Comparison of isotope effects on thermal conductivity of graphene nanoribbons and carbon nanotubes Xiuqiang Li, Jie Chen, Chenxi Yu, and Gang Zhang Citation: Applied Physics Letters 103, 013111 (2013); doi: 10.1063/1.4813111 View online: http://dx.doi.org/10.1063/1.4813111 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/103/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermal transport in folded zigzag and armchair graphene nanoribbons Appl. Phys. Lett. 104, 241908 (2014); 10.1063/1.4884278 The effect of defects on negative differential thermal resistance in symmetric graphene nanoribbons Appl. Phys. Lett. 104, 013106 (2014); 10.1063/1.4861472 Interfacial thermal conductance limit and thermal rectification across vertical carbon nanotube/graphene nanoribbon-silicon interfaces J. Appl. 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Downloaded to IP: 222.66.175.157 On: Sat, 25 Apr 2015 10:10:11 APPLIED PHYSICS LETTERS 103, 013111 (2013) Comparison of isotope effects on thermal conductivity of graphene nanoribbons and carbon nanotubes Xiuqiang Li,1,a) Jie Chen,2,a) Chenxi Yu,1 and Gang Zhang1,3,b) 1 Key Laboratory for the Physics and Chemistry of Nanodevices and Department of Electronics, Peking University, Beijing 100871, People’s Republic of China 2 Department of Physics, Centre for Computational Science and Engineering, and Graphene Research Centre, National University of Singapore, Singapore, Singapore 117542 3 Institute of High Performance Computing, Singapore, Singapore 138632 (Received 12 September 2012; accepted 16 June 2013; published online 3 July 2013) By using molecular dynamics simulation, we explore the isotope effect on thermal conductivity of graphene nanoribbons (GNRs) and carbon nanotubes (CNTs). For both GNRs and CNTs, the lattice thermal conductivity decreases when isotope concentration increases from 0% to 30%. The thermal conductivity reduction ratio in GNRs is less than that in CNTs. For example, thermal conductivity of CNT with 5% 13C concentration is 25% lower than that of pure CNTs; however, the reduction in thermal conductivity of GNRs with the same isotope concentration is only about 12%. Lattice dynamics analysis reveals that these phenomena are related to the phonon C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4813111] localization. V In the past decades, carbon nanotube (CNT) and graphene have attracted lots of research attention due to their promising electronic, mechanical, and thermal properties. In carbon nanotube, the atoms are arranged in a onedimensional lattice, while graphene is a monolayer of graphite with a honeycomb lattice structure. Single walled carbon nanotube (SWCNT) can be seen as a narrow graphene nanoribbon (GNR) rolled seamlessly. Inversely, several recent experiments1–3 report the possibilities of making highly pristine narrow GNRs with controlled edge structures by unzipping SWCNTs. These techniques closely relate onedimensional CNTs with two-dimensional GNRs. It is interesting to find that both CNT and graphene have ultra-high thermal conductivity. Room temperature thermal conductivity of CNT is larger than 3000 W/m K.4 Moreover, single layer graphene has the room temperature thermal conductivity as high as 4800–5300 W/m K, reported from a Raman spectroscope based measurement.5 Several independent experimental studies on graphene based on Raman measurements also reported the high room temperature thermal conductivity, ranging from 1800 W/m K to 5000 W/m K.6–10 Such high thermal conductivities of CNT and graphene are among the highest values of known materials, making them promising structures for highefficiency phononics device.11 Theoretical12–18 and experimental19,20 works have focused on the realization of functional thermal devices with these nanoscale structures. The high thermal conductivity of CNTs and graphenes is for pure and defect-free samples. Actually, both CNTs and graphenes can have natural defects and isotope impurities in the process of fabrication. There are three kinds of carbon isotopes: 12C, 13C, and 14C. They have the same electronic structure but different atomic mass. The isotope atoms increase phonon scattering rate due to the mass-difference, resulting in the reduction of thermal conductivity.21 It has a) X. Li and J. Chen contributed equally to this work. Email: [email protected] b) 0003-6951/2013/103(1)/013111/5/$30.00 been reported theoretically that the thermal conductivity of CNT composed of a 50:50 mixture of 12C and 14C is about 40% of that of isotopically pure 12C nanotube.22 This remarkable isotope effect on thermal conductivity of nanotube has been observed experimentally.23 Furthermore, the impact of isotope impurity on thermal conductivity has also been studied in graphene and other nano materials.24–27 Although it is well known that isotope impurity modifies the thermal conductivity of graphene and carbon nanotube, many important and fundamental questions remain unsolved. For example, does phonon scattering due to isotope have the same impact on thermal conductivity of graphene and CNT? From the theoretical point of view, it is not obvious that phonon scattering due to isotope will affect the thermal conductivity of CNTs more than that of graphenes, or inversely. So far there is no report on the comparison of isotope effect on thermal conductivity of graphene and CNT. Besides, as the calculated thermal conductivity depends on the diameter, chirality of CNT22,28 and the width of GNR,29,30 a model that treats both graphene and CNT on the same level of approximation is desired. Furthermore, in thermal conductivity calculation, there are different empirical potentials to describe inter-atomic interactions. In order to theoretically interpret experiments, it is important to explore the effect of the potential used. Thus, in this work, we treat both CNT and GNR on the same footing to investigate the impact of isotope doping on their thermal conductivities. We use molecular dynamics (MD) and lattice dynamics simulations to study the isotope concentration dependent thermal conductivity of GNR and CNT. Our study extends the understanding of isotope effect on thermal conductivity in one-dimensional and two-dimensional materials. As the electrons in graphene and carbon nanotube only have limited contributions to total thermal conductivity,31 in our simulations, classical MD simulation is adopted to calculate lattice thermal conductivity, by using the LAMMPS package.32 The bonding interactions between carbon atoms are modeled by a Tersoff type potential,33 which implicitly 103, 013111-1 C 2013 AIP Publishing LLC V This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 222.66.175.157 On: Sat, 25 Apr 2015 10:10:11 013111-2 Li et al. Appl. Phys. Lett. 103, 013111 (2013) contains many-body information. The accuracy of this potential is demonstrated by MD simulations of carbon based nano structures.22,28,34,35 For example, it has been reported experimentally that 50% isotope concentration can reduce the thermal conductivity of nanotube by 60%.23 This is in quantitative agreement with the theoretical prediction based on MD simulations using Tersoff potential.22 The structures considered in our study include GNR and CNT, as shown in ˚ and length is 103.7 A ˚, Figure 1. The width of GNR is 25.3 A with a total number of 1008 atoms. When rolling this GNR in the width direction, it becomes the (10, 0) CNT. Before non-equilibrium MD simulation, the canonical ensemble MD simulation with Nose–Hoover36,37 heat bath first runs for 4 106 steps to equilibrate the whole system at room temperature. After structure relaxation, atoms at the two ends in the length direction (x) are fixed to prevent atoms from sublimating. Next to the fixed boundary, Nose–Hoover heat baths with different temperature are applied to the two ends of x direction to establish a temperature gradient along the longitudinal direction. Free boundary condition is used in the y direction and z direction. The velocity Verlet algorithm is used to integrate the differential equations of motion. The non-equilibrium MD simulations are then performed long enough (5 106 time steps, time step is 0.5 fs) to allow the system to reach the non-equilibrium steady state, and time average is performed for another 107 time steps. Thermal conductivity is calculated by the Fourier’s law J ¼ jrT; (1) where J is defined as the energy transported in unit time through unit cross sectional area and rT is the temperature gradient. In our calculations, 13C atoms are added by randomly substituting 12C atoms in isotopically pure 12C CNT and corresponding GNR. To suppress the possible fluctuations arising from random distribution, thermal conductivity in the isotopically doped samples is obtained by averaging over ten independent distributions with the same isotope concentration. We compare in Figure 2 the thermal conductivity of CNTs and the corresponding GNRs with different isotope concentration. The concentration of 13C is defined as N13/(N12 þ N13), where N12(N13) is the number of 12C(13C) atoms. As the thermal conductivity of CNT and GNR depends on the system scale,22,28,29 in this work, our concern is the relative change of thermal conductivity with isotope doping, not the absolute value. The calculated thermal conductivity is normalized by the thermal conductivity of corresponding isotopically pure 12C CNT/GNR. Within the FIG. 1. Schematic picture of the graphene nanoribbon and corresponding carbon nanotube. The last layers of atoms at the two ends (in green) are fixed, and the atoms (in red) next to the fixed layers are placed in thermostats. FIG. 2. Isotopic concentration dependent thermal conductivity of GNR and CNT. The calculated thermal conductivities are normalized by the thermal conductivities of corresponding pure GNR/CNT. concentration range (0%–30%) considered in our study, thermal conductivity decreases with 13C isotope concentration in both CNT and GNR. In the low isotope content region (less than 10% 13C), thermal conductivity of CNT decreases rapidly with increasing isotope concentration. For instance, thermal conductivity of CNT with 5% 13C is 25% lower than that of isotopically pure CNT. While in the high isotope concentration region (10%–30% 13C), the thermal conductivity decreases slowly with further increase in isotope concentration. For a given isotope concentration, the error bars are determined from the standard deviations of the thermal conductivities for ten samples with different isotopic distributions. The physical mechanism for this isotope concentration dependent thermal conductivity is the localization of phonon modes. In the low isotope concentration region, most 13C atoms are far away from each other, the number of localized modes increases rapidly with doping concentration, thus, the thermal conductivity decreases quickly. However, in the high isotope concentration region, the 13C atoms are close to each other. Thus, the number of localized modes increases very slowly with doping concentration, which results in the slow reduction in thermal conductivity.24 Similar phenomenon is also observed in the isotopic concentration dependent thermal conductivity of GNRs. More interestingly, the reduction ratio is much smaller in GNR than that in CNT. Thermal conductivity of GNR is reduced by 12% with the 13C concentration of 5%. Such observation suggests that the effect of isotope impurity on thermal transport is stronger in CNT than that in GNR. To qualitatively understand the physical mechanism for different isotopic concentration dependent thermal conductivity in CNT and GNR, we carried out a vibrational eigenmode analysis. All phonon modes are computed along the axial direction with Tersoff potential by using the “General Utility Lattice Program” (GULP) package.38 Mode localization can be quantitatively characterized by the phonon participation ratio Pk defined for each eigen-mode k as39 P1 k ¼N XX i eia;k eia;k 2 ; (2) a This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 222.66.175.157 On: Sat, 25 Apr 2015 10:10:11 013111-3 Li et al. where N is the total number of atoms and eia,k is the a-th eigenvector component of eigenmode k for the i-th atom. The participation ratio measures the fraction of atoms participating in a given mode and effectively indicates the localized modes with O(1/N) and delocalized modes with O(1). It can provide quantitative information about localization effect for each eigenmode k(x,k), where x is the eigenfrequency and k is the wave vector in the first Brillouin zone. Figure 3(a) shows the phonon participation ratio for each eigenmode k in the isotopically pure 12C (10, 0) CNT, with the red color denoting the localized phonons (Pk 0) and the blue color denoting the delocalized modes (Pk 1). Since the periodic boundary condition is used along the axial direction in the lattice dynamics calculations, there is no dangling bond in CNT. As a result, majority of the phonons in pure 12C (10,0) CNT exhibits a delocalized nature, except for those modes near the Brillouin zone edge. Phonon localization can be visualized in the coordinate space through the spatial distribution of phonon energy.40–42 Further analysis based on phonon energy distribution reveals that no localized phonons exist in pure 12C CNT due to its perfect bonding (inset of Fig. 3(a)). In contrast, when CNT is unzipped to GNR structure, dangling bonds are generated at the edge of GNR. Due to the localized nature of the edge states,43 the calculated energy distribution shows that the localized phonons (Pk < 0.2) only reside at those dangling sites (inset of Fig. 3(b)). Furthermore, Fig. 3(b) shows that the edge scattering causes significant reduction of phonon participation ratio for majority phonon modes in isotopically pure 12C GNR compared to that in corresponding CNT. When (10, 0) CNT is doped with 30% 13C, the phonon scattering due to isotope leads to the obvious reduction of phonon participation ratio (Fig. 3(c)), especially for those Appl. Phys. Lett. 103, 013111 (2013) intermediate and high frequency phonons. Our results are consistent with a recent study on thermal transport in defectengineered CNT.44 It was found that the intermediate and high frequency phonons are filtered out by defect scattering, leading to the reduction of thermal conductivity when the defect concentration increases. Therefore, the enhanced phonon localization effect is responsible for the reduction of thermal conductivity in doped CNT. For the GNR case, as the phonon participation ratio is already greatly suppressed by the edge scattering (Fig. 3(b)), the phonon scattering due to isotope can only induce a much weaker phonon localization effect in GNR (Fig. 3(d)) than that in CNT (Fig. 3(c)). This is the underlying physical mechanism that with the same isotope concentration, the reduction of thermal conductivity in GNR is smaller than that in CNT. It is worth pointing out that as the eigenfrequency of phonon modes may differ from that for the isotopically pure CNT/GNR after doping, it is quite difficult to provide a quantitative comparison of the phonon participation ratio at the same frequency before and after isotope doping. Thus, only qualitative analysis is provided here. Next, we discuss the effect of inter-atomic potential on the calculation results of reduction in thermal conductivity. The adaptive intermolecular reactive empirical bond order (AIREBO)45 potential is used to compare the results calculated from Tersoff potential. This potential is an extension of Brenner’s reactive empirical bond order potential. It is similar to the reactive empirical bond order potential, and it includes an additional term for longer range interactions. In general, the AIREBO potential yields lower thermal conductivity than Tersoff potential does.46 In Figure 4, the thermal conductivity of isotope doped GNR calculated with Tersoff (AIREBO) potentials is normalized by the thermal FIG. 3. Comparison of phonon participation ratio (calculated with Tersoff potential) in both CNT and GNR before and after isotopic doping. The participation ratio for each phonon mode k(x,k) is plotted according to the color bar. (a) Pure (10,0) CNT. The inset shows spatial distribution for delocalized phonons (blue). (b) Pure (10,0) GNR. The inset shows the spatial distribution for delocalized phonons (blue) and localized phonons (red). (c) (10,0) CNT doped with 30% 13C. (d) (10,0) GNR doped with 30% 13C. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 222.66.175.157 On: Sat, 25 Apr 2015 10:10:11 013111-4 Li et al. FIG. 4. Isotope effect on thermal conductivity of GNR calculated with Tersoff and AIREBO potentials. The calculated thermal conductivities are normalized by the thermal conductivities of pure GNR with the same interatomic potentials, respectively. conductivity of isotopically pure 12C GNR calculated with the same potential, respectively. It is interesting to find that the amount of reduction in thermal conductivity predicted with AIREBO potential is much lower than those from Tersoff potential. The calculation results using Tersoff potential show that compared to pure GNR, there is about 30% reduction in thermal conductivity of GNR with 30% isotope concentration. In contrast, there is only 16% reduction in the thermal conductivity calculated by using AIREBO potential at the same isotope concentration. Thus, the choice of interatomic potential plays a significant role in the predicted isotope effect. This phenomenon can be understood from the phonon vibrational eigenmode analysis. In the lattice dynamics Appl. Phys. Lett. 103, 013111 (2013) calculation, Brenner potential is adopted due to its availability in GULP package. As shown in Figure 5, although there are some common traits between the results from Tersoff and Brenner potentials, the phonon participation ratio calculated from different potentials shows quite different features. For instance, the cut-off frequency for the Brenner calculation result is reduced to 2000 cm1, compared to that about 2500 cm1 by using Tersoff potential, consistent with previous study.47 The high frequency phonon modes with Brenner potential is greatly suppressed compared to that with Tersoff potential (Figs. 5(a) and 5(b)). When introducing isotopic scattering which can induce reduction of participation ratio for high frequency phonons, the suppressed phonon mode in high frequency regime of Brenner spectra leads to the weak dependence on isotope doping. This is similar to the reported difference in phonon density of states calculated by using Tersoff and Brenner potentials.48 To compare with experiment,26 we did the additional calculations on thermal conductivity of GNRs composed of a 50:50 mixture of 12C and 13C, using both Tersoff and AIREBO potentials, respectively. For the calculation results using Tersoff potential, compare to isotopically pure 12C GNR, there is 30% reduction in thermal conductivity, while the reduction is about 25% if AIREBO potential is used. The experimental reports26 show that the thermal conductivity of isotopically pure 12C graphene is more than a factor of two higher than the value of thermal conductivity in graphene sheets composed of a 50:50 mixture of 12C and 13C. This discrepancy is partially due to the size effect on thermal conductivity. In experiment, graphene sheets with scale of micrometers are used. However, in the present simulations, ˚ and length is the width of graphene nanoribbon is 25.3 A ˚ 103.7 A, respectively. Although MD simulation can describe phonon-defect and phonon-phonon interactions with FIG. 5. Comparison of phonon participation ratio in GNR with different interatomic potentials. The participation ratio for each phonon mode k(x,k) is plotted according to the color bar. (a) Pure (10,0) GNR with Brenner potential. (b) Pure (10,0) GNR with Tersoff potential. (c) Isotopically doped (10,0) GNR with Brenner potential. (d) Isotopically doped (10,0) GNR with Tersoff potential. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 222.66.175.157 On: Sat, 25 Apr 2015 10:10:11 013111-5 Li et al. reasonable accuracy, it is computationally expensive to simulate samples in scale of micrometer. For the small graphene nanoribbon samples, the phonon participation ratio is already greatly suppressed by the boundary scattering, thus isotope impurity can only induce a much weaker effect in narrow GNRs than that in the large graphene sheets. In summary, we have studied isotope effect on thermal conductivity of carbon nanotube and graphene nanoribbons. With the same isotope concentration, the reduction of thermal conductivity in GNR is smaller than that in CNT. From the vibrational eigenmode analysis, it is obvious that random isotope doping induces localization for phonons with high frequency, while the phonon participation ratio in this frequency regime is already greatly suppressed by the edge scattering in GNR. So in GNR, phonon scattering due to isotope will cause less reduction in thermal conductivity than that in CNT. 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