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SEMI E89-1104 E © SEMI 1999, 2004 6 6.1.6 Once the reproducibility is determined over the desired condi tions, seco ndary m e trics, such as P/ T ratio and SNR can be easily calculated. 6.2 A second major goal of an M SA…

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







)(

(4)

The denominator value of n – 1 is used instead of n to

make the sample variance an unbiased estimator of the

population variance.

5.3.36 signal–to-noise ratio (SNR) — ratio of the

variation in the manufactured product to the precision

of the measurement system (MS).

5.3.36.1 Discussion — Because it is difficult to directly

measure the standard deviation of the product without

including variation due to the measurement instrument,

SNR is generally defined as:

RTotal

SNR





 (5)

where



Total

is an estimate of the total population

variance obtained from appropriate measurements of a

large, representative sample of the product.

5.3.37 stability — absence of additional variability due

to taking measurements over time (typically several

days or longer).

5.3.38 statistical model  mathematical function

relating one or more variables to known and

measurable inputs plus one or more unknown stochastic

(error) terms.

5.3.38.1 Discussion — A statistical model consists of

three parts. The first part is the response variable that is

being modeled. The second part is the deterministic or

the systematic part of the model that includes predictor

variables. Finally, the third part is the random error or

stochastic part of the model, which can be quite

elaborate. An example of a statistical model is:







jiij

exy

(6)

where:

= j

measurement,

p = number of input variables,



= i

input variable,



= its corresponding coefficient, and

= error associated with the j

measurement.

In many cases the error distribution(s) are specified

before the model is fit (e.g., as normal).

5.3.39 tolerance — absolute magnitude of the full

range of the product specification.

5.3.40 total variance (

Total



)  sum of the product

variance and the square of the reproducibility.

5.3.41 uncertainty — parameter, associated with a

measurement, that characterizes the dispersion of

values that can be reasonably attributed to the object

being measured.

5.3.41.1 Discussion — Two types of measurement

uncertainty are:

 Type A: uncertainty components evaluated by

statistical methods and

 Type B: uncertainty components evaluated by

other than statistical methods.

5.3.42 upper specification limit (USL) — value of an

attribute above which a product is said to be

nonconforming.

5.3.43 variable  quantitative or qualitative

characteristic of an object, processes, or state that may

take on more than one value.

5.3.43.1 Discussion — When the values occur

unpredictably, it is a random variable.

5.3.44 variance — population variance (see Section

5.3.22).

6 Goals of and Preliminary Steps in Planning

an MSA

6.1 One major goal of an MSA is to determine MS

reproducibility under the desired operating conditions.

This may range from a simple repeatability study, to a

complex determination of the relative contributions of

many sources of MS variability using factorial

experiments and analysis of variance (ANOVA). Some

of the typical types of variability-determination goals to

be met by an MSA include the following:

6.1.1 Determination of repeatability.

6.1.2 Determination of the effect of loading and

unloading the wafer samples between measurements.

6.1.3 Determination of stability.

NOTE 3: Stability may also have a bias component that can

be confounded with the variability component of stability (see

Section 6.2.3).

6.1.4 Assessment of the multiple factors that affect the

MS variability, which requires the use of factorial

experiments and ANOVA.

6.1.5 Assessment of the largest sources of variability in

order to focus improvement efforts on the most critical

sources. This identification can be made from the

analysis of a properly designed and executed factorial

experiment.

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6.1.6 Once the reproducibility is determined over the

desired conditions, secondary metrics, such as P/T ratio

and SNR can be easily calculated.

6.2 A second major goal of an MSA is to determine

bias and bias-related considerations. In general, bias

studies require standards or “golden wafers” with

“known” true measurement values or agreed-upon

consensus values. Corrective actions ranging from

simple recalibration to repair of the MS may be

required as a result of the bias determinations. Some of

the typical elements of a bias study include:

6.2.1 Determination of bias.

6.2.2 Determination of linearity or whether bias is

constant over the range of values that the metrology

equipment will be used.

NOTE 4: Linearity also may have a variability component.

6.2.3 Determination of the stability or whether the bias

is constant over time.

NOTE 5: Stability also may have a variability component

(see Section 6.1.3).

6.2.4 Determination of matching tolerance.

NOTE 6: In the absence of certified or other reference

materials, as required for a determination of bias, matching

tolerance may be determined by measuring the same set of

test materials on the two MSs whose matching is being

evaluated. Although this alternate method does not provide a

measure of the value of bias, the difference between the mean

values measured on the two MSs is the same as the difference

in bias.

6.3 Before undertaking an MSA, determine and list the

specific goals to be attained by the MSA to be

performed.

6.4 It must be emphasized that all of the procedures in

this guide yield meaningful estimates only on

instruments that are in statistical control. Therefore the

first step in analyzing the capability of any MS is to

establish that it is in statistical control through control

charts or other accepted quality control techniques.

Detailed discussion of these techniques is outside the

scope of this guide (see Section 3.1).

6.5 It is also useful to review prior studies on the MS to

be analyzed to determine (1) what sources of variability

were significant and (2) previously obtained values for

quantities such as bias, repeatability, and

reproducibility.

7 Procedure to Determine Reproducibility

7.1 Identify all the factors of interest that contribute to

non-negligible sources of variability in the MS. These

factors should be made explicit before an experiment

can be appropriately designed or accurately analyzed.

The most common sources include:

 Repeatability

 Load-unload operation

 Time over intervals of day(s) or week(s)

 Wafer handling hardware, including ability to place

the measurement probe at the same location on the

wafer

 Measurement software, including type, revision, or

both

 Setup procedure

 Environmental conditions, including temperature

and humidity

 Shift (may be confounded with environmental con-

ditions or time or both)

 Magnitudes of attribute(s) of test wafer(s) under

investigation

 Operator (usually not significant for automated

MSs)

NOTE 7: The factors listed are all included under the term

“reproducibility.” There is no common agreement as to what

factors should be included in reproducibility. Therefore the

selection and recording of the factors included in a particular

MSA are especially critical.

NOTE 8: Variability associated with factors independent of

the MS, such as true value differences in the item being

measured, should not be included in reproducibility. Given

the design of the MS, it may be necessary to estimate these

sources of variability before they are removed from total

variability. If such factors are included in the MS, as for

example, when different sites on a wafer are measured,

variability associated with these factors should be estimated

and removed from total MS variability. Although variability

due to the measurement process physically altering the

measurand (i.e., the interaction between MS and wafer)

should be included in reproducibility, consideration of this

topic is beyond the scope of this guide (see Section 3.3).

7.2 Identify all factors outside of the MS for which an

interaction with one or more factors within the MS may

exist. Treat these factors as fixed factors and the

interactions as random factors.

NOTE 9: Typically, the only fixed factor considered is

related to the measured object, where making measurements

at different locations, for different true values, or on different

wafer types may be influenced by certain aspects of the MS.

7.3 Identify the nested factors and the factors in which

they are nested. Several levels of nesting may exist for

a given factor.

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NOTE 10: Nesting occurs when the measurement procedure

causes a natural hierarchy of events to exist. For example, if

repeatability is estimated by taking measurements with the

wafer fixed in the gauge, and variability due to the load-

unload operation is estimated by taking a second set of

measurements after the wafer has been removed and reloaded,

repeatability is nested in load-unload. The nesting occurs

because each set of repeated measurements is unique to the

load in which they appear. Nesting may also occur if a

physical hierarchy exists, such as when both measurement

locations (sites) and wafer type are fixed factors in an MSA

(sites being nested in wafer type).

7.4 Select appropriate test wafers for the MSA.

7.4.1 In the most general case, many types of wafers

should be used. These should be representative of the

types and levels to be measured in practice by the MS

under evaluation.

7.4.2 The test wafers should be as uniform as possible

to prevent confounding of handling, loading, or

positioning factors, if these are chosen to be examined.

7.4.3 It is necessary to use only one test wafer of each

type selected.

7.4.4 Use the same test wafers throughout the MSA.

7.5 Prepare a statistical model for the MSA in order to

plan and analyze it properly.

NOTE 11: Statistical models are discussed in Related

Information 3. Eastman

gives some examples of models for

reproducibility studies.

7.6 Select the measurement protocol. Include several

examples of every factor that was selected in Section

7.1.

7.7 Select and implement the appropriate MS setup.

NOTE 12: Usually, only a single setup is used in an MSA to

determine reproducibility. However, if setup procedure has

been chosen as a factor to be evaluated, select the various

setups to be used in the MSA.

NOTE 13: Examples of MSAs are given in Related

Information 4, 5, and 6. Additional examples may be found

in the literature.

7.8 Determine if the MS needs to be calibrated.

7.8.1 Calibrate, if necessary, following the

manufacturer’s recommended procedure.

4 Eastman, S. A., “Evaluating Automated Wafer Measurement

Instruments,” International SEMATECH technology transfer

document 94112638A-XFR (February 28, 1995), Section 5 and case

studies in appendices. However, note that neither the terminology nor

the nomenclature in this document is identical with that given in this

guide. A PDF file of this report can be downloaded from

International SEMATECH’s public web site at

http://www.sematech.org/docubase/wrappers/26.htm

7.8.2 Do not recalibrate during the MSA unless

required by the normal operating procedure.

7.9 Perform the MSA, randomizing both the sample

wafers chosen and the order in which measurements are

made, whenever possible. Record the actual order in

which all measurements are made.

7.10 To correctly estimate the reproducibility and

individual sources of variance, conduct a variance

components analysis.

7.10.1 If the experiment has been designed to include

multiple repeated measurements taken under the same

(repeatability) conditions, it is possible to estimate

repeatability. Without multiple repeated measurements,

repeatability will be confounded with other sources of

variability.

7.10.2 Other factors are those from Section 7.1 that

have been included in the experiment design.

7.10.3 Reproducibility is the square root of the sum of

the individual variance components (see Equation 3 in

Section 5.3.30.1).

NOTE 14: The exact nature of the variance components

analysis depends on the design of the experiment conducted.

For details, see the examples in Related Information 4, 5, and

6, Box, Hunter, and Hunter,

or Montgomery.

In designing

and performing a complete designed experiment and the

associated analysis, it is strongly recommended to utilize

some form of commercially available statistical analysis

software. If such a product is used, it is advisable to confirm

that the assumptions made in the software are congruent with

those in this guide.

8 Procedure to Determine Bias, Including

Linearity, Stability, and Matching Tolerance

8.1 Obtain J appropriate reference materials, preferably

CRMs, where J is between 3 and 10 so that the entire

range of interest of the parameter is covered. If CRMs

are not available, use wafers with well accepted

consensus values covering the range of values of the

attributes to be measured. The reference materials

should have characteristics similar to the wafers to be

measured by the MS being evaluated or their values

should be transferable to such wafers.

NOTE 15: Bias cannot be determined if suitable reference

materials are not available. In such situations, it is necessary

to resort to correlation experiments to establish the systematic

errors between different MSs. If the bias is not known, these

systematic errors may confound the estimated reproducibility.

8.2 Select and implement the appropriate MS setup.

5 Box, G. E. P., Hunter, W. G., and Hunter, J. S., Statistics for

Experimenters (Wiley, New York, 1978).

6 Montgomery, D.C., Design and Analysis of Experiments, 5th Ed.

(Wiley, New York, 2000).