Defect-to-Yield Correlation and Kill Ratio

1. Theory

• Definition: Defect-to-yield correlation overlays in-line defect maps on die sort results to measure how often each defect class kills a die (the kill ratio) and how much yield each class costs (TI; Applied Materials).
• Kill ratio: Killed die over all die carrying a defect type, measured on die with a single defect (TI).
• Baseline correction: Dies fail for other reasons too, so the observed failure of defective dies is corrected by the yield of defect-free dies (Applied Materials).
• Vital few: Only some anomalies are defects; weighting classes by kill probability turns anomaly counts into expected faults and defect-limited yield (Riley).

2. Working Principle

• Overlay: Join adder defects to die bins; group dies by the defect class they carry (Applied Materials; Defect Data Handling with Python and Perl page).
• Estimate: $P_k=N_D/N_T$ on single-defect dies, or $FP=1-(1/c)(N_{good}/N)$ with baseline yield $c$ (TI; Applied Materials).
• Combine: Several defects in one die via $1-\prod (1-C{F}_i)$; per-type yield limits multiplied (Applied Materials; TI).
• Rank: Yield loss per step and bin, $FP\cdot N_b/N$, ranked into a Pareto (Applied Materials).

3. Applications

• Yield-impact Pareto: Which classes and layers to attack first (Applied Materials; Riley).
• Yield prediction: Layer limits from density and kill ratio, as on the Yield Models and Critical Area page (TI).
• Inspection value: Linking inspection to yield and reliability (KLA).

Kill Ratio from Overlay

Name	Formula	Related Formulae	Variables/Constants
Empirical killer probability	$P_{k,i}=\frac{N_{D,i}}{N_{T,i}}$	$\text{single-defect die only}$	$N_{D,i}:killeddie$ $\begin{array}{r}{N}_{T,i}:alldiecarryingdefecttypei;TI:\\ diewithseveraldefectsremovedfirst\end{array}$ $\begin{array}{r}(myreading:theoverlayisajoinof\\ defectandprobetables,asontheDefect\\ DataHandlingwithPythonandPerlpage)\end{array}$
Correcting for baseline yield	$F{P}_b=1-\frac{1}{c}\frac{N_{good,b}}{N_b}$	$FP<0\Rightarrow FP=0$	$\begin{array}{r}c:yieldofdieswithoutdefects;Applied\\ Materials:subsetsofdieswithbin-b\\ defects,addersonly\end{array}$ $\begin{array}{r}(computed:withc=0.9and60of100\\ defectivediegood,FP=0.33against0.40\\ withoutthecorrection)\end{array}$
Several defects per die	$C{F}_{die}=1-{\prod }_i\left(1-C{F}_i\right)$	$Cf=1-(1-P_s)(1-P_o)$	$\begin{array}{r}AppliedMaterials:criticalityfromshort\\ andopenprobabilitiesbinnedintenths\end{array}$ $\begin{array}{r}(computed:defectsof0.3and0.5give\\ 0.65forthedie)\end{array}$

Yield Limits and Loss

Name	Formula	Related Formulae	Variables/Constants
Yield limit per defect type	$Y_i={\left(1+\frac{P_{k,i}{D}_{k,i}A}{\alpha }\right)}^{-\alpha }$	$Y={\prod }_i{Y}_i\cdot Y_C$	$\begin{array}{r}TI:forunclustereddefects,clustered\\ defectsas{Y}_C=1-deaddie/totaldie;\\ YieldModelsandCriticalAreapage\end{array}$ $\begin{array}{r}(computed:P_k=0.3,D=0.5\mathrm{c}{\mathrm{m}}^{-2},A=1\mathrm{c}{\mathrm{m}}^2\\ and\alpha =2giveY=0.865)\end{array}$
Defect and non-defect yield	$Y_{final}=Y_{nd}{Y}_d$	$Y{L}_b=F{P}_b\frac{N_b}{N}$	$\begin{array}{r}AppliedMaterials:yieldlossperstepand\\ bin,averagedoverhistory,addersonly\end{array}$ $\begin{array}{r}(myreading:rankingYLbystepandbin\\ givestheyield-impactPareto)\end{array}$
Single-defect die available	$\frac{P(1)}{P(\ge 1)}=\frac{\lambda e^{-\lambda }}{1-e^{-\lambda }}$	$\lambda =1,2,4:0.58,0.31,0.075$	$\begin{array}{r}TI:about60,25and4percentusable,15\\ timesthesampleat4defectsperdie\end{array}$ $\begin{array}{r}(computed,assumingPoissondefects:the\\ ratioofsamplesizesis7.8)\end{array}$

Vital Few and Defect-Limited Yield

Name	Formula	Related Formulae	Variables/Constants
Weighted kill ratio	$K=\frac{{\sum }_i{p}_i{n}_i}{N},F=KA$	$K=\frac{0+10+10+24}{100}=0.44$	$\begin{array}{r}Riley:classifiedanomalyshares40,10,\\ 20and30percentwithp=0,1,0.5and\\ 0.8,1000anomaliesgiving440faults\end{array}$ $(computed:KandFreproduced)$
Defect-limited yield	$DLY=\frac{e^{-f}{D}_A^{\prime }+D_O}{D_I},f=Ka$	$a=\frac{A}{D_A},\text{clean die}=\frac{D_O}{D_I}$	$D_O:diewithoutanomalies$ $D_I:dieinspected$ $\begin{array}{r}{D}_A:anomalousdie;Riley:aqualitymetric,\\ notayieldpredictor,cumulativeasa\\ product\end{array}$ $\begin{array}{r}(myreading:clean-diefractionisthe\\ lowerbound,DLYtheversionweightedby\\ killratio)\end{array}$
Why it matters	$\text{yield}\leftrightarrow \text{reliability}$	$\text{over-inspect rather than under-inspect}$	$\begin{array}{r}KLA:killerdefectsfailatprobe,latent\\ oneslater;DefectDetectionFundamentals\\ pageforcapturerate\end{array}$ $\begin{array}{r}(myreading:lossfromdetecteddefects\\ shouldbescaledbycapturerateto\\ estimatetotalloss)\end{array}$

Important Questions

Ques: How is a kill ratio measured from inspection and sort data?

By overlaying defect maps on die results. TI’s patent defines the killer probability of type $i$ as $N_{D,i}/N_{T,i}$: die killed over all die carrying that defect type, computed after eliminating die with multiple defects. Applied Materials overlays inline defects with sort yield to deduce fail probabilities by class and uses adders only, defects first seen at the inspected layer. Their per-step, per-bin yield loss is $YL=FP\cdot N_b/N$, with $N_b$ the dies carrying bin-b defects. In my reading, the overlay is a left join of die results to defects, as on the Defect Data Handling with Python and Perl page, followed by grouping by defect class. Reference: N. Atchison and R. Ross (Texas Instruments), ‘Method for the calculation of wafer probe yield limits from in-line defect monitor data’, US Patent 6,324,481 (2001): killer probability Pk_i = N_Di/N_Ti, the killing die over all die with defect type i, calculated after eliminating die with multiple defects, per-type limit Y_i = 1/(1 + Pk_i Dk_i A/alpha)^alpha for unclustered defects, clustered defects such as scratches as Y_C = 1 - dead die from clusters/total die tested, overall yield as the product of the limits, at 1 defect/cm2 on a 1 cm2 die about 63.2 per cent of die defective and about 60 per cent usable, about 25 per cent usable at 2/cm2 and about 4 per cent at 4/cm2, needing 15 times the sample at 4/cm2 than at 1/cm2 (fetch-tool summary); R. Shimshi, Y. Nehmadi, V. Svidenko, A. T. Schwarm and S. Jawaharlal (Applied Materials), ‘Dynamic inline yield analysis and prediction of a defect limited yield using inline inspection defects’, US Patent 7,937,179 (2011): fail probabilities of defect classes deduced empirically by overlaying inline defects with sort yield, Y_final = Y_non-defects Y_defects with Y_defects = 1 - YL_defects, FP(s,b) = 1 - (1/c)(good dies in subset C/all dies in subset C) with c the non-defect yield and negative values set to 0, YL(s,b) = FP(s,b) N(s,b)/N, criticality factor Cf = 1 - (1 - Ps)(1 - Po) from short and open probabilities binned in steps of 0.1, defects in one die combined as CF_die = 1 - prod(1 - CF_i), defect data of adders only, defects originating first in the inspected layer (fetch-tool summary); Defect Data Handling with Python and Perl page of this vault.

Ques: Why must the kill ratio be corrected for baseline yield?

Because defect-free dies also fail. Applied Materials writes the fail probability for dies carrying bin-b defects as $FP=1-(1/c)(N_{good}/N)$, where $c$ is the yield of dies without defects, and sets negative values to zero. Total yield is $Y_{final}=Y_{nd}{Y}_d$. If $c$ is 0.9 and 60 of 100 defective dies pass, FP is 0.33, against 0.40 from the raw failure fraction (computed). The raw figure blames the defects for losses that would have happened anyway. TI’s simple ratio $N_D/N_T$ has no such correction. In my reading, it overstates the kill ratio whenever non-defect yield is well below 1, for example on a new product where parametric or design losses are still large, so the correction matters most early in a ramp. Reference: R. Shimshi, Y. Nehmadi, V. Svidenko, A. T. Schwarm and S. Jawaharlal (Applied Materials), ‘Dynamic inline yield analysis and prediction of a defect limited yield using inline inspection defects’, US Patent 7,937,179 (2011): fail probabilities of defect classes deduced empirically by overlaying inline defects with sort yield, Y_final = Y_non-defects Y_defects with Y_defects = 1 - YL_defects, FP(s,b) = 1 - (1/c)(good dies in subset C/all dies in subset C) with c the non-defect yield and negative values set to 0, YL(s,b) = FP(s,b) N(s,b)/N, criticality factor Cf = 1 - (1 - Ps)(1 - Po) from short and open probabilities binned in steps of 0.1, defects in one die combined as CF_die = 1 - prod(1 - CF_i), defect data of adders only, defects originating first in the inspected layer (fetch-tool summary); N. Atchison and R. Ross (Texas Instruments), ‘Method for the calculation of wafer probe yield limits from in-line defect monitor data’, US Patent 6,324,481 (2001): killer probability Pk_i = N_Di/N_Ti, the killing die over all die with defect type i, calculated after eliminating die with multiple defects, per-type limit Y_i = 1/(1 + Pk_i Dk_i A/alpha)^alpha for unclustered defects, clustered defects such as scratches as Y_C = 1 - dead die from clusters/total die tested, overall yield as the product of the limits, at 1 defect/cm2 on a 1 cm2 die about 63.2 per cent of die defective and about 60 per cent usable, about 25 per cent usable at 2/cm2 and about 4 per cent at 4/cm2, needing 15 times the sample at 4/cm2 than at 1/cm2 (fetch-tool summary).

Ques: How are dies with several defects handled?

Either excluded or combined. TI eliminates die with multiple defects before computing killer probabilities, so each die’s fate can be blamed on one defect. Applied Materials keeps them and combines all defects in a die into one virtual defect with $C{F}_{die}=1-\prod (1-C{F}_i)$, where each defect’s criticality is $1-(1-P_s)(1-P_o)$ from its short and open probabilities. Two defects of 0.3 and 0.5 give a die value of 0.65 (computed). Exclusion is simpler but wastes data, and the waste grows with defect density. Combining uses all dies but needs a criticality value for every defect first, which in turn needs layout or class information. At high defect density, where few dies carry a single defect, combining is the only practical choice (my reading). Reference: N. Atchison and R. Ross (Texas Instruments), ‘Method for the calculation of wafer probe yield limits from in-line defect monitor data’, US Patent 6,324,481 (2001): killer probability Pk_i = N_Di/N_Ti, the killing die over all die with defect type i, calculated after eliminating die with multiple defects, per-type limit Y_i = 1/(1 + Pk_i Dk_i A/alpha)^alpha for unclustered defects, clustered defects such as scratches as Y_C = 1 - dead die from clusters/total die tested, overall yield as the product of the limits, at 1 defect/cm2 on a 1 cm2 die about 63.2 per cent of die defective and about 60 per cent usable, about 25 per cent usable at 2/cm2 and about 4 per cent at 4/cm2, needing 15 times the sample at 4/cm2 than at 1/cm2 (fetch-tool summary); R. Shimshi, Y. Nehmadi, V. Svidenko, A. T. Schwarm and S. Jawaharlal (Applied Materials), ‘Dynamic inline yield analysis and prediction of a defect limited yield using inline inspection defects’, US Patent 7,937,179 (2011): fail probabilities of defect classes deduced empirically by overlaying inline defects with sort yield, Y_final = Y_non-defects Y_defects with Y_defects = 1 - YL_defects, FP(s,b) = 1 - (1/c)(good dies in subset C/all dies in subset C) with c the non-defect yield and negative values set to 0, YL(s,b) = FP(s,b) N(s,b)/N, criticality factor Cf = 1 - (1 - Ps)(1 - Po) from short and open probabilities binned in steps of 0.1, defects in one die combined as CF_die = 1 - prod(1 - CF_i), defect data of adders only, defects originating first in the inspected layer (fetch-tool summary).

Ques: What share of dies is usable for single-defect kill ratios, and which figures are right?

For random defects, about 58, 31 and 7.5 per cent of defective die. TI states that at 1 defect per die about 63.2 per cent of die are defective and about 60 per cent usable, falling to about 25 per cent at 2 and about 4 per cent at 4. It concludes that 15 times the sample is needed at 4 as at 1. For Poisson defects, the single-defect share of defective die is $\lambda e^{-\lambda }/(1-e^{-\lambda })$: 0.58, 0.31 and 0.075, a sample ratio of 7.8 (computed). TI’s first figure and its 63.2 per cent match Poisson, but its 25 and 4 per cent do not. The Poisson values are more likely correct for random defects, and TI’s lower figures probably rest on unstated assumptions (my reading). Reference: N. Atchison and R. Ross (Texas Instruments), ‘Method for the calculation of wafer probe yield limits from in-line defect monitor data’, US Patent 6,324,481 (2001): killer probability Pk_i = N_Di/N_Ti, the killing die over all die with defect type i, calculated after eliminating die with multiple defects, per-type limit Y_i = 1/(1 + Pk_i Dk_i A/alpha)^alpha for unclustered defects, clustered defects such as scratches as Y_C = 1 - dead die from clusters/total die tested, overall yield as the product of the limits, at 1 defect/cm2 on a 1 cm2 die about 63.2 per cent of die defective and about 60 per cent usable, about 25 per cent usable at 2/cm2 and about 4 per cent at 4/cm2, needing 15 times the sample at 4/cm2 than at 1/cm2 (fetch-tool summary).

Ques: How does the vital-few kill ratio turn anomalies into expected faults?

By weighting each class by its kill probability. Riley distinguishes anomalies from defects, the vital few, and defines $K=\sum p_i{n}_i/N$ over classified anomalies, with expected faults $F=KA$. In his example, 1000 anomalies are classed 40, 10, 20 and 30 per cent as types A to D, with kill probabilities 0, 1, 0.5 and 0.8. That gives $K=0.44$ and 440 faults (computed check), from 600 potentially harmful defects. His defect-limited yield, $(e^{-f}{D}_A^{\prime }+D_O)/D_I$ with $f=Ka$, folds this into die counts. He warns that DLY is a quality metric, not a yield predictor, and that cumulative DLY over several levels is the product of the level values. The kill probabilities themselves must come from overlay studies such as those of TI and Applied Materials (my reading). Reference: S. L. Riley, ‘Semiconductor defect management: separating the vital few from the trivial many’, slides (19 November 2009): defects as the vital-few subset of anomalies, weighted kill ratio K = sum(p_i n_i)/N over classified anomalies, faults F = K A, per-type K_i = p_i n_i/N, DLY = (e^(-f) D_A’ + D_O)/D_I with f = K a and a = A/D_A, a mixed random and clustered form, percent clean die D_O/D_I, DLY a quality metric not a yield predictor, cumulative DLY as a product over levels, example of 1000 anomalies with classes A to D at 40, 10, 20 and 30 per cent and p = 0, 1, 0.5 and 0.8 giving 600 defects, K = 0.44 and 440 faults (fetch-tool summary).

Ques: How do kill ratios become a yield-impact Pareto and a yield prediction?

Through per-type yield limits and per-bin losses. TI computes a limit for each type, $Y_i=(1+P_{k,i}{D}_{k,i}A/\alpha {)}^{-\alpha }$, treats clustered defects separately, and multiplies the limits. With $P_k=0.3$, 0.5 $\mathrm{c}{\mathrm{m}}^{-2}$, 1 $\mathrm{c}{\mathrm{m}}^2$ and $\alpha =2$, that is 0.865 (computed). Applied Materials gives losses per step and bin, which ranked form the Pareto (my reading). KLA argues that over-inspecting is more cost-effective than under-inspecting, because yield and reliability defects are linked. In my reading, losses estimated from detected defects should be divided by the capture rate of the Defect Detection Fundamentals page to estimate the total, since undetected defects also kill dies. The resulting limits feed the models of the Yield Models and Critical Area page. Reference: N. Atchison and R. Ross (Texas Instruments), ‘Method for the calculation of wafer probe yield limits from in-line defect monitor data’, US Patent 6,324,481 (2001): killer probability Pk_i = N_Di/N_Ti, the killing die over all die with defect type i, calculated after eliminating die with multiple defects, per-type limit Y_i = 1/(1 + Pk_i Dk_i A/alpha)^alpha for unclustered defects, clustered defects such as scratches as Y_C = 1 - dead die from clusters/total die tested, overall yield as the product of the limits, at 1 defect/cm2 on a 1 cm2 die about 63.2 per cent of die defective and about 60 per cent usable, about 25 per cent usable at 2/cm2 and about 4 per cent at 4/cm2, needing 15 times the sample at 4/cm2 than at 1/cm2 (fetch-tool summary); R. Shimshi, Y. Nehmadi, V. Svidenko, A. T. Schwarm and S. Jawaharlal (Applied Materials), ‘Dynamic inline yield analysis and prediction of a defect limited yield using inline inspection defects’, US Patent 7,937,179 (2011): fail probabilities of defect classes deduced empirically by overlaying inline defects with sort yield, Y_final = Y_non-defects Y_defects with Y_defects = 1 - YL_defects, FP(s,b) = 1 - (1/c)(good dies in subset C/all dies in subset C) with c the non-defect yield and negative values set to 0, YL(s,b) = FP(s,b) N(s,b)/N, criticality factor Cf = 1 - (1 - Ps)(1 - Po) from short and open probabilities binned in steps of 0.1, defects in one die combined as CF_die = 1 - prod(1 - CF_i), defect data of adders only, defects originating first in the inspected layer (fetch-tool summary); KLA-Tencor, ‘Value of process control’, The ConFab slides (May 2015): strong correlation between yield and reliability citing Shirley (Intel, IRPS 1995), killer defects failing at probe and latent defects at burn-in or in the field, always more cost-effective to over-inspect than to under-inspect, cumulative yield falling with design rule unless step yield improves (fetch-tool summary); Defect Detection Fundamentals page of this vault; Yield Models and Critical Area page of this vault.

References

1. N. Atchison and R. Ross, “Method for the calculation of wafer probe yield limits from in-line defect monitor data,” US Patent 6,324,481, Texas Instruments, 2001. Available: https://patents.google.com/patent/US6324481
2. R. Shimshi, Y. Nehmadi, V. Svidenko, A. T. Schwarm and S. Jawaharlal, “Dynamic inline yield analysis and prediction of a defect limited yield using inline inspection defects,” US Patent 7,937,179, Applied Materials, 2011. Available: https://patents.google.com/patent/US7937179
3. S. L. Riley, “Semiconductor defect management: separating the vital few from the trivial many,” slides, 2009. Available: https://www.slideshare.net/slriley/semiconductordefectmanagementseparatingthevitalfewfromthetrivialmany
4. KLA-Tencor, “Value of process control,” The ConFab slides, May 2015. Available: https://sst.semiconductor-digest.com/wp-content/uploads/2015/05/Robert-Cappel-presentation.pdf
5. Yield Models and Critical Area page of this vault.
6. Defect Detection Fundamentals page of this vault.
7. Defect Data Handling with Python and Perl page of this vault.

Related pages in this vault: Yield Models and Critical Area, Defect Detection Fundamentals, Defect Data Handling with Python and Perl, Defect SPC and Excursion Monitoring, Defect Inspection Workflow and Sampling Plans.