Yield Analysis Based on Fault Probability and Kill Ratio, Part 2

David Hu and Milo Koretsky, Oregon State University, Corvallis, Ore.; Manu Rehani and David Abercrombie, LSI Logic, Gresham, Ore. -- Semiconductor International, 12/1/2001

At a Glance In this two-part series on defect-limited yield estimation, we demonstrate that there are only two reliable means of estimating fault probabilities (FPs) — by simulating probable defect mechanisms on the layout for that layer, and by using the defect-limited yield equation. We further show that FPs estimated from the defect-limited yield equation are meaningful and can be readily defined. In fact, the yield predicted by the FPs estimated from the limited yield concept and those estimated from simulation are shown to be comparable to actual yields observed in the fab. The engineers arrive at a method of predicting yield for a given layer, even given more than one defect mechanism or defect type on that layer.

In Part 1 of this article, we compared two ways of estimating fault probability — by analyzing existing bin and defect data and inferring fault probabilities (FPs) through models of the defect distributions, and by simulating the defect distribution using Monte Carlo techniques and the device layout. Using three months of data for a given device, we showed that FPs estimated from bin and defect map data are uniformly greater than the FPs estimated by Monte Carlo simulation.

This and the subsequent analysis indicated that, under typical fab conditions, the assumption of a completely random defect distribution gives accurate fault probabilities. We demonstrated, in fact, that with in-line data, the FP estimated for a given layer or process step equals the weighted average of the FPs of the possible defect mechanisms operating at that layer.

In general, as shown in Part 1 of this series, the agreement between kill ratios (KRs) and the simulated FPs from the device layout is better than that between the estimated FPs and the simulated FPs based on bin and defect inspection data. We determined, however, that these differences are harder to quantify for the larger size bins because of the large confidence intervals, which resulted from small sample sizes at larger size bins. In light of this problem, we chose to not use size bins when comparing estimated FPs and KRs with the simulation FPs (Table 3).

Table 4 shows the results of estimating FP based on the limited yield concept, where we used Equation 12 to solve for p_A(0):

  (12)

 Equation 15 to solve for a:

 (15)

 and Equation 17b to solve for :

 (17b)

To be certain that we are correct in assuming that Equation 11

 (11)

which is only true when the defects are distributed completely randomly, can be equated with Equation 16 under present fab conditions,

 (16)

we can compare the LY estimates using Equations 4 and 11:

 (4)

 (11)

They should be comparable if the assumption is correct, since both equations assume a completely random distribution of the defects. The results are shown in Table 5. We find that all the errors are below 1% except for TN1T, which has a relatively high FP of ~0.15. As shown previously, we expect a less accurate approximation at high FPs. Even so, the approximation is less than 5% off.

Table 3. Simulated FPs Averaged Over All Sizes vs. Estimated KRs and FPs From Bin and Defect Map Data

Inspection step	KR	KRupper	KRlower	FP	FPupper	FPlower	FPsim	Fault mechanism
ISEF	0.029914	0.04509	0.015147	0.1739	0.2074	0.1438	0.0136	Active bridge
M1EF	0.036939	0.059977	0.014788	0.183	0.2245	0.1462	0.0551	M1 bridge
M2EF	0.055396	0.080438	0.031174	0.2	0.2391	0.1648	0.0613	M2 bridge
M3EF	0.005604	0.046474	0	0.19	0.2513	0.1381	0.033	M3 bridge
POLF	0	0	0	0.206	0.2282	0.1851	0.0252	Poly bridge
TN1T	0.185705	0.221953	0.15056	0.1931	0.2465	0.1468	0.029	Contact
TN2T	0.037836	0.078933	0	0.1778	0.2341	0.1301	0.0144	ViaM1M2
TN3T	0.068107	0.114166	0.024554	0.1823	0.2464	0.1289	0.0078	ViaM2M3

Table 4. Estimated FPs Based on Limited Yield Equation vs. Simulated FPs for Different Values of x_o

Defect type	Defect density (defects/die)	pA(0)est (RHS of Eq.15)	a	LYA (RHS of Eq. 17a)	Estimated FP based on Eq. 17	FPsim (xo=0.1)	FPsim (xo=0.5)	FPsim (xo=1)	Fault mechanism (assumed in simulation based on die layout)
ISEF	0.5373	0.7101	0.4066	0.9913	0.0164	0.0005	0.0136	0.0498	Active bridge
M1EF	0.1657	0.8791	0.2662	0.9955	0.0272	0.0023	0.055	0.167	M1 bridge
M2EF	0.1693	0.8696	0.3753	0.9928	0.0432	0.0025	0.0613	0.191	M2 bridge
M3EF	0.0678	0.9376	0.6423	0.9997	0.0052	0.0013	0.0333	0.107	M3 bridge
POLF	0.4538	0.7591	0.2977	1	0	0.001	0.025	0.09	Poly bridge
TN1T	0.0707	0.9461	0.1183	0.99	0.1486	0.0012	0.0294	0.104	Contact
TN2T	0.0909	0.9286	0.1861	0.9973	0.0299	0.0006	0.0144	0.052	ViaM1M2
TN3T	0.0654	0.9424	0.3103	0.9961	0.0605	0.0003	0.0078	0.028	ViaM2M3

Table 5. Comparison of LYs Predicted by Eq. 4 and 11

DefectType	DD	FP	LY (estimated from Eq. 4)	LY (estimated from Eq. 11)	LY % error
ISEF	0.5373	0.0164	0.9912	0.9913	0.0074
MIEF	0.1657	0.0272	0.9955	0.9955	0.0003
M2EF	0.1693	0.0432	0.9927	0.9928	0.0088
M3EF	0.0678	0.0052	0.9996	0.9997	0.0053
POLF	0.4538	0	1	1	0
TN1T	0.0707	0.1486	0.9895	0.99	0.0456
TN2T	0.0909	0.0299	0.9973	0.9973	0.0014
TN3T	0.0654	0.0605	0.9961	0.9961	0.0049

The results from Table 1 (in Part 1 of this article) showed that for the larger size bins, the FPs between the simulation and defect map data agree, but only because the confidence intervals are so broad that the comparison was meaningless. At lower size bins, where the confidence intervals are much narrower, there was a large discrepancy.

In general, the FPs estimated from the bin and defect map data using Equation 5 are uniformly greater than the FPs estimated by computer simulation on die layout. This is probably due to hidden defects that can contribute to increasing the estimated FP over the true FP. In other words, the defects detected at any of the inspection steps may not represent all the defects actually present on the die at the particular inspection step.

Table 6 summarizes possible ways by which defects can go undetected. Column 2 identifies locations where defects that occur between two inspection steps may hide. For example, defects that occur between the ISEF and POLF inspection steps may hide under the poly or nitride deposits when inspected at POLF, but defects that occur between the POLF and SPCE inspection steps have virtually no place to hide when inspected at SPCE.

Table 6. Defect Detection and Fault Mechanism by Layer

Inspection step	Defect can hide under…	Defect seen here may be seen at…	Defects seen here may cause…
ISEF	Nitride (only defects <1 µm)	POLF, SPCE, SL2R	Active bridge, active break (formed at ISEF), poly-M1 short
POLF	Poly (only defects <1 µm)	SPCE, SL2R	Poly bridge, poly break, poly-M1 short, contact
SPCE	Virtually no place to hide	SL2R	Poly bridge, poly break, poly-M1 short, con-tact
SL2R	Virtually no place to hide	Cannot be seen at any subsequent inspection steps	Poly-M1 short, contact
TN1T	PSG, Ti-Ni (only defects <1 µm)	M1MD, M1EF	Contact, M1 bridge, M1 break, poly-M1 short
M1MD	M1, M1 PR	M1EF	M1 bridge, M1 break, M1M2 short, via-M1M2
M1EF	Virtually no place to hide	Cannot be seen at any subsequent inspection steps	M1 bridge, M1 break, M1M2 short, via-M1M2
TN2T	IMD1, Ti-Ni (only defects <1 µm)	Cannot be seen at any subsequent inspection steps	Via, M2 bridge, M2 break, M1M2 short
M2MD	M2, M2 PR	M2EF	M2 bridge, M2 break, M2M3 short, via-M2M3
M2EF	Virtually no place to hide	Cannot be seen at any subsequent inspection steps	M2 bridge, M2 break, M2M3 short, via-M2M3

Column 3 represents the subsequent inspection steps where the defects may be seen. For example, defects seen at M1MD inspection may be seen at M1EF, while defects seen at M1EF cannot be seen at any other subsequent steps (because the IMD1 and Ti-N 2 deposition prior to the next inspection will bury it).

The last column shows the fault mechanism by which a defect may cause a die to fail. This defect may cause a fault at the inspection step or layer at which it is detected, or can cause a fault at a subsequent layer.

Another explanation for the discrepancy between the FP estimated from the bin and defect data and the simulated FP based on die layout is that the fault mechanism upon which the simulated FP is based is not the only fault mechanism that may cause a fault. It is also possible that it is not the correct fault mechanism for the particular layer.

The additional fault mechanisms by which the defects at a particular inspection step may cause a fault are shown in Table 6. For example, at ISEF, not only can the active areas be affected by extra field oxide, causing bridging across the active area, but they can also be affected by missing oxide, causing an active break. In addition, defects detected at ISEF may cause the poly and metal 1 formed at a subsequent step to be shorted together. Thus, for the defects detected at the ISEF inspection step, there may actually be three fault mechanisms at work that cause a die to fail.

By a similar process of looking at each inspection step and the process steps that precede and follow it, we can guess the possible fault mechanisms of each defect type that are detected at a particular inspection step. Table 7 shows the values of the simulated FPs associated with these possible fault mechanisms at some representative layers, compared with the estimated KRs and FPs for each of the inspection steps.

Table 7. Possible Fault Mechanisms for the Defects Detected at Each Layer Compared with Bin Estimated KRs and FPs

Inspection step	Bin estimated				Simulated fault mechanism/FP
	KR	KRupper	KRlower	FP (from Eq. 17)
ISEF	0.0299	0.0451	0.0151	0.0164	Active bridge/0.0136	Active break/0.0092	PolyM1 short/0.303
M1EF	0.0369	0.06	0.0148	0.0272	M1 bridge/0.0551	M1 break/0.0741	M1M2 short/0.416	M1M2 via/0.0144
M2EF	0.0554	0.0804	0.0312	0.0432	M2 bridge/0.0613	M2 break/0.093	M2M3 short/0.358	M2M3 via/0.0078
M3EF	0.0056	0.0465	0	0.0052	M3 bridge/0.033	M3 break/0.0626	M3M4 short/NA	M3M4 via/0.0002
POLF	0	0	0	0	Poly bridge/0.0252	Poly break/0.093	PolyM1 short/0.303	DiffusionM1 contact/0.0293
TN1T	0.1857	0.222	0.1506	0.1486	DiffusionM1 contact/0.029	M1bridge/0.0613	M1 break/0.0741	PolyM1 short/0.303
TN2T	0.0378	0.0789	0	0.0299	M1M2 via/0.0144	M2 bridge/0.0613	M2 break/0.093	M1M2 short/0.416
TN3T	0.0681	0.1142	0.0246	0.0605	M2M3 via/0.0078	M3 bridge/0.033	M3 break/0.0626	M2M3 short/0.358

Examining Table 7, we see that there are several possibly very different values for the FP for a particular layer: the KRs and FPs estimated from the bin and defect data, and the simulated FPs for each possible defect mechanism. In general, given that all the defect types that occur at the particular layer are known, the FP for a particular layer can be calculated by the following expression, for layer l:

 (28)

where t_m1, t_m2,,t_m3,… are the fraction of defects detected at inspection step/layer l that can cause a fault by defect mechanisms m1, m2, m3,…, respectively. What we have been measuring is FP_l. The fractions of defects detected (ts) cannot be determined without classifying the various defects seen at each layer and determining their fault mechanisms and fractions by the use of test structures and failure analysis.⁸ That is, we can only estimate FP_ls from the current classification scheme of the bin and defect data. However, the fault mechanisms can be speculated based upon past experience with similar circuits.

For example, most random defects on a layer with metal lines will cause shorts and opens, or bridges and breaks. Once a defect mechanism is presumed for a defect detected on a layer, its FP can then be estimated by Monte Carlo simulation based on the layout for that layer and the presumed defect mechaism.⁸ Of course, we still cannot determine the ts from the simulation because these are parameters that depend on the process step itself. But for some defect mechanisms, we can rule them out based on the estimated KRs for the layer in which they occur, since we know that the true FP cannot be greater than the estimated KR, as shown previously.

Looking at Table 7, for example, we see that the KR for the TN2T layer is about 0.05. But the simulated FP for the mechanism, M1M2 short, is 0.4. Thus we can say that the fraction of defects that cause M1M2 short is very small, almost negligible, if such defect types exist at all, at the TN2T layer. For the TN2T layer, then, we are left with the bin-estimated KR and FP, and the size-weighted average of the simulated FPs as possible candidates for the average FP of this layer. Thus, we can say that for TN2T,

  (29)

where we have eliminated M1M2 short as a possible mechanism because its FP is so much higher than the estimated KR. By a similar process of elimination of fault mechanisms whose simulated FP value is much larger than the estimated KR value for the same layer, we see that for all the layers in Table 7, the average simulated FP per layer is between ~0 and 0.1. Since the remaining fault mechanisms in Table 7 have comparable FP values, we retain the simulated FPs whose fault mechanisms are also shown in Tables 1-4, for each layer, to represent that layer’s average simulated FP for purposes of comparison.

To further narrow the true FP range per layer, and confirm the previous analysis, we can use the bin-estimated values we have to calculate yields for wafers whose actual yields and defect density per layer are known. We will include simulated FPs based on x₀=0.1 and x₀=1 as well, for the defect mechanisms assumed in Tables 1-4. Table 8 shows a comparison of the estimated yields compared with the actual yields for 11 wafers.

Table 8. Random Yields Calculated Based on Estimates of KR, FP, FP by DLY, and Simulated FPs

			Estimated FP by Ross method	Estimated KR	FP from LY equation	Sim FP (x0=0.1)	Sim FP (x0=0.5)	Sim FP (x0=1.0)
Lot	Wafer	Actual Random Yield	YR	YR	YR	YR	YR	YR
1	24	0.919	0.84	0.964	0.972	0.999	0.965	0.892
2	25	0.926	0.827	0.949	0.961	0.999	0.964	0.89
3	24	0.942	0.852	0.961	0.97	0.999	0.968	0.901
4	24	0.912	0.811	0.959	0.969	0.998	0.955	0.864
5	24	0.918	0.671	0.914	0.94	0.998	0.948	0.835
6	25	0.948	0.667	0.923	0.947	0.998	0.943	0.824
7	25	0.976	0.74	0.927	0.943	0.997	0.934	0.805
8	24	0.939	0.839	0.961	0.968	0.998	0.962	0.882
9	24	0.916	0.916	0.961	0.968	0.999	0.988	0.96
10	24	0.938	0.854	0.964	0.972	0.999	0.966	0.894
11	24	0.949	0.82	0.954	0.99	0.998	0.96	0.877

It must be pointed out that the predicted yields are based on only the steps/layers for which FPs, KRs and defect densities are available. Thus what we are estimating is really a maximum yield because not all steps that have potentially fatal defects are included. As can be readily seen, the predicted yields based on FPs estimated from the Ross method, Equation 5, uniformly under-predict the yield quite a bit. This means the bin-estimated FPs are gross overestimates of the true FP per layer, as anticipated based on the fact that their values are so much higher than the KRs. Since not all the yields per layer are included, we would expect the calculated yields to be significantly higher than the actual yields.

Keeping this in mind, we see that the KRs also tend to overestimate the FPs, more so than the simulated FPs between x₀=0.1 and x₀=0.5. The simulated FPs based on x₀=1.0 is readily seen to be an overestimate of the true FP as well, since they are well below the yields estimated from the KRs.

Thus, from Table 7, we see that the bin-estimated FPs based on Equation 5 are drastic overestimates of the true FPs; the KRs as well are overestimates, though less so. Only the FPs estimated from Equation 17 (derived from the limited yield equation, Equation 11) and the simulation FPs with x₀<0.5 give yield results that are consistent with the actual yields.

From the results shown in Table 8, we see that not only the bin-estimated FPs based on the Ross method, but the KRs as well, are overestimates of the true FPs. But we have shown that the FPs based on the limited yield equation and the FPs based on simulation are better estimates of the true FPs. Of course, from Equation 8 we know that KRs will always be overestimates of FP but, nevertheless, we had hoped that the KRs would provide good approximations to the FPs. That is, the defect density would be low enough that they would approximate FP very closely.

However, as seen from the results in Table 8, the estimated KRs are apparently too high for practical purposes. It seems, then, that the only method to obtain accurate FPs from bin and defect data alone is by using Equation 17 — that is, by equating the DLY based on the KR (Equation 11) with the yield based on the negative binomial equation (Equation 16).

The question that remains is how to be sure that the FPs we are measuring are, in fact, the same FP defined by Equation 28. To answer this question, we can begin by looking at Equation 9 again. Let us suppose we are interested in estimating a KR for layer A. We would need all the values or estimates for the parameters on the RHS of Equation 9. T_A is simple enough to estimate, once we know the spatial probability distribution function of defects distributed on layer A, p_A. Since T_A is the number of die with the number of defects M1 detected at layer A, we can use Equation 12 to estimate T_A,

 (12)

To estimate T_GA, we can start off by using the formula for conditional probabilities:⁹

 (30)

where P(G/A₁), P(G/A₂), P(G/A₃), …., are the conditional probabilities of a die not failing given it has exactly one defect found on layer A, exactly two defects found on layer A, exactly three defects found on layer A, …, respectively. These probabilities are weighted by the probability that a defect will have a certain number n of defects occurring on it, p_A(n); i.e., the probability that exactly n number of defects will occur on the die. For P(G/A₁), we have,

 (31)    

where t_x, t_y, t_z,… are the fraction of defects with fault mechanism x, y, z…(or defect types x, y, z,…) found on layer A. Y_B, Y_C …are the yields for all the other layers B, C,…. , which are independent of the yield of layer A. Y_S is the systematic yield. Defining    as,

 (32)

where t_x + t_y+ t_z +…=1, Equation (31) becomes,

 (33)                   

We can show by a similar process that,

 (34)

and

 (35)

etc. Substituting Equations 33-35 back into Equation 30, we obtain,

 (36)

T_G is simply the number of die that are yielding,

 (37)

Substituting Equations 12, 36 and 37 into the expression for the KR, Equation 9, and simplifying, we have,

 (38)    

Since we know the spatial probability distribution function, p_A, as given by Equation 13 and can estimate Y_A by Equation 16, we can solve Equation 38 for . The s for each layer were calculated according to Equation 38, and these values were the same as those calculated by Equation 17b. Thus we have confirmed that the s that we calculated based on Equation 17b and shown in Table 4, are the average FP for layer A as defined by Equation 32.

From Equation 38, we can further say that only when the defect density is extremely low are   and KR_A  the same. That is, as the defect density approaches zero,

 (39)

while , , … approach zero, and Y_A approaches 1, so that Equation 38 becomes

  (40)

The approach of p_A(0) to 1, and hence the approach of KR_A to , is faster for spatial distributions that follow a more random pattern; i.e., as a in Equation 38, contained in the expressions for Y_A and p_A, becomes larger. This is shown in Figure 3 where Equation 38 is used to plot out KR_A for different values of a and compared to the given .

Conclusions

We see that the only two reliable means of estimating s are by means of simulation of probable defect mechanisms on the layout for that layer, and by the use of the defect-limited yield equation. Furthermore, the s we estimated in Table 4, based on the use of the defect-limited yield equation, are meaningful, in the sense that they are defined by Equation 31. This definition of    is the only definition that would make sense in the yield equation given by Equation 16 because the term   in Equation 16 must refer to the average number of faults per die, , and only if it is defined as in Equation 31 can this term be equivalent to .¹⁰ In other words, this definition of    is the only one that allows us to predict the yield for layer A, even if there is more than one defect mechanism, or defect type, on the layer.

Finally, our analysis shows that the KR_A approximates the only under the conditions of low defect density and low clustering (high cluster factor a), but shows the value of the concept of KR in yield prediction as an estimate of an upper limit for the FP and as an integral part of the limited yield equation.

References

8. A.V. Ferris-Prabhu, Introduction to Semiconductor Device Yield Modeling, 1992, Artech House Inc., Norwood, Mass., p. 43.

9. S.M. Ross, Introduction to Probability Models, Academic Press, 1993.

10. C.H. Stapper, “Modeling of Integrated Circuit Defect Sensitivities,” IBM Journal of Research and Development, Vol. 27, November 1983.

Author Information

David T. Hu worked as an intern at LSI Logic from July 2000 to February 2001. His project involved predicting chip yield from in-line bin and defect data. He is currently an M.S. candidate in the chemical engineering department at Oregon State University, and has a B.S. in chemistry from the University of Portland.
Phone: 1-541-752-8334
e-mail: huth@che.orst.edu

Milo D. Koretsky is an associate professor of chemical engineering at Oregon State University. His research interests are in thin-film materials processing, including plasma etching, chemical vapor deposition, electrochemical processes and chemical process statistics. His teaching interests include the integration of microelectronics unit operations into the core chemical engineering curriculum. He also serves as the chemical engineering advisor to the MECOP internship program. He received B.S. and M.S. degrees in chemical engineering from the University of California at San Diego, and a Ph.D. in chemical engineering from the University of California at Berkeley.
Phone: 1-541-737-4591

Manu Rehani is the section manager of yield data systems at LSI Logic and is responsible for implementing and sustaining the yield analysis and reporting infrastructure. He has been with the company since 1997, holding various defect detection, analysis and production management positions. He has a M.S. in chemical engineering from Oregon State University.

David Abercrombie is the yield engineering manager at LSI Logic. In addition to driving in-line defect and end-of-line wafer sort yield improvement, his team also researches and develops software systems for data warehousing, automation and engineering productivity. He has a B.S. in electrical engineering from Clemson University and a M.S. in electrical engineering from North Carolina State University.

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