Yield Models That Turn Defect Density into Die Cost

A wafer is a fixed cost. The price of turning it into working parts depends on how many dies fit on it and what fraction of them work, and the second factor has a much stronger dependence on die area than the first. Yield modelling exists to make that dependence quantitative, because the decision to enlarge a die, add a core, or accept a larger SRAM array is a decision about cost per working unit, and the cost curve is not linear.

Counting dies and counting working dies

The number of die sites on a wafer is a geometry problem with an edge correction. A wafer is a disc with a flat or a notch; a die is a rectangle of area A. The gross count for a grid-aligned layout is roughly the usable wafer area divided by A, with the fraction lost to partial dies at the edge on the order of one die perimeter’s worth of sites. For a 300 mm wafer, usable area is about seventy thousand square millimetres, so a 100 mm² die fits about seven hundred sites before edge loss and several tens fewer after it.

The defect-limited yield is a separate factor. It comes from the probability that a die contains no fatal defect. Defects land on the wafer at some density, each one lands somewhere, and a defect is fatal if it lands in a critical area, meaning a region where a particle of that size causes a short or an open. The critical area depends on the layout: two wires running close together have a critical area proportional to their length and to the size of particle that would bridge the gap.

The simplest model assumes defects land independently and uniformly. If the expected number of fatal defects on a die is the defect density D times the die area A, and the count follows a Poisson distribution, then the probability of zero defects is exp(−D·A). This is the Poisson yield model, and it predicts a yield of about 37 percent when the expected count reaches one.

import math

def poisson(d0, a):
    return math.exp(-d0 * a)

def murphy(d0, a):
    # integrate the Poisson yield over a triangular distribution of
    # local defect densities, which is what produces the squared term
    k = d0 * a
    if k == 0:
        return 1.0
    return ((1 - math.exp(-k)) / k) ** 2

for a in (0.5, 1.0, 2.0):
    print(a, round(poisson(0.5, a), 3), round(murphy(0.5, a), 3))

At a defect density of half a defect per square centimetre, the models diverge as area grows. For a die of one square centimetre the Poisson estimate is 0.607 and Murphy’s is 0.619; at two square centimetres they are 0.368 and 0.400. Murphy’s clustering is mild because its density distribution is narrow. A negative binomial with a clustering parameter of one, which corresponds to strong clustering, gives 0.667 at one square centimetre and 0.500 at two — a yield about 36 percent higher than Poisson at the larger area, and therefore a cost per working die about a quarter lower. The Poisson model is not wrong as a model of independently placed defects; it is the assumption of independence that is wrong, and the size of the error is set by how strongly the defects cluster.

Clustering and the negative binomial

Defects do not land uniformly. A particle shed by a specific piece of equipment lands on many wafers from that tool, and within a wafer it lands preferentially near the region that tool handled. The result is clustering: some regions have many defects, most have few. Clustering raises the yield compared with the uniform model at the same average density, because the dies that get hit are hit repeatedly and the number of dies with exactly one defect is smaller than independence would predict. A die is lost to the first defect it receives; a second defect in the same die costs nothing additional.

The negative binomial distribution captures this with a clustering parameter. The yield becomes a function of the mean defect count per die divided by the clustering parameter, raised to the negative power of that parameter. As the clustering parameter grows the model approaches Poisson; as it falls, yield improves at fixed average density. The parameter is fitted to measured wafer maps, it is process-specific, and it changes as defect sources are eliminated, so it is a calibration constant rather than a property of the technology node.

Murphy’s model takes a different route to the same conclusion. Rather than assuming a fixed defect density, it assumes a distribution of densities across the wafer, symmetric about a mean and triangular in shape, and integrates the Poisson yield over that distribution. The resulting expression is the square of the ratio of one minus the exponential of the mean count to the mean count. It clusters more than Poisson and less than a low-parameter negative binomial, and it has the property of depending only on the mean count, which made it tractable before the distribution fit was standard practice.

The choice of model matters most at the extrapolation boundary. A fab estimating the yield of a new design from the measured yield of a smaller test chip must decide whether the defect density is constant with area or whether the clustering changes. Fitting a model to one area and predicting another is where the error appears, and the standard practice is to fit to at least two areas, since the two-parameter form has enough freedom to match a curve through multiple points.

Redundancy converts yield into a spare-part problem

A memory array large enough to be worth its area will contain defects. The standard response is to build in spare rows and spare columns, test each array after fabrication, and remap the addresses of a faulty row or column to a spare by blowing fuses or by programming a latch. The die passes if the number of faulty lines is within the number of spares, and the probability of that is the sum over the distribution of fault counts up to the spare count.

The arithmetic is that a large array with the same defect density as logic can pass even when it contains dozens of defects, because a single faulty cell in a column of a thousand bits costs one spare column, not the thousand bits of yield that replacing cells individually would imply. This is why the yield of a memory die can be far higher than the Poisson model applied to its area would suggest, and why the model must be applied to the granularity of the replaceable unit rather than to the die. The relevant question is how many repairable units fail, not how many defects exist.

Repair inverts the area penalty in another way. Adding spares costs area, and the spares themselves must be tested and must be free of fatal defects. The optimum number of spares is where the marginal die yield gained by one more spare equals the yield lost by the extra area it consumes. That optimum is typically small, on the order of a few percent of the array’s rows, and it depends on the defect density: a mature process needs fewer spares, and the same design sold on a less mature process may need more.

The test itself is not free. Finding the faulty lines requires reading and writing every cell, which is the dominant test-time cost for a large array and is discussed in Wafer Probe and the Economics of Test Coverage. Die cost therefore has three terms that move in opposite directions with array size: the area term rises, the repair term falls as a fraction of dies needing it, and the test term rises with the number of cells.

Wafer size, edge loss, and the shape of the cost curve

The move to larger wafers is a packing argument. Doubling the diameter roughly quadruples the area while doubling the perimeter, so the fraction of the wafer consumed by incomplete dies at the edge falls. A 300 mm wafer has about 2.25 times the area of a 200 mm wafer and loses a smaller fraction to partial dies, which is why the industry moved. The gain in usable die per wafer is a few percent beyond the area ratio, and that is the entire economic case for the transition; the tooling change is expensive enough that a few percent matters.

Cost per working die is the wafer cost divided by the product of gross die count and yield. Because yield falls exponentially in area while die count falls inversely with area, the cost curve has a shallow minimum: making dies too small multiplies the number of packages, the number of test insertions, and the number of assembly operations, all of which have their own cost floors, while making them too large lets the yield term dominate. The minimum is not a property of the process alone; it depends on package cost and test cost, which is why a cost-optimised design for a cheap plastic package lands at a different area than one for a large substrate with many interconnects.

The other edge effect is that yield is not uniform across the wafer. Dies near the edge receive less uniform resist coverage during spin coating, experience a different thermal history during anneal, and sit in a region where the lithography focus is harder to control. A wafer map of passing dies typically shows an interior region of good yield and a ring of poorer yield near the edge, with the exact shape determined by which process is currently limiting. Fabs sometimes compensate by accepting different grades from different radial zones, which is one source of parts sold at different frequencies from the same wafer.

Where the model stops being predictive

The models above assume defects are the only failure mode. A design can also fail for parametric reasons: every transistor works but the circuit runs too slowly at the slow corner, or draws too much current at the fast corner. Parametric yield has a different shape, a tail of a distribution rather than an exponential in area, and it is what limits frequency binning. A design binned at three frequencies sells the same die at three prices, and the fraction in each bin is a property of the process variation discussed in Process Corners and Why Timing Sign-off Is a Set of Scenarios, not of the defect density.

The models also assume yield learning is monotonic, which it is not. Introducing a new layer or a new material resets part of the learning curve, and the defect density used to predict mature yield is the mature figure, not the figure at introduction. Estimating a new design’s cost from a mature process’s yield therefore understates the early cost and overstates the mature one, and any decision that depends on the difference between those two should be made with the curve rather than the endpoint.