High-yield, high-performance memory design

Appendix: normal quantile (NQ) plots
Appendix: normal quantile (NQ) plots
A NQ plot is an alternative view of a distribution that facilitates easy comparison to Gaussian distributions, and enables detailed analysis of the distribution’s tails (very useful in high-sigma analysis).

The figure below shows three views of the same distribution. The only difference between each view is the y-axis. Each view has different advantages and disadvantages. Crucially, we can transform among the views in a fairly straightforward fashion.

The top plot in the figure has the PDF (probability density function) on the y-axis. This is the plot that we’ve used in the examples. The area under the PDF must integrate to 1.0. The area under the curve at a given range of x-values is the probability of that range of x-values. The PDF curve is intuitive because its y-value for a range of x values is proportional to the number of samples seen in that range of x-values. “Discretized” PDFs – histograms – are intuitive and easy to compute.

 
But the PDF view has disadvantages. First, lower-probability regions run extremely close to the x-axis, and have such small values that it is difficult to distinguish one low-probability value from another (e.g. 1e-4 from 1e-5), because they are both hugging the x-axis. This is troublesome for high-sigma analysis. Second, the “best behaved” or “typical” distributions, namely Gaussian distributions, have a highly nonlinear bell-shaped curve. This means that small nonlinear distortions of that curve are difficult to identify, and in general the nonlinear curve is hard to work with. Compare this to many subdomains of electrical engineering, from circuit analysis to control theory, where the “best behaved” models or systems are linear or linearized models; and linear techniques are key analysis and design tools. Nonlinear bell-shaped curves are not amenable to such analysis.

The middle plot in the figure has the CDF (cumulative distribution function) on the y-axis. A CDF value at x is the area under the PDF from ∞ to x. That is:



Intuitively, the y-axis can be viewed as the probability p that up to a given x-value can occur. CDFs are highly useful for computing yield: CDF values are equal to yield values for “≤” specs; and equal to 1-yield when for “≥” specs. Accordingly, one may inspect CDF plots to see the tradeoff between yield values and spec values. To go backwards from CDF to PDF, one takes the derivative.

However, CDFs have the same disadvantages of PDFs: hard to distinguish among low-probability regions, and Gaussian distributions have highly nonlinear curves which impedes analysis.

Consider a view of a distribution that overcomes some of the disadvantages described, and keeps key benefits. Specifically, consider a view where:

• The “best behaved” distributions (Gaussian) are linear curves, and the larger the deviation from Gaussian the more nonlinear the curve. Different types of deviations indicate different nonlinearities.
• One can directly see the tradeoff between yield and performance.
• One can easily distinguish among different low-probability values.

Remarkably, such a view of a distribution exists: these are NQ plots. An example NQ plot is shown in the bottom of Figure 4.18. An NQ plot is like a CDF, but the y-axis is warped “just right” such that if the underlying distribution is Gaussian, then the NQ curve would be linear. The specific function to warp with is the “normal quantile” function, also known as the “inverse CDF of the Gaussian”, or “probit” function. It takes in CDF value (a probability p), and outputs a normal quantile (NQ) value. The NQ function is not available in closed form, so must be numerically computed as:


where is the “inverse error function” and is typically provided in software packages and libraries.
The “normal quantile” value has a specific interpretation that aids intuition: the number of standard deviations away from the mean, if the distribution was Gaussian. Sigma is a unit for yield that is often simpler to talk about than percent yield or probability of failure; and one can build intuition about the units. Specifically for intuition: 3 sigma is about 1 failure in 1000, 4 sigma is about 1 in 50 thousand, 5 sigma is about 1 in a million, and 6 sigma is about 1 in a billion. Certain circuit types of circuits typically have particular target ranges of sigma values. For example, non-replicated circuits like analog circuits are typically 3-4 sigma; and bitcells are typically around 6 sigma.

About the author
Trent McConaghy is co-founder and Chief Technology Officer of Solido Design Automation Inc. He was a co-founder and Chief Scientist of Analog Design Automation Inc., which was acquired by Synopsys Inc. in 2004. Prior to that, he did research for the Canadian Department of National Defense. Trent has about 40 peer-reviewed technical papers and patents granted / pending. In 2001, Trent was awarded the 2001 Outstanding Young Alumni Award by the University of Saskatchewan. He received his PhD degree in Electrical Engineering from the Katholieke Universiteit Leuven, Belgium, in 2008. He received a Bachelor’s in Engineering (with great distinction), and a Bachelor’s in Computer Science (with great distinction), both from the University of Saskatchewan, Canada, in 1999.

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