THE SEARCH FOR THE SUPERMODULUS EFFECT

Shefford P. Baker, Martha K. Small, Joost J. Vlassak, Brian 1. Daniels and W.O. Nix

Department of Materials Science and Engineering Peterson Laboratory, Building 550 Stanford University Stanford, CA 94305 USA

ABSTRACf. The supermodulus effect has been reported as an anomalous increase of as much as several hundred percent in the elastic properties of certain compositionally­modulated thin metal films as the wavelength of the composition modulation decreases near 2 nm. Although elastic property variations have been detected by a variety of methods, including mechanical deflection experiments and acoustic measurements, the sign, magni­tude and physical basis for such an effect remain in dispute. We report on the mechanical properties of compositionally-modulated Au-Ni and Ag-Pd thin films as determined by three different mechanical deflection experiments: nanoindentation, microbeam deflection and bulge testing. Au-Ni films were fabricated by alternately sputtering Au and Ni onto oxidized Si substrates and were tested by nanoindentation and microbeam deflection tech­niques. The nanoindentation experiments reveal a decrease of about 15% in the indentation modulus at a composition wavelength near 1.6 nm. The microbeam deflection experiments showed only small variations in in-plane stiffness but did reveal strongly wavelength­dependent substrate interaction stresses in these films. A simple analysis of the bulge test indicates that these stresses can reproduce the major features of the supermodulus effect, as reported from bulge test results, as artifacts of the analysis method. A more thorough evaluation of the bulge test shows that this technique can be used to obtain accurate and reproducible results if the initial and boundary conditions are properly accounted for. Ag­Pd films were also prepared by sputtering and were tested using improved sample prep­aration, bulge testing and data analysis techniques. Variations in the biaxial modulus were small and in agreement with the beam deflection results.

1 . Introduction

The elastic properties of a material are generally considered to be structure insensitive. Thus, reports of significant variations in the elastic properties of compositionally­modulated thin films with composition wavelength, A.., at constant average composition have been controversial and have generated much research interest. Figure 1 shows the variation in biaxial elastic modulus of Au-Ni modulated films with composition wavelength as reported in the seminal paper of Yang, Tsakalakos and Hilliard [1]. For A.. > 3 nm, the biaxial modulus is constant at approximately the value expected from a rule of mixtures calculation based on the bulk elastic properties of the constituents. As the composition wavelength decreases below 3 nm, the measured biaxial modulus increases rapidly,

165

M. Nastasi et al. (eds.), Mechanical Propenies and Deformation Behavior of Materials Having Ultra-Fine Microstructures, 165-192. © 1993 Kluwer Academic Publishers.

166

reaching 230% of the long wave­length value at A = 1.5 nm. This large enhancement has been dubbed the "supermodulus effect".

In the years since this initial report, considerable effort has been brought to bear in an attempt to quantify and understand this effect. Reports of experimental work on this topic in the literature can be divided into two groups based on the funda­mental method by which the elastic properties were determined and into three groups based on the magnitude of the results. In one group of experimental techniques, part of the sample is mechanically deflected and

~500 ';;'400

~ 300

~ -a 200 .~

iE 100

o Biaxial Modulus of

Au-Ni Modulated Films

o o o~ o

Bulge test results of Yang, Tsakalakos and Hilliard (1977)

00

o~~--~~--~--~~--~~

012345678

Composition Wavelength (nm)

Fig. 1. Bulge test results of Yang, Tsakalakos and Hilliard [1] for Au-Ni modulated films.

an elastic modulus is determined either from a force-deflection relationship (bulge test, indentation, tensile test) or through a dynamic technique (torsional pendulum method, vibrating reed method). We refer to these as mechanical deflection methods. In the other group of experimental techniques, phonon velocities are measured and elastic constants can be determined. We refer to these as phonon velocity methods. When partitioning the literature by results, we place reports of enhancements of 100% or more over expected values (the supermodulus effect) in one group, results showing variations of either sign on the order of tens of percent in another group and results showing no variation in elastic properties with composition wavelength in the third.

There is some correlation between these two divisions of the experimental literature. We note that, without exception, the large enhancement results were obtained using mechanical deflection techniques. The elastic properties reported from such measurements are elastic moduli which represent various combinations of elastic constants. In addition to the bulge test results of Yang, Tsakalakos and Hilliard for Au-Ni and Cu-Pd films [1], large enhancements have been reported based on the results of bulge tests of Ag-Pd by Henein and Hilliard [2] and of Cu-Ni by Tsakalakos and Hilliard [3]. An important observation regarding these bulge tests is that nonlinear elastic behavior was reported for all films showing enhanced moduli. Figure 2 shows representative stress-strain data from a bulge test of a Ag-Pd film by Henein and Hilliard [2]. The nonlinear behavior shown was repeated during load/unload cycles and so these authors concluded that the film was behaving in a nonlinear elastic manner and considered the initial slope of the data to represent the biaxial modulus of the film. Other mechanical deflection experiments showing enhancements of 100% or more in elastic properties include vibrating reed [4,5] and torsional pendulum [5] tests of Cu-Ni, as well as tension tests of Cu-Ni [5] and Cu-NiFe [6]. A second feature which is common to all of these reports is that the modulus enhancement could be made to vanish by annealing the film. In particular, most of these authors [1-3,6] reported that the magnitude of the enhancement was proportional to the square of the amplitude of the composition modulation as determined by x-ray methods.

In the literature reported to date, positive or negative variations on the order of 10% in elastic properties with composition wavelength are typical of measurements of acoustic

phonon velocities. A number of such methods have been applied to the study of modulated films including Brillouin light scattering measure­ments of thermally excited phonon velocities [7-16] and measurements of acoustically driven phonon velocities by a variety of methods [9,17-21]. This literature is extensive and only selected representative publications are cited here. Phonon velocity mea­surements generate values for various elastic moduli depending on the par­ticular acoustic mode which is ob­served. The most common measure­ments are of the Rayleigh mode ve­locity and results typically show a de­crease in the corresponding shear

167

0.6

Calculated Results from []

0.5 Bulge Test of Ag-Pd []

0

Modulated Film 0 0 ....... 0.4

[]

<':l [] []

~ 0

Q 0

0 0

en 0.3 0 en 0 0

~ ° 0.2 0 0

t/.l 0

0.1 Data of Henein and Hilliard (1983)

0.05 0.10 0.15 0.20 0.25 0.30

Strain(%)

Fig. 2. Bulge test results of Henein and Hilliard [2].

constant of as much as 40% from the long wavelength value as Il decreases to a few nanometers [7-9,11,16,17]. Similar decreases have been calculated in other elastic con­stants and moduli from measurements of phonon velocities in other modes as well [10,18]. In some cases [12,22], increases of as much as 50% have been detected.

A number of reports have been published in which no variations in elastic properties with composition wavelength were seen. These include results of both mechanical deflec­tion experiments [21,23-26] and phonon velocity measurements [13,14,19-21]. Again, the references cited here are representative, not exhaustive.

It would seem to be of interest to make correlations between the material tested and the experimental results. Such a correlation could be made in that all of the reports of large enhancements were for modulated systems in which the constituents are both fcc and form a complete range of solid solutions in some temperature range, whereas the majority of the reports of variations of order 10% were for fcc-bec systems. Unfortunately, it is not pos­sible to separate this correlation from the correlation between experimental technique and magnitude of results described above. Although there are isolated cases of agreement between results obtained by different techniques in different laboratories, overall results are better correlated by the technique used to obtain them than by the material tested.

We herein report on our own search for the supermodulus effect and provide an expla­nation for the discrepancy between results obtained by mechanical deflection experiments and those obtained by phonon velocity measurements.

2 • Mechanical Properties of Au-Ni Films

We have applied two different mechanical deflection techniques, nanoindentation and micro-cantilever beam deflection, to the characterization of the mechanical properties of composition modulated Au-Ni fIlms. The diffIculties inherent in testing of freestanding thin fIlms are widely recognized [27], and both of these techniques allow direct mechanical testing of a fIlm in situ on its substrate.

168

The nanoindentation experiments were conducted using a commercially available instrument: the Nano Indenter [28]. This device can impose loads in 0.251lN increments and can resolve displacements of about 0.3 nm. Depth-sensing indentation experiments have been well-established [29,30] as reliable methods for obtaining both strength and stiffness properties from thin metallic ftlms.

The deformation in a simple beam-bending experiment is well defined and Weihs et al. [31] and Hong et al. [32] have developed experimental techniques which utilize this geome­try in the testing of thin films. In this method, anisotropic silicon etching technology is used to create a series of micrometer-scale cantilever beams which can then be deflected using a nanoindenter. The force-displacement data obtained are then interpreted to deter­mine the mechanical properties of the beams and of the beam materials. For the experiments reported here, Si<h beam structures were fabricated on silicon wafers as described below. These structures were then used as substrates for the sputter deposition of the modulated films. The result was a series of bilayer cantilever beams consisting of a layer of oxide and a layer of modulated metal ftlm.

In a preliminary report of this work [25], we stated that no variations in mechanical properties with composition wavelength were seen. A new analysis which takes better account of the statistical value of individual data points, of particular importance in the analysis of indentation data, has been developed. This analysis has changed our interpre­tation of the results slightly and is described in limited detail in the Appendix.

2.1. SAMPLE PREPARATION AND CHARACI'ERIZATION

Indentation and beam deflection experiments were conducted in the same samples. To pre­pare a substrate, a 1 jlm thick layer of wet-thermal Si<h is first grown on a lightly-doped p-type (100) Si wafer at 1100·C. An outline of the beam pattern is then created in the Si02 using a standard microelectronics industry photolithography technique and is etched through the oxide using a buffered HF solution. This pattern is aligned such that the edges of the beams are parallel to <110> type directions in the underlying Si. The beams are then fully released by etching the exposed Si away using a solution of ethylenediamine and pyrocatechol. In this solution the etch rate of Si<h is negligible and the etch rate of Si is highly anisotropic, proceeding quickly in <100> directions and relatively slowly in <111> directions. The result is a row of Si<h cantilever beams extending out over a crystallo­graphic etch pit which has {Ill} faces. The oxide beams prepared for this study were nominally 1 Ilm thick, 20 Ilm wide and ranged in length from 10 to over 100 Ilm.

The methods used to deposit and characterize the metal films have been described else­where [33-35] and are only briefly reviewed here. The substrates were attached to a table which was maintained near room temperature «40·C) as it was rotated over the sputtering targets. Argon at a working pressure of 0.67 Pa was used to sputter the target materials. The system base pressure was 6.7IlPa. A layer of Cr about 50 nm thick was deposited first to insure good adhesion between the oxide and the subsequent modulated film. Au and Ni were deposited at a constant rate between 0.1 and 1.0 nm/s for each modulated ftlm. The nominal composition of each sample is AUO.5NiO.5. The composition wavelength, and therefore the thickness, were maintained to within 10% across anyone sample.

The samples were characterized by symmetric x-ray diffraction [33]. The six samples reported here correspond to the "series #89" samples in reference [33]. The 0/20 scans for these samples show a single strong 111 Bragg reflection with satellite reflections due to the

169

modulated structure. No reflections corresponding to other orientations were detected. The composition wavelengths were determined from the satellite spacings. The nominal com­position wavelengths (and total thicknesses including the Cr adhesion layer) for these films are 0.91 (830), 1.30 (823), 1.63 (810), 2.06 (615), 2.86 (800) and 3.98 (860) nm. The average lattice parameter normal to the film plane was found to exhibit a maximum at A "" 2 nm.

Transmission electron microscopy [34] of a different set of Au-Ni films deposited under similar conditions ("series #88" samples in ref. [33]) reveal an equiaxed columnar structure of nodular grains which are convex toward the film surface and which have diameters ranging between 30 and 60 nm. Selected area diffraction and high resolution work suggest that these modulated films are coherent for A < 2.63 nm and show evidence of separate, bulk-like, Au and Ni lattice parameters for larger composition wavelengths.

We observed that the oxide beams, which were flat to within optical resolution before the metals were deposited, were curved once the metal films were in place, indicating a stress interaction between the metal and Si02 layers. Indeed the entire substrate for each sample was curved due to these stresses. The significance of these stresses will be dis­cussed below.

2.2. INDENTATION EXPERIMENTS

In order to obtain sufficient data for statistical analysis, a number of arrays of indentations were made in each sample. Each array consisted of 36 indentations; six indentations to each of six different maximum depths. For each indentation, the indenter was loaded so as to maintain the indenter velocity between 3 and 6 nm/s until the programmed displacement was reached. The load was then held constant until the indenter velocity dropped below 0.1 nm/s and was subsequently removed at a constant rate which was somewhat lower than, but scaled by, the fmalloading rate.

2.2.1. Indentation Hardness. A hardness value was determined for each indentation using the method of Doerner and Nix [29]. In this method, Doerner and Nix adapted Sneddon's solution [36] to the problem of contact between a rigid punch and an isotropic elastic half space by incorporating the compliance of the indenter. The result is a relationship between contact stiffness and contact area given by

s= iJp =~E,..[A iJhelastic .J1i (1)

where S is the contact stiffness and A is the projected contact area. Er contains the elastic constants of the indenter and the sample material and is given by

_1 =(1- ~)+(1- vi) E, E, Es

(2)

where E and v are Young's modulus and Poisson's ratio and the subscripts i and s refer to the indenter and the sample being indented, respectively.

As a means of interpreting nanoindentation data, Doerner and Nix considered the final plastic displacement of the material at the tip of the indenter, the "plastic depth", hp, to be the intercept with the displacement axis of a straight line fit to the load-displacement data

170

during the initial portion of unloading. Since the shape of the indenter tip is known, the projected contact area corresponding to any given plastic depth can be detennined and the hardness, H, is simply defined as the maximum load divided by this contact area. The slope of the fitted line gives the contact stiffness. For these experiments, a least squares fit of a straight line to the first 25% of the unloading data was used to obtain the plastic depth and contact stiffness for each indentation. The correlation coefficient from the fit was used as a screening device. Indentations with correlation coefficients less than 0.99 were not included in further calculations.

Indentations were made to a range of depths in each sample and hardness was found to be virtually constant with plastic depth from about 60 nm to near the film thickness. As plastic depth increases beyond the film thickness, the hardness increases slowly as the substrate (HSi = 16 GPa) begins to influence the measurement. Below 60 nm, the hard­ness drops slightly as the plastic depth approaches the peak to peak surface roughness and the contact between the indenter and the sample becomes irregular. The surface topography and the film thickness thus prescribe limits to the indentation size range which can be used to sample the f:tlm. Indentations must be large enough to avoid surface topography effects yet small enough to avoid the influence of the substrate. Figure 3 shows hardness as a function of composition wavelength reported as an average of results from all indentations meeting the condition that 65 nm < hp < 0.2fJ, where fJ is the film thickness, for each sample. Each data point is presented as the mean and standard deviation of results from 18 to 36 individual indentations, except for the sample at A. = 2.06 nm for which only 12 indentations met this condition.

It is evident from Fig. 3 that these films are quite strong and that the strength is independent of composition wavelength. This invariance of hard­ness with A. is somewhat surprising. For these samples, the number of in­terfaces per unit volume varies by a factor of four. In general, these inter­faces might be considered to act as bar­riers to dislocation motion and would therefore be expected to contribute to the strength of these fums. In a similar study of Cu-Ni fUms, Cammarata et al. [26] were able to make a Hall-Petch type correlation of hardness with com­position wavelength. We conclude that some defect structure other than the in­terfaces must dominate in establishing the strength of these films.

8

Hardness of Au-Ni modulated films

°O~~~1~~~2~~~3~~~4~~5

Modulation Wavelength (nm)

Fig. 3. Hardness of Au-Ni modulated fUms.

2.2.2. Indentation Modulus. To detennine the elastic properties of these fUms, a modified version of the method proposed by Doerner and Nix [29] has been used. Since the elastic properties of the indenter are known, the elastic properties of the sample material can be determined from measurements of the contact compliance and contact area using Equations (1) and (2). The indenting machine itself also has some compliance and this compliance

171

must also be accounted for. For a given increment in applied force, displacements occur in both the sample and the machine so that the total compliance can be written as

C=C + {ii_l_ (3) m 2Er ..fA

where C is the total compliance and Cm is the machine compliance. Doerner and Nix implemented this model by plotting total compliance as a function of the reciprocal square root of the contact area. The data are expected to fallon a straight line, for which the slope is determined by the elastic properties of the sample and the indenter and the intercept is established by the machine compliance.

If it could be assumed that Cm were constant, then it would be possible to obtain a cali­bration value and to use this value in Equation (3). In this case, a measurement of sample elastic properties could be obtained from a single indentation. However, Cm is not con­stant due to variations in the compliance of the sample mounting and of the translator used to position the indentation device with respect to the sample surface. These compliances vary from experiment to experiment but are expected to remain constant during a given ex­periment. In the experiments reported here, compressive stresses in the films caused some samples to bend so that the free surface of the fIlm was convex. When such a sample is adhered to a flat mounting block for testing, it is possible for a segment of the middle of the sample to remain unsupported so that it acts as an edge-supported plate during the indenta­tion experiment. As a result, the machine compliance, as determined from the compliance intercept pf a plot of C vs A -0.5, varied by as much as a factor of 45 in these tests. Another notable feature of a plot of C vs A -0.5 is that the variance in the compliance increases as the indentation size decreases due both to the fInite resolution of the machine and to the fact that the indentation size can approach the scale of features which control the contact stiffness in an inhomogeneous material. In order to take both the variations in machine compliance and in the variance of the contact compliance with depth into account, a weighted fIt to the C vs A -0.5 data was performed and the machine compliance was removed. The analysis proce­dure used is described in the Appendix.

A representative plot of C vs A-O·5 is shown in Figure 4, and Figure 5 shows the result­ing indentation modulus as a function of composition wavelength. The elastic modulus which is reported, herein referred to as the "indentation modulus," corresponds to EsI( 1- Y;) in Eq. (2). In order to obtain a sufficiently wide range of indentation depths for these calculations, indentations with plastic depths between 20 nm and 25% of the film thickness were included in the indentation modulus calculations. While this procedure incorporates data from the depth regime where the surface topography has reduced the hardness somewhat, errors in the calculation of area have a relatively small effect on the indentation modulus and there is no correlation between indentation modulus and film thickness, as would be expected if the surface topography had a signifIcant effect on these results. The indentation modulus displays a small minimum at a composition wavelength of about 1.6 nm; where supermodulus effects have been reported [11 and near the maximum in out-of-plane lattice parameter [331.

It has been noted previously [24-261 that an indentation technique may not accurately detect the characteristics of a supermodulus effect. The indentation modulus represents some unknown average of elastic properties in all directions in the film. Thus, if the super­modulus effect were such that a modulated film were to become more compliant in a direc-

172

Variation of Contact Compliance with Indentation ContacrArea

Au-NiFilm A=4.0nm

Indentation Modulus = 137.0 ± 4.9 GPa

0.001 0.002 0.003 0.004 0.005 0.006 (Contact Area)-l/2 (l/nm)

Fig. 4. Typical variation in Cc with ( A )-0.5 for indentation experiments.

;f; Q.150

'" .E ~140 ~ § 130 .~

f! ~ 120

Indentation Modulus of Au-Ni t Modulated FIlms

t

no 0 ......... ~~IL......~~2L......~~3L......~~4.L..o.-~....,.5

Composition Wavelength (nm)

Fig. 5. Indentation modulus variation with composition wavelength.

tion normal to the film and simultaneously stiffer in the plane with decreasing composition wavelength, then it is conceivable that the particular average of elastic constants which comprises the indentation modulus would not show these effects. To investigate this pos­sibility more fully, we carried out the microbeam deflection experiments described below.

2.3. MICROBE AM DEFLECTION EXPERIMENTS

A schematic of the bilayer beam deflection experiment is shown in Figure 6. The cantilever beams are composed of an oxide layer of thickness to with Young's modulus Eo and Poisson's ratio Vo and a metal layer of thickness tt with Young's modulus Ef and Poisson's ratio vf Both of the layers have width w and the beam is deflected at a distance L from the fixed end. To conduct an experiment, the indenter is first positioned over the site where the beam is to be deflected. A typical example of data from the subsequent experiment is presented in Figure 7. Once the indenter is in position, the indenter tip is moved toward the beam at a constant velocity between 6 and 10 nrn/s. The slope of the load-displacement data collected in this region represents the spring constant, S s, of the suspending springs which support the indenter shaft in the Nano Indenter. When the indenter contacts the beam, the stiffness of the beam is added to the system and the load rate is incremented to maintain the con­stant indenter velocity as the beam is de­flected. The suspending springs and the cantilever beam can be modeled as springs in parallel and so the data in this region again fall on a straight line, the slope of which represents the total stiff­ness, St, of the spring and beam system. In this configuration the stiffnesses sum

Load

Modulated Film , Fig. 6. Schematic of the bilayer beam deflection experiment.

so that the beam stiffness, SIJ, is given by Sb = S, - Ss .

The stiffness of a beam is deter­mined by its geometry and its elastic properties. In this case, the stiffness and the geometry are measured and the elastic properties of the oxide layer are known, so the elastic properties of the metal ftlm can be determined. For each deflection, the stiffness of the indenter springs was determined from a linear least squares fit to the data before the beam was contacted and the total stiff­ness was calculated from a similar fit of the data from the point at which the indenter was in full contact with the beam until the beam had been deflec­

250

~ 200

'-' 150

1 ....l 100

50

Beam Deflection Data

total stiffness, St I. ......... .... 1/

.• ' ,/

... " ....

~( ........ / spring stiffness, S8

.... ..'

..........

173

o~~~~~~~~~~~~ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Displacement (~m)

Fig. 7. Typical data from a beam deflection experiment.

ted 5% of its length. The data for all deflections indicated linear elastic behavior in this range. The analysis used here has been described by Weihs [37] and Hong [32]. In this analysis, the deflection of the beam at the point where the load is applied is assumed to arise from bending only. A simple mechanics of materials model for plane strain bending of a slender prismatic bilayer beam due to a transverse loading yields a solution in the form,

where

and

an2+bn+c=0

Ef(I-~) n Eo(l- Vl) a=tj

3 22 2 4L3tf(l- V;)(St -Ss) b = 4t fto + 6t fto + 4t fto - _-=-.l...-_...!.-_---.:..

wEo

4 4Eto(l- ~)(S,-Ss) c=to

wEo

(4)

To obtain the elastic properties of the metal film, we use our measurements of the beam geometry and stiffness, along with the values of Eo = 64 GPa and Vo = 0.16 reported by Weihs et al. [31], in Equation (4). Many of the constants which describe the sample geometry enter Eq. (4) with significant exponents. Thus, the uncertainties in the four geo­metric terms L, w, to and If, and in the beam stiffness, Sb. were taken into account.

Figure 8 shows the results of these calculations for several values of beam length in !he sample with II, = 1.30 nm. The deflection modulus presented is EjI( 1- v]) in the plane of the film. Three features are immediately evident. First, the error bars are relatively large,

174

amounting typically to 30% of the calculated value. This is primarily due to uncertainty in the film thickness which contributes about 70% of the error bar for a typical deflection. Second, the calculated moduli are much lower than expected. An iso­strain model for uniaxial plane strain deformation in the plane of the m.m yields a value for EI( 1- vj) of 199 GPa using bulk single crystal elastic constants [38]. Finally, we note that the calculated moduli vary systemat­ically with beam length. Clearly there are additional sources of displacement or compliance which are not accounted for in this analysis, at least some of which must be associated with prox­imity to the fixed end of the beam.

1W~~~~~~~~~~~~~ "2 Elastic Modulus Calculated from

~ 1: ~~:. p!~~!,n EXP1erimentts

)~ 8 40 .~

~ 20

°0~~1~0~~2~0~~30~~4~0~~5~0~~6'O

Beam Length (11m)

Fig. 8. Deflection modulus measured at several beam lengths for sample with A. = 1.30 nm

In fact, the simple bending model probably does not describe these experiments well. There are two inaccuracies in the assumptions about the geometry of the beams [25]. First, this analysis assumes the beam to be attached to a rigid support. In fact the highest bending strains are located in the plane of the beam support where the model assumes these strains to be zero. Second, due to variables in the fabrication process, the supporting silicon sub­strate may be etched back beyond the fixed end of the beams. This undercutting was observed to be on the order of a few micrometers for all samples. Furthermore the simple bending model neglects several sources of additional displacement. The largest of these are displacements due to shear stresses in the beam. A crude estimate of these shear displace­ments reveals that they could be of the same order as the displacements due to bending for Llw "" 1. Finally we note that this model requires that the geometry of the problem not be changed by the deformation. This condition is not well approximated for wide beams. An exact solution due to Timoshenko [39] indicates that for homogeneous beams of this aspect ratio, the shear stress varies by nearly a factor of four along the neutral plane across the width of the beam. As a result, cross sections do not remain planar under load and the ge­ometry of the beam could change significantly as it is deflected. All of these errors may be considered to be end effects. For sufficiently long beams, their effects should be negligible. Presumably, if one could continue to deflect a beam further and further from the fixed end, the calculated deflection modulus would approach the expected value. Unfortunately, it was not possible to perform deflections at sufficient beam lengths to avoid these end effects, both because the beam stiffness quickly decreases below the point where it can be detected with the Nano Indenter and because of the curvature of the beams in some samples.,

Despite these limitations it is possible to obtain useful information from these experi­ments. Since any errors apply equally to all of the beam samples in this study, the results are still of comparative value. To eliminate sample to sample variations due to the end effects, the calculated deflection modulus values for beam lengths between 30 and 40 11m were combined using a weighted averaging method for each sample. The mean of beam lengths for each of these averages was 34 ± 211m. The results are shown in Figure 9.

Although these values do not accu­rately reflect the magnitude of the in­plane elastic properties of the metal films for the reasons described above, the relative values are expected to be a very good indicator of any variations in these properties with composition wavelength. As is evident in Fig. 9, there is no consistent trend. If any­thing, these results support the nano­indentation evidence for a small de­crease in elastic stiffness for compo­sition wavelengths near 1.6 nm. There is no evidence for elastic ano­malies on the order of 100% or greater in these samples.

2.4. SUBSTRA1E INTERACTION STRESSES

175

120 ,.-...

= Q,100

1 '" 80

h~ f :s -'8

::E 60

= 0 40 '::2 ~ Deflection Modulus of Au-Ni !i:::: 20

8 Modulated Films °0~~~1~~~2~~~3~~~4~~5

Composition Wavelength (nm)

Fig. 9. Deflection modulus as a function of composition wavelength averaged from deflec­tions at beam lengths between 30 and 40 J.Ull.

It was noted above that the cantilever beams were curved as a result of a stress interaction between the metal ftlm and the oxide substrate. Such stresses are common in thin ftlms and arise whenever the equilibrium in-plane dimensions of the ftlm and substrate change with respect to one another [40]. The curvatures were measured by optical microscopy and were found to vary significantly with composition wavelength. These values are shown as the ftlled diamonds in Figure 10, where the sign of the curvature is positive if the beam curves up and negative if the beam curves down. Each datum represents the mean of at least five measurements of undeflected beams near the deflected beams in each sample. Error bars are too small to be seen in this plot.

The oxide beams were flat in each sample before the mod­ulated ftlm was deposited and were curved once the metal was in place. The magnitude and distribution of the stresses which cause the curvature de­pend on the details of the origin of those stresses. For example, if the stresses arise as a result of differential thermal expan­sion, then one can imagine that the stresses and the curvatures would occur only after the de­position process is complete as the sample is cooled. In this case, the stresses would be partially relaxed by the curva-

-+-Curvature -e-Stress

0.00t-----'c----------t0

-:"8 ~.0.G1

~ -0.02

~ U -0.03

Curvatures and Stresses in

Au-Ni Modulated Films

'2 -200 ~

'-'

'" -400 '" g tI)

-600

-0.04 ~~--'----"--'---'---'--'---'----'----' -800 o 1 2 3 4 5

Composition Wavelength (nm)

Fig. 10. Measured curvatures of Au-Ni bilayer beams and the estimated stresses required to produce those curvatures.

176

ture of the beam and the maximum stress in the metal film would be located at the film! oxide interface. If, on the other hand, the stresses arise as a result of structural evolution during the deposition process, then the beam can curve slightly to partially relax the stress­es induced by that structural evolution as each incremental layer of the film is deposited In this case, the maximum stress in the metal film is located at the top surface. Because these modulated films were all deposited on freestanding beams at a low, nominally identical, temperature, the second model is more appropriate.

To determine the stresses, we again use the simple mechanics of materials model for pure plane-strain bending of the bilayer beam which led to Eq. (4). If the curvatures are allowed to develop as described above, then the relationship between the stress in each infinitesimal layer and the resulting curvature, 1(, can be written [37] as

ICE f ns2 + 2t _\' + t2 [t ]-1

(J- 0 0"' 0 ds - ~1-V;) ! ( .2,' + 4",.,..3 + 6,,;.,2 + 4,}.s+ ': ) (5)

The stress thus obtained is an estimate of the average stress in the ftIm in the regions where the oxide layer is supported by the substrate; that is, the stress which would exist if the beams were not allowed to curve at all.

To calculate the film stress by Equation (5), we must know the elastic properties of both the oxide and film layers. Since the in-plane elastic moduli as measured by beam deflection did not vary dramaticaUy wi~h composition wavelengthi we again used the expected values of 199 GPa for EI( 1- v f) and 66 GPa for EoI( 1- vo ). With the resulting value of n, the quantity in the brackets was integrated numerically for the geometry of each sample. The stresses which result from these calculations are shown by the open squares in Fig. 10. The error bars are standard deviations in which uncertainties in to, 1J and 1( are taken into account

The model selected for the generation of the curvatures allows the stress in each infmitesimallayer to partially relax before the next layer is deposited. Thus, this calculation provides a lower limit to the average stress in the unrelaxed ftIm. For comparison, if the stresses are assumed to arise from differential thermal expansion as described above, then the application of a simple model devised by Weihs [37] indicates that the stress required to produce the measured curvatures would be larger by more than a factor of five. Furthermore, since there are no transverse loadings, the simple beam bending model very accurately reflects the actual condition of these beams. Thus, it is clear that these films sup­port high stresses due to the constraint of the substrate and that these stresses vary strongly with composition wavelength. These stresses go through a compressive maximum near A = 2 nm, corresponding in wavelength to the minimum in indentation modulus and to other effects (maximum in out of plane lattice parameter, supermodulus effect) as reported above.

2.5. BULGE TEST SIMULATIONS

Aside from producing strains which could have an effect on elastic constants, stresses can influence the results of mechanical property measurements by changing the geometry or the initial or boundary conditions of the experiment. Itozaki [41] recognized that the results obtained from a bulge-test experiment are extremely sensitive to the initial configuration of the sample. In particular, he observed that uncertainty in the initial position of a film could

177

lead to large overestimates of its biaxial modulus. Although he was unable to find any rea­son to correlate variations in the initial position of the films with composition wavelength, he observed that both the high modulus values and the nonlinear elastic behavior reported in [1], [2] and [3] could be explained as artifacts resulting from this uncertainty. Our mea­surements of wavelength-dependent substrate stresses provide a way for the initial condi­tions in a bulge test to vary systematically with composition wavelength and so we have investigated the relationship between these stresses, the initial conditions and the resulting calculated biaxial moduli by simulating these experiments.

A schematic of the bulge test is shown in Figure 11. In a typical experiment, the fIlm is removed from the substrate and is clamped over a circular hole ofradius a. A pressure, P, is applied to one side of the film and the fIlm bulges outward a distance h, known as the "bulge height." In a,simple analysis, which was first applied to thin fIlms by Beams [42] and which was used in all of the reports of the supermodulus effect by this technique, the bulged film is assumed to take on the shape of a spherical cap. From this geometry, the pressure vs. height data can be converted to stress vs. strain by the relations

Pa2 a=- (6)

4ht

which results from a force balance applied at the edges of the bulged fIlm, and

2h2 £=~ (7)

3a which results directly from the geometry. In these expressions, a is the biaxial stress, £ is the biaxial strain and t is the thickness of the film.

Fig. 11. Schematic cross section of the bulge test.

. Equations (6) and (7) require that the film be flat and stress free and that the bulge heIght be known absolutely. If the film is slack or if it supports a tensile stress when mounted, then Eq. (6) is unchanged, but, as pointed out by Itozaki, the expression for strain must be modified thus

(8)

178

For a taut film, A = GllY where Go is the tension stress and Y is the biaxial elastic modulus given by Y = E/(1-v). For a slack fllm, an infinitesimal applied pressure causes the fllm to bulge to a height ho. In this case A = -h; and Equations (6) and (8) can be combined to reveal the following relationship between pressure and bulge height

p = :~~ h( h2 - h;) (9)

An initial height, ho, can easily be generated by a compressive interaction stress in the film. The procedure used to mount the films in references [2] and [3] consisted of first attaching a frame to the film, then stripping the fllm (with the frame attached) from the substrate and finally gluing a washer to the fllm. The washer is then placed in the bulge tester and acts as the clamp of the bulging fllm. The film is never unsupported during this mounting procedure and, to the extent that the stresses are not relieved by compliance of the frame or of the adhesive used to attach the washer, the portion of the film which is supported by the adhesive still maintains the substrate stress and the unsupported portion of the fllm in the center of the washer bulges out to fully relieve this stress. The initial bulge height can be determined if the compressive stress, Go, in the supported portion of the mounted fllm is known. The compressive strain in the supported film is given by e = GdY. Combining this with Eq. (7) above, the initial bulge height is given by

h =~3a2Go o 2Y

(10)

Our simulations of bulge test experiments were carried out in the following way: First, we made the assumption that the film stress would be fully maintained when the sample was mounted in the bulge tester. Thus, an initial bulge height, ho, could be determined for each sample from Equation (10) where Go is the measured value of the film stress for that sample and Y = 290 GPa is determined from single crystal elastic constants [38] using a simple isostrain model. We next calculated pressure at 3 j.1m height increments using Equation (9). This results in a set of simulated bulge test P - h "data" which can then be further manipulated.

Yang et al. [1], Henein et al. [2] and Tsakalakos et al. [3] assumed that their fllms were flat at the start of a test [41]. However, if their films supported wavelength-dependent stresses similar to those which we observe, then their experiments would have started from an initial bulge height and they would have obtained P - h data similar to our simulated data. Nonetheless they would have analyzed it using Equations (6) and (7). We have dupli­cated this analysis on our simulated P - h data and the resulting stress-strain curve for the sample with A = 3.98 nm is shown in Figure 12. We note that the result of neglecting the initial height is elastic stress-strain behavior which is apparently nonlinear as described in reports of the supermodulus effect [1-3] (compare Figures 2 and 12). When Yang, Henein, Tsakalakos and their collaborators [1-3] encountered this behavior, they interpreted the bi­axial modulus as the initial slope of this data. If we do the same by performing a simple lin­ear least squares fit to the frrst three points in Fig. 12, we get a value of 14900 GPa-a su­per modulus indeed. Furthermore, when this simulation is repeated for all of the samples in this study, the calculated biaxial modulus is found to vary by nearly two orders ofmagni­tude with a peak at A. = 2.06 nm.

There are three reasons why the biaxial modulus we calculate from the data in Fig. 12 is so high. The first is the arbitrary nature of the spacing of the data points in the simulation. As h ~ ho the derivative of the simulated stress-strain data increases without bound. Thus the value of the slope obtained depends on the spacing of data points and how much data is included in the fit. In our simulation, this fit includes 9 f..IJll of displacement data, which is a reasonable estimate, at least, of the range of experimental data which might be considered for a modulus calculation. Second, the in­teraction stress in these films is com­posed of a wavelength-dependent part

~ 1.8

'" g 1.6

CIl '0 .g 1.4

-3 a 1.2

Simulated Bulge Test 00

0 0 00

00 00

0 0 a A.= 3.98 run

0 0 = 256MPa ho= 206 run

y= 14900GPa

0.01 0.02 0.Q3 0.04

Calculated Strain (%)

179

0.05

Fig. 12. Stress vs. strain calculated from simulated bulge test data assuming an initial bulge height of 221 J.1m.

and a wavelength-independent part. Even if the samples of Yang et al. [1] did support exactly the same wavelength-dependent stresses as we see in our films, the wavelength­independent component was almost certainly more tensile in their films since their films were evaporated at elevated temperature. Finally, we have assumed that all of the substrate stress was maintained during mounting of the sample for testing. This is unlikely to be the case since each step of the mounting process could allow the film to relax somewhat.

We thus considered the conditions required for the generation of "supermodulus" effects of the order of the published results. Since Yang, Tsakalakos and Hilliard [1] did not see a modulus enhancement for A > 3 nm, we can assume that the initial bulge height for their films in this range was zero. Accordingly we adjusted the zero level of stress in Fig. 10 to correspond with the longest wavelength sample by subtracting the measured stress for this sample from the mea­sured stresses for the other samples. Next, we determined the stress level required to produce a peak calculated modulus of twice the expected value, about the size of the modulus en­hancement reported by Yang et al. [1]. For the sample at A = 2.06 nm, an initial height of just 4 J.1m was required to double the modulus. The stress required to produce this initial height was only 0.04% of the mea­sured stress. Thus we completed our simulation by multiplying all of the stresses by this scale factor (0.0004) and recalculating the moduli. The re­sult is shown in Figure 13. The sam­ples at composition wavelengths of

800r----.----.----.----,----.

'2 700 c.. 0600

Biaxial Modulus of Au-Ni Modulated Films from Bulge Test Simulations

'-" o 0

~ 500

~400 ...:; 300 r. r.

o

";j "-.~ 200 input value = 290 GPa

i:E 100

°O~---71----72----~3----~4--~5

Composition Wavelength (nm)

Fig. 13. Simulated "supermodulus effect" obtained by neglecting effects of substrate interaction stresses in bulge test simulations.

180

1.30 and 3.98 nm return the expected value, 290 GPa, because they support virtually the same interaction stress and this is the stress which is subtracted from the measured stress for each sample. The sample at A.. = 0.91 nm returns the expected value since it is in tension. Assuming that the pressure and bulge height are measured accurately, a tensile stress in the fUm will not affect either the linearity or the slope of the stress-strain data.

3 . Analysis of the Bulge Test

It is clear from the simple analysis presented above that the results obtained from a bulge test are very sensitive to initial conditions and sample geometry. In order to determine the efficacy of this technique we have investigated the accuracy of the spherical cap model by means of finite element simulations and have developed new methods for sample prepara­tion and data analysis.

3.1. FINITE ELEMENT MODEL OF THE BULGE TEST

In order to identify possible sources of error in analyzing bulge test data, and to quantify their magnitude, we modeled the deformation behavior of a thin fUm in a bulge test using the method of finite elements. A detailed discussion of the model and results is published elsewhere [43]. A circular fUm geometry was chosen for the model because it has been the most widely used in experiment and analysis. The film was modeled using axisymmetric thin shell elements with an isoparametric formulation. The model formulation allowed for three possible initial states of the fUm: flat and unstressed, taut or slack. The boundary con­ditions in all cases, as shown in Figure 14, are that the center of the film remain at the center and not rotate, so as to maintain a smooth top, and that the edge of the fUm remain fixed and not rotate. A slack fUm is modeled by allowing the edge to slide inward while a small pres­sure is applied to one side of the film (see Figure 15). Upon reaching the desired height, the film is clamped and then unloaded to determine the initial height, ho, of the bulge. The procedure is analogous to fitting a fUm of radius a over a hole of radius a-5, where 5 is small and positive. This models a film, part of which is freestanding, that is in compression on its substrate. A taut film is modeled by simply applying an

z

+ Itniform

center I symmetry axis edge

Fig. 14. Schematic of Finite Element Model boundary conditions. The tick marks across the film indicate the position of the nodes in the model.

initial equi-biaxial stress state. The model was used as a means of simulating bulge test "data," which could then be used to compare various analysis techniques and the effects of errors in the model or in assumptions of initial or boundary conditions.

3.1.1. Initially Flat, Unstressed Film. While in practice it is difficult to fabricate a free­standing fUm that is both flat and stress-free, we look first at this simplest case in order to

Pre-test

z

~t--", ~ ~R

............ - .. ~

P=O

Z

~. '-- a-o R

Fig. 15. Schematic of the procedure for modeling a slack film.

P>O

Z

181

C~R """-_._.""

examine the boundary conditions of the spherical cap model. In particular, it is clear that the film does not remain under equi-biaxial stress and strain as it is bulged, and is therefore not equivalent to a spherical shell under pressure. The film must be clamped and this constraint prevents circumferential strain at the edge. The corresponding stress state at the edge is thus more nearly uniaxial, making the film more compliant overall and changing the dependence of the measured modulus on v. One finds that, for films with Poisson's ratio greater than zero, the spherical cap equations underestimate the film stiffness and that this deviation increases linearly with v, but is independent of E. The result can be summarized in the fol­lowing expression:

Y = Ycalc 1-0.24v

(11)

where Ycalc is the biaxial modulus calculated from the spherical cap equations and Y is a more accurate estimate of the material modulus.

3.1.2. Slack Films. A film in compression on its substrate buckles when a section of the substrate is removed to make it freestanding. Such a film becomes slack overall and wrin­kled around the edge. In modeling the behavior of a slack film in a bulge test, it has been assumed that the wrinkles would be eliminated by the application of an infinitesimal pres­sure [42]. Our finite element results refute that assertion. The stress state in an initially slack film with the elastic properties of aluminum at 0 and 0.2% strain is plotted in Figure 16. The initial bulge height is 51.6 j.lm for a film that is 1 j.lm thick, 7.0 mm in diameter and is under a modest 70 MPa circumferential compressive stress at the edge. The figure shows that, even at stresses likely to have caused yielding, the edge of the film remains in circumferential compression.

Virtually any compressive stress will buckle a thin film, causing it to wrinkle. Our findings indicate that this wrinkling will not be eliminated even by the application of a sub­stantial pressure. This result invalidates bulge test models for slack films based on an axisymmetric shape, which includes all the circular film models used to date. Thus, wrinkled films or films in compression cannot be reliably tested by bulge testing of free standing portions of the film.

3.1.3. Taut Films. Having eliminated the possibility of measuring the properties of slack films, one must turn to films in residual, equi-biaxial tension. As described above, the spherical membrane equations can be modified to model a taut film by replacing the quan­tity A in Eq. (8) with (Yo/Y. The result is a pressure-displacement relation similar to Eq. (9):

182

-CIS

~ '-' til til

g tI)

20 250

0% strain 0.2% strain 200 ---

--~ 0 -, " , '2 150 \

-20 \ ~ " " I , I '-' 100

til , \ til ,

-40 g 50 \

tI) -radial \

-radial - - -circumferential \

-60 0 \ - - -circumferential

-80 -50

0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5

radius (mm) radius (mm)

Fig. 16. Stress distribution in a slack film before applying a pressure and at 0.2% strain at the center of the film.

P - ~h(2Ycalc h2 ) - 2 2 + 0"0 a 3a

(12)

Note that for a linearly elastic material one would expect a plot of Plh vs. h2 to yield a straight line with a slope proportional to the modulus and an intercept proportional to the residual stress in the film. By fitting Equation (12) to simulated data one obtains values for the initial stress in the film and the approximate biaxial modulus, Y calc. The initial stress that one calculates from this procedure is almost identical to the input value. The actual biaxial modulus, Y, is somewhat higher than the calculated value given in Eq. (12). An expression similar to Equation (11), but with a weak dependence on 0"0' can be used to determine Y to within 1% of the correct value [43].

3.2. SQUARE FILMS

It is apparent from Eq. (12) that the bulge properties of a film are extremely sensitive to the film dimensions: the pressure needed to produce a given bulge height varies inversely with the fourth power of the radius, a. Thus, even a 2% error in film radius, which can only be achieved through careful sample preparation and measurement, results in an 8% error in the film modulus. Furthermore, it is difficulrto obtain a perfectly circular film geometry. For these reasons we have developed a new method of sample preparation that takes advantage of standard micromachining techniques, which give substantially greater control over film dimensions. Anisotropic etching of silicon allows us to fabricate square, freestanding SiNx films with very precise dimensions. The SiNx films are then used as substrates onto which metal films are deposited, and the metal-SiNx composites thus obtained are bulge tested. This composite membrane technique, first suggested by Bromley et aL. [44], has a number of advantages that are discussed below. In this section we also discuss the sample preparation process and the derivation of equations for calculating film properties from bulge tests of square films, along with. experimental verification of the accuracy of these equations. A more detailed discussion will be published elsewhere [45].

183

3.2.1. Sample Preparation. Figure 17 depicts the different steps of the sample fabrication process used in this study. First, a SiNx film with a residual tensile stress is deposited by means of low pressure chemical vapor deposition (LPCVD) on both sides of an n-type, (100) Si wafer. SiNx is used because it can be deposited with a residual tensile stress ranging from nearly zero to several GPa depending on the deposition parameters. Second, square windows with sides parallel to the <110> directions of the silicon are patterned in the SiNx on the backside of the wafer by means of photolithography and reactive plasma etching. A KOH:methanol:H20 etchant is used to etch through the silicon to create a free­standing SiNx membrane on a silicon frame [46]. The etch rate of SiNx in this etch ant is negligible compared to that of silicon, so that it can be used both as a mask and as a mem­brane. KOH is an anisotropic etchant that attacks the silicon {100} planes much more rapidly than the {Ill} planes. As a result, the SiNx membranes are virtually perfect squares bounded by {111} planes in the silicon. Finally, the metal film of interest is deposited onto the SiNx membrane and the composite membrane can be tested.

1.

2.

3.

Dry etching: SF6+CF3Br

____ SiN" Anisotropic etching

~ Photoresist ~ Mctalfilm

Fig. 17. The different steps used to fabricate square freestanding metal-SiNx composite films.

3.2.2. Basic Solution/or Square Films. Calculating the deflection of a circular film under a uniform pressure is a difficult problem when the displacements are large. This is even more true for square membranes because the solution is not axisymmetric. However, good ap­proximate solutions can be derived using an energy minimization technique [47]. In this approach, one assumes a displacement field for the film that contains a number of unknown parameters and satisfies the boundary conditions. Using the principle of virtual displace­ments, the unknown parameters are then determined by the condition that the total potential energy of the system be minimum with respect to those parameters. The accuracy of this method depends on the type of displacement field selected and on the number of parame­ters. A displacement field that works particularly well for stress-free films [45] leads to the following expression for the deflection of the center of a square film with sides of length 2a:

184

I

h= f(V{pa4~t-V) J (13)

where t is the film thickness and f(v) is a complicated function of Poisson's ratio which can be approximated by f( v) "" 0.800 + 0.062 v. The form of Equation (13) is the same as that for circularfllms, differing only by the function of Poisson's ratio.

Equation (13) holds only if there are no initial stresses in the fllm. The effect of an equal-biaxial initial stress can be easily accounted for, however, if one assumes that the pressure applied across the film can be resolved into two components, PI and P2, such that PI is balanced by the initial stress in the membrane and P2 by the stretching of the mem­brane as it is bulged. An expression for PI can be derived from the theory of small deflec­tions of membranes [39], and Eq. (13) can be used to calculate P2. The load-deflection relationship for a film with a residual stress 0'0 is then given by

0' t Et 3 P=PI +P2 =cI a~ h+c2 4 h (14)

a (1- v)

where C1 is a constant equal to 3.393, independent of material properties, and C2 is given by f(v)-3. One can determine Young's modulus and the residual stress in a fllm by fitting Equation (14) to bulge test data.

3.2.3. Modifications/or Composite Films. In the composite membrane technique the metal fllm of interest is deposited onto an already freestanding, tensile film of another material, SiNx in this work. If the metal film is in compression, it is often possible to select the deposition parameters of the SiNx film such that the average stress in the composite is tensile and the composite film forms a taut membrane that can be tested without difficulty. This technique has the additional advantages that it can be applied to a wide variety of films without major changes in the sample preparation method, and that the fllms require no handling once deposited.

Equation (14) allows one to obtain the average residual stress and Young's modulus of the composite, provided one knows its Poisson's ratio ve. This last quantity can be calcul­ated readily using the following expression for the in-plane Poisson's ratio of a layered composite, derived from an isostrain model:

_ tsEsVs(l- V~)+tmEmVm(l- 0) Ve - (2) (2) tsEs 1- vm +tmEm 1- Vs

(15)

where the subscripts s and m denote the SiNx and metal film, respectively. Equation (15) requires the knowledge of the Poisson's ratios and Young's moduli of both films. The elastic properties of the metal film are of course not known, but rough estimates are usually sufficient, since C2 is not a very sensitive function of Poisson's ratio. Young's modulus of the film, Em. can then be calculated from the in-plane elastic modulus of the composite, Ee, by solving the following equation, also derived from an isostrain model:

E _ (tsEs + tmEm)2 - (tsEs vm + tmEm Vs )2

e - (ts +tm)(tsEs{l- v~)+ tmEm{l- vi)) (16)

185

The Young's modulus of the metal film can be calculated iteratively for higher accuracy. The residual stress in the metal film, (J'm, is calculated by expressing the average residual stress in the composite, (J'o, as the weighted average of the stresses in the SiNx and metal films:

(17)

3.2.4. Experimental Verification. In order to verify this analysis, aluminum-SiNx compos­ites were fabricated and tested. SiNx films were grown at 785°C in one part NH3 to 5.2 parts SiH2C12 at a total pressure of 40 Pa. Freestanding square windows of the nitride, 4 mm on a side, were created by anisotropic etching of the Si substrate as described above. The average nitride thickness was determined by ellipsometry to be 290 ± 5 nm. In order to use the composite membrane technique, the properties of the substrate film have to be well known. The nitride films were therefore extensively characterized. Finally, 111m aluminum films were evaporated onto a number of these SiNx membranes at 120°C and the resulting composites were tested.

Typical pressure-deflection curves of a SiNx film and a SiNx-AI composite are depicted in Figure 18. Both loading and unloading data are plotted. The hysteresis in the curve for the aluminum film is due to plastic deformation of the aluminum during loading. Plastic deformation occurs mainly near the centers of the edges of the film where the stresses are maximum. The SiNx film, on the other hand, does not show any sign of plasticity. Young's modulus of SiNx is measured to be 222 ± 3 GPa, assuming a Poisson's ratio of 0.28 [45] and the residual stress in the nitride is found to be 124 ± 14 MPa. Using these values in Equations (15) through (17) one finds a Young's mod­ulus of 67 ± 4 GPa for the aluminum '2 25

Pressure-deflection curve; 30 for SiNx and AI+SiNx.

ESiNr 222 GPa I films. These films have a weak (111) ~ 20

EAl=67GPa f

texture. One would, therefore, expect an elastic modulus slightly greater than the polycrystalline value of 69 GPa. The agreement of the bulge test results is nevertheless very good. The residual stress in the aluminum film was found to be 132 ± 8 MPa which also agrees well with thermal stresses in 1 11m aluminum films reported in the literature [48].

4. Bulge Tests of Ag-Pd Films

~ en 15 '-.....Al+SiNx

£ 10

5

o~~~~~~~~~~~~~ o 60 80 100 120

Height (11m)

Fig. 18. Typical pressure-deflection curves for SiNx and SiNx+AI freestanding films

Another modulated film system for which large enhancements in elastic properties were reported on the basis of bulge tests is Ag-Pd [2]. Since this system has been investigated less thoroughly than other "supermodulus" film systems, it was chosen for further investigation using the improved sample preparation, bulge testing and data analysis techniques described above.

186

4.1. SAMPLE PREPARATION AND CHARACTERIZATION

Based on our results for Au-Ni, we expected that Ag-Pd films might be in compression, at least over some range of composition wavelength. We therefore chose deposition condi­tions for SiNx so as to produce films with moderately high tensile stresses, so that the average stresses in the composites would be tensile. Thus, SiNx films were deposited at 785°C in one part NH3 to 3 parts SiH2Cl2 at a total pressure of 48 Pa. The Si substrates were etched to produce nitride windows 2.67 mm on a side. Ag-Pd modulated films were deposited onto these windows using a DC magnetron sputtering system described in detail elsewhere [49]. In this system, shuttered sputtering guns were used for precise control of the composition modulation. The base pressure of the system before deposition was 0.73 IlPa. Argon at 0.40 Pa was used to sputter the Ag and Pd target materials. A seed layer of 17 nm of Pd was deposited at 300°C at a rate of 0.26 nm/sec. This seed layer is necessary to reliably achieve (111) texture in the modulated film. The Ag-Pd films were deposited at 55°C at a rate of 0.29 nm/sec for both materials.

Composition-modulated Ag-Pd films were fabricated with bilayer periods ranging from 1.28 to 4.25 nm as determined by a quartz crystal rate monitor. Equal thicknesses of each material were deposited, with the total film thickness reaching approximately 425 nm. These films were then characterized via x-ray diffraction to determine the texture. Symmetric scans revealed a large 111 peak with a 111 to 200 peak ratio approximately three times larger than that expected for a randomly-oriented thin film specimen. A pole figure of one of the specimens also verified the existence of (111) texture in the ftlm.

4.2. BULGE TEST EXPERIMENTS

A set of 6 nitride films from a single deposition were tested in the bulge test. The biaxial modulus and stress for the nitride was calculated by fitting Eq. (14) to the experimental data, using a Poisson's ratio of 0.28 to calculate C2. From a total of 14 tests, at least two for each sample, the biaxial modulus of the nitride was determined to be 348.2 ± 5.6 OPa. From a total of 5 tests, at least two for each of two samples, the residual ten­sile stress in the nitride was found to be 217 ± 2 MPa. Each of these values is the mean and standard deviation calculated from the measured values. The Young's modulus of this nitride, 250 OPa, is higher than that of the nitride films described in section 3.2 due to the change in the material required to produce the desired stress.

The Ag-Pd films were sputtered onto fresh nitride samples from the same deposition. A typical loading curve from a bulge test of one of the composite samples is shown in Figure 19(a) along with the corresponding plot of Plh vs. h2 in Fig. 19(b). As discussed in section 3.1.3, the latter curve should be linear for a linearly elastic material, with the slope propor­tional to the film stiffness and the intercept to the residual stress in the film. As these repre­sentative plots show, these Ag-Pd films demonstrate linear elastic behavior.

We report here the biaxial moduli of the Ag-Pd films, as opposed to the Young's moduli, in order to be consistent with previous authors. The biaxial modulus of each modulated film was calculated in the following manner to minimize errors due to uncer­tainty in the Poisson's ratio: First the bulge test stiffness, C2EI( 1-v), of the composite was obtained by fitting Eq. (14) to the experimental data, as was done for the nitride films. Next, the rule-of-mixtures value of C2EI( 1-v) of the metal was extracted using the value of this stiffness experimentally measured for the nitride. Finally, C2,Ag-Pd was calculated from an average of the Ag[111] and Pd[111] values of the Poisson's ratio and factored out of the

187

25 0.45

Ag-Pd +SiNx composite 0.40 20 1.=2.5 run

"'" 0.35 ~ "'" ~ loading and unloading 8

15 { 0.30

~ ~ 0.25 '" 10 '-'

'" J: ~ 0.20 5 Loading data showing 0.15

linearly elastic behavior 0 0.10

0 10 20 30 40 50 0 500 1000 1500 2000 2500 a) Displacement <!..lm) b) h2 (llm1 Fig. 19. (a) Loading and unloading data for a SiNx+Ag-Pd composite and (b) the corresponding Plh vs. h2 curve. Each point denotes a measurement.

bulge test stiffness to obtain the biaxial modulus of the modulated film. The resulting values of the biaxial modulus are plotted as a function of composition wavelength in Figure 20. The values calculated by this simple method are in good agreement with those obtained from Equations (15) and (16). Although these results are reported from only one test per sample, based on previous experience with other systems we would expect deviations to be on the order of ±10%. The average modulus of all the films is 176 GPa, which is slightly higher than the rule of mixtures average of 167 GPa calculated from the polycrystalline moduli for silver and palladium layers of equal thickness.

The initial stress in each of the films was calculated using Eq. (17) and is plotted in Figure 21. A conservative estimate of the error in these measurements would be 15% of the

200

190

160

Biaxial moduli of Ag-Pd films Y = 176GPa

ava:.

<> <>

<>

<>

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Composition Wavelength (nm)

Fig. 20. Biaxial moduli of Ag-Pd films plotted vs. composition wavelength. The expected error is approximately ±1O%.

Stress in Ag-Pd films

25 <> <>

o r-____________ ~~t=en=si=on~ compressIOn

-25

<>

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Composition Wavelength (nm)

Fig. 21. Residual stress in Ag-Pd films vs. composition wavelength. The ex­pected error is approximately ±14 MPa.

188

total stress of the composite, or ±14 MPa. From the figure it can be seen that the stress in the films does vary with composition wavelength and is maximum in compression when this quantity is approximately 2 nm.

While more tests must be performed in order to obtain a statistical sampling for error determination, it is clear that the Ag-Pd films tested for this report do not show the super­modulus effect, nor do they exhibit any nonlinear elastic behavior. Further, as with the measurements on Au-Ni films, shown in Fig. 10, there is a maximum compressive stress in these modulated films at the composition wavelength at which the supermodulus effect had previously been observed.

5. Discussion

To our knowledge, the nanoindentation result shown in Fig. 5 is the ftrst report of a small decrease in elastic properties obtained by a mechanical deflection technique which is quanti­tatively similar to the behavior frequently reported from phonon velocity measurements. The analysis used accounts for both the increasing variance of the contact compliance with decreasing indentation depth and for variations in the machine compliance from experiment to experiment. If the variance in the contact compliance is not accounted for, then the scatter in the results obtained using the Doerner and Nix method [29] obscures these elastic prop­erty variations [25]. If the variations in the machine compliance were not taken into account, then a very large decrease in the indentation modulus would be calculated as an artifact of stress-induced sample geometry changes. Each data point in Fig. 5 is the result of between 50 and 100 indentations made at a variety of locations across each sample. Thus, we believe the observed variations in this set of samples to be real. Nonetheless, in tests we conducted on two different sets of Au-Ni films (the "series #87" and "series #88" samples in ref. [33]), no significant variations in indentation modulus with composition wavelength were seen. This is probably due to the fact that these samples were sub­stantially thinner (film thicknesses of about 200 and 450 nm, respectively) and it was not possible to obtain a range of indentations which avoided both surface topography and sub­strate effects. Nanoindentation experiments are relatively easy to perform. Given samples sufftciently thick considering their surface quality and using appropriate data analysis, more work of this type could provide additional useful information regarding the elastic proper­ties of compositionally-modulated ftlms.

Of course, indentation tests are limited in that the influence of individual elastic con­stants is unknown. Until this problem is solved, other deflection geometries must be used in order to obtain known combinations of elastic constants. The microbe am deflection experiment is one method by which this can be accomplished without the requirement of removing the film from its substrate. Unfortunately, the geometry of the beam samples reported here was such that end effects dominated the results and precluded the possibility of obtaining an accurate determination of the film elastic properties. This problem could be resolved by selecting a more appropriate beam geometry. This will, however, require the development of new beam fabrication techniques. Still, the beam deflection results shown in Fig. 9 indicate unambiguously that the in-plane stiffness of these films does not increase dramatically with decreasing composition wavelength. Furthermore, these results suggest that the in-plane stiffness might show a small minimum corresponding in wavelength to the minimum in the indentation modulus.

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Perhaps the most significant result of this study is the discovery of wavelength-depen­dent substrate interaction stresses in Au-Ni and Ag-Pd films. Although measured by differ­ent techniques, the stresses exhibit the same behavior in both modulated film systems: a peak in compression at a composition wavelength near 2 nm. This corresponds in wave­length to the peak of the "supermodulus effect" as previously reported for similar films [1,2]. We have shown how these stresses can account for both the location and general form of biaxial modulus enhancements with composition wavelength and the nonlinear stress-strain behavior reported from bulge tests [1-3] as artifacts of the analysis technique. Although we conducted our simulations based on the average substrate interaction stress in each film, it is important to realize that a stress gradient in the film, or a stress interaction between the modulated portion of the film and any "seed" or "buffer" layer which varies with composition wavelength could produce the same effect even if the film were removed from the substrate and allowed to relax completely before being mounted for testing. With the possible exception of the Au-Ni films of Yang et al. [1], all of the films for which large elastic enhancements were reported also had buffer layers of the order of 50 nm thick. As we have shown, only very small stresses are required to produce very large "supermodulus" effects. The third feature of the "supermodulus" effect, the variation in modulus "enhancement" with composition amplitude, is also explainable as a stress artifact. Since the origin of these stresses is clearly related to the interfaces in these materials and since the amplitude of the composition modulation is adjusted by annealing the films, it is reasonable to assume that the stresses diminish as the films are annealed, and thus the "effect" would be seen to vanish.

Furthermore, we note that wavelength-dependent interaction stresses or stress gradients could have similar effects on the vibrating reed and torsional pendulum experiments, by which "supermodulus" effects were also reported, by changing the geometry of the test sample. In either case only a small curvature across the width of the sample would be required to increase the stiffness of the sample dramatically. Berry [23] conducted vibrating reed experiments on samples prepared by depositing Cu-Ni films on both sides of a silica strip to avoid stress-induced bending and saw no variation in elastic properties with com­position wavelength, whereas both Testardi et al. [4] and Baral et al. [5] tested single Cu­Ni films and reported enhancements.

The remaining technique by which large enhancements were reported is the tension test [5,6]. It is not clear how interaction stresses or stress gradients would influence the results obtained from such experiments. However, Davis et al. [21,50] have recently conducted tension tests of Cu-Pd and Cu-Ni using improved techniques and see no variation in Young's modulus with composition wavelength.

The results obtained from a bulge test, like the results of other mechanical deflection experiments, are quite sensitive to geometry and initial and boundary conditions. The "supermodulus" literature provides ample evidence of this. However, our studies have shown that the bulge test can be a very reliable experimental technique if the geometry and initial and boundary conditions are properly accounted for. Our finite element studies indi­cate that, if the geometry is known precisely for a circular film in tension, the spherical cap model returns the biaxial modulus to within 10% of the actual value and also can be used to determine virtually exactly the value of the tension stress. Furthermore, with an estimate of the Poisson's ratio of the film, an empirical correction [43] can be made to obtain a much more accurate estimate of the biaxial modulus. This analysis also shows clearly that films in compression cannot be tested as freestanding films. In addition, it is very difficult to obtain

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precise circular geometries. The new sample preparation method described above resolves both of these issues and, although the data analysis is somewhat more complicated [45], square composite films can be used to determine very precisely the elastic properties and internal stresses in thin films, even those in moderate compression. This method has been applied to testing of Ag-Pd films. Both the stresses and the biaxial modulus results from these experiments are consistent with the results of our beam deflection experiments in Au­Ni films. No nonlinear elastic behavior was observed.

At this point, all of the modulated film systems for which large enhancements in elastic properties with composition wavelength were reported, i.e. Au-Ni, Cu-Pd, Cu-Ni, and Ag-Pd, have been re-tested using mechanical deflection techniques ([21,26,50] and this study) and, to our knowledge, all but Au-Ni have been tested using phonon velocity mea­surements [14,19-22,50]. No large enhancements in stiffness were found. Only two of these reports ([22] and this study) have shown any significant variation in elastic properties and these were on the order of tens of percent. Although variations of this order have been referred to as "anomalous" in the past, these modulated film systems have extremely high interface densities. It should be expected that the elastic properties of these materials in the small composition wavelength limit, would not be well predicted by simple calculations based on bulk single crystal properties.

6. Conclusions

We have measured composition-wavelength-dependent stresses in Au-Ni and Ag-Pd films and conclude that the "supermodulus effect" seen in mechanical deflection experiments is very likely an artifact of such stresses. We have developed new methods of sample prepa­ration and data analysis for bulge testing which avoid these artifacts and allow accurate measurements of elastic properties and internal stresses. Results of our nanoindentation and micro-beam deflection experiments in Au-Ni and of bulge test experiments in Ag-Pd are consistent with the small variations in elastic properties of modulated films frequently seen in phonon velocity measurements.

Acknowledgments

The authors would like to thank Soonil Hong for preparation of the beam substrates, Alan F. Jankowski for the deposition of the Au-Ni films and Bruce M. Clemens for guidance in the preparation and characterization of the Ag-Pd films. This work was carried out under a grant from the U.S. Department of Energy (DE-FG03-89-ER45387). The work of B.J. Daniels was supported by the National Science Foundation (DMR-9100271)

Appendix

In order to account for variations in machine compliance and in the variance of the contact compliance with indentation depth, the following procedure was used:

1) The total compliance and the contact area were obtained for each indentation as described in section 2.2.2. A data sample was considered to be all of the indentations of the same nominal size from a single array. If fewer than 4 indentations were available for anyone data sample, the sample was considered to be too small and was not considered further.

191

2) The means, C and A of compliance and contact area, along with an estimate of the variance of the compliance, V C, were calculated for each data sample.

3) For each array, the averaged data were then plotted as C vs. (A)-O.5 and a weighted fit of a straight line to these data using the compliance variances as weighting factors was perfonned. The intercept, Cm, and the variance of the intercept, V C ,are easily m detennined from this fit

4) The machine compliance, Cm, for each array was subtracted from the mean total compliance for each data sample from that array. The remaining compliance should be the mean contact compliance, Cc ,for each sample. Because there is some uncer­tainty in the value of Cm for each array, the uncertainty in the value of the mean con­tact compliance for each sample must be increased by this amount. That is, V Cc = V C + V Cm •

5) Once the data from all the arrays in one physical sample have been treated as described in steps 3 and 4, these data are plotted together, now as Cc vs. (A)-O.5, and the maximum likelihood fit described in step 3 is perfonned on this combined data. The slope, p, can then be used in Equations (3) and (2) to obtain a measure of the elastic properties of the sample. The elastic modulus, I, corresponds to Esf(1- vI) in Eq. (2). The standard deviation of the indentation modulus, ..JVi, is calculated from the standard deviation of the slope, FP, using VI = (alla{3)2v /3.

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