3 -scanning thermal microscopeStéphane Lefèvre and Sebastian Volz

Citation: Review of Scientific Instruments 76, 033701 (2005); doi: 10.1063/1.1857151 View online: http://dx.doi.org/10.1063/1.1857151 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/76/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermal conductivity measurement of fluids using the 3 method Rev. Sci. Instrum. 80, 024901 (2009); 10.1063/1.3082036 Thermal conductivity measurement and interface thermal resistance estimation using Si O 2 thin film Rev. Sci. Instrum. 79, 054902 (2008); 10.1063/1.2927253 1 , 2 , and 3 methods for measurements of thermal properties Rev. Sci. Instrum. 76, 124902 (2005); 10.1063/1.2130718 A practical extension of the 3 method to multilayer structures Rev. Sci. Instrum. 76, 053901 (2005); 10.1063/1.1896619 Thermal conductivity of AlAs 0.07 Sb 0.93 and Al 0.9 Ga 0.1 As 0.07 Sb 0.93 alloys and ( AlAs ) 1 /( AlSb ) 11digital-alloy superlattices J. Appl. Phys. 92, 4994 (2002); 10.1063/1.1506194

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

138.195.69.112 On: Tue, 16 Sep 2014 15:13:59

3v-scanning thermal microscopeStéphane LefèvreLaboratoire d’Etude Thermique UMR CNRS 6608, BP40109, 86961 Futuroscope Cedex, France

Sebastian Volza!

Laboratoire d’Energétique Moléculaire et Macroscopique, Combustion, UPR CNRS 288,Ecole Centrale Paris, 92295 Châtenay Malabry, France

sReceived 13 July 2004; accepted 9 November 2004; published online 2 February 2005d

We present the theoretical and experimental calibrations of a scanning thermal microscopesSThMdbased on a hot anemometer wire probe. Under ac heating and due to the linear dependence of theprobe electrical resistance to temperature, the third harmonic of the tip voltage is also linear to thetemperature. This method called “3-v” is currently used to estimate the thermal properties with ahot strip deposited on the sample surface. However, in this last technique, the sensor size allows oneto reach the sample transient thermal behavior whereas the SThM tip does not. We propose athermal model showing what information can in fact be obtained by using the microscope under the3v mode. We finally report an experimental calibration providing the tip/sample contact thermalparameters. ©2005 American Institute of Physics.fDOI: 10.1063/1.1857151g

I. INTRODUCTION

Measuring thermal properties at the nanoscale remainsone of the main unsolved challenges regarding nanoapplica-tions. For instance, the modeling of microelectronics and en-ergy conversion devices and the behavior of insulatorsaero-gels, nanoporous filmsd require the knowledge of thenanoelements properties. The atomic force microscopessAFMd have become the key tools to perform ultra-localtemperature measurements.1,2 The original function of thosesystems was to provide the samples topography with theatomic resolution. Those devices were rapidly developed toalso measure a large variety of local properties—magnetic,electric, elastic, etc. In 1986, Williams and Wickramasingheproposed to mount a thermocouple tip in a conventionalAFM.3 While the temperature was the feedback signal tocontrol the tip height, it is until now used to measure thelocal temperature when the tip is brought in contact with thesurface. Those techniques however require an externalheating4 and the knowledge of the sample geometry to pro-vide local thermal properties.

Other probes consist of electrical resistance that are ther-mally controlled through joule dissipation. The probe tem-perature is directly deduced from its electrical resistance.Those “active” tips also measure the local thermal propertieswithout any external heat source: The input current is con-trolled so that the tip temperature is maintained to a constantvalue, the feedback current then measures the capacity of thesample to conduct heat. The resistive tip were mostly used inthe simple dc mode but the obtained data appeared as non-reproducible, scattered, and relevant for low thermal conduc-tivity materials only.5 A previous work emphasizes the keyrole of the uncertainty due to the tip geometry and to thetip-sample contact resistance and radius.5

Several papers8 claimed that quantitative thermal con-ductivity measurements could however be performed by us-ing the tip in the ac mode and by processing the measure-ments data as done in the hot strip technique.

The aim of this paper is to first prove that the thermalanalysis of the hot strip technique can in no way be appliedto the sSThMd scanning thermal microscope measurementsand second to show what information and improvements arebrought by the ac mode. The present work finally providesthe actual 3v-SThM calibration.

In Sec. II, we present the 3v technique and the experi-mental setup. In Sec. III, we develop the thermal model re-vealing that the conventional 3v analysis cannot be appliedto the SThM. In Sec. IV, we reveal the following improve-ments that can in fact be obtained by using the SThM underthe 3v mode:

• The ac mode allows one to calibrate the tip length andradius before bringing the probe in contact with thesample. The sensitivity of the measurements to thoseparameters is crucial as shown in our previous article.6

• The uncertainty of the ac measurements is several or-ders of magnitude smaller than the one of the dc mode.This uncertainty reduces the thermal conductivity rangethat the microscope can read. This limitation is removedby using the ac technique.

• On the basis of our thermal model, we prove that the dcdata are strongly dependent on the ambient temperaturewhile ac measurements are not.

We then report the 3v-SThM calibration curve that suppliesthe tip/sample contact thermal parameters. Those quantitiescorrespond to the ones of the literature within a satisfyingrange.adElectronic mail: volz@em2c.ecp.fr

REVIEW OF SCIENTIFIC INSTRUMENTS76, 033701s2005d

0034-6748/2005/76~3!/033701/6/$22.50 © 2005 American Institute of Physics76, 033701-1

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

138.195.69.112 On: Tue, 16 Sep 2014 15:13:59

II. SThM SETUP AND THE 3v METHOD

We present local thermal property measurements basedon an ac heating of the resistive probe. This method is called“3-v” and it is currently used to estimate the thermal prop-erties of nanostructures with a hot strip deposited on thesample surface.7 The 3v technique consists of monitoring anac currentI = I0 cossvtd in the probe so that the dissipatedheat fluxJ in the Pt–Rh wire writes

J = RI02

2f1 + coss2vtdg, s1d

whereR is the wollaston wire electrical resistance. The tiptemperatureup can then be related to the dc and the secondharmonic temperature amplitudesuDC andu2v, respectively,as follows:

upstd = uP-DC + uP2vcoss2vt + sd, s2d

wheres is the phase difference between the thermal and theelectrical signals. The tip voltageVp can then be expressed as

Vp = R.I0cossvt + sd = R0s1 + auDCdI0cossvtd

+R0aI0u2v

2cossvt + sd +

R0aI0u2v

2coss3vt + sd.

s3d

Equations3d clearly shows that the third harmonic of the tipvoltage only includes the thermal contribution whereas thefirst harmonic is the sum of the electrical and the thermalsignals.

The 3v-SThM setup is described in Fig. 1. Previousworks8 show the same experimental device. A signal genera-tor provides the current at thev frequency. The thermal volt-age amplitude was found to be one thousand times smallerthan the first harmonic amplitude. We therefore introducedthe tip in a Wheatstone bridge to remove the first harmonicfrom the tip voltage. The output signal is finally processed bya lock-in amplifier whose reference frequency is given by thetriple of the signal generator.

III. THERMAL MODEL

The modeling of the probe is based on the transient finequation including a source term due to the joule dissipation.This equation is written for the second harmonic of the tiptemperature amplitudeuP2v

2iv

aPuP2v =

rI02

2lPSP2 −

hpP

lPSPuP2v +

]2uP2v

]x2 , s4d

where the indexP refers to the probe,a is the thermal dif-fusivity, lP represents the thermal conductivity, andr , S,andp are the wire electrical resistivity, the section area, andthe perimeter, respectively.x is the spatial variable associatedto the wire axis andh is the convection conductance. Equa-tion s4d is derived when setting the temperature solution touP2vstd=uP2vcossvtd. The left-hand sidesLHSd term corre-sponds to the transient contribution and the two first right-hand sidesRHSd terms are the joule and the heat losses con-tributions. As in the dc model, the silver coating can beassimilated to a thermal sink so that the temperature ampli-tude at the silver-platinum junction can be set to zero. On thetip-sample side, we introduce an equivalent thermal conduc-tanceGeq which is the connection of the tip-sample contactconductanceGC and the sample conductanceGS in series

Geq=GCGS

GC + GS. s5d

The heat flux at the sample end then reads

− lPSP]uP2v

]xsx = LPd = GequP2vsx = LPd, s6d

where LP is the platinum wire length. When gathering theheat losses and the joule terms into a single fin parameterm2=hpP/lPSP+2iv /aP, the solution of Eq.s6d leads to theexpression of the measured mean temperature amplitude

uP2v =1

LPE

0

LP

uP2vsxddx

=J0

LPm3

AGeq+ B.GPmLP

fexps2mLPd − 1gGeq+ f1 + exps2mLPdgGPmLP,

s7d

where J0=rI02/2lPSP

2 and theA and B coefficients are de-fined as

A = f− 2 −mLP + 4 expsmLPd − 2 exps2mLPd

+ mLPexps2mLPdg, s8d

B = f1 + mLP − exps2mLPd + mLPexps2mLPdg. s9d

Equation s7d reveals the asymptotic behaviors of the tem-perature amplitude. When the frequency goes to zero, thetemperature amplitude is given by Eq.s7d but with m2

=hpP/lPSP. When the frequency goes to large values, thedependence of the temperature amplitude toGeq disappearsso that

uP2v =J0

LPm3

B

f1 + exps2mLPdg=

J0aP

2vs10d

becausemLP is greater than 1. This behavior corresponds tothe one of a low pass filter as reported in Fig. 2. When thecurrent frequency is larger thanvC=h/CPRP,1 kHz—CP

being the volumetric heat capacity andb is set to 1µm sRef.5d—the thermal inertia of the tip increases and the tip re-sponse diminishes according to av−1 power law. The char-acteristic frequency of the sample thermal dynamics can be

FIG. 1. sColor onlined Schematic of the 3-v experimental setup.

033701-2 S. Lefevre and S. Volz Rev. Sci. Instrum. 76, 033701 ~2005!

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

138.195.69.112 On: Tue, 16 Sep 2014 15:13:59

derived from the heat flux conducted from the tip to thesample

sS= 2lSpbu0S1 + bÎ2iv

aSD , s11d

where the indexS denotes the sample. This expression isobtained when solving the temperature field in a semi-infinite body where the temperatureu0 is imposed on a diskof the system surface. Equations11d directly indicates thatthe characteristic frequency of the sample temperature am-plitude is vS=aS/2b2,1 MHz. We conclude that the tip isnot able to reveal the sample thermal dynamics. This is a keypoint since it proves that the classical hot strip method whichis based on the dynamical response of the sample, cannot bedirectly applied here.

IV. RESULTS AND DISCUSSION

A. Identification of the probe geometric parameters

The first gain of the ac SThM relies on the tipcalibration6 so that theLP andRP parameters can be set. Theprinciple of this calibration consists in measuring the ampli-tudeuP2v versus frequency when the tip is in the air. Thoseexperimental data are then compared to the predictions ofEq. s7d when the equivalent conductanceGeq is set to zero. Astandard identification method is used to set the probe pa-rameters that minimize the difference between measured andcalculated data as shown in Fig. 2sad. This procedure yields a5–10 % deviation compared to the commercial data. Thisdifference leads to a 30–50 % error on the estimation of theamplitudeuP2v. We believe that this calibration is a key con-tribution to the accuracy of the SThM metrology.

B. Uncertainty calculation

The second gain is the lower uncertainty obtained fromthe ac measurements. This uncertainty is crucial because it isrelated to the thermal conductivity range that the SThM canread. Indeed, the tip-sample thermal interaction can be mod-eled by a thermal conductanceGc and by the radiusb but notechnique can directly dissociate the contact conductanceGc

from the sample conductanceGS=plSb. The relevant quan-tity to be measured without ambiguity is the equivalent con-ductanceGeq of Eq. s5d that can be rewritten as

1

Geq=

1

plSb+

1

GC. s12d

Noting d the relative instrument uncertainty on the reverse ofGeq, the thermal conductivity interval that the SThM canread isf0,Gc/dpbg. If lS.Gc/dpb, the term includinglS

on the RHS of Eq.s12d will indeed be smaller than the un-certainty.

The dc relative uncertainty can be derived from5

DP

Pj=

3

4

1

1 + GP/Geq. s13d

Reversing this equation yields

GP/Geq=3

4

Pj

DP− 1. s14d

Knowing the relation between powerPj and voltageU fi-nally leads to

FIG. 2. sColor onlined fsad topg Experimental measurementsspointsd andmodel predictionsslined of the tip temperature amplitude against frequency.Out of contact case.fsbd bottomg Tip temperature amplitude against fre-quency when the tip length varies. In contact model.

033701-3 The 3v-scanning thermal microscope Rev. Sci. Instrum. 76, 033701 ~2005!

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

138.195.69.112 On: Tue, 16 Sep 2014 15:13:59

ddc = 3.U UidUi

Ui2 − U0

2U , s15d

where the subscriptsi ando, respectively, correspond to thein- and out-of-contact voltages. The commercial device pro-vides dUi =0.01 V and Ui ,U0,1 V. The difference be-tween in- and out-of-contact voltages being of the order of0.5 to 0.1 V, the relative instruments uncertaintyddc variesfrom 1 to 5 %.

Let us now calculate the value ofd in the ac mode. TheGeq thermal conductance is measured as the plateau valueshown in Fig. 2sbd. Consequently, Eq.s7d can be studied atthe static limitv=0

V3v = V0 .4sGP/Geqd + 1

sGP/Geqd + 1, s16d

where V0=PjR0I0a / s24GPd=2.7 mVsI0=50 mA,Pj

=3 mWd corresponds to a fictive temperature increaseu0

=Pj /GP when all the joule flux is lost at the tip end atx=LP. A straightforward derivation then brings

dac= U 3dV3vV0

sV3v − V0dsV3v − 4V0dU . s17d

SinceV3v,4 V0, dac=dV3v /V0. The lock-in amplifier pro-vides a stable signal up to 10µV with a very basic Wheat-stone bridge. A reasonable estimate ofdac is therefore 3.10−4.

A striking point is thatdac is increasing when the samplethermal conductivity decreases. Indeed, the tip temperatureu2v andV3v are larger when the sample is an insulator.

In the dc mode withd=0.01, the maximal thermal con-ductivity is lSmax=0.3d−1=30 W m−1 K−1. This estimationemphasizes a severe limitation of the dc metrology. How-ever, the valuelSmax=0.3d−1=1 kW m−1 K−1 in the ac modeindicates that the 3v-SThM is sensitive to all materials. Araw estimation shows that this last metrology is able to de-tect heat flux of the order of the picowatt.

TABLE I. Materials used for the calibration and their corresponding thermalconductivity.

Thermal conductivityMaterials sW m−1 K−1d

Te 3.3Bismuth 7.9

Gadolinium 10.5Scandium 15.8Thalium 16.9Hafnium 23

Antimoine 24.4Rhenium 48

FIG. 3. sad–sdd Topographysleftd and 3v srightd images of scandiumstopd and titaniumsbottomd samples embedded in resin.

033701-4 S. Lefevre and S. Volz Rev. Sci. Instrum. 76, 033701 ~2005!

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

138.195.69.112 On: Tue, 16 Sep 2014 15:13:59

C. Dependence to the ambient temperature

The dependence of theGeq measurement to the ambienttemperatureTa is another stumbling block of the dc tech-nique. The measured voltage can be expressed as

U = R01/2f1 + aDTgÎPj = R0

1/2f1 + aDTgÎsT − Tad. s18d

The uncertainty onU then writesdU=R01/2dTa/2ÎsT−Tad

and the relative uncertainty can be expressed at the first or-der, as

dU

UdTa/2.sT − Tad. s19d

Given a tip temperature of 100 °C andTa=25 °C, a variationof the ambient temperature of 2 °C during a scan leads to adeviation ofdU=0.013 V which is 1.3 times the instrumentuncertainty. In other terms,dTa=2 °C causes a considerableincrease ofddc by a factor 2 to 3 in a conventional STMmapping.

In the ac mode, V3v depends onTa only through theTa

dependences of the thermophysical and electrical propertiesof the sample and the tip:R0, a , lP, aP, lS. which are neg-ligible.

In summary, the ac mode drastically enhances the sensi-tivity of the Geq measurement by at least four orders of mag-nitude. This makes the SThM sensitive to all materials. Italso allows us to get rid of the ambient temperature depen-

dence whereas the dc mode nonaccuracy may be tripled byambient temperature variations. It is also a striking point thatdc and ac sensitivities have opposite behaviors regarding thelow thermal conductivity materials; the ac techniques indeedprovides a better accuracy for insulators.

D. 3v-STM calibration

Equations12d clearly shows that the dependence of theequivalent conductanceGeq to the sample thermal conductiv-ity lS leads to the identification of the contact parametersGc

and b. To estimate those parameters, we scanned a set ofeight samples reported in Table I. They were chosen accord-ing to the fact that the interval of better sensitivity should bein the range of 0–50 W m−1 K−1. The materials were all em-bedded in low thermal conductivity resin.

The uncertainty improvement provided by the ac ap-proach requires a longer measurement duration related to thelock-in amplifier averaging time. We present the results ofthe scanning of scandiumslS=15.8 W m−1 K−1d and tita-nium slS=21.9 W m−1 K−1d samples embedded in low ther-mal conductivity resin. 3v and topography images are pre-sented in Fig. 3. Each scan have lasted about 90 min for apixel resolution of 600 nm. We believe that this resolution isworse than the physical resolution since high contrasts be-tween neighboring pixels can be observed. We presume that

FIG. 4. Calibration curve reporting the equivalent conductance versus the thermal conductivity.Geq=Gs.Gc/ sGs+Gcd whereGs andGc are, respectively, thesample and the contact conductances. The model predictions are represented for different parameters.sad fGc=4.7.10−6 W K−1 and sbd =201 nmg, sbd fGc

=5.10−6 W K−1 and sbd =250 nmg, scd fGc=6.10−6 W K−1 and sbd =50 nmg, sdd fGc=6.10−6 W K−1 and sbd =100 nmg, and sed fGc=5.10−6 W K−1 and sbd=312 nmg.

033701-5 The 3v-scanning thermal microscope Rev. Sci. Instrum. 76, 033701 ~2005!

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

138.195.69.112 On: Tue, 16 Sep 2014 15:13:59

the stains on the scandium or on the titanium samples in the3v images are due to the presence of small sized resinshavings.

Figure 4 reports the equivalent thermal conductance ver-sus the sample thermal conductivity and the equivalent ther-mal resistance versus the reverse oflS. Equations12d indeedproves a linear dependence between the resistance and 1/lS.We suggest that the data scattering is due to the presence ofresin shavings. However, the modelfsad–sed referred linesgcan be fit in order to frame the experimental data. This fittingprovides GC=4.7–6 10−6 W/K and b=100–300 nm. Theconductance value corresponds to those obtained in the dcmode but the contact radius is one order of magnitudesmaller that the one identified in this same mode.5 We sug-gest that the tip temperature may be responsible for such adisagreement. The dc data were obtained at 120 °C whichselects conduction in air as the main heat transfer channel.

Lower temperatures were used in the ac mode so that theconduction in the water meniscus may have been predomi-nant in this case.

1R. B. Dinwiddie, R. J. Pylkki, and P. E. West, inThermal Conductivity,edited by Timothy W. Tong, Vol. 22sTechnomic Publishing Co., Lan-caster, PA, 1994d, pp. 668–677.

2A. Majumdar, Annu. Rev. Mater. Sci.29, 505 s1999d.3C. C. Williams and K. Wickramasinghe, Appl. Phys. Lett.49, 1587s1986d.

4O. Kwon, L. Shi, and A. Majumdar, Trans. ASME, Ser. C: J. Heat Trans-fer 156, 125 s2003d.

5S. Lefèvre, S. Volz, J.-B. Saulnier, C. Fuentes, and N. Trannoy, Rev. Sci.Instrum. 74, 1 s2003d.

6S. Lefèvre, J.-B. Saulnier, C. Fuentes, and S. Volz, Superlattices Micro-struct. 35, 283 s2004d.

7D. G. Cahill and R. O. Pohl, Phys. Rev. B35, 4067s1987d.8G. Fiege, A. Altes, R. Heiderhoof, and L. Balk, J. Phys. D32, L13s1999d.

033701-6 S. Lefevre and S. Volz Rev. Sci. Instrum. 76, 033701 ~2005!

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

138.195.69.112 On: Tue, 16 Sep 2014 15:13:59