Yield Models and Critical Area

1. Theory

• Definition: Yield models predict the fraction of good dies from die area and killer-defect density, and differ in how they treat clustering of defects (Leachman; MIT).
• Poisson: Treats defects as random and independent, $Y=e^{-A{D}_0}$; it is accurate for small dies and low defectivity but pessimistic for large dies (Leachman; NPTEL; Bogdanov et al.).
• Clustering models: Letting defect density vary gives the Murphy, Seeds and negative-binomial models; the cluster parameter $\alpha$ spans Seeds-like to Poisson-like behaviour (Leachman; MIT; Bogdanov et al.).
• Critical area: Only defects whose centre falls in the critical area kill a circuit, so yield integrates critical area against the defect size distribution (MIT; Rakowski and Pleskacz).
• Kill ratio: Not every inspected defect is a killer; the kill ratio is the probability that a defect of a given type and layer kills a chip (Baron et al.).

2. Working Principle

• Measure density: Inline inspection counts defects by layer and type, and a fraction are classified (Baron et al.; Defect Detection Fundamentals page).
• Model size: Defect sizes follow a power law above a peak size near the minimum spacing (Rakowski and Pleskacz; MIT).
• Compute critical area: Layout extraction gives critical area for shorts and opens per layer (Rakowski and Pleskacz; MIT).
• Predict yield: Combine density, kill ratio and critical area per layer and multiply layer yields (Leachman; Rakowski and Pleskacz; Baron et al.).
• Separate losses: Split yield into systematic and random parts from wafer maps (Leachman).

3. Applications

• Yield ramp: Tracking random defect density as yield rises from below 10 per cent to 60 to 80 per cent (NPTEL).
• Design for manufacturing: Critical-area reduction by spacing wires (Rakowski and Pleskacz).
• Inspection prioritization: Ranking layers and defect types by expected killers per chip (Baron et al.).

Die Yield Models

Name	Formula	Related Formulae	Variables/Constants
Poisson yield	$Y=e^{-A{D}_0}$	$Y={\prod }_j{e}^{-A_j{D}_{0,j}}(\text{layers})$	$A:dieorcriticalarea$ $\begin{array}{r}{D}_0:meankiller-defectdensity;Leachman,\\ Eq.2andproductoverlayers;MIT:\\ points,alldefectsfatal,uncorrelated;\\ NPTEL:validatlowdefectivity,slightly\\ pessimistic\end{array}$ $\begin{array}{r}(computed:A{D}_0=0.657gives0.518,\\ matchingLeachma{n}^{\prime }s52percentrandom\\ yield)\end{array}$
Murphy and Seeds	$Y_M={\left[\frac{1-e^{-A{D}_0}}{A{D}_0}\right]}^2$	$Y_S=\frac{1}{1+A{D}_0}$	$\begin{array}{r}Leachman,Eqs.9and10;MIT:Murphyfrom\\ adistributeddensity,Seedsfroman\\ exponentialone;NPTEL:Seedsoptimistic,\\ suitedtoclustering\end{array}$ $\begin{array}{r}(computed:atA{D}_0=1,Y_M=0.400and{Y}_S=\\ 0.500;at2,0.187and0.333)\end{array}$
Negative binomial	$Y={\left(1+\frac{A{D}_0}{\alpha }\right)}^{-\alpha }$	$Y_{BE}={\left(1+A{D}_0\right)}^{-n}$	$\alpha :clusterparameter$ $\begin{array}{r}n:criticallayers;Leachman,Eqs.11and\\ 12;Bogdanovetal.:gamma-mixedPoisson,\\ varianceabovemean,\alpha 0.3to7,\\ clusteringvanishingabove4or5;MIT:\alpha \\ 1.5to2formemoryandmicroprocessors\end{array}$ $\begin{array}{r}(computed:atA{D}_0=1,Y=0.444,0.402\\ and0.386for\alpha =2,5and10)\end{array}$

Critical Area

Name	Formula	Related Formulae	Variables/Constants
Defect size distribution	$f\left(r\right)=\frac{2\left(p-1\right)X_0^{p-1}}{\left(p+1\right)r^p}(r>X_0)$	$f\left(r\right)=\frac{2\left(p-1\right)r}{\left(p+1\right)X_0^2}(r\le X_0)$	$X_0:peakdefectsize$ $\begin{array}{r}p:tailexponent;RakowskiandPleskacz:\\ theseforms,p=3,X_{0}20percentof\\ minimumspacing,D_0=10\mathrm{c}{\mathrm{m}}^{-2};MIT:power\\ lawwithminimumsizeneartheminimum\\ feature;NPTEL:defectsaboveabout10\mu \mathrm{m}\\ rare,lossmainlyfrommediumsizes\end{array}$ $\begin{array}{r}(computed:withp=3thetailfallsas\\ r^{-3},sodoublingsizecutsfrequency8\\ times)\end{array}$
Critical area for shorts	$A_c\left(r\right)=L\left(r-s\right)(r>s)$	${\bar{A}}_c={\int }_s^{\infty }{A}_c\left(r\right)f\left(r\right)dr=\frac{L{X}_0^2}{2s}(p=3)$	$L:parallelrunlength$ $\begin{array}{r}s:spacing;MIT:criticalareaasthe\\ regionwhereadefectcentrecauses\\ failure,computedseparatelyforopensand\\ shorts;RakowskiandPleskacz:shorts\\ only,layout-extracted;two-linegeometry\\ andintegral(computed;myreadingofthe\\ simplestcase)\end{array}$ $\begin{array}{r}(assumed:L=1\mathrm{mm},s=50\mathrm{nm},X_0=10\mathrm{nm};\\ computed:{\bar{A}}_c=1\times 10^{-6}\mathrm{m}{\mathrm{m}}^2,so{D}_0=10\mathrm{c}{\mathrm{m}}^{-2}\\ gives1\times 10^{-7}expectedshorts)\end{array}$
Yield from critical area	$Y_i=\exp \left[-{\int }_0^{\infty }{A}_{cr,i}\left(r\right)D_i\left(r\right)dr\right]$	$Y={\prod }_i{Y}_i,P_{fail}=1-Y$	$\begin{array}{r}{D}_i(r)=D_{0,i}f(r);RakowskiandPleskacz:\\ theserelations,AN3gate{P}_{sh}=1.28\times 10^{-6};\\ MIT:viayieldwith{\lambda }_v=n{p}_v\end{array}$ $\begin{array}{r}(assumed:1\times 10^{8}viaswith{p}_v=1\times 10^{-9};\\ computed:viayield0.905)\end{array}$

Yield from Inspection

Name	Formula	Related Formulae	Variables/Constants
Kill ratio	$pK{R}_{jl}=1-{\psi }_{jl}=1-E\left[e^{-r\left(j\right)a\left(l\right)x}\right]$	$x=\log \left(1+s\right)$	$\begin{array}{r}\psi :survivalprobability,r(j):defect-type\\ factor,a(l):layerfactor\end{array}$ $\begin{array}{r}s:defectsize;Baronetal.,Eqs.1,2\\ and19,multiplicativemodelbest\end{array}$ $\begin{array}{r}(myreading:akillratioof0.2meansone\\ infiveinspecteddefectsofthattypeand\\ layerkillsadie)\end{array}$
Yield from inspection data	$y=\exp \left[-b-{\sum }_j{\sum }_l{\lambda }_{jl}\left(1-{\psi }_{jl}\right)\right]$	${\lambda }_{jl}\left(1-{\psi }_{jl}\right)=\text{expected killers per chip}$	$b:othercauses$ $\begin{array}{r}{\lambda }_{jl}:expecteddefectsoftypejonlayerl\\ perchip;Baronetal.,Eq.35,nearly\\ 1000lots,fewdefectsclassified;Defect\\ DetectionFundamentalspageforcapture\\ andclassification\end{array}$ $\begin{array}{r}(assumed:0.5defectsperchipatkill\\ ratio0.2oneachoffivelayers;\\ computed:y=e^{-0.5}=0.61withb=0)\end{array}$
Random and systematic yield	$Y=Y_S{Y}_R$	$Y_R=e^{-A{D}_0}$	$Y_S:systematicyield$ $\begin{array}{r}{Y}_R:random-defectyield;Leachman:\\ binomial-sigmaestimateof{Y}_{R}fromthebest\\ diesites,130\mathrm{nm}example{Y}_{S}83percent,\\ Y_{R}52percent,D_{0}1.01\mathrm{c}{\mathrm{m}}^{-2}on0.65c{m}^2;\\ NPTEL:startupbelow10percent,mature\\ 60to80percent\end{array}$ $\begin{array}{r}(computed:0.832x0.518=0.431,\\ Leachma{n}^{\prime }s43.1percent;andA{D}_0=0.657)\end{array}$

Important Questions

Ques: How do the Poisson, Murphy, Seeds and negative-binomial models compare?

They differ in how defect density varies. Leachman gives Poisson $e^{-A{D}_0}$, Murphy $[(1-e^{-A{D}_0})/A{D}_0{]}^2$, Seeds $1/(1+A{D}_0)$ and negative binomial $(1+A{D}_0/\alpha {)}^{-\alpha }$. At $A{D}_0$ = 1 they give 0.368, 0.400, 0.500 and, for $\alpha$ = 2, 0.444; at $A{D}_0$ = 2 they give 0.135, 0.187, 0.333 and 0.250 (computed), so the spread widens with die size. Leachman calls Poisson accurate for dies below about 0.25 cm${}^2$ or $A{D}_0$ below 1 and the negative binomial the most flexible for large dies, with Murphy used when $\alpha$ is unknown. NPTEL calls Poisson slightly pessimistic and Seeds optimistic. Bogdanov et al. explain the pessimism: faults cluster, so the variance exceeds the mean. Reference: R. C. Leachman, ‘Yield Modeling and Analysis’, IEOR 130 course document (Spring 2014): Poisson e^(-A D0), Murphy [(1 - e^(-A D0))/(A D0)]^2, Seeds 1/(1 + A D0), negative binomial (1 + A D0/alpha)^(-alpha), Bose-Einstein (1 + A D0)^(-n) over n critical layers, alpha about 10 approximating Poisson, 5 Murphy and 1 Seeds, Poisson accurate for die below about 0.25 cm2 and A D0 below 1, negative binomial most flexible for large die and Murphy used when alpha is unknown, 130 nm example with 0.65 cm2 die, 43 per cent yield, Y_S 83 per cent, Y_R 52 per cent, D0 1.01 per cm2, binomial-sigma method from 372 sites (fetch-tool summary); NPTEL course 103106075 ‘Introduction to Microelectronic Fabrication Processes’, Module 11, Lecture 36 ‘Yield Models and data analysis’: Poisson reasonably valid at low defectivity and slightly pessimistic, Seeds optimistic and suited to clustering, Murphy rarely used, gamma parameter 1 giving Seeds, about 4.2 matching Murphy and infinity Poisson, random defects above about 10 um generally absent, yield loss mainly from medium-size defects, startup yield below 10 per cent rising to 60 to 80 per cent (fetch-tool summary); Yu. I. Bogdanov, N. A. Bogdanova and A. V. Rudnev, ‘Multilevel Clustering Fault Model for IC Manufacture’, arXiv:physics/0310012: Poisson Y = e^-lambda badly pessimistic for large chips because faults cluster, negative binomial Y = (1 + lambda0/a)^-a with variance above the mean, cluster parameter typically 0.3 to 7 with clustering vanishing above about 4 or 5, at 64M complexity and p = 1e-7 yield 0.0012 Poisson against 0.4924 and 0.7388 with one and two clustering levels (fetch-tool summary).

Ques: Which cluster parameter reproduces the Murphy model, 5 or 4.2?

The sources differ. Leachman states that $\alpha$ of about 10 approximates Poisson, 5 Murphy and 1 Seeds, while NPTEL states that a gamma parameter of about 4.2 matches Murphy, with 1 giving Seeds. At $A{D}_0$ = 1, Murphy gives 0.400 and the negative binomial gives 0.402 for $\alpha$ = 5 and 0.408 for 4.2; at $A{D}_0$ = 2, Murphy gives 0.187 against 0.186 and 0.195 (computed). The value 5 stays within 1 per cent of Murphy over this range, whereas 4.2 drifts 4 per cent high at $A{D}_0$ = 2, so Leachman’s 5 is more likely the better match for typical dies (my reading). Bogdanov et al. add that clustering effectively disappears above 4 or 5, and MIT quotes 1.5 to 2 for real memory and microprocessor products. Reference: R. C. Leachman, ‘Yield Modeling and Analysis’, IEOR 130 course document (Spring 2014): Poisson e^(-A D0), Murphy [(1 - e^(-A D0))/(A D0)]^2, Seeds 1/(1 + A D0), negative binomial (1 + A D0/alpha)^(-alpha), Bose-Einstein (1 + A D0)^(-n) over n critical layers, alpha about 10 approximating Poisson, 5 Murphy and 1 Seeds, Poisson accurate for die below about 0.25 cm2 and A D0 below 1, negative binomial most flexible for large die and Murphy used when alpha is unknown, 130 nm example with 0.65 cm2 die, 43 per cent yield, Y_S 83 per cent, Y_R 52 per cent, D0 1.01 per cm2, binomial-sigma method from 372 sites (fetch-tool summary); NPTEL course 103106075 ‘Introduction to Microelectronic Fabrication Processes’, Module 11, Lecture 36 ‘Yield Models and data analysis’: Poisson reasonably valid at low defectivity and slightly pessimistic, Seeds optimistic and suited to clustering, Murphy rarely used, gamma parameter 1 giving Seeds, about 4.2 matching Murphy and infinity Poisson, random defects above about 10 um generally absent, yield loss mainly from medium-size defects, startup yield below 10 per cent rising to 60 to 80 per cent (fetch-tool summary); Yu. I. Bogdanov, N. A. Bogdanova and A. V. Rudnev, ‘Multilevel Clustering Fault Model for IC Manufacture’, arXiv:physics/0310012: Poisson Y = e^-lambda badly pessimistic for large chips because faults cluster, negative binomial Y = (1 + lambda0/a)^-a with variance above the mean, cluster parameter typically 0.3 to 7 with clustering vanishing above about 4 or 5, at 64M complexity and p = 1e-7 yield 0.0012 Poisson against 0.4924 and 0.7388 with one and two clustering levels (fetch-tool summary); MIT OpenCourseWare 2.830J Control of Manufacturing Processes, Lecture 10 ‘Yield Modeling’ (11 March 2008): Poisson with critical area and defect density, Murphy with distributed defect density, Seeds with exponential density, negative binomial with gamma density and cluster parameter alpha, high alpha little clustering, approaching Poisson for large and Seeds for small alpha, typical memory and microprocessors alpha 1.5 to 2, critical area as region where a defect centre causes failure computed separately for opens and shorts, power-law size distribution with minimum size about the minimum feature, via yield from per-via failure probability with lambda = n p_v (fetch-tool summary).

Ques: What is critical area and how is it combined with the defect size distribution?

It is where a defect centre must land to cause a fault. MIT defines it this way and computes it separately for shorts and opens. Rakowski and Pleskacz use a size distribution rising linearly to a peak $X_0$, 20 per cent of minimum spacing, and falling as $r^{-3}$ above it, with yield $\exp [-\int A_{cr}(r)D(r)dr]$ multiplied over defect types. For two parallel lines of run length L and spacing s, a defect of size r shorts them over $L(r-s)$ (computed; my reading), and integrating gives $L{X}_0^2/(2s)$; for an assumed 1 $\mathrm{mm}$ run at 50 $\mathrm{nm}$ spacing with $X_0$ = 10 $\mathrm{nm}$ that is $10^{-6}$ $\mathrm{m}{\mathrm{m}}^2$ (computed). Doubling the spacing halves it, which is why wire spreading improves yield. NPTEL notes that medium-size defects cause most loss. Reference: MIT OpenCourseWare 2.830J Control of Manufacturing Processes, Lecture 10 ‘Yield Modeling’ (11 March 2008): Poisson with critical area and defect density, Murphy with distributed defect density, Seeds with exponential density, negative binomial with gamma density and cluster parameter alpha, high alpha little clustering, approaching Poisson for large and Seeds for small alpha, typical memory and microprocessors alpha 1.5 to 2, critical area as region where a defect centre causes failure computed separately for opens and shorts, power-law size distribution with minimum size about the minimum feature, via yield from per-via failure probability with lambda = n p_v (fetch-tool summary); M. Rakowski and W. A. Pleskacz (Warsaw University of Technology), ‘The influence of yield model parameters on the probability of defect occurrence’: size distribution 2(p - 1) r/((p + 1) X0^2) below X0 and 2(p - 1) X0^(p - 1)/((p + 1) r^p) above, p = 3, X0 20 per cent of minimum spacing, D0 = 10 per cm2, Y_i = exp(-integral A_cr D_i dr), Y = product over defect types, P_sh = 1 - Y, 0.8 um AN3 gate example P_sh = 1.28e-6 (fetch-tool summary); NPTEL course 103106075 ‘Introduction to Microelectronic Fabrication Processes’, Module 11, Lecture 36 ‘Yield Models and data analysis’: Poisson reasonably valid at low defectivity and slightly pessimistic, Seeds optimistic and suited to clustering, Murphy rarely used, gamma parameter 1 giving Seeds, about 4.2 matching Murphy and infinity Poisson, random defects above about 10 um generally absent, yield loss mainly from medium-size defects, startup yield below 10 per cent rising to 60 to 80 per cent (fetch-tool summary).

Ques: How is yield predicted from inspection data with kill ratios?

By weighting defect counts by the probability that each kills. Baron et al. define the kill ratio of a defect type on a layer as the probability that it kills a chip, $1-\psi$, with survival modelled as $\exp [-r(j)a(l)\log (1+s)]$, and predict yield as $\exp [-b-\sum {\lambda }_{jl}(1-{\psi }_{jl})]$, where ${\lambda }_{jl}$ is the expected number of such defects per chip. With an assumed 0.5 defects per chip at kill ratio 0.2 on each of five layers, the expected killers total 0.5 and yield is 0.61 (computed). They fitted nearly 1000 lots while classifying only a small portion of defects, so capture and classification quality limit such forecasts (Defect Detection Fundamentals page; my reading). Reference: M. Baron, A. Takken, E. Yashchin and M. Lanzerotti, ‘Factorial Analysis and Forecasting of Integrated-Circuit Yield’, IBM Research Report RC23386 (2004): kill ratio as the probability that a defect of type j on layer l kills a chip, pKR = 1 - psi = 1 - E exp(-r(j) a(l) x) with x = log(1 + s), yield y = exp(-b - sum lambda_jl (1 - psi_jl)), multiplicative model best, nearly 1000 lots, only a small portion of defects classified (fetch-tool summary); Defect Detection Fundamentals page of this vault.

Ques: How are systematic and random yield losses separated?

By comparing the best die sites with the average. Leachman writes yield as systematic times random, $Y=Y_S{Y}_R$, and estimates $Y_R$ from the highest yields across stacked wafer maps using a binomial-sigma correction: with 372 sites and a best site at 57 per cent, $Y_R$ is 51.8 per cent, so a 43.1 per cent average implies $Y_S$ of 83.2 per cent. For the 130 $\mathrm{nm}$ example on a 0.65 cm${}^2$ die, $Y_R$ of 52 per cent corresponds to $D_0$ of 1.01 $\mathrm{c}{\mathrm{m}}^{-2}$; $0.832\times 0.518$ = 0.431 and $A{D}_0$ = 0.657 confirm the numbers (computed). NPTEL notes that startup yields can be below 10 per cent and mature ones 60 to 80 per cent. Reference: R. C. Leachman, ‘Yield Modeling and Analysis’, IEOR 130 course document (Spring 2014): Poisson e^(-A D0), Murphy [(1 - e^(-A D0))/(A D0)]^2, Seeds 1/(1 + A D0), negative binomial (1 + A D0/alpha)^(-alpha), Bose-Einstein (1 + A D0)^(-n) over n critical layers, alpha about 10 approximating Poisson, 5 Murphy and 1 Seeds, Poisson accurate for die below about 0.25 cm2 and A D0 below 1, negative binomial most flexible for large die and Murphy used when alpha is unknown, 130 nm example with 0.65 cm2 die, 43 per cent yield, Y_S 83 per cent, Y_R 52 per cent, D0 1.01 per cm2, binomial-sigma method from 372 sites (fetch-tool summary); NPTEL course 103106075 ‘Introduction to Microelectronic Fabrication Processes’, Module 11, Lecture 36 ‘Yield Models and data analysis’: Poisson reasonably valid at low defectivity and slightly pessimistic, Seeds optimistic and suited to clustering, Murphy rarely used, gamma parameter 1 giving Seeds, about 4.2 matching Murphy and infinity Poisson, random defects above about 10 um generally absent, yield loss mainly from medium-size defects, startup yield below 10 per cent rising to 60 to 80 per cent (fetch-tool summary).

References

1. R. C. Leachman, “Yield modeling and analysis,” IEOR 130 Methods of Manufacturing Improvement, course document, 2014. Available: https://fog.misty.com/perry/cod/references/yield_models.pdf
2. MIT OpenCourseWare, 2.830J/6.780J Control of Manufacturing Processes, Lecture 10, “Yield modeling,” 2008. Available: https://www.ocw.mit.edu/courses/2-830j-control-of-manufacturing-processes-sma-6303-spring-2008/4aff1e21de13870355ef44dbe71f45c6_lecture10.pdf
3. NPTEL, “Introduction to Microelectronic Fabrication Processes,” Module 11, Lecture 36, “Yield models and data analysis.” Available: https://archive.nptel.ac.in/content/storage2/courses/103106075/Courses/Lecture36.html
4. Yu. I. Bogdanov, N. A. Bogdanova and A. V. Rudnev, “Multilevel clustering fault model for IC manufacture,” arXiv:physics/0310012. Available: https://arxiv.org/pdf/physics/0310012
5. M. Rakowski and W. A. Pleskacz, “The influence of yield model parameters on the probability of defect occurrence,” Warsaw University of Technology. Available: https://yadda.icm.edu.pl/baztech/element/bwmeta1.element.baztech-article-BAT8-0009-0076/c/httpwww_itl_waw_plczasopismajtit20073101.pdf
6. M. Baron, A. Takken, E. Yashchin and M. Lanzerotti, “Factorial analysis and forecasting of integrated-circuit yield,” IBM Research Report RC23386, 2004. Available: https://dominoweb.draco.res.ibm.com/reports/rc23386.pdf
7. Defect Detection Fundamentals page of this vault.

Related pages in this vault: Defect Detection Fundamentals, Statistical Process Control (SPC), Patterned-Wafer Optical Inspection, Weibull Distribution, Resolution Enhancement and Multiple Patterning, Defect Sources and Classification, Wafer-Map Spatial Signature Analysis, Defect SPC and Excursion Monitoring, Defect Data Handling with Python and Perl, Defect-to-Yield Correlation and Kill Ratio.