2.
Measurement Process Characterization
2.6. Case studies 2.6.3. Evaluation of type A uncertainty


Purpose of this page  The purpose of this page is to outline an analysis of data taken during a gauge study to quantify the type A uncertainty component for resistivity (ohm.cm) measurements on silicon wafers made with a gauge that was part of the initial study.  
Summary of standard deviations at three levels 
The level1,
level2, and
level3 standard deviations
for the uncertainty analysis are summarized in the table below from the
gauge case study.
Standard deviations for probe #2362 Level Symbol Estimate DF Level1 s_{1} 0.0729 300 Level2 s_{2} 0.0362 50 Level3 s_{3} 0.0197 5 

Calculation of individual components for days and runs 
The standard deviation that estimates the
day effect is
The standard deviation that estimates the run effect is


Calculation of the standard deviation of the certified value showing sensitivity coefficients 
The certified value for each wafer is the average of N = 6
repeatability measurements at the center of the wafer on M = 1
days and over P = 1 runs. Notice that N, M and
P are not necessarily the same as the number of measurements in
the gauge study per wafer; namely, J, K and L.
The standard deviation of a certified value (for timedependent sources
of error), is


The equation must be rewritten to calculate degrees of freedom 
Degrees of freedom cannot be calculated from the equation above because
the calculations for the individual components involve differences among
variances. The table of
sensitivity coefficients for a 3level design shows that for
Then the degrees of freedom can be approximated using the WelchSatterthwaite method. 

Probe bias  Graphs of probe biases  A graphical analysis shows the relative biases among the 5 probes. For each wafer, differences from the wafer average by probe are plotted versus wafer number. The graphs verify that probe #2362 (coded as 5) is biased low relative to the other probes. The bias shows up more strongly after the probes have been in use (run 2).  
How to deal with bias due to the probe 
Probe #2362 was chosen for the certification process because of its
superior precision, but its bias relative to the other probes creates
a problem. There are two possibilities for handling this problem:
The best strategy, as followed in the certification process, is to correct all measurements for the average bias of probe #2362 and take the standard deviation of the correction as a type A component of uncertainty. 

Correction for bias or probe #2362 and uncertainty 
Biases by probe and wafer are shown in the
gauge case study. Biases for
probe #2362 are summarized in table below for the two runs. The
correction is taken to be the negative of the average bias. The
standard deviation of the correction is the standard deviation of
the average of the ten biases.
Estimated biases for probe #2362 Wafer Probe Run 1 Run 2 All 138 2362 0.0372 0.0507 139 2362 0.0094 0.0657 140 2362 0.0261 0.0398 141 2362 0.0252 0.0534 142 2362 0.0383 0.0469 Average 0.0272 0.0513 0.0393 Standard deviation 0.0162 (10 values) 

Configurations Database and plot of differences  Measurements on the check wafers were made with the probe wired in two different configurations (A, B). A plot of differences between configuration A and configuration B shows no bias between the two configurations.  
Test for difference between configurations 
This finding is consistent over runs 1 and 2
and is confirmed by the
tstatistics in the
table below where the average differences and standard deviations are
computed from 6 days of measurements on 5 wafers. A tstatistic < 2
indicates no significant difference. The conclusion is that there is
no bias due to wiring configuration and no contribution to uncertainty
from this source.
Differences between configurations Status Average Std dev DF t Pre 0.00858 0.0242 29 1.9 Post 0.0110 0.0354 29 1.7 

Error budget showing sensitivity coefficients, standard deviations and degrees of freedom 
The error budget
showing sensitivity coefficients for computing the standard
uncertainty and degrees of freedom is outlined below.


Standard uncertainty includes components for repeatability, days, runs and probe 
The standard uncertainty is computed from the
error budget as
\( \displaystyle \large{ u = \sqrt{ \sum_i a_i^2 \cdot s_i^2} = \sqrt{\frac{5}{6} s_2^2 + s_3^2 + \frac{1}{10} s_{probe}^2 } = 0.0388 \,\, \mbox{ohm.cm} } \) 

Approximate degrees of freedom and expanded uncertainty 
The degrees of freedom associated with u are
approximated by the WelchSatterthwaite formula as:
The critical value at the 0.05 significance level with 42 degrees of freedom, from the ttable, is 2.018 so the expanded uncertainty is
