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2. Measurement Process Characterization
2.6. Case studies
2.6.1. Gauge study of resistivity probes

2.6.1.2.

Analysis and interpretation

Graphs of probe effect on repeatability A graphical analysis shows repeatability standard deviations plotted by wafer and probe. Probes are coded by numbers with probe #2362 coded as #5. The plots show that for both runs the precision of this probe is better than for the other probes.

Probe #2362, because of its superior precision, was chosen as the tool for measuring all 100 ohm.cm resistivity wafers at NIST. Therefore, the remainder of the analysis focuses on this probe.

Plot of repeatability standard deviations for probe #2362 from the nested design over days, wafers, runs The precision of probe #2362 is first checked for consistency by plotting the repeatability standard deviations over days, wafers and runs. Days are coded by letter. The plots verify that, for both runs, probe repeatability is not dependent on wafers or days although the standard deviations on days D, E, and F of run 2 are larger in some instances than for the other days. This is not surprising because repeated probing on the wafer surfaces can cause slight degradation. Then the repeatability standard deviations are pooled over:
  • K = 6 days for K(J - 1) = 30 degrees of freedom
  • L = 2 runs for LK(J - 1) = 60 degrees of freedom
  • Q = 5 wafers for QLK(J - 1) = 300 degrees of freedom
The results of pooling are shown below. Intermediate steps are not shown, but the section on repeatability standard deviations shows an example of pooling over wafers.
Pooled level-1 standard deviations (ohm.cm)

  Probe   Run 1    DF      Run 2     DF    Pooled   DF

  2362.   0.0658  150      0.0758   150    0.0710  300
Graphs of reproducibility and stability for probe #2362 Averages of the 6 center measurements on each wafer are plotted on a single graph for each wafer. The points (connected by lines) on the left side of each graph are averages at the wafer center plotted over 5 days; the points on the right are the same measurements repeated after one month as a check on the stability of the measurement process. The plots show day-to-day variability as well as slight variability from run-to-run.

Earlier work discounts long-term drift in the gauge as the cause of these changes. A reasonable conclusion is that day-to-day and run-to-run variations come from random fluctuations in the measurement process.

Level-2 (reproducibility) standard deviations computed from day averages and pooled over wafers and runs Level-2 standard deviations (with K - 1 = 5 degrees of freedom each) are computed from the daily averages that are recorded in the database. Then the level-2 standard deviations are pooled over:
  • L = 2 runs for L(K - 1) = 10 degrees of freedom
  • Q = 5 wafers for QL(K - 1) = 50 degrees of freedom
as shown in the table below. The table shows that the level-2 standard deviations are consistent over wafers and runs.
Level-2 standard deviations (ohm.cm) for 5 wafers

                    Run 1                   Run 2
 Wafer  Probe   Average  Stddev  DF    Average Stddev  DF

  138.  2362.  95.0928   0.0359  5    95.1243  0.0453  5
  139.  2362.  99.3060   0.0472  5    99.3098  0.0215  5
  140.  2362.  96.0357   0.0273  5    96.0765  0.0276  5
  141.  2362. 101.0602   0.0232  5   101.0790  0.0537  5
  142.  2362.  94.2148   0.0274  5    94.2438  0.0370  5

        2362.   Pooled   0.0333 25             0.0388 25

              (over 2 runs)                    0.0362 50
Level-3 (stability) standard deviations computed from run averages and pooled over wafers Level-3 standard deviations are computed from the averages of the two runs. Then the level-3 standard deviations are pooled over the five wafers to obtain a standard deviation with 5 degrees of freedom as shown in the table below.
Level-3 standard deviations (ohm.cm) for 5 wafers
               Run 1      Run 2
Wafer Probe  Average     Average      Diff     Stddev  DF

138.  2362.  95.0928     95.1243     -0.0315   0.0223   1
139.  2362.  99.3060     99.3098     -0.0038   0.0027   1
140.  2362.  96.0357     96.0765     -0.0408   0.0289   1
141.  2362. 101.0602    101.0790     -0.0188   0.0133   1
142.  2362.  94.2148     94.2438     -0.0290   0.0205   1

2362. Pooled 0.0197 5

Graphs of probe biases A graphical analysis shows the relative biases among the 5 probes. For each wafer, differences from the wafer average by probe are plotted versus wafer number. The graphs verify that probe #2362 (coded as 5) is biased low relative to the other probes. The bias shows up more strongly after the probes have been in use (run 2).
Formulas for computation of biases for probe #2362 Biases by probe are shown in the following table.
Differences from the mean for each wafer
     Wafer Probe    Run 1     Run 2

      138.    1.   0.0248   -0.0119
      138.  281.   0.0108    0.0323
      138.  283.   0.0193   -0.0258
      138. 2062.  -0.0175    0.0561
      138. 2362.  -0.0372   -0.0507

      139.    1.  -0.0036   -0.0007
      139.  281.   0.0394    0.0050
      139.  283.   0.0057    0.0239
      139. 2062.  -0.0323    0.0373
      139. 2362.  -0.0094   -0.0657

      140.    1.   0.0400    0.0109
      140.  281.   0.0187    0.0106
      140.  283.  -0.0201    0.0003
      140. 2062.  -0.0126    0.0182
      140. 2362.  -0.0261   -0.0398

      141.    1.   0.0394    0.0324
      141.  281.  -0.0107   -0.0037
      141.  283.   0.0246   -0.0191
      141. 2062.  -0.0280    0.0436
      141. 2362.  -0.0252   -0.0534

      142.    1.   0.0062    0.0093
      142.  281.   0.0376    0.0174
      142.  283.  -0.0044    0.0192
      142. 2062.  -0.0011    0.0008
      142. 2362.  -0.0383   -0.0469
How to deal with bias due to the probe Probe #2362 was chosen for the certification process because of its superior precision, but its bias relative to the other probes creates a problem. There are two possibilities for handling this problem:
  1. Correct all measurements made with probe #2362 to the average of the probes.
  2. Include the standard deviation for the difference among probes in the uncertainty budget.
The better choice is (1) if we can assume that the probes in the study represent a random sample of probes of this type. This is particularly true when the unit (resistivity) is defined by a test method.
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