Diffusion Processes in Advanced Technological Materials || Measurement of Stresses in Thin Films and...

8 Measurement of Stresses in ThinFilms and Their Relaxation

Oliver Kraft

Institut für Materialforschung II, Forschungszentrum Karlsruhe und Institut für Zuverlässigkeit von Bauteilen und Systemen

Universität Karlsruhe, Germany

Huajian Gao

Max-Planck-Institut für Metallforschung Stuttgart, Germany

8.1 Introduction

Thin films are used in many technological applications because theyprovide special chemical, mechanical, and electrical properties, as andwhere needed, using only small amounts of materials. For example, thehigh density and fast performance of modern computers have been possi-ble due to the incorporation of multilayer thin-film structures coupledwith submicron photolithography on the back of the active devices formedin Si chips. Because thin-film structures are always deposited onto rela-tively rigid substrates, they are heavily stressed and undergo relaxationwhen subjected to fabrication processes involving higher temperaturesthan those used for the deposition. This chapter discusses techniques formeasuring stresses in thin metal films attached to substrates and theirrelaxation by plastic deformation and diffusional creep.

Figure 8.1 shows the origin of stresses in thin films as described byNix.[1] As a start, we consider a film/substrate composite that is free ofstress [Fig. 8.1(a)]. If we remove the film from the substrate, its lateraldimensions will match those of the substrate [Fig. 8.1(b)] and the film can be reattached without causing any stress. However, if we assume thatthe film is subject to a dilatational strain relative to the substrate, then thelateral dimensions of the film no longer match those of the substrate [Fig. 8.1(c)]. A stress must be applied to the film in order to make it fit tothe substrate again [Fig. 8.1(d)]. Now the film can be rigidly reattached to the substrate and the applied stress removed. As a result, shear stressesare produced near the edge of the film/substrate composite, which main-tains a biaxial stress in the film [Fig. 8.1(e)]. In addition, the shear

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366 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

Figure 8.1 Illustration origin of biaxial stress in a thin film: (a) stress-free film onthe substrate; (b) remove film from the substrate; (c) film dimension change rela-tive to the substrate; (d) external stress is applied to the film to return to the sub-strate dimension; (e) film is reattached to the substrate and external stresses areremoved. As a result, (1) an internal biaxial stress remains in the film; (2) the inho-mogeneous stress state near the edges leads to a bending moment, which causesthe substrate to bend; and (3) a homogeneous stress in the substrate is devel-oped, which can be neglected for the thin-film limit. After Nix.[1]

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STRESSES IN THIN FILMS AND THEIR RELAXATION, KRAFT, GAO 367

stresses cause a bending moment that will lead to a curvature of the sub-strate. As discussed below, the measurement of this curvature is a com-mon technique to determine the average stress in the film. It should benoted that, under the assumed condition that the film thickness tf is muchsmaller than the thickness ts of the substrate (100 tf ts), stresses in thesubstrate are of minor importance.

For pure dilatational strain, that is, a volume change of the film with respectto the substrate, with the strain components being ∆exx ∆eyy ∆ezz ∆e andconsidering only elastic deformation for an isotropic material, the stress statein the film is given by:

s sxx syy 1

Ef

nf

e and szz 0, (1)

where Ef is Young’s modulus and nf is Poisson’s ratio of the film material.The stress component szz perpendicular to the film surface equals zerobecause the film is free to expand or contract in this direction. There are manypossible reasons for a relative volume change of the film material with respectto the substrate. The most important is change of temperature if the thermalexpansion coefficients of film and substrate are different:

e (af as)T aT, (2)

where ∆T is the imposed temperature change, and af and as are the thermalexpansion coefficients of the film and the substrate, respectively. Othermechanisms that cause dilatational strains include the annihilation of latticedefects such as excess vacancies, dislocations, and grain boundaries; phasetransformations; and composition changes. These mechanisms typicallyoccur during film growth or during an initial heat treatment until a stablemicrostructure has evolved. For the special case of heteroepitaxial films, thefilm is strained to accommodate the mismatch of the lattice parameters.

For crystalline materials with an elastic anisotropy, the film stressdepends on the orientation of a single crystalline film or the texture of apolycrystalline film. For cubic materials, grains oriented with both (111)and (100) directions parallel to the film normal are biaxially isotropic inthe film plane. For single crystal films with one of these orientations, auniform elastic strain leads to a uniform stress state, and Eq. (1) becomes:

s111 Y111e and s100 Y100e, (3)

with

Y111 C

6

1

C

1

4

4(C

21

C1

1

2

2C

41

C2)

44

(4)

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and

Y100 C11 C12 , (5)

where Y111 and Y100 are the biaxial moduli for (111) and (100) out-of-planeoriented crystals, respectively, and C11, C12, and C44 are the anisotropicstiffness coefficients of the cubic material. For instance, for Al and Cu atroom temperature, the biaxial moduli calculate to Y111 116 and 261GPa, and Y100 100 and 115 GPa, so that Y111Y100 1.2 and 2.3, respec-tively. From Eq. (3), it is evident that for a given ∆e, the ratio of stressesfor the two orientations is s111s100 Y111Y100. For a polycrystallinefilm, however, stress continuity requires s111 s100 at the grain bound-aries.[2] Let us consider stress and strain states in a film with a mixed tex-ture of (100) and (111) grains as shown in Fig. 8.2. In large grains, thestress in the interior of the grains reaches the expected value for therespective orientation [Fig. 8.2(a)]. Near the grain boundaries, therequired stress variation is confined to roughly the film thickness.[3]

Thus, in the limit of d tf (d is the grain size), the volume fraction ofthe film with inhomogeneous strain is small (isostrain condition). In fine-grained films [Fig. 8.2(b)], nearly complete stress accommodation isrequired, the stress will approach a uniform value everywhere (isostresscondition), and the strains will follow e100e111 Y111Y100. Note that nearthe interface, if perfect adhesion is assumed, a stress singularity wouldarise and shear stresses must develop on the grain boundaries. In a realpolycrystalline metal film, the grain size is typically on the order of thefilm thickness and, as a result, the average stresses in each orientation areexpected to lie between the isostrain and isostress limits. Finally, notethat the considerations above are valid for purely elastic behavior and,once plastic deformation of the film material is involved, an even broaderstress distribution can be expected.

8.2 Measurement Techniques

Stresses can never be measured directly. Any experimental determinationof a stress involves the measurement of an elastic strain and the use of appro-priate elastic constants to calculate the stress. Two main methods are used todetermine the stresses or strains in thin films. The first method is based on themeasurement of the curvature of the substrate, which is bent as a result of thefilm stress. The second method is based on the measurement of the elasticstrain directly in the film by x-ray diffraction. For the first technique, the elas-tic properties of the substrate material need to be known to determine the filmstress; for the second, those of the film material itself need to be known.

2C212

C11


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Figure 8.2 Schematics showing the stress and strain distributions during elasticdeformation for a film microstructure consisting of (a) large (111) and (100) grains(d W tf), and (b) alternating narrow (111) and (100) grains (d ≈ tf). In the limit oflarge grains, the average strains in both components are the same (isostraincase). In the limit of small grains, we assume that the grains are narrower than thefilm thickness such that the stress is the same in both orientations (isostresscase). After Baker et al.[2]

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8.2.1 Substrate Curvature

The substrate curvature technique makes use of the fact that the radiusof curvature, R, of the substrate is proportional to the stress s in the film,as quantitatively expressed by Stoney’s equation:[4]

s , (6)

where Es and ns are the Young’s modulus and Poisson’s ratio of the sub-strate material; tf and ts are the thickness of the film and the substrate,respectively; and tf must be much less than ts for Eq. (6) to be valid. Itshould also be noted that the substrate is usually not perfectly flat and Ro

is the radius of curvature of the substrate without the film. As Eq. (6)shows, an advantage of this method is that the only film property that mustbe known to evaluate the stress is the film thickness; knowledge of theelastic properties of the film is not required. However, the value measuredfor s represents the average stress in the film, and a statement on the uni-formity of the stress is not possible.

Another sample geometry, which is also frequently used, consists of acantilever beam with one fixed end. Then the film stress is given by:

s Es u, (7)

where L is the length and u is the deflection of the free end of the can-tilever beam.

To determine stresses accurately, the radius of curvature needs to bemeasured over a range from a few meters to several kilometers, or the dis-placement of the cantilever beam must be determined in the micrometerregime. This has been achieved experimentally in many different ways;for example, by optical methods;[5] profilometric, interferometric, or capac-itive methods;[6] and, for different conditions, during film deposition,[7, 8]

in vacuum, or at elevated temperatures.[9–12]

Figure 8.3 shows an experimental setup that has been used by a numberof authors[5, 12, 13] to measure film stresses as a function of temperature for dif-ferent thin-film materials. A laser beam is used to scan the surface of the sam-ple while it is heated. The curvature is determined from the orientation of thesample surface as measured by the angle of reflection of the laser beam usinga position sensitive detector. Figure 8.4 shows some typical results for thestress as a function of temperature for a thin metal film (a) and a thin glass

t s2

3L2tf

1Ro

1R

ts2

6tf

Es1 ns


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Figure 8.3 Schematic showing an experimental setup for wafer curvaturemeasurements.

Figure 8.4 Film stress as a function of temperature as determined from wafer cur-vature measurements for (a) a 1.0-µm-thick Cu film and (b) SiO2 films made bychemical vapor deposition on Si substrates.

Notes: 1. Data for (a) from Keller et al.[13]

2. Data for (b) from C. A. Volkert, unpublished work (1997)

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film (b) on Si substrates. At room temperature, the metal film (500-nm-thickCu film) is in a biaxial tensile stress state. On heating, the film tends toexpand more than the substrate and the tensile stress is reduced. At roughly150°C, the stress equals zero, and on further heating, the film stress becomescompressive. Up to a temperature of about 250°C, the film is deformed elas-tically. The slope of the curve is given by ∆aEf(1 nf), which is obtainedby combining Eqs. (1) and (2). Above 250°C, the film is deformed plasticallyand the compressive stress does not increase further. On cooling from themaximum temperature of 500°C, the film contracts more than the substrateand, as a result, the film stress becomes tensile. Again, the film is firstdeformed elastically and then plastically at temperatures below 400°C. Thegeneral trends of the described behavior are typical for many metal films (forexample, Al,[10] Cu,[14] or Ag[15]) because metals usually have a much higherthermal expansion coefficient compared to substrate materials such as Si orglass. In contrast, glass films on Si show a different behavior because the rela-tion of the thermal expansion coefficients is reversed. Figure 8.4(b) shows thecooling curves of two different glass films on a Si substrate. At high temper-ature, the films cannot support any stress because the glass transition tem-perature is exceeded and the film flows viscously. Reaching the glass transi-tion temperature, the films are deformed elastically and stresses are produced.Here, the films tend to contract less than the substrate and compressivestresses arise. Note that the stress temperature curve is not linear because thethermal expansion coefficient and Young’s modulus are not constant in thewide range of temperatures investigated. This example shows nicely that themeasurement of film stresses is a powerful tool to study film properties. Forexample, from the curves in Fig. 8.4(b), the glass transition temperature forthe two CVD-SiO2 films with different dopants can be determined.

The elastic or reversible portions of the thermal cycling data can be usedto estimate the product ∆aEf(1 nf). If, for instance, two different substrateswith known but different thermal expansion coefficients are used, the valuesof af and Ef(1 nf) can be determined independently for the film.[16]

The interpretation of the stress temperature behavior to obtain infor-mation about plasticity is not straightforward. Because the temperatureand strain are changed simultaneously, the measured stress is a complicatedfunction of film strength, time- and temperature-dependent creep processes,and possibly strain hardening, voiding, or cracking events. Nevertheless,useful information has been gained from thermal cycling experiments bysimply approximating the measured stress as the film strength at a giventemperature.[9] Another common experiment is to interrupt the tempera-ture cycle at a certain temperature and to observe the stress relaxation.[10, 11]

In this situation, the total film strain is held constant but elastic strain ischanged into plastic strain corresponding to a stress relaxation. Figure 8.5illustrates such an experiment, showing the stress relaxation at three dif-


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ferent temperatures for the same film/substrate specimen after coolingfrom a maximum temperature of 500°C. It can be seen that a strong initialstress drop is observed, followed by a much slower further decrease instress. Mechanisms for plastic deformation and stress relaxation in metalthin films are discussed in Sec. 8.3.

8.2.2 X-Ray Diffraction

The most common x-ray technique to determine stresses in thin filmsis the so-called sin2ψ method.[17–19] This method is based on the measure-ment of lattice spacings with a resolution better than 104 Å. This is illus-trated in Figure 8.6, which shows a thin crystalline film stressed in tension[Fig. 8.6(a)]. The spacing of lattice planes that are perpendicular to thefilm plane is increased, due to this tension, while the spacing of planesparallel to the film is decreased, due to Poisson’s contraction. For an equi-

Figure 8.5 Stress relaxation in a 0.6-µm-thick Cu film on a Si substrate at varioustemperatures as measured by the wafer curvature technique. Data from Keller et al.[11]

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biaxial stress state, which is often present in thin films, the lattice spacingdψ depends linearly on sin2ψ [Fig. 8.6(b)], where ψ is the angle betweenthe normals of the measured lattice planes and the film surface.Comparing the strained lattice spacings dx and dz for ψ 1 and ψ 0,respectively, to the unstrained lattice spacing do allows us to determine thestrains exx eyy and ezz in and out of the plane of the film:

exx (8a)

ezz . (8b)dz do

do

dx do

do


Figure 8.6 Schematic representation of the sin2ψ method for a textured thin film.

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Furthermore, it is possible to calculate the stress state of the film using theelastic constants of the film material. This method works particularly wellfor single-crystalline or textured films with an orientation that results in anisotropic biaxial modulus of the film. This is the case for cubic materialswith (111) and (100) planes parallel to the surface. For the latter, thestresses are given by:

sxx C11exx C12eyy C12ezz (9a)

syy C12exx C11eyy C12ezz (9b)

szz C12exx C12eyy C11ezz. (9c)

Note that for a (100)-oriented crystal, the Cartesian coordinate systems of thecrystal and the sample have the same orientation. This is no longer the casefor a (111)-oriented crystal or texture. Therefore, the stiffness coefficientsneed to be transformed from the crystal to the sample coordinate system.Then the stress components in the sample coordinate system are calculatedto be:

sxx C11exx C12eyy C13ezz, (10a)

syy C12exx C11eyy C13ezz, (10b)

szz C13exx C13eyy C33ezz, (10c)

where C11,C12,C13, and C33 are the transformed stiffness coefficients.[19]

It is obvious from Eq. (8) that the accuracy of the measurementdepends very strongly on the value used for the unstrained-lattice spacing do.Since this value depends sensitively on temperature and material impurities,it is advantageous to determine do for each stress measurement. Thisleads also to a reduction of the influence of systematic errors in theexperimental setup related to the alignment and calibration of thegoniometer and detector. For a thin film with a strong texture, it is rea-sonable to assume that the out-of-plane stress component szz equals zero.Assuming an equi-biaxial stress state, it can be shown that the unstrainedlattice spacing is given for sin2ψo 2C13(C33 2C13), where theappropriate stiffness coefficients for the sample coordinate system areused. This is illustrated in Fig. 8.7(a), which shows a plot of d versussin2ψ from measurements on a thin Al film on a Si substrate[20, 21] at tem-peratures of 20, 200, and 440°C. The curves with the negative slopeswere obtained on heating, and the ones with positive slopes on cooling,indicating compressive and tensile film stresses, respectively. The stressas a function of temperature obtained from these measurements is shownin Fig. 8.7(b).

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For films with a mixed texture, such as (111) and (100), as oftenobserved in Cu thin films, measurements for each texture componentbecome necessary.[2, 14, 22] Assuming only minor stress interactions betweenthe components, the average stress for each orientation can be determined.Figure 8.8 compares results of wafer curvature and x-ray measurementson identical samples.[2, 13] A passivated Cu film with 1.0 µm thickness anda mixed (111) and (100) texture was subjected to a temperature cycle upto 600°C. As shown, the x-ray method allows the average stress for thetwo texture components to be measured, while the wafer curvature tech-niques give the average film stress.

The details of the stress analysis for films that have other mixed tex-tures or other crystal structures can be more complicated. A more detailedtreatment of the sin2ψ method including grain interaction and elasticanisotropy is given in Clemens and Bain[18] and Van Leeuwen et al.[23]

Finally, it should be pointed out that x-ray measurements can be used todetermine more complex stress states such as those present in narrow con-ductor lines.[19, 20, 24]

The x-ray measurements described so far typically use x-raybeams with a size on the order of mm. As a result, the measured stressis averaged over many grains. This allows us to study the stress behavior


Figure 8.7. (a) Spacing of the (422) lattice planes in a 0.7-µm-thick Al film with a(111) texture as a function of sin2ψ for three different measurements on heatingand cooling. On heating, the film is under compression (lines with negative slope);on cooling, it is under tension (lines with positive slope). The intersection of thelines is associated with the unstrained lattice spacing for each temperature.(b) Stress as a function of temperature as obtained from the sin2ψ method. Datafrom Kraft and Nix.[21]

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of different texture components, but not stress variations from grain tograin. More recently, several techniques have been developed to meas-ure stresses with very high spatial resolution.[25–27] For some of thosemeasurements, performed with a submicrometer resolution, white syn-chrotron x-ray radiation was used. In order to determine stresses, thedeterioration of the Laue diffraction patterns obtained was analyzed.As an example of such a measurement, Fig. 8.9 shows the stress dis-tribution in an area of approximately 15 by 15 µm2 in a thin Al film asa function of temperature.[28] It can be seen that there are very strongvariations of more than 100 MPa in stress from grain to grain. Thesevariations may correspond to different orientations, but even for grainswith the same orientation, significant differences in stress areobserved.

Figure 8.8 Stress-temperature evolution in a 1-µm-thick passivated Cu film:Comparison between the film stress as measured by the wafer curvature tech-nique and the stress as measured by x-rays in (111)- and (100)-oriented grains.Data from Baker et al.[2] and Keller et al.[13]

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8.3 Stress Relaxation

As described above, the elastic behavior of thin films is basicallyunderstood and does not differ from bulk behavior. In contrast, the plasticbehavior of thin metal films is not well understood and is fundamentallydifferent compared to that of bulk materials. This section summarizes thecurrent understanding of thin-film plasticity. Contributions from disloca-tion plasticity and creep deformation to the relaxation of stresses are dis-cussed, as well as the experimental evidence.

8.3.1 Experimental Observations

Figure 8.10 shows the stress evolution in Cu thin films on a Si substrateand exemplifies typical behavior of metal films on low-thermal-expansion-


Figure 8.9 Thermal cycling results of a 15 15 µm2 area of an Al(Cu) bond pad(blanket film) showing an averaged biaxial stress component vs. temperature. Theinsets show details of the stress distribution in the film at different temperatures.(The two-dimensional maps are plots of measurements of the in-plane stress.)Note the large stress inhomogeneities from grain to grain and even within individ-ual grains. From Tamura et al.,[28] with permission.

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coefficient substrates. These films have a silicon nitride underlayer andare either capped by an aluminum oxide layer, which was formed by self-passivation of a Cu-1 at.% Al film,[29–31] or are uncapped. The stresseswere measured by the wafer curvature technique. At the beginning of theexperiment, after sputter deposition and annealing at 600°C, the films areunder tensile stress at room temperature. Because of the larger thermalexpansion coefficient of the film compared to the substrate, the stressdecreases linearly with increasing temperature. The film undergoes bothelastic and plastic deformation during thermal cycling. In general, thestresses in the capped film are much higher than in the uncapped onethroughout the experiment. At high temperature, this difference has beenattributed to diffusional creep,[13, 30, 31] which is active in the uncapped filmand is described in detail in Sec. 8.3.3. The plastic deformation in thecapped film can be assumed to be mediated only by dislocation motionsince the presence of the passivation layer suppresses surface diffusion,which is required for diffusional creep (Fig. 8.10 and Sec. 3.3). Often, thestress-temperature evolution is regarded as a measure of film strength asa function of temperature.[9] However, this picture is not completely accu-rate because the film stress relaxes quite substantially when temperatureis held constant.[10, 32] Therefore, the time dependence of the plastic defor-mation needs also to be taken into account. Furthermore, the stress-temperature evolution is influenced by strain hardening[13, 33, 34] as the film

Figure 8.10 Wafer curvature experiments have shown stress temperature curvesthat are strongly dependent on the surface passivation of Cu thin films on sub-strate. Data are given for pure Cu and Cu-1 at.% Al films where Al tends to seg-regate toward the surface and form an oxide cap. The film stress was measuredby the wafer curvature technique. Data from Weiss et al.[31]

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is plastically strained; for instance, the total thermal strain for the filmsshown in Fig. 8.10 is 0.7% and the plastic strain is about 0.5%.

Beside thermal straining, mechanical properties of thin metal filmshave been measured by micro-tensile testing; for example, results havebeen reported for Ni,[35] Al,[36] Cu,[35, 37] and multilayered films.[38, 39] Theseexperiments confirmed that thin films exhibit very high yield strength,which usually increases with decreasing film thickness, grain size, and/orlayer thickness in multilayered thin films. As an example, Fig. 8.11(a)shows the stress-strain curve of freestanding 1.1-µm-thick Cu film.However, in such tests on freestanding films, the deformation is not con-strained by a substrate and, furthermore, testing in compression is not pos-sible. It has been shown that this can be overcome by testing thin films ona polyimide substrate.[40] Figure 8.11(b) shows the stress-strain curveof such an experiment on a 0.7-µm-thick Cu film obtained using x-raymeasurements. On loading, the film deforms first elastically until yieldingat about 250 MPa is observed. Then, on straining to 0.5%, the flow stressincreases to 400 Mpa, indicating the presence of strain-hardening effects.On unloading of the specimen, the contracting elastic substrate com-presses the film, which then undergoes plastic deformation in the oppositedirection. It appears that on reverse loading, the yield strength is some-what smaller compared to the initial loading. For interpreting mechanical


Figure 8.11 Stress evolution as a function of applied strain during tensile tests for(a) a freestanding Cu film with a thickness of 1.1 µm and (b) for a 1.0-µm-thick Cufilm on a polymer substrate.The film stress for (b) was measured by x-ray diffraction.

Notes: 1. Data for (a) from Read.[37]

2. Data for (b) from M. Hommel, O. Kraft, and E. Arzt, “A new method to study cyclic deforma-tion of thin films in tension and compression,” J. Mater. Res., 14(6):2373–2376 (1999).

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behavior, the advantage of this test method over thermal straining isrelated to the fact that the temperature is not altered during the experi-ment. In particular, this allows us to attribute the observed asymmetry inyield stress to hardening effects since any influence from the temperaturechange in the thermal cycling experiments is eliminated.

8.3.2 Dislocation Plasticity

As a start, we consider the motion of a single dislocation in a single-crystalline thin film attached to a substrate, as shown in Fig. 8.12(a) and

Figure 8.12 Schematic representation of the motion of a single dislocation in asingle-crystalline film, according to the model of Nix and Freund[1, 45] for a film with(a) a free surface and (b) a cap layer. It indicates in (a) that a dislocation segmentwith the length dx needs to be created when the dislocation moves by dx.

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(b) for a free and an impenetrable surface, respectively. As the dislocationsadvance in the film by “channeling,” they create additional dislocationsegments along the bottom and the top surface if it is impenetrable. Thismechanism has been observed for the motion of dislocation in large-grained polycrystalline Al films on amorphous substrates[9] and in epitax-ial Al films on Si[41] or Al2O3 substrates.[42] For this last example, Fig. 8.13shows a dislocation, which has channeled over a distance of severalmicrometers. It appears that there is a repulsive force from the interfaceon the dislocation as it stands off by as much as 100 nm. The contrastclose to the interface indicates the presence of other dislocations, whichmay be related to the epitaxial misfit.

In contrast, it was observed that in polycrystalline films with grainsize of the order of the film thickness, dislocations did not channel.Kobrinsky and Thompson[32] describe the dislocation motion below150°C in thin Ag films (200 nm in thickness) as jerky. As observedduring in situ transmission electron microscopy (TEM), dislocationsare pinned by obstacles and do not move most of the time, until a sud-den jump of a dislocation segment between two pinning points isobserved. The typical pinning point and jump distances were bothfound to be between 50 and 100 nm, which are significantly smallerthan the film thickness or grain size. These observations are confirmedby in situ TEM, both in cross section as well as in plan view, on poly-crystalline Cu films that were deposited onto amorphous SiNx layerson Si substrates.[42, 43] Figure 8.14 shows the typical dislocation distri-bution after cooling from 600 to 130°C in such a film. At tempera-


Figure 8.13 Dislocation in a 350-nm-thick Al film that was grown epitaxially on a(0001)-oriented Al2O3 substrate. Glide of a dislocation on the Al plane, which isinclined ∼70 degrees to the (111)Al G (0001)Al2O3 interface, created a dislocationsegment nearly parallel to the interface. Contrast near the film/substrate interfaceindicates the presence of other, possibly misfit, dislocations. The cross-sectionalTEM image is from Dehm et al.[42] with permission.

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tures below 220°C, short segments of tangled dislocations and ajerky dislocation motion were observed. It was also pointed out thatno interfacial dislocation segments were found, which would beexpected for the dislocation motion described by Fig. 8.12.Furthermore, cross-sectional in situ TEM revealed that dislocationsare attracted rather than repelled by the Cu/SiNx interface and, again,no evidence for interfacial dislocation segments was found.[43] Incontrast, Shen et al.[34] and Weihnacht and Brückner[44] observedinterfacial dislocations and dislocation pileup in plan-view andcross-sectional TEM, respectively.

Nix[1] and Freund[45] have derived the following formalism for themotion of a single dislocation in a thin film, as shown in Fig. 8.12. As thedislocation channels through the film, an interface dislocation segment iscreated. As a result, the critical resolved shear stress ty to move a disloca-tion in a thin film depends on its thickness h and is given by:

ty, Nix 2p(

b1Geff

n)

sinh

j, (11)

Figure 8.14 Dislocations in Cu grains at (a) 600°C and (b) 130°C. No interfacialdislocations deposited by advancing threading dislocations are discernable in theimages. At elevated temperatures (a), dislocations appear to be longer and moremobile than at lower temperatures (b), where dislocation motion became jerky anddislocation tangles formed. Plan-view TEM images are from Dehm et al.,[42] withpermission.

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where

Geff lnxbsh; (12)

j is the angle between the glide plane normal and the film normal; b is themagnitude of the Burgers vector; n is Poisson’s ratio; G and Gs are theshear moduli of the film and the substrate, respectively; and xs is a numer-ical constant close to unity that defines the cutoff radius of the stress fieldof the dislocation at the film/substrate interface. Extending this model topolycrystalline thin films, Thompson[46] suggested that additional disloca-tion segments must be created at the grain boundaries. This reasoningresults in the following critical shear stress:

ty,Th sin

hj

, (13)

with

Wd 4p(

G1

b

2

n) ln , (14)

where d is the grain size and Wd is the line energy of the dislocation seg-ments along the grain boundaries and the film/substrate interface where,for the sake of simplicity, Thompson assumed that the effective modulusequals the film modulus (G Geff). As can be seen from Eqs. (11) and(13), these models predict nearly a 1/h and/or a 1/d dependence of theyield strength. A similar result has been also obtained by Chaudhari.[47]

An experimental validation of these models has been given byVenkatraman and Bravman[9] for coarse-grained Al films or by Dehmet al.[48] for epitaxial thin Al films on Al2O3 substrates. However, for mostpolycrystalline fine-grained films, the model tends to underestimate theyield strength.[13, 22] Kobrinsky and Thompson[32] pointed out that it is notadequate to use the simple picture shown in Fig. 8.12 for dislocationmotion in thin films to predict their strength. This is because a much morecomplex dislocation behavior is observed in TEM, including the jerkymotion of dislocations at low temperatures and the absence of interfacedislocations. Also, time-dependent stress relaxation in thin films is notaddressed by this model. Therefore, Kobrinsky and Thompson suggestthat dislocation-mediated plasticity in thin metal films is significantly

db

2d

Wdb

GGsG Gs


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affected by thermal activation of the dislocation glide. Note that thismechanism was previously discussed by Flinn et al.[5] and Volkert et al.[10]

to control the stress-temperature behavior of Al thin films. The thermally activated glide of dislocations implies that the applied

shear stress is not large enough to drive a dislocation through an array ofobstacles, such as forest dislocations or particles. As a result, dislocations arepinned at these obstacles. However, thermal activation may provide the addi-tional energy to help a dislocation segment to overcome the obstacle. Thisresults in the observed jerky motion as a dislocation moves step by step fromobstacle to obstacle. A constitutive law for the thermally activated glide canbe given if a certain obstacle “shape” is assumed. For rectangular obstacles,the plastic deformation rate e is given by Frost and Ashby [49, p. 8]:

e eo exp 1 sts, (15)

where eo is a characteristic constant, ∆F is the activation energy at zerostress, t^ is the critical shear stress without thermal activation, and s is theSchmid factor.

Figure 8.15 compares this approach and experimental data for self-passivated 0.5 and 1.0 µm thick Cu 1 at.% Al films. The starting points on

FkT

Figure 8.15 Stress evolution as a function of temperature for Cu films with a caplayer and a thickness of (a) 1.0 mm and (b) 0.5 µm. The film stress was measuredby the wafer curvature technique.[29] The dashed lines represent modeled curvesfor thermally activated dislocation glide. Parameters used were for bulk Cu:[49]

eo 1 106 s1, ∆F 3.5 1019 J.The Schmid s 0.27 is for (111)-oriented grains.t^ was adjusted to 150 and 235 MPa for the 1.0- and 0.5-µm-thick films, respectively.

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heating were adjusted to the experimental value. It can be seen that a reason-able agreement is obtained, although the compressive plateau is not well rep-resented. The agreement for the two film thicknesses was obtained by adjust-ing t^ to 150 and 235 MPa, respectively. Using Orowan’s classical result thatthe critical shear stress depends on the pinning point distance L as t^ GbL,the corresponding pinning point distances are 80 and 50 nm, respectively.These values are indeed much smaller than the film thickness and of the sameorder of magnitude as the ones determined by in situ TEM on Ag films.[15, 43]

However, the approach presented here does not take into account that the pin-ning point distance changes during heating and cooling (see Fig. 8.14) as aresult of strain hardening and possibly recovery processes.

Strong kinematic strain hardening is a competing approach to modelthe stress-temperature curves for explaining some of the observedeffects. Shen et al.[34] suggested that the increase in flow stress on cool-ing is related to the buildup of a back stress sb, which lowers the yieldstrength on reverse loading. This behavior is represented in Fig. 8.16:


Figure 8.16 Stress-strain curve given by the kinematic strain-hardening model inShen et al.[34] under cyclic loading, compared to the experimental data from Fig. 8.2.

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After the stress reaches sy on loading, plastic deformation occurs with alinear increase in flow stress. Due to this increase, a back stress sb

builds up. On reverse loading, the yield strength in compression islowered by this back stress and the film yields at sreverse sy sb.Figure 8.16 also shows that this predicted behavior is in agreement withthe cyclic stress-strain behavior of the Cu films on polyimide substrates.Shen et al. have shown in their original work that their model can beapplied fairly well to model thermal stress cycles of Cu films when atemperature-dependent yield strength is assumed. The incrementalincrease in stress, that is, the hardening rate, in the elastic-plastic regimeis given by:

∂s ,

(16)

where M is the biaxial modulus of the film, e* is the hardening parame-ter, and sy(T) is the temperature-dependent yield strength. The yieldstrength was assumed to decrease linearly with temperature, that is,sy(T ) so(1 TT*), where so is the yield strength at T 0 K and T*

is the temperature at which the yield strength becomes zero. Hence, e*,so, and T* can be used to fit the model empirically to experimental data.Nix and Leung[50] have adapted this approach and tried to reduce theempirical fit parameters. They suggested that the yield strength isrelated to the dislocation motion in the film, as given by Eq. (11).Therefore, its temperature dependence is related to the temperaturedependence of the shear modulus, which can be assumed to be G Go(1 c*(T 300)Tm), where Go is the shear modulus at room tem-perature, c* is a constant close to 0.5 for most FCC metals, and Tm is themelting temperature.

Figure 8.17 shows a quantitative comparison of this approach tothe data from Fig. 8.15. Again, a reasonable agreement betweenexperiment and model can be seen. In particular, the plateau in com-pression is well reproduced. Note that the hardening rate, which isadjusted to fit the experimental data, is of the order of the shearmodulus of Cu, which is about two orders of magnitude larger thanthe hardening rate in bulk Cu. Furthermore, this approach does notallow us to account for any time-dependent deformation; therefore, itis not possible to account for strain rate effects or stress relaxation.More recently, Kraft et al.[51] suggested that both thermally activateddislocation glide and strain hardening be included by modifying

a∂T

M

1

sey(

*

T)

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Eq. (15) as:

e eo exp 1 s(s

t^sb) , (17)

where sb is the back stress acting on dislocations in the active slip system,which can be, for example, described to depend linearly on the plastic

strain epl: sb ts

^

ee

p*l, as suggested by Shen et al. [see Eq. (16)]. The influence

of the back stress is also illustrated in Fig. 8.18, where the behavior lead-ing to the two stress-temperature curves is identical except for the pres-ence (solid curve) or absence (dashed line) of the back stress. Similar toFig. 8.17, the effect of the back stress leads to the asymmetry in the mag-nitude of the stresses on heating and cooling, and the high stresses areobtained by adjusting L to 50 nm for t^ GbL. (Compare to Fig. 8.15.)However, the approach presented here does not take into account that thepinning point distance changes during heating and cooling (see Fig. 8.14)as a result of strain hardening or recovery processes.

Baker et al.[52] pointed out that channeling of a single dislocation, asshown in Fig. 8.12, is completely reversible. As a result, a reduction of an

FkT


Figure 8.17 Experimental data from Fig. 8.15 compared to the kinematic strain-hardening model.[34] The solid line represents the model using the following param-eters: G 45 GPa, e* 0.01, M 220 GPa. (This value was adapted to fit thethermoelastic region.) The dashed lines indicate the yield strength as a function oftemperature.

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applied stress below the critical stress, given by Eq. (11), should lead tospontaneous removal of the dislocation. This phenomenon leads to a plas-tic deformation in the reverse direction to the applied stress. Such a behav-ior has been observed in a variety of face-centered-cubic metal films,which were contaminated by oxygen, with the role of the oxygen not yetunderstood. This effect should also influence dislocation interactionmechanisms, which may lead to the strong (kinematic) hardening effectsin thin films. Figure 8.19 shows three scenarios that lead to hardeningbehavior comparable to the experimental observations: (a) A channelingdislocation has to overcome interfacial dislocations.[53] (b) The interfacialdislocation of a channeling dislocation is deposited in an existing array ofparallel interface dislocations.[44] (c) The nucleation of dislocation occursat a Frank-Read source located in the interior of a grain. The repeated

Figure 8.18 Stress vs. temperature change for a film on a substrate, illustrating thedifference between response according to the standard plastic rate equation forthermally activated dislocation glide [Eq. (15), dashed line] and a rate equationmodified by inclusion of a back stress, depending on plastic strain [Eq. (17), solidline]. The parameters used were the same as in Fig. 8.15 and e* 0.005.

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emission of dislocation loops leads to a back-stress onto the dislocationsource,[54, 55] making the nucleation event more difficult from time to time.

So far, the models presented, which are intended to describe thestrength and hardening behavior of thin metal films, are based on the con-cept of misfit dislocations; they imply that the misfit dislocation at theinterface to the amorphous substrate still supplies a stress field.


Figure 8.19 Schematic representation of possible dislocation arrangements lead-ing to strain hardening in a thin film on a substrate.

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Dislocation core structure is known to play an important role in thestrength of solids. The classical dislocation core models are based on aline defect in a crystalline structure. Such models do not apply in the caseof amorphous solids. Since, in many cases, misfit dislocations are locatedat an interface between a crystalline film and an amorphous substrate (oran adhesive or functional layer between film and substrate), a fundamen-tal question arises: What is the equilibrium core structure of dislocationsat an interface between crystalline and amorphous materials? At a crystalline-amorphous interface, as in the case of Al or Cu films deposited on anaSiOx or aSiNx substrate, the core of a misfit dislocation may spread alongthe interface, as illustrated in Fig. 8.20. The substrate has no simple crys-tallographic relationship to the film and hence may be thought of as acontinuum. Dashed lines have been drawn on the substrate to mark theoriginal positions of the atomic planes in the film. Figure 8.20 depicts anedge dislocation with Burgers vector b parallel to the interface, climbingtoward the substrate under an applied load and then spreading its corealong the interface. The dispersed dislocation core may be modeled as an

Figure 8.20 Schematic of dislocation core spreading at an incoherent interfacebetween a crystalline and an amorphous solid. (a) The dislocation climbs downtoward the substrate but cannot penetrate it. (b) If it is possible for sliding to occurat the interface, the dislocation core may spread into the interface to a width 2c.

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array of infinitesimal dislocations, as described by Eshelby.[56] The extentof spreading can be characterized by the half-width, c, of the slippedregion.

The dislocation core spreading illustrated in Fig. 8.20 requires that thematerial on one side of the interface be able to slide by small amounts (b)relative to the material on the other side of the interface. This cannot occurover large distances in real coherent interfaces. However, for incoherentinterfaces between crystalline and amorphous solids, small relative displace-ments are possible, and dislocation core spreading may occur over large dis-tances. When a shear strain is applied across the interface, we can imaginethat some film-substrate bonds are broken and new, energetically morefavorable bonds are formed in their place. Since only a few bonds move, themean position of the plane can move by an amount that is much less than b.

A wide range of film-substrate systems have incoherent interfaces andmay be susceptible to dislocation core spreading following a mechanismlike that illustrated in Fig. 8.20. Transmission electron microscopy (TEM)investigations have shown dislocations disappearing at the interfacebetween an Al film and its native Al2O3 passivation[57] and at the interfacebetween a Cu film and a SiNx barrier layer.[43]

Gao et al.[58] have developed a mathematical model to describe thetime-dependent process of dislocation core spreading along a crystalline-amorphous interface. The model is briefly described here. We consideredge dislocations spreading along an interface characterized by a shearadhesive strength, t0, below which no core spreading occurs and abovewhich spreading takes place in a viscous manner. The equilibrium corewidth and the rate of core spreading for single dislocations and a periodicarray of dislocations can be calculated as a function of interface adhesion.

We consider an edge dislocation spreading along an interface betweena crystalline and an amorphous solid, as depicted in Fig. 8.20. At timet 0, the dislocation core structure is compact. Subsequently, deformationnear the film-substrate interface is assumed to be described by a viscousmodel. This problem has been treated in detail by Gao et al.[58] for a singleor planar array of dislocations in a homogeneous bulk material or at theinterface between a thin film and a semi-infinite substrate, where the filmand substrate may have the same, or different, elastic constants. Some ofthe solutions are analytic and others are based on an implicit finite differ-ence method. For an edge dislocation spreading along an interface betweena crystal and an amorphous material with the same elastic constants, thesolution shows that the equilibrium core width for a single dislocation is:

ce , (18)Eb

8(1 n2)t0


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and for a period array of edge dislocations with spacing L is:

ce sin1tanh . (19)

We observe that the equilibrium core width scales inversely with the inter-face adhesion t0. The stronger the interface, the more compact the core. Theanalyses by Gao et al.[58] and Baker et al.[59] show that the viscous spreadingof a single dislocation core can be described by the change in the half width,

c(t) 1 0.3634 exp t, (20)

of the slipped zone, where h and ti are the viscosity and the thickness ofthe interface, respectively. At t 0, the half-width

c(0) 4p(1

E

bn2)t0

(21)

defines a region over which viscous slip occurs immediately because theshear stress associated with a compact dislocation core exceeds the interfaceadhesion t0. Although the slipping width c(t) is somewhat ad hoc in repre-senting the core width, the evolution Eq. (20) provides the insight that thetime scale over which the core spreading occurs is largely determined by theratio ht0. Since viscosity usually decreases with temperature, we concludethat core spreading occurs faster at higher temperatures and stronger inter-faces, and slower at lower temperatures and weaker interfaces. It can be fur-ther argued that dislocation core spreading reduces the effectiveness ofhardening mechanisms, as discussed in Figs. 8.12 and 8.19.[58] As a result,dislocation-mediated plasticity in thin films is immediately related to diffu-sional processes as well as to the properties of surfaces and interfaces.However, a complete understanding of dislocation plasticity is still notachieved and an incorporation of thermally activated core spreading in thedislocation models described above has not yet been attempted.

8.3.3 Diffusional Creep

As discussed in Sec. 3.1 and shown in Fig. 8.10, wafer curvatureexperiments have shown that the thermal mechanical behavior of Cu

9t0tihb

Eb8(1 n2)t0

pEb8(1 n2)t0L

Lp

Ch_08.qxd 11/29/04 6:45 PM Page 393

thin films on substrates depends strongly on the passivation of the filmsurface. This phenomenon has recently been explained by a specialkind of Coble-creep in thin films on substrates, often referred to as theconstrained diffusional creep.[31, 60] The continuum model for con-strained diffusional creep developed has adopted the hypothesis thatuncapped, pure Cu films are subject to active surface and grain boundarydiffusion, which causes transport of material between the film surfaceand the grain boundaries.[60] Diffusion in thin films on substrates is fun-damentally different in nature from that in bulk materials. A constraintis imposed by the usually perfect bonding between film and substrate(for example, material sequence Cu-aSiNx-aSiOx-Si,[61–63] whichimplies that no sliding occurs at the film-substrate interface. In contrastto previously proposed models of diffusion in thin films,[64] the con-straint that no sliding occurs at the film-substrate interface renders thediffusion in thin films an inherently transient problem. Steady-statesolutions frequently used to describe grain boundary diffusion may notbe used. An additional constraint is that material transport cannot pro-ceed in the substrate and diffusion has to stop at the film-substrateinterface. Constrained diffusional creep leads to formation of a newmaterial defect called the grain boundary diffusion wedge, which alsoplays an important role in dislocation nucleation processes in thinfilms.[60] This process corresponds to the classical Coble creep in a thin-film structure, except that the roots of the grain boundaries are con-strained not to slide with respect to the substrate. In the remainder ofthis section, we briefly describe the theory of constrained diffusionalcreep as well as the grain boundary diffusion wedge and its associateddislocation mechanisms.

In the model of Gao et al.,[60] constrained diffusional creep under aconstant nominal stress in the film is considered and diffusion is modeledas dislocation climb in the grain boundary. The solution for a single edgedislocation is used as the Green’s function to construct a solution withinfinitesimal Voltera edge dislocations. For an inserted grain boundarywedge of width 2u(z,t) in a film with thickness h(0 z h), the stressalong the grain boundary is given by:

sgb(z,t) s0 h

0

S(z, z )∂u

∂(zz,t)

dz, (22)

where S(z,z) is a Green’s function kernel for the continuous dislocationproblem (Cauchy kernel), G is the shear modulus, n is Poisson’s ratio, ands0 is the stress in the absence of diffusion. Balancing the diffusive atomicflux with the displacement rate (mass conservation), the diffusion equation

G2p(1 n)


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based on Coble creep,[65]

∂u

∂(zt,t)

, (23)

is woven into the problem. Here, dgbDgb denotes the product of grainboundary thickness and atomic diffusivity, t is time, Ω is the atomic vol-ume, and kT has the usual meaning. Combining Eqs. (22) and (23) yieldsan integro-differential governing equation for constrained diffusionalcreep:

h0

S(z,z) ∂3s

∂g

zb(

3

z,t) dz, (24)

which describes the time evolution of grain boundary stress under the fol-lowing boundary conditions:

sgb(0,t) 0, sgb(h,t) sgb(h,t) 0 (25)

and an initial condition

sgb(z,0) s0. (26)

The boundary conditions in Eq. (25) correspond to continuity ofchemical potential at the junction between the film surface with the grainboundary, as well as no mass flux and no sliding at the film-substrateinterface. The initial condition in Eq. (26) defines a Green’s functionsolution to a step loading on the thin film. The solution can be used laterto construct more general solutions to an arbitrary time-dependent load-ing s0(t) via a convolution scheme.[31] The solution to Eq. (26) shows thatthe insertion of a material wedge into the grain boundary causes expo-nential stress decay with a characteristic time scaling with the cube of thefilm thickness,

t0 . (27)

The diffusion wedge represents columns of atoms corresponding to anarray of climb-edge dislocations that have been transported into the grain

2p(1 n)kTh3

GdgbDgbΩ

GdgbDgbΩ2p(1 n)kT

∂sgb(z,t)

∂t

∂2sgb(z,t)

∂z2

dgbDgbΩ

2kT

Ch_08.qxd 11/29/04 6:45 PM Page 395

boundary. With respect to the lattice distortion around the diffusionwedge, the dislocations in the grain boundary exemplify geometricallynecessary dislocations[66, 67] that are usually associated with nonuniformplastic deformation. Gao et al.[60] used an eigenmode expansion to obtainan integro-differential eigenvalue problem that is subsequently trans-formed into a standard Cauchy-type singular equation to be solved by aGauss-Chebyshev quadrature scheme. Gao et al. provide details of thenumerical procedure.[60] A simplified solution,

sgb(z,t) s0eltt0 f (zh), (28)

can be used for all practical purposes, where the function f(x) isdefined as:

f (x) 31 x2, (29)

the parameter l is related to the film thickness h and grain size d as:

l 11.72 1.69(hd) 73.09(hd)2 0 hd 0.28.10 30.65(hd) 0.2 hd 10. (30)

Averaging Eq. (28) over the film thickness shows that the average stresson the grain boundary decays exponentially as:

sgb(t) s0eltt0. (31)

Therefore, on the scale of a characteristic time t0l, the traction along thegrain boundary is fully relaxed and the diffusion wedge exhibits the sametraction-free condition as a crack, indicating a singular stress field nearthe root of the grain boundary. This leads to extraordinarily largeresolved shear stresses on planes parallel and close to the film-substrateinterface and, in the (111)-textured films, can cause emission of parallelglide dislocations. Although the resolved shear stresses on the parallelglide planes induced by a diffusion wedge are similar to those caused bya crack, differences in the dislocation nucleation process may arise. Themechanisms involved are still not well understood and are currentlybeing investigated.

Experimentally, diffusional creep as a major stress relaxation mecha-nism in uncapped thin copper films has been verified in a number ofstudies.[11, 15, 31, 61, 63] Experiments have indicated that, while plastic deformation

l3p


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in passivated or relatively thick copper films is characterized by the“classical” threading dislocations, parallel glide dislocations becomeincreasingly dominant in unpassivated thin films with thicknesses of lessthan 400 nm.[61–63] The experimental observations of parallel glide dislo-cation emission strongly indicate that constrained grain boundary diffu-sion is a dominant stress relaxation mechanism for unpassivated thin filmson substrate. Figure 8.21 shows an in situ TEM observation of parallelglide dislocations being emitted from a grain boundary source during thecooling portion of the second thermal cycle of the Cu-aSiNx-aSiOx-Si(100) system. The experiments have also indicated that nucleation ofparallel glide dislocations is very selective to the type of grain boundary.Grain boundaries with 6- and 66-degree tilt angles seem to be preferrednucleation sites for parallel glide dislocations. The smallest film thicknessinvestigated is about 35 nm.

In modeling a thermal cycling experiment, both the nominal stress s0

and temperature (hence the characteristic time t0 also) vary with time, andEq. (31) cannot be directly used. Weiss et al.[31] have developed an inte-gral convolution scheme to calculate the grain boundary stress and haveused it to determine the average stress over the entire film using an equation

Figure 8.21 During the cooling portion of the second thermal cycle of the Cu-aSiNx-aSiOx-Si(100) film-on-substrate system, parallel glide dislocations were emitted fromthe grain boundary source. (a) At 170°C, the first dislocation has been emitted andis pinned in the middle. (b) The second emitted dislocation has quickly overcome thepinning point and has pushed the first dislocation forward by the time 150°C isreached. (c) At 135°C, the third dislocation is pinned, while the first two haveadvanced far into the grain (180 and 300 nm). From Dehm et al.,[63] with permission.

Ch_08.qxd 11/29/04 6:45 PM Page 397

by Xia and Hutchinson:[68]

s s0 (s0 sgb) tanh . (32)

Weiss et al.[31] took the experimentally measured stress in passivatedCu 1 at.% Al film as the nominal stress s0 and found that the stress s cal-culated from Eq. (32) is in reasonable agreement with the measured stressin unpassivated pure Cu film. Therefore, they concluded that the observeddifference in thermomechanical behavior between passivated and unpas-sivated films can be attributed to constrained diffusional creep.

By implicitly assuming that surface diffusivity is much faster thangrain boundary diffusivity, the model of Gao et al.[60] does not explicitlyconsider mass transport along the surface of the film. Depending on mate-rials, this assumption may not always be valid. Zhang and Gao[69] recentlystudied coupled surface and grain boundary diffusion and concluded thatthe analyses by Weiss et al.[31] and Gao et al.[60] remain valid even when sur-face diffusivity is comparable to grain boundary diffusivity. Specifically,Zhang and Gao[69] have shown that the results in[31] and[60] become gener-ally valid when an effective diffusivity

deff Deff (33)

is used to replace the grain boundary diffusivity dgbDgb, where dsDs

denotes the surface diffusivity. Note that, due to the large constant coeffi-cient of 12p, the effect of surface diffusion becomes significant only whenthe grain boundary diffusivity is much larger than the surface diffusivity.

8.4 Conclusion

In this chapter, techniques for the measurement of stresses in metal-lic thin films held on substrates and their relaxation by plastic deforma-tion and diffusional creep are described. All studies mentioned indicateclearly that the motion of dislocations is constrained in thin metal filmsby the presence of interfaces and grain boundaries. As a result, themechanical behavior of thin metal films can be qualitatively summarizedby saying that smaller is stronger. On the other hand, diffusional processesplay a more important role with decreasing dimensions as the densityof fast diffusion pathways increases. Therefore, it would be helpful to

12pdsDsdgb Dgb12pdsDs dgb Dgb

d4tf

4tfd


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describe the deformation behavior of thin films using deformation mech-anism maps[70] similar to the ones established for bulk materials by Frostand Ashby,[49] which include both dislocation-mediated and diffusion-mediated mechanisms. Unfortunately, clear, quantitative descriptionshave not yet been established for thin films, neither empirically nor the-oretically, and the use of quantitative descriptions obtained from bulkmaterials does not appear to be adequate. This is further accentuated bythe fact that subtle changes in the chemistry of the interfaces, whichmight be beyond experimental control, can dramatically influence thethin-film mechanical behavior. The models discussed for diffusionalcreep and dislocation core spreading will contribute to a framework thatwill lead eventually to a more complete description of thin-film mechanicalbehavior that takes into account the interaction of dislocation plasticityand diffusional processes. For this, a unified view of the experimentalobservations, including mechanical testing as well as in situ electronmicroscopy, and of the theory is indispensable.

In most technological applications, thin films are rigidly attached toa substrate that is typically much thicker than the film itself. As a resultof this configuration, the thin films are commonly subject to high inter-nal stresses, which may endanger the operation of small-scale devices.Failure mechanisms discussed in the literature include the following.(1) Thermal-stress-induced voiding has been a problem for many yearsfor Al or Cu conductor lines.[71, 72] This phenomenon is driven by highhydrostatic tensile stresses within interconnect lines under a cappinglayer during cooling. Even in blanket films, where two-dimensional in-plane stresses are observed, a few cases of stress-induced voiding havebeen reported.[73, 74] (2) Stresses may also cause the delamination orcracking of thin films.[75] It has been shown that the plasticity in thinmetal layers affects the interface fracture resistance in thin-film inter-connect structures.[76] Specifically, the TaN/SiO2 interface fracture energywas measured in thin-film Cu/TaN/SiO2 structures in which the Cu layerwas varied over a wide range of thickness. It was found that in a regimeof 0.25 to 2.5 µm, the delaminating resistance is dominated by the con-tribution of plastic deformation in the metallic layer. (3) More recently,the behavior of thin metal films under cyclic loading conditions has beengiven experimental as well as theoretical consideration. In particular,cracking of a brittle layer caused by ratcheting in an adjacent ductilelayer has been observed in a thin-film system under cyclic thermal load-ing.[77] Moreover, cyclic mechanical or thermomechanical loading canalso lead to the formation of voids and extrusions in thin metalfilms.[78–80] From these observations, it has been concluded that the roleof point defects and diffusion is much more important for fatigue in thinfilms than in bulk materials.

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In summary, continuum theories of plasticity or lifetime models thatincorporate dislocation plasticity and diffusional processes with respect toa physical or microstructural length scale may turn out to be very helpfulif they can be validated by experiments. The predictive capability of suchtheories will be very useful in device simulation codes for microelectron-ics, MEMS, and other applications.

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