Effect of temperature gradient on simultaneously experimental determination ofthermal expansion coefficients and elastic modulus of thin film materialsTei-Chen Chen, Wen-Jong Lin, and Dao-Long Chen Citation: Journal of Applied Physics 96, 3800 (2004); doi: 10.1063/1.1789629 View online: http://dx.doi.org/10.1063/1.1789629 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/96/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Bidirectional thermal expansion measurement for evaluating Poisson’s ratio of thin films Appl. Phys. Lett. 89, 031913 (2006); 10.1063/1.2227524 A simple approach to determine five thermomechanical properties of thin ductile films on an elastic substrate Appl. Phys. Lett. 85, 6173 (2004); 10.1063/1.1840125 On-wafer characterization of thermomechanical properties of dielectric thin films by a bending beam technique J. Appl. Phys. 88, 3029 (2000); 10.1063/1.1287771 Simultaneous measurement of Young’s modulus, Poisson ratio, and coefficient of thermal expansion of thin filmson substrates J. Appl. Phys. 87, 1575 (2000); 10.1063/1.372054 Measurement of elastic modulus, Poisson ratio, and coefficient of thermal expansion of on-wafer submicron films J. Appl. Phys. 85, 6421 (1999); 10.1063/1.370146

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Effect of temperature gradient on simultaneously experimentaldetermination of thermal expansion coefficientsand elastic modulus of thin film materials

Tei-Chen Chen,a) Wen-Jong Lin, and Dao-Long ChenDepartment of Mechanical Engineering, National Cheng Kung University, Tainan 701,Taiwan, Republic of China

(Received 29 March 2004; accepted 14 July 2004)

Some specific experimental methods to simultaneously determine the thermal expansion coefficientsaF and biaxial elastic modulusEF / s1−nFd of thin film materials have been reported recently. Inthese methods, the deflections or the curvature change of the thin films, deposited on two differenttypes of circular disks with known material properties, generally can be measured with a variety ofoptical techniques. The temperature-dependent deflection behaviors of thin films are then obtainedby heating the samples in the range from room temperature to a slightly higher temperature level atwhich the physical properties and microstructures of thin film materials still remain unchanged. Byusing the relations between stress, deflection, and temperature, the physical properties of thin filmscan be finally calculated by using the slopes of two lines in the stress versus temperature plot. Theserelations, however, are formulated under the condition of uniform temperature rise. If the heatingprocesses of samples are conducted in the condition that there exists a small steady-statetemperature gradient along the thickness of samples due to the effect of natural heat convection onthe upper surface of thin film, the formulation mentioned above shall be modified. It is found thatthe deflection of sample induced by the small temperature gradient along the thickness due to naturalheat convection is very significant and comparable to that induced by uniform temperature rise.Consequently, if the effect of this temperature gradient is carelessly disregarded in physicalmodeling, a significantly different value of elastic modulus may be misleadingly obtained. Somecases are exemplified and illustrated to show the influence of temperature gradient on the evaluationof material properties. ©2004 American Institute of Physics. [DOI: 10.1063/1.1789629]

I. INTRODUCTION

Thin film materials have wide applications in a varietyof electronic devices and hard coating technology of ma-chine parts such as antireflection coatings, opticalwaveguides, metal oxide semiconductor devices, insulatorsin electronic devices, and narrow-bandpass filters. It is wellknown that the physical properties of thin film material arequite different from bulk material.1–3 The discrepancy be-tween them may be as high as one order in magnitude. Con-sequently, an accurate determination of mechanical proper-ties of thin film materials, such as thermal expansioncoefficient and elastic modulus, becomes extremely impor-tant and has attracted extensive attention in recent years.However, only few results have been reported.

The total residual stress on a thin film material is gener-ally composed of intrinsic stress and thermal stress. Theformer is generated during film growth due to relativelycomplicated microscopic mechanisms, such as the differentspacing of atoms in a growing film, the incorporation ofexcess vacancies, the presence of impurities, and bombard-ment by energetic particles, while the latter results from thedifference in the thermal expansion coefficients between ad-jacent layers. Residual stresses induced in the thin film ma-

terial significantly influence not only the mechanical perfor-mance of coatings such as spallation resistance, thermalcycling life, and fatigue properties but also the optical, elec-trical, and magnetic behaviors of layer devices due to thecracking, interfacial delamination, and the change of physi-cal properties due to their stress and deformation dependencein nature. A reliable evaluation of intrinsic and thermalstresses is dependent on the accurate values of thermal ex-pansion coefficient and Young’s modulus of thin film mate-rials.

Generally, elastic modulus and thermal expansion coef-ficient of thin film materials can be individually and indepen-dently determined by different experimental methods. Forinstance, a variety of methods, such as the tensile testmethod,4 the bending tests,5 the mechanical deflectionmethod,6 the suspended flexural vibration method,7 the reso-nance frequency method,8 and the photoacoustic measuringtechnique,9 have been proposed to determine the elasticmodulus of the coating materials. On the other hand, theoptical levered laser technique,10 the bending beamtechnique,11 and the capacitance cell method12 have been re-ported to determine the thermal expansion coefficient of thinfilm materials. Recently, the method to use a sample curva-ture technique to measure thermal stresses of thin films ofthe same material deposited on two different substrates withknown properties has been reported.13–17 The temperature-dependent deflection behaviors of thin films are obtained by

a)Author to whom correspondence should be addressed; electronic mail:ctcx831@mail.ncku.edu.tw

JOURNAL OF APPLIED PHYSICS VOLUME 96, NUMBER 7 1 OCTOBER 2004

0021-8979/2004/96(7)/3800/7/$22.00 © 2004 American Institute of Physics3800

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heating the samples in the range from room temperature to aslightly higher temperature level at which the physical prop-erties and microstructures of thin film materials still remainunchanged. By analyzing the thermal stress data as a func-tion of temperature, the thermal expansion coefficientsaF

and biaxial elastic modulusEF / s1−nFd of thin filmmaterials,13–15 even the thermal expansion coefficientsaF,the Poisson’s rationF, and Young’s modulusEF of thin filmmaterials,16,17 can be simultaneously determined from theslopes of two lines in the stress versus temperature plot.

II. MECHANICAL MODEL

For a structure with a very small film to substrate thick-ness ratio, generally, it can be satisfactorily assumed that anisotropic homogeneous stress distribution is induced in thefilm layer. The internal stress in thin film layer,s, can thenbe related to the deflection of the circular substrate at a dis-tanceR from the center of the substrate by using the modi-fied Stoney’s formula18–20

s =ESdS

2d

3s1 − nSdR2dF, s1d

wheres is the stress in the thin film which includes both theintrinsic stresssi, produced during film deposition, and thethermal stresssth due to thermal expansion mismatch be-tween the film and the substrate;d is the average deflectionof the substrate measured at the radiusR on the substrate dueto the action of stresss; ES andnS are the Young’s modulusand Poisson’s ratio of the substrate, respectively;dS and dF

are the thicknesses of the substrate and the thin film layer,respectively.

A. Samples with uniform temperature distributionsalong thickness

If the temperature is uniformly distributed in both thesubstrate and the thin film, the stress in the thin film can bewritten as

s = si + sth = si + saS− aFdEF

1 − nFsT2 − T1d, s2d

wherea is the thermal expansion coefficients; subscriptsSandF denote the physical properties of the substrate and thefilm layer, respectively; andT2 and T1 are the temperaturesof stress measurement and film deposition, respectively.Therefore, ifd represents the thermal deflection of substrateonly due to uniform temperature rise fromT1 and T2, byusing Eqs.(1) and (2), its relation with material propertiesand temperature can be written as

d = F3dFEF/s1 − nFddS

2ES/s1 − nSd GfsaS− aFdsT2 − T1dg. s3d

If all the physical properties, as shown in Eq.(2), aretemperature independent, the slope of the measured stress-temperature curve is equal to

ds

dT=

dsth

dT= saS− aFd

EF

1 − nF. s4d

Therefore, in the absence of any information aboutaF orEF / s1−nFd, these two unknown properties of the thin filmmaterial can be easily obtained by determiningds /dT ontwo different substrates with known mechanical properties,as shown in Eq.(4), and then solving these two algebraicequations directly.

B. Samples with nonuniform temperature distributionsalong the thickness

All the above relations, however, are formulated underthe assumption of uniform temperature distribution along thethickness of sample. If the samples are heated from the bot-tom surface of the substrate, as shown in Fig. 1(a), ratherthan being located in a small closed space capable of main-taining uniform temperature and then heating uniformly fromall surfaces, the cooling effect due to natural heat convectionwill occur on the upper surface of thin film as well as theside surfaces of samples. Consequently, a temperature gradi-ent along the thickness of the sample will be inherently in-troduced even in the condition of steady-state heat conduc-tion and result in an extra thermal deflection due tononuniform temperature distribution along the thickness.Therefore, to accurately estimate the physical properties ofthe thin film materials, we shall simultaneously measure thetemperatures at both the upper surface of thin film and thebottom surface of substrate. It can be shown in the followingsection that if the influence of temperature gradient along thethickness of the sample is neglected and the formulas(1)–(4)

FIG. 1. (a) Schematic diagram of experimental apparatus,(b) coordinates offilm and substrate layered structure.

J. Appl. Phys., Vol. 96, No. 7, 1 October 2004 Chen, Lin, and Chen 3801

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are misused directly, a very significant error in estimatingphysical properties of thin film material will be probablytaken place.

1. Theoretical temperature distributions along thethickness of the sample

a. Prediction of overall heat transfer coefficient hC. Pre-diction of hC in natural convection on a vertical surface ofsubstrate can be accurately performed by using the well-known Squire-Eckert formulation and the equations pro-posed by Churchill and Chu in the field of natural heat con-vection. The average overall heat transfer coefficienthC

proposed by Churchill-Chu correlation equation is given by21

hC = NuLk

L, s5ad

where

NuL = 0.68 + 0.67 RaL1/4F1 +S0.492

PrD9/16G−4/9

, s5bd

RaL =gbDTL3

na, s5cd

and

Pr =n

a, s5dd

in which NuL is the average Nusselt number;L is the verticallength of the plate; RaL is the Reynolds number; Pr is thePrandtl number;n is the kinematic viscosity;a is the thermaldiffusivity; b is the reciprocal of absolute ambient tempera-ture; g is the gravitational acceleration; andk is the thermalconductivity.

b. Evaluation of steady-state temperature distributionsalong the thickness of sample.If the sample is originally atuniform temperatureT1 and then heated from the bottomsurface of the substrate, the temperature distributions alongthe thickness of the film and the substrate can be accuratelyevaluated by solving the following steady-state heat conduc-tion equation and associated thermal boundary conditions inwhich the effect of interfacial thermal contact resistance be-tween the thin film and the substrate is included. The one-dimensional steady-state heat conduction equation of thetwo-layer sample, as shown in Fig. 1(b), along thickness di-rection,z, can be expressed as

d2TS

dz2 = 0, 0ø z ø dS;d2TF

dz2 = 0, dS, z ø dS+ dF;

s6d

where TSszd and TFszd are the temperature distributionsalong the substrate and the thin film, respectively; the axiszis along the thickness of the layered plate and originated atthe bottom surface of the substrate, as shown in Fig. 1(b).

The associated thermal boundary conditions are as fol-lows:

− kSdTS

dz= q0 at z = 0, s7ad

kSdTS

dz= kF

dTF

dz=

TS− TF

Rthat z = dS, s7bd

− kFdTF

dz= hCsTF − TAd at z = dS+ dF, s7cd

TF = sT2dU at z = dS+ dF, s7dd

TS= sT2dB at z = 0, s7ed

whereq0 is the heat flux prescribed on the bottom surface ofsubstrate by the hot storage;Rth denotes the interfacial ther-mal contact resistance between the thin film and the sub-strate;TA,sT2dU, and sT2dB are the ambient temperature, theupper surface temperature of thin film layer, and the bottomsurface temperature of substrate, respectively.

The solutions of Eq.(6) that satisfy the boundary condi-tions (7a)–(7e) can be easily obtained as

TSszd = a1 + a2z, 0 ø z ø dS;

TFszd = a3 + a4z, dS, z ø dS+ dF; s8ad

where

a1 = sT2dU + hCfsT2dU − TAgsdF/kF + dS/kSd

+ RthhCfsT2dU − TAg

= sT2dB, s8bd

a2 = − hCfsT2dU − TAg/kS, s8cd

a3 = T2 + hCfsT2dU − TAgsdF + dSd/kF, s8dd

a4 = − hCfsT2dU − TAg/kF. s8ed

If the thickness of film layer is much thinner than that of thesubstrate, i.e.,dF!dS, and the value of interfacial thermalcontact resistance between the film and the substrate,Rth, ismuch smaller thandS/kS in Eq. (8b), then the temperaturegradient in the substrate,a2, as shown in Eq.(8c), can bebriefly written as

a2 = dTS/dz> fsT2dU − sT2dBg/dS. s8fd

In convenience of deflection and stress calculations, an-other coordinatez, which is parallel toz axis with relationz=z−e (wheree is distance between the bottom surface andthe neutral surface of the structure), is adopted in analysis.Therefore, the temperature distributions in terms ofz can beexpressed as

TSszd = b1 + b2z, − eø zø dS− e; s9ad

TFszd = b3 + b4z, dS− e, zø dS+ dF − e; s9bd

where

b1 = a1 + a2e, b2 = a2, b3 = a3 + a4e, b4 = a4. s9cd

3802 J. Appl. Phys., Vol. 96, No. 7, 1 October 2004 Chen, Lin, and Chen

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2. Deflection of two-layer circular sample due tononuniform temperature rise

The mechanical equilibrium equation of a circularsample with radiusR subjected to a lateral loadingpsrd canbe formulated and expressed as follows:22

S d2

dr2 +1

r

d

drDSd2w

dr2 +1

r

dw

drD =

psrdD

. s10ad

The associated mechanical boundary conditions are22

ws0d = 0,d2w

dr2 +ns

r

dw

dr= −

MT

Dat r = R, s10bd

where wsrd is the flexure of the circular plate;psrd is thelateral loading density alongz axis and equal to zero in ourproblem;MT is the thermal moment induced in the specimenas the temperature at the upper surface of thin film is in-creased fromT1 to T2 [i.e., sT2dU and sT2dB at the uppersurface of this film and the bottom surface of substrate, re-spectively]. D is the flexure rigidity of the plate having theform

D =E−e

dS−e ES

1 − nS2z2dz+E

dS−e

dS+dF−e EF

1 − nF2 z2dz

= S EFdF

1 − nF2DFdF

2

3+ dFsdS− ed + sdS− ed2G

+ S ESdS

1 − nS2DSdS

2

3+ dSe+ e2D s11ad

where

e=sdF

2EFd/s1 − nFd + sdS2ESd/s1 − nSd

2fsdFEFd/s1 − nFd + sdSESd/s1 − nSdg. s11bd

If the thickness of film layer is much thinner than that of thesubstrate, i.e.,dF!dS, then Eqs.(11a) and (11b) can be, re-spectively, reduced to

D >1

12S ESdS

3

1 − nS2D +

1

4SEFdFdS

2

1 − nF2 D, e>

dS

2F1 −SdF

dSDEF

ESG .

s12d

Consequently, by using the boundary conditions in Eq.(10b),the solution of Eq.(10a) can be obtained and written as

wsrd = −MTr2

2Ds1 + nSd. s13d

The thermal bending momentMT induced in the specimen asthe temperature at the upper surface of thin film is increasedfrom T1 to T2 can be written as

MT =E−e

dS−e

zS ES

1 − nSDhaSfTSszd − T1gjdz

+EdS−e

dS+dF−e

zS EF

1 − nFDhaFfTFszd − T1gjdz

= sMTdnon+ sMTduni, s14ad

where

sMTdnon>b2dS

3

12S ESaS

1 − nSD >

fsT2dU − sT2dBgdS2

12S ESaS

1 − nSD ,

s14bd

sMTduni > S EF

1 − nFDSdSdF

2DhsaS− aFdfsT2dU − T1gj,

s14cd

in which sMTdnon and sMTduni represent the thermal bendingmoment contributed due to temperature gradient(nonuni-form temperature distribution) along the thickness of thesubstrate and uniform temperature rise in the sample, respec-tively. Moreover, it can be seen that these two componentshave the same order in magnitude and shall be consideredsimultaneously in analysis.

Therefore, the deflection of specimen atr =R due to ther-mal bending moment can be obtained by substituting Eqs.(15) into Eqs.(14) and expressed as

d = wsRd = dnon+ duni, s15ad

where

dnon>b2aSR

2

2>

fsT2dU − sT2dBgaSR2

2dS, s15bd

TABLE I. Material physical properties of two different substratesa.

Young’s modulus Poisson’s Thermal expansion Thermal conductivitySubstrate ES sGPad ratio nS coefficientaS s°C−1d KS sW/m Kd

BK-7 81.0 0.208 7.40310−6 1.5Pyrex 62.7 0.200 3.25310−6 1.3

aSee Ref. 14.

TABLE II. Temperature distributions and temperature gradient of substrate.

sT2dU s°Cd sT2dB s°Cd dTS/dz s°C/mmd

31 31.032 0.021BK-7 40 40.112 0.075substrate 52 52.239 0.159

70 70.458 0.30531 31.038 0.025

Pyrex 40 40.129 0.086substrate 52 52.276 0.184

70 70.528 0.352

J. Appl. Phys., Vol. 96, No. 7, 1 October 2004 Chen, Lin, and Chen 3803

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duni > F3dFEF/s1 − nFddS

2ES/s1 − nSd GhsaS− aFdfsT2dU − T1gR2j.

s15cd

It is satisfactorily assumed that the stress in the thin filmlayer is uniformly distributed due to its very thin thicknessand is calculated by using the following relation:

sth >FT

dF=

2MT/dS

dF=

2sMTdnon

dSdF+

2sMTduni

dSdF

= ssthdnon+ ssthduni, s16ad

where

ssthdnon=b2dS

2

6dFS ESaS

1 − nSD >

fsT2dU − sT2dBgdS

6dFS ESaS

1 − nSD ,

s16bd

ssthduni = S EF

1 − nFDhsaS− aSdfsT2dU − T1gj

=ESdS

2duni

3s1 − nSdR2dF, s16cd

whereFT is the resultant thermal force exerted on the thinfilm layer. In other words, the relations between displace-ment, thermal stress, and temperature under the condition ofnonuniform temperature distributions along the thicknessshall be modified from Eqs.(1) and (3) to (15) and (16),respectively.

It shall be emphasized that since the parameterb2 inssthdnon is nonlinearly dependent on temperature rise due tothe effect of natural heat convection, only the uniform partsof thermal deflection,duni, and thermal stress,ssthduni, arelinearly related to temperature rise and are easily adopted todetermine the material properties of thin film layer. Theseuniform terms can be evaluated by subtracting individuallythe nonuniform terms from the total thermal deflection andtotal thermal stress, respectively. In other words, the Eq.(4)shall be modified by usingssthduni instead ofsth as follows:

dssthduni

dT= saS− aFd

EF

1 − nF. s17d

Consequently, two kinds of physical properties of thin filmlayer, i.e.,EF / s1−nFd and aF, can be easily evaluated byusing the samples with the same thin film material depositedon two different substrate materials through Eq.(17). For thespecific case of uniform temperature distribution withouttemperature gradient along the thickness of the substrate, i.e.,b2=0 [i.e., sT2dU=sT2dB=T2], thendnon=ssthdnon=0, the cor-responding maximum deflection of plate atr =R and stress inthe thin film layer then completely become identical to Eqs.(1) and (3), respectively.

III. EXAMPLES UNDER INVESTIGATION

To evaluate the effects of temperature gradient, the lit-erature, which simultaneously determines the thermal expan-sion coefficient and the elastic modulus of Ta2O5 thin filmusing phase shifting interferometry,14 is exemplified andstudied again. In this reference, the samples, which weredeposited with the same thin film material upon two differentsubstrates, were heated from the bottom of the substrate by ahot stage linked with an electric temperature controller;while a thermocouple was attached to the upper surface ofthin film layer to sense the temperature variation over there.Moreover, a phase shifting Twyman-Green interferometerapparatus was installed in front of the upper surface of thinfilm layer to optically detect the deformation fringes ofsamples. Since all the experiments were conducted in anopen room space, the upper surface of the thin film wasmainly cooled by the effect of natural heat convection. Con-sequently, a temperature gradient along the thickness of thesamples will be inherently introduced even in the conditionof steady-state heat conduction. The physical properties oftwo different kinds of substrate are listed in Table I. Experi-mental results show that the shape of the bare BK-7 substrateis upwardly concave, while the bare Pyrex substrate is con-vex. After film deposition, the shape of the BK-7 coatedsubstrate is less concave and thus presents compressive filmstress. On the other hand, the shape of the Pyrex coated

TABLE III. Thermal deflection of specimen and thermal stress of thin film.

BK-7 substrateTF s°Cd hC sW/m2 Kd d smmda dnonsmmd duni smmd sth sGPad ssthduni sGPad

26 … −0.155 0 −0.155 −0.398 −0.39831 6.4 −0.142 0.008 −0.150 −0.365 −0.38540 8.0 −0.115 0.028 −0.143 −0.295 −0.36752 9.2 −0.080 0.059 −0.139 −0.206 −0.35770 10.4 −0.025 0.110 −0.134 −0.054 −0.344

PyrexTF s°Cd hC sW/m2 Kd d smmda dnonsmmd duni smmd sth sGPad ssthduni sGPad

26 … −0.199 0 −0.199 −0.391 −0.39131 6.4 −0.198 0.004 −0.202 −0.389 −0.39740 8.0 −0.194 0.014 −0.208 −0.382 −0.40952 9.2 −0.181 0.030 −0.213 −0.360 −0.41970 10.4 −0.173 0.056 −0.229 −0.338 −0.450

aReference 14.

3804 J. Appl. Phys., Vol. 96, No. 7, 1 October 2004 Chen, Lin, and Chen

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substrate is more convex and thus also shows the compres-sive film stress. The thicknesses of substrate and thin filmlayer are 1.5 mm and 0.299µm, respectively. There is onlyfew information on the interfacial thermal contact resistanceRth between the film and the substrate; its value is, in general,within the range of 1310−6–1310−9 m2 K/W.23,24 In thepresent study a maximum value of 1310−6 m2 K/W, whichis still much smaller thandS/kSs>1310−3 m2 K/Wd, isadopted in the temperature calculation. It can be seen that theinfluence of this maximum interfacial thermal contact resis-tance on temperature distributions is still less than 0.2% andcan be satisfactorily disregarded in the calculation in thiscase.

If the value of thermal conductivity of the thin film isapproximately the same order as the bulk, the temperaturedistributions, temperature gradients, thermal deflection, andstress due to uniform temperature distribution, i.e.,sT2dU ,sT2dB,dTS/dz,duni, andssthduni are evaluated based onEqs. (8), (15), and (16) and shown as in Tables II and III,respectively.

The thermal deflections and thermal stresses versus tem-perature for two different substrate materials are shown inFigs. 2 and 3, respectively. Only the parts of thermal deflec-tion and thermal stress induced by uniform temperature risecan be adopted to determine the physical properties of thinfilm layer. A simple linearization scheme of least squaremethod is adopted to evaluate the slopesdssthduni/dT of twodifferent substrate materials. The physical properties of thinfilm material Ta2O5 evaluated by using Eqs.(17) instead ofEq. (4) in the present study are shown as in Table IV.

However, if we did not take account of the effect of

FIG. 2. Thermal stress and deflection of substrate BK-7 under different filmtemperatures.

FIG. 3. Thermal stress and deflection of substrate Pyrex under different filmtemperatures.

J. Appl. Phys., Vol. 96, No. 7, 1 October 2004 Chen, Lin, and Chen 3805

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temperature gradient along the thickness of the sample anddirectly adopt Eqs.(1), (3), and(4) under the assumption ofuniform temperature rise, the values of biaxial elastic modu-lus and coefficient of thermal expansion become 1549.4 GPaands2.42310−6d °C−1, respectively.14 The value of the elas-tic biaxial modulus becomes too high and is almost 15 timesmore than the bulk or three times more than that obtained inthe present study.

It was reported that the physical properties of thin filmmaterials are closely related to the interface conditions be-tween thin film layer and substrate, microscopic structures,porosity ratio, defects, and thickness of thin film layer. Thevalue of elastic biaxial modulus of Ta2O5 bulk material isapproximately equal to the order of 100 GPa. Although thepacking density of Ta2O5 thin film layer deposited by ionbeam sputter deposition in the thickness 0.299µm is veryhigh, it is still unreasonable to achieve the improvement inelastic modulus by five times higher than bulk material, com-pared to the original 15 times evaluated by the model ofuniform temperature rise.1–3 Normally, the thin film modulusis lower than that of the bulk material because of the porositymentioned above. For different materials produced by differ-ent processes, however, the argument on this issue somewhatstill exists. The error of the dual-substrate method may comefrom several sources. One of the most negative characteris-tics of this method is that the slope of the measured stress-temperature curve, as shown in Eq.(17), is extremely small.For instance, if the values ofsaS−aFd andEF / s1−nFd are ofthe order of 10−6 °C and 102 GPa, respectively, the slope ofstress-temperature curve is only about 10−4 GPa/ °C. Inother words, a 50 °C temperature rise in sample only ap-proximately results in 5310−3 GPa increase in film stress or10−2 mm change in thermal deflection. Consequently, the de-duced thin film properties are very sensitive to the errors ofexperimental measurement and substrate property. For in-stance, if the Young’s modulus and thermal expansion coef-ficient are 15% higher for BK-7 glass and 15% lower forPyrex glass than those listed in Table I, the values of biaxialelastic modulus and coefficient of thermal expansion willbecome 156.6 GPa ands7.68310−6d °C−1, respectively.Consequently, it is highly desirable to use single crystalstructures, with which Mother Nature ensures a minimumvariation of substrate structures. The structure and propertyvariations of BK-7 and Pyrex glasses, however, are not easyto predict during their forming processes. Even with singlecrystal substrates, the measured substrate property data errorscan propagate into the thin film properties. The errors of the

substrate properties are probably major contributors to theunreasonable results; the deduced biaxial modulus listed inthis paper, therefore, may not be reliable.

IV. CONCLUDING REMARKS

In this paper, modified deflection, thermal stress, andtemperature relations due to the existence of temperature gra-dient along the thickness of samples are proposed and for-mulated in analytical modeling of some experimental meth-ods to simultaneously determine the thermal expansioncoefficientsaF and biaxial elastic modulusEF / s1−nFd ofthin film materials. It is seen that the small temperature gra-dient induced along the thickness of samples has a very sig-nificant influence on the accurate determination of elasticbiaxial modulus and thermal expansion coefficient of thinfilm materials and shall be taken account of in analysis. If theeffect of this temperature gradient is carelessly disregardedin physical modeling, a significantly different value of elasticmodulus may be misleadingly obtained.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theNational Science of Taiwan, ROC(Grant No. NSC 89-2212-E-006-011) and valuable comments of reviewer.

1M. Ohring, The Materials Science of Thin Films(Academic, San Diego,1992).

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TABLE IV. Physical properties of thin film material Ta2O5.

Physical properties EF / s1−nFd sGPad aF s°C−1d

597.7 5.43310−6

3806 J. Appl. Phys., Vol. 96, No. 7, 1 October 2004 Chen, Lin, and Chen

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