Stress development and relaxation in copper films during thermal cycling

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Stress development and relaxation in copper films during thermalcyclingM. D. Thouless, J. Gupta,a) and J. M. E. HarperIBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York 10598

(Received 9 November 1992; accepted 7 April 1993)

The reliability of integrated-circuit wiring depends strongly on the development andrelaxation of stresses that promote void and hillock formation. In this paper an analysisbased on existing models of creep is presented that predicts the stresses developed inthin blanket films of copper on Si wafers subjected to thermal cycling. The results areportrayed on deformation-mechanism maps that identify the dominant mechanismsexpected to operate during thermal cycling. These predictions are compared withtemperature-ramped and isothermal stress measurements for a 1 /xm-thick sputteredCu film in the temperature range 25-450 °C. The models successfully predict boththe rate of stress relaxation when the film is held at a constant temperature and thestress-temperature hysteresis generated during thermal cycling. For 1 /u,m-thick Cufilms cycled in the temperature range 25-450 °C, the deformation maps indicate thatgrain-boundary diffusion controls the stress relief at higher temperatures (>300 °C)when only a low stress can be sustained in the films, power-law creep is important atintermediate temperatures and determines the maximum compressive stress, and that ifyield by dislocation glide (low-temperature plasticity) occurs, it will do so only at thelowest temperatures (<100 °C). This last mechanism did not appear to be operating inthe film studied for this project.

I. INTRODUCTION

Recently, there has been renewed interest in the ef-fects of stress in thin metal films used in microelectronicchips and packaging.1"9 In these systems, metal linesare mechanically constrained by the silicon substrate,encapsulating layers of silica, or ceramic packagingmaterials. Owing to differences in thermal expansion,high stresses can develop during the thermal excursionsexperienced during processing steps or service. Anyattempt to quantify or model the effect of these stresseson the processing or reliability of microelectronic mate-rials must incorporate an understanding of the relaxationmechanisms that are appropriate for the alloy under theconditions of interest. One approach used to investigatethese mechanisms is to measure the development ofstresses during thermal cycling of a metal film depositedon a substrate. The temperature dependences of stressesin both copper and aluminum lines have been measuredand modeled by Flinn et al.,5'6 who interpreted theoverall shapes of the stress-temperature curves in termsof a temperature-dependent flow stress. However, asis graphically demonstrated for bulk materials in thedeformation-mechanism maps of Frost and Ashby,10 anassumption of a single dominant relaxation mechanismcan be an oversimplification when modeling creep over

a)Present address: Bain & Co., Copley Place, Boston, Massachu-setts 02117.

a range of temperatures and stresses. These deformationmaps, which are based on models and empirical rela-tionships that have been found satisfactory in explainingor describing creep behavior, illustrate the conditions forwhich each of several relaxation mechanisms is expectedto dominate.

A straightforward approach to modeling the stressrelaxation of thin films is to assume that the creep prop-erties can be described by those of the material in a bulkform.11 Any major discrepancy between the resultingpredictions and experimental observations would thenindicate the possibility of additional microstructural ormechanistic effects peculiar to the thin-film geometry.In this paper the behavior of a copper film has beenanalyzed using the mechanisms described by Frost andAshby,10 and the predictions have been compared withexperimental measurements made on a pure copperfilm deposited on a silicon substrate. The comparisonindicated that the salient features of the response ofthe film to thermal cycling were in excellent agreementwith the simple assumption that the creep propertiescould be described by existing models for creep of bulkpolycrystalline copper.

II. ANALYSIS

To establish a relationship among the stress, tem-perature, and relaxation rate, the film is modeled asa Maxwell element consisting of a spring in series

J. Mater. Res., Vol. 8, No. 8, Aug 1993 1993 Materials Research Society 1845

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M.D. Thouless, J. Gupta, and J.M.E. Harper: Stress development and relaxation in copper films

with linear and nonlinear dashpots. The elastic re-sponse is dictated by the spring while the properties ofthe dashpots incorporate the various relaxation mecha-nisms of the film. If the substrate is elastic and muchthicker than the film, then a change in temperature,AT, induces an increment of strain in the film of

Ae = A a A 7 (1)

where A a is the difference in the coefficients of thermalexpansion between the film and substrate. This strainis treated as a boundary condition for the Maxwellelement. The instantaneous response to such an ap-plied strain is elastic. With time, the elastic strain,elastic, is accommodated by the development of a

plastic strain, eplastic, but the total strain remains con-stant. The net stress, a, acting on the film is dic-tated by the current level of the elastic strain:

a — Eeelastic (2)

where E is the appropriate modulus given by E/{\ — v)for isotropic, equibiaxial stressed films, E is Young'smodulus, and v is Poisson's ratio. The rate at whichplastic deformation takes place is of the general form,

^plastic (3)

where C, and nt are material parameters that depend onthe appropriate relaxation mechanisms. It is important tonote that the plastic-strain rate depends on the instan-taneous stress in the film and the constitutive behaviorof the relaxation mechanisms; it is not dictated by theheating or cooling rate.

A. Mechanisms

In general, several concurrent processes are expectedto contribute to the relaxation of a film. For the purposesof this paper, the mechanisms considered are the steady-state ones described on the deformation maps by Frostand Ashby10: yield, power-law creep, and creep bygrain-boundary and lattice diffusion. Furthermore, it isassumed that these mechanisms are perfectly symmet-rical with respect to tensile and compressive stresses.The details of the constitutive equations can be foundin Ref. 10; however, one important modification mustbe made for a film deforming by grain-boundary andlattice diffusion. In the original formulation of Cobleand Nabarro-Herring creep, atoms are assumed to flowbetween neighboring grain boundaries subjected to dif-ferent normal stresses [Fig. l(a)]. Although this willoccur if the film is loaded in a nonbiaxial fashion, itcannot operate when the in-plane stresses are equibiaxial,as in an isotropic system loaded by thermal stresses[Fig. l(b)]. Under these conditions, the equations fordiffusional creep have to be modified slightly to accountfor the fact that matter must flow between the grain

boundaries and the free surface rather than betweenneighboring grain boundaries.12 It can be shown that thecontribution of grain-boundary diffusion to the plastic-strain rate is approximately12'13:

ex = eo((r/ai) exp(-Qb/kT) (4a)

where e0 is a normalizing constant set to 1 s"1, a isthe stress in the film, Qb is the activation energy forgrain-boundary diffusion, T is the absolute tempera-ture, and

kTdh2i0

* n ' (4b)

where 8Dob is the pre-exponential coefficient for grain-boundary diffusion, D, is the atomic volume of thediffusing species, k is Boltzmann's constant, d is theaverage grain size of the film, and h is the film thickness.Note that the film thickness enters the expression forthe creep rate, instead of a higher power of the grainsize. The additional path between the surface and grainboundaries which is provided by the grains themselvesresults in a contribution to the plastic-strain rate of

e2 = €o(cr/(T2) exp(—Qv/kT) (5a)

where the parameter <x2 can be approximated as12'13

a2 = hk (5b)

(a) uniaxial stress

(b) equibiaxial stress

FIG. 1. Schematic illustrations of the possible paths along whichatoms can diffuse to relax stresses in (a) a uniaxial stressed film, (b) anequibiaxially stressed film. (Diffusion from the grain boundaries to thesurface is, of course, a mechanism that will also operate in uniaxialgeometries, as well as the biaxial ones.)

1846 J. Mater. Res., Vol. 8, No. 8, Aug 1993



M. D. Thouless, J. Gupta, and J. M. E. Harper: Stress development and relaxation in copper films

and Qv and Dov are the activation energy and pre-exponential coefficient for lattice diffusion, respectively.Equations (4) and (5) are based on the assumption thatsurface diffusion is sufficiently rapid that it does notinfluence the creep rate. A more rigorous analysis mustincorporate the effects of finite surface-diffusion rateswhich will tend to retard the creep; it will be verysensitive to the details of the surface geometry andhistory of the creep.12 Another crucial influence ofsurface diffusion is its role in cavity growth; if surfacediffusion is low, then cavity growth can occur in theform of narrow voids which may cause failure of thefilm.12'14 The retarding influence of surface diffusion oncreep has been ignored in this paper.

Expressions for other relaxation mechanisms whichinvolve the motion of dislocations can be adopted with-out modification from Frost and Ashby.10 The notationsin that reference have been adopted to make comparisonsmore apparent, and the equations have been re-expressedin terms of principal stresses and strains. At low temper-atures yielding occurs by obstacle-controlled dislocationglide. Ignoring work-hardening, the expression for thislow-temperature plasticity is10

where f is the shear strength at 0 K, y0 is a pre-exponential constant, and Qg is the activation energyfor dislocation glide.

Power-law creep in which the plastic-strain ratehas a nonlinear dependence on the stress occurs whendislocations have enough thermal energy to climb andescape from obstacles on the glide planes. At hightemperatures the climb is controlled by diffusion fromthe dislocations through the lattice; at low temperaturesit is controlled by diffusion along the dislocation cores.However, it appears that a simple power-law relationshipbetween the strain rate and stress does not satisfactorilydescribe experimental observations at high stresses. Frostand Ashby have presented formulations that describethe behavior (termed "power-law breakdown") in thistransition from a regime in which the dislocation motionis controlled by glide to one in which it is controlledby climb. Their expression for high-temperature creepin which climb is controlled by lattice diffusion is10

= eo

X

1a<JL04

where

0"4 = /JLADovfib]

(7a)

(7b)

fi is the shear modulus which is assumed to be tem-perature dependent, b is an appropriate Burgers vector,and A, n, and a' are empirical fitting parameters. Forlow-temperature creep in which climb is controlled bycore diffusion10:

a'

X sinh f f JVV3/4/J

JL0-5

\J exp(-ec/kr) (8a)

where

I" 3kTbe0

lOAacDocti(8b)

and Qc and acD0C are the activation energy and pre-exponential coefficient for core diffusion, respectively.If the magnitude of a is less than about <J3/JL/CX', thesetwo expressions reduce to simple power laws:

i5 = exp(-Qc/kD (9)

B. Stress development in thermal cycling

Parameters for copper used in Eqs. (4) to (9) canbe obtained from Table 4.1 in Ref. 10, and the resultantexpressions are summarized in the Appendix. Theseexpressions can be used in conjunction with Eqs. (1)to (3) to make predictions for the stress-temperaturerelationships of copper films deposited on silicon. Thecalculations were performed by assuming an initiallystress-free state at 25 °C, incrementally changing thetemperature and associated strain, calculating the amountof strain relaxed during the increment of time corre-sponding to the chosen heating/cooling rate, and thencomputing the new stress caused by the remaining com-ponent of the elastic strain. All of the mechanisms listedin the previous section were assumed to be capable ofcontributing to the stress relaxation. However, followingthe discussion in Chap. 3 of Ref. 10, some relaxationmechanisms were considered to be competing whileothers were assumed to act in concert, so that

ePiastic = (<b + e3) + greatest of [(e4 + e5) or(10)

After one heating cycle to 450 °C, the stress followeda steady-state curve for subsequent cycles between thesame temperatures. It is this steady-state cycle that hasbeen plotted in Fig. 2 for 1 fim thick copper films cycledat ± 1 and ±10 °C/min. Superimposed on these plotsare boundaries indicating regions in which a particularmechanism makes the largest contribution to the relax-ation rate. Therefore, the predicted cycles have been

J. Mater. Res., Vol. 8, No. 8, Aug 1993 1847



M.D. Thouless, J. Gupta, and J .M.E. Harper: Stress development and relaxation in copper films

grain size - 0.1 fim.« 1°C/minute• 10°C/minute

-100

-200

POWER-LAW BREAKDOWN

(Ion-temperature creep)

J I I i_

0 50 100 150 200 250 300 350 400 450 500Temperature /°C

1 fim copper film on silicon

(a)

grain size = 1 fim .

* 1°C/minute

• 10°C/minute

POWER-LAW CREEP

ow-temperature creep) I —

POWER-LAW CREEP f -(low-temperature creep)

0 50 100 150 200 250 300 350 400 450 500Temperature /°C


(b)

grain size = 10 fim.

<• 1°C/minute

• 10°C/minute

POWER-LAW CREEP(low-temperature creep) I "

0 50 100 150 200 250 300 350 400 450 500Temperature /°C


(C)

FIG. 2. Predictions of the stress-temperature plots for a 1 fim copperfilm on silicon with a grain size of (a) 0.1 fim, (b) 1 fim, and(c) 10 fim. These predictions were made using the data given inRef. 10 and are plotted on deformation-mechanism maps showing thedominant relaxation mechanisms.10

plotted on deformation maps that are essentially identicalto those presented in Ref. 10.

Several important features can be observed fromFig. 2. The dominant mechanisms of stress relief arepredicted to change as a copper film is cycled through thetemperatures that might realistically be expected in theprocessing or service of integrated circuits. In particular,while grain-boundary diffusion is important at highertemperatures, various forms of dislocation motion aredominant over a wide range of temperatures. It shouldalso be noted that if yield, in the conventional senseof low-temperature plasticity, occurs, it manifests itselfonly on the cooling cycle at relatively low temperatures.For the value of yield stress assumed in constructingFig. 2, the copper does just yield at low temperatures.However, the yield stress is very sensitive to microstruc-tural features such as the grain size and the degree ofwork hardening; therefore, it is conceivable that theyield stress might be vastly different from the valueassumed for the deformation maps, and more or lessyielding may occur. A significant feature in all the stress-temperature curves is the point of maximum compressivestress achieved during the heating cycles. These maximaare not manifestations of a sudden onset of yield; theymerely represent the temperatures at which the rateof plastic relaxation exceeds the applied loading rate.There is no mechanism change, and the positions andmagnitudes of the maxima are rate-dependent. Finally, itshould also be noted that the initial portion of the heatingcycle at low temperatures is essentially elastic. This is,of course, identical to what happens in conventionaltensile/compression tests, and it means that the slopeof such lines can be interpreted as a direct measure ofthe appropriate elastic modulus. Similar elastic responseswere observed in simulations in which the heating andcooling rates were suddenly changed while a stress wasbeing supported by the film; an example can be seen inthe high temperature behavior of Fig. 2(c).

III. EXPERIMENTAL

Copper thin-film samples were deposited with athickness of 1 fim on 125 mm diameter oxidizedSi(100) wafers. Sputter deposition was performed inan MRC 643 magnetron sputtering system from a99.9998% pure Cu target with a base pressure of5-8 X 10~7 Torr and a deposition pressure of 4 mTorrAr. Temperature-ramped stress measurements were madeusing a Flexus F2408 full-wafer stress gauge15 atnominal heating rates of 2 and 10 °C/min. The coolingwas uncontrolled, and measurements indicated that itwas approximately 4.4 °C/min at 450 °C, but less thanabout 1.5 °C/min at 100 °C. The curvature of the samplewas monitored optically during heating in a furnacewhich contained a flowing gettered helium atmosphere.




M. D. Thouless, J. Gupta, and J. M. E. Harper: Stress development and relaxation in copper films

The film stress was calculated using Stoney's equation.16

Temperature calibration of the furnace thermometer wasmade using a thermocouple bonded to a Si wafer andcompared with the melting temperatures of variousreference materials. The temperature measurements wereestimated to be accurate to within ± 5 °C with theexception of an initial lag at the lower temperaturesin the heating cycle. The measured stress for a 1 jj,mcopper film heated twice to 450 °C at 10 °C/min isshown as a function of temperature in Fig. 3(a). Thefirst cycle was accompanied by grain growth from amean size of about 200 nm to about 500 nm17; furthergrowth did not occur during subsequent temperaturecycles. The second temperature cycle is representativeof the sample in its 450 °C-annealed condition. Theslopes at low temperatures of the heating portion ofthe steady-state experimental curves are within 5% of

D

Oo

100 200 300 400

Temperature / degrees C

(a)

5 0 0

o

oo

100 200 300 400Temperature / degrees C

500

(b)FIG. 3. Experimental observations of stress versus temperature for1 fim sputtered copper on silicon at nominal heating rates of(a) 10 °C/min and (b) 2 °C/min. Grain growth is associated with thefirst heating shown in (a). No grain growth occurred in the subsequentcycles or when the same sample was used to generate the data for (b).

the value of 2 MPa/°C predicted from the appropriatevalues of the elastic parameters given in the Appendix.The shallow slope of the first heating curve suggeststhat there was considerable porosity in the as-depositedfilm which was eliminated during the first cycle upto 450 °C. Subsequent temperature cycles at a heatingrate of 2 °C/min for the same sample are shown inFig. 3(b). The shapes of the curves in both Figs. 3(a)and 3(b) (i.e., the lack of a discontinuity in the slopeupon cooling) indicate that yield by "low-temperatureplasticity" did not occur when the sample was cooled toroom temperature.

The sample was also used to make isothermal mea-surements of stress relief over periods of up to 22 hat a sequence of different temperatures: 187, 130, 112,and 74 °C. Prior to each measurement, the sample washeated to 450 °C at 10 °C/min, and then cooled to therelevant temperature. Plots of the stresses as they relaxedwith time are shown in Fig. 4. The time period beforethe start of each data set shows the time spent heatingto 450 °C and cooling to the measurement temperature.

IV. DISCUSSION

Figure 5(a) shows the predictions for the responseof a 1 fim copper film with a grain size of 500 nmwhich is heated and cooled at rates corresponding tothe experimental conditions. In particular, the detailsof the cooling rate were closely mimicked in constructingthe curves. The isothermal plots of Fig. 5(b) have beencalculated using initial stresses given by the experimentaldata rather than those predicted from Fig. 5(a). A com-parison between these two figures and the experimentalplots shows good qualitative agreement, suggesting that,to within a reasonable approximation, the parameters forthe creep of bulk polycrystalline copper are appropriateto characterize the relaxation of copper films. However,

2.5

2.0

"3"

I «I 1.0

Temperature

• V - ...7

(C)

mmtt

1 12

I

1 130

10 15

Time (hours)

20 25

FIG. 4. Plots of the isothermal stress relaxation in 1 fj,m thick sput-tered copper films on silicon after annealing at 450 °C.





-100

-200

i 1 1 1 1grain size = 500 nm

• heating rate = 10 "C/rmin

• heating rate = 2 "C/min

50 100 150 200 250 300 350 400 450 500Temperature / °C

(a)

10 15Time / hours

1 /Aim copper film on silicon

(b)FIG. 5. Predictions of stress as a function of (a) temperature and(b) time during isothermal holds, using the parameters for relaxationmechanisms given in Ref. 10, and the grain size and heating andcooling rates appropriate for the experimental conditions.

some discrepancies exist in the details of the compar-isons. In particular, the predicted relaxation rates appearto be too low and the resultant stresses are therefore toohigh; this is particularly marked in the compressive stressregion. In addition, the predicted isothermal curves lackthe curvature shown in the measured curves.

Among the various mechanisms that contribute tothe relaxation of the films there are enough parametersthat, given sufficient flexibility in choosing their values,a good fit to the data could probably be obtained.Such an exercise would be of rather limited utilityand negate the approach of this paper in seeking amechanistic-based interpretation rather than a purelyempirical one. However, since the deformation mapsof Fig. 2 suggest that the most important mechanismis power-law breakdown, it is of interest to examine

the sensitivity of the predictions to changes in someof the parameters involved in this mechanism. (At thisjuncture, it should be noted that attempts to model thebehavior assuming a relaxation mechanism with a lineardependence on stress, such as grain-boundary diffusion,gave results that were qualitatively different from theobserved responses and confirmed the predictions of theFrost-Ashby deformation maps that a power-law typeof mechanism was more appropriate.) Figure 6 showsthe predictions for one set of parameters that gave avery satisfactory fit to the data for the isothermal holds(Fig. 4). The power-law exponent, n, was reduced from4.8 to 3.0, the activation energy for core diffusion, Qc,was increased from 117 kJ/mole to 125 kJ/mole, andthe constant in front of the expression for the creeprate was increased by a factor of 500, which is wellwithin the range of coefficients used in bulk deformationmodels.10 In constructing Fig. 6(a), which shows thepredicted response to thermal cycling, the yield stress

400

300

200

100

0

-100

-200

grain size = 500 nm« heating rate - 10 °C/min• heating rate = 2 °C/min

50 100 150 200 250 300 350 400 450 500Temperature / °C

(a)

10 15Time / hours

1 fj,m copper film on silicon

(b)

FIG. 6. Predictions of stress as a function of (a) temperature and(b) time during isothermal holds, using values for the grain size andheating and cooling rates appropriate for the experimental conditions.The parameters for power-law breakdown have been adjusted to givea fit to the isothermal stress measurements.





was increased to a level sufficient to ensure that low-temperature plasticity did not operate, consistent with thedata of Fig. 3. The plots of Fig. 6(b) were, like thoseof Fig. 5(b), calculated to indicate relaxation from thestarting stresses measured in the experiments, althoughit is evident that the predicted stresses for these tem-peratures [obtained from the cooling curve of Fig. 6(a)]are quite close to the experimental values. It will beobserved that, while these small changes in the valuesof some of the parameters give predictions for theisothermal holds that are in excellent agreement withthe experiments, capturing nuances that are completelylacking in the previous predictions [Fig. 5(b)], there arestill some discrepancies between the details of Fig. 6(a)and those of the experimental results. Further attemptsto reconcile the details are probably unjustified, sinceit is quite likely that impurities in the films, unevengrain size distributions, and surface inhomogeneities willhave some effects on the stress relaxation mechanisms.Overall, however, the agreement between the predictedstress and experimental behavior is excellent.

V. CONCLUSIONS

An analysis of the stress relaxation of a thin copperfilm constrained on a silicon wafer during tempera-ture cycling has been presented. The parameters anddeformation mechanisms appropriate for bulk copperwere used, and the resultant stress-temperature plotswere superimposed on deformation-mechanism mapsthat delineate the major deformation mechanisms. Usingthis approach, it appears that, for temperature cycling ofa 1 /im-thick Cu film with a grain size of about 1 jttmat 1-10 °C/min between 25 °C and 450 °C:

(1) Deformation by lattice diffusion is negligible.(2) Grain-boundary diffusion dominates at the

higher temperatures (>300 °C).(3) Creep by temperature-assisted dislocation mo-

tion is important at intermediate temperatures and de-termines the value of the maximum compressive stress.

(4) Yield by low-temperature plasticity is predictedto occur, it at all, only at the lowest temperatures(<100 °C) and highest stress as (>250 MPa).

Stress-temperature measurements made on a sput-tered Cu film agreed very well with the calculatedcurves and demonstrate the major predicted features.This agreement suggests that the relaxation of copperfilms may be adequately described by the use ofbulk deformation mechanisms, and that approachessimilar to that outlined here could be used foranalyzing more complicated metallization geometriesin integrated circuits and packaging. In general, finite-element methods would have to be used for computingthe deviatoric stresses associated with the relaxationprocesses. As has been emphasized in this study, any

attempts to model the development of stress and itseffects in other materials must be accompanied by anunderstanding of the dominant relaxation mechanismsappropriate to the stresses and temperatures of interest.

ACKNOWLEDGMENT

The authors thank Cyril Cabral, Jr. for assistancewith the stress-temperature measurements and filmdeposition.

REFERENCES1. M. Murakami, Thin Solid Films 59, 105-116 (1979).2. M. Murakami, T. S. Kuan, and I. A. Blech, in Treatise on Materi-

als Science and Technology (Academic Press, New York, 1982),Vol. 24, pp. 163-210.

3. D. S. Gardner and P. A. Flinn, IEEE Trans. Electron Devices 35,2160 (1988).

4. M.A. Korhonen, P. B0rgesen, and C-Y. Li, Mater. Res. Bull.XVII, 61 (1992).

5. P. A. Flinn, J. Mater. Res. 6, 1498-1501 (1991).6. P. A. Flinn, D. S. Gardner, and W.D. Nix, IEEE Trans. Electron

Devices ED-34, 689-699 (1987).7. P. L. Pai and C. H. Ting, Proc. 1989 IEEE VMIC Conference,

258 (Institute of Electrical and Electronic Engineers, New York,1989).

8. T. Kwok, K. K. Chan, H. Chan, and J. Simko, J. Vac. Sci. Technol.A9 (4), 2523 (1991).

9. J. E. Sanchez, Jr. and J. W. Morris, Jr., in Materials ReliabilityIssues in Microelectronics, edited by J. R. Lloyd, P. S. Ho, C. T.Sah, and F. Yost (Mater. Res. Soc. Symp. Proc. 225, Pittsburgh,PA, 1991), p. 53.

10. H.J. Frost and M.F. Ashby, Deformation-Mechanisms Maps(Pergamon Press, Oxford, 1982).

11. V.M. Koleshko, V.F. Belitsky, and I.V. Kiryushin, Thin SolidFilms 142, 199-212 (1986).

12. M.D. Thouless, Acta Metall. Mater. 41, 1057-1064 (1993).13. G.B. Gibbs, Philos. Mag. 13, 589-593 (1966).14. T-J. Chuang, K. I. Kagawa, J. R. Rice, and L. B. Sills, Acta Metall.

27, 265-284 (1979).15. Flexus, Inc., Sunnyvale, CA.16. G. Stoney, Proc. R. Soc. London A 82, 172 (1909).17. J. Gupta, J. M. E. Harper, J. L. Mauer, IV, P. G. Blauner, and D. A.

Smith, Appl. Phys. Lett. 61, 663-665 (1992).18. CRC Handbook of Chemistry and Physics (CRC Press Inc., Boca

Raton, FL, 1992).

APPENDIX

The rate equations (assuming a film thickness ofused for copper in Fig. 2 were10

1.282 X IP"8

Yd5.127 X 10"5

6 2 " ' Tdi3 = 5.774 X 105

f 2.557 X 104

X exp 1

a exp(-1.251 X 104/r)

a exp(-2.369 X IO4/T)

- 2.177 X 10"V)





/458.4o-

X exp(-2.369 X W4/T)

sinh

X exp(-1.407 X IO4/T)

The following material parameters were used for thecopper film10:

= 4.713 X 1010[l - 3.557 X 10~4r],

E =1 - v

= 4.102^1,

and the thermal expansion mismatch between the copperand the silicon substrate was taken to be18:

Aa = 1.18 X 10"7°C.



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Stress development and relaxation in copper films during thermal cycling