Intrinsic strength and failure behaviors of ultra-small single-walled carbon nanotubes NguyenTuanHunga,∗,DoVanTruongb,c,VuongVanThanhb,c,RiichiroSaitoa aDepartmentofPhysics,TohokuUniversity,Sendai980-8578,Japan bInternationalInstituteforComputationalScienceandEngineering,HanoiUniversityofScienceandTechnology,Hanoi,Vietnam cDepartmentofDesignofMachineryandRobot,HanoiUniversityofScienceandTechnology,Hanoi,Vietnam Abstract 6Theintrinsicmechanicalstrengthofsingle-walledcarbonnanotubes(SWNTs)withinthediameterrangeof0.3-0.8nmhasbeen 1studiedbasedonabinitiodensityfunctionaltheorycalculations. Incontrasttopredicting“smallerisstrongerandmoreelastic”in 0 nanomaterials,thestrengthoftheSWNTsissignificantlyreducedwhendecreasingthetubediameter. Theresultsobtainedshow 2 thattheYoung’smodulusEsignificantlyreducedintheultra-smallSWNTswiththediameterlessthan0.4nmoriginatesfromtheir nverylargecurvatureeffect,whileitisaconstantofabout1.0TPa,andindependentofthediameterandchiralindexforthelarge a tube. WefindthatthePoisson’sratio,idealstrengthandidealstrainaredependentonthediameterandchiralindex. Furthermore, J therelationsbetween E andidealstrengthindicatethatGriffith’sestimateofbrittlefracturecouldbreakdowninthesmallest(2, 5 2)nanotube,withthebreakingstrengthof15%of E. Ourresultsprovideimportantinsightsintointrinsicmechanicalbehaviorof ] ultra-smallSWNTsundertheircurvatureeffect. i c Keywords: Carbonnanotubes,Stress-straincurve,Intrinsicstrength,Densityfunctionaltheory s - l r t1. Introduction strengtharekeyfactorsrelatingtothestabilityandlifetimeof m devices.Forthesereasons,studyingthemechanicalresponseof . Due to one-dimensional (1D) structures [1], the single- t smallSWNTsunderstrainshouldbeanecessarytaskinorder a walledcarbonnanotubes(SWNTs)aretheidealmaterialfora m toimprovethefutureSWNTs-baseddevices. variety of applications relating to tensile strain. The SWNTs d-and graphene are known as the strongest materials with ul- TheSWNTstructureisuniqueduetothestrongbondingbe- ntrahigh axial Young’s modulus of about 1.0 TPa and tensile tweenthecarbons(sp2 hybridizationoftheatomicorbitals)of ostrengthapproaching100-130GPa[3,4,2,5,6,7,8,9]. Both the curved graphene sheet, which is stronger than in diamond cexperimental and theoretical studies [5, 10, 26] showed that with sp3 hybridization because of the difference in C-C bond [the diameter of the large SWNTs does not significantly affect lengths (0.142 and 0.154 nm for graphene and diamond re- 1their mechanical properties. However, the physical and me- spectively) [1]. The changes in the C-C bond structure such vchanical properties of ultra-small SWNTs with the diameters as defects, grain boundaries, chemical substitutions or curva- 7smaller than 0.4 nm expected are different from those larger ture effects are the main causes to make changes in mechani- 8 7than that due to their very large curvature effect. Many ef- cal properties of SWNTs and graphene. The results obtained 0forts have been made to synthesize the ultra-small SWNTs in by the density function theory (DFT) and molecular dynamic 0recent years. The smallest stable (2, 2) SWNT with a diam- (MD) calculations showed that the Young’s modulus and ten- 1.eter of 0.3 nm observed by Zhao et al. [11] could be grown silestrengthofSWNTs[19,20,21,22]andgraphene[23]with 0insidemulti-wallcarbonnanotubes(MWNTs). The(2,2)nan- vacancy-relateddefectsdependontheconcentrationofdefects 6otube investigated by the first principle calculations is tunable anddefectcharacteristics.Zhangetal.[24]investigatedthatthe 1between metallic and semiconducting properties by changing grainboundaries(GBs)aresignificantlyreducedthemechani- : vtheFermilevel[12]. Infact,theultra-smallnanotubesareless cal strength of graphene. Mortazavi et al. [25] reported that Xistable than the large nanotubes [13, 14]. However, if we fab- the Young’s modulus of a nitrogen doped in a graphene is al- ricatednicelyinsomespecialgeometries,wecanmeasurethe mostindependentofnitrogenatomconcentration,butthesub- r avaluesoftheultra-smallSWNTs. Although,manystudieshave stituted nitrogen atoms are decreased the tensile strength and focused on synthesis, physical and chemical properties of the ductilefailurebehaviorofgraphene. Fortheperfect(5,5),(6, smallSWNTs[11,12,15,16,17,18],theirmechanicalproper- 3) and (8, 0) SWNTs, the tight binding (TB) and DFT calcu- tieshaveyettobeclarified. Moreover,theintrinsicmechanical lationsshowedthatSWNTscanreachtheYoung’smodulusof properties such as Young’s modulus, Poisson’s ratio and ideal 1.0 TPa and a maximum tensile strength of 100 GPa with no chiraldependence[26]. However, thecriticaltensilestrainfor breakinghasachiraldependence. Theexperiment[5]hasused ∗Correspondingauthor.Tel.:+81227957754;Fax:+81227956447. theopticalcharacterizationwithamagneticactuationtechnique Emailaddress:[email protected](NguyenTuan Hung) to measure the stiffness of the (17, 12), (17, 10) and (18, 10) PreprintsubmittedtoComputationalMaterialsScience January6,2016 SWNTsandfoundthattheYoung’smodulusisnotdependent method[38]was1×1×k,inwhichkdependsonthelengthof onthenanotubechiralindex,andhasanaveragevalueof0.97 theSWNTs. ± 0.16 TPa. This mechanical response is also observed in the To simulate the effect of tensile strain in the SWNTs, first, graphene. Both the experiment and DFT calculation reported the models were fully relaxed by using the Broyden-Fretcher- the Young’s modulus of 1.0 TPa for both the zigzag and the Goldfarb-Shanno(BFGS)minimizationmethodfortheatomic armchairtensilestraindirections[3,28]. In1920,Griffith[31] positions,andcelldimensionsinthezdirection. Thesemodels extrapolatedanmaximumintrinsicstrengthσ ofaboutE/9for were considered equilibrium until all the Hellmann-Feynman I the fracture of brittle material, where E is the Young’s modu- forces and the normal component of the stress σ less than zz lusofthematerialunderuniaxialtension. Thisestimateisstill 5.0×10−4 Ry/a.u. and1.0×10−2 GPa,respectively. Thenthe valid for the brittle material in nano-scale. Both the experi- loadingstrainwasappliedtothemodelsbyelongatingthecell mentandtheoryshowedthatσ /Eisapproximately0.1forthe alongthezdirectionwithanincrementof0.02.Atnearthefrac- I grapheneandnanotubes[3,28,10]. Therefore,itisinteresting turepoint,thestrainwasrefinedwithaverysmallincrementof toinvestigatethatthemechanicalresponseofthelargeSWNTs 0.005. Aftereachincrementofthestrain, theatomicstructure and graphene is consistent with the small SWNTs, which are was fully relaxed under fixed cell dimensions. Here, the ten- dominatedbytheirverylargecurvatureeffect. silestrainisdefinedasε ≡ ∆L/L ,where L isthelengthof zz 0 0 Inthispaper,wepresentthefirst-principlestoinvestigatethe theunitcellatgeometryoptimizationand∆Listheincrement structuralandmechanicalpropertiesofthesmallSWNTswith of the length under tension [Fig. 1(a)]. We also investigated thediameterintherangefrom0.3to0.8nmunderuniaxialten- themechanicalresponseofgrapheneundertensilestraininthe sion. The paper is organized as follows. Section 2 describes zigzagandarmchairdirectionstoelucidatecurvatureeffect,as the setup of the DFT calculations and the detailed simulation showninFig. 1(b). procedure. Section3describesYoung’smodulus,Poisson’sra- tio,idealstrengthandfracturemechanismoftheSWNTsunder (a) " (b) zz tensilestrain. Finally,section4summarizestheresults. armchair 2. Methodology zigzag ofFsimrsat-llprdiniacmipeleter(asbingilnei-twioa)llesdimcualrabtioonnsnafnoortutbeenssil(eSWstNraTinss) l1 l1 l2 was performed. We used Quantum-ESPRESSO (QE) pack- l3 l2 l3 age [32] for the first-principle calculations, which is a full density functional theory [33, 34] simulation package using a plane-wave basic set with pseudopotentials. The Rabe- z y Rappe-Kaxiras-Joannopoulos(RRKJ)[35]ultrasoftpseudopo- tentials was used to calculate the pseudopotential plane-wave " with an energy cutoff of 60 Ry for the wave function. The x y zz x exchange-correlationenergywasevaluatedbygeneral-gradient approximation(GGA) [36] using the Perdew-Burke-Ernzerhof Figure1: (Coloronline)(a)Thesimulationmodelofsmallnanotubesunder (PBE)[37]function. tensilestraininzdirection. (b)Themodelofgrapheneundertensilestrainin We examined three models of a series of small diameter zigzagandarmchairdirections.Thenotationsofl1tol3arethreebondlengths aroundCatomic. single-walled carbon nanotubes: the armchair type (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) SWNTs; the zigzag type (3, 0), (4, 0), (5, 0), (6, 0), (7, 0) SWNTs and the chiral type (3, 1), (3, 2), (4, 1), (4, 2), (5, 2) SWNTs, which have the diameters in the 3. Resultsanddiscussions range from 0.3 to 0.8 nm. Here, the SWNT structure in our notationisdenotedbyasetofintegers(n,m)whichisashort- The scanning of the potential energy surface (PES) is per- handforthechiralvectorC =na +ma ,wherea anda are formed for the smallest (2, 2) SWNT to search for the ground h 1 2 1 2 the unit vectors of an unrolled graphene sheet [1]. The chiral statewiththelengthL rangingfrom0.24to0.27nmbycalcu- 0 vectorC definesthecircumferentialdirectionoftherolled-up latingthetotalenergyperatomwithdifferentlengths. Fig.2(a) h grapheneintoacylinder,givingthediameterD . Thecrosssec- showstheobservedminimumofthetotalenergyatL =0.257 0 0 tionalareaofaSWNTlayerwascalculatedusingtheinterlayer nm. In Fig. 2(b), we present the illustrations of atomic struc- distance of a MWNT (0.34 nm) [29] as its thickness. Since a turesforthe(2,2)SWNTatthegroundstate,whereL =0.257 0 periodic boundary condition was applied for three dimensions nm, D = 0.282nm, andl = l = 0.149nm(l = 0.139nm) 0 1 3 2 in all models, the thickness of the vacuum region was set at representthelength,thediameter,andtheC–Cbondlengthpar- 12 Å perpendicular to the tube axis to avoid the undesirable allelto(perpendicularto)thetubeaxis,respectively. Thesame interactions from the neighboring SWNTs. The k-point grids optimizationisperformedfortheothernanotubes. Theequilib- intheBrillouin-zoneselectedaccordingtotheMonkhorst-Pack riumconfigurationsarelistedinTable1forallSWNTsinthis 2 Table1:Equilibriumconfigurationsincludingnumberofatominunitcelln,lengthL0,diameterD0,bondlengthl1,2.,3,andbindingenergyEbandYoung’smodulus E,Poisson’sratioν,idealstrainsεI,andidealstrengthσIfordifferentSWNTs. SWNT n L (nm) D (nm) l (nm) l (nm) l (nm) E (eV) E(TPa) ν ε σ (GPa) σ /E 0 0 1 2 3 b I I I (2,2) 8 0.257 0.282 0.149 0.139 0.149 7.770 0.576 0.147 0.270 86.44 0.150 (3,3) 12 0.246 0.420 0.143 0.144 0.143 8.365 0.954 0.067 0.305 97.90 0.103 (4,4) 16 0.246 0.553 0.143 0.143 0.143 8.578 0.965 0.108 0.325 101.44 0.105 (5,5) 20 0.246 0.686 0.143 0.143 0.143 8.676 0.981 0.104 0.320 103.80 0.106 (6,6) 24 0.246 0.821 0.142 0.142 0.142 8.729 0.978 0.093 0.310 105.44 0.108 (3,0) 12 0.421 0.263 0.149 0.141 0.149 7.682 0.765 0.100 0.235 102.59 0.131 (4,0) 16 0.421 0.337 0.148 0.138 0.148 8.090 0.842 0.124 0.215 102.58 0.120 (5,0) 20 0.426 0.407 0.145 0.141 0.145 8.343 0.914 0.112 0.185 89.53 0.096 (6,0) 24 0.426 0.483 0.144 0.141 0.144 8.493 0.936 0.106 0.205 102.65 0.108 (7,0) 28 0.426 0.559 0.143 0.141 0.143 8.586 0.997 0.085 0.180 102.29 0.101 (3,1) 52 1.536 0.303 0.145 0.145 0.145 7.894 0.817 0.087 0.175 83.58 0.102 (3,2) 76 1.865 0.356 0.144 0.143 0.144 8.201 0.887 0.093 0.215 81.56 0.092 (4,1) 28 0.649 0.357 0.145 0.144 0.142 8.253 0.927 0.112 0.210 93.27 0.101 (4,2) 56 1.127 0.428 0.143 0.143 0.144 8.390 0.935 0.093 0.235 77.36 0.083 (5,2) 52 0.887 0.500 0.143 0.143 0.143 8.517 0.929 0.119 0.210 95.82 0.103 (a) (b) ) a V) 153.94 TP 1.0 (5,5)(7,0) (3,3) y/atom(e ��153.95 zx ymn 941.0 0L0.1.203597 nnmm0.149 nm Edulus( 00..89 (6,6) (4,4(5),(26),(04),2()5,0()4,1()(34,,20))(3,1) (3,0) erg View along x axis mo n 0.7 e 153.96 s Total �153.97 y D0.0282 nm Young’ 0.6 (2,2) � 0.24 0.25 0.26 0.27 z x 0.1 0.2 0.3 0.4 Length L0 (nm) View along z axis Inverse diameter D 1 (nm 1) 0− − Figure2: (Coloronline)(a)Totalenergyperatomofthe(2,2)SWNTplotted Figure 3: (Color online) Young’s modulus of the single-walled carbon nan- asafunctionofthetubelength. (b)Unitcellofthe(2,2)SWNTshowingthe otubesasafunctionofinverseoftubediameter. Thedashlineisfittedbya lengthL0,thediameterD0,theC-Cbondperpendicularthetubeaxis,andthe second-orderpolynomial. thatalongtothetubeaxis. Young’smodulusE,whichisdefinedas (cid:12) study. TheresultsshowthattheC-Cbondlengthsofthesmall E = 1 ∂2U(cid:12)(cid:12)(cid:12) (1) nanotubestendtobelongerthanthoseofthelargenanotubes. V0 ∂ε2zz(cid:12)(cid:12)εzz=0 Inthecalculationofthebindingenergy,wetaketheenergyof whereU isthestrainenergyandε istheuniaxialstrain. We anisolatedCatom(E )asthereferenceenergies,withE be- zz C tot appliedthesmallstrains(±0.005,±0.01,±0.015,±0.02),which ingthetotalenergyofthesystemcontainingnCatomsinthe stayintheharmonicregime. Here,thenominalplatethickness unit cell. The binding energy per C atom, E = E − nE , b tot C d assumed was independent of ε . The nanotubes with the is summarized in Table 1. It is found that the binding energy 0 zz diameter D > d and D ≤ d wereconsideredasthehollow becomeslargerwhenthediameteroftheSWNTsincreases. In 0 0 0 0 andsolidcylinders,respectively. ThevolumeatequilibriumV other words, the large SWNTs are more stable than the small 0 isdefinedas SWNTs. (cid:40) πL D d if D >d V = 0 0 0 0 0 (2) The most basic mechanical property of the SWNTs is the 0 πL (D /2+d /2)2 if D ≤d 0 0 0 0 0 3 Fromthepointofviewofelasticitytheory,itiswell-recognized 0.15 thatthevalueof E isrelatedtod ofthetube. Thewallthick- 0 ness is considered as the interlayer spacing of graphite and ν (2,2) multi-walled carbon nanotubes in nature based on the van der tio 0.13 (5,2) (4,0) a Waals interactions [28], in which d0 assumed is independent r (4,1) of the strain. Both the experimental and the theoretical stud- n’s 0.11 (5,5) (4,4)(6,0) (5,0) ies[3,28]haveusedtheconstantthicknessof0.334nmtocal- o culatethemechanicalpropertiesofthegraphene. Inthisstudy, oiss 0.09 (6,6) (4,2) (3,2) (3,0) d0 of0.34nm, whichwasobservedinexperimentalimagesof P (7,0) (3,1) MWNT[29],isusedtoestimateE oftheSWNTs. Asshowin 0.07 (3,3) Table 1, the geometric structures of the (2, 2), (3, 0), and (3, 0.1 0.2 0.3 0.4 1) nanotubes are the solid cylinders with D < d . While the 0 0 remainingnanotubesarethehollowcylinderswithD0 >d0. Inverse diameter D0−1 (nm−1) Figure3showstheobtainedYoung’smodulusE ofthenan- otubes, in which they are a constant of around 1 TPa and in- Figure4:(Coloronline)Poisson’sratioofthesingle-walledcarbonnanotubes dependent of the inverse of tube diameter 1/D and the chiral asafunctionofinverseoftubediameter. Thedashlineisfittedbyasecond- 0 orderpolynomial. index(n,m)forthelargeSWNTswithD >0.4nm. Thesere- 0 sultsareingoodagreementwithboththepreviousexperiment andtheoryones[5,9,10,26,27]. ForthesmallSWNTswith D ≤0.4nm,Einvestigateddependsonthetubediameter[Fig. 0 Brennerpotentials(ν=0.19). 3]. Itiswell-knownthatinnanomaterials,thesmallertheyare thestrongerandmoreelastic[39,40], howevertheresultsob- Figure 5 shows the stress–strain curves of the armchair, tainedshowthattheruleisbrokenwhenD ≤0.4nm. Eofthe zigzag, and chiral nanotubes. The stress computed from QE 0 (2,2)nanotubeisdecreasedabout41%comparingwiththe(6, package [32] is automatically evaluated over the entire super- 6) nanotube. The significant reduction is due to the curvature cell volume Vcell. Therefore, we need to rescale the supercell of the small nanotubes. In Fig. 1, since the l2 sigma bond is stress by Vcell/V0 to obtain the stress of the SWNTs. The re- perpendiculartothetubeaxis,theintrinsicstrengthofthearm- lationship between V0 and d0 in Eq. 2 shows that the stress is chairnanotubesoriginatesfromthel andl sigmabonds. The inversely proportional to the constant thickness. That means 1 3 l andl inthe(3,3),(4,4),(5,5)and(6,6)nanotubesarevery that,ingeneral,thestressbecomeslargerwhenthemagnitude 1 3 close to the value 0.142 nm of the graphene, and their E are of d0 decreases, and vice versa. For the armchair nanotubes similar to the experimental value of about 1 TPa [3, 5], while [Fig. 5(a)], the ideal strength σI (maximum tensile strength) affected by the curvature, the l and l in the (2, 2) nanotube reaches about 100 GPa at an ideal tensile strain εI of around 1 3 arelongerthanthoseanditsEisonly0.587TPa[Table1]. For 0.30, which are consistent with the earlier DFT estimate of thezigzagnanotubes,becausethel sigmabondisparallelwith 114.6GPaatεI of0.295forthearmchairnanotube[10],andin 2 thetubeaxis,thetensilestrengthismainlycharacterizedbythe agreementwiththeexperimentalobservations(150±45GPa) l bond. However, when the diameter of the tubes is smaller foradefect-freeMWCNTusingthetransmissionelectronmi- 2 than 0.4 nm (for the (3, 0) and (4, 0) nanotubes), the strength croscope(TEM)[30]. Inthearmchairdirection[Fig. 5(a)],the also depends on the l and l bonds [Table 1]. For the chiral grapheneissomewhatstrongerthanthenanotubes, withσI of 1 3 nanotubes, the tensile strength investigated is governed by not 113.4 GPa. The ideal strength of graphene between 110 and only the l , l and l sigma bonds, and the angles between the 130 GPa have been also predicted by both the experiment [3] 1 2 3 bonds and tube axis (for the (3, 1), (3, 2), (4, 1), (4, 2), (5, 2) and the DFT calculation [28]. For the zigzag nanotubes, the nanotubes)butalsothecurvatureofthetube(forthe(3,1)and idealstrengthofSWNTsissimilartothatofthearmchairnan- (3,2)nanotubes). otubesandgrapheneinthezigzagdirectionwithσI ofaround Figure4showstheobtainedPoisson’sratioνoftheSWNTs 100 GPa [Fig. 5 (b)]. However, εI of around 0.20 is smaller as a function of the inverse of tube diameter. Here, ν is given than that of the armchair nanotubes. These ideal strength and as: idealstrainarecompatiblewiththeearlierDFTcalculationof ν≡−L0 ∆D (3) 107.4GPaand0.208foranzigzagnanotube,respectively[10]. ∆LD +d Forthechiralnanotubes,σ andε arefoundapproximately90 0 0 I I The value of ν is from 0.07 to 0.15 and has a downward ten- GPa and 0.20, respectively. The results obtained above show dencywiththelargeSWNTs. Forthesmallones,thePoisson’s that σI and εI depend not only on the tube diameter, but also ratioislargecomparedwiththatofthelargernanotubes,espe- onthetubechiralindex. Theidealstrengthandtheidealstrain cially for the (2, 2) nanotube. Since the thickness of wall of evaluatedfromthestress–strainrelationship[Fig. 5]arelisted tube is correlated to the Poisson’s ratio, we estimated ν for an inTable1. expandeddiameterD +d [Eq.3]insteadofthetubediameter, TherelationshipbetweentheYoung’smodulusandtheideal 0 0 D ,asthepreviousstudies[41,42]. Therefore,ourresultsare strength has been known as Griffith’s estimate of brittle frac- 0 smallerthanthevaluesobtainedbyPortaletal.[41]withDFT ture [31]. Due to σ from 90 to 100 GPa and E ∼ 0.1 TPa I calculations(ν=0.14),andYakobsonetal.[42]usingTersoff- for the large nanotubes, the ratio σ /E has the value between I 4 (a) Armchair 4. Conclusion 120 Insummary,abinitiodensity-functionaltheorycalculations 100 withthegeneralgradientapproximationhavebeencarriedout ) a to investigate the intrinsic mechanical strength of the single- P G 80 walledcarbonnanotubeswiththedifferentchiralitiesanddiam- ( etersundertensilestrain. Theresultsobtainedrevealsthatthe zz 60 (2,2) σ intrinsic strength in the nanotubes originates from the sigma (3,3) ess 40 (4,4) bonds. The atomic structures and the bond lengths of small tr (5,5) SWNTs(withdiameter<0.4nm)aresignificantlychangedun- S 20 (6,6) dertheirverylargecurvatureeffect. Thestrengthofthesmall Graphene SWNTs is significantly weaker than the large ones. This is in 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 contrast with the phenomenon “smaller is stronger and more Tensile strain ε elastic” in nanomaterials [39, 40]. For the large SWNTs, the zz Young’s modulus E ∼ 1.0 TPa is independent of the diam- (b) Zigzag eter and the chiral index. These results are in good agree- 120 ment with the previous experimental and the theoretical stud- ies [5, 30, 10, 3, 28]. The Poisson’s ratio ν has a noticeable 100 ) downward trend with the large SWNTs. For the small nan- a P G 80 otubes, ν is large compared with that of the larger nanotubes, ( especiallyforthe(2, 2)nanotube. Theidealstrengthobtained σzz 60 (3,0) from 90 to 100 GPa and the ideal strain from 0.20 to 0.30 de- (4,0) s pendsonthediameterandthechiralindex.Furthermore,there- es 40 (5,0) r lationshipsbetweentheYoung’smodulusandtheidealstrength t (6,0) S 20 (7,0) indicate that Griffith’s estimate of brittle fracture could break Graphene downinthesmallest(2,2)nanotube,withthebreakingstrength 0 of15%ofE. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Tensile strain ε zz (c) Chiral 5. Acknowledgements 120 N.T.H. acknowledges the support of the Interdepartmental DoctoralDegreeProgramforMulti-dimensionalMaterialsSci- 100 a) enceLeadersofTohokuUniversity,R.S.acknowledgesMEXT P G 80 Grants Nos. 25107005 and 25286005, and D.V.T. and V.V.T. ( acknowledgesNAFOSTEDNo. 107.02.2012.20. zz 60 σ (3,1) s es 40 (3,2) References r t (4,1) S 20 (4,2) References (5,2) 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 [1] R.Saito,G.Dresselhaus,M.S.Dresselhaus,PhysicalPropertiesofCar- bonNanotubes,ImperialCollegePress,London,1998. 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