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Equation 15 · Patterning at the Limit: What Actually Happens When a Chip Is Manufactured

What does this equation mean?

Y=(1+DSc)−c,Y = \left(1 + \frac{D S}{c}\right)^{-c},

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Start withD S
Divide byc
This relates toY
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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YY

Symbol Y

the widely used negative binomial form.

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DD

Symbol D

the defect density.

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SS

Symbol S

the die area.

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cc

Symbol c

the cluster parameter [ 16 ].

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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addition

addition

Add the term after the plus sign to the term or group before it.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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DSD S

Numerator: D S

The complete quantity above the fraction bar.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

Yield modelling then converts defect density into economics. The widely used negative binomial form is Y=(1+DSc)−cY = \left(1 + \frac{D S}{c}\right)^{-c}. with D the defect density, S the die area, and c a cluster parameter [ 16 ] . As c →\to ∞\infty the expression tends to the Poisson model Y = e−DSe^{-D S} , which assumes defects fall independently and uniformly.

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Sources cited in the surrounding passage

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