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SEMI M56-1103 © SEMI 2003 9 R3-6 For a p rocess characteristic with both uppe r and lower specifications lim i ts, the formulae for α and β are: () ( ) ( ) () ( ) ( ) LSL USL G USL x G L SL x f x dx G USL x G LSL x f x d…

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SEMI M56-1103 © SEMI 2003 8
RELATED INFORMATION 3
BACKGROUND OF THE METHODOLOGY
NOTICE: This related information is not an official part of SEMI M56. It was derived from task force
deliberations during the development of the document. This related information was approved for publication by
full letter ballot procedures on September 3, 2003.
R3-1 The model in this practice is used to assign
probabilities (risks) to specification-related events. In
general, there are two probabilities that are of interest.
The first is the probability of getting a measurement
outside of the specification range for a conforming
item. This is often called the producer’s risk because
conforming product would be needlessly rejected in this
situation. The second probability is that of getting a
measurement within specifications for a non-
conforming item. Consumer’s risk is the term applied
to this probability, since there is a chance of non-
conforming product being accepted.
R3-1.1 For the purposes of this standard, the event of
rejecting an item whose true value lies inside of the
specification limits is considered a Type I error with
error rate (probability)
α
.
R3-1.2 The event of accepting an item whose true
value lies outside of the specification limits is
considered a Type II error with error rate (probability)
β
.
R3-1.3 One of the difficulties in formulating the
statistical model is that there are two or more
distributions involved, possibly in an interdependent
fashion. Given a true underlying (fixed) value, x, a
single measurement of that value produces the obser-
vation Y = x + U, where U is a random and unobs-
ervable value attributable to measurement variability.
If one takes n measurements of the same value x, i.e.,
Y
i
= x + U
i
, where i = 1,…, n, Y and U are treated as
random variables from the observation and
measurement variability distributions, respectively.
The situation is further complicated if several x values
are measured. In this situation, the x values come from
the process (or characteristic) distribution X such that
Y
ij
= X
j
+ U
ij
, where i = 1,…, n and j = 1,…, m. The
relationship between X and U is known as a
convolution. Additional distributions may be involved
if one assumes that certain characteristics of the
measurement process do not remain constant over the
measurement range. For example, one may assume that
bias is a function of the measured value. To model this
a third distribution would be needed, but the scope of
this practice is limited to the use of only two
distributions (characteristic and measurement
variability).
R3-2 The symbols in the equations derived for
α
and
β
have the following meanings:
X = random value attributable to the process
characteristic,
U = random and unobservable value attributable
to measurement variability,
f (x) = process characteristic PDF,
G(u) = measurement variability CDF,
USL = upper specification limit,
LSL = lower specification limit, and
Φ(u) = Gaussian CDF.
R3-3 The process characteristic PDF and measurement
variability CDF are assumed to be independent of each
other.
R3-4 For a process characteristic with only a USL:
R3-4.1 Define
α
as a joint probability:
[
]
()()
,
1
USL
PX USLX U USL
GUSL x f xdx
α
−∞
=≤ +>
=− −


∫
R3-4.2 Similarly, define
β
as:
[
]
()()
,
USL
PX USLX U USL
GUSL x f xdx
β
∞
=> +≤
=−
∫
R3-5 For a process characteristic with only an LSL, the
formulae for
α
and
β
are:
()()
LSL
GLSL x f xdx
a
•
=-
Ú
()()
1
LSL
GLSL x f xdx
b
-•
È˘
=- -
Î˚
Ú
SEMI M56-1103 © SEMI 2003 9
R3-6 For a process characteristic with both upper and
lower specifications limits, the formulae for
α
and
β
are:
()()()
()()()
LSL
USL
G USL x G LSL x f x dx
G USL x G LSL x f x dx
b
-•
•
È˘
=---+
Î˚
È˘
-- -
Î˚
Ú
Ú
()()
()()
1
USL
LSL
USL
LSL
GUSL x f xdx
GLSL x f xdx
a
È˘
=- - +
Î˚
-
Ú
Ú
R3-7 To use the above functions in practice, additional
assumptions must be made:
R3-7.1 Equipment bias is treated as constant
throughout the measurement range. This is needed to
avoid a third function in the integral.
R3-7.2 In general, the distribution of the measurement
variability is assumed Gaussian (or normal), with mean,
µ
, and variance,
σ
2
. For measurement gauges,
µ
=
δ
,
the bias, and
σ
2
=
σ
M
2
, the measurement system
variance.
R3-7.3 The distribution of the parameter of interest has
been characterized. This may be done by taking
multiple measurements at each point in the
measurement range and averaging, or by deconvolving
F(x) and G(u).
R3-8 Given the assumptions above about the mea-
surement system, the following relationships apply:
R3-8.1 When only a USL is given, the formulae for
α
and
β
are:
∫
∞−
+−
Φ−=
USL
M
xxf
xUSL
d)(1
σ
δ
α
∫
∞
+−
Φ=
USL
M
xxf
xUSL
d)(
σ
δ
β
R3-8.2 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
R3-8.3 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
R3.9 The equations used in Section 7.4 are those in
Sections R3-8.1 through R3-8.3.
SEMI M56-1103 © SEMI 2003 10
RELATED INFORMATION 4
EXAMPLE OF CALCULATION ACCORDING TO PROCEDURE
NOTICE: This related information is not an official part of SEMI M56. It was derived from task force
deliberations during the development of the document. This related information was approved for publication by
full letter ballot procedures on September 3, 2003.
R4-1 This example relates to the calculation of the cost
components due to misclassification caused by
measurement variability in flatness measurement during
wafer manufacturing. The following premises are
made:
R4-1.1 A wafer manufacturer must supply a customer
with a specific number of wafers that conform to a site
flatness specification of 0.13 µm
, SFQR as defined in
SEMI M1.
R4-1.2 Most wafers produced do conform to the
specification. Due to variability in the manufacturing
process, some wafers do not conform to the
specification.
R4-1.3 The flatness measurement step is near the end
of the wafer manufacturing process. It follows surface
polishing and precedes final cleaning, surface
inspection, and packaging.
R4-1.4 All of the wafers are measured for site flatness.
The measurement data is used to determine if wafers
are passed for further processing and shipment or if
they are failed and scrapped without further processing.
This example premises no rework for failed wafers.
R4-1.5 The customer is always able to detect a non-
conforming wafer. When a non-conforming wafer is
detected, it is returned to the manufacturer and a
replacement wafer must be supplied to satisfy the
specified order quantity.
R4-2 The following steps follow the procedures
outlined in Section 6.
R4-2.1
Estimation of Process Distribution — A total of
17,698 observations was made on flatness
measurements. An empirical estimate of the flatness
CDF was calculated from these data and is shown in the
column of Table R4-1 labeled F(x).
R4.2.2 Estimation of Bias and Variance for Each
Gauge Compared — Two gauges are available. Gauge
A has zero bias and a measurement variability of
σ
A
= 0.00433, which results in a P/T ratio of 10%,
calculated in accordance with SEMI M27 for a single
sided distribution. Gauge B also has zero bias, but has
a measurement variability of
σ
B
= 0.00867 or a P/T
ratio of 20%. Let the distribution of the measurement
variability for the two gauges be normally distributed
with a mean of zero and standard deviations given
above.
R4-2.3 Calculation of
α
and
β
— To calculate
α
and
β
,
one needs to estimate
f(x) and Φ(z), where
z = (USL
−
x)/
σ
M
.
The values in the column labeled f(x) are derived by
dividing the value in the count column by the total
number of observations. σ(z) is calculated from the
normal CDF using the numbers in the first column as x,
0.13 as the USL, and either σ
A
= 0.00433 or
σ
B
= 0.00867 for σ
M
, depending on whether Gauge A or
Gauge B is used. To calculate
α
, for each row multiply
f(x) by 1
−
Ф(z) and sum over all rows at or below the
specification limit. To calculate β, for each row
multiply f(x) by Ф(z) and sum over all rows above the
USL. The calculations for
α
and
β
for the two P/T
ratios of 10% and 20% can also be found in Table R4-1.
R4-2.4 Decision Model — One of four situations may
occur depending on the true flatness of the wafer,
conforming or nonconforming, and the decision made
based on the output from the metrology equipment,
pass or fail. Two correspond to correct actions: passing
a conforming item and failing a nonconforming item.
Zero incremental cost is associated with correct
classification. The two situations associated with
misclassification, passing a nonconforming item and
failing a conforming item, have actual incremental
costs.
R4-2.5
Estimation of Costs — For purposes of
illustration, let the cost of failing a conforming wafer
(c
fc
) be $300 and the cost of passing a nonconforming
(c
pn
) wafer be $220. The cost of ownership due to mea-
surement variability and bias can then be calculated as:
cost = c
fc
P(fail,conforming)+c
pn
P(pass, nonconforming)
= c
fc
α
+ c
pn
β
R4-2.6 Calculation of Costs — The cost components
due to measurement variability for Gauge A and Gauge
B are:
cost
A
= c
fc
α
A
+ c
pn
β
A
=
(300)(0.01042530) + (220)(0.00011698) = 3.1533
cost
B
= c
fc
α
B
+ c
pn
β
B
=
(300)(0.01428055) + (220)(0.00145950) = 4.6053