Understand Probe-Card Measurement Variations

This article assumes a knowledge of statistics.

Many test engineers assume the tools they use to analyze a process or inspect a product are always accurate. In reality, every tool exhibits errors that affect measurements. Errors may arise from the tool itself, the operators, the environment, or other factors. Before you use a measuring instrument, you must determine whether the magnitude of the instrument’s error meets the tolerance range required by the customer. Basic experiment techniques and statistical methods can help you draw the proper conclusions.

Figure 1. A typical probe card must accurately place minute electrical contacts on a wafer or die undergoing testing. Basic mechanical measurements and statistical tools help you determine the placement accuracy.

 Our company, a supplier of probe cards and test equipment for semiconductor manufacturers, uses gage repeatability and reproducibility (Gage-R&R) studies to understand and quantify one of the major components of measurement error. We’ll describe how we use the Gage-R&R method to characterize probe-tip location using a probe-card analyzer. But the procedures are general, and you can apply them to any measurement instrument. Our method is similar to the one described by the Automotive Industry Action Group^.1

 The errors within a measurement process have two components: bias and variation. Bias is the tendency for a measurement to always be offset on the high or low side of a value. You can determine a bias only by comparing measured values to calibration standards such as those traceable to NIST. Unfortunately, no standards exist for aligning probe tips on a probe card. The only comparison that takes place is between a supplier’s final-inspection measurements of probe tips and the measurements made by a customer at incoming inspection. Differences in measurement results can prove difficult to resolve. 

Variation is best described as the distribution of results that occurs when you repeatedly measure the same characteristic, on the same item, using the same instrument. Variation has two components, repeatability and reproducibility.

In simple terms, repeatability indicates an instrument’s ability to produce the same results during multiple measurements on the same part. Reproducibility indicates the ability of different operators to produce similar results on the same part. Measurement professionals define these errors in these two measurements in terms of standard deviation of all the measurement values collected for each. You obtain the total measurement error, sR&R, from the following equation:

You quantify the measuring capability of an instrument by using a precision-to-tolerance ratio, (P/T).²  In this ratio, P represents 6 x s_R&R, and T represents a tolerance specified by the customer.

When you achieve a P/T ratio of £0.1, the instrument will make acceptable measurements. When the ratio lies between >0.1 and <0.3, the instrument will make marginal measurements. And if the ratio exceeds 0.3 then the measurements are unacceptable. These interpretations of the ratio are standards accepted in industry.³

The ratio also tells you what tolerance you have to work with in manufacturing. If the ratio is 0.23, for example, the measurement variation amounts to 23% of the total tolerance range, leaving only 77% of the tolerance available for variation in the manufacturing process. This means that the manufacturing processes must be 23% better than the specified tolerances to make up for the variation in the measurement process. You always want to reduce the measurement variation so the manufacturing processes can “use” most of the tolerance range.

The ANOVA Method Yields Results
We used the standard Analysis of Variance (ANOVA) statistical method to analyze the Gage-R&R data. The method calculates s_{Repeatability} and s_{Reproducibility}, the quantities you need to calculate P/T. Applying the ANOVA method lets us partition the total variation in the data into various components such as variations due to the gage and variations due to the operators. You can acquire a good working knowledge of the ANOVA method from books on statistics.⁴ Many statistical software packages can run the ANOVA analyses.

Now that you understand how measurement errors can affect manufacturing and testing, you need to know more about how to determine those errors. You can perform two types of studies to measure the variation of an instrument; a gage-repeatability study and a Gage-R&R study. In a typical gage-repeatability study, one operator performs multiple tests without changing the setup between tests. This study provides a quick and rough estimate of repeatability. Knowing the repeatability helps you identify any major problems with the measurement system before you add other possible sources of variability.

Test equipment that operates automatically, such as a probe-card analyzer, represents the only class of equipment suitable for gage-repeatability studies. That’s because after the operator performs the initial setup, he or she has no further interaction with the measurement process. When we perform a gage-repeatability study on a probe-card analyzer, we calibrate the system and have the operator follow a detailed setup procedure. After setup, the operator simply pushes the start button. After we obtain the results of the gage-repeatability study, we conduct the Gage-R&R study—a study of repeatability and reproducibility. By comparing the results of the two studies, we can identify any increase in variation caused by having different operators conduct the test.

Specifically, we performed a gage-repeatability test by having an operator make multiple measurements of the x axis location (align (x)) of probe tips on a single probe card (Fig. 1). In a typical test, one operator ran five trials without changing the setup. The data in Table 1 shows the raw align(x) data for a set of 40 probes on a single card along with the data for the standard deviation (S). We calculated the P/T ratio by using the following equation:

                (Eq. 1)

The value of ô is S/c₄  where S represents the average of the standard deviations and c₄ equals 0.9400, a constant based on five measurements per probe. Such values have been computed for different sample sizes and are readily available.⁵ The value of USL – LSL represents the difference between the upper and lower specification limits.

In this example, S=0.0262 and P/T = 0.2358. We assumed the customer specified a tolerance of 0.7092 mils, or ±9 mM. This P/T ratio reflects the true measurement-system repeatability. In fact, it represents the minimum attainable P/T because the study includes no other source of variation such as the variation due to different operators. So, before we studied other factors of variation, the result told us the system would make marginal measurements. Remember that a P/T ratio of £0.1 means an instrument will make acceptable measurements. A ratio between >0.1 and <0.3 means an instrument will make marginal measurements. A ratio above 0.3 is unacceptable.

Next, we ran a complete Gage-R&R study in which two operators performed three trials using the same probe card from the gage-repeatability study. Two operators ran three tests for each probe and each operator set up the equipment independent of the other. This independence helps determine the interaction between the operators and the measurement system. The data in Table 2 shows the align(X) measurements for the 40 probes. The ANOVA method yielded the components of variance, which include gage, operators, parts, and the interaction between the operators and the measurement process.

Operators Cause Variability
Having different operators perform the setup of the measurement system causes the most variability and, in the above example, is identified by the interaction component of variance. The setup process includes mounting a fixture to the probe-card analyzer and tightening bolts to a specified torque value and in a specific sequence. Mounting the probe card into the fixture and tightening additional bolts follows this step.

Any variation in the setup process used by each operator may result in a large interaction component. Studies have proven that strictly following systematic process instruction makes the interaction between operators and the measurement process insignificant and reduces operator variability. You can combine an insignificant interaction with the gage error or repeatability. If during testing you find the interaction between the operators and the measurement process to be statistically significant, then you must take action to correct a bad gage, properly train operators, or correct your measurement study procedures.⁶

The ANOVA data in Table 3 lists the different sources of variation such as operators, parts, interaction between operators and the measurement process, and gage error. It also includes other statistical data such as the sum of squares, degrees of freedom, mean squares, and the F-ratio for the operator and the measurement process interaction. (The details about how we calculated this other statistical data are beyond the scope of this article. To learn about these calculations, read, Design and Analysis of Experiments.⁷)

Operators Interact with the Process
The first step in interpreting the ANOVA results is to determine whether there is a significant interaction between operators and the measurement process. To do this, we compared the magnitude of the calculated ratio F(Cal) to that of the theoretical ratio F(Tab) given in the calculated ANOVA data (Table 3). Comparing these values lets us know whether the interaction is significant or not. These theoretical values have been computed for various degrees of freedom and levels of significance and are available in statistical tables. The topic of degrees of freedom is beyond the scope of this article. In simple terms, it serves as a means for getting the theoretical value from the statistical tables.

The level of significance indicates the risk we will assume. For example, for a 5% level of significance, we have a 95% (100% – 5%) confidence about our decision. Typically an experimenter chooses the confidence level before he or she starts an experiment. When analyzing the data, the experimenter can make a decision about the significance of the interaction. Many commercial statistical software packages can take care of these calculations. We used 5% as the level of significance for our study.

We compare the distribution of the data collected with the theoretical distribution to arrive at a conclusion regarding the significance of the interaction. If the magnitude of the calculated value is less than the theoretical value, then we conclude that the interaction is insignificant. Otherwise the interaction is significant.

In our example, the F(Cal) value (0.46145) is less than the tabulated value (1.475) at 5% level of significance, so we conclude that the interaction effect between the operators and the measurement process is insignificant.

The next step is to calculate the P/T ratio based on the combined repeatability and reproducibility sigma value as shown previously in Equation 1:

The value in parentheses represents the combined standard derivation of both repeatability and reproducibility errors computed from our Gage-R&R studies.

The customer defined the tolerance range, T, as 0.7092 mils or ±9 mm. The P/T ratio reflects both the repeatability and reproducibility components of error in the measurement process. Since the ratio is slightly less than 0.3, we concluded that the measurement tool is marginally acceptable.

The P/T ratio depends upon the value of the tolerance. A tolerance less than 0.7092 mils yields a larger ratio that may go above 0.3. On the other hand, a greater tolerance makes the P/T ratio smaller. Such a change in the ratio does not mean that the measurement system has gotten better, it simply means that the customer has specified a larger tolerance. If you lack a tolerance, or if tolerances for the products tested are different, you can use the total measurement error, which is the sum of the errors due to the gage, operators, and the parts.⁸

The graphs in Figure 2 summarize the results of a Gage-R&R study. Figure 2a provides a repeatability-and-reproducibility plot. Each rectangle represents an operator, and the horizontal line within the rectangle indicates the average for that operator. The height of the rectangle measures the variability for each operator. The three small circles connected by vertical lines correspond to the readings recorded for each probe.

If repeatability is perfect, the graph shows only a single circle instead of three circles connected by vertical lines. In the same way, if reproducibility is perfect the horizontal lines within rectangles will be at the same level.

As an alternative, Figure 2b plots the averages of three measurements for each probe for two operators. The consistent pattern of averages across probes indicates the reproducibility of the measurement system.

The graph in Figure 2c provides another summary of average measurements for the trials and probes. The ends of the vertical lines represent the minimum and maximum measurement for all probes for each operator. The shaded box shows the boundaries that include the middle 50% of the measurements, and the small box inside the shaded box shows the median for all the values. The location of the small square near the center of the shaded box indicates that the distribution is close to normal. While you might choose other methods to determine whether or not the measurements are normally distributed, this graph provides a simple way to verify that assumption.

The Gage R&R study is the only statistical technique available that quantifies the magnitude of the two components of variation and compares them to the tolerances of the product to be measured. The Gage R&R study allows you to determine the suitability of the measurement process for its intended purpose. You should conduct the study prior to the purchase of new measurement instruments, and periodically throughout their life to verify stability of the measurement process. T&MW

FOOTNOTES
1. Brown, L.A., B.R. Daugherty, and V.W. Lowe, “Measurement Systems Analysis—Reference Manual,” Auto Industry Action Group (AIAG), Troy, MI. October 1990.
2. Montgomery, D.C., Introduction to Statistical Quality Control, John Wiley & Sons, New York, NY. 1996.
3. Introduction to Measurement Capability Analysis, 91090709A-ENG, SEMATECH, September 1991.
4. Box, G.E.P., W.G. Hunter, and J.S. Hunter, Statistics for Experimenters: An Introduction to Design, Data Analysis, and Modeling, John Wiley & Sons, New York, NY. 1978.
5. Montgomery, D.C., op. cit.
6. Ermer, D.S., and P.E Prond, “Geometrical Analysis and Comparison of Measurement Studies,” ASQC Second Annual Measurements Conference, NIST. 1993.
7. Montgomery, D.C., Design and Analysis of Experiments, John Wiley & Sons, New York, NY. 1996.
8. Ermer, D.S., et. al., op. cit.

Chander M. Sekar has been the corporate statistician at Cerprobe since 1996. He holds a Ph.D. in statistics from the University of Madras (India), and he has more than 20 years of teaching experience. Sekar is a Certified Quality Engineer.

Henry P. Scutoski is the vice president of quality at Cerprobe and has more than 30 years of experience in quality measurement. Scutoski received an M.B.A. from the University of Dallas and a B.S. in industrial engineering from Bradley University. He is also a Certified Quality Engineer.