2.6.3.2. Analysis and interpretation

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

2.6.3.2. Analysis and interpretation

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 level-1, level-2, and level-3 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
 Level-1      s₁       0.0729     300
 Level-2      s₂       0.0362      50
 Level-3      s₃       0.0197       5

Calculation of individual components for days and runs

The standard deviation that estimates the day effect is

\( \displaystyle \large{ s_{days} = \sqrt{s_2^2 - \frac{1}{6} s_1^2} = 0.0217 \,\, \mbox{ohm.cm} } \)

The standard deviation that estimates the run effect is

\( \displaystyle \large{ s_{runs} = \sqrt{s_3^2 - \frac{1}{6} s_2^2} = 0.0130 \,\, \mbox{ohm.cm} } \)

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 time-dependent sources of error), is

\( \displaystyle \large{ s = \sqrt{s_{runs}^2 + s_{days}^2 - \frac{1}{6} s_1^2} } \) Standard deviations for days and runs are included in this calculation, even though there were no replications over days or runs for the certification measurements. These factors contribute to the overall uncertainty of the measurement process even though they are not sampled for the particular measurements of interest.

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 3-level design shows that for

N = J, M = 1, P = 1

the equation above can be rewritten in the form

\( \displaystyle \large{ s = \sqrt{\frac{5}{6} s_2^2 + s_3^2} } \)

Then the degrees of freedom can be approximated using the Welch-Satterthwaite 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:

Correct all measurements made with probe #2362 to the average of the probes.
Include the standard deviation for the difference among probes in the uncertainty budget.

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 t-statistics in the table below where the average differences and standard deviations are computed from 6 days of measurements on 5 wafers. A t-statistic < 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.

Error budget for resistivity (ohm.cm)
Source Type Sensitivity Standard
Deviation DF

Repeatability A \( a_1 = 0 \) 0.0729 300

Reproducibility A \( a_2 = \sqrt{5/6} \) 0.0362 50

Run-to-run A \( a_3 = 1 \) 0.0197 5

Probe #2362 A \( a_4 = \sqrt{1/10} \) 0.0162 5

Wiring
Configuration A A \( a_5 = 1 \) 0 --

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 Welch-Satterthwaite formula as:

\( \displaystyle \large{ \nu = \frac{u^4}{ \sum_{i=1}^5 \frac{a_i^4 \cdot s_i^4}{\nu_i} } = 42 } \) where the \( \nu_i \) are the degrees of freedom given in the rightmost column of the table.

The critical value at the 0.05 significance level with 42 degrees of freedom, from the t-table, is 2.018 so the expanded uncertainty is

\( U = 2.018 \cdot u = 0.078 \,\, \mbox{ohm.cm} \)