DTIC ADA560663: Quasi-Ballistic Thermal Transport from Nanoscale Interfaces Observed Using Ultrafast Coherent Soft X-ray Beams

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LETTERS
PUBLISHEDONLINE:8NOVEMBER2009|DOI:10.1038/NMAT2568
Quasi-ballistic thermal transport from nanoscale
interfaces observed using ultrafast coherent soft
X-ray beams
MarkE.Siemens1*,QingLi1,RongguiYang2,KeithA.Nelson3,ErikH.Anderson4,
MargaretM.Murnane1 andHenryC.Kapteyn1
Fourier theory of thermal transport considers heat transport the reduction in thermal flux for a nanoscale heat source, but
as a diffusive process where energy flow is driven by a there is significant variation in their predictions17–19. One model,
temperature gradient. However, this is not valid at length the ballistic–diffusive equation model17, provides clear insight by
scalessmallerthanthemeanfreepathfortheenergycarriersin separating the size-dependent heat flux into contributions from
amaterial,whichcanbehundredsofnanometresincrystalline ballisticanddiffusivecomponents,similartowhatisdoneinthe
materials at room temperature. In this case, heat flow will case of rarefied gas20 or electron flow21. Figure2 illustrates the
become ‘ballistic’—driven by direct point-to-point transport predictions of this model qualitatively: at large length scales, the
ofenergyquanta1.Pastexperimentshavedemonstratedsize- thermaltransportisdominatedbytheFouriercomponent,butas
dependentballisticthermaltransportthroughnanostructures thedimensionofthehotspotisreduced,ballistictransportbecomes
such as thin films, superlattices, nanowires and carbon increasinglyimportant.Othermodelssuggestthatthetransportis
nanotubes1–8. The Fourier law should also break down in the determinedbydifferenteffects,suchasnon-propagatingphonon
caseofheatdissipationfromananoscaleheatsourceintothe modes22 and interface effects19. This extensive modelling effort
bulk.However,despiteconsiderabletheoreticaldiscussionand has, however, taken place in an absence of real experimental
directapplicationtothermalmanagementinnanoelectronics2, data with which to benchmark the models. Given the variations
nano-enabled energy systems9,10 and nanomedicine11, this betweenthesemodels,quantitativeexperimentsareveryimportant,
non-Fourier heat dissipation has not been experimentally bothforafundamentalunderstandingofthebasicandimportant
observed so far. Here, we report the first observation and phenomenonofheatflow,andtobenchmarkmodelsofnanoscale
quantitativemeasurementsofthistransitionfromdiffusiveto heattransferfornanosystemdesign.
ballistic thermal transport from a nanoscale hotspot, finding In many practical nanosystems, an interface separates a small
a significant (as much as three times) decrease in energy heatedregionfromaheat-sinksubstrate,andthethermalresistance
transportawayfromthenanoscaleheatsourcecomparedwith of this interface must also be taken into account. The heat
Fourier-lawpredictions. flow from the heated nanostructure, as shown in Fig.1a, can
WhenheatflowsfromahotspotwithadimensionLsmallerthan then be characterized by an effective resistance acting at the
themeanfreepathofphononsintheheatsinkΛ,energy-carrying interface—with contributions from both a ballistic correction to
phononstravelballisticallyawayfromthesourceforasignificant theFourierlawandtheconventionalthermalboundaryresistance
distance before experiencing collisions. However, in a Fourier (TBR) characterizing interfacial transport at bulk length scales.
pictureofthermaltransport,heatflowisassumedtobediffusive TBRismeasuredinthinfilmsusingtime-resolvedopticalpump–
(experiencingmanycollisions),withthethermalfluxqdetermined probe methods such as transient thermoreflectance4,23 (TTR).
bythetemperaturegradient(q∼∇T).Whenthereisasignificant However,thediffractionlimitforthevisibleprobelightlimitsthis
energydensitychangeoveradistancelessthanΛ,thereisnolocal techniquetostructureswithlinewidthsgreaterthan∼1µm—too
phononequilibriumandthereforetemperaturecannotbedefined, large in scale to observe ballistic size effects from a nanoscale
as illustrated in Fig.1a. Under these circumstances, the Fourier heat source at room temperature. Using a specialized geometry
law, which assumes local thermal equilibrium, overestimates the that allowed confined heating without a heterogeneous material
thermaltransport1,12,13. interface, Sverdrup et al. used electrical resistance thermometry
In the intermediate case when the hotspot dimension L to measure steady-state temperature distributions in a thin 5µm
is of the order of Λ, transport is neither fully ballistic nor silicon membrane near a 300-nm-wide highly doped silicon
diffusive. The full Boltzmann transport equation for phonons resistor. At temperatures below 200K, they observed a slight
mustthenbesolvedtomodelthermaltransport1.Unfortunately, deviation from the prediction of Fourier heat conduction22.
an analytical solution of the Boltzmann transport equation for However, this experiment did not allow for any systematic and
real systems is usually impractical without some simplifying quantitativecomparisonbetweenexperimentandtheoryowingto
assumptions, and numerical simulations vary widely in their the limitations posed by the sample geometry. More recent work
prediction of the strength of this nanoscale hotspot effect14–16. did not observe any ballistic size effect at room temperature24.
Severalphenomenologicalmodelshavebeenproposedtoexplain It is worth noting however, that all of these measurements were
1JILA,UniversityofColoradoatBoulder,Boulder,Colorado80309,USA,2DepartmentofMechanicalEngineering,UniversityofColorado,Boulder,
Colorado80309,USA,3DepartmentofChemistry,MIT,Cambridge,Massachusetts02139,USA,4LawrenceBerkeleyLabsandCenterforX-rayOptics,
Berkeley,California94720,USA.*e-mail:[email protected].
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Quasi-ballistic thermal transport from nanoscale interfaces observed
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a L 1 b
h
Normaliz Sporoftb Xe-ray RReesssffflllooofffeeettttccc tttttXXXeeoo-- drCayCsD
Λ e
d tempera 120 μ m Dsoiffftr Xac-treadys
Diffusive phonon Quasi-ballistic ture 120 μ m
transport: Λ << L transport: Λ > L 0 Infrared pump
c
Ti:sapphire amplifier
2 kHz, 2 mJ, 25 fs Infrared pump beam
Nanostructured
Ni/sapphire sample
Beam splitter
Time-delay stage Soft X-ray
probe beam
Al filter CCD
Hollow waveguide
for HHG Toroidal mirror
Figure1|Experimentalset-upformeasuringthermaltransportacrossananoscaleinterface.a,Schematicillustratingthedifferencebetweendiffusive
andquasi-ballisticthermaltransportacrossaninterface.Whenthesizeoftheinterfaceislessthanthephononmeanpathinthesubstrate,theFourier
diffusiveheatequationbreaksdown.b,SamplegeometryshowinginfraredlaserilluminationandsoftX-raydetectionscheme.c,Experimental
geometry:aninfraredlaserbeamat800nmheatsthenanostructure,andasoftX-raybeamat29nmmonitorstheheatflowfromthenanostructureinto
thesubstrate.
done using steady-state techniques poorly suited to capturing Phonon-transport regime
short-timescaleballisticdynamics.
Ballistic Quasi-ballistic Diffusive
In this work, we use a beam of ultrafast coherent soft
10
X-rays, at a wavelength of 29nm, to directly observe the
csauohroreloianutgendddinnyagnnsa.omsBtiyrcusicntotuefrrefearuosnminanegtordsiccifaafllrleayctmhioeoanntiotsoforsuinorfgctedXiis-nprtaloayceliimtgshetnb,tuwilnke d to rFourier 8
can detect dynamic temperature changes more sensitively than ze
would be possible using either ultrashort optical pulse probing mali 6
or steady-state measurements. This ultrafast short-wavelength nor rBallistic
y
terxapnesriiemnte-ndtiaflfrcaacptaiobnilitmyeththaotdalololowgsyursetpormesaepntosuatnnaenwolsycadleetvheelormpeadl stivit 4 rFourier + rBallistic
si
transportoverarangeofnanostructuresizes. al re
Figure1aillustratesthedifferencebetweendiffusiveandballistic m 2
heattransfer,andFig.1b,cshowsthesamplegeometryandoverall Ther rFourier
experimentalset-up.Periodicnickellineswithaheightof20nm,
lengthof120µmandwithwidthsLvaryingfrom65to2,000nm 0
0.1 1 10
were fabricated on a sapphire substrate using electron beam Inverse Knudsen number = L /Λ
lithography and a lift-off process (at a fixed linewidth-to-period
ratioof25%).Anidenticalnickelnanostructurewaspreparedon Figure2|AnalyticalpredictionoftheFourierandballisticcomponentsof
a fused-silica reference substrate. As discussed in Supplementary theresistivityforthermaltransportawayfromahalf-cylinderof
SectionS1,comparedwithfusedsilica(whichhasanaveragedmean diameterL.WhenthelinewidthLismuchlargerthanthephononmean
freepathforheat-carryingphononsof∼2nm),theaveragedmean freepathΛ,theFouriercomponent(blue)isthelargest,butforL<Λ,the
free path of heat-carrying phonons in sapphire is of the order of ballisticresistivity(red)isdominant.Figure4showsthefirstmeasurement
100–150nmatroomtemperature.Allexperimentswerecarriedout oftheballisticcorrectiontotheFourierresistivity.
atroomtemperatureandinhighvacuum(∼10−6torr).Thenickel
lines were impulsively heated by ultrashort (25fs) pulses from a waveguidefilledwithargongas.Theresultingnonlinearinteraction
high-power Ti:sapphire laser-amplifier system. Pump pulses at a betweentheintenselaserpulsesandtheatomsinthegasgenerates
wavelength of 800nm were focused to a fluence of 2.5mJcm−2, odd harmonics of the driving laser frequency. Absorption of the
in a spot roughly five times the size of the nanostructured nickel argongasandaluminiumfilters(usedtoblocktheresidual800nm
sampletoensureuniformheating.Owingtothehightransmission light) eliminate all but the 25th, 27th and 29th harmonic orders,
of the sapphire and fused-silica substrates at 800nm, only the with a spectrum peaked at 29nm. The full spatial coherence and
nanostructurewasheated. short wavelength of the soft X-ray beams from the waveguide
To probe the heat flow dynamics, an ultrafast, coherent soft makethemidealforhighlysensitiveinterferometricmeasurements
X-ray beam is diffracted from the nanostructure at a variable of nanoscale surface deformation26. Furthermore, the very short
pump–probe time delay. The soft X-ray beam is generated duration of the HHG pulses (sub-10fs; ref. 27) makes possible
throughthehigh-orderharmonicgeneration(HHG)process25 by unprecedentedtemporalresolutionfordynamicsexperiments.The
focusing∼1mJpulsesfromtheTi:sapphireamplifierintoahollow softX-rayharmonicsarerefocusedtoa∼100µmspotonthesample
NATUREMATERIALS|VOL9|JANUARY2010|www.nature.com/naturematerials 27

©2010 Macmillan Publishers Limited. All rights reserved.
LETTERS NATUREMATERIALS
DOI:10.1038/NMAT2568
a al1.0 surfacestraincausedbythepatternedheatinggeneratestheSAW;
gn L = 810 nm weconfirmthatthisisthesourceoftheoscillationsbycomparing
n si Measured data themeasuredfrequencyofoscillationwiththeRayleighvelocityfor
actio RReessiissttiivviittyy ((bbeuslkt vfiat)lu =e )2 =.1 2.0 SAWpropagationinthesubstrate28.
diffr0.5 To interpret our results, we use an analysis similar to that
Δ used for TTR measurements of interfacial thermal resistivity. To
d
ze extract the cooling dynamics of the nickel nanostructure from
mali data shown in Fig.3, we implemented a multiphysics model
Nor 0 includingthermaltransport,thermomechanicsandFresneloptical
0 2,000 4,000 6,000 8,000 propagation (see Supplementary Section S2). The model takes
Time (ps)
the full two-dimensional geometry into account. First, thermal
b nal 1.0 transport across the interface is governed by the conservation of
sig L = 350 nm energyacrossthesurface,wherex andy arealongandawayfrom
ction MReesaisstuivreitdy d(baetast fit) = 2.9 thesurfacerespectively
Δ d diffra0.5 Resistivity (bulk value) = 2.0 ρ h C dTNi(x,t)=−(TNi(x,t)−Ts(x,y=0,t))
ze Ni Ni Ni dt r
ali Effective
m
or 0
N wherer istheinterfaceresistivitythatwillbemodifiedlater
0 2,000 4,000 6,000 to incluEdffeectibveallistic effects away from the interface, ρ and C
Time (ps) Ni Ni
arethedensityandvolumetricspecificheatinthenanostructure,
c gnal 1.0 respectivelyand,TNi andTs arethetime-andposition-dependent
on si L = 1M90ea nsmured data temperaturesinthenanostructureandsubstrate,respectively.Heat
cti Resistivity (best fit) = 3.1 propagationismodelledbynumericallysolvingtheheatequation
diffra 0.5 Resistivity (bulk value) = 2.0
Δ ∂T(x,y,t) k
alized s ∂t =Css∇2Ts(x,y,t) (1)
m
or 0
N
inthesubstratebeneathasinglenickellineusingperiodicboundary
0 1,000 2,000 3,000 4,000
conditionstoaccountfortheneighbouringlines.Inequation(1),
Time (ps)
k and C are the thermal conductivity and volumetric specific
d nal 1.0 hseatofthsesubstraterespectively.Second,thermalexpansionofthe
on sig L = M80e ansmured data substratesurface(cid:49)hsiscalculatedfromthethermoelasticequation
cti Resistivity (best fit) = 4.4 assuming a stress-free interface29, which involves a weighted
diffra 0.5 Resistivity (bulk value) = 2.0 summationoverthetemperaturedistributionateachpoint
Δ
d
alize (cid:49)h(x)=2(1+νs)αs(cid:90) (cid:90) T(x ,y) ydxdy
m s 3π 1 (x−x )2+y2
Nor 0 x1 y 1
0 500 1,000 1,500 2,000
where α is the coefficient of linear thermal expansion of the
Time (ps) s
substrateandν isPoisson’sratio.Thermalexpansionofthenickel
s
Figure3|NormalizeddynamicsoftX-raydiffractionsignalforNilineson lines is also accounted for using (cid:49)h (x)=α T (x). Finally,
Ni Ni Ni
sapphire.a–d,Nilinewidthsare810nm(a),350nm(b),190nm(c)and Fresnel optical propagation and diffraction from the resulting
80nm(d).Thesignalineachcaseconsistsofasharpriseowingto surface profile gives the relative signal contributions from the
impulsivelaserheating,athermaldecayowingtointerfacialthermal nickellinesandthesubstrate.Weparameterizetheresultsofthis
transportandanoscillationowingtosurfaceacousticwavepropagation. numericalmodelintermsofaneffectiveresistivityr thatacts
Effective
TheredandbluecurvesshowthefitusingtheFourierlawforthermal attheinterfaceandincludesbothconventionalthermalboundary
transportinthesubstrateandeitherlarge-scaleinterfacialresistance resistivityandsize-dependentballisticeffectsinthesubstrateowing
(blue)oramodifiedeffectiveinterfacialresistance(rEffective,unitsof to the confined heat source, and fit to the time-resolved thermal
10−9Km2W−1),whichaccountsforballisticeffectsfromananoscale decay data. We use the parameter resistivity (r, units m2KW−1)
heatsource(red).Thedeviationbetweenthetwofitsincreaseswith ratherthantheresistance(R,unitsKW−1),asthelatterdependson
decreasinglinewidth. thecontactareaofthenanostructuredinterface.
To qualitatively understand the nature of the ballistic thermal
usingagrazing-incidencetoroidalmirror.TheHHGbeamreflected transport,wemodifiedthemodelproposedbyChenforheatflow
anddiffractedfromthenanostructureisthenrecordedusingaCCD from a nanoscale sphere into an infinite substrate13, which sepa-
(charge-coupleddevice)camera(Fig.1b). ratelycalculatesthediffusiveandballisticcontributionstothetotal
ThechangeinsoftX-rayprobediffractionasafunctionofdelay flux17. We implemented the model using a cylindrical geometry
aftertheinfraredheatingpulseforvaryingnickellinewidthisshown and extracted the resistivity, which is inversely proportional to
in Fig.3. The signal decays as the heat in the nickel nano-lines the heat flux. Details of our model are given in Supplementary
dissipatesintothesubstrate.Therateofthisthermaldecaydepends Section S3. The results, shown in Fig.2, illustrate the nature of
onthethermaltransportacrossthenickel/substrateinterfaceandin the quasi-ballistic heat-transfer effect as a function of the inverse
thesubstrate.Therapidoscillationsinthesignalareduetosurface Knudsen number L/Λ. For linewidths larger than the mean free
acoustic wave (SAW) propagation in the nanostructure. Periodic path,theresistivitypredictedbytheFourierlawdominatesandthe
28 NATUREMATERIALS|VOL9|JANUARY2010|www.nature.com/naturematerials

©2010 Macmillan Publishers Limited. All rights reserved.
NATUREMATERIALS LETTERS
DOI:10.1038/NMAT2568
7 of the average phonon mean free paths of Λ =2nm (upper
FS
limit of ∼5nm beyond which a good fit is no longer possible)
–1W) Fused-silica substrate andΛ =120±10nm.AsdiscussedaboveandinSupplementary
2 mK 6 SectionSaS4,thesevaluescomparewelltotheexpectedvalueofΛ.
¬910 5 of Opuhronmoenasutrraenmsepnotrstdefrmomonstaratneatnhoastctahleephheyasticaslofuurncdeamcaenntables
x
vity r(Interface 4 Sapphire substrate wtfurhneheedilceephrnaspgtthotrho:oLpsdco→asbleeysLotc+fhotahnΛtesF.ihdToeeuharretiinesegorffuterarcactnmievstepionorbiermesticlusaatmrinvgiebthreyetcchaootaerrnrdreetchcrtteeeigodpinhobonyinnrmoethqnoiumsdirceeialna1s3nge,
e resisti 3 rBC issho(1w+nΛas/tLh)e,wrehdicdhasihsetdhelinexepienriFmige.n4t.alTdheispeimndpelniecsethwaetohbesaetrfvloedw,
c
erfa 2 fromananoscaleheatsourcesmallerthanthephononmeanfree
nt pathcanbethoughtofasemergingfromalargerspotcorresponding
e i
ctiv 1 rTBR rTBR tothesizeofthephononmeanfreepathinthesubstrate.
e This is the first direct and systematic measurement of quasi-
Eff
ballisticthermaltransporteffectsfromananoscaleheatedregion.
0 We observe the breakdown of conventional Fourier heat con-
100 1,000
duction from interfaces smaller than the phonon mean free path
Interface contact size L (nm)
Λ. We find excellent agreement between the experimental data
Figure4|Measuredeffectivethermalresistivityfornickel andananalyticalmodelthatdeterminesthethermaltransportby
nanostructuresofwidthLdepositedonfused-silicaandsapphire summingtheballisticanddiffusivecontributions,anddemonstrate
substrates.Theblueandreddottedhorizontallinesshowthelarge-scale that the Fourier law can still be used if corrected with an extra
resistivityrTBRfordatafromthefused-silicaandsapphiresubstrates.The size-dependent resistance, proportional to the Knudsen number
blueandreddashedcurvesshowmodelpredictionsfortheballistic Λ/L, accounting for ballistic effects. Ballistic effects are expected
resistivitycorrectionrBC∼Λ/L,withΛFS=2nmandΛSa=120nm, to get even stronger as nanostructures shrink in size or for sili-
respectively.Theerrorbarsindicatethestandarddeviationinthefits con substrates with even longer mean free paths (Λ ∼250nm).
Si
tothedata. Theseexperimentsthusadvancetheunderstandingofheat-transfer
fundamentals,andareimportantforthedesignandmanipulation
ballistic resistivity is negligible. However, when the dimension of ofnanoscalethermaltransportincircuits,thermoelectrics,photo-
theheatedinterfaceiscomparabletoorsmallerthanthephonon voltaicsandotherstructuresofinterestinnanotechnology.
mean free path in the substrate, the Fourier law significantly
Received22July2009;accepted6October2009;
under-predictstheresistivity(orover-predictstheheatflux).The
under-prediction in the Fourier law is given by r /r , so publishedonline8November2009
Ballistic Fourier
we introduce a ballistic correction resistivity rBC proportional to References
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