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. Phys. 113, 064311 (2013); 10.1063/1.4790367
The effects of diameter and chirality on the thermal transport in free-standing and supported carbon-nanotubes
Appl. Phys. Lett. 100, 233105 (2012); 10.1063/1.4725194
Isotopic effects on the thermal conductivity of graphene nanoribbons: Localization mechanism
J. Appl. Phys. 107, 054314 (2010); 10.1063/1.3329541
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
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.
Furthermore, we find that the amount of reduction induced by
isotope effect in thermal conductivity calculated with
AIREBO potential is smaller than that calculated with Tersoff
potential. These results extend the understanding of isotope
effect, which is one of the commonly used approaches to
manipulate thermal conductivity of nano materials.
This work was supported by National Natural Science
Foundation of China (Grant No. 11274011), the Ministry of
Education of China (Grant No. 20110001120133), the
Ministry of Science and Technology of China (Grant No.
2011CB933001).
1
L. Jiao, L. Zhang, X. Wang, G. Diankov, and H. Dai, Nature 458, 877
(2009).
2
Z. Zhang, Z. Sun, J. Yao, D. V. Kosynkin, and J. M. Tour, J. Am. Chem.
Soc. 131, 13460 (2009).
3
X. Wang and H. Dai, Nature Chem. 2, 661 (2010).
4
P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, Phys. Rev. Lett. 87,
215502 (2001).
5
A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao,
and C. N. Lau, Nano Lett. 8, 902 (2008).
6
J. H. Seol, I. Jo, A. L. Moore, L. Lindsay, Z. H. Aitken, M. T. Pettes, X.
Li, Z. Yao, R. Huang, D. Broido, N. Mingo, R. S. Ruoff, and L. Shi,
Science 328, 213 (2010).
7
W. Cai, A. L. Moore, Y. Zhu, X. Li, S. Chen, L. Shi, and R. S. Ruoff,
Nano Lett. 10, 1645 (2010).
8
S. Ghosh, W. Bao, D. L. Nika, S. Subrina, E. P. Pokatilov, C. N. Lau, and
A. A. Balandin, Nature Mater. 9, 555 (2010).
9
S. Chen, A. L. Moore, W. W. Cai, J. Suk, J. An, C. Mishra, C. Amos, A. W.
Magnuson, J. Y. Kang, L. Shi, and R. S. Ruoff, ACS Nano 5, 321 (2011).
10
J.-U. Lee, D. Yoon, H. Kim, S. W. Lee, and H. Cheong, Phys. Rev. B 83,
081419 (2011).
Appl. Phys. Lett. 103, 013111 (2013)
11
N. Li, J. Ren, L. Wang, G. Zhang, P. H€anggi, and B. Li, Rev. Mod. Phys.
84, 1045 (2012).
G. Wu and B. Li, Phys. Rev. B 76, 085424 (2007).
13
N. Yang, G. Zhang, and B. Li, Appl. Phys. Lett. 93, 243111 (2008).
14
N. Yang, G. Zhang, and B. Li, Appl. Phys. Lett. 95, 033107 (2009).
15
J. Hu, X. Ruan, and Y. P. Chen, Nano Lett. 9, 2730 (2009).
16
G. Zhang and H. Zhang, Nanoscale 3, 4604 (2011).
17
K. G. S. H. Gunawardana, K. Mullen, J. Hu, Y. P. Chen, and X. Ruan,
Phys. Rev. B 85, 245417 (2012).
18
H. Y. Cao, Z.-X. Guo, H. Xiang, and X. G. Gong, Phys. Lett. A 376, 525
(2012).
19
C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, Science 314, 1121
(2006).
20
H. Tian, D. Xie, Y. Yang, T.-L. Ren, G. Zhang, Y. Wang, C. Zhou, P.
Peng, L. Wang, and L. Liu, Sci. Rep. 2, 523 (2012).
21
G. Zhang and B. Li, NanoScale 2, 1058 (2010).
22
G. Zhang and B. Li, J. Chem. Phys. 123, 114714 (2005).
23
C. W. Chang, A. M. Fennimore, A. Afanasiev, D. Okawa, T. Ikuno, H.
Garcia, D. Li, A. Majumdar, and A. Zettl, Phys. Rev. Lett. 97, 085901
(2006).
24
J. W. Jiang, J. H. Lan, J. S. Wang, and B. Li, J. Appl. Phys. 107, 054314
(2010).
25
J. Hu, S. Schiffli, A. Vallabhaneni, X. L. Ruan, and Y. P. Chen, Appl.
Phys. Lett. 97, 133107 (2010).
26
S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. J. Cho, W. Cai, A. A.
Balandin, and R. S. Ruoff, Nature Mater. 11, 203 (2012).
27
G. Balasubramanian, I. K. Puri, M. Bohm, and F. Leroy, Nanoscale 3,
3714 (2011).
28
B. Qiu, Y. Wang, Q. Zhao, and X. Ruan, Appl. Phys. Lett. 100, 233105
(2012).
29
Z. Guo, D. Zhang, and X. G. Gong, Appl. Phys. Lett. 95, 163103 (2009).
30
Y. Xu, J.-S. Wang, W. H. Duan, B. L. Gu, and B. Li, Phys. Rev. B 78,
224303 (2008).
31
T. Yamamoto, S. Watanabe, and K. Watanabe, Phys. Rev. Lett. 92,
075502 (2004).
32
S. Plimpton, J. Comput. Phys. 117, 1 (1995).
33
J. Tersoff, Phys. Rev. B 39, 5566 (1989).
34
J. Chen, G. Zhang, and B. Li, Nanoscale 5, 532 (2013).
35
C. Yu and G. Zhang, J. Appl. Phys. 113, 044306 (2013).
36
S. J. Nose, J. Chem. Phys. 81, 511 (1984).
37
W. G. Hoover, Phys. Rev. A 31, 1695 (1985).
38
J. D. Gale, J. Chem. Soc., Faraday Trans. 93, 629 (1997).
39
A. Bodapati, P. K. Schelling, S. R. Phillpot, and K. Keblinski, Phys. Rev. B
74, 245207 (2006).
40
J. Chen, G. Zhang, and B. Li, Nano Lett. 10, 3978 (2010).
41
J. Chen, G. Zhang, and B. Li, J. Chem. Phys. 135, 204705 (2011).
42
J. Chen, G. Zhang, and B. Li, J. Chem. Phys. 135, 104508 (2011).
43
J.-W. Jiang, J. Chen, J.-S. Wang, and B. Li, Phys. Rev. B 80, 052301
(2009).
44
C. Sevik, H. Sevincli, G. Cuniberti, and T. Cagin, Nano Lett. 11, 4971
(2011).
45
S. J. Stuart, A. B. Tutein, and J. A. Harrison, J. Chem. Phys. 112, 6472
(2000).
46
Q. X. Pei, Z. D. Sha, and Y. W. Zhang, Carbon 49, 4752 (2011).
47
L. Lindsay and D. A. Broido, Phys. Rev. B 81, 205441 (2010).
48
J. W. Lee, A. J. Meade, E. V. Barrera, and J. A. Templeton, J. Nanoeng.
Nanosyst. 224, 41 (2010).
12
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