2.
Measurement Process Characterization
2.6. Case studies


Focus of this case study  The purpose of this case study is to demonstrate uncertainty analysis using statistical techniques coupled with type B analyses and propagation of error. It is a continuation of the case study of type A uncertainties.  
Background  description of measurements and constraints 
The measurements in question are volume resistivities (ohm.cm)
of silicon wafers which have the following definition:
\( X = voltage/current \,\, (ohm) \) \( t = thickness_{wafer}(cm) = 0.628 \,\, cm \) \( K_a = factor_{electrical} = 4.50 \, ohm.cm\) \( F_T = correction_{temp} \approx 1 \,\, C \) \( F_{t/s} = factor_{thickness/separation} \approx 1.0 \) 

Type A evaluations  The resistivity measurements, discussed in the case study of type A evaluations, were replicated to cover the following sources of uncertainty in the measurement process, and the associated uncertainties are reported in units of resistivity (ohm.cm).  
Need for propagation of error 
Not all factors could be replicated during the gauge
experiment. Wafer thickness and measurements required for the scale
corrections were measured offline. Thus, the type B evaluation of
uncertainty is computed using propagation of error. The
propagation of error formula in
units of resistivity is as follows:


Standard deviations for type B evaluations  Standard deviations for the type B components are summarized here. For a complete explanation, see the publication (Ehrstein and Croarkin).  
Electrical measurements 
There are two basic sources of uncertainty for the electrical
measurements. The first is the leastcount of the digital volt meter
in the measurement of X with a maximum bound of
which is assumed to be the halfwidth of a uniform distribution. The second is the uncertainty of the electrical scale factor. This has two sources of uncertainty:
The maximum bounds to these errors are assumed to be halfwidths of
\( \displaystyle s_{Ka} = \sqrt{\frac{0.0001^2}{3} + \frac{0.00038^2}{3}} = 0.000227 \,\, ohm.cm \) 

Thickness 
The standard deviation for thickness, t,
accounts for two sources of uncertainty:


Temperature correction 
The standard deviation for the temperature correction is calculated from
its defining equation as shown below. Thus, the standard deviation for
the correction is the standard deviation associated with the measurement
of temperature multiplied by the temperature coefficient,
C(t) = 0.0083. The maximum bound to the
error of the temperature measurement is assumed to be the halfwidth


Thickness scale factor  The standard deviation for the thickness scale factor is negligible.  
Associated sensitivity coefficients 
Sensitivity coefficients for translating the standard deviations for
the type B components into units of resistivity (ohm.cm) from the
propagation of error equation are listed below and in the error budget.
The sensitivity coefficient for a source is the multiplicative factor
associated with the standard deviation in the
formula above; i.e., the partial
derivative with respect to that variable from the
propagation of error
equation.
\( a_7 = ( \rho / K_a) = 100/4.50 = 22.222 \) \( a_8 = ( \rho / t) = 100/0.628 = 159.24 \) \( a_9 = ( \rho / F_T) = 100 \) \( a_{10} = (\rho / F_{t/s}) = 100 \)


Sensitivity coefficients and degrees of freedom  Sensitivity coefficients for the type A components are shown in the case study of type A uncertainty analysis and repeated below. Degrees of freedom for type B uncertainties based on assumed distributions, according to the convention, are assumed to be infinite.  
Error budget showing sensitivity coefficients, standard deviations and degrees of freedom 
The error budget showing sensitivity
coefficients for computing the relative standard uncertainty of volume
resistivity (ohm.cm) with degrees of freedom is outlined below.


Standard uncertainty 
The standard uncertainty is computed as:
 
Approximate degrees of freedom and expanded uncertainty 
The degrees of freedom associated with u are
approximated by the WelchSatterthwaite formula as:
