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SEMI E89-1104 E © SEMI 1999, 2004 5 n i i x x n s 1 2 2 ) ( 1 1 (4) The denominator value of n – 1 is used i nstead of n to make the sample variance an unbiased estimator of the population va riance. 5.3.36 si …
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measurement process is in a state of statistical control,
the precision of the process has no meaning. Since the
precision is poorer for greater dispersion of the test
results, specific measures of variability (such as
repeatability and reproducibility) are actually direct
measures of the imprecision of the measurement
process.
5.3.24 precision-to-tolerance (P/T) ratio — ratio of the
precision of a measurement system (MS) to the
tolerance (i.e., absolute magnitude of the full range of
the product specification).
5.3.24.1 Discussion — If the variability associated with
the measurement of a parameter by an MS is very small
compared with the width of the specification range, the
probability of obtaining a test result outside the
specification limits when the value of the parameter
actually lies within the specification limits (or
conversely) is quite small. On the other hand, if the
ratio is too large, the probability of obtaining a false test
result is much greater.
5.3.25 predictor variable — variable that can contribute
to the explanation of the outcome of a designed
experiment. Also called “input variable,” “descriptor
variable,” and “explanatory variable.”
5.3.25.1 Discussion — The term “independent variable”
is not recommended as a synonym due to potential
confusion with independence.
5.3.26 product standard deviation (
Product
)
population standard deviation associated with the
distribution of values of all possible realizations of a
property of an entity manufactured under specified
conditions.
5.3.26.1 Discussion — The product variability may be
estimated by taking a representative sample from the
population and calculating its sample standard deviation
(s
Product
) taking suitable account of MS variation (see
Section 10.2).
5.3.27 reference material — material or substance, one
or more of whose property values are sufficiently
homogeneous and well established to be used for the
calibration of a MS, for the assessment of a measure-
ment method or for assigning values to materials.
5.3.28 repeatability (
r
) — variability associated with
repeated measurements taken under repeatability
conditions.
5.3.29 repeatability conditions — test conditions
involving acquisition of a series of test results with the
same test protocol and MS setup in the same laboratory
by the same operator on the same equipment in the
shortest practical period of time on the same test wafer
without explicit recalibration.
5.3.29.1 Discussion — The acquisition of test data
under repeatability conditions is intended to avoid
influences of long-term drift, operator or MS
differences, material variability, and the like.
Recalibration of the MS is expected to cause
discontinuous differences in test results. However, if
recalibration is required by the test protocol or is
internal to the MS, it is considered to be an allowable
variation in determination of repeatability.
5.3.30 reproducibility (
R
) — variability associated
with the measurement system (MS) when
measurements are made under different (but typical)
conditions.
5.3.30.1 Discussion — Changes associated with
subsystems or test conditions are potential sources of
variation to be estimated. Repeatability is one source of
variation. Other relevant sources of variability may
include time, operator, setup procedure, wafer (of like
variety), measurement location, test instrumentation,
environmental conditions, etc. Although the total
number of contributors to the variance can be
exceedingly large, one typically focuses on a subset that
accounts for a significant portion of the expected MS
variability. For clarity, the selected subset should be
reported together with the reproducibility. If q different
conditions introduce variability into the measurement
independently from one another, the variances add
directly
22
3
2
2
2
1 qR
K
(3)
and they may be separated by the use of judiciously
designed experiments.
5.3.31 response variable — variable representing the
outcome of a designed experiment. Also called “output
variable.”
5.3.31.1 Discussion — The term “dependent variable”
is not recommended as a synonym due to potential
confusion with independence.
5.3.32 root sum of squares (RSS) difference — square
root of the difference of the squares of two numbers.
5.3.33 root sum of squares (RSS) sum — square root of
the sums of the squares of two or more numbers.
5.3.34 sample standard deviation (s) square root of
the sample variance.
5.3.35 sample variance (s
2
) — measure of dispersion
given by the average squared deviation from the mean
for a set of numbers.
5.3.35.1 Discussion — If x
i
is an individual
measurement,
x is the average across all
measurements, and n is the number of measurements,
then
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n
i
i
xx
n
s
1
22
)(
1
1
(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:
2
22
R
RTotal
SNR
(5)
where
Total
2
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:
p
i
jiij
exy
1
(6)
where:
y
j
= j
th
measurement,
p = number of input variables,
x
i
= i
th
input variable,
i
= its corresponding coefficient, and
e
j
= error associated with the j
th
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 (
2
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.