Semiconductor Wafer Yield Calculator

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Introduction to defect-limited die yield on a silicon wafer

A finished 300 mm wafer leaves a leading-edge fab carrying somewhere between a few dozen and several thousand identical die, and only some fraction of them will pass electrical test. That fraction is the die yield, and together with the number of die the reticle grid managed to place on the wafer it determines the only number a product team really cares about: the cost of one working chip. This calculator does the two halves of that problem properly and keeps them separate. First it counts gross die per wafer by actually laying a rectangular stepper grid over the usable area of the wafer and counting the sites that fit — not by dividing areas. Then it applies four named, published defect-limited yield models to that count, reports all four side by side, and lets you pick which one drives the headline cost figure.

The reason the page insists on naming the model is that the models genuinely disagree. At a low defect-area product they agree to within a fraction of a point and the choice does not matter. At the defect-area products that a modern 800 mm² accelerator die actually operates at, the Poisson model and the Seeds model differ by ten percentage points of absolute yield, which on a sixteen-thousand-dollar wafer is a swing of roughly ninety dollars in the cost of every single chip. A wafer yield calculator that offers one unnamed exponential and calls it "the" yield is not giving you an answer, it is giving you one of four answers and hiding which.

Everything on this page runs in your browser and nothing is transmitted anywhere, which matters because die dimensions, defect density and wafer price are among the most closely held numbers a semiconductor company owns.

How to use the wafer inputs, and which units each one wants

The form is deliberately dimensional rather than area-based, because die packing depends on the aspect ratio of the die and not only on its area — a 4 mm by 208 mm sliver and a 29 mm by 28.7 mm square have almost the same area and wildly different die counts.

Wafer diameter in millimetres. SEMI M1, the specification for polished single-crystal silicon wafers, is the standard that fixes the nominal diameters used in production; 200 mm and 300 mm are the two volume sizes, with 150 mm still common for analogue, power and MEMS work.
Die width and die height in millimetres, measured on the die itself, excluding the scribe. The single-exposure reticle field on a 193 nm immersion scanner is 26 mm by 33 mm, so a monolithic die larger than 858 mm² requires stitching or a different exposure strategy.
Scribe (street) width in millimetres. This is the sacrificial lane between neighbouring die that the saw or the laser consumes, and it typically holds process control monitors and alignment structures. It adds to the stepping pitch but not to the die area, which is why it hurts the die count without helping yield.
Edge exclusion in millimetres. Film thickness, resist coating and etch uniformity all degrade in a ring near the wafer edge, and fabs simply refuse to count die whose area intrudes into that ring. A 3 mm exclusion is a representative production value.
Defect density in defects per square centimetre. This is the single most common place to slip a unit on this page: die dimensions are millimetres, defect density is per square centimetre, and a square centimetre is one hundred square millimetres, not ten. The calculator prints the die area in both units and prints the dimensionless defect-area product so you can sanity-check the conversion yourself.
Clustering parameter for the negative binomial model, dimensionless. Leave it at 2 if you have no data.
Wafer cost in dollars for a fully processed wafer at wafer-out, before probe, dicing, packaging and final test.

Press the calculate button and the page returns the gross die count, the defect-area product, all four model yields, expected good die, cost per good die, silicon utilisation, a scale drawing of the actual die placement it counted, and a sensitivity sweep across defect density. Every input is validated in JavaScript rather than by the browser, so an impossible combination gets a specific message instead of a silent failure or a stale number.

The gross die per wafer formula, and why area division is not it

Start with the geometry. Let $d$ be the wafer diameter, $e$ the edge exclusion, $w$ and $h$ the die width and height, and $s$ the scribe width. The usable radius and the stepping pitch are

Formula: R = d / 2 − e, p_x = w + s, p_y = h + s

$R = \frac{d}{2} - e, p_{x} = w + s, p_{y} = h + s$

A die whose lower-left corner sits at $(x_{0}, y_{0})$ is counted only if all four of its corners lie inside the usable circle, which reduces to a single condition on the corner farthest from the wafer centre:

Formula: (max(|x_0|,|x_0+w|))^2 + (max(|y_0|,|y_0+h|))^2 ≤ R^2

${(\max (| x_{0} |, | x_{0} + w |))}^{2} + {(\max (| y_{0} |, | y_{0} + h |))}^{2} \leq R^{2}$

The calculator sweeps the origin of the grid over a full pitch in both axes, counts the fitting sites at each alignment in closed form row by row, and reports the best alignment found. That is what a stepper job actually does when the layout team optimises the shot map, and it is why the answer here is normally a few die better than a smooth formula.

Two closed-form expressions are reported alongside the placement count so you can see the size of the correction. Writing $S$ for the die area, the naive area ratio is an unreachable upper bound, and the classic circumference-corrected approximation is the form most often quoted in industry spreadsheets:

Formula: N_area = (π (d/2)^2) / S, N_approx = (π d^2) / (4 S) − (π d) / (sqrt(2 S))

$N_{a r e a} = \frac{π {(d / 2)}^{2}}{S}, N_{a p p r o x} = \frac{π d^{2}}{4 S} - \frac{π d}{\sqrt{2 S}}$

The second term is the edge correction: it is proportional to the wafer circumference and inversely proportional to the linear size of the die, which is exactly the way partial die accumulate around the rim. De Vries compared several such formulas against exact counts and showed that their accuracy depends on both die area and die aspect ratio, and that the mean of die width and die height is the right parameter for capturing the aspect-ratio dependence — which is the reason this page asks for width and height separately and then counts rather than approximates.

Four named yield formulas, and why they disagree

Once the gross count is fixed, the yield question is separate and purely statistical. Convert the die area to square centimetres so that it matches the units of the defect density, and form the dimensionless defect-area product:

Formula: A = (w ⋅ h) / 100 cm², λ = A ⋅ D_0

$A = \frac{w \cdot h}{100} cm², λ = A \cdot D_{0}$

If fatal defects fall independently and uniformly at random, the count of defects on one die is Poisson with mean $λ$ and the die works only when that count is zero:

Formula: Y_Poisson = e^−AD_0

$Y_{P o i s s o n} = e^{- A D_{0}}$

Real defect densities are not constant across a wafer, a lot or a fab; defects cluster. Murphy's response was to let the density itself be a random variable with density $f (D)$ of mean $D_{0}$ and average the exponential over it:

Formula: Y = ∫_0^∞ e^−AD f(D) d D

$Y = \int_{0}^{\infty} e^{- A D} f (D) d D$

Each choice of $f (D)$ collapses that integral to a closed form, and each closed form carries a name. A symmetric triangular density on $[0, 2 D_{0}]$ gives the Murphy model; a uniform density on the same interval gives a lesser-used variant; an exponential density gives the Seeds model:

Formula: Y_Murphy = ((1−e^−AD_0)/(AD_0))^2, Y_unif = (1 − e^−2AD_0) / (2 A D_0), Y_Seeds = 1 / (1 + A D_0)

$Y_{M u r p h y} = {(\frac{1 - e^{- A D_{0}}}{A D_{0}})}^{2}, Y_{u n i f} = \frac{1 - e^{- 2 A D_{0}}}{2 A D_{0}}, Y_{S e e d s} = \frac{1}{1 + A D_{0}}$

A gamma density gives the negative binomial model, which is the one high-volume manufacturing actually fits to wafer map data, and which subsumes all of the above through its clustering parameter $α$ :

Formula: Y_NB = (1+(AD_0)/α)^−α

$Y_{N B} = {(1 + \frac{A D_{0}}{α})}^{- α}$

Small $α$ means heavy clustering: defects pile onto a few die and leave the rest clean, so yield rises. As $α$ grows the gamma density tightens onto its mean and the model converges on Poisson. Leachman's course notes give the standard correspondences — the negative binomial is essentially Poisson for $α \geq 10$ , close to Murphy at $α = 5$ and close to Seeds at $α = 1$ — and the estimator to use when you do have defect counts per die, with sample mean $μ$ and sample variance $σ^{2}$ :

Formula: α = μ^2 / (σ^2 − μ)

$α = \frac{μ^{2}}{σ^{2} - μ}$

Finally the economics. With $N_{g}$ gross die, model yield $Y$ and wafer cost $C_{w}$ :

Formula: N_good = N_g Y, C_die = C_w / N_good, U = (N_g S) / (π (d/2)^2)

$N_{g o o d} = N_{g} Y, C_{d i e} = \frac{C_{w}}{N_{g o o d}}, U = \frac{N_{g} S}{π {(d / 2)}^{2}}$

One property of the Poisson form is worth keeping in view because it is the basis of nearly all fab-floor yield accounting: defect densities from separate layers or steps add, so the yields multiply, which lets you attribute a yield loss to the layer that caused it.

Formula: Y = ∏ i = 1 n e^−AD_i = e^−A∑i=1nD_i

$Y = \prod_{i = 1}^{n} e^{- A D_{i}} = e^{- A \sum_{i = 1}^{n} D_{i}}$

Running the same identity backwards turns a measured yield into an implied defect density, which is how fabs compare processes across products of different die size: $D_{0} = - \frac{\ln Y}{A}$ . Note that this inversion is model-specific; feeding a Poisson inversion a yield that was really produced by clustered defects will understate the defect density.

A worked example: an 832 mm² accelerator die on a 300 mm wafer

Take the values the form loads by default, which are representative of a large AI accelerator built at an advanced logic node: a 300 mm wafer, a die measuring 26.0 mm by 32.0 mm, a 0.10 mm scribe, a 3 mm edge exclusion, a defect density of 0.09 per cm², a clustering parameter of 2, and a wafer price of $16,000.

The usable radius is 150 − 3 = 147 mm and the stepping pitch is 26.1 mm by 32.1 mm. Sweeping the grid origin over one pitch in each axis, the best alignment fits 65 whole die inside the 147 mm circle. For comparison, the area ratio would allow π × 150² ÷ 832 = 85.0 die, and the circumference-corrected approximation predicts π × 300² ÷ (4 × 832) − π × 300 ÷ √1664 = 85.0 − 23.1 = 61.9. The exact placement therefore beats the closed-form approximation by three die, about five per cent of the wafer's output, while the area ratio overstates it by twenty.

The die area is 26.0 × 32.0 = 832 mm², which is 8.32 cm², so the defect-area product is 8.32 × 0.09 = 0.7488 — comfortably into the regime where the models separate. Substituting into each formula:

Poisson: $e^{- 0.7488} = 0.4729$ , so 47.29% yield and 30.7 good die.
Murphy: ${(0.5271 / 0.7488)}^{2} = 0.4955$ , so 49.55% yield and 32.2 good die.
Seeds: $1 / 1.7488 = 0.5718$ , so 57.18% yield and 37.2 good die.
Negative binomial at $α = 2$ : ${1.3744}^{- 2} = 0.5294$ , so 52.94% yield and 34.4 good die.

At $16,000 per wafer the cost of one working die is $520.48 under Poisson, $496.82 under Murphy, $464.98 under the negative binomial and $430.47 under Seeds. That is a $90 spread — more than seventeen per cent — produced by nothing but the choice of yield model. Silicon utilisation is 65 × 832 ÷ 70,686 = 76.5%, meaning almost a quarter of the polished wafer never becomes product, which is the geometric penalty for a die this large.

Interpreting the result: gross die, net die and good die

Three different die counts get called "die per wafer" in casual conversation, and confusing them is the fastest way to get a cost model wrong by ten per cent.

Gross die per wafer is what this calculator counts: every complete die site the stepper grid places entirely inside the usable radius. Net die per wafer subtracts the sites a fab gives up for reasons that have nothing to do with random defects — process control monitor sites, alignment and overlay structures, drop-in test chips, and any site sacrificed to the notch or to wafer-ID marking. Depending on the product this is anywhere from zero to a handful of sites, and it is a fixed subtraction that no yield model will predict for you. Good die per wafer is the subset of net die that passes wafer probe, and it is the product of the die count and the yield.

The yield the models return is also only the defect-limited or random component. Overall die yield is conventionally decomposed as a product of a systematic-limited yield and this random component, $Y = Y_{S} \cdot Y_{R}$ , where the systematic term captures excursions, edge loss, parametric fallout and design-process interactions that are not random point defects. If you are calibrating against a real fab number, expect the measured yield to sit below the value here, and expect the gap to be the systematic term rather than an error in the defect density.

As a rule of thumb on when the model choice matters: below a defect-area product of about 0.2 all four models sit within a point of each other and the argument is academic. Between 0.2 and 1 the spread opens to several points. Above 1 the models diverge sharply and using Poisson without saying so will materially understate the yield of a large die, because Poisson is exactly the model that assumes away the clustering that saves large die.

Symbols, units and where each one enters

Every quantity used by the wafer yield and die-per-wafer calculation
Symbol	Quantity	Unit	Role
$d$	Wafer diameter	mm	Input; SEMI M1 nominal sizes are 150, 200 and 300 mm
$e$	Edge exclusion	mm	Input; shrinks the usable radius, does not change yield
$R$	Usable radius	mm	Derived as $d / 2 - e$
$w, h$	Die width and height	mm	Input; excludes the scribe
$s$	Scribe (street) width	mm	Input; adds to pitch, not to die area
$S$	Die area	mm²	Derived as $w \cdot h$
$A$	Die area in yield units	cm²	Derived as $S / 100$
$D_{0}$	Mean fatal defect density	defects/cm²	Input; the dominant uncertainty in the whole model
$λ$	Defect-area product	dimensionless	Derived as $A D_{0}$ ; sets how far the models diverge
$α$	Clustering parameter	dimensionless	Input; negative binomial only, smaller means more clustered
$N_{g}$	Gross die per wafer	die	Counted by grid placement, not by area division
$Y$	Defect-limited die yield	fraction	Poisson, Murphy, Seeds or negative binomial
$C_{w}$	Processed wafer cost	$	Input; wafer-out, before probe and packaging
$C_{d i e}$	Cost per good die	$	Output; wafer cost over expected good die

Limitations and assumptions you should carry with the number

This is a first-order planning model, and it is worth being explicit about what it does not do.

Only random defects are modelled. All four yield formulas describe the defect-limited component. Systematic loss — excursions, edge effects beyond the exclusion ring, parametric fallout, design-process marginality — is not included, so the good-die number is an upper bound on what a real lot will produce.
Defect density is an assumption, not a measurement. Foundries do not publish per-node defect densities, and the value dominates the answer. Use the sensitivity sweep rather than a point estimate, and treat any figure you did not measure on your own wafers with suspicion.
The clustering parameter is unidentifiable from yield alone. Given only a die area and a yield, there is no way to separate $D_{0}$ from $α$ . Fitting both requires wafer map data across several die areas, for example by the windowing technique.
The placement count ignores the notch and the wafer flat. Production wafers carry a notch or flat for orientation, which removes a small amount of usable area near one point on the rim. The count also assumes a single rectangular die stepped on a regular grid; multi-project reticles, mixed die sizes and stitched fields are not represented.
Grid alignment is optimised, not exhaustive. The origin sweep samples a grid of offsets over one pitch and reports the best count found. For very small die relative to the wafer the sampled optimum can sit one die below the true optimum.
Cost is silicon only. The cost per good die divides the wafer price by expected good die. It excludes mask amortisation, probe and test time, dicing, packaging, assembly yield, burn-in and final test yield, all of which are real and none of which scale with die area the same way.
Redundancy and repair are not modelled. Memory and many large logic products ship with spare rows, columns or cores and a repair step, which lifts effective yield well above any defect-limited figure. Chiplet partitioning has the same effect by construction, since it converts one large die into several small ones.

Common questions about wafer yield and die per wafer

Which yield model does this wafer yield calculator use?

It computes all four named models at once and lets you choose which one drives the headline good-die and cost figures. The negative binomial model is selected by default because it is the model the industry actually uses for large die, and because Poisson, Murphy and Seeds are all recoverable from it by fixing the clustering parameter at roughly 10, 5 and 1 respectively.

Why is gross die per wafer larger than the classic approximation formula?

The approximation subtracts a fixed edge-loss term proportional to the wafer circumference, which is a smooth average over all die shapes and grid alignments. This calculator instead places a real rectangular grid and counts the die that fit entirely inside the usable radius, then shifts the grid origin to find the best alignment. A good alignment routinely recovers several die that the averaged formula gives away.

What defect density should I enter for a modern process?

Defect density is entered in defects per square centimetre, and a mature high-volume logic process typically sits somewhere around 0.05 to 0.2 per square centimetre, while a node in early ramp can be several times worse. Foundries rarely publish these numbers, so treat any figure you did not measure yourself as an assumption and run the sensitivity sweep rather than trusting a single value.

What is the clustering parameter alpha in the negative binomial model?

Alpha describes how strongly defects bunch together instead of scattering independently. A small alpha means heavy clustering, which leaves more completely clean die and therefore raises yield; a large alpha means near-random defects and the model converges on Poisson. If you have wafer map data, alpha can be estimated as the squared mean number of defects per die divided by the variance minus the mean.

How do gross die, net die and good die differ?

Gross die is every complete die the reticle grid places inside the usable area. Net die subtracts sites given up to process control monitors, alignment structures and any partial die at the edge, so it is the number of die actually sent to probe. Good die is the subset that passes electrical test, which is what the yield model estimates and what the cost per die is divided by.

Does edge exclusion change the yield or only the die count?

In this calculator edge exclusion only shrinks the usable radius and therefore the gross die count, because the defect-limited yield models depend on die area and defect density alone. Real edge loss is a systematic mechanism rather than a random one, so raising the exclusion ring is a way of removing known-bad sites from the count instead of letting them drag the fitted defect density upward.

Sources for the yield models and wafer geometry

Sources. The four yield formulas, the compound-distribution derivation, the clustering-parameter estimator and the systematic/random decomposition on this page follow the published literature below. Formulas were checked against the primary and institutional sources, not against other calculators.

Murphy, B. T. (1964). Cost-size optima of monolithic integrated circuits. Proceedings of the IEEE, 52(12), 1537–1545. doi:10.1109/PROC.1964.3442 — the compound-distribution integral and the triangular-density model.
Seeds, R. B. (1967). Yield and cost analysis of bipolar LSI. IEEE International Electron Devices Meeting, Washington DC — the exponential-density model.
Stapper, C. H. (1973). Defect density distribution for LSI yield calculations. IEEE Transactions on Electron Devices, 20(7), 655–657. doi:10.1109/T-ED.1973.17719.
Stapper, C. H. (1989). Large-area fault clusters and fault tolerance in VLSI circuits: a review. IBM Journal of Research and Development, 33(2), 162–173. doi:10.1147/rd.332.0162 — negative binomial yield and defect clustering.
Leachman, R. C. (2014). Yield Modeling and Analysis, IEOR 130 Methods of Manufacturing Improvement, University of California, Berkeley. Course notes (PDF) — the source used to verify every formula reproduced above, including the Poisson, uniform, Murphy, Seeds, Bose-Einstein and negative binomial closed forms, the clustering-parameter estimator, the α ≈ 10 / 5 / 1 correspondences, and the systematic-versus-random decomposition.
de Vries, D. K. (2005). Investigation of gross die per wafer formulas. IEEE Transactions on Semiconductor Manufacturing, 18(1), 136–139. doi:10.1109/TSM.2004.836656 — the comparison of closed-form gross-die formulas against exact counts, and the finding that die aspect ratio matters.
SEMI. SEMI M1, Specification for Polished Single Crystal Silicon Wafers. SEMI standards store — the standard that fixes nominal wafer diameters and the fixed quality area. SEMI M1 is a paid standard and its dimensional tables were not read directly for this page; the diameters offered here (150, 200 and 300 mm) are the widely used nominal sizes it covers, and the edge exclusion is left as a user input rather than quoted from the standard.

Note on the edge exclusion default: 3 mm is used because Leachman's report describes a fab calibrating its defect density inside a 3 mm wafer edge exclusion. It is a representative value, not a standardised one — set it to whatever your own process specifies.

Wafer diameter (mm) SEMI M1 nominal sizes: 150, 200 or 300 mm.

Die width (mm) Die only, without the scribe lane.

Die height (mm) Single-exposure reticle field is 26 × 33 mm.

Scribe / street width (mm) Sawing lane between die; 0.05–0.15 mm is typical.

Edge exclusion (mm) Unusable rim ring. 2–3 mm is representative.

Defect density (defects per cm²) Per square centimetre, not per square millimetre.

Headline yield model All four are always reported; this one drives the cost.

Clustering parameter α Negative binomial only. Lower = more clustered.

Processed wafer cost ($) Wafer-out, before probe and packaging.

Enter the wafer and die geometry, then calculate to place the die grid and estimate yield.

Scale drawing of the counted die grid on the wafer.

Die placement: exact count versus the closed-form formulas

The four yield models at your defect-area product

Sensitivity: what happens if the defect density is wrong

Fab Yield Sprint Mini-Game

Sweep a cleaning beam across a spinning wafer and scrub particles before they reach die sites. Defect spawn pressure is driven by the defect density you entered above, and the die penalty by your die size, so the game gets harder exactly where the yield model says it should.

Click to Play

Sweep the beam to scrub defects before they reach die sites. Hold the pointer down, or focus the wafer and hold space, to pulse cleaning energy.

Keyboard: arrow keys or WASD steer the beam, space or enter pulses it, and R restarts. Keys only act while the wafer itself has focus, so the calculator above keeps working normally.

Yield --

Good Dies --

Dies Lost --

Run Time 0.0 s

Best Yield --

Wafer Value Saved --

Enter wafer size, die area, and defect density above—then play to feel how each parameter shifts defect pressure and cost.

Semiconductor Wafer Yield Calculator

Introduction to defect-limited die yield on a silicon wafer

How to use the wafer inputs, and which units each one wants

The gross die per wafer formula, and why area division is not it

Four named yield formulas, and why they disagree

A worked example: an 832 mm² accelerator die on a 300 mm wafer

Interpreting the result: gross die, net die and good die

Symbols, units and where each one enters

Limitations and assumptions you should carry with the number

Common questions about wafer yield and die per wafer

Sources for the yield models and wafer geometry

Die placement: exact count versus the closed-form formulas

The four yield models at your defect-area product

Sensitivity: what happens if the defect density is wrong

Fab Yield Sprint Mini-Game

Click to Play

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