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SEMI E89-1104 E © SEMI 1999, 2004 8 8.3 Determine if the MS needs to be calibrated. Calibrate, if necessary, following the manufacturer’s recommended pr ocedure. Do not recalibrate during th e bias determi n ation unless…
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NOTE 10: Nesting occurs when the measurement procedure
causes a natural hierarchy of events to exist. For example, if
repeatability is estimated by taking measurements with the
wafer fixed in the gauge, and variability due to the load-
unload operation is estimated by taking a second set of
measurements after the wafer has been removed and reloaded,
repeatability is nested in load-unload. The nesting occurs
because each set of repeated measurements is unique to the
load in which they appear. Nesting may also occur if a
physical hierarchy exists, such as when both measurement
locations (sites) and wafer type are fixed factors in an MSA
(sites being nested in wafer type).
7.4 Select appropriate test wafers for the MSA.
7.4.1 In the most general case, many types of wafers
should be used. These should be representative of the
types and levels to be measured in practice by the MS
under evaluation.
7.4.2 The test wafers should be as uniform as possible
to prevent confounding of handling, loading, or
positioning factors, if these are chosen to be examined.
7.4.3 It is necessary to use only one test wafer of each
type selected.
7.4.4 Use the same test wafers throughout the MSA.
7.5 Prepare a statistical model for the MSA in order to
plan and analyze it properly.
NOTE 11: Statistical models are discussed in Related
Information 3. Eastman
4
gives some examples of models for
reproducibility studies.
7.6 Select the measurement protocol. Include several
examples of every factor that was selected in Section
7.1.
7.7 Select and implement the appropriate MS setup.
NOTE 12: Usually, only a single setup is used in an MSA to
determine reproducibility. However, if setup procedure has
been chosen as a factor to be evaluated, select the various
setups to be used in the MSA.
NOTE 13: Examples of MSAs are given in Related
Information 4, 5, and 6. Additional examples may be found
in the literature.
4
7.8 Determine if the MS needs to be calibrated.
7.8.1 Calibrate, if necessary, following the
manufacturer’s recommended procedure.
4 Eastman, S. A., “Evaluating Automated Wafer Measurement
Instruments,” International SEMATECH technology transfer
document 94112638A-XFR (February 28, 1995), Section 5 and case
studies in appendices. However, note that neither the terminology nor
the nomenclature in this document is identical with that given in this
guide. A PDF file of this report can be downloaded from
International SEMATECH’s public web site at
http://www.sematech.org/docubase/wrappers/26.htm
.
7.8.2 Do not recalibrate during the MSA unless
required by the normal operating procedure.
7.9 Perform the MSA, randomizing both the sample
wafers chosen and the order in which measurements are
made, whenever possible. Record the actual order in
which all measurements are made.
7.10 To correctly estimate the reproducibility and
individual sources of variance, conduct a variance
components analysis.
7.10.1 If the experiment has been designed to include
multiple repeated measurements taken under the same
(repeatability) conditions, it is possible to estimate
repeatability. Without multiple repeated measurements,
repeatability will be confounded with other sources of
variability.
7.10.2 Other factors are those from Section 7.1 that
have been included in the experiment design.
7.10.3 Reproducibility is the square root of the sum of
the individual variance components (see Equation 3 in
Section 5.3.30.1).
NOTE 14: The exact nature of the variance components
analysis depends on the design of the experiment conducted.
For details, see the examples in Related Information 4, 5, and
6, Box, Hunter, and Hunter,
5
or Montgomery.
6
In designing
and performing a complete designed experiment and the
associated analysis, it is strongly recommended to utilize
some form of commercially available statistical analysis
software. If such a product is used, it is advisable to confirm
that the assumptions made in the software are congruent with
those in this guide.
8 Procedure to Determine Bias, Including
Linearity, Stability, and Matching Tolerance
8.1 Obtain J appropriate reference materials, preferably
CRMs, where J is between 3 and 10 so that the entire
range of interest of the parameter is covered. If CRMs
are not available, use wafers with well accepted
consensus values covering the range of values of the
attributes to be measured. The reference materials
should have characteristics similar to the wafers to be
measured by the MS being evaluated or their values
should be transferable to such wafers.
NOTE 15: Bias cannot be determined if suitable reference
materials are not available. In such situations, it is necessary
to resort to correlation experiments to establish the systematic
errors between different MSs. If the bias is not known, these
systematic errors may confound the estimated reproducibility.
8.2 Select and implement the appropriate MS setup.
5 Box, G. E. P., Hunter, W. G., and Hunter, J. S., Statistics for
Experimenters (Wiley, New York, 1978).
6 Montgomery, D.C., Design and Analysis of Experiments, 5th Ed.
(Wiley, New York, 2000).
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8.3 Determine if the MS needs to be calibrated.
Calibrate, if necessary, following the manufacturer’s
recommended procedure. Do not recalibrate during the
bias determination unless required by the normal
operating procedure.
8.4 Chose the minimum number of measurements to be
made on each of the J reference materials, n, as the
smallest integer greater than (4
/
)
2
where
is the
estimate of reproducibility (
R
, see Section 7) and
is
the bias shift to be detected.
NOTE 16: In this relationship, the assumption is made that
the maximum acceptable probability of calibrating when
unnecessary (Type 1 error) is 0.1 and the maximum tolerable
probability of not calibrating when necessary (Type 2 error) is
0.01. If use of other values of these errors is desired, the more
complete formula for minimum sample size is the smallest
integer greater than [
(z
1
+ z
2
)/
]
2
where:
z
1
= value above which p
1
/2 percent of the standard normal
distribution falls where p
1
is the largest acceptable risk
of calibrating when unnecessary,
z
2
= value above which p
2
/2 percent of the standard normal
distribution falls where p
2
is the largest acceptable risk
of not calibrating when necessary,
= estimate of the variability as defined in Section 8.4,
and
= shift (bias) to be detected.
8.4.1 If no information on MS variability is available,
use a minimum of 16 measurements on a reference
wafer to estimate bias.
8.5 Perform the measurements on one or more days,
depending if stability information is desired or not. To
minimize systematic errors, randomize the sequence of
measurements among the reference wafers for each
daily measurement. If the MS is performed over
several days, make an approximately equal number of
measurements on each day with at least two
measurements made per day.
8.6 Analysis of the Measurement Data
8.6.1 This analysis can be done with most
commercially available statistical analysis software
packages. The procedures given in Section 8.7 are
intended only for those without access to such software.
8.6.2 The full model to determine the effect of bias over
time and measurement range is:
ijiiij
xabyE ][ (7)
where
E[y
ij
]
= expected response for reference wafer j on day
i,
b
i
= bias on day i,
x
ij
= certified value for reference wafer j on day i,
and
a
i
= slope for day i.
NOTE 17: If multiple measurements are made for a given
reference wafer on a given day, it is assumed that
][][
ijijk
yEyE
(8)
where k is any one of the multiple measurements made on
reference wafer j on day i.
8.6.3 When bias and slope do not differ significantly
over days (i.e., a
i
a and b
i
b for all i), the reduced
model is:
jj
axbyE ][
(9)
8.6.4 Linearity — The calibration coefficient a is a
linear constant that relates the rate at which bias
changes for every unit change in the measurement
range. This value should not be significantly different
from one. Otherwise, the MS exhibits nonlinearity.
8.6.5 Stability — Test for a significant difference
between the full and reduced models given by
Equations 7 and 9, respectively. If there is no
significant difference, continue. Otherwise, bias is not
stable over time.
8.6.6 Test for model lack of fit. Significant lack of fit
indicates a possible nonlinear relationship involving
bias that is not captured by the reduced model
8.6.7 If the test for lack of fit is not significant, estimate
a and b.
8.6.7.1 If a is not statistically different from one and b
is statistically different from zero, b is an estimate of
the bias.
8.6.7.2 If a is statistically different from one, bias is not
constant. A linear relationship exists between bias and
measurement range.
8.6.7.3 If b is not statistically different from zero, the
estimate of bias is zero.
8.7 Detailed Analysis Procedures — If the measure-
ments are made over a single day only, skip Sections
8.7.1 and 8.7.2 and begin the analysis with Section
8.7.3. Also skip Section 8.7.5. In this case, no
information about time variation of bias (stability) can
be obtained from the MSA. Otherwise begin the
analysis with Section 8.7.1.
8.7.1 Estimate the slopes (
i
a
~
) and biases (
i
b
~
) for each
day’s measurements to determine the effect of time and
measurement range on the bias:
J
j
n
k
iijk
J
j
n
k
iijk
i
ij
ij
i
x
ijk
x
xxyy
a
11
2
11
)(
)()(
~
(10)
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iiii
xayb
~
~
(11)
where:
J = number of reference wafers,
n
ij
= number of measurement results for reference
wafer j on day i,
y
ijk
= result of measurement k on reference wafer j
on day i,
i
y = mean of the measurement results on day i,
x
ijk
= value of the reference parameter on day i, and
i
x = mean of the values of the reference parameter
on day i.
8.7.2 Calculate the expected response on each reference
wafer j on each day i, E[y
ij
]:
jiiiij
xabyE
~
~
(12)
where:
x
ij
= value of the reference parameter for wafer j of
day i, and
i
a
~
and
i
b
~
are the slope and intercept (bias) for day i as
found from Equations 10 and 11, respectively,
in Section 8.7.1.
NOTE 18: When multiple measurements are made for a
given reference wafer on a given day, it is assumed for all k
that
ijijk
yEyE (13)
where k is any one of the multiple measurements made on the
i day on the j reference wafer.
8.7.3 Estimate the slopes (
i
a
ˆ
) and biases (
i
b
ˆ
) for the
entire data set:
D
i
J
j
n
k
ijk
D
i
J
j
n
k
ijkijk
ij
ij
xx
xxyy
a
111
2
111
)(
))((
ˆ
(14)
xayb
ˆ
ˆ
(15)
where:
D = number of days over which measurements are
made,
x = grand mean of the values of the reference
parameter,
y = grand mean of the measurement results, and
the other parameters are defined in Section 8.7.1.
NOTE 19: This procedure makes the assumption that the bias
and slope do not differ significantly over days. This
assumption is tested in Section 8.7.5.
8.7.4 Calculate the expected response on each wafer j,
E[y
j
]:
jj
xabyE
ˆ
ˆ
(16)
8.7.5 Determine whether bias and slope differ
significantly over days or not as follows:
8.7.5.1 Calculate the Sum of Squares for the difference
between the two cases, SS
M
:
2
11
DJ
Mijj
ij
SS Ey Ey
(17)
8.7.5.2 Calculate the Error Sum of Squares (SS
e
):
MYe
SSSSSSSS
(18)
where
2
111
()
ij
n
DJ
Y ijk
ijk
SS y y
(19)
and
111
()()
ij
n
DJ
ijk ijk
ijk
SS y y x x
(20)
8.7.5.3 Calculate probability associated with the
function F
M
that follows a Fisher’s F distribution with
2D 1 and n 2D degrees of freedom under the
hypothesis that there is no difference between the two
cases:
(2)
(2 1)
M
M
e
nDSS
F
DSS
(21)
NOTE 20: This probability can be calculated using the Excel
function, FDIST(F
M
,2D1, n2D).
8.7.5.4 Reject the hypothesis that there is no difference
between the two cases when the probability associated
with F
M
is small (e.g., 0.05 or below). In this case, bias
is not stable over time and it is not possible to establish
a value for it.
8.7.6 Test for lack of fit as follows:
8.7.6.1 Calculate the Sum of Squares for Pure Error
(SS
p
):
2
111
()
ij
n
DJ
pijkj
ijk
SS y y
(22)
where all the symbols have been defined previously.
8.7.6.2 Calculate the lack of fit F test
))((
)2(
pe
p
LOF
SSSSJn
SSJ
F
(23)
where, again, all the symbols have been defined
previously.