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SEMI M56-1103 © SEMI 2003 2 2.8 The output of this practice is the cost com ponent due to measurement variability and bias for each process characteristic and metrol ogy system evaluated. 2.9 This practice does not consi…

SEMI M56-1103 © SEMI 2003 1

SEMI M56-1103

PRACTICE FOR DETERMINING COST COMPONENTS FOR

METROLOGY EQUIPMENT DUE TO MEASUREMENT VARIABILITY

AND BIAS

This practice was technically approved by the Global Silicon Wafer Committee and is the direct

responsibility of the North American Silicon Wafer Committee. Current edition approved by the North

American Regional Standards Committee on September 3, 2003. Initially available at www.semi.org

October 2003; to be published November 2003.

1 Purpose

1.1 In silicon manufacturing product disposition is

typically contingent on metrology equipment output.

For example, a piece of work may be shipped,

scrapped, or reworked based on information from one

or more measured output characteristics. A cost model

provides a mechanism for estimating the costs

associated with operating metrology equipment under

certain sets of assumptions and conditions.

1.2 This practice provides a standard methodology for

metrology equipment suppliers and users to determine

the cost due to misclassification of product because of

measurement variability and bias during testing of the

product for conformance to a specification.

1.3 This practice is intended to be useful for pre-

purchase evaluation of measurement equipment with

different P/T (Precision/Tolerance) 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 practice can help users select the optimum

cost solution based on their specific process capability

and requirements.

1.4 Neither traditional cost elements (e.g., equipment,

maintenance, operational, and staffing cost) nor product

cost data are included in the practice; it is intended that

each user should utilize appropriate data based on the

applicable business model for the facility.

2 Scope

2.1 This practice covers a methodology to determine

the costs associated with misclassification of product

due to variability and bias of measurement equipment.

These costs are associated with failing product that

conforms with specifications and passing product that

does not conform with specifications.

2.2 This practice can be applied to make relative cost

comparisons 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 practice can also be used to compare the relative

costs of a single instrument under different operating

conditions.

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.

2.3 This practice can be applied to any metrology

equipment or to the metrology portion of any

equipment that includes metrology.

2.4 This practice covers the case in which the

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 practice.

2.5 To apply the formulae in this practice, 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.

2.6 To apply the formulae in this practice, the user

must be able to estimate all biases and variances

associated with the 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).

2.7 To apply the methodology of this practice, the user

must estimate the costs of classification and

misclassification due to decisions made as a result of

measurements. Estimates of such costs are obtained

from the business model used to describe the

manufacturing process in which the measurements are

performed. Consequently, different users may arrive at

different estimates of costs because their computations

may contain different elements.

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

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,

— 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,

— the probability of a beta

error, also called the beta error rate.

5.4.5 bias,

— 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.

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,

— 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):

∑

−

)(

where:

n = number of data values,

= value of the i

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

and

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:

uuG

)(

exp

)()(













−

−=Φ=

∫

∞−

πσ

(1)

where:

= bias,

= standard deviation of the measurement

distribution.

NOTE 4: The quantity σ

includes the effects of the change

in bias over the time interval in which σ

has been

established.