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SEMI E89-1104 E © SEMI 1999, 2004 18 R4-2.10.2 Because this is far above the generally acceptable maximum value of 30% for a suitable MS, it might not be suitable for som e applications. Improveme nt activities would beg…
SEMI E89-1104
E
© SEMI 1999, 2004 17
R4-2.3 Reproducibility (
R
) is calculated as the square
root of the sum of the estimates, i.e.,
12.1348 0.1256 0 462.9869 0.8446
21.82
R
(R4-2)
R4-2.4 Repeatability (
r
) is estimated by the variance
component corresponding to repeat and is equal to zero.
R4-2.5 Stability is estimated by the square root of the
Day component of the MSA,
stability
= 3.4835.
R4-2.6 Figure R4-1 shows a Pareto chart of the sigmas.
0 5 10 15 20
Sigma
Repeat
Cycle
Residual
Day
Day x W afer
Figure R4-1
Pareto Chart for Effects Sigmas (All Five Wafers)
R4-2.7 Linearity is more difficult to estimate. In
general, it is measured by the differences in variability
associated with the different wafer types. The large
wafer-by-day interaction suggests that there may be a
linearity problem. An examination of the standard
deviations for each group by day (see Figure R4-2)
indicates that Wafer Type 5 (Deep UV Resist) may be
responsible. These standard deviations were calculated
from the four measurements made on every wafer each
day.
R4-2.8
When the variance components analysis was
rerun, excluding Wafer Type 5 data, the new estimates
obtained confirmed that Wafer Type 5 was problematic.
An investigation into the potential cause suggested that
the gauge was degrading the resist at the point of
measurement, causing the readings to decrease over
time.
R4-2.9
If one were interested in measuring only Wafer
Types 1 through 4 (employing a different gauge for
Wafer Type 5, for example), the improved variance
components estimates from Table R4-3 could be used.
R4-2.10 The new estimates for
r
and
R
, are 0 and
1.565, respectively. Stability, as measured by day-to-
day variability, improves to 0.6450. The new Pareto
chart of the effects sigmas is shown in Figure R4-3. The
specification range for the product being measured was
given as 10 Å, thus the P/T ratio for the MS for Wafer
Types 1 to 4 is
61.565
100 93.9%
10
P
T
(R4-3)
Table R4-3 Variance Components Estimates for
Wafers 1 – 4
Variance
Component
Estimated
Variance
Estimated
Sigma
Day 0.4160 0.6450
Cycle 0.0203 0.1426
Repeat 0 0
Day x Wafer 1.4918 1.2214
Residual 0.1937 0.4402
012345678
Day
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Standard Deviation
Wafer 1
Wafer 2
Wafer 3
Wafer 4
Wafer 5
Figure R4-2
Wafer Sigmas by Day
0.0 0.4 0.8 1.2
Sigma
Repeat
Cycle
Residual
Day
Day x W afer
Figure R4-3
Pareto Chart for Effects Sigmas
(Wafer Type 5 Removed)
SEMI E89-1104
E
© SEMI 1999, 2004 18
R4-2.10.2 Because this is far above the generally
acceptable maximum value of 30% for a suitable MS, it
might not be suitable for some applications.
Improvement activities would begin with an
investigation of why the Wafer Day sigma is so large.
This term indicates differences between the ways
certain wafers vary from day to day (stability
differences between wafers).
R4-2.11 The actual measurement data obtained in this
example MSA is provided in Table R4-4. It may be
useful for working through the calculations in this MSA
either using statistical analysis software or manually to
confirm understanding.
Table R4-4 Measurement Data (Average of Measurements Taken at Three Points Each Time) in Example MSA
Day Cycle Repeat Wafer 1 Wafer 2 Wafer 3 Wafer 4 Wafer 5
1 1 1 48.747 980.007 7,907.71 2,576.78 13,061.50
1 1 2 49.603 980.050 7,906.78 2,576.66 13,059.92
1 2 1 49.273 979.623 7,906.18 2,577.14 13,064.28
1 2 2 49.200 979.770 7,906.87 2,577.06 13,062.84
2 1 1 47.187 980.260 7,906.94 2,576.50 13,034.58
2 1 2 47.613 980.303 7,906.60 2,576.59 13,032.42
2 2 1 48.500 980.840 7,906.85 2,576.87 13,033.23
2 2 2 48.560 980.567 7,906.35 2,576.74 13,032.46
3 1 1 47.270 983.120 7,907.46 2,577.35 13,017.89
3 1 2 47.367 983.083 7,909.03 2,577.58 13,016.72
3 2 1 47.550 982.987 7,909.45 2,577.68 13,020.23
3 2 2 47.530 982.870 7,909.67 2,577.48 13,019.89
4 1 1 46.373 982.420 7,909.09 2,576.53 13,000.94
4 1 2 45.913 982.520 7,908.90 2,576.89 12,999.77
4 2 1 47.380 982.210 7,908.35 2,576.84 12,996.04
4 2 2 47.340 982.220 7,908.96 2,577.26 12,995.51
5 1 1 44.457 982.080 7,907.22 2,576.54 12,963.97
5 1 2 44.690 982.020 7,907.36 2,576.21 12,962.29
5 2 1 44.563 981.963 7,907.77 2,576.67 12,963.52
5 2 2 44.613 982.120 7,906.02 2,576.58 12,962.83
6 1 1 45.753 982.200 7,906.16 2,576.42 12,949.45
6 1 2 45.317 982.207 7,904.74 2,576.29 12,948.40
6 2 1 45.030 982.013 7,903.42 2,576.39 12,949.32
6 2 2 45.880 981.837 7,904.56 2,576.33 12,948.96
7 1 1 42.460 981.393 7,904.80 2,575.98 12,938.10
7 1 2 43.350 981.743 7,904.26 2,576.16 12,936.95
7 2 1 44.060 981.770 7,903.59 2,576.05 12,936.02
7 2 2 43.983 981.720 7,903.71 2,576.30 12,934.25
8 1 1 45.783 982.300 7,905.23 2,576.07 12,934.40
8 1 2 46.440 982.420 7,905.62 2,576.56 12,932.71
8 2 1 45.907 982.653 7,905.69 2,576.30 12,930.05
8 2 2 46.573 982.320 7,905.61 2,576.29 12,927.99
SEMI E89-1104
E
© SEMI 1999, 2004 19
RELATED INFORMATION 5
EXAMPLE OF A MEASUREMENT SYSTEM ANALYSIS WITH DAY AND
LOAD ONLY
NOTICE: This related information is not an official part of SEMI E89. It was derived from task force deliberations
during the revision of SEMI E89-0999 in 2001-2003. This related information was approved by full letter ballot
procedures and was approved for publication by the NA RSC on August 16, 2004.
R5-1 Introduction
R5-1.1 The following example illustrates how an MSA
is performed for a hypothetical measurement instrument
where only day and load are considered. Hand
calculations are provided and possible because the data
are balanced (i.e., each load has the same number of
repeated measurements). Rounding of value may cause
results may vary slightly. The data are shown in Table
R5-1.
Table R5-1 Data for MSA with Load and Repeat
Repeat Load 1 Load 2 Load 3
1 593.46 593.43 594.11
2 593.15 594.22 594.03
3 592.87 593.63 593.92
4 593.25 592.73 593.2
5 593.43 592.98 593.16
6 593.21 593.18 593.41
7 593.08 592.74 593.57
8 593.68 592.65 593.46
9 593.06 592.69 592.7
10 592.68 593.38 593.42
11 593.20 593.86 593.5
12 593.00 593.27 593.39
The model for the MSA is
Y
ij
=
+ l
i
+ r
ij
(R5-1)
Y
ij
= measurement on the i
th
load and j
th
repeat,
= true value of the measurand,
l
i
= error term associated with the i
th
load, and
r
ij
= error term associated with the i
th
load, j
th
repeat.
R5-1.2 Load and Repeat are treated as random effects.
Repeat is nested within Load.
R5-2 Analysis
R5-2.1 Calculate the mean over all observations (the
grand mean,
Y
) and the mean for each load (
i
Y
).
Table R5-2 Mean by Load and Grand Mean
N Obs Mean
Load 1 12 593.17
Load 2 12 593.23
Load 3 12 593.49
Grand Mean 36 593.30
R5-2.2 Calculate the Sum of Squares for Load (SS
L
):
3
2
1
12 0.6830
Li
i
SS Y Y
(R5-2)
R5-2.2.1 In general, the formula for SS
L
is
2
1
L
Li
i
SS n Y Y
(R5-3)
where
n = number of repeats per load and,
L = number of loads.
R5-2.3 Calculate the Sum of Squares for Repeatability
(SS
r
):
12 3
2
11
5.2911
riji
ji
SS Y Y
(R5-4)
where:
ij
Y = observation for the i
th
load, j
th
repeat.
R5-2.3.1 In general, the formula for SS
r
is
2
11
nL
riji
ji
SS Y Y
(R5-5)
R5-2.4 Calculate the degrees of freedom for Load (df
L
)
as L – 1. Calculate the degrees of freedom for
repeatability (df
r
) as L(n – 1).