= Poisson defect yield
{c}
{title2=$Y=e^{-DA}$}
Consider a deliberately simplified model: fatal defects form a spatially uniform <Poisson point process> with intensity $D$ per unit <area>. A <semiconductor die> of <area> $A$ works exactly when it contains no fatal defect. Its defect count $N$ has a <Poisson distribution> with <expected value> $DA$, hence its probability of working, called its yield, is
$$
Y=\Pr(N=0)=\frac{(DA)^0}{0!}e^{-DA}=e^{-DA}.
$$
The product $DA$ is dimensionless. The formula concerns fatal defects, not every visible imperfection. Spatial uniformity, independent defect counts in disjoint <areas>, and failure from even one fatal defect are model assumptions. Clustering, shared processing failures, repairable regions and defects that affect only some functions can invalidate them. This is a chosen probability model, not historical yield data.
Choose $D=0.005\,\mathrm{mm}^{-2}$. For <semiconductor die> <areas> $25$, $100$ and $200\,\mathrm{mm}^2$, the model gives $Y=e^{-0.125}=0.882497$, $e^{-0.5}=0.606531$ and $e^{-1}=0.367879$. Doubling the <area> squares the yield: $Y(2A)=Y(A)^2$. This follows either from the <exponential function> or by splitting the larger <area> into two disjoint parts and using <independence>. More functions on one <semiconductor die> can save assembly while making the no-defect requirement harder to satisfy.
\Image[/integrated-circuit-poisson-yield-cost.png]
{title=A chosen <Poisson defect yield> model connects die <area> to expected output}
{description=Original model calculations with $D=0.005\,\mathrm{mm}^{-2}$. Left: $Y=e^{-DA}$ and three worked points. Right: expected good <semiconductor dies> per circular $200\,\mathrm{mm}$ <semiconductor wafer>, comparing an area-only gross count with the declared approximate edge correction. No scribe lanes, packaging losses or measured yields are included. The annotation uses an assumed cost of $1000$ units per wafer.}
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