1. Samsung Austin Semiconductor
Metrology Site Reduction Analysis of All Combinations Using NChooseK Matrix JSL Function
Dan Sutton
Samsung Austin Semiconductor
ABSTRACT
METHODS (continued)
Metrology on semiconductor wafers is necessary to understand process targeting and sources of variation. For new device introduction and initial volume production, wafer metrology may use a large number of sites for establishing targets, setting up APC (Automated Process Control), and determine sources of variation across the wafer. However, ongoing metrology on production is often viewed as non-value added. As manufacturing volume increases and the process is maturing, the question arises: what is the purpose of the continuing metrology in relation to the number of sites? Can the number of sites be reduced to increase throughput at metrology? In the past, sites may have been removed based on current maps with reduced sites on other production, and the impact of the new map analyzed with the current site data. Utilizing the built in JSL function NChooseKMatrix(n,k) which generates all possible combinations for Choose K sites from N sites, and the Prediction Profiler, JMP scripts were created that could quickly analyze and weigh all combinations of varying K, and allow the user to pick the best reduced site map based on various requirements. Examples for different scenarios will be shown.
13 choose 12: choose 1: 13 Include 9 sites: Exclude 4 sites:
13 choose 11: choose 2: 78 [ ] [ ] combination 1
13 choose 10: choose 3: 286 [ ] [ ] combination 2
13 choose 9: choose 4: 715 [ ] [ ] combination 3
13 choose 8: choose 5: 1287 [ ] [ ] combination 4
13 choose 7: choose 6: 1716 … … …
[ ] [ ] combination 715
For a few select maps, one could calculate the averages for N sites vs. the K sites, and compare using linear regression in software such as Excel, Minitab, or JMP. However, to calculate all combinations, this can take hours to days. Instead, matrix operations were used to speed up the process. JSL (JMP Scripting Language) was used because there are already built-in functions called NChooseK(n,k), which calculates the number of combinations, and NChooseK Matrix(n,k), which returns the matrix of all the combinations. In the case of 13Choose9, the matrix is 715 rows and 4 columns. The exclusion matrix 13Choose4 was chosen over the inclusion 13Choose9 matrix, since software such as JMP is optimized for data filtering, in that it has built in exclusion processes for large datasets.
Matrix operations are used in JMP to quickly calculate statistics such as R-square [2], because of the speed of calculations compared to manipulating spreadsheets, data tables, and JMP reports. This same algorithm was used to help determine optimal site reduced maps for various scenarios.
METHODS
In semiconductor manufacturing, a typical wafer may have 70 to 100 shots or exposures of the reticle. Measuring every shot is not practical for volume manufacturing. For metrology coverage of the wafer surface, a reduced pattern such as a 13 site map is often used (Figure 1a). A common task to improve throughput may involve site reduction of 2 or more sites (Figure 1b). The original data can be analyzed to show what the trend would have been for less sites.
Fig. 1a: 13 site Fig. 1b: 9 site Fig. 2: JMP Add-in Calculator
To determine an optimal 9 site map, one would need to look at all possible combinations of the existing 13 sites. The formula for combinations can be written as:
where n is the number of things to choose from, and we choose k of them. This is often called “n choose k” (such as “13 choose 9”). For 13 sites that would be reduced to 9 sites, this becomes a total of 715 combinations. The inclusion of 9 sites can also be viewed as excluding 4 sites, thus “13 exclude 4” gives the same number of combinations. To see this in action, refer to the JMP add-in “Combinations and Permutations Calculator”, Figure 2 [1].
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2. Samsung Austin Semiconductor
Metrology Site Reduction Analysis of All Combinations Using NChooseK Matrix JSL Function
Dan Sutton
Samsung Austin Semiconductor
RESULTS
RESULTS (continued)
RESULTS (continued)
Other authors have reported attempts to build the best model of the wafer from the remaining sites [3],[4]. But often for volume manufacturing, the process has been well characterized and a model for across the wafer is no longer needed. Metrology is used instead to center the average, and to monitor for changes in variation across the wafer. For centering purposes, even just one site could be used, but what if that site had a change in variation? Due to this concern, metrology will still use some number of multiple sites. Each scenario can allow some site reduction to maintain centering while monitoring for changes in variation from the remaining sites.
Example 1: What would be the optimal maps for reduction in 13 sites, reduced to 9, 8, 7, 6, 5 sites?
In this example, metrology wants to know how reducing number of sites can decrease R-squared (Figure 3). In this case, 13 sites will be an R-square of 1 for the baseline. This roll off curve for less sites is a classic example of NChooseK applications. The JMP script can determine the top R-squared maps, but the maximum R-squared will still decrease with decreasing number of sites.
Fig. 3: R-square vs. # of sites for all combinations
Example 2: What are the optimal maps for a parameter that will minimize any shifts?
In this example, the engineers have already centered the process based on a parameter. They desire that the site reduction will not significantly change the centering, such that they have to update charts and software settings. If similar top R-square value maps are compared, which ones have the minimal impact on the centering parameter? Likewise, are there top R-square value maps that also do not increase long term standard deviation, which is used in Pp/Ppk calculations (long term Process Performance indices)?
The JMP Profiler can be used with weighting factors determined by user to satisfy the optimal maps. Figure 4 shows all maps for the absolute value of the change in the centering parameter vs. R-square and long-term standard deviation. Figure 5a and 5b zoom in to the left side of the graphs in Figure 4. In this example 6 of the top maps that meet the criteria are selected and highlighted on the graphs.
Fig. 4: R-square and change in standard deviation vs. change in centering parameter
Fig. 5a and Fig. 5b: Zoom in on left side of Figure 4.

3. Samsung Austin Semiconductor
Metrology Site Reduction Analysis of All Combinations Using NChooseK Matrix JSL Function
Dan Sutton
Samsung Austin Semiconductor
RESULTS (continued)
RESULTS (continued)
Example 3: For metrology tools that use one site map for many applications, what would be the optimal map that reduces impact? In this example, a metrology map is used many times for ease of setup or to correlate sites at multiple metrology steps. In this case, one map may not be optimal at all metrology steps. In Fig. 6, the first site reduction map is shown as such an example where it is at least the 7th best map on some step, but 713th at some other step. The JMP script can determine the relative rankings and pick the best candidate maps that minimizes the impact on the specified desirabilities. Map 167 in contrast has many steps where it is the top pick for a site reduction map. Further analysis will be done by the user to determine if this map is sufficient. Fig. 6. Distribution of rankings for typical map #1 vs. top 4 maps
Example 4: Across wafer variation.
While no one map with site reduction can detect all sources of across wafer changes, the goal of the wafer fab should be to have enough different maps to cover various locations on the wafer. For example, if sites 2, 7, 8, 11 are excluded on one map, another map should hopefully include these sites. Low volume metrology are not usually candidates for site reduction, since they will not significantly improve overall metrology capacity, so they will typically maintain the monitoring of the standard sites. Fig. 7 represents a theoretical distribution of what percentage of each site is measured at multiple steps.
Fig. 7: Percent each site is sampled after different site reduction maps are chosen at different steps
CONCLUSIONS
By utilizing the JMP Scripting Language, a complete analysis can be made of all possible combinations for metrology site reduction, utilizing built in matrix functions and other analysis platforms available in JMP. The analysis can be reduced from days and hours to just minutes, allowing the user to review the proposed maps and determine optimal maps based on knowledge and expertise of the application using the metrology.
REFERENCES
[1] M. Stephens, M. Bailey, “Combinations and permutations calculator: A tasty new add-in”, JMP Blog, (2014). [2] SAS Institute Inc JMP® 14 Scripting Guide. Cary, NC: SAS Institute Inc. “Regression Example”, pg [3] P. Prakash, B. Honari, A. Johnston, and S. McLoone, “Optimal wafer site selection using forward selection component analysis,” in Advanced Semiconductor Manufacturing Conference (ASMC), rd Annual SEMI, pp. 91–96, IEEE, [4] T. L. Vincent, J. B. Stirton, and K. Poolla, “Metrology sampling strategies for process monitoring applications,” IEEE Transactions on Semiconductor Manufacturing, vol. 24, no. 4, pp. 489–498, 2011.