Yield Fundamentals

Lesson 6 covered how defects are detected and classified. This lesson covers why they matter in a single, quantified way: yield, the fraction of manufactured die that actually work, and the mathematical models that relate yield to defect density and die area. These models matter well beyond the fab floor; they are also the reasoning behind design decisions, like keeping die area down or adding redundancy, covered later in this path and elsewhere on this site.

Reading time: about 9 minutes.

7.1 Defect density and the Poisson model

Random, "killer" defects (lesson 6.3's particle contamination and certain pattern defects) are assumed to land randomly across a wafer, at some average density per unit area, called the defect density, commonly written D0, in defects per unit area (often per square centimeter). Given a die area A and defect density D0, the simplest and most widely taught yield model, the Poisson (or Seeds) model, predicts the fraction of die with zero defects, and therefore (under this simple model's assumption that any one defect kills the die it lands on) the fraction of die that are good:

Y = e^(-A * D0)

This comes directly from the Poisson distribution's probability of observing zero events (here, zero defects landing on a given die) when the expected number of events is A times D0, the average number of defects expected per die of that area, given that density. The equation's shape already tells the central story of yield: yield falls off exponentially as either die area or defect density increases, so a larger die at the same defect density, or the same die area at a higher (worse) defect density, both yield worse, and the relationship is strongly nonlinear, not a simple proportional loss.

7.2 Worked example: comparing two defect densities

7.3 Beyond the simple Poisson model

Real yield data across many different products and fabs consistently yields somewhat better than the simple Poisson model predicts, especially at larger values of A times D0, because the Poisson model's core assumption, that defects land purely randomly and independently across every wafer with the same average density everywhere, does not hold perfectly in practice: real defect density itself varies somewhat from wafer to wafer, and sometimes systematically across a wafer (for example, slightly higher near the wafer edge). Two more refined models account for this:

• Murphy's model assumes defect density itself follows a distribution across wafers (commonly modeled as roughly triangular or otherwise clustered around a mean, rather than every wafer having the exact same D0), and integrates the simple Poisson yield formula over that distribution, generally predicting somewhat higher yield than the simple Poisson model at the same mean defect density, because it captures the fact that some wafers have better-than-average, not just worse-than-average, defect density.
• The negative binomial model goes further, explicitly modeling defect clustering, the empirical observation that real defects are not purely independently, randomly scattered but tend to cluster somewhat (from a single contamination event producing several nearby defects, for example), using a clustering parameter (commonly denoted alpha) that, in the limit of very little clustering, reduces back toward the simple Poisson model's prediction, and in the limit of strong clustering, predicts meaningfully higher yield than Poisson would, for the same mean defect density, because clustered defects concentrate their damage onto fewer dies rather than spreading evenly across more dies.

Yield models progressively refined this way (Poisson, then Murphy, then negative binomial) are a recurring pattern in engineering modeling generally: the simplest model captures the first-order physics and is the right place to start reasoning from, but real data consistently motivates adding the next layer of refinement once the simple model's specific, identifiable simplifying assumption (here, perfectly uniform, perfectly independent defect placement) is shown not to hold.

7.4 Interview questions

Write the Poisson (Seeds) yield equation and explain each term. Y = e^(-A x D0): A is die area, D0 is defect density, and Y is the predicted fraction of good die, derived from the Poisson probability of zero defects landing on a die given an expected A x D0 defects per die.

Why does yield fall off exponentially, not linearly, with die area? Because the Poisson model's underlying probability of zero defects on a die is itself an exponential function of the expected defect count (A x D0), so yield loss accelerates as either die area or defect density grows, not just scales proportionally.

Why does real measured yield typically come in higher than the simple Poisson model predicts? The simple Poisson model assumes perfectly uniform, independent defect placement across every wafer at the same density; real defect density varies wafer to wafer and defects tend to cluster, both effects that more refined models (Murphy, negative binomial) account for, generally predicting higher yield than simple Poisson at the same mean defect density.

What does the negative binomial model's clustering parameter represent? How strongly real defects cluster together rather than scattering independently; in the limit of little clustering the model approaches the simple Poisson prediction, and with strong clustering it predicts meaningfully higher yield for the same mean defect density, since clustered defects concentrate damage onto fewer dies.

In the worked example, why did tripling defect density more than double yield loss rather than simply tripling it? Because yield depends exponentially on the product A x D0, not linearly; the nonlinearity of the exponential function means the yield-loss impact of a given defect-density change grows faster than proportionally as the baseline A x D0 value increases.