- Email: [email protected]
Description of brittle failure of non-uniform MEMS geometries
Thin Solid Films 515 (2007) 3267 – 3276 www.elsevier.com/locate/tsf
Description of brittle failure of non-uniform MEMS geometries A. McCarty a , I. Chasiotis b,⁎ a
Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, U.S.A. Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.
b
Available online 28 February 2006
Abstract The description of probability of failure of polysilicon micromachined components with complex geometries using a single set of Weibull parameters was investigated. Strength data from both uniform tension and from twelve non-uniform specimen geometries with central perforations were employed. These perforations allowed for twelve different combinations of stress concentration factors and radii of curvature. Two methods were applied to determine the Weibull parameters: in the first method, only the strength data from uniform tension specimens were used to determine the Weibull modulus and the material scale parameter. In the second approach, the strength data from all non-uniform tension geometries were used to calculate the material scale parameter and the Weibull modulus using the maximum likelihood method. The non-uniform stress state in each perforated specimen was taken into account through an elasticity finite element model and the use of the integral form of the Weibull probability function. Using the first method, an analysis considering active flaw populations at the top specimen surface or the specimen sidewalls indicated that the active flaw population is not the same at all scales: for 1–3 μm radius perforations and small stress concentration factor (K = 3) the active flaw population was located at the specimen top surface, e.g. surface roughness, which, as the analysis indicated, was also the case for uniform tension specimens. For higher stress concentration factors (nominal K = 6 and 8) the analysis indicated that the active flaw population was located at the hole sidewall surface. As a result, for a given material the geometry of the specimen and the local state of stress determine the active flaw population from all flaws generated during fabrication and processing. Therefore, it is important that all potential failure modes be considered when extrapolations to other geometries and loadings are made using the Weibull probability function. © 2006 Elsevier B.V. All rights reserved. Keywords: Weibull; Stress concentrations; Fracture strength; Polysilicon
1. Introduction Brittle failure of uniform tension specimens, as characterized by significant scatter in the strength data, is often described by a two or three parameter Weibull distribution function. This probability of failure function is based on the weakest link assumption (WLA); the weakest material element will initiate catastrophic failure [1,2]. The advantage of this method of analysis is that Weibull parameters (i.e. Weibull modulus and characteristic strength) measured at one scale could be used to analyze self-similar geometries at another scale. To allow for scalability in the description of material failure three conditions must be satisfied: (a) the active flaw population must be invariant between different size specimens, (b) failure initiated by a flaw must be catastrophic, and (c) the stress field must be the same for all specimens. The last condition implies that, ⁎ Corresponding author. E-mail address: [email protected] (I. Chasiotis). 0040-6090/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2006.01.039
although the material might be the same, different loading profiles would invalidate previously determined Weibull parameters unique to that particular geometry. From a fracture mechanics viewpoint, component strength is affected by a change in mode of loading for a given set of flaws; this is because different stress distributions affect the probability of larger or smaller size flaws being critically loaded. Upon modification of boundary conditions, cracks originally loaded in mode I are then loaded in mixed mode and vice versa. Under these circumstances new Weibull parameters are required to describe the material strength. Although the description of brittle failure using an energy release rate criterion instead of the material strength is more deterministic, this is true only for homogeneous and isotropic materials. For instance, for polycrystalline silicon used in Microelectromechanical Systems (MEMS), one can hardly speak of a single value for the fracture toughness; the latter varies between 0.8 and 1.2 GPa due to material anisotropy [3,4]. The use of such fracture data in device design requires detailed
3268
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
modeling of the material microstructure. Instead, from a device design viewpoint it is valuable to examine the possibility of an efficient use of strength data combined with appropriate modeling in order to obtain failure predictions for MEMS components. Among the advantages of the Weibull probability function is the potential to determine the location of process-related flaws, e.g. whether the most detrimental flaws reside on a particular specimen surface, edge, or they are uniformly distributed in the entire component volume. For any of these flaw distributions the probability of failure can be calculated using parameters that are characteristic for the specific material. Work by Chasiotis and Knauss [5] indicated that proper specimen design could reveal the source of failure in microscale components and help to distinguish between surface and volume active flaw populations, which in turn could be related to the fabrication conditions. An extension of their work using a variety of perforated polysilicon specimens fabricated by Multi-User MEMS Processes (MUMPs) 35 indicated that the characteristic strength scaled inversely with the size of the perforation [6,7]. Other studies have also been conducted to characterize brittle failure of microscale polysilicon specimens under uniform stress [8]. When different geometries (or stress fields) are used the Weibull analysis can be conducted using the integral form of the probability function. Johnson et al. [9] used the integral form of the Weibull function to derive analytically the Weibull parameters from MEMS scale beams that were subject to bending. Bagdahn et al. [10] used the Ceramics Analysis and Reliability Evaluation of Structures (CARES) design software [11] with material scale parameter and Weibull modulus derived from uniform tension specimens to predict failure of specimens with 2.5 μm edge or central circular perforations fabricated by MUMPs 42. The authors evaluated their approach by examining the agreement between the predicted and the experimental characteristic strength that agreed within 9%. In that study, the sidewall surfaces of the notched specimens were identified as the location of the active flaw population, as reported also before by Chasiotis and Knauss for MUMPs35 [5]. In the work by Bagdahn et al. [10] the failure of the uniform tension specimens scaled with the top specimen surface while the failure of specimens with round holes, for which data from uniform tension specimens were used to make predictions, scaled with the hole sidewall surface. The present work examined the applicability of a unique set of Weibull parameters to describe the probability of failure of a wide variety of specimens with internal holes of different radii of curvature and stress concentration factors. To remove the geometric factor of specimen strength from the Weibull probability function and compute a pair of geometry independent parameters two approaches were compared in terms of their accuracy in describing the probability of failure of non-uniform geometries. The first approach, also employed in [10], used data from uniform tension specimens of the same material to predict the strength parameters of nonuniform specimens. In the second method that is introduced
in this work, the Weibull parameters were obtained from the combined experimental data from all non-uniform geometries using an optimization approach. In each case a single pair of Weibull parameters was derived to compute the strength of all non-uniform geometries. The effectiveness of the two different approaches was determined using as metrics the predictions of the characteristic strength and the entire probability of failure curve for each perforated specimen geometry. 2. Method of analysis The probability of failure of a uniform tension specimen, Pf, under stress, σ, is described by the Weibull probability distribution function r−ru m Pf ðrÞ ¼ 1−exp − ð1Þ rc where the shape parameter or Weibull modulus, m, indicates the scatter of strength values and for most ceramics m = 5–10. Lower values are possible but when m = 3, or smaller, the probability density function becomes increasingly skewed. The location parameter (or stress threshold), σu, describes the stress below which failure will not occur. The characteristic strength, σc, is the stress for which the probability of failure is 63.2% when σu = 0. In order to simplify the determination of the Weibull modulus and characteristic strength, a conservative description of strength is selected where the location parameter in Eq. (1) is set equal to zero. Eq. (1) can be implemented using an ordinal probability estimator and linear regression. This method is useful because the only requirement to determine the probability of failure for a strength datum point is the rank of the measurement with respect to the rest of the data. However, the choice of ordinal probability estimator can greatly affect the values of the Weibull parameters. This choice hinges upon the size of the data set. Important considerations are summarized in [5]. An alternative approach is the maximum likelihood method. In evaluating data as a random sample of a population, the maximum likelihood determines the statistical parameters of a distribution that are most probable to describe the population [12]. The likelihood function is the product of the probability density function evaluated at each of the data points as LðhÞ ¼
n Y
Pðri ; hÞ
ð2Þ
i¼1
where P(σi,θ) is the probability density function, σi is the result from the ith test, and θ represents the parameters to be determined for the data set. For the case of the Weibull probability distribution function m n Y mrm−1 ri i Lðm; rc Þ ¼ exp − ð3Þ m r r c c i¼1 By setting the partial derivative of the likelihood function with respect to the parameters m, σc equal to zero, a system of
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
equations is developed to calculate the maximum of the likelihood function. From Eq. (3) the following system of equations is determined " #1=m̂ n 1 X ̂ m r̂ c ¼ r ð4Þ n i¼1 i where m̂ ¼
n ð1=r̂ c Þ
n m̂ P i¼1
̂ rm i lnri −
n P
ð5Þ lnri
i¼1
Eqs. (4) and (5) are coupled and are solved numerically for mˆ , σ ˆ c, which are estimates of the Weibull modulus and the characteristic strength, respectively. A general form of Eq. (1) (with σu = 0) can be used to account for variable stresses in a specimen by assuming, for instance, that the entire volume of the material contributes to the failure probability Z rðx; y; zÞ m Pf ðrÞ ¼ 1−exp − dV ð6Þ r0 This form of distribution function does not require the material to be under uniform tension and it depends on the material scale parameter, σ0, which, together with m, are, in principle, independent of specimen size and stress distribution. Thus, σ0 and m could be determined from any specimen geometry. In contrast, the characteristic strength depends on the size and geometry of the specimen and the applied stress field. Limitations on the geometry invariability of σ0 and m are discussed in the Results and discussion section. In order to determine the Weibull parameters from Eq. (6) the stress as a function of location must be known analytically or numerically. Note that Eq. (6) is in agreement with WLA: it represents the combined probability of failure for a large number of “links” each weighted by the local stress. The “links” that are near the perforation have considerably higher probability to fail compared to those in the far field. By defining a quantity that contains the effects of specimen geometry and stress distribution, σ0 and m can be determined. The stress state is represented by the stress value at an arbitrary material point, e.g. the maximum stress, σmax Z rmax m rðx; y; zÞ m Pf ðrÞ ¼ 1−exp − dV rmax r0
Veff ¼
rðx; y; zÞ rmax
ð7Þ
m dV
and σ0 are constant with respect to location, Eq. (7) can be written as rmax m Pf ðrÞ ¼ 1−exp −Veff ð9Þ r0 The geometry factor, Veff, can be calculated numerically using a Finite Element (FE) model to determine σ(x, y, z) for the given boundary conditions. However, Veff depends on the Weibull modulus, and the three parameters, Veff, m, and σ0 must be determined simultaneously. Veff can be combined with σ0 as 1=m 1 rc ¼ r0 ð10Þ Veff Replacing in Eq. (9) the characteristic strength for the material scale parameter and the effective volume as described by Eq. (10) rmax m ð11Þ Pf ðrÞ ¼ 1−exp − rc Eq. (11) can be used with linear regression or the maximum likelihood method, to determine the Weibull modulus and the characteristic strength. If linear regression is selected, then the ordinal probability must be determined. For a small sample sizes, as the one used here, the best description of the ordinal probability according to [13] is Pfi ¼
i ¼ 3=8 n þ 1=4
ð12Þ
where n is the number of data points and i is the ordinal rank of a datum point. The formulation of Eqs. (6)–(11) was done under the assumption that the active flaw population was uniformly distributed in the entire specimen volume. If the flaws causing failure reside at a specimen surface, the same analysis should be carried out to find an effective surface area, Aeff, instead of Veff. Since the determination of the location of the catastrophic flaw is not always a straightforward task, a surface and volume Weibull analysis combined with fractographic studies should be applied. 3. Specimen preparation and experimental procedure
The integral of the relative stress distribution in Eq. (7) has units of volume and is defined as the effective volume (or geometry factor), Veff Z
3269
ð8Þ
The effective volume can be viewed as the volume of a uniform tension specimen that has the same probability of failure as the non-uniform tension specimen. Because σmax, m,
The polysilicon test samples were dog-bone shaped tensile specimens with various geometries of circular/elliptical perforations centrally located at the gauge section. They were fabricated at Cronos (former MCNC, North Carolina) using the MUMPS 35 run. The average grain size was 300 nm. The dimensions of the gauge section (L × W × T) varied from 250 × 30 × 2 to 700 × 340 × 2 μm depending on the size of the perforation. The variations in gauge section and the perforation geometry produced specimens with nominal stress concentration factors of 3, 6, and 8. Because of the finite width specimens the exact stress concentration factors were determined by a FE analysis and they varied between 3 and 11. For the purpose of
3270
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
the present discussion reference to each of the geometries is done according to its nominal radius of curvature and stress concentration factor. For each stress concentration factor, specimens with similar geometry and nominal radii of curvature of 1, 2, 3, and 8 μm were fabricated. These specimens were cofabricated on each die together with two 50 μm wide uniform tension specimens. Due to limitations in fabricating large numbers of every specimen geometry on a single die, we tested a total of 12 samples per non-uniform geometry and 24 uniform tension samples. From those tests, 17 successful tests of uniform tension specimens and 8–11 successful tests per nonuniform geometry were used in the analysis. Fig. 1 shows a typical uniaxial tension specimen and three examples of perforations whose exact dimensions were measured by a Scanning Electron Microscope (SEM).
The specimens were subjected to uniaxial tension as described in [14]. The large paddle shown in Fig. 1(a) was gripped via a method developed by this group that employs a flat glass grip coated with a UV-curable adhesive [15]. A piezoelectrically driven actuator was used to impose global displacements while the resulting applied force was measured using a load cell, which, in turn, provided the far-field stress at failure. A FE analysis implemented in ANSYS was used to determine the stress field at failure in each specimen. More details of the experimental procedures are presented in [5]. In that work a total of 100 data points were used, while for the present analysis we tested 31 additional specimens from the same fabrication run (i.e. 131 data points in total.) It should be noted that the Pf–σf trends were the same as those in [5] after the addition of the new data points.
Fig. 1. (a) Uniaxial tension specimen containing a perforation at the center of the gauge section. Typical hole geometries located at the gauge section that resulted in nominal stress concentration factors, (b) K = 3, (c) K = 6, and (d) K = 8. In total, 12 different perforated specimen geometries were fabricated.
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
4. Implementation of Weibull analysis to small scale specimens with variable geometries For specimen geometries that do not allow for an analytical description of the stress state in Eq. (8), a numerical method is necessary. A FE model of each specimen was generated in ANSYS to determine the ratio of the local average stress in each element of the discretized specimen geometry to the maximum stress in the entire specimen. This ratio was raised to the power of the Weibull modulus and the integral in Eq. (8) became a summation over all of the elements in the FE model. The elements used in Eq. (8) may be those of the entire specimen volume, the top surface, the sidewall surface, or any other region where the active flaw population resides. The implementation of the Weibull analysis was done in Matlab. Among the different locations for the active flaw population considered here, a top surface-only analysis was conducted based on prior surface roughness data for the two specimen sides [16]. For MUMPs polysilicon the top and bottom film surfaces have distinctly different surface roughness. As shown in Fig. 2(a), the top surface roughness is very irregular, while the surface roughness of the bottom surface (Fig. 2(b)) is dominated by hemispherical protrusions. The difference between the two surfaces is due to the different degree of crystallinity and the considerably smaller grain size at the bottom of the film. The profile of the top surface roughness follows the grain structure. In principle, the Weibull parameters σ0 and m can be determined from any specimen geometry. In this work two
3271
different approaches were explored. In the first, σ0 and m were determined from the uniform tension specimens, as reported before in [10], but now applied to predict the probability of failure of a significantly larger number of specimen geometries. In the second case, they were determined via an optimization scheme where the values of σ0 were averaged for all nonuniform specimen geometries. Note that averaging σ0 is appropriate, as, given certain constraints, σ0 can be conceived a “material property.” The same is not true for σc. In both methods described in the next section, the characteristic strength and the Weibull modulus for each of the non-uniform specimens were determined using the maximum likelihood method which is appropriate even for small specimen populations [17]. The two sets of data used to compute the Weibull parameters for MUMPs polysilicon were the strength results from the uniform tension specimens (method #1) and the perforated specimens (method #2). 4.1. Calculation of Weibull parameters from specimens with uniform geometry First, predictions of the characteristic strength for nonuniform specimens using the material scale parameter and the Weibull modulus computed via the maximum likelihood method from uniform tension specimen strength data were obtained. Specifically, using FE results and Eq. (8) for each of the specimen geometries, the effective volume (or surface) was determined using the Weibull modulus of the uniform tension specimens. Then, the effective volume (surface) for uniform gauge section specimens was used to determine the material scale parameter using Eq. (10). The material scale parameter from uniform gauge specimens and the effective volume (surface) for each of the non-uniform geometries allowed for calculation of the characteristic strength of the perforated geometries using Eq. (10). The latter was compared to that calculated from the experimental data as a metric to determine the accuracy of failure predictions based on Weibull parameters derived from uniform tension specimen strength data. 4.2. Calculation of Weibull parameters from specimens with non-uniform geometries
Fig. 2. (a) Roughness of the top surface and (b) the bottom surface of a polysilicon specimen [16].
We also introduced a new method of analysis that employed strength data from all non-uniform specimen geometries to determine the Weibull modulus and the material scale parameter that provided the best description of failure for all perforated specimens. A parametric study of Weibull modulus was performed to determine the set of Weibull constants that described most accurately the probability of failure of all specimen geometries. For various Weibull moduli, the characteristic strength of each specimen was calculated using Eq. (4), and the effective volume (surface) was computed using Eq. (8). These values were substituted in Eq. (10) to determine the material scale parameter for every geometry for a chosen Weibull modulus. The material scale parameters of all specimen geometries were then averaged to determine a representative
3272
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
value of σ0 for a Weibull modulus. This procedure was repeated for various values of m. A metric was defined to evaluate the quality of predictions for each (m, σ0) pair and determine the pair that gives the best agreement with the experimental data. Each (m, σ0) pair was used to calculate the probability of failure for each of the geometries and for all experimental failure strengths. The absolute values of the difference between the ordinal probability and the Weibull-predicted probability were summed for all data points and for all specimen geometries. This procedure was repeated for all (m, σ0) pairs. The pair that produced the smallest sum of residuals gave the best descriptions for the strength of the non-uniform specimens. In order to evaluate the relative quality of predictions from the two methods employed to determine m and σ0, two metrics were used: (a) the percent difference between the predicted characteristic strength and the characteristic strength determined from an analysis of the experimental data for every specimen geometry, (b) the sum of the probability residuals calculated as described in the parametric study above. While σc is an important parameter in describing the Weibull distribution function, m can greatly alter the probability predictions while keeping σc constant. For this reason, the most accurate Weibull parameters should predict the probability of failure of the entire experimental data set (method (b)) instead of only the value of the characteristic strength. Furthermore, the residuals of failure probabilities were compared for different considerations of active flaw populations (top surface vs. side wall.) The location of flaw population that provided the minimum sum of residuals described the specimen failure in the most accurate way. 5. Results and discussion The data from each of the specimen geometries were analyzed using the maximum likelihood method to determine the characteristic strength. Using the first approach described in Section 4.1, the effective volume and surface were found for each specimen geometry using the Weibull modulus determined from the uniform specimen data. Various analyses considered the active flaw population being uniformly distributed over the entire volume (V ), the top surface (Ta), the sidewall surface of the entire sample (Sw), and finally, the hole sidewall surface (Ha). The percent difference in the characteristic strength between the predictions and the values determined from the experimental data for non-uniform tension specimens are given in Table 1. The values of (Sw) and (Ha) are identical because of the overwhelming probability for failure to start at the stress concentration. For instance, doubling the specimen length in the presence of a stress concentration did not change our results as long as the location of the boundary conditions did not affect the stress field around the hole. As done in previous works [5,10], the analysis that resulted in the closest agreement between predictions and experimental results provided an indication of the location of the active flaw population. In the present work, a particular analysis (Ta, or Ha) did not uniquely provide the best results for all of the
sample geometries. The (Ta) characteristic strength predictions for specimens with small holes and the smallest stress concentration factor (K = 3 and R = 1, 2, 3 μm) or specimens with holes with the smallest radius of curvature (R = 1 μm) and the largest stress concentration factor (K = 8) were the closest to the experimentally determined, indicating that the active flaw population resided at the top specimen surface rather than the hole sidewalls. These cases are highlighted in Table 1. For the remaining specimen geometries, the best agreement was achieved when the Weibull analysis assumed that failure was initiated at the perforation sidewall (Ha). The latter agrees with the results in [10] where 2.5 μm circular holes were considered. The two different dependencies, (Ta) vs. (Ha), are interpreted as follows: The severity of (Ha) flaws was smaller for holes with small radii of curvature and small stress concentration factors. High stresses were concentrated at a small perforation sidewall surface and, thus, a smaller flaw population of sidewall defects was active (provided that the small size of the hole did not alter the local flaw population density.) In comparison, significant part of the specimen top surface was still at high stress and the flaw population residing at the top surface (surface roughness) was active. This argument also explains the trends in the characteristic strength reported previously for the subset of data seen in Fig. 3 [5]. As marked by the dashed circle, the specimens for which failure was initiated at the specimen surface demonstrated the highest local strengths and were separated from the rest of the data. The values of σc clustered according to the smallest computed values of (Ta) as also presented in Table 1. For the rest of the specimen geometries the hole sidewall surface (Ha) provided the best overall results predicting the characteristic strength within 3.1–17.6%. The
Table 1 Percent difference in characteristic strength (predicted characteristic strength using the Weibull parameters from uniform tension specimens minus the characteristic strength determined from experimental strength data) for various flaw populations: volume (V ), top surface (Ta), entire sidewall surface (Sw), interior hole surface (Ha)
K/ #Specimens
R (mm)
V (%)
Ta (%)
Sw (%)
Ha (%)
3/10 3/10 3/8 3/9
1 2 3 8
14.5 18.2 17.1 39.8
7.7 15.5 15.6 40.1
20.2 16.8 18.5 4.7
20.2 16.8 18.5 4.7
6/11 6/10 6/8 6/11
1 2 3 8
31.7 50.7 57.1 51.6
29.3 50.7 53.8 51.6
12.1 3.1 6.8 12.7
12.1 3.1 6.8 12.7
8/8 8/9 8/9 8/11
1 2 3 8
10.1 51.8 53.3 58.3
10.1 51.7 53.4 58.4
26.3 6.5 6.3 17.6
26.3 6.5 6.3 17.6
The results are presented as a function of the stress concentration factor, K, and the local radius of curvature, R. The first column includes the number of specimens tested for each geometry. The shaded cells point to the flaw analysis that resulted in the closest prediction of the characteristic strength.
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
Fig. 3. Characteristic strength calculated from experimental data from various specimen geometries. The geometries for which, according to the analysis, the active flaw population was located at the specimen top surface supported higher characteristic strength (K = 3, R = 1, 2, 3 μm and K = 8, R = 1 μm.) The dashed line includes the specimen geometries for which failure was initiated at the specimen top surface compared to the interior hole sidewall surface for the rest of the specimens.
Weibull modulus used in this first method of analysis, as determined from the uniform tension specimens, was m = 11.6, and the material scale parameter was σ0 = 149 MPa m2 / 11.6 (Sw and Ha), σ0 = 185 MPa m2 / 11.6 (Ta), and σ0 = 60 MPa m3 / 11.6 (V). The second method for the calculation of m and σ0 aimed at providing a more accurate description of the entire probability curve of the complete set of perforated specimens. The failure description using this approach was compared to the predictions obtained using the first method of calculating m and σ0 from uniform tension data. The second method was based on two parametric analyses of the strength data from non-uniform tension specimens: one using the top specimen surface and a second using the interior hole surface. A study based on the volume of the specimens yielded almost the same results as the top surface analysis. Figs. 4 and 5 show
Fig. 4. Sum of residuals from a parametric study that considered the top specimen surface as the location of the detrimental flaws. The sum of residuals is the sum of differences of the probability of failure predicted by the parametric study minus the ordinal probability of failure obtained from strength data. The open square is the Weibull modulus, m = 11.6, determined from uniform tension specimens. The minimum of the plot is at m = 9.4.
3273
Fig. 5. Sum of residuals from a parametric study that considered the interior hole sidewall as the location of the detrimental flaws. The sum of residuals is the sum of differences of the probability of failure predicted by the parametric study minus the ordinal probability of failure obtained from strength data. The open square is the Weibull modulus, m = 11.6, determined from uniform tension specimens. The minimum of the plot is at m = 5.
plots of the sum of the probability of failure residuals as a function of Weibull modulus for the top specimen surface and the interior hole surface, respectively. The minimum sum of the probability of failure residuals for the top surface was found for m = 9.4 (close to that determined from the uniform tension experiments) while for the hole surface analysis the minimum value of the residuals of the probability of failure occurred for m = 5. Table 2 Residuals of experimental vs. calculated probabilities of failure computed using Weibull parameters determined from strength data from uniform tension specimens (Method 1) and strength data from non-uniform gauge section specimens (Method 2)
Specimen geometry
Method 1: Predictions based on m and σ 0 determined from uniform tension specimens
Method 2: Predictions based on m and σ 0 from a parametric study using non-uniform specimen geometries
K
R (μm)
Top surface analysis
Interior hole surface
Top surface
Interior hole surface
3 3 3 3
1 2 3 8
1.65 2.74 1.89 4.10
1.47 1.54 1.45 1.30
4.85 4.28 3.04 1.56
0.64 0.84 1.21 1.35
6 6 6 6
1 2 3 8
4.83 4.89 3.96 5.38
2.31 1.44 1.45 3.12
3.20 1.86 3.84 2.40
0.85 1.36 1.28 0.94
8 8 8 8
1 2 3 8
2.26 4.42 4.43 5.44
3.99 1.85 1.89 3.81
2.69 4.37 3.81 3.72
2.25 1.58 1.80 1.09
Sum of residuals
46.0
25.6
39.6
15.2
The residuals of failure predictions were computed over the entire cumulative density curve, as opposed at a particular point (characteristic strength) in Table 1.
3274
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
Table 2 summarizes the sum of the residuals between the experimental ordinal probability of failure and the calculated probability of failure using both methods to compute m and
σ0 (e.g. uniform and perforated tension specimens). As expected, using the second method the minimum sum of residuals for the hole surface analysis was considerably less
Fig. 6. Calculated probabilities of failure based on the optimum value m = 5 of the parametric study for the interior hole surface analysis (see Fig. 5) for (a) K = 3, R = 1 μm, (b) K = 6, R = 1 μm, and (c) K = 8, R = 8 μm.
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
than that of the top surface analysis (39.6 vs. 15.2, respectively) which points to an active flaw population at the inner hole surface. Based on the interior hole surface analysis the Weibull parameters were m = 5 and σ0 = 8.67 MPa m2/5. Contrary to using the characteristic strength as a criterion for the location of the active flaw population, this second method of failure description provided the best description of strength for all perforated geometries using the interior hole surface. Examples of the quality of probability of failure descriptions using the second approach and the same set of m and σ0 are given in Fig. 6(a)–(c) with good agreement with the experimental results for each of the three stress concentration factors, K = 3, 6, and 8. The use of the second metric (Table 2) to evaluate the material domain where the active flaw population resides using data from uniform tension specimens pointed to a surface flaw dependence only for K = 8 and R = 1 μm holes. The difference between (Ha) and (Ta) residuals for (K = 3 and R = 1, 2, 3 μm) using the first method of calculating m and σ0 was small compared to other geometries and perhaps not sufficient to obtain firm conclusions. A detailed fractographic analysis would provide additional evidence to determine the location of the fracture initiating flaws. Based on specimen scaling data for MUMPs polysilicon the active flaw population in uniform tension specimens resides at the top specimen surface [10,16] due to pitting and degradation of the surface when exposed to hydrofluoric acid release (in the presence of Au metallization) [18]. When notches are fabricated by Reactive Ion Etching (RIE) the stresses are localized at the notches and the probability for active large surface pits decreases. In this case, vertical ridges at the hole sidewalls become the dominant source of failure, as was reported in [5,10]. Strictly speaking, uniform tension strength data may be used to obtain Weibull predictions only if the flaw population remains of the same nature at the global and the local scale after the fabrication of non-uniform specimens. If the analysis presented here points to the right location of the active flaw population for perforated specimens (hole sidewalls) then the parametric study of non-uniform specimen strength data provides an appropriate description for the failure of these geometries. As a last note, it should be mentioned that the parameters m, σ0 in Eq. (6) are “material constants” given the first constraint for the applicability of the Weibull probability function. In other words, the local material elements used in the series implementation of the integral in Eq. (6) must have the same flaw distribution. Scaling of failure cannot be extended indefinitely to smaller specimen sizes and there is a minimum specimen size for which scaling could be applied. This minimum specimen size is a function of the maximum flaw size and flaw distribution, i.e. a representative volume (surface) element. Similarly, high stress concentrations that generate sharp stress gradients are also a limiting factor when using Eq. (6) in a discretized material volume (surface). Additional material length scales, such as the grain size, should be considered in determining the smallest specimen volume (surface) that this analysis can be applied to.
3275
6. Conclusions This study used two basic geometric variables: the radius of curvature and the stress concentration factor. The variation of the radius of curvature and the stress concentration factor accentuated the effect of specimen scaling in brittle fracture. In principle, any non-singular device geometry can be described locally using these two geometry variables. The preceding analysis indicated that a pair of (m, σ0) Weibull parameters may describe sufficiently well the probability of failure of a variety of MEMS geometries with different stress distributions provided that the active flaw population is properly identified. This is true when the material scale parameter can be considered a “material property”, i.e. invariant for a wide range of scales and stress profiles. The parametric optimization to determine m and σ0 produced, as expected, a more accurate description of the probability of failure compared to the use of strength data only from uniform tension specimens. The latter may not always follow the same scaling in complex specimen geometries even though the material remains the same. The analysis of failure using strength data from all non-uniform specimen geometries, invariably indicated that the closest description of the probability of failure was obtained assuming that the hole sidewall population was responsible for catastrophic failure. In this case, the active flaws were attributed to the use of RIE to pattern polysilicon. On the other hand, for some specimen geometries with holes of small radii of curvature or small stress concentration factors, an analysis based on Weibull parameters derived from uniform tension specimens pointed to the top specimen surface near the perforation as the location of the active flaw population. Thus, “drilling” a hole in a MEMS specimen via RIE does not guarantee that the failure behavior of the MEMS component will retain the same failure characteristics as a component under uniform tension. Therefore, it is important that all potential failure modes be considered when extrapolations to other geometries and loadings are made using the Weibull probability function. The optimization approach using the sum of the probability of failure residuals as a metric to determine an optimum Weibull modulus, m, allowed for better description of the entire shape of the probability density function, instead of one point (characteristic strength) of the same curve. This optimization approach always pointed to the same active flaw population compared to using only the characteristic strength as a criterion for the quality of description of the probability density function. Detailed fractographic imaging combined with the present Weibull analyses would allow for unequivocal determination of the location of failure initiating flaws. Acknowledgements The authors gratefully acknowledge the support by the Air Force Office of Scientific Research (AFOSR) through grant F49620-03-1-0080 with Dr. B.L. Lee as monitor, and the
3276
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
support by the National Science Foundation (NSF) under grant CMS-0301584. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
F.T. Peirce, J. Text. Inst. 17 (1926) 355. W. Weibull, J. Appl. Mech. 18 (1951) 293. R. Ballarini, R.L. Mullen, A.H. Heuer, Int. J. Fract. 95 (1999) 19. I. Chasiotis, S. Cho, K. Jonnalagadda, Proc. Mat. Res. Soc. 854E (2005) U.10.6.1. I. Chasiotis, W.G. Knauss, J. Mech. Phys. Solids 51 (2003) 1551. I. Chasiotis, W.G. Knauss, Proc. Mat. Res. Soc. 687 (2002) B2.4.1. W.G. Knauss, I. Chasiotis, Y. Huang, Mech. Mater. 35 (2003) 217. D.A. LaVan, T. Tsuchiya, G. Coles, W.G. Knauss, I. Chasiotis, D. Read, ASTM STP 1413 (2001) 1. G.C. Johnson, P.T. Jones, M.T. Wu, T. Honda, Mechanical Properties of Structural Films, ASTM STP, vol. 1413, 2001, p. 1.
[10] J. Bagdahn, W.N. Sharpe, O. Jadaan, J. MEMS 2 (2003) 302. [11] N.N. Nemeth, L.A. Janosik, J.L. Palko, NAFEMS ′99 World Congress, 1999, p. 1. [12] M. Evans, N. Hastings, B. Peacock, Statistical Distributions, Second ed. John Wiley & Sons, Inc., NY, 1993. [13] J.D. Sullivan, P.H. Lauson, Mat. Sci. Lett. 5 (1980) 1245. [14] I. Chasiotis, W.G. Knauss, Exp. Mech. 42 (2002) 51. [15] I. Chasiotis, W.G. Knauss, Proc. of IX Int. Cong. of Soc. Exp. Mech., 2000, p. 56. [16] I. Chasiotis, W.G. Knauss. “Experimentation at the Micron- and Submicron Scale”, in: W. Gerberich, W. Yang (volume Eds.), Comprehensive Structural Integrity, vol. 8. Interfacial and Nanoscale Failure, Elsevier Science, 2003, p. 41. [17] M.R. Gurvich, A.T. Dinbenedetto, A. Pegoretti, J. Mater. Sci. 32 (1997) 3716. [18] I. Chasiotis, W.G. Knauss, J. Mech. Phys. Solids 51 (2003) 1533.
Description of brittle failure of non-uniform MEMS geometries A. McCarty a , I. Chasiotis b,⁎ a
Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, U.S.A. Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.
b
Available online 28 February 2006
Abstract The description of probability of failure of polysilicon micromachined components with complex geometries using a single set of Weibull parameters was investigated. Strength data from both uniform tension and from twelve non-uniform specimen geometries with central perforations were employed. These perforations allowed for twelve different combinations of stress concentration factors and radii of curvature. Two methods were applied to determine the Weibull parameters: in the first method, only the strength data from uniform tension specimens were used to determine the Weibull modulus and the material scale parameter. In the second approach, the strength data from all non-uniform tension geometries were used to calculate the material scale parameter and the Weibull modulus using the maximum likelihood method. The non-uniform stress state in each perforated specimen was taken into account through an elasticity finite element model and the use of the integral form of the Weibull probability function. Using the first method, an analysis considering active flaw populations at the top specimen surface or the specimen sidewalls indicated that the active flaw population is not the same at all scales: for 1–3 μm radius perforations and small stress concentration factor (K = 3) the active flaw population was located at the specimen top surface, e.g. surface roughness, which, as the analysis indicated, was also the case for uniform tension specimens. For higher stress concentration factors (nominal K = 6 and 8) the analysis indicated that the active flaw population was located at the hole sidewall surface. As a result, for a given material the geometry of the specimen and the local state of stress determine the active flaw population from all flaws generated during fabrication and processing. Therefore, it is important that all potential failure modes be considered when extrapolations to other geometries and loadings are made using the Weibull probability function. © 2006 Elsevier B.V. All rights reserved. Keywords: Weibull; Stress concentrations; Fracture strength; Polysilicon
1. Introduction Brittle failure of uniform tension specimens, as characterized by significant scatter in the strength data, is often described by a two or three parameter Weibull distribution function. This probability of failure function is based on the weakest link assumption (WLA); the weakest material element will initiate catastrophic failure [1,2]. The advantage of this method of analysis is that Weibull parameters (i.e. Weibull modulus and characteristic strength) measured at one scale could be used to analyze self-similar geometries at another scale. To allow for scalability in the description of material failure three conditions must be satisfied: (a) the active flaw population must be invariant between different size specimens, (b) failure initiated by a flaw must be catastrophic, and (c) the stress field must be the same for all specimens. The last condition implies that, ⁎ Corresponding author. E-mail address: [email protected] (I. Chasiotis). 0040-6090/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2006.01.039
although the material might be the same, different loading profiles would invalidate previously determined Weibull parameters unique to that particular geometry. From a fracture mechanics viewpoint, component strength is affected by a change in mode of loading for a given set of flaws; this is because different stress distributions affect the probability of larger or smaller size flaws being critically loaded. Upon modification of boundary conditions, cracks originally loaded in mode I are then loaded in mixed mode and vice versa. Under these circumstances new Weibull parameters are required to describe the material strength. Although the description of brittle failure using an energy release rate criterion instead of the material strength is more deterministic, this is true only for homogeneous and isotropic materials. For instance, for polycrystalline silicon used in Microelectromechanical Systems (MEMS), one can hardly speak of a single value for the fracture toughness; the latter varies between 0.8 and 1.2 GPa due to material anisotropy [3,4]. The use of such fracture data in device design requires detailed
3268
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
modeling of the material microstructure. Instead, from a device design viewpoint it is valuable to examine the possibility of an efficient use of strength data combined with appropriate modeling in order to obtain failure predictions for MEMS components. Among the advantages of the Weibull probability function is the potential to determine the location of process-related flaws, e.g. whether the most detrimental flaws reside on a particular specimen surface, edge, or they are uniformly distributed in the entire component volume. For any of these flaw distributions the probability of failure can be calculated using parameters that are characteristic for the specific material. Work by Chasiotis and Knauss [5] indicated that proper specimen design could reveal the source of failure in microscale components and help to distinguish between surface and volume active flaw populations, which in turn could be related to the fabrication conditions. An extension of their work using a variety of perforated polysilicon specimens fabricated by Multi-User MEMS Processes (MUMPs) 35 indicated that the characteristic strength scaled inversely with the size of the perforation [6,7]. Other studies have also been conducted to characterize brittle failure of microscale polysilicon specimens under uniform stress [8]. When different geometries (or stress fields) are used the Weibull analysis can be conducted using the integral form of the probability function. Johnson et al. [9] used the integral form of the Weibull function to derive analytically the Weibull parameters from MEMS scale beams that were subject to bending. Bagdahn et al. [10] used the Ceramics Analysis and Reliability Evaluation of Structures (CARES) design software [11] with material scale parameter and Weibull modulus derived from uniform tension specimens to predict failure of specimens with 2.5 μm edge or central circular perforations fabricated by MUMPs 42. The authors evaluated their approach by examining the agreement between the predicted and the experimental characteristic strength that agreed within 9%. In that study, the sidewall surfaces of the notched specimens were identified as the location of the active flaw population, as reported also before by Chasiotis and Knauss for MUMPs35 [5]. In the work by Bagdahn et al. [10] the failure of the uniform tension specimens scaled with the top specimen surface while the failure of specimens with round holes, for which data from uniform tension specimens were used to make predictions, scaled with the hole sidewall surface. The present work examined the applicability of a unique set of Weibull parameters to describe the probability of failure of a wide variety of specimens with internal holes of different radii of curvature and stress concentration factors. To remove the geometric factor of specimen strength from the Weibull probability function and compute a pair of geometry independent parameters two approaches were compared in terms of their accuracy in describing the probability of failure of non-uniform geometries. The first approach, also employed in [10], used data from uniform tension specimens of the same material to predict the strength parameters of nonuniform specimens. In the second method that is introduced
in this work, the Weibull parameters were obtained from the combined experimental data from all non-uniform geometries using an optimization approach. In each case a single pair of Weibull parameters was derived to compute the strength of all non-uniform geometries. The effectiveness of the two different approaches was determined using as metrics the predictions of the characteristic strength and the entire probability of failure curve for each perforated specimen geometry. 2. Method of analysis The probability of failure of a uniform tension specimen, Pf, under stress, σ, is described by the Weibull probability distribution function r−ru m Pf ðrÞ ¼ 1−exp − ð1Þ rc where the shape parameter or Weibull modulus, m, indicates the scatter of strength values and for most ceramics m = 5–10. Lower values are possible but when m = 3, or smaller, the probability density function becomes increasingly skewed. The location parameter (or stress threshold), σu, describes the stress below which failure will not occur. The characteristic strength, σc, is the stress for which the probability of failure is 63.2% when σu = 0. In order to simplify the determination of the Weibull modulus and characteristic strength, a conservative description of strength is selected where the location parameter in Eq. (1) is set equal to zero. Eq. (1) can be implemented using an ordinal probability estimator and linear regression. This method is useful because the only requirement to determine the probability of failure for a strength datum point is the rank of the measurement with respect to the rest of the data. However, the choice of ordinal probability estimator can greatly affect the values of the Weibull parameters. This choice hinges upon the size of the data set. Important considerations are summarized in [5]. An alternative approach is the maximum likelihood method. In evaluating data as a random sample of a population, the maximum likelihood determines the statistical parameters of a distribution that are most probable to describe the population [12]. The likelihood function is the product of the probability density function evaluated at each of the data points as LðhÞ ¼
n Y
Pðri ; hÞ
ð2Þ
i¼1
where P(σi,θ) is the probability density function, σi is the result from the ith test, and θ represents the parameters to be determined for the data set. For the case of the Weibull probability distribution function m n Y mrm−1 ri i Lðm; rc Þ ¼ exp − ð3Þ m r r c c i¼1 By setting the partial derivative of the likelihood function with respect to the parameters m, σc equal to zero, a system of
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
equations is developed to calculate the maximum of the likelihood function. From Eq. (3) the following system of equations is determined " #1=m̂ n 1 X ̂ m r̂ c ¼ r ð4Þ n i¼1 i where m̂ ¼
n ð1=r̂ c Þ
n m̂ P i¼1
̂ rm i lnri −
n P
ð5Þ lnri
i¼1
Eqs. (4) and (5) are coupled and are solved numerically for mˆ , σ ˆ c, which are estimates of the Weibull modulus and the characteristic strength, respectively. A general form of Eq. (1) (with σu = 0) can be used to account for variable stresses in a specimen by assuming, for instance, that the entire volume of the material contributes to the failure probability Z rðx; y; zÞ m Pf ðrÞ ¼ 1−exp − dV ð6Þ r0 This form of distribution function does not require the material to be under uniform tension and it depends on the material scale parameter, σ0, which, together with m, are, in principle, independent of specimen size and stress distribution. Thus, σ0 and m could be determined from any specimen geometry. In contrast, the characteristic strength depends on the size and geometry of the specimen and the applied stress field. Limitations on the geometry invariability of σ0 and m are discussed in the Results and discussion section. In order to determine the Weibull parameters from Eq. (6) the stress as a function of location must be known analytically or numerically. Note that Eq. (6) is in agreement with WLA: it represents the combined probability of failure for a large number of “links” each weighted by the local stress. The “links” that are near the perforation have considerably higher probability to fail compared to those in the far field. By defining a quantity that contains the effects of specimen geometry and stress distribution, σ0 and m can be determined. The stress state is represented by the stress value at an arbitrary material point, e.g. the maximum stress, σmax Z rmax m rðx; y; zÞ m Pf ðrÞ ¼ 1−exp − dV rmax r0
Veff ¼
rðx; y; zÞ rmax
ð7Þ
m dV
and σ0 are constant with respect to location, Eq. (7) can be written as rmax m Pf ðrÞ ¼ 1−exp −Veff ð9Þ r0 The geometry factor, Veff, can be calculated numerically using a Finite Element (FE) model to determine σ(x, y, z) for the given boundary conditions. However, Veff depends on the Weibull modulus, and the three parameters, Veff, m, and σ0 must be determined simultaneously. Veff can be combined with σ0 as 1=m 1 rc ¼ r0 ð10Þ Veff Replacing in Eq. (9) the characteristic strength for the material scale parameter and the effective volume as described by Eq. (10) rmax m ð11Þ Pf ðrÞ ¼ 1−exp − rc Eq. (11) can be used with linear regression or the maximum likelihood method, to determine the Weibull modulus and the characteristic strength. If linear regression is selected, then the ordinal probability must be determined. For a small sample sizes, as the one used here, the best description of the ordinal probability according to [13] is Pfi ¼
i ¼ 3=8 n þ 1=4
ð12Þ
where n is the number of data points and i is the ordinal rank of a datum point. The formulation of Eqs. (6)–(11) was done under the assumption that the active flaw population was uniformly distributed in the entire specimen volume. If the flaws causing failure reside at a specimen surface, the same analysis should be carried out to find an effective surface area, Aeff, instead of Veff. Since the determination of the location of the catastrophic flaw is not always a straightforward task, a surface and volume Weibull analysis combined with fractographic studies should be applied. 3. Specimen preparation and experimental procedure
The integral of the relative stress distribution in Eq. (7) has units of volume and is defined as the effective volume (or geometry factor), Veff Z
3269
ð8Þ
The effective volume can be viewed as the volume of a uniform tension specimen that has the same probability of failure as the non-uniform tension specimen. Because σmax, m,
The polysilicon test samples were dog-bone shaped tensile specimens with various geometries of circular/elliptical perforations centrally located at the gauge section. They were fabricated at Cronos (former MCNC, North Carolina) using the MUMPS 35 run. The average grain size was 300 nm. The dimensions of the gauge section (L × W × T) varied from 250 × 30 × 2 to 700 × 340 × 2 μm depending on the size of the perforation. The variations in gauge section and the perforation geometry produced specimens with nominal stress concentration factors of 3, 6, and 8. Because of the finite width specimens the exact stress concentration factors were determined by a FE analysis and they varied between 3 and 11. For the purpose of
3270
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
the present discussion reference to each of the geometries is done according to its nominal radius of curvature and stress concentration factor. For each stress concentration factor, specimens with similar geometry and nominal radii of curvature of 1, 2, 3, and 8 μm were fabricated. These specimens were cofabricated on each die together with two 50 μm wide uniform tension specimens. Due to limitations in fabricating large numbers of every specimen geometry on a single die, we tested a total of 12 samples per non-uniform geometry and 24 uniform tension samples. From those tests, 17 successful tests of uniform tension specimens and 8–11 successful tests per nonuniform geometry were used in the analysis. Fig. 1 shows a typical uniaxial tension specimen and three examples of perforations whose exact dimensions were measured by a Scanning Electron Microscope (SEM).
The specimens were subjected to uniaxial tension as described in [14]. The large paddle shown in Fig. 1(a) was gripped via a method developed by this group that employs a flat glass grip coated with a UV-curable adhesive [15]. A piezoelectrically driven actuator was used to impose global displacements while the resulting applied force was measured using a load cell, which, in turn, provided the far-field stress at failure. A FE analysis implemented in ANSYS was used to determine the stress field at failure in each specimen. More details of the experimental procedures are presented in [5]. In that work a total of 100 data points were used, while for the present analysis we tested 31 additional specimens from the same fabrication run (i.e. 131 data points in total.) It should be noted that the Pf–σf trends were the same as those in [5] after the addition of the new data points.
Fig. 1. (a) Uniaxial tension specimen containing a perforation at the center of the gauge section. Typical hole geometries located at the gauge section that resulted in nominal stress concentration factors, (b) K = 3, (c) K = 6, and (d) K = 8. In total, 12 different perforated specimen geometries were fabricated.
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
4. Implementation of Weibull analysis to small scale specimens with variable geometries For specimen geometries that do not allow for an analytical description of the stress state in Eq. (8), a numerical method is necessary. A FE model of each specimen was generated in ANSYS to determine the ratio of the local average stress in each element of the discretized specimen geometry to the maximum stress in the entire specimen. This ratio was raised to the power of the Weibull modulus and the integral in Eq. (8) became a summation over all of the elements in the FE model. The elements used in Eq. (8) may be those of the entire specimen volume, the top surface, the sidewall surface, or any other region where the active flaw population resides. The implementation of the Weibull analysis was done in Matlab. Among the different locations for the active flaw population considered here, a top surface-only analysis was conducted based on prior surface roughness data for the two specimen sides [16]. For MUMPs polysilicon the top and bottom film surfaces have distinctly different surface roughness. As shown in Fig. 2(a), the top surface roughness is very irregular, while the surface roughness of the bottom surface (Fig. 2(b)) is dominated by hemispherical protrusions. The difference between the two surfaces is due to the different degree of crystallinity and the considerably smaller grain size at the bottom of the film. The profile of the top surface roughness follows the grain structure. In principle, the Weibull parameters σ0 and m can be determined from any specimen geometry. In this work two
3271
different approaches were explored. In the first, σ0 and m were determined from the uniform tension specimens, as reported before in [10], but now applied to predict the probability of failure of a significantly larger number of specimen geometries. In the second case, they were determined via an optimization scheme where the values of σ0 were averaged for all nonuniform specimen geometries. Note that averaging σ0 is appropriate, as, given certain constraints, σ0 can be conceived a “material property.” The same is not true for σc. In both methods described in the next section, the characteristic strength and the Weibull modulus for each of the non-uniform specimens were determined using the maximum likelihood method which is appropriate even for small specimen populations [17]. The two sets of data used to compute the Weibull parameters for MUMPs polysilicon were the strength results from the uniform tension specimens (method #1) and the perforated specimens (method #2). 4.1. Calculation of Weibull parameters from specimens with uniform geometry First, predictions of the characteristic strength for nonuniform specimens using the material scale parameter and the Weibull modulus computed via the maximum likelihood method from uniform tension specimen strength data were obtained. Specifically, using FE results and Eq. (8) for each of the specimen geometries, the effective volume (or surface) was determined using the Weibull modulus of the uniform tension specimens. Then, the effective volume (surface) for uniform gauge section specimens was used to determine the material scale parameter using Eq. (10). The material scale parameter from uniform gauge specimens and the effective volume (surface) for each of the non-uniform geometries allowed for calculation of the characteristic strength of the perforated geometries using Eq. (10). The latter was compared to that calculated from the experimental data as a metric to determine the accuracy of failure predictions based on Weibull parameters derived from uniform tension specimen strength data. 4.2. Calculation of Weibull parameters from specimens with non-uniform geometries
Fig. 2. (a) Roughness of the top surface and (b) the bottom surface of a polysilicon specimen [16].
We also introduced a new method of analysis that employed strength data from all non-uniform specimen geometries to determine the Weibull modulus and the material scale parameter that provided the best description of failure for all perforated specimens. A parametric study of Weibull modulus was performed to determine the set of Weibull constants that described most accurately the probability of failure of all specimen geometries. For various Weibull moduli, the characteristic strength of each specimen was calculated using Eq. (4), and the effective volume (surface) was computed using Eq. (8). These values were substituted in Eq. (10) to determine the material scale parameter for every geometry for a chosen Weibull modulus. The material scale parameters of all specimen geometries were then averaged to determine a representative
3272
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
value of σ0 for a Weibull modulus. This procedure was repeated for various values of m. A metric was defined to evaluate the quality of predictions for each (m, σ0) pair and determine the pair that gives the best agreement with the experimental data. Each (m, σ0) pair was used to calculate the probability of failure for each of the geometries and for all experimental failure strengths. The absolute values of the difference between the ordinal probability and the Weibull-predicted probability were summed for all data points and for all specimen geometries. This procedure was repeated for all (m, σ0) pairs. The pair that produced the smallest sum of residuals gave the best descriptions for the strength of the non-uniform specimens. In order to evaluate the relative quality of predictions from the two methods employed to determine m and σ0, two metrics were used: (a) the percent difference between the predicted characteristic strength and the characteristic strength determined from an analysis of the experimental data for every specimen geometry, (b) the sum of the probability residuals calculated as described in the parametric study above. While σc is an important parameter in describing the Weibull distribution function, m can greatly alter the probability predictions while keeping σc constant. For this reason, the most accurate Weibull parameters should predict the probability of failure of the entire experimental data set (method (b)) instead of only the value of the characteristic strength. Furthermore, the residuals of failure probabilities were compared for different considerations of active flaw populations (top surface vs. side wall.) The location of flaw population that provided the minimum sum of residuals described the specimen failure in the most accurate way. 5. Results and discussion The data from each of the specimen geometries were analyzed using the maximum likelihood method to determine the characteristic strength. Using the first approach described in Section 4.1, the effective volume and surface were found for each specimen geometry using the Weibull modulus determined from the uniform specimen data. Various analyses considered the active flaw population being uniformly distributed over the entire volume (V ), the top surface (Ta), the sidewall surface of the entire sample (Sw), and finally, the hole sidewall surface (Ha). The percent difference in the characteristic strength between the predictions and the values determined from the experimental data for non-uniform tension specimens are given in Table 1. The values of (Sw) and (Ha) are identical because of the overwhelming probability for failure to start at the stress concentration. For instance, doubling the specimen length in the presence of a stress concentration did not change our results as long as the location of the boundary conditions did not affect the stress field around the hole. As done in previous works [5,10], the analysis that resulted in the closest agreement between predictions and experimental results provided an indication of the location of the active flaw population. In the present work, a particular analysis (Ta, or Ha) did not uniquely provide the best results for all of the
sample geometries. The (Ta) characteristic strength predictions for specimens with small holes and the smallest stress concentration factor (K = 3 and R = 1, 2, 3 μm) or specimens with holes with the smallest radius of curvature (R = 1 μm) and the largest stress concentration factor (K = 8) were the closest to the experimentally determined, indicating that the active flaw population resided at the top specimen surface rather than the hole sidewalls. These cases are highlighted in Table 1. For the remaining specimen geometries, the best agreement was achieved when the Weibull analysis assumed that failure was initiated at the perforation sidewall (Ha). The latter agrees with the results in [10] where 2.5 μm circular holes were considered. The two different dependencies, (Ta) vs. (Ha), are interpreted as follows: The severity of (Ha) flaws was smaller for holes with small radii of curvature and small stress concentration factors. High stresses were concentrated at a small perforation sidewall surface and, thus, a smaller flaw population of sidewall defects was active (provided that the small size of the hole did not alter the local flaw population density.) In comparison, significant part of the specimen top surface was still at high stress and the flaw population residing at the top surface (surface roughness) was active. This argument also explains the trends in the characteristic strength reported previously for the subset of data seen in Fig. 3 [5]. As marked by the dashed circle, the specimens for which failure was initiated at the specimen surface demonstrated the highest local strengths and were separated from the rest of the data. The values of σc clustered according to the smallest computed values of (Ta) as also presented in Table 1. For the rest of the specimen geometries the hole sidewall surface (Ha) provided the best overall results predicting the characteristic strength within 3.1–17.6%. The
Table 1 Percent difference in characteristic strength (predicted characteristic strength using the Weibull parameters from uniform tension specimens minus the characteristic strength determined from experimental strength data) for various flaw populations: volume (V ), top surface (Ta), entire sidewall surface (Sw), interior hole surface (Ha)
K/ #Specimens
R (mm)
V (%)
Ta (%)
Sw (%)
Ha (%)
3/10 3/10 3/8 3/9
1 2 3 8
14.5 18.2 17.1 39.8
7.7 15.5 15.6 40.1
20.2 16.8 18.5 4.7
20.2 16.8 18.5 4.7
6/11 6/10 6/8 6/11
1 2 3 8
31.7 50.7 57.1 51.6
29.3 50.7 53.8 51.6
12.1 3.1 6.8 12.7
12.1 3.1 6.8 12.7
8/8 8/9 8/9 8/11
1 2 3 8
10.1 51.8 53.3 58.3
10.1 51.7 53.4 58.4
26.3 6.5 6.3 17.6
26.3 6.5 6.3 17.6
The results are presented as a function of the stress concentration factor, K, and the local radius of curvature, R. The first column includes the number of specimens tested for each geometry. The shaded cells point to the flaw analysis that resulted in the closest prediction of the characteristic strength.
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
Fig. 3. Characteristic strength calculated from experimental data from various specimen geometries. The geometries for which, according to the analysis, the active flaw population was located at the specimen top surface supported higher characteristic strength (K = 3, R = 1, 2, 3 μm and K = 8, R = 1 μm.) The dashed line includes the specimen geometries for which failure was initiated at the specimen top surface compared to the interior hole sidewall surface for the rest of the specimens.
Weibull modulus used in this first method of analysis, as determined from the uniform tension specimens, was m = 11.6, and the material scale parameter was σ0 = 149 MPa m2 / 11.6 (Sw and Ha), σ0 = 185 MPa m2 / 11.6 (Ta), and σ0 = 60 MPa m3 / 11.6 (V). The second method for the calculation of m and σ0 aimed at providing a more accurate description of the entire probability curve of the complete set of perforated specimens. The failure description using this approach was compared to the predictions obtained using the first method of calculating m and σ0 from uniform tension data. The second method was based on two parametric analyses of the strength data from non-uniform tension specimens: one using the top specimen surface and a second using the interior hole surface. A study based on the volume of the specimens yielded almost the same results as the top surface analysis. Figs. 4 and 5 show
Fig. 4. Sum of residuals from a parametric study that considered the top specimen surface as the location of the detrimental flaws. The sum of residuals is the sum of differences of the probability of failure predicted by the parametric study minus the ordinal probability of failure obtained from strength data. The open square is the Weibull modulus, m = 11.6, determined from uniform tension specimens. The minimum of the plot is at m = 9.4.
3273
Fig. 5. Sum of residuals from a parametric study that considered the interior hole sidewall as the location of the detrimental flaws. The sum of residuals is the sum of differences of the probability of failure predicted by the parametric study minus the ordinal probability of failure obtained from strength data. The open square is the Weibull modulus, m = 11.6, determined from uniform tension specimens. The minimum of the plot is at m = 5.
plots of the sum of the probability of failure residuals as a function of Weibull modulus for the top specimen surface and the interior hole surface, respectively. The minimum sum of the probability of failure residuals for the top surface was found for m = 9.4 (close to that determined from the uniform tension experiments) while for the hole surface analysis the minimum value of the residuals of the probability of failure occurred for m = 5. Table 2 Residuals of experimental vs. calculated probabilities of failure computed using Weibull parameters determined from strength data from uniform tension specimens (Method 1) and strength data from non-uniform gauge section specimens (Method 2)
Specimen geometry
Method 1: Predictions based on m and σ 0 determined from uniform tension specimens
Method 2: Predictions based on m and σ 0 from a parametric study using non-uniform specimen geometries
K
R (μm)
Top surface analysis
Interior hole surface
Top surface
Interior hole surface
3 3 3 3
1 2 3 8
1.65 2.74 1.89 4.10
1.47 1.54 1.45 1.30
4.85 4.28 3.04 1.56
0.64 0.84 1.21 1.35
6 6 6 6
1 2 3 8
4.83 4.89 3.96 5.38
2.31 1.44 1.45 3.12
3.20 1.86 3.84 2.40
0.85 1.36 1.28 0.94
8 8 8 8
1 2 3 8
2.26 4.42 4.43 5.44
3.99 1.85 1.89 3.81
2.69 4.37 3.81 3.72
2.25 1.58 1.80 1.09
Sum of residuals
46.0
25.6
39.6
15.2
The residuals of failure predictions were computed over the entire cumulative density curve, as opposed at a particular point (characteristic strength) in Table 1.
3274
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
Table 2 summarizes the sum of the residuals between the experimental ordinal probability of failure and the calculated probability of failure using both methods to compute m and
σ0 (e.g. uniform and perforated tension specimens). As expected, using the second method the minimum sum of residuals for the hole surface analysis was considerably less
Fig. 6. Calculated probabilities of failure based on the optimum value m = 5 of the parametric study for the interior hole surface analysis (see Fig. 5) for (a) K = 3, R = 1 μm, (b) K = 6, R = 1 μm, and (c) K = 8, R = 8 μm.
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
than that of the top surface analysis (39.6 vs. 15.2, respectively) which points to an active flaw population at the inner hole surface. Based on the interior hole surface analysis the Weibull parameters were m = 5 and σ0 = 8.67 MPa m2/5. Contrary to using the characteristic strength as a criterion for the location of the active flaw population, this second method of failure description provided the best description of strength for all perforated geometries using the interior hole surface. Examples of the quality of probability of failure descriptions using the second approach and the same set of m and σ0 are given in Fig. 6(a)–(c) with good agreement with the experimental results for each of the three stress concentration factors, K = 3, 6, and 8. The use of the second metric (Table 2) to evaluate the material domain where the active flaw population resides using data from uniform tension specimens pointed to a surface flaw dependence only for K = 8 and R = 1 μm holes. The difference between (Ha) and (Ta) residuals for (K = 3 and R = 1, 2, 3 μm) using the first method of calculating m and σ0 was small compared to other geometries and perhaps not sufficient to obtain firm conclusions. A detailed fractographic analysis would provide additional evidence to determine the location of the fracture initiating flaws. Based on specimen scaling data for MUMPs polysilicon the active flaw population in uniform tension specimens resides at the top specimen surface [10,16] due to pitting and degradation of the surface when exposed to hydrofluoric acid release (in the presence of Au metallization) [18]. When notches are fabricated by Reactive Ion Etching (RIE) the stresses are localized at the notches and the probability for active large surface pits decreases. In this case, vertical ridges at the hole sidewalls become the dominant source of failure, as was reported in [5,10]. Strictly speaking, uniform tension strength data may be used to obtain Weibull predictions only if the flaw population remains of the same nature at the global and the local scale after the fabrication of non-uniform specimens. If the analysis presented here points to the right location of the active flaw population for perforated specimens (hole sidewalls) then the parametric study of non-uniform specimen strength data provides an appropriate description for the failure of these geometries. As a last note, it should be mentioned that the parameters m, σ0 in Eq. (6) are “material constants” given the first constraint for the applicability of the Weibull probability function. In other words, the local material elements used in the series implementation of the integral in Eq. (6) must have the same flaw distribution. Scaling of failure cannot be extended indefinitely to smaller specimen sizes and there is a minimum specimen size for which scaling could be applied. This minimum specimen size is a function of the maximum flaw size and flaw distribution, i.e. a representative volume (surface) element. Similarly, high stress concentrations that generate sharp stress gradients are also a limiting factor when using Eq. (6) in a discretized material volume (surface). Additional material length scales, such as the grain size, should be considered in determining the smallest specimen volume (surface) that this analysis can be applied to.
3275
6. Conclusions This study used two basic geometric variables: the radius of curvature and the stress concentration factor. The variation of the radius of curvature and the stress concentration factor accentuated the effect of specimen scaling in brittle fracture. In principle, any non-singular device geometry can be described locally using these two geometry variables. The preceding analysis indicated that a pair of (m, σ0) Weibull parameters may describe sufficiently well the probability of failure of a variety of MEMS geometries with different stress distributions provided that the active flaw population is properly identified. This is true when the material scale parameter can be considered a “material property”, i.e. invariant for a wide range of scales and stress profiles. The parametric optimization to determine m and σ0 produced, as expected, a more accurate description of the probability of failure compared to the use of strength data only from uniform tension specimens. The latter may not always follow the same scaling in complex specimen geometries even though the material remains the same. The analysis of failure using strength data from all non-uniform specimen geometries, invariably indicated that the closest description of the probability of failure was obtained assuming that the hole sidewall population was responsible for catastrophic failure. In this case, the active flaws were attributed to the use of RIE to pattern polysilicon. On the other hand, for some specimen geometries with holes of small radii of curvature or small stress concentration factors, an analysis based on Weibull parameters derived from uniform tension specimens pointed to the top specimen surface near the perforation as the location of the active flaw population. Thus, “drilling” a hole in a MEMS specimen via RIE does not guarantee that the failure behavior of the MEMS component will retain the same failure characteristics as a component under uniform tension. Therefore, it is important that all potential failure modes be considered when extrapolations to other geometries and loadings are made using the Weibull probability function. The optimization approach using the sum of the probability of failure residuals as a metric to determine an optimum Weibull modulus, m, allowed for better description of the entire shape of the probability density function, instead of one point (characteristic strength) of the same curve. This optimization approach always pointed to the same active flaw population compared to using only the characteristic strength as a criterion for the quality of description of the probability density function. Detailed fractographic imaging combined with the present Weibull analyses would allow for unequivocal determination of the location of failure initiating flaws. Acknowledgements The authors gratefully acknowledge the support by the Air Force Office of Scientific Research (AFOSR) through grant F49620-03-1-0080 with Dr. B.L. Lee as monitor, and the
3276
A. McCarty, I. Chasiotis / Thin Solid Films 515 (2007) 3267–3276
support by the National Science Foundation (NSF) under grant CMS-0301584. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
F.T. Peirce, J. Text. Inst. 17 (1926) 355. W. Weibull, J. Appl. Mech. 18 (1951) 293. R. Ballarini, R.L. Mullen, A.H. Heuer, Int. J. Fract. 95 (1999) 19. I. Chasiotis, S. Cho, K. Jonnalagadda, Proc. Mat. Res. Soc. 854E (2005) U.10.6.1. I. Chasiotis, W.G. Knauss, J. Mech. Phys. Solids 51 (2003) 1551. I. Chasiotis, W.G. Knauss, Proc. Mat. Res. Soc. 687 (2002) B2.4.1. W.G. Knauss, I. Chasiotis, Y. Huang, Mech. Mater. 35 (2003) 217. D.A. LaVan, T. Tsuchiya, G. Coles, W.G. Knauss, I. Chasiotis, D. Read, ASTM STP 1413 (2001) 1. G.C. Johnson, P.T. Jones, M.T. Wu, T. Honda, Mechanical Properties of Structural Films, ASTM STP, vol. 1413, 2001, p. 1.
[10] J. Bagdahn, W.N. Sharpe, O. Jadaan, J. MEMS 2 (2003) 302. [11] N.N. Nemeth, L.A. Janosik, J.L. Palko, NAFEMS ′99 World Congress, 1999, p. 1. [12] M. Evans, N. Hastings, B. Peacock, Statistical Distributions, Second ed. John Wiley & Sons, Inc., NY, 1993. [13] J.D. Sullivan, P.H. Lauson, Mat. Sci. Lett. 5 (1980) 1245. [14] I. Chasiotis, W.G. Knauss, Exp. Mech. 42 (2002) 51. [15] I. Chasiotis, W.G. Knauss, Proc. of IX Int. Cong. of Soc. Exp. Mech., 2000, p. 56. [16] I. Chasiotis, W.G. Knauss. “Experimentation at the Micron- and Submicron Scale”, in: W. Gerberich, W. Yang (volume Eds.), Comprehensive Structural Integrity, vol. 8. Interfacial and Nanoscale Failure, Elsevier Science, 2003, p. 41. [17] M.R. Gurvich, A.T. Dinbenedetto, A. Pegoretti, J. Mater. Sci. 32 (1997) 3716. [18] I. Chasiotis, W.G. Knauss, J. Mech. Phys. Solids 51 (2003) 1533.