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SEMI E35-0305 © SEMI 1995, 2005 13 A1-3 Limitations A1-3.1 Application of this appendix requires that all m easurement variances, biases, and process c h aracteristic distributions be accurately characteri zed. If this i…
SEMI E35-0305 © SEMI 1995, 2005 12
APPENDIX 1
METHODOLOGY FOR DETERMINING ALPHA AND BETA ERRORS
NOTICE: The material in this appendix is an official part of SEMI E35 and was approved by full letter ballot
procedures on December 10, 2004.
A1-1 Purpose
A1-1.1 In silicon manufacturing product disposition is typically contingent on metrology equipment output. For
example, a unit may be shipped, scrapped, or reworked based on information from one or more measured output
characteristics.
A1-1.2 This appendix provides a standard methodology to include the cost due to misclassification of product
because of measurement variability and bias during testing of the product for conformance to a specification.
A1-1.3 This appendix is also intended to be useful for pre-purchase evaluation of pieces of measurement equipment
with different P/T ratios. It is also intended to be useful when comparing the operation of a single measurement
instrument in different throughput modes where P/T ratio changes with the throughput mode utilized or other
equipment setup factors. This appendix can help users select the optimum cost solution based on their specific
process capability and requirements.
A1-2 Scope
A1-2.1 This appendix covers a methodology to determine the alpha and beta probabilities associated with
misclassification of product due to variability and bias of measurement equipment. Alpha probability (α) is
associated with failing units that conform to specifications. Beta probability (β) is associated with passing units that
do not conform to specifications.
A1-2.2 This appendix can be applied to make relative misclassification cost comparisons as part of a COO analysis
between two or more measurement gauges operating under conditions specified by the user. These conditions may
include any aspects of the measurement system affecting the measurement results directly or indirectly (e.g.,
ambient conditions, throughput, recipe). This appendix can also be used to compare the relative costs of a single
instrument under different operating conditions.
A1-2.2.1 A measurement gauge is defined by a specific realization of systems and subsystems required to make
measurements. Gauges that differ in at least one system or subsystem may be considered different for the purpose
of comparison.
A1-2.3 This appendix can be applied to any piece of metrology equipment or to the metrology portion of any piece
of equipment that includes metrology.
A1-2.4 This appendix covers the case in which the piece of metrology equipment is used to make binary decisions
about the item being measured, that is, only two outcomes are possible (e.g., pass/fail, go/no-go, ship/scrap).
Models for decisions with more than two outcomes, such as the binning of data, are beyond the scope of this
Appendix.
A1-2.5 To apply the formulae in this appendix, the user must estimate either the probability density function (PDF)
or cumulative distribution function (CDF) of the process characteristic or characteristics of interest. The
distributions can be based on a theoretical model or on empirical information obtained from manufacturing data.
A1-2.6 To apply the formulae in this appendix, the user must be able to estimate all biases and variances associated
with the piece of equipment being compared under the specified operating conditions and with respect to the process
characteristic(s) of interest. These values can be obtained from a measurement system capability analysis (see SEMI
E89).
A1-2.7 This appendix does not consider costs associated with other components of uncertainty outside of
measurement variability and bias.
SEMI E35-0305 © SEMI 1995, 2005 13
A1-3 Limitations
A1-3.1 Application of this appendix requires that all measurement variances, biases, and process characteristic
distributions be accurately characterized. If this is not done, the resulting calculation may be in error.
A1-3.2 Application of this appendix 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.
A1-3.3 This appendix 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 appendix.
A1-3.4 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 provided in §A1-5.
A1-4 Procedure
A1-4.1 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).
A1-4.2 Unless already known, establish the bias and standard deviation for each measurement gauge to be
compared in accordance with SEMI E89.
A1-4.3 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
(1)
where:
= bias,
M
= standard deviation of the measurement distribution.
NOTE 1: The quantity σ
M
includes the effects of the change in bias over the time interval in which σ
M
has been established.
A1-4.4 Calculate
and
as follows. Note that the symbols in the equations for
and
have the following
meanings:
f(x) = PDF of process characteristic x,
USL = upper specification limit,
LSL = lower specification limit, and
(u) = Gaussian CDF (see Equation (1) in ¶A1-4.3).
A1-4.4.1 For a characteristic with only a USL, use the following equations to calculate
and
:
xxf
xUSL
USL
M
d1
(2)
xxf
xUSL
USL
M
d
(3)
A1-4.4.2 For a characteristic with only an LSL, use the following equations to calculate
and
:
xxf
xLSL
LSL
M
d
(4)
SEMI E35-0305 © SEMI 1995, 2005 14
xxf
xLSL
LSL
M
d1
(5)
A1-4.4.3 For a characteristic with both upper and lower specifications limits, use the following equations to
calculate
and
:
USL
LSL
M
USL
LSL
M
xxf
xLSL
xxf
xUSL
d
d1
(6)
xxf
xLSLxUSL
xxf
xLSLxUSL
USL
MM
LSL
MM
d
d
(7)
A1-5 Extensions of the Methodology to More Complex Situations
A1-5.1 In general, when a single characteristic on a unit is measured once on a single gauge and 100% sampling is
employed, the model will take the form described earlier. If the situation is more complex, the nature of the model
will be different. Factors that can affect the nature of the model include the following:
number of units examined (lot acceptance sampling vs. 100% sampling),
number of times a unit is inspected (single vs. multiple),
effect of the inspection process on the unit (destructive vs. non-destructive),
number of item characteristics examined for a single decision (one vs. many), and
cost functions associated with the business decisions (fixed vs. variable).
A1-5.2 Extension to Multiple Gauges
A1-5.2.1 To extend the model to multiple gauges, one must make the additional assumption that all measuring
gauges are measuring the same characteristic.
A1-5.2.1.1 In addition, define a conforming item as one that all gauges show the measured characteristic to be in
specification.
A1-5.2.1.2 A nonconforming item is taken to be one in which at least one gauge shows the measured characteristic
to be outside of specification.
A1-5.2.2 It is then possible to define a set of
and
error rates for each gauge. Let
1
,
2
, …,
n
be the
values
and
1
,
2
, …,
n
be the
values associated with the n different gauges.
A1-5.2.3 The overall
value,
T
, is calculated from the equation:
T
=
n
i
i
1
1
(9)
where:
xxf
USL
d
, and
f(x) = PDF of the characteristic being measured.
NOTE 2: This equation is based on the probability P that the item is conforming but that one or more gauges give a conforming
result:
T
= P[Item is conforming, 1 gauges show nonconforming]
= P[Item is conforming]