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SEMI M56-1103 © SEMI 2003 3 5.4.6 cumulative distribution functio n ( CDF ) — a mathematical formula that describes th e probability a measurable event occurs at or below a specific value. 5.4.7 distribution — a characte…
SEMI M56-1103 © SEMI 2003 2
2.8 The output of this practice is the cost component
due to measurement variability and bias for each
process characteristic and metrology system evaluated.
2.9 This practice does not consider costs associated
with other components of uncertainty outside of
measurement variability and bias.
2.10 Although it is focused on the issue of misclassifi-
cation, the practice may be extended to include costs
associated with correct classification by the metrology
equipment.
2.11 This practice was developed to evaluate
metrology equipment used in silicon wafer
manufacturing, but it may be extended to other types of
metrology equipment by making suitable modifications.
NOTICE: This practice does not purport to address
safety issues, if any, associated with its use. It is the
responsibility of the users of this standard to establish
appropriate safety health practices and determine the
applicability of regulatory or other limitations prior to
use.
3 Limitations
3.1 Application of this practice requires that all
measurement variances and biases, process
characteristic distributions, and costs of classification
and misclassification be accurately characterized. If
this is not done, the resulting calculation may be in
error.
3.2 Application of this practice also requires that
measurement variances and biases be constant over the
measurement range interval of interest. If this is not the
case, the resulting calculation may be in error.
3.3 This practice considers only the case of two
convolved distributions, process characteristic
variability and measurement variability; consideration
of additional distributions (such as would be required if
the measurement bias or variance were assumed to be a
function of the measured value) is beyond the scope of
this practice.
3.4 Extension of the model to the cases of decisions
based on (1) measurements on multiple gauges, (2)
multiple inspections with the same gauge or (3) on
multiple characteristics is outside the scope of this
practice, but information on such extensions is provided
in Related Information 1 for the convenience of the
user.
3.5 This model should not be confused with a cost-of-
ownership model, such as that developed in SEMI E35,
which explicitly includes information on the cost
elements involved.
4 Referenced Standards
4.1 SEMI Standards
SEMI E35 — Cost of Ownership for Semiconductor
Manufacturing Equipment Metrics
SEMI E89 — Guide for Measurement System
Capability Analysis
SEMI M1 — Specifications for Polished
Monocrystalline Silicon Wafers
SEMI M27 — Practice for Determining the Precision
over Tolerance Ratio of Test Equipment
4.2 ISO Document
1
International Vocabulary of Basic and General Terms
in Metrology, Second Edition [VIM]
NOTICE: Unless otherwise indicated, all documents
cited shall be the latest published versions.
5 Terminology
5.1 General terminology related to metrology is
defined in VIM.
5.2 Terminology relating to measurement system
capability analysis is defined in SEMI E89.
5.3 Terminology related to silicon wafer technology is
defined in SEMI M1.
5.4 Other Definitions
5.4.1 alpha error — the error that occurs when a
conforming item is incorrectly reported as non-
conforming. This is also called Type I error.
5.4.2 alpha probability,
α
— the probability of an
alpha error, also called the alpha error rate.
5.4.3 beta error — the error that occurs when a non-
conforming item is incorrectly reported as conforming.
This is also called Type II error.
5.4.4 beta probability,
β
— the probability of a beta
error, also called the beta error rate.
5.4.5 bias,
δ
— the difference between the mean value
of measurements made on the same object and a true
value.
NOTE 1: Bias is illustrated in Related Information 2 (Figure
R2-1). However, in many cases the true value is unknown. A
value established by reference gauges or a consensus value
may be used as a substitute.
1
International Organization for Standardization: ISO Central
Secretariat, 1, rue de Varembé, Case postale 56, CH-1211 Geneva 20,
Switzerland, Telephone: 41.22.749.01.11; Fax: 41.22.733.34.30, Web
site: www.iso.ch.
SEMI M56-1103 © SEMI 2003 3
5.4.6 cumulative distribution function (CDF) — a
mathematical formula that describes the probability a
measurable event occurs at or below a specific value.
5.4.7 distribution — a characterization of the
probability of realization for a measurable event over
the range of values that the measurements may assume.
5.4.8 joint probability — a probability density or
cumulative distribution function comprised of two or
more random variables.
5.4.9 lower specification limit (LSL) — value of a char-
acteristic below which a product is said to be non-
conforming.
5.4.10 measurand — particular attribute of a phe-
nomenon, body, or substance subject to measurement
[VIM].
5.4.11 measurement variability — differences associ-
ated with making multiple measurements on a given
measurand under specific conditions.
NOTE 2: Measurement variability (its distribution) is
illustrated in Related Information 2 (Figure R2-1). Common
(general) estimators of measurement variability are the
variance, standard deviation, and variance components.
Specific estimators include repeatability and reproducibility.
5.4.12 probability density function (PDF) — a
mathematical formula that specifies the relationship
between values that a random variable may assume and
their likelihood of occurrence. It is the first derivative
of the CDF.
5.4.13 random variable — a measurable event
occurring such that any value from its distribution is
equally likely to take place.
5.4.14 standard deviation,
σ
— the positive square
root of the variance.
NOTE 3: The standard deviation of a population may be
estimated from experimentally obtained data by the sample
standard deviation (s):
∑
=
−
−
=
n
i
i
xx
n
s
1
2
)(
1
1
where:
n = number of data values,
x
i
= value of the i
th
data point, and
x = mean of the data distribution.
5.4.15 upper specification limit (USL) — value of a
characteristic, above which a product is said to be non-
conforming.
5.4.16 variance — a statistical estimator that quantifies
spread around the mean of a PDF.
6 Summary of Practice
6.1 The process distribution for the characteristic of
interest is estimated or determined.
6.2 The bias and standard deviation for the
characteristic of interest are determined for each mea-
surement instrument to be evaluated or compared.
6.3 The quantities
α
and
β
are calculated from the
appropriate formula depending on the nature of the
specification (LSL only, USL only, or both).
6.4 Costs, based on the applicable business model, are
assigned to each of the four possible measurement out-
comes.
6.5 The cost components arising from measurement
variability are calculated for each measurement
instrument being evaluated or compared.
6.6 The costs due to measurement variability are
compared to establish the most cost effective
measurement solution for the application.
7 Procedure
7.1 Estimate the PDF for the process characteristic to
be studied. This can be done using empirical data that
represents the process. Although actual data may be
used to create a discrete PDF, it is sometimes
convenient to use the data to parametrically fit a PDF
model (e.g., log normal).
7.2 Unless already known, establish the bias and
standard deviation for each measurement gauge to be
compared in accordance with SEMI E89.
7.3 In all cases take the measurement influence into
account so that it does not broaden the PDF. This may
be done by taking repeated measurements at each point
in the measurement range and calculating the mean, or
by deconvolving the process characteristic PDF,
f(x),
and the measurement variability CDF, G(u), generally
assumed to be a Gaussian (or normal) distribution with
arithmetic mean equal to the bias, so that:
x
x
uuG
M
u
M
d
2
)(
exp
2
1
)()(
2
2
−
−=Φ=
∫
∞−
σ
δ
πσ
(1)
where:
δ
= bias,
σ
M
= standard deviation of the measurement
distribution.
NOTE 4: The quantity σ
M
includes the effects of the change
in bias over the time interval in which σ
M
has been
established.
SEMI M56-1103 © SEMI 2003 4
7.4 Calculate
α
and
β
as follows (Note 4). Note that
the symbols in the equations for
α
and
β
have the
following meanings:
f (x) = PDF of process characteristic x,
USL = upper specification limit,
LSL = lower specification limit, and
Φ(u) = Gaussian CDF (see Equation (1) in Section
7.3).
7.4.1 For a characteristic with only a USL, use the
following equations to calculate
α
and
β
:
∫
∞−
+−
Φ−=
USL
M
xxf
xUSL
d)(1
σ
δ
α
∫
∞
+−
Φ=
USL
M
xxf
xUSL
d)(
σ
δ
β
7.4.2 For a characteristic with only an LSL, use the
following equations to calculate
α
and
β
:
∫
∞
+−
Φ=
LSL
M
xxf
xLSL
d)(
σ
δ
α
∫
∞−
+−
Φ−=
LSL
M
xxf
xLSL
d)(1
σ
δ
β
7.4.3 For a characteristic with both upper and lower
specifications limits, use the following equations to
calculate
α
and
β
:
xxf
xLSL
xxf
xUSL
USL
LSL
M
USL
LSL
M
d)(
d)(1
∫
∫
+−
Φ
+
+−
Φ−=
σ
δ
σ
δ
α
xxf
xLSLxUSL
xxf
xLSLxUSL
USL
MM
LSL
MM
d)(
d)(
∫
∫
∞
∞−
+−
Φ−
+−
Φ
+
+−
Φ−
+−
Φ=
σ
δ
σ
δ
σ
δ
σ
δ
β
NOTE 5: Background information related to the calculation
of
α
and
β
is given in Related Information 3.
7.5 Use a binary decision model. If measurement-
based decisions are labeled as pass or fail and items are
inherently conforming or nonconforming, there are only
four outcomes:
• pass a conforming item,
• fail a conforming item (
α
error),
• pass a nonconforming item (
β
error), and
• fail a nonconforming item.
The probabilities associated with these outcomes are
1−
α
,
α
,
β
, and 1−
β
, respectively.
7.6 To define the cost model, assign costs to each of
the four decisions above on the basis of the business
model used for the manufacturing process as follows:
• c
pc
: cost of passing a conforming item,
• c
fc
: cost of failing a conforming item (
α
error),
• c
pn
: cost of passing a non-conforming item (
β
error), and
• c
fn
: cost of failing a non-conforming item.
7.6.1 Assign zero incremental cost to the two correct
outcomes (c
pc
and c
fn
).
7.6.2 Assign the incremental costs for the error out-
comes, c
fc
and c
pn
, on the basis of the business model
used for the manufacturing process.
7.7 Calculate the cost due to misclassification resulting
from measurement variability based on the assigned
incremental costs given in Section 7.6 and the
frequency of occurrence of
α
and
β
errors as follows:
cost = c
fc
α
+ c
pn
β
, (2)
using the appropriate equations for α and β as given in
Section 7.4, depending on the nature of the
specification (LSL only, USL only, or both).
NOTE 6: See Related Information 4 for an example of this
calculation.
7.8 If it is desired to include costs for correct as well as
incorrect classification, calculate the total cost resulting
from measurement variability based on the assigned
incremental costs given in Section 7.6 and the
frequency of occurrence of
α
and
β
errors as follows:
cost = c
pc
(1
−α
) +
c
fc
α
+
c
pn
β
+ c
fn
(1−
β
), (3)
using the appropriate equations for α and β as given in
Section 7.4, depending on the nature of the
specification (LSL only, USL only, or both).