Semiconductor Wafer DPW, Poisson & Murphy Yield Formula

Account for circular wafer edge loss and non-uniform fab defect density!

Dividing a 300 mm silicon wafer’s area PI() * R_eff^2 by Rectangular Die Area A_die (mm^2) overcounts usable chips because rectangular dies clip around the circular wafer perimeter! Subtracting the perimeter-clipping term PI() * 2 * R_eff / SQRT(2 * A_die) and flooring via INT(...) gives Gross Dies per Wafer (DPW). Next, multiplying Defect Density D_0 (defects/cm^2) by Die Area in cm^2 (A_cm2 = A_die / 100) gives Expected Faults per Die lambda = D_0 * A_cm2: (1) Pure random Poisson statistics predict Yield_Poisson = EXP(-lambda) (too pessimistic across wafer-to-wafer variation!); (2) Bernard Murphy (1964, Bell Labs) integrated a symmetric triangular defect distribution to derive Yield_Murphy = ((1 - EXP(-lambda)) / lambda)^2; and (3) R.B. Seeds (1967) integrated an exponential defect distribution to derive Yield_Seeds = 1 / (1 + lambda)!

Try it on Effective_Wafer_Radius R_eff = 147 mm (300 mm wafer minus 3 mm edge exclusion, B2), Die_Area A_die = 400 mm^2 (20x20 mm = 4.00 cm^2 SoC, B3), Defect_Density D_0 = 0.10 defects/cm^2 (B4 -> lambda = 0.40 faults/die), and Processed_Wafer_Cost C_wafer = $12,000 (B5): Gross DPW is INT(67,886.68 / 400 - 923.63 / 28.284) = INT(169.72 - 32.66) = 137 gross dies (E2); Poisson Yield is EXP(-0.40) = 0.6703 (E3); Murphy Yield is ((1 - EXP(-0.40)) / 0.40)^2 = 0.6793 (E4); and Good-Die Silicon Cost under Murphy Yield is $12,000 / (137 * 0.6793) = $128.94/die (E5)! Test pristine D_0 = 0 in cell B4: an unguarded Murphy formula divides 0 / 0 (#DIV/0!), whereas our IF(lambda = 0, 1, ...) guard cleanly returns 1.0000 (100% yield, $87.59/die)! In a shared Arte Sheets fab-costing workbook, ask the in-app Pro Assistant for a hint: "Check E4 when Defect_Density B4 = 0—how do we apply L’Hopital’s limit 1.0000 without #DIV/0!?"

Singularity-Safe Bernard Murphy Wafer Yield Formula

Guard lambda = 0 explicitly so zero defect density evaluates to its exact L’Hopital limit ((lambda) / lambda)^2 = 1.0000!

=LET(r_ef, B2, a_mm, B3, d_0, B4, c_wf, B5, IF(OR(r_ef <= 0, a_mm <= 0, a_mm >= PI() * r_ef ^ 2, d_0 < 0, c_wf <= 0), "Invalid Wafer Geometry", LET(lm_d, d_0 * (a_mm / 100), IF(lm_d = 0, 1, ROUND(((1 - EXP(-lm_d)) / lm_d) ^ 2, 4)))))

Cost per Known-Good Die Formula

Use unrounded Murphy yield inside C_wafer / (DPW * Yield_Murphy) so fab gross-margin models avoid cent-level drift!

=LET(lm_d, B4 * (B3 / 100), yl_m, IF(lm_d = 0, 1, ((1 - EXP(-lm_d)) / lm_d) ^ 2), IF(OR(ISTEXT(E4), E2 <= 0), "Invalid Wafer Geometry", ROUND(B5 / (E2 * yl_m), 2)))

Further reading

• Arte Sheets Lesson — Williams-Waterman-Patterson Roofline Compute vs Memory Bound Formula
• Arte Sheets Lesson — Wright Learning Curve Unit Cost & Cumulative Batch Average