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Yield Prediction by Sampling

Critical area estimation (and hence yield prediction) by sampling is based on the statistics of survey sampling [1]. Properties of a population can be estimated, with bounds on the error of estimation, by taking a number of random samples from the population. For example, the estimate of the whole population mean, $ \mu$ is given by the sample average $ \overline{{y}}$,
$\displaystyle \hat{{\mu}}$ = $\displaystyle \overline{{y}}$ = $\displaystyle {\frac{{\displaystyle \sum_{ i=1}^{n} y_{i}}}{{ n}}}$ (1)
The estimated variance of $ \overline{{y}}$ is,
$\displaystyle \hat{{V}}$($\displaystyle \overline{{y}}$) = $\displaystyle {\frac{{s^2}}{{n}}}$$\displaystyle \left(\vphantom{\frac{N-n}{N}}\right.$$\displaystyle {\frac{{N-n}}{{N}}}$$\displaystyle \left.\vphantom{\frac{N-n}{N}}\right)$ (2)
where N is the population size, n is the number of samples and s2 is the sample variance,
s2 = $\displaystyle {\frac{{\displaystyle \sum_{ i=1}^{n} (y_{i}-\overline{y})^2}}{{ n-1}}}$
The bound on the error of estimation is,
2$\displaystyle \sqrt{{\hat{V}(\overline{y})}}$ = 2$\displaystyle \sqrt{{\frac{s^2}{n}\left(\frac{N-n}{N}\right)}}$ (3)
An IC chip layout can be viewed as a population which has regions with different susceptibilities to defects. By randomly sampling the chip layout, an estimate of the critical areas for the whole chip can be obtained. Survey sampling is particularly suited to IC yield prediction since, for large populations, the error bound on estimates does not depend on the population size but on the variance of the population. This implies that the number of samples required to characterise very large chips does not increase with chip area or even its complexity but only with the variation of the fault sensitivity over the chip. It is this property that enables sampling to be used for even the largest layouts, as the variation of layout fault sensitivity within a chip is not related to its size.

Stratified Sampling

In many cases an improved method of estimating population parameters can be obtain using stratified random sampling [1]. A stratified sample is generated by separating the population into a number of non-overlapping regions, called strata. A simple random sample, as described above, is then selected from each region. Stratified sampling can increase the accuracy of population estimates where the selected strata have less variance than the population as a whole.

The estimator of the whole population mean, $ \mu$ for a stratified survey is given by the stratified sample average $ \overline{{y}}_{{st}}^{}$,

$\displaystyle \hat{{\mu}}$ = $\displaystyle \overline{{y}}_{{st}}^{}$ = $\displaystyle {\frac{{1}}{{N}}}$$\displaystyle \sum_{{ i=1}}^{{L}}$Ni$\displaystyle \overline{{y}}_{{i}}^{}$ (4)
the estimated variance of $ \overline{{y}}_{{st}}^{}$:
$\displaystyle \hat{{V}}$($\displaystyle \overline{{y}}_{{st}}^{}$) = $\displaystyle {\frac{{1}}{{N^2}}}$$\displaystyle \sum_{{ i=1}}^{{L}}$Ni2$\displaystyle \hat{{V}}$($\displaystyle \overline{{y}}_{{i}}^{}$) (5)
where Ni is the population size of the ith of L strata.

This technique can be usefully applied to the estimation of IC critical areas by dividing up the layout area into a number of regions (strata) for which the critical area is estimated using either simple random sampling or systematic sampling [1]. Since IC layouts are usually composed of large circuit blocks of similar layout types there is often less variation locally than over the chip as a whole. Consequently a stratified sampling scheme will nearly always result in a more accurate estimate of total chip critical area with smaller bounds on the error of estimation than an equivalent simple random sample.

Sample Error Bounds

The average number of faults for each defect type can be estimated by sampling the IC layout. Typically as many as 4000 samples would be used to obtain an accurate measurement.

The chip yield is determined by combining a yield model with the average number of faults per chip for each defect type. The bounds on the error of the yield prediction can be calculated by substituting the lower/upper error bounds of the average number of faults into the yield model to determine the upper/lower error bound on the yield. However, this may result in inaccurate error bounds, since critical areas are normally correlated to some degree.

More accurate estimates of error bounds can generated by making use of a property of the Poisson model. This model allows a single term $ \lambda_{{dev}}^{}$, the average number of chip faults, to be derived from the sum of the average number of faults for each defect type.

YTot = Y0e-$\scriptstyle \lambda_{{dev}}$ (6)
$\displaystyle \lambda_{{dev}}^{}$ = $\displaystyle \sum_{{j=1}}^{{k}}$$\displaystyle \lambda_{j}^{}$ (7)


The sampled mean of $ \lambda_{{dev}}^{}$, $\displaystyle \sum_{{i=1}}^{n}$$\displaystyle {\frac{{\lambda_{dev(i)}}}{{n}}}$ (where n is the number of samples), is equal to the sum of the sample means of $ \lambda_{j}^{}$, $\displaystyle \sum_{{j=1}}^{k}$$\displaystyle \overline{{\lambda_{j}}}$ (where k is the number of defect types), but the error bounds on $ \overline{{\lambda_{dev}}}$ are not the same and are typically smaller. This method has the effect of reducing the variability of the sampled population, which is why the error bounds improve. The sum of all the defect mechanisms fault probabilities in a sample region is, in general, less variable than that of a single defect mechanism. For example, transistor gate area does not normally coincide with contacts. By combining the fault mechanisms of transistor gate oxide pinholes and contact failures the overall variability between sample regions is reduced.

When the Poisson yield model is used to predict yield, the error bounds of $ \lambda_{{dev}}^{}$ can be used directly to give the bounds of the predicted yield. For other models the result are applied indirectly. Assuming the error bounds are small a new yield model can be generated from the prediction, YTot, of the existing model. The new model (equation (9)), is based on the Poisson model of equation (6), and gives the same yield prediction using the term $ \lambda_{{dev}}^{}$ and a constant clustering factor Cf.

Cf = $\displaystyle {\frac{{ -\ln \frac{Y_{Tot}}{ Y_0}}}{{\lambda_{dev}}}}$ (8)
YTot = Y0e-Cf$\scriptstyle \lambda_{{dev}}$ (9)

This model is then used to give the error bounds on the original prediction.

Reference

1
W. Mendenhall, L. Ott, and R. L. Scheaffer,
Elementary Survey Sampling,
Wadsworth Publishing Company Inc, Belmont, California, 1991.



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