Measurement of the electron-phonon couplingfactor dependence on film thickness and grainsize in Au, Cr, and Al

John L. Hostetler, Andrew N. Smith, Daniel M. Czajkowsky, and Pamela M. Norris

Femtosecond thermoreflectance data for thin films and bulk quantities of Au, Cr, and Al are comparedwith the parabolic two-step thermal diffusion model for the purpose of determining the electron-phononcoupling factor. The thin films were evaporated and sputtered onto different substrates to produce filmsthat vary structurally. The measurement of the electron-phonon coupling factor is shown to be sensitiveto grain size and film thickness. The thin-film thermoreflectance data are compared with that of thecorresponding bulk material and to a theoretical model relating the coupling rate to the grain-boundaryscattering and size effects on the mean free path of the relevant energy carrier. © 1999 Optical Societyof America

OCIS codes: 120.5700, 240.0310, 260.3910, 350.5340.

l

1. Introduction

Nonequilibrium distributions between electrons andphonons in metals and superconductors are no longerlimited to optical excitations, but are becoming in-creasing relevant in microelectronic devices process-ing high-frequency signals in the terahertz andsubterahertz regime. Fast microwave devices suchas thermal bolometers,1 heterojunction bipolar tran-sistors and heterojunction field-effect transistors2

that exhibit nonequilibrium phenomena possesshighly mobile free electrons that can counteract sig-nal degradation and enlarge the region of latticeheating, thereby reducing peak lattice temperatures.This phenomena is seen in laser irradiation of mir-rors where ultrashort laser pulses ~,2 ps! can havehigher fluences than longer pulses without irrevers-ibly damaging the structure of the surface. Thesenonequilibrium features promise a future for tera-hertz devices, and accurate knowledge of the heat

J. L. Hostetler, A. N. Smith, and P. M. Norris are with theDepartment of Mechanical and Aerospace Engineering, Universityof Virginia, Charlottesville, Virginia 22903. D. M. Czajkowsky iswith the Department of Biophysics, University of Virginia, Char-lottesville, Virginia 22903.

Received 26 October 1998; revised manuscript received 11 Feb-ruary 1999.

0003-6935y99y163614-07$15.00y0© 1999 Optical Society of America

3614 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

transfer mechanisms involved is imperative for thecore design of such technology.

Generation of hot electrons in microwave devicescan occur thermionically by the utilization of singleor double potential barriers or through tunnelingeffects as in the tunneling electron hot electron am-plifier.2 Injection of hot electrons into the control-ling region ~the base! of microwave devices resultsin ballistic and quasi-ballistic electron transportthrough the base and collector. Quasi-ballisticelectrons are hot electrons that suffer scatteringduring flight through the base whose length is typ-ically less than the mean free path of the electron.Such devices have charging times as short as 500 fs,which can generate nonequilibrium distributionsbetween the free electrons and the lattice. Char-acterization of the efficiency and performance ofthese devices is based on determining the amount ofquasi-ballistic electrons in the distribution that suf-fer inelastic collisions that degrade mobility andthermalize the free carriers with the lattice.Knowledge of the rate of electron-phonon thermal-ization is a key component to characterizing thenature of quasi-ballistic electron distributions inmicrowave devices and is also used for determiningpeak lattice temperatures in mirrors used for ultra-short laser-pulse irradiation.

Nonequilibrium heating of metals at room temper-ature can be described by the parabolic two-step~PTS! thermal diffusion model. For ultrashort pulseaser heating, the PTS model accounts for photon

w

ereat

na

absorption by free electrons and subsequent electron-lattice diffusion. The two coupled equations aregiven by3

Ce~Te!]Te

]t5 ¹@K~Te!¹Te# 2 G~Te 2 Tl! 1 S, (1)

Cl

]Tl

]t5 G~Te 2 Tl!, (2)

here Te and Tl are the electron and lattice temper-atures, respectively; Ce~Te! and Cl are the volumetricheat capacity for the electron and lattice, respective-ly; K~Te! is the electron thermal conductivity; G is thelectron-phonon coupling parameter that governs theate of energy transfer to the lattice from the hotlectrons; and S is the laser source term representingGaussian temporal pulse shape with constant op-

ical properties.4Measurement of the electron-phonon coupling pa-

rameter is typically accomplished through use ofthermoreflectance methods, where the sample’s re-flectivity is monitored after the incidence of an ultra-short laser pulse.5,6 The linear relationshipbetween reflectivity and temperature results in a re-flectance trace that can be compared with thermalrelaxation models. This method is employed in thisinvestigation to further study the dependence of theelectron-phonon coupling factor on crystallinity andgrain size and thickness for Au, Cr, and Al. Thethermoreflectance responses of corresponding bulkmaterials were examined to compare with the thin-film results. Furthermore, a theoretical formulationof the coupling factor’s dependence on scatteringmechanisms developed by Qiu and Tien7 is comparedwith experimental results and discussed.

2. Experimental Technique

A pump and probe experimental setup,5,6,8,9 shown inFig. 1, was employed to monitor the thermoreflec-tance response of each sample after the absorption ofan ultrashort laser pulse ~200 fs FWHM! of photonenergy at 1.55 eV ~l 5 790 nm!. The pulses from a76-MHz Ti:sapphire laser are separated into twobeams with an intensity ratio of 95:5 by a nonpolar-izing beam splitter. The intense pump beam ~;6nJypulse! is used to heat the film while the low-powerprobe beam is used to monitor the reflectivity. Thepump beam passes through an acousto-optic modu-

Fig. 1. Experimental setup. AOM, acousto-optic modulator.

lator that effectively chops the beam on and off at afrequency of 1 MHz. The pump beam is focused to adiameter of ;40 mm at an incident angle of 30° offnormal with an estimated fluence of ,10 Jym2.This creates a transient reflectance occurring at 1MHz that is detected using a lock-in amplifier. Theprobe beam passes through a dovetail prism mountedon a movable stage that is used to increase the opticalpath length of the probe beam and hence the timedelay between the pump and the probe pulses. Theprobe, which was centered in the heated area, is fo-cused to ;5 mm at near-normal incidence to minimizethe illuminated area. A half-wave retardation plateis used to rotate the pump beam’s polarization to beorthogonal with the probe polarization. A polarizerwith an extinction ratio of ;1026 is placed in front ofthe detector to filter out the scattered pump beam.A lock-in amplifier monitors the detector’s responseas the probe pulse is delayed in time, resulting in thetemporal relaxation profile of the sample.

3. Experimental Considerations

Equations ~1! and ~2! were solved numerically usingthe Crank–Nicolson scheme with a uniform gridspacing of 0.5 nm and insulated boundary conditions.The temperature dependence of the electron heat ca-pacity Ce~Te! and the thermal conductivity K~Te! aregiven by

Ce~Te! 5 gTe, (3a)

K~Te! 5 Keq

Te

Tl, (3b)

where g is the electron-specific heat constant10 andKeq is the thermal conductivity at equilibrium. Theassumption of an insulated boundary at the film andsubstrate interface is validated by considering onlythe data within the first 2 ps, where the substrateeffects have not yet affected the surface reflectivity.By monitoring only the center of the heated regionwith the probe, the radial diffusion in the film isneglected in the model. The one-dimensional as-sumption is valid in this investigation because theratio of the width of the heated region to the pene-tration depth is ;103. To extrapolate the electron-phonon coupling parameter G from the data, themodel is first multiplied by a constant that matchesthe magnitude of the thermoreflectance data to thePTS model and then the value of G is incrementedusing a curve-fitting routine written in LabView untila least-squares minimum is obtained. Further-more, the PTS model is averaged over a Gaussian of200 fs ~FWHM! to account for the shape of the prob-ing pulse’s effect on the phenomena measured.

It is well known that the thermal conductivityK~Te! of thin metallic films suffers degradation be-cause of size and structural effects. To overcome theneed for accurate values of the reduced thermal con-ductivity, the diffusion term in Eq. ~1! can be made

egligible by fabricating the films with thicknessespproximately two times the optical penetration

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3615

n

~tsl

aeets~e

Einss

e

fts

Seista

laTtagbt

cosstg;lttsims

3

depth ~10–40 nm!. This ensures that the film isvolumetrically heated from the front surface to thefilm and substrate interface, and thermal diffusionwithin the film can be ignored. Although for nearlyoptically thin samples the diffusion term can be ne-glected, this is not the case for the bulk materials.This is accounted for in this study by assuming thatthe bulk samples of Au, Cr, and Al possess thermaldiffusivities of published pure samples.

The assumption that pure ballistic transport of freeelectrons is negligible is based on the fact that thefree electrons almost instantaneously reach the backside of the film, and the dominant process of relax-ation is electron-phonon interaction. In bulk mate-rials at low temperatures this assumption is subjectto question; however, because of the high ambienttemperature of the experiment ~300 K!, the effect is

egligible.The dependence of the optical properties on bound

lattice! and free electrons plays a significant role inhe absorption and relaxation processes involved inhort-pulse laser heating. The resulting thermal re-axation trace can be a convolution of both Te and Tl,

which is described by Brorson et al.5 to be a linearrelationship:

DR 5 aDTe 1 bDTl, (4)

where the parameters a and b describe how much thefree and bound electrons affect reflectivity. It is wellknown that the reflectivity of metals is a function ofphoton energy, and contributions from free electronsthrough interband transitions and bound electronsthrough interband transitions create structure in thereflectance spectra.11,12 This indicates that a and bre functions of photon energy and the complex di-lectric function. If the pump beam is of a photonnergy that excites both bound and free electrons,hen absorption will occur in both systems. Themall free electron to lattice heat capacity ratio;0.01! will ensure nonequilibrium heating; how-ver, the amount of nonequilibrium, i.e., Te 2 Tl, will

be affected. Determining the parameters a and b inq. ~4! is the focus of a planned study by the authors

n which wavelengths ranging from the UV to theear-IR will be used to compare theoretical expres-ions with experimental data. For purposes of mea-uring the parameter G, this obstacle can be

overcome in two ways. The ratio ayb can be esti-mated by comparing the peak with the thermaliza-tion point after the fast free-electron transient hasdecayed away, or by considering only the data duringthe fast transient for curve-fitting purposes and com-paring with Te only. The second approach is validbecause, immediately following the laser absorption,the fast transient is largely due to free-electron exci-tation and relaxation. The lattice temperature issmall during this period and does not play a role untila few picoseconds after the absorption. The secondapproach is used in this investigation, and the resultsagree well with previous studies5,6 where ayb wasstimated. However, Al shows no fast transient

616 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

ree-electron spike in the early picoseconds of excita-ion, and the result is that aluminum’s response tohort-pulse heating is similar to the Tl profile. This

aspect is discussed further in Section 5 where thealuminum data are presented.

After absorption of the laser pulse, a thermal ex-pansion results that generates acoustic waves ofultrasonic frequency; therefore the resulting reflec-tance that is due to the thermal relaxation is alsosuperimposed with the strain effects on reflectivity.9These strain effects can be significant depending onthe material’s thermal expansion, the photon energy,and the fluence of the heating beam. Furthermore,these effects can be seen in the first few picosecondsafter absorption and can affect the measured G value.

tudies were performed to characterize the strainffects for different fluence levels for Au and Cr for anncident photon of 1.55 eV. Gold shows little to notrain effects, even at higher fluences, but Cr showshat the strain effects are minimal at fluences aroundnd below ;9 Jym2. The strain generation’s depen-

dence on material properties and photon wavelengthindicates that this effect must be characterized foreach sample before comparing data with thermal re-laxation models. For example, ultrasonic effectswere seen in the Al samples at fluences below 9 Jym2,but were not present during the first few picoseconds,thus allowing the extrapolation of G.

4. Sample Preparation

The thin-film fabrication involved e-beam evapora-tion at 1027 Torr and sputtering at relatively highpressures ~1023 Torr! onto several different sub-strates. All substrates were scrubbed with ethanol,trichloroacetic acid, and methanol prior to O2 plasmacleaning and were placed immediately in the evapo-ration or sputtering chamber. The thin-film thick-nesses were determined using a profilometerpostdeposition and compared with the in situ oscil-ating crystal method, whereas grain size was char-cterized using an atomic force microscope ~AFM!.he AFM maps, shown in Figs. 2~a! and 2~b! two of

he Au films and Fig. 3 for the Al film, indicate thatll the films were polycrystalline. The averagerain diameter was found by dividing a specified areay the number of grains in that area and assuminghe grain’s cross section to be circular.

Two sets of Au films were deposited onto single-rystalline Si and amorphous silica glass using evap-ration and sputtering techniques. The evaporatedamples were deposited to a thickness of 30 ~65! nmimultaneously at a rate of 1 Åys. Figure 2~a! showshe topography of the evaporated Au on Si, whererowth of smaller grains with an average diameter of40 nm near the substrate surface and a layer of

arger columnar grains ~;100 nm! sparsely grown onop form two distinct stratified layers. An examina-ion of the surface in several spots showed that theseurface characteristics were constant in the region ofnterest, i.e., in the region of the optical measure-

ent. The Au on the glass sample showed the sametratified layers and grain sizes, but had a larger

fwc

number density for the large columnar grains grownnear the free surface.

The sputtered samples were deposited simulta-neously at a rate of 100 Åymin onto the same sub-strate materials as the evaporated samples to athickness of 38 ~65! nm, as measured by a profilome-ter postdeposition. Figure 2~b! shows the AFM re-sults for the sputtered Au on the glass sample. Bothof the sputtered films showed no stratified effect, andthe average grain diameter ~;25–35 nm! was smallerthan the evaporated samples.

The Cr and Al films were deposited by evaporationonto three different substrates simultaneously:single-crystalline Si, amorphous silica glass, andthermally grown SiO2. The thermally grown SiO2was deposited on a single-crystalline Si wafer, andthe thickness ~634 nm! was measured using an inter-erometric technique13 before the layer of Cr and Alas deposited. Profilometry measurements indi-

ate that the Cr film thicknesses were ;51 ~65! nmwhereas the crystal method indicated a thickness of

Fig. 2. AFM contrast maps of ~a! evapor

Fig. 3. AFM contrast map of evaporated Al on SiO2.

30 nm; however, the profilometer indicates an Alsample thickness of 13 ~65! nm, which agrees wellwith the crystal measurement of 12 nm. The pro-filometry measurements were used in calculationsbecause of the frequent and large errors that vibrat-ing crystals may produce. Figure 3 shows the evap-orated Al on thermally grown SiO2 sample where theaverage grain size is ;70 nm. The average graindiameter and film thickness for each sample are re-ported in Table 1 in Section 5.

Thick samples of highly pure Au ~99.99%!, Cr~99.97%!, and Al ~99.999%! were investigated to com-pare the thin-film thermoreflectance responses withthose of the corresponding bulk materials. Theblocks of Au, Cr, and Al were mechanically polishedusing a high-speed buffing wheel and jeweler’s rougeand finishing polish until the surface had a near mir-ror finish. The samples were cleaned with acetoneand methanol after polishing.

5. Results

The resulting values for G and the correspondingthin-film parameters are summarized in Table 1.Five scans were taken for each sample, and the valuereported for G is the average and standard deviationof the extrapolated value. The fluence for each sam-ple was approximately 7.5 ~61! Jym2. The tracesare normalized at the peak to compare the shape ofthe trace instead of the magnitude.

The thermoreflectance traces for the evaporatedand bulk Au samples and the corresponding curve fitsare shown in Fig. 4. During the first 200 fs, theabsorption creates nonuniform electron temperaturedistributions, and the PTS is subject to question dur-ing this time.14 For this reason, the data used in thecurve-fitting routine was from the peak to ;1.5 psand was fit with only the Te profile. Although thethin-film traces are similar to the PTS model duringthis time, the traces for the bulk samples differ. Theoptically thin films are heated nearly uniformly dur-

Au on Si and ~b! sputtered Au on glass.

ated

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3617

EivTC

fc

Table 1. Measured Value of Thickness, Grain Size, and

3

ing the absorption, and this is believed to be thereason for the similar shapes; however, this is not thecase in the bulk materials, and the results show de-viations from the model during the absorption. Thesputtered sample’s response looks similar to theevaporated samples, so the data are not shown, butthe value of G extrapolated is listed in Table 1. In-spection of the traces indicates that the heating foreach sample is similar; however, the traces differduring the relaxation. The evaporated AuySi has ahigher average G value of 4.4 ~60.2! 3 1016 Wym3 K,evident by the steeper slope present just past thepeak. This is believed to be due to the smaller av-erage grain size ~D ; 35 nm! of the sample as com-pared with the evaporated Auyglass sample wherethe value of G extrapolated was 3.6 ~60.1! 3 1016

Wym3 K for a grain diameter of D ; 45 nm. Thesame relationship can be seen with the sputteredsamples where the Auyglass has a lower G value thanthe sputtered AuySi, indicating a dependence of scat-tering mechanisms on substrate as well as depositiontechnique.

The reflectance trace for the bulk Au sample showsthat the transient decay occurs on a shorter time

Electron-Phonon Coupling Factor

MaterialDepositionSubstrate

Thickness~nm!

d ~65!

Grain size~nm!

D ~610!Gfilm 3 1016

~Wym3 K!

Au Evaporated, glass 30 45 3.6 ~60.1!Evaporated, Si 30 37 4.4 ~60.2!Sputtered, glass 38 35 4.0 ~60.2!Sputtered, Si 38 27 5.0 ~60.2!Bulk — — 2.9 ~60.3!

Cr Evaporated, glass 51 45 13.1 ~61.4!Evaporated, Si 51 35 13.6 ~61.8!Evaporated, SiO2 51 55 10.0 ~60.4!

Al Evaporated, glass 13 55 22.5 ~61.1!Evaporated, Si 13 35 20.1 ~60.8!Evaporated, SiO2 13 70 22.7 ~60.4!Bulk — — 24.5 ~61.4!

Fig. 4. Thermoreflectance traces of evaporated thin film and bulkAu where G is given in units of 1016 Wym3 K.

618 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

scale than in the thin-film samples. This is expectedas the diffusion term in Eq. ~1! is no longer negligibleand plays a significant role in the coupling of energybetween the free electrons and the lattice by enlarg-ing the affected region. The extrapolated value of Gfor the bulk sample is 2.9 ~60.3! 3 1016 Wym3 K.

The thin-film values for Au agree well with valuesreported by Elsayed-Ali et al.6 and Orlande et al.15

where it was found that polycrystalline Au films hada G of ;4.0 3 1016 Wym3 K. The bulk measurementof 2.9 3 1016 Wym3 K agrees well with Brorson et al.,5but does not agree with the single-crystalline Auvalue reported by Elsayed-Ali et al. of 3.5 3 1016

Wym3 K. This discrepancy could be due to the largenumber of dislocation densities reported by Elsayed-Ali et al. in the single-crystalline samples. The bulksample in this investigation was a rolled sheet of Au,which typically possess large grains of the order of ;1mm in diameter, resulting in virtually no grain-boundary scattering.

Figure 5 shows the traces of the Cr samples onthree different substrates. The thickness of the Crfilms was found to be ;50 nm. Because the opticalpenetration depth is ;15 nm, there may be somediffusion occurring within the thin film. This wasaccounted for by including the diffusion term in Eq.~1! with a reduced value for conductivity that wascalculated from a measurement of the electrical re-sistivity for evaporated Cr films on quartz substrates.Fouad et al.16 reported a ratio of the thin-film resis-tivity to the bulk resistivity rfyrb of 1.15 for a 50-nmfilm. Assuming that the Wiedemann–Franz lawholds in the thin-film regime,17 the thermal conduc-tivity ratio KfyKb yielded a conductivity of 83 Wym K.

xtrapolating the G value for the reduced conductiv-ty yielded a value that was reduced by 4% of thealue using the bulk conductivity of 93.7 Wym K.he reduced value of conductivity was used in all ther thin-film analyses.The fast transient data considered in the curve fit

or the Cr samples was from the peak to ;0.7 ps andompared with Te only. The value of the coupling

parameter measured here for all the thin-film Crsamples is lower than the value reported by Brorsonet al.5 of 42 3 1016 Wym3 K. As can be seen in Fig.

Fig. 5. Thermoreflectance traces and corresponding curve fits forevaporated thin film and bulk Cr where values of G are given inunits of 1016 Wym3 K.

sT

pteptt

lm

a

twct

e

a

ap

w

fd

6, the extrapolated G is 3.65 3 1019 Wym3 K for thebulk sample, which indicates that the excitation andrelaxation are similar to the parabolic one-stepmodel, where it is assumed that the laser energy isinstantaneously absorbed by the lattice. This couldbe an artifact of the laser-pulse width, which may beconsiderably longer than the thermalization time forthis sample. This enhancement of electron-phononinteraction could be due to the mechanical polishingof the bulk sample that may have introduced a thinlayer of defects that significantly affected the initialabsorption and dissipation of the radiation energy.For this reason the G value extrapolated for the bulkample is subject to question and is not reported inable 1.The thermoreflectance traces for the four Al sam-

les are shown in Fig. 6, and, as mentioned above,here is no fast transient free-electron spike. How-ver, the resulting traces differ significantly from thearabolic one-step, where radiation energy is instan-aneously absorbed by the lattice. This indicateshat nonequilibrium heating is present.

There is a temporal lag of energy transfer from theaser pulse to the lattice, and the resulting traces are

ore similar to Tl than to Te. The Fermi energy ofAl ~11.7 eV! is considerably higher than Au and Cr,and this results in a small amount of Fermi smearinginduced by a photon of energy 1.55 eV, meaning thatthe reflectance is weakly dependent on Te. There islso an interband transition peak18 centered at the

photon energy resulting in the absorption of radiationby the lattice as well as the creation of free electronsor excitons during the heating time, which are be-lieved to create a loss during the excitation. In theAu and Cr analysis, G was measured by consideringhe energy loss from the free-electron system,hereas in the Al case, the value of G is measured by

onsidering the gain of energy into the lattice. Forhese reasons the data were fit with the Tl profile and

the value of G obtained as before. The data consid-red in this case are from 0 to ;2 ps so that substrate

effects could be neglected and the ultrasonic effectsare not yet significant. The three thin-film traces,

Fig. 6. Thermoreflectance traces and corresponding curve fits forevaporated thin film and bulk Al where values of G are given inunits of 1016 Wym3 K.

as well as the bulk sample shown, yielded similar

values for G, ;21 3 1016 Wym3 K, indicating that thedifferences in traces are largely due to thin-filmthickness rather than grain-boundary scattering.This is logical given the small film thickness ~13 nm!nd large grain diameter ~;70 nm!, indicating that

the film is most likely one grain thick.

6. Discussion

The electron-phonon coupling factor for metal filmsGf has been related to the mean free path of the freeelectron Lf through the consideration of the phononemission and absorption probabilities and is given bythe following expression7:

Gf 5916

nk2TD2nF

Lf Tl EF, (5)

where n is the number density of electrons, k is Bolt-zmann’s constant, TD is the Debye temperature, nF isthe Fermi velocity, and EF is the Fermi energy. Toevaluate the effects of free-electron scattering off offilm and grain boundaries, the mean free path of thefilm is related to the thin-film thickness d and theverage grain diameter D through the following ap-roximated expression7:

Lf 5Le,b

1 13Le,b

8d~1 2 P! 1

75 S Rg

1 2 RgD Le,b

D

, (6)

here Le,b is the electron mean free path during elec-trical conduction in bulk materials, P is the specularreflection parameter of electrons at film surfaces, andRg is the electron reflection coefficient at grain bound-aries. The value of P is taken to be zero because ofthe very small wavelength of the free electron ~;10Å! that results in diffusive scattering only at thin-filmboundaries. The value of Rg for Au is taken fromempirical studies and is 0.17.19 Using constantsrom Kittel,10 Eq. ~5! is plotted for Au in Fig. 7, where

5 33 nm and Le,b 5 40 nm. The data predict themagnitude and trend well, and it can be seen that fordecreasing average grain diameter the electron-phonon coupling factor increases. The measuredbulk value of G 5 2.9 3 1016 Wym3 K, not plotted,

Fig. 7. Electron-phonon coupling factor for Au versus averagegrain diameter.

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3619

fywp

f

2. K. S. Yngvesson, Microwave Semiconductor Devices, 1st ed.

3

indicates that the slope of Gf versus D flattens out asthe grain diameter increases as predicted by Eqs. ~5!and ~6!.

Because of the wide variability of published valuesof crucial parameters needed for Eqs. ~5! and ~6!, theprevious analysis is not possible for Cr at this time.However, the thermalization time tc or the character-istic time of energy exchange between the free elec-trons and lattice can be estimated through thefollowing simple formula20:

tc 5Ce

G, (7)

where Ce 5 1.4 3 104 Jym3 K.10 The average valueor the Cr measurements of G 5 12.2 3 1016 Wym3 Kields a thermalization time of 0.11 ps, which agreesell with the value reported by Qiu and Tien of 0.10s.20

The predicted G value of Al from Eqs. ~5! and ~6!yields a value of 69.9 3 1016 Wym3 K for a grain sizeD ; 60 nm and a film thickness of d 5 13 nm. Theempirical value of Rg 5 0.23 ~Ref. 19! and constantsrom Kittel10 were used in the calculation. The re-

sulting thermalization time for the Al samples is;0.19 ps, where Ce 5 4.1 3 105 Jym3 K.10

The higher values of G for Cr and Al as comparedwith Au confirm the traditional belief that the chro-mium congener family and the trivalent main groupmetals have stronger electron-lattice interactionthan the noble metals such as Au.

7. Conclusions

A thermoreflectance method was employed to inves-tigate the dependency of the coupling of thermal en-ergy from the free-electron system to the lattice inAu, Cr, and Al thin films on structural propertiessuch as thickness and grain diameter. The presenceof strain and its effect on the measurement wereaddressed. Nonequilibrium effects were seen in Alin which the resulting traces differ from the parabolicone-step model and behave similar to the lattice pro-file of the PTS diffusion model. The measured val-ues of the electron-phonon coupling factor werecompared with theory which accounts for size effectsand grain-boundary scattering during thermal con-duction, where the strength of coupling is seen toincrease for decreasing average grain diameter.

References1. P. Das, Lasers and Optical Engineering, 1st ed. ~Springer-

Verlag, New York, 1991!.

620 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

~Kluwer Academic, Norwell, Mass., 1991!.3. S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, “Electron

emission from metal surfaces exposed to ultrashort laser puls-es,” Sov. Phys. JETP 39, 375–377 ~1974!.

4. T. Q. Qiu and C. L. Tien, “Femtosecond laser heating of multi-layer metals. I. Analysis,” Int. J. Heat Mass Transfer 37,2789–2797 ~1994!.

5. S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K.Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus,“Femtosecond room-temperature measurement of theelectron-phonon coupling constant l in metallic superconduc-tors,” Phys. Rev. Lett. 64, 2172–2175 ~1990!.

6. H. E. Elsayed-Ali, T. Juhasz, G. O. Smith, and W. E. Bron,“Femtosecond thermoreflectivity of polycrystalline and single-crystalline gold films,” Phys. Rev. B 43, 4488–4491 ~1991!.

7. T. Q. Qiu and C. L. Tien, “Size effects on nonequilibrium laserheating of metal films,” J. Heat Transfer 115, 842–847 ~1993!.

8. C. A. Paddock and G. L. Eesley, “Transient thermoreflectancefrom thin metal films,” J. Appl. Phys. 60, 285–290 ~1986!.

9. J. L. Hostetler, A. N. Smith, and P. M. Norris, “Thin-filmthermal conductivity and thickness measurements using pico-second ultrasonics,” Micro. Thermophys. Eng. 1, 237–244~1997!.

10. C. Kittel, Introduction to Solid State Physics, 7th ed. ~Wiley,New York, 1996!.

11. B. R. Cooper and H. Ehrenreich, “Optical properties of noblemetals II,” Phys. Rev. 138, A494–A507 ~1965!.

12. H. Ehrenreich and H. R. Philip, “Optical properties of Ag andCu,” Phys. Rev. 28, 1622–1629 ~1962!.

13. J. L. Hostetler, D. Stewart, C. E. Daitch, C. S. Ashley, andP. M. Norris, “Optical polarized reflectance characterization ofthin aerogel and xerogel films,” J. Non-Cryst. Solids 225,19–23 ~1998!.

14. D. Y. Tzou, Macro- to Microscale Heat Transfer ~Taylor &Francis, Bristol, Pa., 1997!.

15. H. R. B. Orlande, M. N. Ozisik, and D. Y. Tzou, “Inverseanalysis for estimating the electron-phonon coupling factor inthin metal films,” J. Appl. Phys. 78, 1843–1849 ~1995!.

16. S. S. Fouad, A. H. Ammar, and M. H. El-Fazary, “A newapproach to the correlation of the electrical properties withinterband and intraband transitions of thin Cr films,” Phys.Status Solidi, 187, 99–108 ~1995!.

17. S. Kumar and G. C. Vradis, “Thermal conductivity of thinmetallic films,” J. Heat Transfer 116, 28–34 ~1994!.

18. H. Ehrenreich, “The optical properties of metals,” IEEE Spec-trum 2, 162–170 ~1965!.

19. J. W. C. De Vries, “Temperature and thickness dependence ofthe resistivity of thin polycrystalline aluminum, cobalt, nickel,palladium, silver and gold films,” Thin Solid Films 167, 25–32~1988!.

20. T. Q. Qiu and C. L. Tien, “Heat transfer mechanisms duringshort-pulse laser heating of metals,” J. Heat Transfer 115,835–841 ~1993!.