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SEMI E89-1104 E © SEMI 1999, 2004 9 i i i i x a y b ~ ~ (11) where: J = number of reference wafers, n ij = number of measurement res ults for reference wafer j on day i , y ijk = result of measurement k on reference …
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 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)
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.