poisson yield model
**Poisson Yield Model** is the **simplest mathematical framework for estimating semiconductor die yield from defect density, assuming that killer defects occur randomly and independently across the wafer surface — providing the foundational yield equation Y = exp(−D₀ × A) where Y is yield, D₀ is defect density, and A is chip area** — the starting point for every yield engineer's analysis and the baseline against which more sophisticated yield models are benchmarked.
**What Is the Poisson Yield Model?**
- **Definition**: A yield model based on the Poisson probability distribution, which describes the probability of a given number of independent random events occurring in a fixed area. Die yield equals the probability of zero killer defects landing on a die: Y = P(0 defects) = exp(−D₀ × A).
- **Assumptions**: Defects are randomly distributed (no clustering), each defect independently kills the die, defect density D₀ is uniform across the wafer, and all defects are killer defects.
- **Parameters**: D₀ (defect density, defects/cm²) and A (die area, cm²). The product D₀ × A represents the average number of defects per die.
- **Simplicity**: Only two parameters — makes it easy to calculate, communicate, and use for quick estimates during process development.
**Why the Poisson Yield Model Matters**
- **First-Order Estimation**: Provides a quick, intuitive yield estimate that captures the fundamental relationship between defect density, die area, and yield — useful for initial process assessments.
- **Process Comparison**: Comparing D₀ values across process generations, equipment sets, or fabs provides a normalized defectivity metric independent of die size.
- **Yield Sensitivity Analysis**: The exponential dependence on D₀ × A immediately reveals that large die are exponentially more sensitive to defect density — quantifying the area-yield trade-off.
- **Cost Modeling**: Die cost = wafer cost / (dies per wafer × yield) — Poisson yield feeds directly into manufacturing cost models for product pricing and technology ROI.
- **Teaching Tool**: The Poisson model builds intuition for yield engineering — students and new engineers learn the fundamental D₀ × A relationship before encountering more complex models.
**Poisson Yield Model Derivation**
**Statistical Foundation**:
- Poisson distribution: P(k defects) = (λᵏ × e⁻λ) / k!, where λ = D₀ × A is the average defect count per die.
- Die yield = P(0 defects) = e⁻λ = exp(−D₀ × A).
- For D₀ = 0.5/cm² and A = 1 cm²: Y = exp(−0.5) = 60.7%.
- For D₀ = 0.1/cm² and A = 1 cm²: Y = exp(−0.1) = 90.5%.
**Yield Sensitivity to Parameters**:
| D₀ (def/cm²) | A = 0.5 cm² | A = 1.0 cm² | A = 2.0 cm² |
|---------------|-------------|-------------|-------------|
| 0.1 | 95.1% | 90.5% | 81.9% |
| 0.5 | 77.9% | 60.7% | 36.8% |
| 1.0 | 60.7% | 36.8% | 13.5% |
| 2.0 | 36.8% | 13.5% | 1.8% |
**Limitations of the Poisson Model**
- **No Clustering**: Real defects cluster spatially (particles, scratches, equipment issues) — clustering means some die get many defects while others get none, actually improving yield vs. Poisson prediction.
- **Overly Pessimistic for Large Die**: The random assumption spreads defects uniformly — real clustering leaves more defect-free areas than Poisson predicts.
- **Ignores Systematic Defects**: Pattern-dependent, layout-sensitive, and process-integration defects are not random — they affect specific die locations systematically.
- **Single Defect Type**: Real fabs have multiple defect types (particles, pattern defects, electrical defects) with different densities and kill ratios.
Poisson Yield Model is **the foundational equation of semiconductor yield engineering** — providing the essential intuition that yield decreases exponentially with defect density and die area, serving as the starting point from which more accurate models (negative binomial, compound Poisson) are developed to capture the clustering and systematic effects present in real manufacturing.