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Appl.
Phys.
A74,339–343(2002)/DigitalObjectIdentier(DOI)10.
1007/s003390201277AppliedPhysicsAMaterialsScience&ProcessingThermalpropertiesofcarbonnanotubesandnanotube-basedmaterialsJ.
Hone1,,M.
C.
Llaguno1,M.
J.
Biercuk1,A.
T.
Johnson1,B.
Batlogg2,Z.
Benes3,J.
E.
Fischer31DepartmentofPhysicsandAstronomyandLaboratoryforResearchontheStructureofMatter,UniversityofPennsylvania,PhiladelphiaPA19104-6272,USA2BellLaboratories,LucentTechnologies,MurrayHill,NJ079743,USA3DepartmentofMaterialsScienceandEngineeringandLaboratoryforResearchontheStructureofMatter,UniversityofPennsylvania,PhiladelphiaPA19104-6272,USAReceived:17October2001/Accepted:3December2001/Publishedonline:4March2002–Springer-Verlag2002Abstract.
Thethermalpropertiesofcarbonnanotubesaredi-rectlyrelatedtotheiruniquestructureandsmallsize.
Becauseoftheseproperties,nanotubesmayprovetobeanidealmate-rialforthestudyoflow-dimensionalphononphysics,andforthermalmanagement,bothonthemacro-andthemicro-scale.
Wehavebeguntoexplorethethermalpropertiesofnano-tubesbymeasuringthespecicheatandthermalconductivityofbulkSWNTsamples.
Inaddition,wehavesynthesizednanotube-basedcompositematerialsandmeasuredtheirther-malconductivity.
Themeasuredspecicheatofsingle-wallednanotubesdiffersfromthatofboth2Dgrapheneand3Dgraphite,es-peciallyatlowtemperatures,where1Dquantizationofthephononbandstructureisobserved.
Themeasuredspecicheatshowsonlyweakeffectsofintertubecouplinginnanotubebundling,suggestingthatthiscouplingisweakerthanex-pected.
Thethermalconductivityofnanotubesislarge,eveninbulksamples:alignedbundlesofSWNTsshowathermalconductivityof>200W/mKatroomtemperature.
AlinearK(T)uptoapproximately40Kmaybedueto1Dquantiza-tion;measurementofK(T)ofsampleswithdifferentaveragenanotubediameterssupportsthisinterpretation.
Nanotube–epoxyblendsshowsignicantlyenhancedthermalconductivity,showingthatnanotube-basedcompos-itesmaybeusefulnotonlyfortheirpotentiallyhighstrength,butalsofortheirpotentiallyhighthermalconductivity.
PACS:62.
25.
+g;63.
22.
+m;61.
46.
+w1SpecicHeatAcarbonnanotubecanbethoughtofasasinglegraphenesheetthatiswrappedintoacylinder.
Wrappingthesheethastwomajoreffectsonthephononbandstructure.
Firstly,the2DCorrespondingauthor.
Presentaddress:DepartmentofPhysics,CaliforniaInstituteofTechnology,PasadenaCA91125,USA(Fax:+1-626/683-9060,E-mail:hone@caltech.
edu)phononbandstructureofthesheet"folds"intothe1Dband-structureofthetube.
Secondly,thecylindricalshapeofthetuberendersitstifferthanthesheet,changingthedispersionofthelowest-lyingmodes.
Figure1showsthetheoreticallyderivedlow-energyphononbandstructureofanisolated(10,10)nanotube[1].
The1Dquantizednatureofthebandstructureisevident:thereareaseriesof1D"sub-bands"separatedatthezonecenterbyenergiesofafewmeV.
Therearefouracousticbands;onelongitudinal(LA),twodegeneratetransverse(TA),andone"twist",allofwhichhavelineardispersionatlowenergy.
Thehighphononbandvelocity(vLA=24km/s,vTA=15km/s,0510152000.
10.
20.
30.
4E(meV)k(1/)Fig.
1.
Low-energyphononbandstructureofa(10,10)carbonnanotube[1].
Theinsetshowsthephonondensityofstates(PDOS)ofanisolatednano-tube(solidline)comparedtothePDOSofgraphene(dot–dashedline)andgraphite(dashedline)340vtwist=9km/s),coupledwiththesmalldiameterofthenano-tube,causestherelativelylargesub-bandsplitting.
Theinsetshowsthelow-energyphonondensityofstates(PDOS)ofa(10,10)nanotube(solidline)comparedtothatof2Dgraphene(dot-dashedline)and3Dgraphite(dashedline).
ThenanotubePDOSisconstantatthelowestenergies,andthenincreasesstepwisewiththeentryofhighersub-bands.
Asthesystemis1D,thereisavanHovesingularityateachbandedge.
ThegraphenePDOSislargeinthisenergyrangebecauseoftheexistenceofanout-of-planetransversemodewithquadraticdispersion.
ThePDOSofgraphite,how-ever,issignicantlyreducedbecauseoftheaddedcouplingbetweenneighboringlayers,whichrendersthesystem3Dratherthan2D,andshiftslow-energyspectralweightupwardinenergy.
Figure2showsthecalculatedlow-temperatureheatspe-cicheatofanisolatednanotube.
BecausethePDOSiscon-stantatlowenergy,thespecicheatdisplayslineartempera-turedependenceatlowtemperaturebecauseonlytheacousticmodesarepopulated[2].
Above6K,theslopeofC(T)increasesastheopticalsub-bandsbecomepopulated.
Thislinearbehavior,withanincreaseinslopenear6K,istheexpectedsignaturefora1Dquantizedphononspectruminsingle-wallednanotubes.
Ascanbeseeninthecaseofgrapheneandgraphite,changingthedimensionalityofasystemcangreatlyaffectthelow-energyPDOSandthereforethelow-temperaturespecicheat.
Ingraphite,therangeofthiseffectisrelatedtotheDe-byeenergyoftheinterlayermodes,approximately10meV.
Innanotubes,itmightbeexpectedthatbundlingtubesinto3Dcrystallinearrays("ropes")wouldreducethelow-energyPDOS.
Mizeletal.
[3]havecalculatedthelow-energyphononbandstructureofSWNTbundles,andndarelativelyhighintertubeDebyeenergyE⊥D,approximately5meV,forthecasewhereneighboringtubeshavegraphite-like("strong")coupling.
Figure3showsthepredictedspecicheatof2Dgraphene,3Dgraphite,1DisolatedSWNTsand3DSWNTropes,show-ingthedramaticeffectsofdifferinginterlayercoupling.
Athightemperatures,allofthespecicheatsareidenticalandmostlyreectthephononstructureoftheconstituent2DFig.
2.
Calculatedsingle-nanotubespecicheatC(mJ/gK)Fig.
3.
Predictedspecicheatofagraphenesheet,anisolatednanotube,graphite,andananotuberopegraphenesheets.
Comparing2Dgrapheneto3Dgraphite,wecanseethatbelowapproximately60K,interlayercouplingcausesC(T)todecreasemuchmorestronglywithtempera-ture.
TheisolatedSWNTdisplaysasmallerC(T)atlowtemperaturecomparedtothegraphenesheetduetotheab-senceofthequadraticout-of-planemode-tubes,whicharestiffertobendingthansheets.
Finally,thestronglycoupledropecurvedivergesbelowthesingle-tubeC(T)below30K,followingacurvesimilartothatofgraphite.
ItisclearfromFig.
3thatgraphite-likecouplingbetweenneighboringtubesinaSWNTbundleshouldcausethesignatureof1Dquan-tizationinthespecicheattobeobscured.
However,inthecaseofweakercoupling,single-tubebehaviormightpersisttolowertemperatures.
TheSWNTsamplesusedformeasurementofspe-cicheat[4]wereobtainedfrompuriedSWNTsuspen-sions(tubes@rice),andsubsequentlypuriedandvacuum-annealed.
ThesamplewascomposedoflargebundlesofSWNTswithanaveragetubediameterof1.
25nmandtheresidualcatalystconcentrationwasapproximately2at.
%,asdeterminedbyX-raydiffraction(XRD),high-resolutiontransmissionelectronmicroscopy(HRTEM)andenergy-dispersiveX-rayspectroscopy(EDX)measurements.
Afteranalysis,thesampleswerebakedat300Cunderdynamicvacuumforthreedaystoremoveatmosphericcontaminants,andthenkeptundervacuumuntilminutesbeforethemeas-urementswerebegun.
Specicheatwasmeasuredfrom300Kto2Kusingarelaxationtechnique.
Figure4showsthemeasuredspecicheatofaSWNTsamplefrom2Kto300K.
Thehollowcirclesrepresenttherawdata.
Thesolidlinerepresentsthecontributionfromthecatalyst,basedontheknownspecicheatofNiandCo.
Fi-nally,thesolidcirclesrepresentthecorrectedspecicheatoftheSWNTs.
Figure5showsthemeasuredspecicheatonalogarithmicscale,comparedtothepredictedspecicheatofgraphene,graphite,isolatedSWNTsandstronglycoupledropes.
ThemeasureddatafollowstheisolatedSWNTcurveallthewaydownto4K.
Thus,existingmodelsofthephononbandstructureofSWNTsarelargelyconsistentwiththemeas-ureddata.
Thedatadivergebelowthesingle-tubecurveat4K,ratherthanat30K,asexpectedforthecaseofgraphite-341C(mJ/gK)Fig.
4.
MeasuredspecicheatofapuriedSWNTsample[4].
TherawdataiscorrectedforthecontributionfromthecatalystimpuritiestoobtainthecontributionfromtheSWNTs10C(mJ/gK)Fig.
5.
Measuredspecicheatcomparedtotheoreticalmodels[4].
ThedataagreewiththepredictedC(T)ofanisolatednanotubedownto4K,imply-ingaweakintertubeinteractionlikecoupling.
Therefore,weconcludethatSWNTsaremuchmoreweaklycoupledmechanicallythanmightbeexpected;itshouldbepossibletoobservethe1D–2Dtransitioncharacter-isticofsingletubes.
ThesolidpointsinFig.
6showthelow-temperaturemeas-uredspecicheatonalinearscale.
Thedataclearlyshowalineartemperaturedependencefrom2Kto8K,withanincreaseinslopeabove8K.
Thisbehaviorisdirectconrma-tionofa1D-quantizedphononspectruminSWNTs.
How-ever,thelinearslopedoesnotextrapolatetozeroatT=0,aswouldbeexpectedforisolatedSWNTs.
Weattributethisdis-crepancytointertubecoupling,whichshouldcauseT3-likebehavioratlowtemperature.
ThelinesinFig.
6showtheresultsofemployingasim-plemodeltosimulatethebehaviorofweaklycoupledSWNTropes.
Inthismodel,theacousticmodesarecollapsedontoasinglemodewithDebyeenergyEDintheon-tubedirection,andtransverseDebyeenergyE⊥D,withaspe-cicheatrepresentedbythedashedline.
AsingleopticalmodeentersatEsub,withspecicheatrepresentedbythe02468C(mJ/gK)Fig.
6.
Measuredspecicheatatlowtemperature;tusingasimplemodeltoaccountforweakintertubeinteractions[4]dot-dashedline;thesolidlinerepresentsthesumofthetwocontributions.
ED,E⊥D,andEsubaretakenasindepen-dentttingparameters,andadjustedtogivethebestttothemeasureddata.
ThevaluesobtainedareED=92meV,ED⊥=1.
4meV,andEsub=4.
1meV.
ThetheoreticalacousticmodevelocitiesforaSWNT[1]translateintoaneffectiveDebyeenergyof103meV,onlyslightlyhigherthanthetted92meV.
OurttedEsub(4.
1meV),however,isconsiderablylargerthanthetheoreticalsingle-tubevalueof2.
7meV.
Therstopticalsub-bandcor-respondstotubeattening,andshouldrequiresignicantlymoreenergyinaropesincetubesareconstrainedbytheirneighbors;theoreticalcalculations[5]thattakeradialtube-tubeinteractionsintoaccountshowexcellentagreementwiththeexperimentalvalue.
Theexperimentaltube–tubecoup-ling,measuredbyED⊥=1.
2meV,issignicantlysmallerthanthetheoreticalvalueofapproximately5meV[3]ob-tainedusingcouplingconstantsderivedfromgraphite.
Thedifferencemayberelatedtothelackofcommensurabilitybetweenneighboringtubes,whichwouldimplyadramaticweakeningofthecorrugationintheintertubepotential,sothattubesinarealropemayslideortwistmorefreelythanexpected.
Themeasuredhighon-tubeDebyeenergyconrms,inabulksample,thehighYoung'smoduluspreviouslyobservedforindividualtubes[6].
Theweaktube–tubecoupling,how-ever,impliesthatthemechanicalstrengthofSWNTropeswillberelativelypoor.
Itmaybenecessarytocrosslinktubeswithinarope,ortoseparatethemcompletely,inordertore-alizetheirnear-idealpropertiesinhigh-strengthcomposites.
However,weakcouplingmaybeanadvantageforhighther-malconductivity.
Berberetal.
[7]ndthatstrongtube–tubecouplingdecreasesthehigh-temperaturethermalconductiv-ityofSWNTbundlesbyanorderofmagnituderelativetoisolatedtubes;weakcouplingmayimplynosignicantre-ductioninthethermalconductivitywhentubesarebundledintoropes.
Similarly,incomposites,theinnertubesinaropeshouldberelativelyunperturbedbythesurroundingmatrix,whichcouldalsobeanadvantageforhighthermalconduc-tivity.
Theissuesofcommensurabilitythatwereraisedasanexplanationfortheweaktube–tubemechanicalcouplingalso342haveimplicationsfortheelectroniccouplingbetweenneigh-boringSWNTsinarope[8].
2ThermalconductivityAsdiamondandgraphitedisplaythehighestknownther-malconductivityatmoderatetemperatures,itislikelythatnanotubesshouldbeoutstandinginthisregardaswell.
In-deed,recenttheoreticalwork[7]haspredictedthattheroom-temperaturethermalconductivityofnanotubesisashighas6600W/mK.
Inaddition,atlowtemperature,thethermalconductivityshouldshowtheeffectsof1Dquantizationjustasisseeninthespecicheat.
Thethermalconductivityinahighlyanisotropicmaterialismostsensitivetothehigh-velocityandhigh-scattering-lengthphonons.
Therefore,itislikelythateveninnanotubebundles,thethermalconductiv-ityshoulddirectlyprobeon-tubephononsandbeinsensitivetointer-tubecoupling.
Figure7showsthemeasuredtemperature-dependentther-malconductivityofbulksamplesofSWNTsthathavebeenalignedbyltrationinahighmagneticeld[9].
Inthealign-mentdirection,theroom-temperaturethermalconductivityisgreaterthan200W/mK,whichiscomparabletoagoodmetalandwithinanorderofmagnitudeofthatofhighlycrys-tallinegraphiteordiamond.
Thethermalconductivityofun-alignedsamplesisaboutoneorderofmagnitudesmaller[10].
However,thetemperaturedependenceofthethermalcon-ductivityisroughlythesameinbothtypesofsample.
Also,inbothtypesofsample,simultaneousmeasurementoftheelectricalandthermalconductivityshowsthattheelectroniccontributiontoK(T)isnegligibleatalltemperatures.
Belowapproximately40K,thethermalconductivitydis-playsastrictlylineartemperaturedependenceinallsamples.
Thistemperaturedependenceislikelytobedueto1Dquan-K(W/m-K)Fig.
7.
ThermalconductivityofabulksampleofSWNTsinwhichthetubesarealignedbyltrationinastrongmagneticeld[9].
Themeasurementistakeninthedirectionparalleltothetubestization,inwhichonlytheacousticmodesofthetubecarryanyheatow.
However,theroleofintertubecontactsonthetemperaturedependenceofK(T)isunknown.
InordertomoredenitivelydeterminewhetherthelinearK(T)isdueto1Dquantization,wehavemeasuredK(T)forsampleswithdifferentnanotubediameters.
Becausethephononsub-bandsplittingincreaseswithdecreasingtubediameter,weexpectthatthelinearK(T)shouldextendtohighertemperaturesinsampleswithasmalleraveragetubediameter.
Figure8showsthethermalconductivitydividedbytem-peratureforfourSWNTsamples[11].
Thesamplesweresynthesizedbylaserablationatdifferingoventemperaturesinordertoproducedifferentaveragenanotubediameters.
Twosamplessynthesizedat1100C,withanaveragediameterof1.
4nm,andtwosamplessynthesizedat1200C,withanaveragediameterof1.
2nm,weremeasured.
Allfoursam-plesshowalinearK(T)atlowtemperature,asshownbytheconstantvalueofK/T(normalizedto1hereforallsam-ples).
Forthe1.
4nmdiametersamples(opensymbols),K/Tbeginstoincreaseatapproximately35K,whileasimilarin-creaseisnotseenforthe1.
2nmsamples(lledsymbols)untilapproximately40K.
Thisbehaviorisconsistentwithourexpectationsfora1Dquantizedthermalconductivity.
Apuz-zlinginconsistency,however,isthatthelinearK(T)extendstoapproximately40KwhilethelinearC(T)extendsonlytoapproximately8K.
Ifthephononscatteringtimeisrela-tivelyconstantforallmodes,thesetemperaturesshouldberoughlyequal.
Onepossibleexplanationforthisdiscrepancyisthattheopticalsub-bandsofthenanotubescattermuchmorestronglythantheacousticsub-bands,sothattheirin-uenceonthethermalconductivityissuppresseduntilhighertemperaturesarereached.
Clearly,moreexperimentalandtheoreticalworkisnecessaryinordertofullyunderstandthisbehavior.
Compositematerialshavinghighthermalconductivityhaveanumberofpotentialapplications,particularlyinheatsinkingforelectronicsandmotors.
ToexplorethepotentialK/T(arb.
units)Fig.
8.
Thermalconductivitydividedbytemperature,K/T,ofSWNTsam-pleswithdifferentaveragediameters[11].
TherangeoflinearK(T),i.
e.
constantK/T,extendstohighertemperaturesinsampleswithasmallerdiameter,aswouldbeexpectedforascenarioof1Dquantizationofthephononstructure343Fig.
9.
ThermalconductivityenhancementforepoxysampleswithvaryingloadingofSWNTsandvapor-growncarbonbers(VGCF)[12]ofusingnanotubesforsuchapplications,wehavesynthe-sizednanotube-basedcompositesbymixingas-grownnano-tubesootintoindustrialepoxy(ShellChemicalsEpon862epoxyresin)[12].
Asacomparison,highlygraphiticvapor-growncarbonbers(VGCF)weremixedintothesameresin.
Figure9showsthemeasuredroom-temperaturethermalcon-ductivityenhancementforsampleswith0–1wt%nanotubes,and0–2wt%VGCF.
Thenanotubesamplesshowanincreas-ingthermalconductivityenhancementwithincreasingload-ing,witha120%enhancementat1%loading.
Inaddition,nanotubesseemtobesuperiortoVGCFasallermaterial.
Thisinitialresultdemonstratesthatnanotubesare,infact,anexcellentllerformakinghigh-thermal-conductivitycom-posites.
Acknowledgements.
ThisworkwassupportedbyNSFGrantNo.
DMR-9802560,DOEGrantNo.
DEFG02-98ER45701andtheLaboratoryforResearchontheStructureofMatterMRSEC,No.
DMR00-79909.
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