 Integrated circuit cost per good die

Let a circular semiconductor wafer have diameter $d$ and processed cost $C_w$. For square semiconductor dies of area $A$, ignoring all edge losses gives the ideal gross count $N_0=\pi d^2/(4A)$. For illustration adopt the geometric edge approximationHere $G$ is the good-die count. The correction scales as wafer perimeter divided by die side length: only a boundary strip can lose partially fitting squares. Its coefficient is an approximation depending on layout; the expression is not an exact packing theorem and is useful only when $\sqrt{A}\ll d$. Counts from it need not be integers. Actual layouts, edge exclusion and separation lanes determine an integer gross count. The expected value of $G$ follows by linearity of expectation; independence between semiconductor dies is not required for that step if each has the same yield.

For the original example $d=200\mathrm{mm}$, $A=100{\mathrm{mm}}^2$, $D=0.005{\mathrm{mm}}^{-2}$ and $C_w=1000$ cost units, $N_0=314.159$ and $N_g\approx 269.730$. The predicted good count is $163.600$ per wafer, and the processed cost is $6.11248$ units per good die. This is total processed cost divided by expected output, appropriate to a large repeated batch; it is not the expected value of a single-wafer random ratio $C_w/G$, which is undefined when $G=0$. Test, packaging, design and equipment costs are excluded.

A hypothetical test readout makes the inference explicit. Suppose $600$ of $1000$ tested semiconductor dies, each of area $100{\mathrm{mm}}^2$, pass a specified functional test. These are invented counts for calculation, not reported measurements. The pass fraction is $\hat{Y}=0.600$; within the Poisson defect yield assumptions it impliesIf pass indicators are independent random variables with common success probability $Y$, their sum has a binomial distribution. The fitted standard deviation of the pass fraction is $\sqrt{0.6(1-0.6)/1000}=0.01549$. A test readout estimates pass probability for its tested functions and conditions; it cannot alone establish spatially independent defects, detect every latent failure or distinguish fabrication losses from a faulty test. The logarithm only translates the observed fraction into a defect intensity after those additional assumptions are adopted.