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Choosing an Appropriate Distribution to Analyze Process Variations

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Michael J. Varnau
Delphi Electronics and Safety

This article describes a reliability analysis of various process variations for aluminum wire bonds used in hybrid electronic modules. It illustrates the importance of selecting an appropriate distribution for data analysis based on an understanding of the real world factors driving the data and calculating the mathematical best fit to the distribution.

Theoretical Overview
There are many times that real world data are non-Gaussian (i.e. not Normally distributed). However, because of difficulty of analysis, or lack of training, Normality is often assumed. This assumption becomes more reasonable when samples are accumulated across multiple lots because of the Central Limit Theorem. The Normality assumption is also less critical if the only use of the data is to compare means of the samples. However, if the data are not Normally distributed, and one wants to make accurate predictions near the tails of the distribution, it is necessary to choose the correct statistical model for the distribution. Weibull++'s Distribution Wizard and built-in statistical models can easily handle this otherwise difficult task.

Background
It is common in hybrid electronic modules to make aluminum wire bonds directly to thick film conductors on the substrate. However, when the hybrid module is going to be potted or overmolded, it is necessary in high reliability applications to increase the strength of the wire bond. This is typically done by soldering an aluminum clad copper wire bond pad to the substrate. This structure provides a higher reliability bond because it eliminates the brittle intermetallics associated with a traditional aluminum bond to thick film conductor. However, this is done at the expense of space, cost and process complexity.

One of the added process complexities is controlling the thickness of the solder joint under the bond pad. This is important since the effective coefficient of thermal expansion of the bond pad is significantly higher than the alumina oxide ceramic substrate. If the joint is too thin, the solder joint can crack in temperature cycling. If the joint is too thick, there is a strong tendency for the bond pad to tilt, which adversely impacts the quality of the bond of the aluminum wire to the bond pad.

The electronics industry typically specifies the minimum average solder joint thickness for small devices such as these bond pads. However, one of the major customers for this product family specifies the absolute minimum acceptable solder joint thickness for all devices in their modules at a -4.5 sigma capability. The process in use exceeded the reliability requirements of all customers, but was not capable of meeting the minimum specified solder joint thickness and constraining the bond pad tilt within the wire bonding process requirements. The wire bonding process capability begins to deteriorate when the tilt exceeds 5 degrees and becomes unacceptable when the tilt exceeds 7 degrees.

Figure 1: Electronic ignition module subassembly, aluminum clad copper bond pad construction (side view) and illustration of solder thickness and tilt angle

Process Development and Data Analysis
Various process variations were evaluated with changes in the bond pad and substrate designs, as well as the solder deposition and reflow processes. The bond pad is rectangular and is much more susceptible to tilt in the short dimension. A process was successfully developed to meet the conflicting requirements in the long dimension. However, across the narrow width of the bond pad, the tilting was much more severe and the process was not capable for either tilt or thickness based on Gaussian statistics.

The Bond Pad Solder Thickness histogram (Figure 2) shows that the distribution of Bond Pad Solder Minimum Thickness is both peaked and skewed to the right. This visual observation is validated by statistical analysis. The data analysis indicates that the average -3 sigma value of the distribution is a negative number, which is physically impossible. The calculated Skew and Kurtosis values shown in Table 1 also validate that a Gaussian model is not a good fit to this data set. The values shown in red indicate that the distribution is non-Gaussian.

Figure 2: Histograms of solder thickness and tilt

Table 1: Data Summary

One of the features of Weibull++ is the ability to quickly determine an appropriate distribution for analysis of data using the Distribution Wizard. The results of the Distribution Wizard for one of the potential improved processes indicates that the best fit is a 3- parameter Weibull and the second best fit is a Lognormal distribution. Probability plots for both distributions are shown in Figures 3 and 4.

Figure 3: Probability plot for Weibull analysis

Figure 4: Probability plot for Lognormal analysis

Both distributions have a mean value of 34.7 microns thickness. The 3-parameter Weibull distribution has ρ = 0.9918 and the Lognormal has ρ = 0.9910 but they make very different predictions near the left tail of the distribution. While the 3-parameter Weibull makes predictions that are closer to the customer expectations, its choice for this physical situation would be difficult to justify. Other experiments clearly show that it is possible to get minimum solder thicknesses much lower than the calculated γ value. There was no change of process parameter between Process 1 and Process 2 that inherently assures a minimum thickness condition. The other consideration is that the distributions of the other process variations in the experimental space are a better fit to the Lognormal distribution. The choice of the Lognormal distribution in this case allows the comparison of all the process variations on the same plot. This is otherwise not possible since the scales of the Lognormal and Weibull distribution plots are different.

Many process variations were evaluated using Design of Experiment methodologies. Two of the most interesting process variations are shown along with the original process in Figure 5. The mean values of Processes 1, 2 and 3 are 8.3, 34.8 and 14.9 microns, respectively. However, the predicted values of mean -4.5 sigma are 2.5, 5.2 and 7.0 microns, respectively. Process 3 has a much lower mean value, but has much less variation, which is represented by a steeper slope in the plotted Lognormal distribution. The differences in process variation are highlighted in the contour plot comparing the three different distributions (Figure 6). The plot shows that the mean values of the distributions are significantly different. The plot highlights the difference in variation between processes and indicates that the amount of variation in Process 1 is not statistically significantly different than Processes 2 or 3. However, the amount of variation in Process 3 is statistically significantly lower than Process 2.

Figure 5: Multiple plot for solder thickness comparison

Figure 6: Contour plot for solder thickness

Similar analyses can (and should) be made for the bond pad tilt for each process. The Distribution Wizard indicates that the best model for bond pad tilt is either a Normal or 2- parameter Weibull distribution, depending on the specific process variation being analyzed. On the whole, the 2-parameter Weibull distribution was the best compromise across the experimental space. The multiple plot and contour plot for Processes 1, 2 and 3 are shown in Figures 7 and 8. The tilt evaluation of Process 3 is shown to be not as good as Process 2, but is acceptable. The plot format chosen here is the "Unreliability vs. Time." The plot has been re-labeled to match its actual use. This format accommodates multiple distributions and gives the user a better feel for how the distribution will behave near the right tail than other plot formats. The contour plot for the tilt evaluation validates our conclusions from the previous plot. Again, Process 3 has more tilt than Process 2, but the mean value is less than Process 1.

Figure 7: Multiple plot for tilt comparison

Figure 8: Contour plot for bond pad tilt

Summary of Data Analysis
This example highlights the fact that choosing a representative distribution to model a process should be based on an understanding of the real world factors driving the data in addition to calculating a mathematical best fit to the distribution. The 3-parameter Weibull distribution was an excellent fit to the data, but did not meet the test for reasonableness. This example also shows that choosing the process with the best mean value is not always the best choice for the process. It is important to evaluate the expected values near the limits that are most critical. In this case, the much lower process variation of Process 3 gives the highest safety margin near the design limit, even though Process 2 has a mean value more than twice as high as Process 3. It is also good practice to choose the process with the lowest variation if all other things are equal.

Summary of Analysis Process
Weibull++ provides a quick, easy and accurate tool for the analysis and plotting of many distributions including those that are non-Gaussian. The Distribution Wizard facilitates the fitting of an appropriate distribution to the data. The suite of plotting tools provides the ability to quickly and easily visualize and compare different data sets. The plotting tools provide for easy annotation of the plots and copying or exporting the plots for inclusion in other software tools. The Quick Calculation Pad allows easy computation of specific values of interest.

The analysis described in this article was performed in ReliaSoft’s Weibull++ 6. If Weibull++ is installed on your computer, you can download the Data Folio from the Web by right-clicking the following link and selecting Save Target As: Weibull++ 6 Data Folio (*.rw6, 398 KB).

 About the Author Michael J. Varnau is a Technical Fellow and Department Manager - Microelectronics Packaging for Delphi Electronics and Safety in Kokomo, Indiana. Mike has been with Delphi for 31 years and has been involved with electronics packaging for most of his career. He is currently leading the flip chip process development and IC and sensor packaging activities at Delphi.

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