# Die Yield Calculator: Poisson, Murphy and Negative Binomial

> Source: https://netray.co/tools/die-yield-poisson-murphy-negative-binomial-calculator
> Publisher: Netray (netray.co), AI and integration engineering for ERP and manufacturing systems.
> Section: Free Tools

## Short answer

Compare Poisson, Murphy and negative binomial die yield models from die size and defect density D0.

## Category

electronics

## Industry

Semiconductors

## Audience

- Yield engineer
- Process integration
- Product engineer
- Fab manager

## Tool

**Kind:** calculator

**Inputs:**

#### Die width

**Key:** die_w

**Type:** number

**Default:** 8

**Min:** 0.5

**Step:** 0.1

**Unit:** mm

#### Die height

**Key:** die_h

**Type:** number

**Default:** 8

**Min:** 0.5

**Step:** 0.1

**Unit:** mm

#### Defect density D0

**Key:** d0

**Type:** number

**Default:** 0.2

**Min:** 0

**Step:** 0.01

**Unit:** defects/cm2

**Help:** Killer defect density, not total defect count.

#### Clustering parameter alpha

**Key:** alpha

**Type:** number

**Default:** 3

**Min:** 0.5

**Step:** 0.5

**Help:** Lower alpha means more clustering. Many fabs fit values from 1 to 5.

**Outputs:**

#### Die area

**Key:** area

**Expr:** die_w * die_h / 100

**Format:** decimal

**Description:** Die area in cm2.

#### Mean defects per die (A x D0)

**Key:** lambda

**Expr:** area * d0

**Format:** decimal

#### Poisson yield

**Key:** poisson

**Expr:** exp(-lambda) * 100

**Format:** percent

**Description:** Y = exp(-A D0). Assumes random, uniform defects, so it is pessimistic for clustered defects.

#### Murphy yield

**Key:** murphy

**Expr:** if(gt(lambda, 0.000001), pow((1 - exp(-lambda)) / lambda, 2), 1) * 100

**Format:** percent

**Primary:** true

**Description:** Y = ((1 - exp(-A D0)) / (A D0))^2, a triangular defect density distribution.

#### Negative binomial yield

**Key:** nb

**Expr:** exp(-alpha * ln(1 + lambda / alpha)) * 100

**Format:** percent

**Primary:** true

**Description:** Y = (1 + A D0 / alpha)^-alpha, fits clustered defects.

**Chart:**

#### Die yield vs defect density D0

**Type:** line

**X:** d0

**From:** 0.05

**To:** 1.5

**Steps:** 30

**Series:**

- poisson
- murphy
- nb

**Disclaimer:** Yield models are fitted approximations. Calibrate D0 and alpha against your own wafer sort data and use only killer defects for D0.

## Content

**Intro:** Yield models turn a defect density and a die size into an expected probability that a die works. This calculator shows three classic models side by side so you can see how much the choice of model changes the forecast.

**Sections:**

#### How to use it

**Body:** Enter the die dimensions in millimeters and the killer defect density D0 in defects per square centimeter. Set alpha for the negative binomial model, using a fitted value if you have one, otherwise start near 3. The calculator reports die area, the mean number of defects per die and the yield from each model. The chart sweeps D0 from 0.05 to 1.5 so you can see where the models diverge. At low defectivity all three agree closely, and they separate as the defect count per die grows.

#### The three models

**Body:** Poisson assumes defects land independently and uniformly, giving Y = exp(-A D0). Real defects cluster, so Poisson understates yield for large die. Murphy assumes defect density varies across the wafer with a triangular distribution and gives Y = ((1 - exp(-A D0)) / (A D0))^2. The negative binomial model, Y = (1 + A D0 / alpha)^-alpha, adds a clustering parameter and reduces to Poisson as alpha grows large. Seeds and Bose-Einstein variants exist too, but these three cover most day to day work.

**Bullets:**

- Poisson: random defects, most pessimistic
- Murphy: simple, no fitted parameter
- Negative binomial: best when you can fit alpha

#### Reading results and common mistakes

**Body:** A 0.64 cm2 die at D0 of 0.2 gives about 0.13 mean defects per die, so every model lands near 88 percent. Make the die 4 times bigger and the models split by several points. Common mistakes: using total inspection defects instead of killer defects, mixing mm2 and cm2, applying one D0 to every layer, and forgetting that line yield, parametric loss and test escapes sit on top of random defect yield.

**Cta:** Netray can connect inspection, defect and sort data held in your ERP or MES to a private model that tracks D0 by product and layer without sending anything outside your network.

**Faq:**

#### Which yield model should I use?

**Answer:** Murphy is a reasonable default when you have no fit data. If you have wafer sort history, fit the negative binomial alpha, since it follows clustered defects best. Poisson is the conservative lower bound.

#### What is a typical D0?

**Answer:** It varies a lot by node, maturity and layer count. Mature processes can run below 0.1 per cm2 while new nodes ramp from several per cm2 down. Use your own fitted value, not a published average.

#### Why is my real yield lower than the model?

**Answer:** Random defect yield is only one factor. Systematic yield loss, parametric failures, probe issues, edge die and test escapes multiply on top. Compare the model to functional yield at wafer sort, not to final yield.

#### What does alpha mean?

**Answer:** Alpha is the clustering parameter. A small alpha means defects cluster strongly and yield is higher than Poisson predicts. As alpha grows, defects look random and the model approaches Poisson.

## Search terms

die yield calculator, poisson yield model, murphy yield model, negative binomial yield model, defect density yield calculator, semiconductor yield formula, wafer yield calculator, D0 defect density

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