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SEMI E89-1104 E © SEMI 1999, 2004 16 RELATED INFORMATION 4 EXAMPLE OF A MEASUREM ENT SYSTEM ANALYSIS NOTICE : This related information is not an o fficial pa rt of SEMI E89. It was deri ved from task force deliberations …

SEMI E89-1104

R3-2.6 When error terms are known to have

nonnormal distributions, special methods and/or

software (beyond the scope of this guide) are needed.

NOTE 1: The model might also include a wafer-by-day

random error term wd

hki

, if it is suspected that measurements

made from day to day on some wafers might vary differently

than measurements made from day to day on other wafers,

and it is desired to obtain an estimate of this component of

reproducibility.

R3-2.7 Reproducibility includes everything but wafer-

to-wafer true thickness variability, so

222

rpdR



 (R3-2)

R3-2.8

The ANOVA for this MSA provides all the

variance estimates needed to calculate repeatability,

reproducibility and precision. If the wafer term w

random, the ANOVA will also estimate



, which is

useful for characterizing the wafer population

variability.

NOTE 2: When inputting this model into a statistical analysis

program, the factors “day” and “wafer” are said to be crossed,

since every wafer is measured on every day. The factor

“positioning” is nested within wafer and day, and

“repeatability” is the residual error term and is nested within

positioning, wafer and day. A factor “A” is nested within

another factor “B” if the levels or values of “A” are different

for every level or value of “B.”

R3-2.9 This example shows how a fairly complicated

MSA can be set up. Each particular situation can lead

to a different model and experimental design,

depending upon the goals and agreed upon sources of

(possibly significant) variation.

SEMI E89-1104

RELATED INFORMATION 4

EXAMPLE OF A MEASUREMENT SYSTEM ANALYSIS

NOTICE: This related information is not an official part of SEMI E89. It was derived from task force deliberations

during the revision of SEMI E89-0999 in 2001-2003. This related information was approved by full letter ballot

procedures and was approved for publication by the NA RSC on August 16, 2004.

R4-1 Introduction

R4-1.1 The following example illustrates how an MSA

was performed on an automated wafer film thickness

MS for the purpose of determining the reproducibility

and variance components. The MS was calibrated prior

to the MSA, but the presence of bias would not affect

the estimation of variance components as long as the

amount of bias does not change over the course of the

MSA. Different wafer types were selected to span the

range of both possible wafer types and typical

thicknesses. Five standard wafers were manufactured

with characteristics as shown in Table R4-1.

Table R4-1 Wafers Used for Analysis

Film Type

Total Nominal Film

Thickness (Å)

Oxide (wafers 1, 2, 3) 50, 980, 7900

(for wafers 1, 2, 3,

respectively)

Polysilicon over Oxide

(wafer 4)

2,575

Deep UV Resist over

ARC over Oxide

(wafer 5)

13,000

UV  Ultra Violet, ARC  Anti Reflective Coating.

R4-1.2 Wafers were measured on eight days, evenly

spaced over a two-week period. On each day of the

MSA each wafer was chosen twice, loaded each time

into the gauge and measured twice without unloading.

The order of selection was random. Therefore, on a

given day, each wafer was measured exactly four times.

Measurements were taken at three points on the wafers

and averaged. The model (see Related Information 3)

used was

hkji



+ d

+ w

+ wd

+ c

hkj

+ r

hkji

+ e

hij

(R4-1)

where:

hkji

= measurement result (average of measurements

taken at three points each time) on day h,

wafer k, cycle j, and repeat i,



= grand mean of film thickness value across all

conditions,

= effect or contribution associated with the h

day (due to day-to-day stability variation),

= offset from the average due to the true

thickness of the film on wafer k,

= effect or contribution due to wafer-by-day

(wafer  day) interaction,

hkj

= effect or contribution due to cycle j, nested in

wafer k and day h,

hkji

= effect or contribution due to repeat i, nested in

cycle j, wafer k, and day h, and

hij

= residual effect due to all terms not included in

the model.

R4-1.2.1 The variabilities contributed by d

, wd

, and

hkj

are part of reproducibility. The variability

contributed by r

hkji

is the repeatability.

R4-2 Analysis

R4-2.1 Most statistical analysis software programs

report estimates as variances and produce the variance

components analysis shown in Table R4-2. (Methods

of moments estimates were used. Negative variance

components were set to zero). Sources of variation,

such as



and



, are standard deviations found by

taking the square roots of the appropriate variance

component or sum of components.

Table R4-2 Variance Components Estimates — All 5

Wafers

Variance

Component

Estimated

Variance

Estimated

Sigma

Day 12.1348 3.4835

Cycle 0.1256 0.3544

Repeat 0 0.0000

Wafer  Day

462.9869 21.5171

Residual 0.8446 0.9190

R4-2.2 Wafer variability (



wafer

) was not reported

because it is associated with a fixed effect. Variance

components are only estimated for random effects. If,

for example, the five wafers were randomly selected

from a distribution for a single wafer type, wafer

variability would be included in the table and could be

used to estimate the SNR. However, wafer variability

would still not be part of



. For further details, see

Related Information 3.

SEMI E89-1104

R4-2.3 Reproducibility (



) is calculated as the square

root of the sum of the estimates, i.e.,

12.1348 0.1256 0 462.9869 0.8446

21.82







(R4-2)

R4-2.4 Repeatability (



) is estimated by the variance

component corresponding to repeat and is equal to zero.

R4-2.5 Stability is estimated by the square root of the

Day component of the MSA,



stability

= 3.4835.

R4-2.6 Figure R4-1 shows a Pareto chart of the sigmas.

0 5 10 15 20

Sigma

Repeat

Cycle

Residual

Day

Day x W afer

Figure R4-1

Pareto Chart for Effects Sigmas (All Five Wafers)

R4-2.7 Linearity is more difficult to estimate. In

general, it is measured by the differences in variability

associated with the different wafer types. The large

wafer-by-day interaction suggests that there may be a

linearity problem. An examination of the standard

deviations for each group by day (see Figure R4-2)

indicates that Wafer Type 5 (Deep UV Resist) may be

responsible. These standard deviations were calculated

from the four measurements made on every wafer each

day.

R4-2.8

When the variance components analysis was

rerun, excluding Wafer Type 5 data, the new estimates

obtained confirmed that Wafer Type 5 was problematic.

An investigation into the potential cause suggested that

the gauge was degrading the resist at the point of

measurement, causing the readings to decrease over

time.

R4-2.9

If one were interested in measuring only Wafer

Types 1 through 4 (employing a different gauge for

Wafer Type 5, for example), the improved variance

components estimates from Table R4-3 could be used.

R4-2.10 The new estimates for



and



, are 0 and

1.565, respectively. Stability, as measured by day-to-

day variability, improves to 0.6450. The new Pareto

chart of the effects sigmas is shown in Figure R4-3. The

specification range for the product being measured was

given as 10 Å, thus the P/T ratio for the MS for Wafer

Types 1 to 4 is

61.565

100 93.9%





(R4-3)

Table R4-3 Variance Components Estimates for

Wafers 1 – 4

Variance

Component

Estimated

Variance

Estimated

Sigma

Day 0.4160 0.6450

Cycle 0.0203 0.1426

Repeat 0 0

Day x Wafer 1.4918 1.2214

Residual 0.1937 0.4402

012345678

Day

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Standard Deviation

Wafer 1

Wafer 2

Wafer 3

Wafer 4

Wafer 5

Figure R4-2

Wafer Sigmas by Day

0.0 0.4 0.8 1.2

Sigma

Repeat

Cycle

Residual

Day

Day x W afer

Figure R4-3

Pareto Chart for Effects Sigmas

(Wafer Type 5 Removed)