributed to the introduction of sp3 bonds into the structure, which can be regarded as randomly distributed defects in a sp2 plane matrix to scatter phonons [32,45]. With increasing hydrogen coverage from 0% to 30%, the sp2 planar network gradually diminishes and deteriorates. As a consequence, the phonon scattering increases and the thermal conductivity decreases. It is interesting to notice that when the hydrogen coverage is above 30%, the obtained thermal conductivity is basically insensitive to the coverage. This phenomenon can be explained by percolation theory. It is known that the site percolation threshold for a honeycomb lattice is 0.697 [46], suggesting that the sp2 bond network is disrupted when the H-coverage reaches approximately 30%. As a result, the heat transport is controlled by the sp3 domains (low conduction domains) when the coverage is above 30%, making the thermal conductivity insensitive to further H-coverage increase.

To explore the physical origin of the hydrogenation-induced reduction in thermal conductivity, we have calculated the phonon spectra of both pristine and hydrogenated graphene sheets. The calculation is performed using the code FixPhonon, which was developed recently by Kong et al. [47,48]. In the code, the phonon spectra are computed by evaluating the eigenvalues of the dynamical matrix constructed by observing the displacement fluctuations of atoms during molecular dynamics simulations, based on the fluctuation– dissipation theorem. The calculated phonon spectra for different hydrogen coverages are shown in Fig. 3. For pristine graphene, the location of G-band (the higher frequency peak) in our calculation shows a good agreement with previous results [38,49] calculated from Fourier transformation of the velocity autocorrelation function. For graphane, the location of G-band in our calculation agrees well with the recent calculation result in [50]. It can be seen from Fig. 3 that the hydrogenation softens the G-band of the phonon spectra remarkably, which causes a reduction in the phonon group velocities. Hence, the thermal conductivity is reduced according to the classical lattice thermal transport theory X ð5Þ j¼ Ctm l where m is the phonon mode at a specific temperature; C, tm and l are the specific heat, group velocity and mean free path of phonon, respectively. It can also be seen that the shift of the G-band is very sensitive to the hydrogenation for coverage less than 30%, which supports our MD simulation results that the thermal conductivity is very sensitive to the hydrogenation at small coverage.

3.2. Patterned hydrogenation with stripes perpendicular to heat flux We turn our attention to the effect of patterned hydrogenation on the thermal conductivity. The effect of patterned

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hydrogenation on the electronic properties has been studied recently [16–18]. The results show that graphene with patterned hydrogenation is more likely to be used in nanoelectronics. We first study cases where hydrogen atoms are arranged in regular stripes perpendicular to the heat flux. Fig. 4a–c show the atomic configurations with the hydrogen coverage of 4.2%, 16.8% and 50%, respectively. For the patterned hydrogenation at different coverages, the obtained thermal conductivities are shown in Fig. 4d. Compared to random hydrogenation, it can be seen that the patterned hydrogenation results in a much larger drop of thermal conductivity at small hydrogen coverages. For example, at a coverage of about 5%, the thermal conductivity is only about 40% of that of pristine graphene. However, with further increasing hydrogen coverage, the thermal conductivity becomes less sensitive to coverage. The thermal conductivities of both armchair and zigzag graphene sheets show the same behavior upon the patterned hydrogenation. The drastic reduction of thermal conductivity even at a small hydrogen coverage can be understood as follows. Since these hydrogenated stripes cover the whole width of graphene sheets, the high thermal conducting sp2 network is disrupted by the stripes of sp3 bonding even at a small coverage. As the thermal transport ability of hydrogenated re-

Fig. 4 – Atomistic configurations of an armchair graphene with hydrogenated strips perpendicular to the heat flux direction for hydrogenation at (a) 4.2% coverage, (b) 16.8% coverage, and (c) 50% coverage. (d) Variation of the thermal conductivity with hydrogen coverage for both armchair and zigzag graphene sheets. The inset in (d) shows the variation of thermal conductivity with the number of stripes.

gion (sp3 C–H bonds) is much lower than un-hydrogenated region (sp2 C–C bonds), the thermal conduction is primarily controlled by the hydrogenated region. As a result, the thermal conductivity shows a sharp drop at a small coverage.

3.3. flux

Patterned hydrogenation with stripes parallel to heat

Now we study cases where hydrogen atoms are arranged in regular stripes parallel to the heat flux. Fig. 5a–c show the atomic configurations with the hydrogen coverage of 5%, 20% and 60%, respectively. The calculated thermal conductivities at different hydrogen coverages are shown in Fig. 5d. It can be seen that the thermal conductivity decreases gradually with increasing coverage from 0% to 100%, which is in strong contrast to the cases with hydrogenated stripes perpendicular to the heat flux. This is because these hydrogenated stripes are parallel to the heat flux and thus the hydrogenation does not break the sp2 network along the heat flux direction. With an increase in hydrogen coverage, the width of un-hydrogenated domains gradually diminishes, as a result, the thermal conductivity decreases gradually.

Fig. 5 – Atomistic configurations of an armchair graphene with hydrogenated strips parallel to the heat flux direction for hydrogenation at (a) 5% coverage, (b) 20% coverage, and (c) 60% coverage. (d) Variation of thermal conductivity with hydrogen coverage for both armchair and zigzag graphene sheets. The inset in (d) shows the variation of thermal conductivity with the number of stripes.

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Fig. 5d also shows that the thermal conductivity of the hydrogenated armchair graphene is lower than that of the hydrogenated zigzag graphene. For the patterned hydrogenation, there are interfaces between graphene and graphane stripes. Hence, our results suggest that a patterned hydrogenated graphene with armchair interfaces has lower thermal conductivity than that with zigzag interfaces for heat transport along the interface direction. This difference in thermal conductivity between hydrogenated armchair and zigzag graphene sheets is similar to previous results of graphene nanoribbons [23,25]. It was found that armchair graphene nanoribbons showed a lower thermal conductivity than zigzag graphene nanoribbons, which was attributed to the different phonon scattering rates at armchair and zigzag edges [25]. Our simulation results show that hydrogenation greatly reduces the thermal conductivity of graphene. The amount of reduction, however, depends on the distribution and coverage of hydrogen atoms. It is therefore possible to tune the thermal conductivity of graphene by changing hydrogen distribution and coverage. Such nanoscale engineering of graphene structures provides a potential route for controlling thermal transport in graphene-based nanodevices. Besides, the substantially lower thermal conductivities of hydrogenated graphene may be of interest for thermoelectric applications.

3.4.

Applicability of the rule of mixtures

The hydrogenated graphene sheets with graphene and graphane stripe patterns studied above provide an ideal model to check the applicability of the rule of mixtures for thermal conductivities. The rule of mixtures has been used to predict the thermal conductivities in composites [51,52]. In the following, we will discuss whether the rule of mixtures can be used to calculate the thermal conductivity of hydrogenated graphene. For a structure consisting of graphene and graphane stripes parallel to the heat flux, it is expected that the thermal conductivity of the structure should decrease linearly with the increase of hydrogenation coverage, that is, k ¼ ke ð1  cÞ þ ka  c

ð6Þ

and for a structure consisting of graphene and graphane stripes perpendicular to the heat flux, it is expected that the thermal conductivity of the composite structure should follow, k ¼ 1=½ð1  cÞ=ke þ c=ka 

ð7Þ

where ke and ka are the thermal conductivities of graphene and graphane, respectively; and c is the hydrogen coverage. Fig. 6 shows the comparison of thermal conductivity of the hydrogenated graphene vs. H-coverage for both MD calculations and the predictions from the rule of mixtures. For the parallel case, the thermal conductivity of hydrogenated graphene from MD calculations is much lower than that predicted by the rule of mixtures at the same H-coverage; while for the perpendicular case, the MD calculated thermal conductivities are also far below the predictions from the rule of mixtures, indicating the failure of the rule of mixtures. It is known that there are interfaces between graphene and graphane stripes in those composite nanostructures.

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Fig. 6 – Comparison of thermal conductivities calculated from MD simulations with those calculated from the rule of mixtures for hydrogenated graphene sheets with patterned stripes both parallel and perpendicular to heat flux.

Since interfaces were found to reduce the thermal conductivity of composites due to interface phonon scattering [53], it is expected that these interfaces play an important role for the observed discrepancy here. To further support this assertion, we performed additional NEMD simulations for the same coverage of hydrogen distributed in different numbers of patterned stripes. The results shown in the insets of Figs. 4d and 5d clearly show that the more interfaces are, the larger reduction of thermal conductivity is, confirming that interfaces are responsible for the deviation between the MD results and the predictions from the rule of mixtures. The reason why the interfaces play such an important role in reducing the thermal conductivity is that the number of interface atoms in the graphene–graphane composite is comparable to the number of atoms in either the graphene or the graphane. Therefore, the interface thermal resistance exhibits strong influence on the heat conduction in the composite, causing the thermal conductivity to be lower than that calculated from the rule of mixtures. Finally, we discuss briefly the thermal stability of the designed hydrogenation patterns on graphene. It was reported that the migration energy barrier of an isolated H atom on pristine graphene is about 0.3 eV [54], which is relatively low and may result in diffusion of H atoms on graphene at room temperature. However, for the designed hydrogenated patterns, the H atoms are distributed alternatively on both sides of the graphene sheet, which is a more stable configuration [5]. Besides, it is shown that the graphene/graphane interfaces of the hydrogenated patterns enhance the stability of H atoms significantly [55]. Therefore, we expect that our designed patterns are stable at room temperature. In fact, no diffusion of H atoms was observed in our MD simulations.

4.

Conclusions

We have performed NEMD simulations to investigate the thermal conductivity of graphene, focusing on the effect of hydrogenation. We have found that the thermal conductivity of hydrogenated graphene depends greatly on the hydrogen distribution and coverage. For random hydrogenation, the

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thermal conductivity decreases rapidly with the increase of coverage from 0% to 30%, and then becomes insensitive with further increasing coverage. For patterned hydrogenation with stripes perpendicular to the heat flux, a small coverage causes a sharp drop of the thermal conductivity. For patterned hydrogenation with stripes parallel to the heat flux, the thermal conductivity gradually decreases with the increase of coverage from 0% to 100%. Percolation theory for a honeycomb network can be used to explain the variation of thermal conductivity at different hydrogenation distributions and coverages. The physical origin for the reduction of thermal conductivity is found to be the softening of the G-bands phonon modes due to the sp2-to-sp3 bonding transition upon hydrogenation. We have also found that the rule of mixtures for thermal conductivity is not applicable to the graphenegraphane composites due to the presence of interfaces. Our work reveals the possibility of using hydrogenation to tune and control the thermal conduction of graphene. These findings may be useful for the applications of graphene in nanoscale devices.

Acknowledgement This work was supported by the Agency for Science, Technology and Research (A*STAR), Singapore.

R E F E R E N C E S

[1] Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, et al. Electric field effect in atomically thin carbon films. Science 2004;306(5659):666–9. [2] Soldano C, Mahmood A, Dujardin E. Production, properties and potential of graphene. Carbon 2010;48(8):2127–50. [3] Geim AK, Novoselov KS. The rise of graphene. Nat Mater 2007;6(3):183–91. [4] Geim AK, Kim P. Carbon wonderland. Sci Am 2008;298(4):90–7. [5] Boukhvalov DW, Katsnelson MI. Chemical functionalization of graphene. J Phys Condens Matter 2009;21(34):344205–12. [6] Lee YL, Kim S, Park C, Ihm J, Son YW. Controlling halfmetallicity of graphene nanoribbons by using a ferroelectric polymer. ACS Nano 2010;4(3):1345–50. [7] Loh KP, Bao QL, Ang PK, Yang JX. The chemistry of graphene. J Mater Chem 2010;20(12):2277–89. [8] Wang ZF, Li QX, Zheng HX, Ren H, Su HB, Shi QW, et al. Tuning the electronic structure of graphene nanoribbons through chemical edge modification: a theoretical study. Phys Rev B 2007;75(11):113406–10. [9] Boukhvalov DW, Katsnelson MI, Lichtenstein AI. Hydrogen on graphene: electronic structure, total energy, structural distortions and magnetism from first-principles calculations. Phys Rev B 2008;77(3):035427–14. [10] Jones JD, Mahajan KK, Williams WH, Ecton PA, Mo Y, Perez JM. Formation of graphane and partially hydrogenated graphene by electron irradiation of adsorbates on graphene. Carbon 2010;48(8):2335–40. [11] Sofo JO, Chaudhari AS, Barber GD. Graphane: a twodimensional hydrocarbon. Phys Rev B 2007;75(15):153401–4. [12] Ryu S, Han MY, Maultzsch J, Heinz TF, Kim P, Steigerwald ML, et al. Reversible basal plane hydrogenation of graphene. Nano Lett 2008;8(12):4597–602.

[13] Samarakoon DK, Wang XQ. Chair and twist-boat membranes in hydrogenated graphene. ACS Nano 2009;3(12):4017–22. [14] Elias DC, Nair RR, Mohiuddin TMG, Morozov SV, Blake P, Halsall MP, et al. Control of graphene’s properties by reversible hydrogenation: evidence for graphane. Science 2009;323(5914):610–3. [15] Roman T, Dino WA, Nakanishi H, Kasai H, Sugimoto T, Tange K. Hydrogen pairing on graphene. Carbon 2007;45(1):218–20. [16] Singh AK, Yakobson BI. Electronics and magnetism of patterned graphene nanoroads. Nano Lett 2009;9(4):1540–3. [17] Balog R, Jorgensen B, Nilsson L, Andersen M, Rienks E, Bianchi M, et al. Bandgap opening in graphene induced by patterned hydrogen adsorption. Nat Mater 2010;9(4):315–9. [18] Fiori G, Lebegue S, Betti A, Michetti P, Klintenberg M, Eriksson O, et al. Simulation of hydrogenated graphene field-effect transistors through a multiscale approach. Phys Rev B 2010;82(15):153404–8. [19] Balandin AA, Ghosh S, Bao WZ, Calizo I, Teweldebrhan D, Miao F, et al. Superior thermal conductivity of single-layer graphene. Nano Lett 2008;8(3):902–7. [20] Cai WW, Moore AL, Zhu YW, Li XS, Chen SS, Shi L, et al. Thermal transport in suspended and supported monolayer graphene grown by chemical vapor deposition. Nano Lett 2010;10(5):1645–51. [21] Ghosh S, Calizo I, Teweldebrhan D, Pokatilov EP, Nika DL, Balandin AA, et al. Extremely high thermal conductivity of graphene: prospects for thermal management applications in nanoelectronic circuits. Appl Phys Lett 2008;92(15):151911–3. [22] Pop E, Mann D, Wang Q, Goodson K, Dai H. Thermal conductance of an individual single-wall carbon nanotube above room temperature. Nano Lett 2006;6(1):96–100. [23] Guo ZX, Zhang D, Gong XG. Thermal conductivity of graphene nanoribbons. Appl Phys Lett 2009;95(16):163103–6. [24] Lan J, Wang JS, Gan CK, Chin SK. Edge effects on quantum thermal transport in graphene nanoribbons: tight-binding calculations. Phys Rev B 2009;79(11):115401–5. [25] Hu JN, Ruan XL, Chen YP. Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study. Nano Lett 2009;9(7):2730–5. [26] Jiang JW, Wang JW, Li B. Thermal conductance of graphene and dimerite. Phys Rev B 2009:79(20):205418–14. [27] Varshney V, Patnaik SS, Roy AK, Froudakis G, Farmer BL. Modeling of thermal transport in pillared-graphene architectures. ACS Nano 2010;4(2):1153–61. [28] Wei Z, Ni Z, Bi K, Chen M, Chen Y. In-plane lattice thermal conductivities of multilayer graphene films. Carbon 2011;49(8):2653–8. [29] Zhang G, Li B. Thermal conductivity of nanotubes revisited: effects of chirality, isotope impurity, tube length, and temperature. J Chem Phys 2005;123(11):114714–8. [30] Wang Z, Tang D, Zheng X, Zhang W, Zhu Y. Length-dependent thermal conductivity of single-wall carbon nanotubes: prediction and measurements. Nanotechnology 2007;18(47):475714–8. [31] Maruyama S. A molecular dynamics simulation of heat conduction in finite length SWNTs. Physica B 2002;323(1):193–5. [32] Chien SK, Yang YT, Chen CK. Influence of hydrogen functionalization on thermal conductivity of graphene: nonequilibrium molecular dynamics simulations. Appl Phys Lett 2011;98(3):033107–10. [33] Muller-Plathe F. A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity. J Chem Phys 1997;106(14):6082–5. [34] Che J, Cagin T, Goddard WA. Thermal conductivity of carbon nanotubes. Nanotechnology 2000;11(2):65–9.

CARBON

4 9 ( 20 1 1 ) 4 7 5 2–47 5 9

[35] Stuart SJ, Tutein AB, Harrison JA. A reactive potential for hydrocarbons with intermolecular interactions. J Chem Phys 2000;112(14):6472–86. [36] Plimpton SJ. Fast parallel algorithms for short-range molecular dynamics. J Comp Phys 1995;117(1):1–19. [37] Brenner DW. The art and science of an analytic potential. Phys Status Solidi B 2000;217(1):23–40. [38] Wei N, Xu L, Wang HQ, Zheng JC. Strain engineering of thermal conductivity in graphene sheets and nanoribbons: a demonstration of magic flexibility. Nanotechnology 2011;22(10):105705–11. [39] Xu ZP, Buehler MJ. Nanoengineering heat transfer performance at carbon nanotube interfaces. ACS Nano 2009;3(9):2767–75. [40] Pei QX, Zhang YW, Shenoy VB. A molecular dynamics study of the mechanical properties of hydrogen functionalized graphene. Carbon 2010;48(3):898–904. [41] Strachan A, Kober EM, van Duin ACT, Oxgaard J, Goddard WA. Thermal decomposition of RDX from reactive molecular dynamics. J Chem Phys 2005;122(5):054502–10. [42] Sen D, Novoselov KS, Reis PM, Buehler MJ. Tearing graphene sheets from adhesive substrates produces tapered nanoribbons. Small 2010;6(10):1108–16. [43] Jiang JW, Wang JS, Li B. Topological effect on thermal conductivity in graphene. J Appl Phys 2010;108(6):064307–12. [44] Pei QX, Zhang YW, Shenoy VB. Mechanical properties of methyl functionalized graphene. Nanotechnology 2010;21(11):115709–16. [45] Pan R, Xu Z, Zhu Z, Wang Z. Thermal conductivity of functionalized single-wall carbon nanotubes. Nanotechnology 2007;18(28):285704–8.

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[46] Suding PN, Ziff RM. Site percolation thresholds for archimedean lattices. Phys Rev E 1999;60(1):275–83. [47] Kong LT. Phonon dispersion measured directly from molecular dynamics simulations. Comput Phys Commun 2011, doi: 10.1016/j.cpc.2011.04.019. [48] Kong LT, Lewis LJ. Surface diffusion coefficients: substrate dynamics matters. Phys Rev B 2008;77(16):165422–5. [49] Xu Z, Buehler MJ. Strain controlled thermomutability of single-walled carbon nanotubes. Nanotechnology 2009;20(18):185701–6. [50] Peelaers H, Hernandez-Nieves AD, Leenaerts O, Partoens B, Peeters FM. Vibrational properties of graphene fluoride and graphane. Appl Phys Lett 2011;98:051914–7. [51] Gaier JR, YoderVandenberg Y, Berkebile S, Stueben H, Balagadde F. The electrical and thermal conductivity of woven pristine and intercalated graphite fiber–polymer composites. Carbon 2003;41(12):2187–93. [52] Nayak R, Dora PT, Satapathy A. A computational and experimental investigation on thermal conductivity of particle reinforced epoxy composites. Comput Mater Sci 2010;48(3):576–81. [53] Cahill DG, Ford WK, Goodson KE, Mahan GD, Majumdar A, Maris HJ, et al. Nanoscale thermal transport. J Appl Phys 2003;93(2):793–818. [54] Boukhvalov DW. Modeling of hydrogen and hydroxyl group migration on graphene. Phys Chem 2010;12(47):15367–71. [55] Ao ZM, Hernandez-Nieves AD, Peeters FM, Li S. Enhanced stability of hydrogen atoms at the graphene/graphane interface of nanoribbons. Appl Phys Lett 2010;97(23):233109– 12.