Thermal diffusivity and effusivity of thin layers using time-domain thermoreflectance

Jean-Luc Battaglia and Andrzej Kusiak*Ecole Nationale Supérieure d’Arts et Métiers, 33405 Talence Cedex, France

Clément Rossignol and Nicolay Chigarev†

Laboratoire de Mécanique Physique, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France�Received 2 April 2007; revised manuscript received 18 August 2007; published 12 November 2007�

Analytical expressions for the thermal diffusivity and effusivity of thin layers starting from the time-domainthermoreflectance are proposed. These relations rest on the analytical solution of one-dimensional heat transferin the medium using integral transforms. For noncapped layers, asymptotic behaviors of the impulse responselead to the analytical expression of the thermal diffusivity according to the optical properties of the medium. Incase of metals, the two-temperature model shows that the capacitance effect at the small times is essentiallygoverned by the electronic contribution. For capped layers with an aluminum film, an analytical expression ofthe thermal effusivity of the layer is derived. The particular influence of the heat penetration depth in thealuminum film during the thermalization between the electron gas and the lattice is demonstrated.

DOI: 10.1103/PhysRevB.76.184110 PACS number�s�: 65.40.�b

I. INTRODUCTION

The time-domain thermoreflectance �TDTR� is widelyused in the field of mechanical and thermal characterizationsof thin layers at the nano- and microscale. As for all thecontactless methods used in the thermal characterizationframework, the TDTR probes the transient average tempera-ture at the surface of the sample which is heated from aphotothermal source. Within this objective, a laser, called thepump, is used to deliver a pulse of very short duration at thesurface of the sample. The pulse generates acoustic wavesinside the medium that lead to deform locally the shape atthe surface. According to the optical properties of the me-dium, local heating occurs through classical scattering be-tween phonons, electrons, and defects. This allows the pres-ence of a transient temperature field inside the medium andat the surface. A second laser of weak magnitude, called theprobe, is focused on the heated area. The reflected beam ofthe probe depends on the deformed shape and it is thus re-lated indirectly to the average temperature on the aimedzone. The reflected intensity is measured along the time inorder to probe the thermal response to the pulse, which isalso called the impulse response. As it has been shown byRef. 1 in case of a metal, the measured reflectivity change atthe surface of the sample depends on the temperature rise forboth the electron gas and the lattice when time is of the orderof the pulse duration. After this time, the reflectivity changeis mainly due to the lattice cooling. During the pulse, theelectronic distribution rapidly changes, leaving the latticetemperature unchanged. During a few picoseconds a scatter-ing process between hot electrons and the lattice in order toretrieve an equilibrium distribution of the excess energy be-tween these two subsystems is observed. It is called the ther-malization process. This phenomenon has been extensivelystudied during the past years and a model has been proposedby Anisimov et al.2 starting from thermophysical propertiesof the electron gas and the lattice: the two-temperaturemodel.3–6

The thermal properties of the layer are estimated startingfrom the heat diffusion equation with source term, also called

the one-temperature model, which is available as from themoment when the thermalization process stops. First resultshave been obtained by Paddock and Eesley7 in order to esti-mate the thermal diffusivity of thin metal films and someother works have followed, thanks to significant improve-ments of the experimental setup.8–10 Thermal conductivitymeasurements11–14 on thin films have also been reported. Themain difficulty encountered in such an experiment, with re-gard to the estimation of the thermal properties, is that itdoes not lead to easily measure the absorbed energy and theaverage temperature on the heated area. Nevertheless, themodulated thermoreflectance approach15–19 can be then effi-ciently implemented to partially overcome this problem.

The source term, which occurs in the one-temperaturemodel, depends on the optical and electronic properties ofthe medium. For insulating and semiconducting materials,the electronic contribution for heat transfer in the mediumcan be neglected in front of that of phonons �lattice�. In sucha case, the heat penetration depth is equal to the optical oneof the pump. Unfortunately, the main part of such materialsare very bad candidates with respect to the thermoreflectancephenomenon. In practice, it is then recommended to deposita metal film, such as aluminum, on the studied layer. It thenraises the question to know what will be the thickness of themetal film which will take part in the absorption of thepump. In the literature, it is observed that this thickness istaken sufficiently small to be regarded as thermally thin. Thecapacitive heating effect observed at the small times after thepump pulse is then related to the film thickness.11,12 Giventhat the optical absorption depth depends on deposition con-ditions during the growth of the Al film, a more rigorousstudy has been realized by Capinski and Maris8 by measur-ing the dependence of the GaAs thermal conductivity forsamples with different Al thicknesses. The expected value forthe GaAs conductivity leads to discriminate the appropriateAl film thickness for future investigations.

In this paper, we demonstrate how the heat absorptiondepth in a metal, which occurs in the source term for theone-temperature model, is related to the optical absorptiondepth, which occurs in the source term for the two-

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temperature model. We propose to derive an analytical ex-pression for the thermal diffusivity and effusivity of the layeraccording to the sample configuration: capped or noncappedwith an aluminum film. In Sec. III, the influence of the op-tical and electronic properties of a noncapped layer on theheat penetration depth during the thermalization stage be-tween the electron gas and the lattice is investigated. In thisanalysis the layer has the behavior of a semi-infinite me-dium. The analysis is carried out starting from the two-temperature model that leads to a reliable expression of theheat source term in the one-temperature model. Consideringthe asymptotic behaviors for the impulse response, an ana-lytical expression of the thermal diffusivity for the layer isproposed. Applications to aluminum and to the Sb2Te3 semi-conducting material are considered. In Sec. IV, the configu-ration of a capped layer with an aluminum film is investi-gated. Using the asymptotic behaviors of the impulseresponse an analytical expression for the thermal effusivityof the layer is obtained. The heat penetration depth reachedduring the thermalization process is measured and comparedto that calculated from the two-temperature model in Sec. IIIwithin the aluminum sample. This calibration is performedon a SiO2 layer capped with an aluminum film. An applica-tion on a Sb2Te3 layer capped with an aluminum film is thenpresented.

II. EXPERIMENTAL SETUP

In our experimental setup, represented in Fig. 1, the pico-second thermoreflectance technique is based on a time re-solved pump-probe setup using ultrashort laser pulses�795 nm, �l=100 fs, I0=5 nJ� generated by a Ti:sapphirelaser.10 The time profile of the laser pulse is f�t�=1/�l cosh�1.76t /�l�2. Radiation of pump is doubled by abeta-BaB2O4 �BBO� nonlinear optical crystal. The probepulse is delayed according to the pump pulse up to 7 ns witha temporal precision of a few tens of femtoseconds by meansof a variable optical path. The pump beam, for which theoptical path length remains constant during the experiment,is modulated at a given frequency of 0.3 MHz by an acousto-optic modulator �AOM�. To increase the signal-to-noise ra-tio, a lock-in amplifier synchronized with the modulation fre-quency is used. Probe and pump beams have a Gaussian

profile and are focused at the surface of the sample within aspot of, respectively, 2rp and 2rh in diameter at normal inci-dence by means of objectives. In this study, the size of thepump-probe correlation is 2 �m. The intensity of the re-flected probe beam is measured using a silicon photodiode�PD�.

III. SEMI-INFINITE MEDIUM

A. Case of metals

The TDTR has been performed on a 1 �m thick alumi-num layer on a Mylar substrate. Thermophysical propertiesof aluminum in the bulk are denoted k, �, and cp, for thethermal conductivity, the density, and the specific heat, re-spectively �Cp=�cp is the specific heat per unit volume�. Themeasured time dependent normalized reflectivity is repre-sented in Fig. 2. The transmittance of the medium is T�

=0.07 and 1/�h= �

4���=6.56 nm is the optical penetration

depth of the pump beam in the medium, where ��=4.85denotes the extinction coefficient of the material at the pumpwavelength:20 �=400 nm. The two-temperature model de-scribes the time dependent electron and lattice temperature,Te and Tl, respectively, during the thermalization process as

Ce�Te��Te

�t= �r,z · �ke�Te,Tl��r,zTe� − G�Te − Tl� + S ,

Cl�Tl

�t= G�Te − Tl� . �1�

In these equations Ce and Cl=Cp are the electronic andlattice specific heat per unit volume and ke=k is the elec-tronic thermal conductivity that can be assimilated to the

FIG. 1. �Color online� Schematic representation of the femto-second thermoreflectance experimental setup �PD: photodiode,AOM: acousto-optic modulator, BBO: beta-BaB2O4 is a nonlinearoptical crystal�.

� � �

�

� � � � � � � � � �

� � � � � � � � �

���

� ��������

� � � � � � �

� � � � � � �� � � � � � � �

��� � � � � � �

�

� � � �

� � �

� � � �

� � �

� � � �

� � �

� � � �

� �

� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �� � � �

FIG. 2. Measured TDTR signal for a 1 �m thick Al layer on aMylar substrate, with expected slope −1/2 at long times. Impulseresponse at long times is also represented according to 1/�t. Thelinear fit is represented with the regression coefficient.

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thermal conductivity of the bulk for metals. These two non-linear equations are coupled through the electron-phononcoupling constant G that has been clearly defined in Ref. 23starting from the Boltzmann transport equation for both theelectrons and the phonons. Principal ideas concerning thedefinition of this last parameter can be found in Refs. 24–26.The first relation in Eq. �1� means that, after the pulse, hotelectrons will move inside the medium while losing theirenergy to the lattice. Let us insist on the fact that this modelhas a physical meaning only during the thermalization pro-cess given that the second relation in Eq. �1� shows that thelattice temperature remains constant when Te=Tl or, in otherwords, when the thermalization process ends.

The heat source in relation �1� is

S =T��h

�rh2 e−�hze−�r/rh�2

I0f�t� . �2�

The finite element method is used in order to solvethe two-temperature model starting from parameters givenin the literature for aluminum4 �G=4.91017 W m−3 K−1,ke=k0Te /Tl with k0=160 W m−1 K−1, Cl=2.443106 J kg−1 K−1, Ce=Te, where =92 J K−2 m−3 is theelectronic specific heat coefficient�. It must be noted that thissimulation remains coherent with the definition of tempera-ture for the electron gas and the lattice since the mean freepath for the electrons in aluminum is approximately 10 nm.27

In other words, the minimum distance between two nodes ofthe mesh is at least more than this critical length. The timedependent temperature for the lattice and the electron gas isreported in Fig. 3. The electron gas temperature increasesvery quickly and reaches its maximum at 50 fs. The tempera-ture of the lattice begins to increase at 20 fs and reaches theelectron gas temperature at �t=200 fs. Then, as also reportedby Ref. 21, the calculation shows the undercooling of the

electrons relative to the lattice at the surface. This undercool-ing comes from the high value of the coupling factor Gfor aluminum; it is also observed for gold1 or copper22

whose coupling factors are smaller �2.31016 and 101016 W m−3 K−1, respectively� but it is less pronouncedthan for aluminum. As viewed on the figure, the completethermalization between the electron gas and the lattice isreached at time �c between 25 and 30 ps. Considering thediffusion coefficient for the electron gas, that is, ae=ke /Ce,and the time �t where the two temperatures cross, it is thuspossible to obtain an evaluation of the heat penetration depthin the medium as 1/ �̃h=�ae�t�34 nm. As expected, the heatpenetration depth reached during thermalization is greaterthan the optical penetration depth 1/�h. As the thermaliza-tion stage ends, the TDTR is only sensitive to the latticecooling and it is then appropriate to use the classical one-temperature model in order to describe heat diffusion insidethe medium. Let us note, however, that the two-temperaturemodel does not degenerate naturally toward the one-temperature model when t��c. As one can see in relation�1�, the second equality leads to dTl /dt=0 when Te=Tl=Tand it is thus necessary to replace Ce by Cp=Ce+Cl and keby k=k0 in the first equality in order to retrieve the one-temperature model as

Cp�T

�t= k

�2T

�z2 + �e−�̃hze−�r/rh�2 �t� . �3�

The time profile f�t� of the pulse is replaced by the Diracfunction �t� in the source term in order to retrieve the sametemperature profile that is calculated from the two-temperature model at the end of the thermalization stage. Themain change, regarding to the expression of the heat sourcein the one-temperature model, is that the heat absorptiondepth 1/ �̃h accounts for heat transport by hot electrons dur-ing the thermalization process. In this relation the only hy-pothesis is that we assume the same shape for the heat ab-sorption depth at the end of the thermalization process thanthat of the optical penetration depth in relation �2�, i.e.,� exp�−�̃hz�, where � is an unknown parameter. This hypoth-esis is validated by comparing the temperature profile alongz at r=0 calculated from the two-temperature model at theend of the thermalization with the function � exp�−�̃hz�where � has been estimated in order to adjust the magnitude.The result is presented in Fig. 4 and it clearly appears thatthe gap between the two curves is weak. Measuring the pa-rameter � is not required since, as we will see it in the fol-lowing, all our matter rests on the ratio of the asymptoticbehaviors of the impulse response. Using the Laplacetransform28 on relation �3�, it is found that the temperature ateach point in the medium is expressed in the frequency do-main ��=2�f being the angular frequency� as

T�r,z, j�� =�

k��̃h2 − 2�

� �̃h

e−z − e−�̃hz�e−�r/rh�2

, =� j�

a.

�4�

In this relation a=k /Cp denotes the thermal diffusivity. It isassumed that the measured change of reflectivity at the

FIG. 3. Two-temperature model simulation for the aluminumsample using the finite element method with I0=5 pJ, �L=100 fs,and T�=0.07. Line with circles represents the lattice temperature,and plain line is the electron gas temperature. Lagrangian quadraticelements have been used and the linear system is solved startingfrom the conjugate gradient method.

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heated surface is linearly related to the average temperatureon the aimed surface. From relation �4� the following ana-lytical expression of the averaged temperature is obtained:

T0�j�� =2

rp2�

0

rp

T�r,z = 0, j��re−�r/rh�2dr

=�

k��̃h2 − 2�

� �̃h

− 1��rh,rp

. �5�

The constant �rh,rp= �rh

2 /rp2��1−exp�−rh

2 /rh2�� is only de-

pendent on the radii of the two laser beams. From relation�5� the following asymptotic behavior of the impulse re-sponse at the small times is obtained:

T0t0→

�rh,rp�

Cp, t → 0�� → �� . �6�

This relation means that, after the pulse, the temperaturein the medium remains constant during a time that is propor-tional to the heat penetration depth in the medium, 1/ �̃h, andon the specific heat per unit volume of the material. Further-more, the following asymptotic semi-infinite behavior at thelong times is obtained:

T0t�→

�rh,rp�̃h�

�kCp��t

, t → ��� → 0� . �7�

According to relations �6� and �7�, the normalized TDTRsignal is expressed as the ratio of the temperature at the longtime and the maximum temperature, which is equal to thetemperature at the small times, as

TDTRt�=

T0t�

T0t0

=1

�̃h�a��

1�t

= �1�t

. �8�

The following analytical expression of the thermal diffusiv-ity is obtained:

a =1

�2�̃h2�

. �9�

According to the thermal behavior at long times, given byrelation �7�, the slope −1/2 fits the TDTR signal in Fig. 2. Itleads us to measure the time constant of the capacitive effectat short times to be 28 ps. It is therefore in good agreementwith the thermalization time �c previously calculated fromthe two-temperature model. The data at the long time arealso represented according to 1/�t in Fig. 2. It lead us tomeasure the linear regression coefficient as �=4.410−6.Using the calculated value for 1 / �̃h as well as the measuredvalue of �, the thermal diffusivity 1.910−5 m2 s−1 is ob-tained. This value is 3.5 times less than the expected thermaldiffusivity for the bulk �6.5810−5 m2 s−1�.29 Even if thereal value of 1 / �̃h must differ slightly from the calculatedone, that does not call basically in question the estimate ofthe thermal diffusivity. As it is known, the properties of alu-minum films are strongly dependent on the structure of thelayer �function of the deposition parameters� and of oxida-tion on the surface. So, one must expect that the thermalproperties of the film, in particular, its thermal conductivity,differ from those of the bulk.

B. Case of insulating and semiconducting materials

The same experiment has been performed on a 400 nmthick Sb2Te3 semiconducting film deposited on a Si sub-strate. A reliable experimental TDTR signal is obtainedwithin this material, which is generally not the case withinthe framework of semiconductors. For semiconducting ma-terials the heat transport by electrons is completely negli-gible in front of the lattice �phonons� contribution. Thus, theheat absorption depth at the small time is equal to the opticalpenetration depth for the pump. The measured normalizedTDTR response is reported in Fig. 5. The semi-infinite be-havior is clearly highlighted with the slope −1/2 in the log-log representation. Representing the data at the long time inthe same figure according to 1/�t leads us to measure thelinear regression coefficient as �=110−5. Then, applyingrelation �9�, with 1/ �̃h=1/�h=12 nm, we obtain a=1/����h�2=4.5810−7 m2 s−1, which is very close to theexpected value �0.6/6440/194=4.810−7 m2 s−1� given byseveral references.29–32

IV. ALUMINUM CAP FILM ON ASEMI-INFINITE MEDIUM

An aluminum film �denoted Al, with CpAl=900

2700 J kg−1 m−3� is generally deposited on the layer �de-noted L� that will be characterized in order to increase thesignal-to-noise ratio during the TDTR.11,13,14 As demon-strated experimentally by Ref. 14, the estimation of the layer

FIG. 4. Temperature profile along z at r=0 calculated from thetwo-temperature model using the finite element method �plain line�and function � exp�−�̃hz� �line with circles� that expresses the heatsource term vs z for the one-temperature model.

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thermal conductivity mainly rests on the aluminum filmthickness. It comes from the fact that for these authors theentire aluminum film is considered as thermally thin. Resultsobtained in the previous section show that it is the case onlyif the film thickness is equal to the heat penetration depth1/ �̃h. Nevertheless, this condition is not required since thethermal properties can be identified considering only theasymptotic behaviors of the impulse response. Assumingone-dimensional heat transfer, the temperature in the layer issolution of

k�2TL�t,z�

�z2 = Cp�TL�t,z�

�twhen z � 0 for t � 0. �10�

Assuming no heat loss with the ambient, the heat flux inthe layer is equal to the rate at which the aluminum filmlosses energy:

k�TL�t,z�

�z=

CpAl

�̃h

�TAl�t��t

at z = 0, for t � 0. �11�

The following thermal resistance RK, also called theKapitza resistance, at the film-layer interface is also consid-ered:

k�TL�t,z�

�z=

TF�t,z� − TAl�t�RK

at z = 0, for t � 0. �12�

It must be noticed that for an insulating layer, the Kapitzaresistance can be neglected given that the thermal resistancefor the layer �which is defined as the ratio of the thermalconductivity k and the heat penetration depth inside the layerduring the experiment� is lower, and relation �12� becomes

TL=TAl. In that case, as demonstrated in Ref. 11, the appli-cation of Laplace transform on relations �10� and �11� leadsto the expression of the normalized temperature of the alu-minum film, which corresponds to the normalized TDTR sig-nal, as

TDTR =TAl�t�TAl�0�

= e�2t erfc���t� . �13�

In this relation �=E�̃h /CpAl, where E=�kCp is the effu-

sivity of the layer. The following analytical expression of thethermal effusivity is obtained:

E =�CpAl

�̃h

. �14�

Relation �13� remains completely consistent with theasymptotic behaviors at the small times �capacitive effect�and the long times �semi-infinite behavior� whatever the con-dition at the film-layer interface. It is thus judicious to esti-mate � by privileging the TDTR measurements carried out inthese two temporal fields. As seen in Eq. �14�, the effusivityof the layer mainly depends on the 1/ �̃h value. In order tohave a more accurate value than that found numerically inthe previous section, an experiment is carried out consideringa SiO2 layer whose thermal properties have been measuredby a different technique. The thermal conductivity was mea-sured by the 3� technique �1.4 W m−1 K−1� and we assumedthe values of the density and the specific heat of the bulk��=2200 kg m−3 and cp=744 J kg−1 K−1�. A 55 nm thick alu-minum film is deposited on the SiO2 layer. The measuredimpulse response and the simulated one with the optimalvalue of �=21 470 are reported in Fig. 6. From relation �14�we found 1/ �̃h=29 nm which is not so far away from thevalue only we had obtained numerically from the two-temperature model and also from the value obtained experi-mentally for the aluminum layer. It is also remarked that the

� � �

�

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� � � � � � � � �

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� � � � � � �

� � � � � � � � � � � � �

��� � � � � � �

� � �

� � �

� � �

� � �

� �

� �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � �� � � �

���

� ��������

FIG. 5. Normalized TDTR experimental response �line� for aSb2Te3 layer �400 nm�. The slope −1/2 �dotted line� appearsclearly at long times. The slope crosses the x coordinates at the time�t which is the thermalization time between the electron gas and thelattice.

FIG. 6. Normalized TDTR response �line� for a SiO2 layer cov-ered with a 50 nm thick aluminum layer. The simulated response�dotted line� is obtained with the optimized value of �=21 470. Itleads to the heat penetration depth in the aluminum film: 29 nm.The simulation of the complete model by the finite element methodis represented by black filled circles. It shows the heat diffusion inthe part of the aluminum film that is not affected during the ther-malization process.

THERMAL DIFFUSIVITY AND EFFUSIVITY OF THIN… PHYSICAL REVIEW B 76, 184110 �2007�

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impulse response decreases quickly at the end of the thermal-ization process, which suggests that there is still a tempera-ture gradient in the aluminum film. This result was expectedsince the thickness of aluminum is larger than the heat pen-etration depth during the thermalization stage. As repre-sented in Fig. 6, this fact is rigorously demonstrated from thecalculation of the response from the finite element method.However, relation �13� is well adapted for an accurate evalu-ation of � that leads to the effusivity of the layer with rela-tion �14�.

In a second application a 55 nm thick aluminum film isdeposited on a 400 nm thick Sb2Te3 layer. The measurednormalized TDTR response is reported in Fig. 7. As for theprevious experiment, the fast cooling after the thermalizationwhich is significant to heat diffusion in the aluminum film isviewed. The optimal value of � that allows the best fittingbetween the measurements and relation �13� is 12 052. Usingthe previous measured value for 1 / �̃h and relation �14�, itleads us to the following thermal effusivity for the Sb2Te3layer: ESbTe=849 W m−2 K−1 s−1/2 whereas the expectedvalue is �0.66440194=866 W m−2 K−1 s−1/2, as re-ported in several references,29–32 which means less than 5%relative error.

V. CONCLUSION

The thermal properties of a thin layer are identified fromthe measured impulse response obtained during a TDTR ex-periment as well as from the classical one-temperature modelwith source term. We demonstrate how to express accuratelythe source term by considering the simulation of the two-temperature model that describes the thermalization processbetween electrons and the lattice during and just after thepulse. Analytical expressions of the thermal properties of athin layer within the configuration of capped or noncappedwith aluminum film have been proposed. They rest on theasymptotic behaviors of the impulse response calculatedfrom the one-temperature model and therefore they do notrequire measuring the absolute average temperature rise aswell as the absorbed energy. It was demonstrated that theheat penetration depth reached during the thermalization pro-cess between electrons and the lattice in metals is a funda-mental parameter with respect to the accuracy of the thermalproperty estimation. Indeed, this parameter occurs in the ex-pression of the thermal diffusivity for noncapped layers andalso in that of the thermal effusivity for capped layers withan aluminum film. As shown, the thermalization time can bemeasured accurately from the TDTR response by represent-ing it in a log-log scale and considering the asymptotic be-havior at long times that is characterized by a slope −1/2.For capped layer, as the semiconducting materials that do nothave generally a good thermoreflectance coefficient, the ther-mal effusivity of the layer with a good accuracy from rela-tion �13� is obtained. The heat penetration depth in the alu-minum film has been measured accurately considering alayer whose thermal properties are known �SiO2�. The mea-sured depth is in good agreement with that obtained from thetwo-temperature model and from that obtained experimen-tally on the aluminum sample. In case of noncapped semi-conducting materials, as the Sb2Te3 sample, the heat andoptical penetration depths are equal, which makes the ana-lytical solution easily exploitable of the moment when oneprecisely knows the optical extinction coefficient of the ma-terial. In such a case, the thermal diffusivity of the layer isreached with a good accuracy from relation �8�.

*jean-luc.battaglia@bordeaux.ensam.fr; URL: www.trefle.u-bordeaux1.fr

†c.rossignol@lmp.u-bordeaux1.fr1 S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K.

Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus,Phys. Rev. Lett. 64, 2172 �1990�.

2 S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel’man, Zh. Eksp.Teor. Fiz. 66, 776 �1974� �Sov. Phys. JETP 39, 375 �1975��.

3 R. W. Schoenlein, W. Z. Lin, J. G. Fujimoto, and G. L. Eesley,Phys. Rev. Lett. 58, 1680 �1987�.

4 G. Tas and H. J. Maris, Phys. Rev. B 49, 15046 �1994�.5 L. Jiang and H.-L. Tsai, J. Heat Transfer 127, 1167 �2005�.6 E. Carpene, Phys. Rev. B 74, 024301 �2006�.

7 C. A. Paddock and G. L. Eesley, J. Appl. Phys. 60, 285 �1986�.8 W. S. Capinski and H. J. Maris, Rev. Sci. Instrum. 67, 2720

�1996�.9 B. Bonello, B. Perrin, and C. Rossignol, J. Appl. Phys. 83, 3081

�1998�.10 C. Rossignol, J. M. Rampnoux, T. Dehoux, S. Dilhaire, and B.

Audoin, Rev. Sci. Instrum. 77, 033101 �2006�.11 B. C. Daly, H. J. Maris, W. K. Ford, G. A. Antonelli, L. Wong,

and E. Andideh, J. Appl. Phys. 92, 6005 �2002�.12 B. C. Daly, H. J. Maris, A. V. Nurmikko, M. Kuball, and J. Han,

J. Appl. Phys. 92, 3820 �2002�.13 C. J. Morath, H. J. Maris, J. J. Cuomo, D. L. Pappas, A. Grill, V.

V. Patel, J. P. Doyle, and K. L. Saenger, J. Appl. Phys. 76, 2636

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FIG. 7. Normalized TDTR response �line� for a Sb2Te3 layercovered with 50 nm thick aluminum. The simulated response �dot-ted line� is obtained with the optimized value of �=12 052.

BATTAGLIA et al. PHYSICAL REVIEW B 76, 184110 �2007�

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�1994�.14 W. S. Capinski, H. J. Maris, T. Ruf, M. Cardona, K. Ploog, and D.

S. Katzer, Phys. Rev. B 59, 8105 �1999�.15 B. Li, L. Pottier, J. P. Roger, and D. Fournier, Thin Solid Films

352, 91 �1999�.16 R. M. Costescu, M. A. Wall, and D. G. Cahill, Phys. Rev. B 67,

054302 �2003�.17 S. Dilhaire, S. Grauby, W. Claeys, and J.-C. Batsale, Microelec-

tron. J. 35, 811 �2004�.18 S. Huxtable, D. G. Cahill, V. Fauconnier, J. O. White, and J.-C.

Zhao, Nat. Mater. 3, 298 �2004�.19 D. G. Cahill, Rev. Sci. Instrum. 75, 5119 �2004�.20 Handbook of Optical Constants of Solids, edited by Edward D.

Palik �Academic, New York, 1985�.21 G. L. Eesley, Phys. Rev. B 33, 2144 �1986�.22 H. E. Elsayed-Ali, T. B. Norris, M. A. Pessot, and G. A. Mourou,

Phys. Rev. Lett. 58, 1212 �1987�.23 P. B. Allen, Phys. Rev. Lett. 59, 1460 �1987�, G=3��2 /�kBT,

with ��2=112 meV2 for W and 304 meV2 for Al.24 J. G. Fujimoto, J. M. Liu, E. P. Ippen, and N. Bloembergen, Phys.

Rev. Lett. 53, 1837 �1984�.25 N. Del Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christo-

filos, and F. Vallée, Phys. Rev. B 61, 16956 �2000�.26 B. Rethfeld, A. Kaiser, M. Vicanek, and G. Simon, Phys. Rev. B

65, 214303 �2002�.27 With respect to the quantum definition, temperature is defined

over the mean free path of the heat carriers, which are thephonons and the electrons. If two regions of space have differentdistributions of electrons and phonons, then they have differenttemperatures. Each distribution is defined by a characteristic fre-quency �p�q� where q is the wave vector and p the polarization.In that case it is then possible to introduce the notion of tem-perature gradient and therefore of thermal conductivity in thesense of Fourier’s law. Using the Sommerfeld theory, the meanfree path of electrons in a metal can be approximated by l= �rs /a0�292 Å/��, with rs /a0=2.07 and ��=3.55 �� cm.

28 J.-L. Battaglia, A. Kusiak, and J.-C. Batsale, J. Heat Transfer129, 756 �2007�.

29 CRC Handbook of Chemistry and Physics, 87th ed. �Taylor &Francis, London, 2006�.

30 Y. Kim, A. DiVenere, G. K. L. Wong, J. B. Ketterson, S. Cho, andJ. R. Meyer, J. Appl. Phys. 91, 715 �2002�.

31 V. Giraud, J. Clusel, V. Sousa, A. Jacquot, A. Dauscher, B. Le-noir, H. Sherrer, and S. Romer, J. Appl. Phys. 98, 013520�2005�.

32 H.-K. Lyeo, D. G. Cahill, B.-S. Lee, J. R. Abelson, M.-H. Know,K.-B. Kim, S. G. Bishop, and B.-K. Cheong, Appl. Phys. Lett.89, 151904 �2006�.

THERMAL DIFFUSIVITY AND EFFUSIVITY OF THIN… PHYSICAL REVIEW B 76, 184110 �2007�

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