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Analysis and Metrology—Yield ManagementImproving yield in IC manufacturing by statistical analysis of a large databaseFrançois Bergeret and Yves Chandon, Motorola Semiconductor Important benefits can be achieved by using well-known analytical methods--that is, the statistical analysis of data from databases. An investigation of problems that arose at the Motorola IC fab in Toulouse, France, demonstrated that data analysis methods are very useful for quality and yield improvement. That research involves all the statistical methods of analyzing data from databases, including those not covered by control charts and design of experiments (DOE). Analysis of variance, nonparametric tests, multiple regression, and generalized linear models were found to be efficient methods for confronting concrete problems such as the nonnormal distribution of electrical tests. Control charts and DOE are efficient methods of controlling and optimizing processes, as illustrated in the literature on the theoretical and practical aspects of statistical quality control (SQC).1—3 However, while control charts are needed in large-volume manufacturing areas to detect and react quickly to drift, they deal with process variables and are often incapable of detecting complex phenomena that eventually affect products' final characteristics. While design of experiments is needed to develop new technologies, implement new processes, and solve problems, it cannot take into account hundreds of process variables. Data analysis is the link between process variables; it is needed to systematically check correlations.
Output I: A statistical dashboard (asterisks indicate alarm values). The first part of this article deals with the precautions that are necessary to prevent drawing false conclusions from data analysis. The second part presents two applications of equipment control through analysis of variance (ANOVA) and nonparametric tests, methods that have been useful in improving yield by detecting problems in etch machines and in reducing defect density by identifying equipment generating too many particles on wafers. The third part is devoted to regression analysis and a generalized linear model (GLM). Regression analysis is used to quantify the effect on yield of the number of particles measured at different process stages. By means of regression analysis, an objective specification for an acceptable number of particles can be set. GLM is used to analyze the reject rate of electrical tests with process variables. Finally, the strong nonnormality of the response variable is discussed. Is it Reasonable to Use Data Analysis? The literature gives several reasons for acting cautiously in applying regression models to arbitrary data.2 However, ANOVA and regression analysis can be successfully applied during production when data are collected over a long period of time or if a high volume of data are collected, when there are no strong correlations between input variables, when orthogonality is maintained, and when teamwork and design of experiments are implemented. To generate sufficient and consistent data, it is necessary to collect data over a long period in which there are changes in calibration and standards. At high volume levels it is possible to collect a large sample in a short time. For example, a key device can generate a sample of 100 lots in 2 weeks. In this case the danger of inconsistent data is minimal. Because the cost of power computing and storage is continually decreasing, it is possible to maintain large databases at low cost. However, since much of what is recorded in databases is not relevant to later statistical analysis, it is the statistician's job to influence what data are stored. For example, it may be useful to study data on the operation of crucial cleaning machines in order to check for drift. All process variables have specification limits, and these variables are controlled by control charts so that they do not fall outside these limits. The specification limits have to be set so that any variation of the variable within the limits has negligible impact on the final characteristics of the device. Such properties are called response variables. One way to set these limits is to use tolerance design during process development. Process variables are forced to a low and a high level to determine their impact on response variables. The use of screening design is then strongly recommended, because this makes it possible to analyze the effects of several process variables during a limited number of runs. We used this method to develop our present technology, employing 14 process variables during 24 runs. Although these 14 variables were critical in our study, there are actually more than 100 process variables for the technology under investigation. The use of stepwise regression is necessary to determine whether process variables are affecting response variables, as was the case in an electrical test investigated in the third part of this article. Because the complexity of the process makes it impossible to detect all the effects of process variables during technology development, all wafer fabrication units have yield enhancement groups. Screening design must be used during development and data analysis during production. Before performing regressions, statisticians must ensure through principal component analysis that there are no strong correlations between input variables. In conjunction with process and device engineers they choose the right input variables without loss of information. If a lurking variable exists, it often provides a clue to solving the process problem under investigation--for example, in the case of a severe yield hit caused by process equipment drift. The lack of orthogonality is another factor that may compromise data analysis. Orthogonality means that all input variables are independent of one another. Orthogonality is always lacking when performing an ANOVA on data from a database. To minimize the effects of nonorthogonality, the analysis must be performed on a large sample. Assume that there are two factors (for example, the two process stages S1 and S2) with two levels (for example, two machines for each step, M11 and M12 for S1, and M21 and M22 for S2). Assume further that lots are randomly assigned to the machines, which is generally the case. The same proportion of lots processed by M11, in a large sample, will be processed by M21 and M22. The same applies to M12, so that orthogonality is approached. Teamwork and design of experiments are useful methods for establishing correlations and causal links between processes and drift. Data analysis is the first step in a larger analytical procedure that can be improved by design of experiments. Thus, although applying data analysis to production can be problematic, there are several ways to avoid false conclusions. It is crucial to confirm by design of experiments. Statistical tests are powerful tools for analyzing data from high-volume manufacturing areas to detect drift and improve yield. Systematic Equipment Control During IC manufacturing, lots are processed at different stages. There are often more than 100 operations, and for each several machines can be used. After several weeks the lots undergo a probe test, which consists of several electrical tests. If a die fails a test, it is rejected. The proportion of good dies, called the probe yield or simply the yield, is one of the most important variables in the industry. According to device engineers who are responsible for yield, more than 50% of yield problems are caused by process equipment. Thus, we decided to systematically analyze equipment having a negative impact on yield. This analysis resulted in an output we call the statistical dashboard. Statistical Dashboard for Yield Improvement. ANOVA is an efficient method of determining whether there is a statistically significant difference between machines. The analysis method compares the averages of a response variable for the different levels of input variables, called factors, and determines if there is a statistically significant difference between these levels. In our investigation the input factors affecting the dashboard included:
The response variables for the dashboard's main device (a device very sensitive to yield detractors with a high production volume of more than 30 lots per week) included:
Output I shows a statistical dashboard. Line i, column j represents the probability value of the ANOVA. When the prob-value was lower than 5%, an alarm was activated (represented in Output I by asterisks). For example, at the implant stage there was a significant difference in yield between the machines. Test 800 also showed a difference. With an
Two kinds of graphs are usually created when a significant difference is observed at a given process step: comparative histograms and time slides. We conducted our analysis and generated graphs with SAS software, a statistical package used widely in the semiconductor manufacturing industry. Comparative histograms are used to analyze machine distributions. Figure 1 shows that the test 800 distribution at implant for P42800, one of the most important yield detractors, was not normal. Time slides, graphs used to check causality, plot response variables versus the date the lots were processed at a given step. Time slides are not control charts because plotted response variables are not measured during production. These variables are usually the result of yield or electrical tests conducted during wafer tests. They are useful engineering plots for determining the date of drift and whether a machine was running alone during a poor yield period. For example, the comparative histograms in Figure 1 showed a significant reject rate for GSD1. The time slide in Figure 2 shows clearly that this machine had been running only during the drift period. Thus, it was impossible to assign the test 800 losses to this tool.
The Etcher Problem. Yield is always modified to ensure confidentiality. One important achievement of the statistical dashboard was that it revealed a consistent yield difference between two etchers producing five lots at different times. Figure 3 shows the two etchers' yield histograms, which indicate the first instance the dashboard detected a significant 5.5% difference for the second machine. As a result, this machine was shut down for several weeks to pinpoint the problem. During this period a dramatic yield drop occurred, as illustrated in Figure 4. Yield recovered after April 26 without explanation, which often signifies that future yield losses are likely. Several analyses failed to detect that the second etcher was less productive than the first at any given process step. Since etch had previously been identified as a critical factor in yield loss, the second etcher was analyzed using an exponentially weighted moving average (EWMA) control chart, an efficient tool for on-line statistical process control because it detects small drifts by retaining the measurements of several previous lots, the most recent having the greatest weight.1 Figure 5 is an EWMA time slide showing all the process events. Preventive maintenance was performed on the etcher on April 23, just before yield recovered--further evidence that the yield loss involved etch.
An EWMA time slide of the remaining-oxide-thickness process parameter associated with etch is shown in Figure 6. On about April 18 the remaining thickness increased dramatically. There is strong evidence that the etcher was involved in the yield drop, because the oxide-thickness process parameter correlated with yield. This correlation was significant because it was limited to a specific time frame, as shown in the EWMA time slide, but it was of no further importance because there was no correlation outside the critical period. In the standard time slide of the remaining oxide thickness shown in Figure 7, it is more difficult to see the drift between April 18 and 24. The control limits of the time slide are established by the software. These limits, although they indicate the evolution of the variable under investigation, are not used for SPC. While it is preferable to use EWMA time slides to investigate yield losses related to whole process steps, standard time slides are used to compare equipment, because each EWMA point is an average of several lots.
Several measures were taken as a result of the statistical analysis. A new frequency of 3000 wafers instead of 10,000 was eventually defined for the preventive maintenance of the etchers so that they would produce the same results and remain under control. The dashboard also revealed that wafer quality itself can be a yield detractor. When it was discovered that one supplier's wafers produced a higher yield than those of the other suppliers, the higher-yielding wafers were chosen for all the lots of key devices. Moreover, the dashboard helped to confirm defectivity-related problems. It also helped to uncover a serious problem in epitaxy by revealing during electrical testing that one reactor's yield was off by 5%. We found that the statistical dashboard can help to increase yield by several percentage points. Effects as small as 2% can be detected if the dashboard is run periodically. Thus, it serves to protect against equipment-related problems.
Particle Dashboard. The statistical dashboard helps considerably to improve yield. Because of the duration of process cycles, however, an early machine-caused drift is only detected after a delay of several weeks during a probe test. Yield loss can be caused by parametric phenomena (such as critical dimensions, thickness, or resistivity) or defectivity problems (such as dust or particles). Given the present state of technology, defectivity problems are the most prevalent cause of yield loss. At different process steps the numbers of particles on the wafers are measured with a complex inspection tool that enables on-line control of defectivity. Drift in the average number of particles can be detected by means of a particle dashboard, which works like a yield dashboard but measures the average number of particles at the beginning, the middle, and near the end of a process. The resulting dashboard contains empty cells representing machines whose particle levels have already been measured. Figure 8 is a histogram of particle data.
Output II: Regression output for yield. Because the number of particles was not distributed normally, we were unable to use ANOVA to test the effect of the equipment on the particles. We could have used a GLM, but this method does not indicate particle distribution.4,5 Thus, we used the nonparametric Kruskal-Wallis test, which compares distribution across K samples without assuming the distribution of the response variable. Since this test is sensitive to differences between the medians of the K samples, it is very effective on particle data.
The particle dashboard alerted us to the fact that one furnace had many more particles than the other, as illustrated in Figure 9. Further investigation showed that the quartz part of the machine was generating particles. After the quartz was changed, the furnace responded with good defectivity results during subsequent periodic tests. The yield dashboard was run again several weeks later, at which time the lots processed in the furnaces were tested. Figure 10 indicates a 1% yield difference, which clearly shows the relationship between the number of particles and yield. An effective means to detect equipment that increases the wafer defectivity rate, the particle dashboard is used twice a month to improve defect density.
Output III: Model with all the repressors.
Output IV: Final model. Two Regression Analyses This section presents two examples of regression analysis. The first, a multiple regression analysis used to quantify the effect of particles on yield, defines specification limits for the number of wafer particles measured on-line and ranks the process steps involved in defectivity. The second presents a process parameter correlated with the reject rate of an electrical test, whose nonnormal distribution required the use of a GLM to perform significance tests. Multiple regression is used to explain a response variable by several input variables, where it is assumed that the response variable is normal. The output of a multiple regression includes a linear equation relating the response variable and the input variables as well as significance tests to determine whether the input variables have a statistically significant effect on the response variable. GLM is an extension of multiple regression in which two main factors—linearity and the normality of the response variable—are not assumed. Several sources deal with GLM.4,6—9 Multiple Regression Analysis. In performing the multiple regression analysis, the input variables were the particle levels and the response variable was wafer yield. The three main objectives of this analysis were:
Since we analyzed a truncated yield in which all the wafers below the threshold were removed, we assumed that yield distribution was normal. PARTPOLY was the most significant contributor to yield loss, PARTMET1 the second most significant, and PARTAA the third. From the graphs alone it is possible to conclude that PARTAA had no effect on yield, which was not the case. All the regressors were statistically significant. The results are illustrated in Figures 11, 12, and 13. Figure 14 shows that significant and important effects can be completely invisible on scatter plots: Although Y was defined by Y = PARTAA — 2 PARTPOLY + 3 PARTMET1 no relationship was discernible on the scatter plot of Y on PARTMET1. The variable Y was not real but the result of a calculation. Its scale was related to the scale of the variables used in the calculation. The multiple regression analysis presented in Output II helped to determine statistically significant effects. For quality control reasons wafers with very low yield at probe are rejected because electrical tests cannot detect all rejects. Some dies that have passed the test can have hidden problems. Thus, to protect customers, all the dies of low-yielding wafers are rejected. On the basis of this cautious rule, it is possible to set specification limits for the number of particles measured in process. Wafers with more than the specified number of particles can be scrapped far before the probe, saving time and money. It is possible to set a maxi-mum number of particles at each step using this regression equation: YIELD = 85.27 — 0.066 PARTMET1 — 0.044 PARTPOLY — 0.008 PARTAA. If it is assumed that 100 particles have been measured at an active area (PARTAA = 100) and 10 particles at poly deposition (PARTPOLY = 10), we have the equation: YIELD = 84.03 — 0.066 PARTMET1. If it is assumed that a wafer with a probe yield <75% must be rejected at probe, we have: 84.03 — 0.066 PARTMET1 > 75, which gives: PARTMET1 < 136. Since wafers with more than 136 particles are likely to be rejected at probe, it is better to reject them at metal-1 deposition. The same calculations can be applied to PARTAA and PARTPOLY to define specification limits for the number of particles on inspected wafers.
Output II shows that the adjusted R2 for the probe yield was 17%, indicating that other parameters were influencing yield. Because of capacity and manufacturing feasibility, particles are not measured at each process step. Thus, the number of particles at unmeasured steps may also influence the probe yield and are not included in this model. The partitioning technique, which consists of measuring wafer particles at several process steps, can be very useful in increasing the quality of the model and can better explain yield losses related to defectivity. The value of R2 may be low because not all particles cause killer defects. Particle impact depends on particle size and location. In addition, non-defectivity-related phenomena, such as parametric problems, also influence yield. Figure 14 indicates that particles act as a maximum yield limiter; wafers containing an excessive number of particles cannot achieve an acceptable yield.
GLM Analysis. When the ultimate production technology, a high-volume production device, was affected by dramatic yield losses, we decided to analyze the main reject test. This procedure has two advantages: First, electrical tests can be interpreted by the device engineer and more easily related to process deficiencies; and second, statistical tests are more powerful than electrical tests, which are far less variable than yield. In our investigation, the main contributor to the Pareto chart of test rejects was P42004, whose distribution was y. In performing a GLM it is necessary to define:
For P42004 we selected a y distribution, which can handle several types of response variables, including variables with a log-normal-like distribution. In addition, the log link always produces positive estimates of the reject rate. The predictors are critical dimensions because they are very important variables for this technology and possibly related to P42004. Output III shows the parameter estimates for the whole model. The analysis of deviance generalizes the ANOVA. The GLM calculations differ from those of the multiple regression analysis because in the GLM the normality of the response variable is not assumed. The principle, however, remains the same: to calculate a test statistic to determine whether the input variables have a significant effect on the response variable. Since several parameters were not significant at the 5% level, they were removed from the model. The model eventually selected in Output IV includes three critical dimensions, P0397, P0424, and P0374. P0397 is the most important because of the value of its x2 statistic compared to the other regressors. These critical dimensions represent polysilicon masking (poly 1 and poly 2, the main regressors), and metal masking. A standard multiple regression analysis was also performed. After a stepwise selection, only one regressor, P0397, remained. P0377 and P0424 were not significant at the 5% level. Thus, the GLM allowed us to detect relationships that the multiple regression analysis could not detect.
One critical dimension, P0395, was not included in the final model although it was physically related to P0397 and also very significant. P0395 was excluded because the analysis was not a DOE. There was a strong correlation between P0397 and P0395, as shown in Figure 15, which led to a lack of orthogonality. Thus, process engineers were easily able to control P0395, so that this regressor was selected for further analysis as a factor of a DOE. Figure 16 shows that if the critical dimension is above 1.35 µm, most of the lots have a very low reject rate for P42004. If the critical dimension is between 1.3 and 1.35 µm, roughly half of the lots have an important reject rate for P42004. Below 1.3 µm, most of the lots have a very important reject rate for P42004. It is significant that GLM was used to perform accurate statistical tests, because the conclusions derived from data analysis must determine the process variables involved in yield drift. False conclusions result in a loss of time and effort. Conclusion Probe yield at the Motorola plant was increased by several percentage points thanks to data analysis techniques. Such techniques can be put to powerful use as the third pillar of a statistical quality and yield improvement approach to complement control charts and design of experiments. Teamwork among process, device, and maintenance engineers is also critical to understanding the phenomena highlighted by statistics. With these caveats, we believe that the use of data analysis tools in the semiconductor industry should be generalized to analyze information from ever-more-powerful databases. Acknowledgments The authors wish to thank Henri Flechner of Motorola for the idea of the statistical dashboard and for his constant support of this tool. We also wish to thank Kevin Marble of Motorola for his helpful comments on this article. Finally, we would like to thank Medhi Azza, Laurence Buada, François Nanot, and other students at Paul Sabatier University for creating SAS programs. References 1. T Ryan, Statistical Methods for Quality Improvement (New York: Wiley, 1989). 2. G Box, W Hunter, and J Hunter, "Hazards of Fitting Regression Equations to Happenstance Data," in Statistics for Experimenters (New York: Wiley, 1978), 487—498. 3. R Myers and D Montgomery, Response Surface Methodology (New York: Wiley, 1995). 4. P McCullagh, and J Nelder, Generalized Linear Models (London: Chapman and Hall, 1989). 5. D Michelson, "A Data-Driven Method for Calculating Limits for Particle Control Charts," in Proceedings of the International Society for Optical Engineering (SPIE) (Bellingham, WA: SPIE, 1996), 38—44. 6. J Nelder and R Wedderburn, "Generalized Linear Models," Journal of the Royal Statistical Society A 132 (1972), 370—384. 7. R Myers and D Montgomery, "A Tutorial on Generalized Linear Models," Journal of Quality Technology 29, no. 3 (1997), 274—291. 8. M Tobin et al., "Design of Experiment to Test Sigma Reduction," Process Characterization and Optimization (New York: Wiley, 1993), 91—113. 9. J Nelder, The GLIM Manual (Oxford, UK: Numerical Algorithm Group, 1974). François Bergeret, PhD, is a statistician at the Motorola MOS-20 wafer fab in Toulouse, France. He is responsible for data analysis to help solve yield issues, training, and consulting on design of experiments and statistical programs development. He is also responsible for two statistical courses at Paul Sabatier University of Toulouse and works with university researchers to develop new statistical methods for the semiconductor industry. He earned an MS in economics and statistics and completed his PhD in statistics at Paul Sabatier University, specializing in reducing IC test time. (Bergeret can be reached at +33 5 61199445 or r21445@email.sps.mot.com.)
Yves Chandon is defect reduction and SPC section manager at the Motorola MOS-20 wafer fab. He has worked at Motorola for 19 years in process and device engineering and holds an engineering degree in physics from the National Institute of Applied Sciences in Rennes, France. He is the author of seven papers on the use of statistical techniques for quality improvement and cost reduction in wafer processing and is a six sigma black belt. (Chandon can be reached at +33 5 61199521 or ttt442@email.sps.mot.com.)
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risk of 5% the number of false alarms can be significant. If 100 process stages are included in the dashboard, an average of five false alarms will sound. Thus, for five process stages the dashboard will indicate a yield difference between devices although there actually is none. This problem can be solved by setting the 
