Scaling of Resistance and Electron Mean Free Path of Single

ScalingofResistanceandElectronMeanFreePathofSingle-WalledCarbon

Nanotubes

MeninderPurewal,ByungHeeHong,AnirudhhRavi,BhupeshChandra,JamesHone,andPhilipKim

122332

3

DepartmentofAppliedPhysics,ColumbiaUniversity,NewYork,NewYork10027

DepartmentofPhysics,ColumbiaUniversity,NewYork,NewYork10027and

DepartmentofMechanicalEngineering,ColumbiaUniversity,NewYork,NewYork10027

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Weprentanexperimentalinvestigationonthescalingofresistanceinindividualsinglewalled

carbonnanotubedeviceswithchannellengthsthatvaryfourordersofmagnitudeonthesame

sample.Theelectronmeanfreepathisobtainedfromthelinearscalingofresistancewithlengthat

varioustemperatures.Thelowtemperaturemeanfreepathisdeterminedbyimpurityscattering,

whileathightemperaturethemeanfreepathdecreaswithincreasingtemperature,indicatingthat

itislimitedbyelectron-phononscattering.Anunusuallylongmeanfreepathatroomtemperature

hasbeenexperimentallyconfirmed.Exponentiallyincreasingresistancewithlengthatextremely

longlengthscalessuggestsanomalouslocalizationeffects.

Singlewalledcarbonnanotubes(SWNTs)are1Dcon-

ductorsthatexhibitarichvarietyoflowdimensional

chargetransportphenomena[1],includingballisticcon-

duction[2,3,4,5,6],localization[7]and1Dvariable

rangehopping[8].Theelectronmeanfreepath,L,is

m

oneoftheimportantlengthscalesthatcharacterizethe

different1Dtransportregimes.Onemethodofdetermin-

ingLinSWNTsistomeasureballisticconductionfora

m

givendevicechannellength.However,thismethodyields

alowerboundofL,andworksonlyatlowtemperature

m

[2,3,4,5]orathighertemperatureforsmalllengthscales

(<60nm)[6].AnotherapproachtoobtainLatroom

m

temperatureistoemployscanningprobemicroscopyto

measurethelinearscalingofthechannelresistance[9],or

unon-invasivemulti-terminalmeasurements[10].Due

totheexperimentallimitationsoftheapproaches,the

characterizationofLforthesameSWNTsoverarange

m

oftemperaturesisyettoberealized.

Recentadvancesinthegrowthofextremelylong

SWNTs(>1mm)[11]nowallowforanintensivestudy

ontheirintrinsicproperties.Inthisletter,weprent

experimentalmeasurementsonthescalingbehaviorof

resistanceinindividual,millimeterlongSWNTsforthe

temperaturerangeof1.6-300K.Fromthelinearscaling

ofresistance,thetemperaturedependentelectronmean

freepathiscalculatedforeachtemperature.Beyondthe

linearscalingregime,weobrvethattheresistancein-

creasexponentiallywithlength,indicatinglocalization

behavior.

MacroscopicallylongandstraightindividualSWNTs

weregrownonadegeneratelydopedSi/SiOsubstrate

2

(t=500nm)usingthechemicalvapordeposition

ox

methoddescribedinRef.[11].Thiswasfollowedbythe

fabricationofmultiplePdelectrodeswithvariouspa-

rations(200nm-400µm)(Fig.1(a)).Pdelectrodeswere

chontocreatehighlytransparentSWNT-electrodecon-

tacts[4].ThediametersoftheSWNTsweremeasured

byatomicforcemicroscope(AFM).WechoSWNTs

withdiameterdlessthan2.5nmtoexcludeanypossi-

bilityofincludingmultiwallednanotubes(MWNT)in

20ȝm

(a)

󰂑

100

󰂅

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I

s

d

(

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A

)

¯

10

󰂑

V

U

1

0.1

󰂅

{

¯

V

U

0.8 ȝm

3ȝm

7ȝm

10 ȝm

20 ȝm

50 ȝm

M1

(b)

015-15

V(V)

g

100

󰂑

(c)

󰂅

{

I

s

d

(

n

A

)

10

1

󰂑

󰂅

{

¯

V

U

¯

V

1ȝm

1.5 ȝm

5ȝm

10ȝm

25 ȝm

55 ȝm

U

SC3

0.1

500ȝm

-30-150

V(V)

g

FIG.1:(a)OpticalimageshowingtypicalSWNTdeviceswith

multiplePdelectrodes.(Int)Scanningelectronmicroscope

imageofanisolatedSWNTcontactedwiththeelectrodes.

RoomtemperatureI(V)oflectedchannellengthsfor(b)

SDg

metallicSWNT(M1)and(c)miconductingSWNT(SC3)

withV=6.4and2.7mV,respectively.

SD

thisstudy.Inaddition,weconfirmedthatthehigh

biassaturationcurrentis<30µAforallSWNTsstud-

ied[12],assuringthatthesamplesconsistedofsingle

tubesratherthansmallbundlesorMWNTs.Thesub-

stratewasudasagateelectrodetotunethechemical

potentialofthesamplebytheapplicationofagatevolt-

ageV.Asmalldcsource-drainbiasvoltage(<10mV),

g

V,wasappliedbetweenpairsofconcutiveelectrodes,

SD

andthetwo-terminallinearresponconductancewas

determinedfromthemeasuredsource-draincurrentI.

SD

Fig.1(b-c)showsthemeasuredIasafunctionof

SD

Vforlectedchannellengthctionsontworepre-

SD

ntativeSWNTs.Allcurvesexhibita‘gap’likefea-

ture-arangeofVwhereIissuppresd.On

gSD

thesameSWNT,everydevice(pairofconcutiveelec-

trodes)showsasimilarI(V)uptoalength-dependent

SDg

multiplicativefactor,oncewealignthecentersofthe

gapregionforeachcurve.ThesimilarityoftheI(V)

SDg

behaviorindifferentctionsforeachSWNTsample

indicatesthatthecorresponding‘gap’featuresarede-

rivedfromtheintrinsicelectronicstructureoftheSWNT

ratherthantheeffectsofrandomlocalvariation.

WeuthequalitativelydifferentI(V)behaviorsof

SDg

differentSWNTstocategorizethemasmetallic(M-NT)

ormiconductingnanotubes(S-NT).TypicalS-NTs

(Fig.1(c))exhibitanoffcurrentregionI<10A

SD

−10

whentheFermienergyEliesntheenergygap[13,14].

F

Ontheotherhand,aweakersuppressionofI(V)is

SDg

obrvedinthe‘smallgap’regioninM-NTs(Fig.1(b)).

The‘smallgap’inM-NTshasbeenattributedtothe

curvature-inducedenergygapE<100meV[15],which

g

isdistinguishedfromtheS-NTenergygap,whichscales

withdiameterasE∼1/d(nm)[1].Amongthe11

g

SWNTswestudiedinthisletter,wefound4M-NTs

and7S-NTs.EachoftheSWNTsexhibitagapcen-

teredatV>0,indicatingtheirp-dopednature.At

g

largenegativegatevoltage(V<−20V),Elieswell

gF

outsideofthegapregionandI(V)saturatestoI,

SDg

sat

whovaluedependsonlyontheappliedVandchan-

SD

SD

nellengthLoftheSWNTction.Thetwo-terminal

resistanceoftheSWNTctionisthenobtainedfrom

R(L)=V/I.Wenotethatfour-terminalresistance

SD

SD

sat

measurementsarepossibleforeachctionbyutilizing

theavailablemultipleelectrodeconfiguration.However,

inourexperiment,thefourterminalmeasurementsyield

esntiallysimilarresultstothetwoterminalR(L),which

preventsparationofthe‘contact’resistancecontribu-

tionfromR(L).Suchinparablecontactresistancebe-

tweenSWNT-metalelectrodeswasreportedtobecaud

bytheinvasivenessofmetalcontacts[16].

WedesignedmanypairsofelectrodeswithdifferentL

oneachSWNTsothatthescalingofR(L)canbestudied

foraspecificsampleatagiventemperatureT.Fig.2(a)

showR(L)ofareprentativeSWNTmeasuredinthe

temperaturerangeof1.6-300KandwithanLrangeof

200nm-50µm.Intheranges,R(L)increaslinearly

andappearstoconvergetoafinitevalueforsmallL(in-

ttoFig.2(a)).Wefoundthatthisscalingbehaviorcan

bedescribedwellbyasimplelineardependencewithan

offt:R(L)=ρL+R,whereρandRareinterpreted

cc

asthe1Dresistivityandcontactresistance,respectively.

ThesolidlinesinFig.2(a)arethetwoparameterline

fitsofthedatapointsatagivenTvalue.Fromthe

fits,R(T)andρ(T)areobtainedasshowninFig.2(b)

c

andFig.2(c),respectively.Forthissample,Rremains

c

fairlyconstantat∼8kΩandρ(T)exhibitstypicalmetal-

licbehavior,i.e.itdecreaswithTandsaturatestoa

valueρatlowtemperatures.Similarscalingbehavior

sat

ofR(L)isobrvedinotherSWNTs,fromwhichbothR

c

andρ(T)areextractedwithinthelinearscalingregime.

TableIsummarizesd,R,andρforthe4M-NTs

csat

2

300 K

)

400

200 K

ȍ

k

(

R

110 K

)

200

ȍ

k

(

100

50 K

R

0

02040

L (ȝm)

1.65 K

10

(a)

1

L (ȝm)

10

10

10

)

m

)

ȝ

ȍ

/

k

ȍ

(

k

(

5

c

5

R

ȡ

(b)

(c)

0

100

200

300

0

100

T (K)

T (K)

200

300

FIG.2:(a)(Int)R(L)forsampleM1atlecttemperatures

rangingfrom1.6-300K.(Main)Alog-logplothighlightsthe

behaviorsatdifferentlengthsscaling3ordersofmagnitude.

Fromthelinearfits(solidlines)ofthedatapoints,weob-

tainthe1Dresistivity(b)andthecontactresistance(c)at

differenttemperatures.Thedashedlinein(c)reprentsR.

Q

and7S-NTsconsideredinthisstudy.Tounderstandthe

scalingofR(L)inFig.2,webeginwiththetwo-terminal

Landauer-ButtikerformulaappliedtoSWNTs[9].Ifwe

consider4low-energychannelsintheSWNT,2eachfor

spinandbanddegeneracy,thenthescalingofresistance

isgivenbyR(L)=(h/4e)(L/L+1)+R,wheree

2

mnc

andhareelectronchargeandPlankconstantandL

m

andRaretheelectronmeanfreepathandthenon-

nc

transparentcontactresistance,respectively.Notethat

weparateoutthecontributionofRfromthetotal

nc

contactresistanceR,sothatthecontactresistancebe-

c

comesthequantumresistanceR=h/4ewhenthecon-

Q

2

tactsbecomefullytransparent.Fromtheexperimentally

obtainedρ(T)andR,wecandeduceL=R/ρ(T)

cmQ

andR=R−RforeachofourSWNTsamples.In

nccQ

particular,wenotethatRRforthemajorityof

ncQ

<

∼

oursamples,suggestingthatthebarrieratthecontacts

isverythinandaddsonlyanegligiblecontributionwhen

Lbecomessubstantiallylarge.

Wenowdiscussthetemperaturedependentbehavior

ofthemeanfreepath.Fig.3isthecentralresultof

thisletter,showingL(T)oftheSWNTslistedinTa-

m

bleI.Overall,L(T)exhibitsdifferentbehaviorsin

m

tworegimesparatedbyT:(i)thehightempera-

cr

tureregime(T>T)whereL∼T(dashedline

crm

−1

inFig.3),whichindicatesthatinelasticscatteringbe-

3

TABLEI:DevicecharacteristicsforSWNTsudinthisstudy.ThecharacterM(SC)isdesignatedformetallic(miconduct-

ing)SWNTs.

M2M4SC2SC4SC6

2.0±.21.7±.61.6±.41.9±.42.2±.22.2±.2

7.9±.88.3±2.510.2±4.510.4±.925.4±4.221.8±14

0.76±.020.93±.012.95±.054.64±.018.13±.3116.3±.13

8.56±.237.07±.082.24±.041.40±.010.80±.030.40±.01

strongsampledependentbehavior,generallywefound

L>>Linalltemperatureranges,withthetempera-

cm

turedependenceexhibitingatrendofincreasingLwith

c

increasingT(upperintstoFig.4).Thisobrvedbe-

haviorofL(T)excludesthequantuminterferencere-

c

latedtostronglocalizationeffectssuchasAndersonLo-

calization[7]fromthepossiblescenarios.Inparticular,in

thehightemperatureregime(T>T),thephacoher-

cr

encelengthLislimitedbythepha-breakingelectron-

φ

phononscattering,andthusL∼L<<L,inviting

φmc

studytoelucidatetheobrvedfurtherlocalizationbe-

haviorbeyondthestronglocalizationlimit[22,23].

Inconclusion,wedeterminethelengthdependentre-

sistanceforSWNTswithchannellengthsranged200nm

-400µm.Fromthescalingbehaviorweevaluatethe

electronmeanfreepathandlocalizationlengthofthe

SWNTforarangeoftemperatures.Whilethelowtem-

peraturemeanfreepathisdeterminedbytheimpurity

scattering,anunusuallylongmeanfreepathisdemon-

stratedatroomtemperature,evenwiththedominant

electron-phononscattering.

WethankI.Aleiner,B.Altshuler,andP.Jarillo-

Herreroforhelpfuldiscussions.Thisworkissupported

bytheNSFNIRT(ECS0507111),CAREER(DMR-

0349232),NSEC(CHE-0117752),andtheNewYork

StateOfficeofScience,Technology,andAcademicRe-

arch(NYSTAR).

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