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Printed in the United Stales of America
Joumalof
Computational and Theoretical Nanoscience
Vol. 8.81 4-8 19.2011
Boundary Condition and Strain Effects on the Quality
Factors of Single Walled Carbon Nanotubes
Sung Youb Kim and Harold S. Park*
University 01 Colorado. 80ufde r, CO 80309. Uniled Slales 01 America
띠」u』’￡4
zuI**
띠띠매t
We utilize classical molecular dynamics to study energy dissipation (the Q-Iactors) 이 carbon
nanotube-based nanoresona10rs undergoing Ilexural oscillations. Specifically. we have studied the
difference in Q-factors 01 nanotubes wi1h lixedllixed and lixedlfree bounda깨 conditions. In doing
so. we have lound that lixedlfixed nanotubes have signilicantly higher Q-Iactors. particularly at low
temperatures. Furthermore. we have lound that mechanical strain can be utilized to enhance the Qfactors 01 lixedllixed nanotubes by lactors 01 2-4 across a range 01 temperatures lor tensile strains
ranging Irom 0 to 6%. Th e res비ts collectively indicate that fixeψlixed carbon nanotubes sho비d
be prelerable lor NEMS applications at low temperature due to a combination 01 inherently higher
Q-factors , and the lact that the Q-Iactors can be lurther improved through the application 01 tensile
strain
Keywords: Nanotubes , Strain, Quality Factor, Boundary Conditions.
1. INTRODUCTl ON
Carbon nanotubes (CNTs) have been amongst the
most-studied nanomaterials ,1 with recent applications in
nanoeleclromechanical syslems (NEMS). 2.3 Due 10 their
combination of high sliffness and low weight , CNTs have
been investigated for NEMS-based sensing applications ,
with a particular interest in ultrasensitive mass sensing4. !i
and force sensing. 6. 7
For enhanced sensilivity to environmental (i.e.. forces
and masses) changes , the quality (Q)-factors of NEMS
are of particular imeres t. In particular, high Q-factors are
desired for NEMS applications as Ihis indicates smaller
energy dissipa1ion per vibrational cycle , or equivalently
smaller linewidlhs during the resonance of the NEMS.
Regardless of the physical imerpκtation ， higher Q-factors
lead to greater sensitivity and resolution of adsorbed
masses or surrounding forces , and thus are highly desired.
Th e resonance , and Q-factors of both fixedlfree ,“ and
fixedlfixed 3. 9- 12 CNT oscillators have been studied experimentally by various researchers. Q-factors ranging from
159 to 1000' were found. Particularly relevant to the
present work is the study of Pu rcell et al.," who appJied
tension to the CNTs using an electric field , Ihereby tuning
the resonant frequency of Ihe CNT; they reported Q-factors
as high as 2500, though the dependence of Q with tensile
•Author to whom correspondcnce should be addressed
814
J.
Comput. Th80f. Nanosci. 20f1 , VoJ. 8, NO. 5
strain was not discussed. More recently, Q-fac10rS as high
as 10' were found by Huttel et al. 12 in fixedlfixed carbon
nanotubes
The nature of the boundary conditions that are used
to fix or clamp the ends of the CNTs during resonance
are of Înterest for various reasons , though a cornprehensive experimental study delineating the effects of boundary condition , i. e .. fixedlfree versus fixedlfixed ends on the
Q-factors of the CNTs has not been performed. The boundary condition is imponant for the following reasons. First,
fixedlfree nanotubes will have a higher dynamic range ,'
which means that they can undergo larger deformations
before nonlinear effects are excited. Furthermore , exirinsic clamping losses 13 through interaction with the fixed
substrate will be minimized for fixedlfree boundary conditions , due to being clamped at only one end.
In contrast , while fixedlfixed nanotubes may suffer enhanced extrinsic c1 amping losses , evidence exists
that. because mechanical strain can more easily be
applied experimentally in the fixedlfixed configuration ,
that the Q-factors of various nanostructures , including
mctal nanowires. 14 graphenc monolaycrs. IS silicon a~~ sil
ícon nitríde nanowi-res ,16.17 MEMS heterostructuresl 8 and
carbon nanotubes 19 can be tuned and , more importantly,
enhanced through the application of tensile mechanical
str3m.
Energy dissipation in oscillating CNTs has been sludied
using c1 assical molec비 ar dynamics (MD) for both single
1546- 19SS12011 1&18 141OO6
doi:10.1166Ij ctn.201 1.17!i g
흩‘
Boundary Condition and Strain EtTects on the Quality Factors of Single Walled Carbon Nanotubes
Kim and Park
the Q-factors reponed here for fixedlfixed (5 ,5) CNTs and
the fixedlfree (5 ,5) CNTs reponed by Ji. ng et .1.2<J
3.
RESULTS
3.1. Boundary Condition Effects on Q
We first discuss the effect of boundary condition on the
Q-factors of the CNTs. Ag.in , we h.ve ca1c ulated only the
fixedlfixed (5 ,5) single-w.J1 ed CNTs; the data for comparison was taken from the identical (5 ,5) fÌ<edlfree CNTs of
Jiang et al. 20
Similar to Jiang et a l. .'o we show in Figure 1 the
extemal energy (EE) time history for the fixedlfixed (5 ,5)
CNT at different temperatures , ranging from 0.05 K to
1.6
,.
;
,,,
1.4
‘.
Extemal energy at 8 K
2 , SIMULATION DETAILS
Classical MD simulations were performed on (5 ,5)
fixedlfixed single-wa1led CNTs , whcre both ends of the
CNT were fixed. The (5 ,5) fixeψ떠ed CNT had 240 carbon .toms and a length of 28.37 A, which is identical to
the single-wa 1led CNT considered by Jiang et al. 20
We utilized the second generation Brenner potenti.l (REBO- Il)" for the carbon-;;.rbon interactions; this
ootenti.l has been shown to .ccurately κproduce binding energies. force constants and elastic properties of
램phene. For .11 simul.tions , the CNT w.s first equilibr.ted .t • specified temperature using • Nose-Hoover
thermostat'6 for 10α)()() steps with a 1 femtosecond (fs)
time step within .n NVT ensemble. After the initi.l thermal equilibration , a sinusoidal velocity was applied to the
CNT which c.used the CNT to osc i11 ate; the osc i11 ation
w.s performed within .n energy-conserving NVE ensemble. The velocity profile was zero .t the fixed ends of
the CNT, .nd sinusoidally increased to • maximum at the
center of the CNT, which caused the magnitude ~f oscillation of the center of the CNT to be about 0.70 A, which
is about 2 .4 7% of the length of the CNT. We emphasize that the energy of the CNT increased only 1.1 6 eV
Icompared to .n origin.l energy of 1695 eV) due to the
imp ed sinusoid.l velocity profile , or less th.n 0.1 % of
the total energy of the CNT, such that no띠 inear vibralion어 effects would not be spuriously introduced in the
simulations.
We performed simulations only on fixedlfixed CNTs in
thc present work , as re5ults f~r_ the Q-factors_of fixedlfr::
cNÌ's were obt.ined in • 2004 paper by Jiang et .1.'0
A direct comparison between the fixedlfixed geometries in
the present work to the fixedlfree results of Jiang et a1. is
feasible .s Ji.ng et a1. uscd not only the s.me potential
IREBO- lI), but also the same geometry, i. e. , (5 ,5) CNTs
파h 240 atoms. Therefore , we are .ble to isol.te boundary
condition effects as the cause of .ny diffcrences between
‘
J. Comput. Theo r.
Nano.따
8, 81 4--8 19, 2011
r
‘
‘
‘
‘
‘
1.2
(
>
으 1.0
l
f
i
a
’
@
~
0.8
얻
종
w
0.6
l
j·
‘
0.4
0.2
100
200
3∞
400
500
400
5∞
nme (psec)
Ext ernal energy at 80 K
1.6
1.4
:>
오 1.0
e
§ o8
ãi
훌 0.6
X
w
0.4
0.2
0.0
o
100
200
300
Time (psec)
Fig. 1. Intrinsic energy dissipation for fixedlfi x.cd CNT at different tempernture. no applied stro. in
815
「m
I며녁-n
피m”m〉피n工 ‘
waJl ed (SW) CNTs.'o and multi wa Jl ed (M W) CNTs ,20-21
and more recently for graphene mono .nd mu 1tilayers. 15 •23
Of these , Jiang et al. 'o studied the temperature dependence
of Q-factor degradation in both fiKedlfree SWCNTs and
MWCNTs , while other works 2l • 22 focused on frictional
effects on energy dissipation in MWCNTs.
Therefore , the purpose of the present work is to utilize
c1 .ssical MD to study the f0 1l0wing. First , we will determine the effec lS of boundary condition on the intrinsic
O-factors of single-w.1I ed CNTs. Second , we will deter미ne the ‘Jtility of mechanical strain in enh.ncing the
intrinsic Q-factors of fixeψfixed CNTs , where by focusing on intrinsic loss mechanisms , i. e. , due to thermoelastic
di~sipation，24 we neglcct extrinsic energy loss mcchanisms
such as gas damping effects .nd c1 amping losses on the
CNT Q-factors.
‘
Bounφry
Condition and Strain Effccts on the Quality Factors of Single Walled Carbon Nanotubes
293 K. We note that the EE is defined as the difference of
the potential energy (PE) before and after the sinusoidal
velocity profile, which causes the subsequent oscillation of
the CNT, is applied.
As can be seen in Figure 1, the energy dissipation
clearly increases with an increase in temperature, which
demon5trates the thermoelastic dissipation that is characteristic in oscillating structures. We further quantify the
nature of the temperature-dependent energy dissipation by
plotting in Figure 2 the Q-factor versus temperature for
both tixedlfixed and tixedlfree (5 ,5) CNTs.
The first trend we note is that, particularly at low temperatures , the Q-factors of tixedlfixed CNTs are significantly higher than those of fixedlfree CNTs; for example ,
at I K , the Q-factors of fixedlfixed CNTs are about
360,000, while the Q-faclors of fixedlfree CNTs are about
13 ,000 , for an increase in Q of nearly a factor of 30 simply
by changing the boundaη condition.
Despite the significant increase in Q-factor at low
temperatures , Figure 2 demonstrates that, as temperature
increases , the Q-factors of fixedltixed CNTs degrade more
quickly Ihan do the Q-factors of tixedlfree CNTs. Thus , at
293 K , the Q-factor of the fixedlfixed CNT has reduced
10 2200, while the Q-factors of the tixedlfree CNT reduce
to aboUl 1500. Th e reason for this is due to their different exponent which relates the Q-factor and temperature
For example , Jiang et al. 20 determined that Q "" IfT o.)6
for fixedlfree CNTs. In the present work , we determine for
fixedlfixed CNTs that the relationship between Q-factor
and temperature is Q"" 1f T o.9I
The exponent relating Q-factor and temperature is
different between fixedlfixed and fixedlfree boundary
conditions for one key rea50n. Before discussing this , we
、
10'
、
‘
、
‘
、
Rx-Fi…
Fix.Free
、
*
、
10'
、
、
、‘
s
‘m
2m
、‘
、
j
‘
、
10'
、
‘、i
、
、
그
。
、
‘
i
、
10-1
100
101
、
1어
‘
、
10'‘
‘‘
‘‘
10"
Temperature (K)
Fig.2. Q-factor as a function oftemperature and boundary condition for
CNTs; fixed-f，πe data taken from Jiang el 3 1. Reprinted wilh perrnission
from [2이. H. Jiang et al., Phys. Rev. Lell. 93. t85501 (2004). <() 2004.
American Physical Society
816
Kim and Park
note that the exponent of 0.91 for the fix때lfixed CNTs is
essentially the same as the thermoelastic damping exponent of 1. 0 that is expected for b비 k materials.27 Th e exponent of 0 .3 6 for the fixedlfree CNTs obtained by Jiang
et al. 20 reflects the fact that fixedlfree CNTs have one unrestrained end where the carbon atoms at the unfixed end
have dangling bonds; these dangling bonds induce a surface effect which causes the thermoelastic damping exponent to deviate sharply from thε bulk value. In contrast,
all undercoordinated atoms at the ends of the CNT are
fixed in the fixeψfixed CNTs; because of this , all surface
effects have been removed , which cau5es the thermoelaslic damping exponent to essentially mimic that of a bulk
material.
3.2. Strain Effects on Q-Factors of FixedIFixed CNTs
The results of the previous section suggest that there
may be advantages to using fixedlfixed CNTs for lowtemperature NEMS applications. while the situation is less
clear at realistic operating temperatures like room temperature , both due 10 the similarity in Q-factor, and also due
10 the fact that extrinsic damping losses , like gas damping ,
have not been considered
However, as discussed in the introduction. one advantage of the fixedlfixed configuration is that it is possibJe experimcntaJJy to impose strain , both ten5 i1 e and
compressive, on the CNT or other nanostructures , as
has been demonstrated by various researchers. 11 ‘ 17. 18.28
Furthermore. because mechanical strain, and in particular tensile mechanicaI strain has been successfuJJy utilized in the past to enhance the Q-factors of various
nanostructures ,l4. ".17. J8 including carbon-based nan<•
structures such as graphene ,15 we now investigate the
effect of strain on the Q-factors of fixedlfixed CNTs.
To study the Q-factors of fixedlfixed CNTs under strain,
we foJJow a similar procedure as described previousJy,
except that first , the CNT is stretched uniaxiaJJy under
either tension or compression with strain increments of
1% to the desired tensile or compressive strain state al
o K foJJowing an energy minimization algorithm. At that
point, the CNT is equilibrated at the desired temperature ,
and then the sinusoidal velocity field is applied in order to
cause the CNT to osciJJate.
Figure 3 ilJ ustrates the effect of applied tensile and compressive strain on the Q-factors of fixedlfixed CNTs at the
same temperature. In partic비 ar， it dcmonstrates Ihe significantly different response in energy dissipation at 20 K at
different Jevels of slrain. It is clcarly observed that there
is significantJy less energy dissipation at 5% tensile strain
than at 3% compressive strain. Th is resu It is consistent
with previous studies on metal nanowires 14 and graphene
monolayers" where tensile strain was also found to mitigate intrinsic energy dissipation.
To quantify the effects of tensile and compressive stram
on the Q-factors of fixedlfixed CNTs as a function of
J. Comput. Th eor. Nanosci. 8, 814-8 19. 2011
‘
“
Boundary Co ndition and Strain Effects on the Quality
Kim olld ft rk
--+--OOIK
•
002K
...
OOBK
•
020 K
- .• .-OBOK
40K
-- -‘(f- -- 293 K
1.2E+06
1.4
1.0E+06
1.2
엄
오 1.0
‘ε6 . 0E+05
a
떠
‘
a 4OE+05
@
:ii 0.8
m
x
• ’
(; 8.0E+05
>
i:i5
Single Walled Carbon N.notubes
I.4 E.06
E:xlerna! energy (20 K, -3 .0% strain)
1.6
i
factoπ 。 f
0.6
2.0E+05
0.4
O.OE +OO
0.2
-2.0E+05
w
r
o
-2
-4
2
4
‘‘
,‘ ,.,‘,
-‘,‘
6
·i
Strain (%)
0.0
o
1∞
200
300
400
Time (psec)
5∞
8.0E+04
•
080K
- .....-140K
External energy (20 K. +5.0% strain)
6.0E+04
1.6
1.4
a
1.2
‘a、
:>
a
엄
4.0E+04
;즙
3.0E.04
혼@ 0.8
@
i
l니
+04
1.0E+04
어
t
,‘ ’
-*’;-,‘-,
-;
li·
•.
‘,‘ ,.
;‘‘
-
5.0E+04
a 2.0E
요 1.0
1 ·;i*-
--‘-020K
7.0E+04
..
1
O.OE+OO
0.6
-1.0E+04
0.4
o
-2
-4
,--r-! : ::::
2
4
6
Strain (%)
‘
0.2
100
200
300
Ti me (psec)
400
Fig. 4_ (Top) Q-fac or as a functìon of tcmperature and strain for
fi ，eψfi ，ed CNT. (B이tom) Extended view 10 deJineate the Q-factor vari4
alion with temperature and strain.
5∞
Fig. J. Intrinsic energy dissipation for.~xed/fixed CNT at same lemperdifferent amounts of applied tens i1 e or compressive st띠in
at~re.
“‘
remperalurc , we plot in Figure 4 the Q-faclors 없
a S a ftun
t ion of 잉
s tηr띠atn
따
15 ranging “
flro
。아m 4% compressive 10 6% tensile
for temperalures ranging from 1 K to 140 K. It can be
observed thal for aU temperalures , lhere is a nearly linear increase in Q-faclor with applied lensile strain; however, further inspection of Figure 4 also indicales thal , with
incre.sing lcmperalure, the lincar constanl of Q enhancement also decreases.
To quanlify this further, we assume a linear relationship
belween Q-fac lO r and slr.in for a given temperaLUre as
(n
“
Q=AEQ，。
”
whcrc E is the imposed slrain , Qo is lhe Q-factor for
a given lemperature .1 0% applied slrain , and A is an
J. Comput. Theo r.
Nano양i.
8, 814-8 19, 2011
unknown conslanl thal depends upon the lemperalure. In
Table 1, we Ii sl the Q-faClors for various lemperatures as
a function of slrain, and also give the constanl A.
As can be seen , whilc Ihe approximalely linear relalionship holds across the range of lemperaturcs between
Q-faclor and lensile slrain , lhe conslanl A decreases significantly wilh an increase in lemperalure from A = 0.55 at
Table 1. Q-fact。π as a runction of strain and temperature for fi.x:eψfi.x:ed
(5.5) single-wal1 ed CNT. A!so gi ‘ es the constant A in the equation Q::::;:;
AE Q.。
T ("K)
Qo (0% Strain)
Q
(3% Strain)
360 ， αm
g
640.αm
66 ， αm
110.αm
20
140
293
24 ， αm
41.000
6.100
2.600
3, 400
2, 200
Q (6% Strain)
A
1.200, 000
2tO, 000
0.55
71 ， αm
0 .49
8, 400
0 .4 1
3.200
0.24
0 .5 3
817
Boundary Condition and Strain Effects 00 the Qua 1i ty Factors of Single Walled Carbon Nanotubes
ω」u」·￠〈ZQ￠〈띠m띠￠
1 K to A = 0.24 at 293 K. This indicates that while tensile
strain does increase the Q-factors of fixe d/fixed CNTs at
elevated temperatures , its effectlveness diminishes wilh an
In crcase 10 temperature.
It is .l50 relevant to discuss the amount of tensile strains
we have considered in the present work. The m.ximum
tensile strain we applied was 6%; for comparison , previous
MD simulations by Bclytschko et al. 2. of the tensile loading of single walled carbon nanombe found failure strains
ranging from 10- 15%. Overall , these results indicate that
the amount of tensile strain (0-셔6%) we have utilized in the
present work should not lead to any instabilities or defects
in the fixedlfixed CNT.
We c1 0se by discussing two items. First. we discuss
Figure 5, where we plot the depcndence of the Q-factor on
temperature and both tensile and compressive strain. What
is most interesting about Figure 5 is that regardless of the
amount of tensile strain that is applied to the fìxe d/fixed
CNT, the thermoeI a5tic damping exponent remains coostant at 0.91 , i.e. , Q"" IfT o,.,. Finally, we note that when
compressive strain is applied , the thermoelastic damping
exponent decreases dramatically, with a decrease in expc•
nent with an increase in strain
Second, an interesting future research path concems
the application of teosile strain, and the effects on the
Q-factors of MWCNTs. As shown by Jiang et al. 20
for MWCNTs , and by Kim and Park for multilayer
graphene ," frictional interactions between the carbon layers cause a significant decrease in the Q-factors of these
multi-walled or multilayer structures. It seems likely that
tensile strain will improve the Q-factor degradation due t。
these interlayer frictional effects , though the nature of the
effect and the factors controlling it has not been quantified
tc• date.
•
---""
.-------•
---
10'
‘
。
-
꽁 1바
ε
-3.0% (data)
-3.0%{fiued)
0.0% (d:ua)
O.O%(fiued)
3.0% (data)
3.0% (fiued)
6.0% (data)
- - - 6.0% (fitted)
‘
•
m
j
。
4. CONCLUSIONS
In conclusion , we have utilized classical MD to study
boundary condition and strain effects on the Q-faclors
of single-walled CNTs. We have found thal , excluding
extrinsic damping effecIs , fixe d/fixed CNTs generally have
higher Q-factors than do fixedlfree CNTs , though the difference in Q-factor due to boundary condition decreases
with increasing temperature. We further demonstrated that
the Q-factors of fixe d/fixed CNTs can be enhanced by
factors of 2-4 through the application of tensile mechanical strain , though again , the effectivene5s of the Q-factor
enhancement decreases with increa5ing temperature. The
results collectively indicate that fixeψfixed CNTs should
be preferable for NEMS applications at lower temperature5
due to a combination of their inherently higher Q-띠ctors，
and the fact that the Q-factors can be further improved
through application of tensile strain. However, near room
temperature , with mechanical strain , the intrin5ic Q-factors
of fixe d/fìxed CNTs are about double those of fixedlfree
CNTs , which may or may not be sufficient to overcome the
1055 in Q-factors that result due to the increase in eXlrinsic
damping for fixedlfixed nanostructures.
Acknowledgments: Sung Youb Kim and Harold S.
Park both gralefully acknowledge the support of DARPA
through grant HROOll-08-l-0047. The authors note that
the contenl of this article does not necessarily reflect the
position or the policy of the govemment , and Ihat no
official govemment endorsement of the re5ults should be
infeπ'ed. HSP also acknowlcdges support from NSF grant
CMMI-0750395.
References
1. S. I;j; ma. Nalu" 354 , 56 (1991)
2. R. H. Baughman. A. A. Zak. hidov, and W. A. de Heer, Science
297. 787 (2002)
3. V. Sazonov，ι Y. Yaish, H. Ustunel, D. Roundy, T. A. Arias. and P. L
McEuen. Natuπ 431.284 (2001)
4. K. Jensen , K. Kim. and A. Ze ttl, Nar. Nanotechnol. 3, 533 (2008)
5. H.-Y. Chiu. P. Hung. H. W. C. Postmι and M. Bockrath‘ Nono Lett.
8, 4342 (2008)
6. H. J. Mamin and D. Rugar. Appl. Phys. μ 11. 79.3358 (20tH)
7. T. D. Slowe, K.. Yasumur.ι T. W. Kenny, D. Botkin. K. Wago. and
D. Rugar, Appl. Phys , LeII, 71 , 288 (1997)
8. P. Poncharnl , Z L. Wang, D. Ugarte. 8nd W. A. de Heer. Science
283. 1513 (1999)
9. H. B. Peng, C. W. Chang, s. Aloni , T. D. Yuzvinsky, and A. ze 11 ,
Phys. R~‘’ μ'11， 97 , 0&7203 (2006)
10. B. Wilkamp. M. Poot. and H. 5. J. van der Zant. NOllo uu. 6.2904
(2006)
11. S T. Pu πell. P. Vi ncen t, C. Joumel, nnd V. T. Binh. Phys. Re‘’ ùtt
89. 276103 (2002)
2. A. K. Huttel. G. A. SleeJe. B. Wi tkamp , M. Poot. L. P.
Kouwenhoven, and H. S. J. Viln der Za nt, Nano μ'11. 9, 2547 (2여)9)‘
13. K. Y. Yasumura. T. D. Stowe. E. M. Chow. T. Pfafman. T. W. Kenny,
B. C. Stipe. and D. Rugar, Journa[ oJ Microeleclrom l'chanical SyrIems 9, 117 (2000)
14. S. Y. K;m and H. S. Park , Phys, Rev. UII. 101 , 215502 (2008)‘
‘
10‘
‘
10'
10。
10’
Temperature (K)
10'
Fig.5. Q-factor as a function of temperalure and strain for fixedlfixed
CNT,
818
Kùn and Park
,
’
J.
Compu t. Theor. Nanosci.
8, 814-8 19, 2011
i
Kim alld Park
Boundary Condition and Strain E tTects on thc Quality Factors of Single Walled Carbon Nanotubes
15. S. Y. Kim and H. S. Park, Nano μ tt. 9, 969 (2009)
16. S. S. Verbridge , J. M. Paφlι R. B. Reichenbach, L. M. Bellan , and
H. G. Craighead. J. Appl. Phys. 99 , 1243여 (2006)
17. S. S. Verbridge, D. F. Shapiro, H. G. Craighead , and J. M ‘ Parpia.
Nano utt. 7, 1728 (2007)
18. V. Cimall a. C. Foerster. F. Wi1I. K τ。nisch. K. Brueckner.
R. Stephan , M. E. Hein. O. Ambacher, and E. Aperathitis. Appl.
Phys. utt. 88 , 253501 (2006)
19. M. C. Oai. K. Eom. and C.-W. Kìm. Appf. Phys. Lelt. 95. 203104
(2009)
20. H. Jiang. M. F. Yu , B. Liu , and Y. H lang. Phys. Re\~ Lett. 93 , 185501
(2004)
2 1. W. Guo. Y. Ouo. H. Gao. Q. Zheng, and W. Zhong, Phys. Re\~ rr.
91. 125501 (2003)
‘
u
22. Y. Zhao , c. -C. M a. G. Chen , and Q. Jiang, Phys. Rα Lett.
91 , 175504 (2003)
23 , S. Y. Kim and H. S Park , Appl. Phys. lι tI. 94. 101918
(2009)
24. R. Lifshitz and M. L. Roukes, Phy.s. Re~~ B 6 1, 5600 (2이삐)
25. D. W. Brenner, 0 ‘ A. Shenderov，ι J. A. Harrison , S. J. Stuart, B. Ni ,
and S. B. Sinnott, J. Phys.: Condens. Matler 14 , 783 (2002)
26. w. G‘ Hoover, Phys. Rα A 31.1695 (1985)
27. P. Mohanty, D. A. Harrington, K. L. Ekinci , Y. T. Yang, M. J
Murphy. and M. L. Roukes, Phys. Rev. B 66 , 085416 (2002)
28. M.- F. Yu , O. Lo urie , M ‘ J. Dyer, K. Moloni , T. F. Ke lly, and R. S
Ru。π" Science 287. 637 (2000)
29. T. Belytschko, S. P. Xiao, G. C. Schatz, and R. S. Ruoff, Phys. Rev. B
65 , 235430 (2애2)
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Received: 13 September 2009. Accepted: 10 January 2010.
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Comput. Theor. Nanosci. 8 , 814-819, 2011
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