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SEMI M32-0998 © SEMI 1998, 2004 10 A4-3.2 No control m ethod can hold a process per f ectly still. Even the best real-time control methods can allow a process to drift as m u ch as ± 1.5 si gma. For this reason, a 6-sigm…

SEMI M32-0998 © SEMI 1998, 2004 9

500

1000

1500

2000

2500

0.0 1.0 2.0 2.9 3.9 4.8 5.8 6.7

Figure A4-2

Normal vs. Lognormal Shape

A4-1.7 Statistical specifications quantify the quality

level so it is not an abstract assumption left to

conjecture. This allows both the user and supplier to

optimize their control systems to that level. Capability

based sampling is one example. Simply put, this means

to use process capability information to help define the

sampling plan for a process. A detailed discussion of

this subject is beyond the scope of this document, but

the basic concept is easy to understand. If a process is

very capable and controlled it does not need as much

sampling as one that is not. Implementing this strategy

is a big step ahead of standard sampling theory. It might

be the only way to rigorously transfer from inspection

based quality to process control based quality, without

incurring high risks. Statistical specifications create the

environment which facilitates this improved strategy.

Hopefully, the quality levels would be defined in ppm,

but any level agreed upon by both the user and the

supplier would work.

A4-2 Tolerance Specifications

LSL to USL, or Target ± xx

A4-2.1 The main problem with a conventional

tolerance is that many people perceive it as a goal post

that requires 100% of the material to fall inside. This

document has shown in detail that this is simply a

wrong perception. Stating the quality level in the

specification, instead of just implying it, corrects this

problem.

A4-2.2 Another common misconception is that

tolerances and tail probabilities divert attention away

from centering the process. Of course, centering is

important. Considering the impact of poor quality on all

customers, it is clear that a well centered process is far

more efficient than one that is not. It is also generally

true that the suitability of the product is not

significantly different immediately on one side of a

tolerance line compared to the other. However, there

are a number of issues that warrant discussion.

A4-2.2.1 First, centering is usually not difficult, so it

usually does not need a major emphasis. Those who are

experienced in quality improvement techniques would

certainly not forget to center the process. Simply

monitoring the mean trend gives an excellent measure

of centering, and it keeps the focus on the process

rather than the specification. Industry carries a long

history where specifications prevented the attainment of

good process control. Many of the issues have been

explained earlier in this document. To summarize,

process controls are the only way to truly maintain a

centered process, and these controls can only work if

there is enough tolerance in the specification to

accommodate the sensitivity limitations of the control

methods.

A4-2.2.2 The second issue is that it might not be

possible to center. This is clearly the case in one-sided

processes that are bounded by zero. Optimizing

multiple parameters might also dictate that some

parameters get worse.

A4-2.2.3 The final, and most important, issue is the

challenge of teaching statistical methods to the factory

population. It is very easy to teach averages and

centering since they tend to be well understood already.

It is not as easy to teach sigma and the fact that sigma

trends must be thoroughly understood before much of

anything can be said about averages. Confirming that

the R or S chart is controlled before studying the Xbar

chart and doing an F-test before doing a t-test are just

two examples. These are very important concepts that

must be fully understood in order to apply statistical

methods. Over-stressing centering de-emphasizes sigma

and creates an environment where it is not appreciated.

Thinking in terms of sigma is the most important

concept that most people will learn about process

control methods. It deserves a major emphasis.

A4-2.2.4 Of course, centering would be a concern if

mixing and matching of lots is the control method, but

this can be easily spotted in the statistical

characterization process. The fact that some of the

alternatives to tolerances are much more likely to cause

this practice will be further explained in Paragraphs A4-

3 and A4-4.

A4-3 Multiple Distributional Percentile

Specifications

≤

A, %

≤

B, and % ≤ C

A4-3.1 Errors in the 10% or higher range can be

generated if a specification in the central portion of the

distribution is used. An example of such a specification

is “50% ≤ 1.2 microns”.

SEMI M32-0998 © SEMI 1998, 2004 10

A4-3.2 No control method can hold a process perfectly

still. Even the best real-time control methods can allow

a process to drift as much as ± 1.5 sigma. For this

reason, a 6-sigma design was defined as 4.5 sigma to

the nearest tolerance.

A4-3.3 Sampling issues are a major concern. Many

kinds of variation can be quantified by rigorously

studying the appropriate samplings (within batch

samples for batch-to-batch variation, monthly samples

for month-to-month variation, etc.). However, there are

practical limits on how many kinds of variation can be

continuously monitored. There is a large amount of

work involved, and there are hundreds of potential

sources of variation. Resource restrictions regularly

force suppliers to concentrate on the known key

variables and leave the rest as error in the system (at

least for now). As sample periods get longer, there is

more statistical sensitivity, but also more probability

that the drift will be confounded with user-driven

process improvements. For example, month-to-month

random drift is hard to quantify because production

processes are often improved before many months of

data can be observed. Measurement error is another

main contributor which further complicates the other

sensitivity issues. These factors all combine to allow a

certain amount of undetectable process drift. This

process drift needs to be considered when defining the

specification approach.

A4-3.4 These variations cannot be detected because of

statistical sensitivity limitations. Table A4-1 shows the

sample sizes needed to detect different levels of process

shift. Delta will be defined as the number of sigma of

undetectable mean drift. For simplicity, a one-way shift

is shown, but in most practical cases, the shift could go

plus or minus. Using the old standard of 5% supplier

risk and 10% user risk, 72 samples are needed to detect

a delta of 0.25 sigma. This example (0.25 sigma of

mean drift) has been used throughout this document.

Since most of the discussion will relate to process drift,

the terms “drift” and “shift” will be used

interchangeably. Clearly, 72 is too large a sampling for

most real-time control systems, but it is still an

understatement of the problem. Most users currently

expect their risk to be much smaller than 10%. The last

row shows that the sample size would need to be 311 to

reach 0.1% user and supplier risks. It is generally

agreed that 0.1% (i.e., 1000 ppm) is the lowest level

that is feasible using standard probability theory. For

lower levels, it is necessary to move farther out on the

distribution tails where it is necessary to rely on more

than just stndard sampling probabilities.

Table A4-1 Sampling Requirements

(a = supplier risk, b = user risk)

Detectable

Mean Shift

(in Sigma)

0.125 0.25 0.50 0.75 1.0

Min N

a = 20%

b = 20%

92 24 7 4 4

Min N

a = 5%

b = 10%

280 72 20 10 7

Min N

a = 1%

b = 1%

696 177 47 23 15

Min N

a = 0.1%

b = 0.1%

1228 311 82 40 25

A4-3.5 Figures A4-3, A4-4, and A4-5 show the error

generated by 0.25 sigma of undetectable mean drift.

Again, the number of sigma of undetectable mean drift

will be defined as delta. Notice how the error grows as

the point of interest moves toward the center of the

distribution. At the four sigma point (out on the tail)

there is very little error, so suppliers can be generous

with internal buffer specifications to protect the user.

This is not as feasible in the center of the distribution

because there is so much material at stake. A percentile

specification in the center of the process shown would

have about 1000 times more undetected, out-of-spec

product than a specification at the 4 sigma point. Since

the shift could go either way, this could also be 1000

times more material that was rejected and should not

have been.

100

8 1013151820232628

Figure A4-3

Error at 0 Sigma

A4-3.6 The left curve in Figure A4-3 shows a

distribution with an expected mean of 18. The right

curve shows the same distribution with an undetectable

drift of 0.25 sigma. If a specification limit is placed at

18, then 50% of the material is below the specification

for the nominal distribution. However, only 40.13% is

below the specification for the distribution that has

drifted. The error, shown by the area between the

vertical lines and below the right curve, is 9.87% or

98,700 ppm.

SEMI M32-0998 © SEMI 1998, 2004 11

100

26 28

Figure A4-4

Error at 2 Sigma

A4-3.7 The error when the specification is 2 sigma

from the mean, shown in Figure A4-4 as the area

between the vertical lines and below the right curve, is

1.73% or 17,300 ppm.

100

26 28

Figure A4-5

Error at 4 Sigma

A4-3.8 The error when the specification is 4 sigma

from the mean, shown in Figure A4-5 as the area

between the vertical lines and below the right curve

(but indistinguishable), is 0.01% or 100 ppm.

A4-3.9 Figure A4-4 further quantifies this error.

Percent-less-than calculations are shown for each curve

at increments of 1 sigma from the mean. The starting

mean is 18, and the sigma is 2.

100

26 28

Value

Description

Left

Curve %

Right

Curve %

Error

10 - 4 s 0.01 0.00 0.01

12 - 3 s 0.13 0.06 0.07

14 - 2 s 2.28 1.22 1.06

16 - 1 s 15.87 10.56 5.31

18 mean 50.00 40.13 9.87

20 + 1 s 84.13 77.34 6.79

22 + 2 s 97.72 95.99 1.73

24 + 3 s 99.87 99.70 0.17

26 + 4 s 100.00 99.99 0.01

Figure A4-6

Percentage Error

A4-3.10 Please note that this is a very conservative

example. The delta and the resulting error can get much

larger.

A4-3.11 A major concern is what to do when the

specification is not met. Users would certainly not want

suppliers to reject material in the center of the

distribution. Alternatives are to (1) reject the whole

shipment, (2) reject nearby lots, (3) mix and match lots,

or (4) translate the percentage hypothetically lost in the

center back out to the tail in order to reject the worst

material. The first two alternatives are really more like

penalties and would lead to unnecessary loss of good

material. The third is the worst form of inspecting in

quality. It is generally agreed that this is the main thing

to avoid. Translating the error back to the tail is

possible but not necessarily a good approach. This

would involve translating a reject percentage with a

large potential error back out to a region that would

have had a very small error. A multiple percentile

specification can also be quite difficult in a production

environment. Each of the percentile criteria must be

evaluated to determine which one requires the highest

reject rate. That reject rate must then be translated back

to the tail, and 100% sample data must be used to

determine which actual pieces to reject. If the statistical

distribution is defined, one specification would give the

same amount of information with much less work. If

only one criterion is used, and it is well out on the tail

(hopefully in the ppm range) then this method is very

similar to a statistical specification.

A4-3.12 A commonly stated goal of multiple percentile

specifications is to focus on centering rather than

distribution tails. All of the discussion in Section A4-

2.2 concerning centering is equally applicable here.

Multiple percentile specifications are more likely to

cause inspected-in quality which detracts from process

control based centering.

A4-3.13 Ease of use is another stated goal, but for all

of the reasons stated above, this method is one of the

most difficult to use. Statistical calculations can be

done in milliseconds while the potential of large

samplings, raw data manipulation, and duplicate effort

associated with multiple distributional percentile

specifications create much more work. Characterizing

with percentiles is an alternative for processes that have

not yet been statistically characterized, but once that

hurdle is crossed, everything gets many times easier.

A4-4 Mean and Sigma Specifications

Mean

≤

Value, Sigm

≤ Value, o

Low to High Mean, Sigma ≤ Value