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SEMI E89-1104 E © SEMI 1999, 2004 10 8.7.6.3 Under the assu mption of no l ack of fit, the ratio F LOF follows Fisher’s F distributio n. Calculate the probability associated with this ratio. NOTE 21: This probability can…
SEMI E89-1104
E
© SEMI 1999, 2004 9
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.
SEMI E89-1104
E
© SEMI 1999, 2004 10
8.7.6.3 Under the assumption of no lack of fit, the ratio
F
LOF
follows Fisher’s F distribution. Calculate the
probability associated with this ratio.
NOTE 21: This probability can be calculated using the Excel
function: FDIST(F
LOF
,nJ,J2).
8.7.6.4 Reject the assumption of no lack of fit if this
probability is small (e.g., 0.01 or less). In this case,
nonlinearity is probably present in the MS.
8.7.7 If there is no evidence of lack of fit, calculate the
approximate 95% confidence interval for a
ˆ
:
2
111
/( )
ˆ
2
()
ij
e
n
DJ
ijk j
ijk
SS n J
a
xx
v
(24)
where:
j
x
v
= (mean) value of the reference parameter from
reference wafer j.
8.7.7.1 If the interval contains the number one, a
ˆ
is not
statistically different from one. If this is the case and
also b
ˆ
is significantly different from zero, b
ˆ
is the
estimate of the bias.
8.7.7.2 If the interval does not contain the number one,
a
ˆ
is significantly different from one and bias is not
constant with parameter level.
8.7.7.3 If
b
ˆ
is not significantly different from zero, the
estimate of bias is zero.
8.8 Matching Tolerance
8.8.1 Determine the bias of two MSs of the same kind
under conditions of reproducibility.
8.8.2
If the results give a stable bias determination for
each MS and if each MS has acceptable linearity,
subtract the two biases to obtain the matching tolerance,
m
:
21
biasbias
m
(25)
where:
bias
1
= bias of the first MS and
bias
2
= bias of the second MS.
9 Procedure to Determine P/T Ratio
9.1 Estimate the precision of the MS, P, as 6s
R
if the
specification is symmetrically two-sided or as 3s
R
if the
specification is asymmetrical or one-sided.
9.2 Take the tolerance, T, as follows:
9.2.1 The difference between the USL and the LSL for
a symmetrical two-sided specification,
9.2.2 The smaller difference between the USL (or LSL)
and the target value for an asymmetrical two-sided
specification, or
9.2.3 The difference between the USL (or LSL) and the
median of the expected distribution for a one-sided
specification.
NOTE 22: If a symmetrical two-sided specification is given
as
Target Tolerance,
double the stated tolerance to obtain the correct value.
9.3 Calculate the P/T ratio, in percent, as follows:
100(%)/
T
P
TP
(26)
rounding to the nearest percent.
NOTE 23: Generally, a P/T ratio should be 30% or less.
10 Procedure to Determine SNR
10.1 Estimate the precision of the MS, P, as 6s
R
if the
process distribution is two-sided or as 3s
R
if the process
distribution is asymmetrical or one-sided.
10.2 Estimate the population standard deviation of the
process distribution from process data as
process
.
Because it is difficult to directly measure the standard
deviation of the product without including variation due
to the MS,
process
is generally defined as:
22
RTotalprocess
(27)
where:
2
Total
= variance obtained by measuring a large
representative sample of the product and
2
R
= variance determined from the reproducibility
(see Section 7).
NOTE 24: Detailed procedures for obtaining the process
distribution and the associated standard deviation are outside
the scope of this guide.
10.3 Calculate the SNR, in percent, as follows:
100(%)
R
process
SNR (28)
rounding to the nearest percent.
NOTE 25: In general, an SNR of 10 or more generally means
the MS is suitable for measuring the product, while an SNR of
less than 3 or 4 may be a concern in a particular measurement
process.
11 Related Documents
11.1 The following documents describe MSA
methodologies similar to those advocated in this guide.
SEMI E89-1104
E
© SEMI 1999, 2004 11
Ballard, D. H., McCormack, D. W., Jr., Moore, T. L.,
Pore, M., Prins, J., Tobias, P. A. (1998). “A
Comparison of Gauge Study Practices,” Proceedings
from the 1997 Joint Statistical Meetings of the
American Statistical Association, Quality and
Productivity Section.
John, Peter, “Alternative Models for Gauge Studies,”
International SEMATECH technology transfer
document 93081755A-TR (February 24, 1994); PDF
file can be downloaded from International
SEMATECH’s public web site at http://www.
sematech.org/docubase/wrappers/26.htm.
Montgomery, D.C. and Runger G. C. (1993), “Gauge
Capability and Designed Experiments. Part I: Basic
Methods,”
Quality Engineering 6(1), 115–135.
Montgomery, D.C. and Runger G. C. (1993), “Gauge
Capability and Designed Experiments. Part II:
Experimental Design Models and Variance Component
Estimation,” Quality Engineering 6(2), 289–305.
Potter, R.W. (1991), “Measurement System Capability
Analysis,” IEEE/SEMI Advanced Semiconductor
Manufacturing Conference, pp. 121–125.
---, Measurement Systems Analysis Reference Manual,
Third Edition, (2002); Daimler Chrysler Corp., Ford
Motor Co, and General Motors Corp. Automotive
Industry Action Group (AIAG);
http://www.aiag.org.
---, “NIST/SEMATECH e-Handbook of Statistical
Methods,”
http://www.itl.nist.gov/div898/handbook.