A Nonparametric Approach to Estimate System Burn-in ... - IEEE Xplore

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IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 9, NO. 3, AUGUST 1996

A Nonparametric Approach to Estimate System Burn-in Time

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Wei-Ting Kary Chien and Way Kuo

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5

read data

Abstract-System burn-in can get rid of more residual defects than component and subsystem burn-ins because incompatibility exists not only among components, but also among different subsystems and at the system level. There are two major disadvantages for performing the system burn-in: the high burn-in cost and the complicated failure rate function. This paper proposes a nonparametric approach to estimate the optimal system burn-in time. The Anderson-Darling statistic is used to check the constant failure rate (CFR), and the pool-adjacent-violator (PAV) algorithm is applied to “unimodalize” the failure rate curve. Given experimental data, the system burn-in time can be determined easily without going through complex parameter estimation and curve fittings.

calculate

I. INTRODUCTION Bum-in is a technique to weed-out infant mortality by applying higher than usual levels of stress (e.g., temperature, pressure, humidity, voltage, corrosion, and vibration) to speed-up the deterioration of materials or electronic components so that the analysts can collect information more quickly. The importance of burn-in is increasing because the failure time of a product becomes so long under the present advanced manufacturing technology. Therefore, bum-in is widely used by the semiconductor industry and is especially important for the high-density integrated circuits (IC) [ 3 ] ,[6], [IO]. Two important features for designing a burn-in test are burn-in duration and temperature. Based on experiences, cost factors, and the characteristics of the components, burn-in time can be set to either minimize the cost per time [9] or maximize the resultant reliability. Component bum-in is usually set to have a one-week bum-in time (i.e., 168 h). The burn-in temperature should be chosen to detect defective items faster and, at the same time, to avoid damaging the good ones; 150°C is quite common in industry for buming-in IC’s, which is called component burn-in to distinguish from burning-in a PC board (subsystem burn-in), for example, and a whole system (system bum-in). Bum-in at different levels is considered in [4] and [13]. Although component bum-in can be performed at high temperature (and therefore can apply higher stress on samples to have shorter experimental duration), it is recognized that higher level burnin (e.g., system burn-in) is more efficient than the lower ones (e.g., subsystem and component bum-ins) in removing the incompatibility that results from bad workmanship, welding joints, design, and inconsistent components. The compatibility factors analyzed by Chien and Kuo 141 show that system bum-in can remove a large portion of the potential defects at an acceptable cost. However, system behavior is difficult to model by using any parametric approach because many components are contained in the system. This is especially true for the system which is in its early failure stage. Manuscript received March 17, 1994; revised October 12, 1995. This research was solicited and supported in part by an IBM Headquarters manufacturing project. Dr. Kuo was supported in part by the Fulbright Foundation through the Senior Fulbright Scholarship,and the National Science Foundation project #DMI-9400051. The authors are with the Texas A&M University, College Station, TX 77843 USA.

Publisher Item Identifier S 0894-6507(96)05684-9.

algorithm to “unimodalize ‘

r”l find MRL

9 find

Fig. 1. Analysis flow chart This paper suggests a nonparametric procedure to investigate the bum-in effectiveness by using the Anderson-Darling statistic, the pooled-adjacent-violator (PAV) algorithm [ 11, and a graphic technique. Unlike some parametric and Bayesian approaches which must specify a distribution for system failure mechanism to estimate the parameter(s) of the model and to select a suitable prior distribution for Bayesian analysis the proposed method is very straightforward and can be easily understood by practitioners. It is worthwhile to point out that it is not possible to describe a system’s behavior by a single probability distribution because each is composed of many components and subsystems which have fairly different characteristics. In other words, although parametric methods can provide more information (e.g., quantile, mode, and skewness) and usually have easier derivations than nonparametric analyses, these merits can only be possessed if the parametric form is correct. Furthermore, in most cases, parametric approaches are applied under strong assumptions,

08944507/96$05.00 0 1996 IEEE


462

Hazard Rate Function h(t) x Simulation ............... ...... PAV adjust *

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Fig. 2. The simulated and modified system hazard rates. 11. PRELIMINARIES

TABLE I PARAMETERS OF

EACHCOMPONENT’

3,000 13,000 6,000 8,000

0.35 0.40 0.25 0.40 1.00 0.30

IN A SERIES SYSTEM

T 12.0 14.0

8.0 10.0 16.0 10.0

II 36.0 40.0 40.0 50.0 70.0 46.0

A. U-Shaped Failure Rate Function

73,000

1.1

88,000 97,000 69,000 10 39,000

3.5 2.2 1.6 1.o 2.6

‘Both Xf and X f are in the units of FIT, a failure in lo9 device hours. T,” and TF are in 1000 h. which usually neglect the incompatibility mentioned previously, on components’ failure mechanisms and system structures. Some basic probability concepts are briefly reviewed in Section 11. If the components have U-shaped failure rates (it should be noted here that the proposed method can also be applied to non-U-shaped system failure rates, which are even easier than the U-shaped case), we can obtain the system burn-in time by the method in Section I11 and the cost function in Section IV. An example is designed in Section V to illustrate the proposed method, which is also demonstrated by the flow chart in Fig. 1, where data are generated by simulation. Section VI gives some conclusions and discussions.

A U-shaped failure rate function is assumed for electronic components including: infant mortality (which has decreasing failure rate (DFX)), useful life (which has constant failure rate (CFR)), and wearout (which has increasing failure rate (IFR)). Weibull distribution is usually assumed to describe the infant mortality for electronic components [2]. However, little research has been done on the electronic wear-out process. To simplify the analysis, Weibull is used to portray the wear-out behavior. The probability density function (pdf, f(t)), the cumulative distribution function (CDF, F ( t ) ) , and the failure rate function ( h ( t ) )of the Weibull distribution with scale parameter X and shape parameter /? are (for t 2 0)

B. System Failure Rate The series system structure dominates most of the electronic apparatus. Suppose a series system has r components and the


Segment 29 Segment 30

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1012.9204 1738.7085

1095.9015 1890.0299

1100.9216 1936.5501

1158.6342 1957.2296

4O%g% 5 % % 948.1353 961.0366 1437.1414 1440.8340

5%% 975.6393 1457.3135

6%?3% 1019.1879 1543.0679

1038.8641 1608.2305

1150.8560 1722.9728

1251.0032 1723.3047

&:!%

?&%:

334.535i

3V1.4LU4

1346.1227

1586.7233

%

~ 1415.2717 %‘E : ~ ~

1271.0037 1855 4687

‘Failure times start from the beginning of the corresponding segment.

components function independently, then

1 - F,(t) = P ( X , 2 t )

where X , is the system failure time and H ( t ) is the cumulative failure rate at time t and the subscripts i and s are used to denote component z and system, respectively. If a system contains components that follow the Weibull distribution with common shape parameter /?, then the series system failure rate becomes ptP-1

2

A.!

Z=1

The life distribution of a series system will not be Weibull if the pz’s are not the same; this somewhat elucidates the difficulty of modeling the system failure behavior. The series system is considered only for simulating system failure mechanism; the proposed approach is independent of the system configuration. 111. METHODS A U-shaped system failure rate is generated by Monte-Carlo simulation to illustrate the proposed method. The Anderson-Darling statistic is then used to test exponentiality in order to find the change ) CFR to IFR ( t ~ Finally, ~ ) . points from DFR to CFR ( t ~and~ from

the total costs and the mean residual lives (MRL’s) under different burn-in times are calculated. The tL1 and t~~ can he used to set bum-in time, warranty plan, releasing time of the product, and to construct the life-cycle model. For example, the bum-in time longer than t~~ won’t be economical are able , because the system is already in its useful life. From t ~we ~ to know when the product starts to wear-out. We use the superscripts D and I to represent the X and /3 in the DFR and IFR regions, respectively.

A. Generating a U-Shaped Failure Rate Curve The area under the curve in h ( t )versus t plotting, i.e., H ( t ) , can be obtained by summing all the small “strips” decomposed under the curve. The length of the DFR region ( T D )and that of the CFR region ( T c ) as well as the parameters of the Weibull distributions assumed for DFR and IFR have to be specified. The failure rate of the CFR region, hC, can be expressed as

The failure rate in the wear-out section is first determined by (1) under different t and then shifted upward by hC as determined by (3).

B. Simulation Time is divided into small segments with time length At. There are T S I M / A t testing periods if TSIMrepresents the total simulating

~

~


464

SIMULATION RESULTS3

Segment (IC) 1 2 3 4 (tL.,)5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 (tL2125 26 27 28 29 30

Starting time (1,000 hr) 0 2 4 6 8

Ending time

Sample Failure size number

(1,000 hr)

10

12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

18

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

Test exponen(fk) (h,(t)) tiality 43 00003583 1 20 0.0001667 1 16 0.0001333 1 7 0.0000875 1 4 0.0000667 0 5 0.0001250 0 2 00000500 0 1 00000250 0 5 0.0001250 0 2 O.bO00500 0 4 0.0001000 0 4 0.0001000 0 2 00000500 0 3 0.0000750 0 3 0.0000750 0 2 0.0000500 0 4 0.0001000 0 2 0 0000500 0 3 0.0000750 0 3 00000750 0 0 0 0000000 2 4 0.0001000 0 3 0.0000750 0 4 0.0001000 0 4 00000500 1 18 0.0001800 1 21 00002100 1 20 0.0002000 1 25 0.0002500 1 35 0.0002917 1

(Nk)

2 4 6 8 10 12 14 16

Hazard rate

60 60 60 40 30 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 40 50 50 50 50 60

Modified Weibull hs(t) testing (h:(t)) 0 0003583 I 00001667 1 0.0001333 0 0.0000875 1 0.0000658 0 0.0000658 0 0.0000658 0 0.0000658 1 0 0000658 0 0 0000658 0 0.0000658 0 0.0000658 0 0.0000658 0 0.0000658 0 0.0000658 0 0.0000658 0 0.0000658 0 0.0000658 0 00000658 0 0.0000658 0 0.0000658 2 0.0000658 0 0.0000658 0 00000658 0 0.0000658 0 0.0001800 0 0.0002049 0 0.0002049 1 0.0002500 1 0.0002917 1

obtained by UNDERDIFFERENT BURN-INTIMES

Burn-in Burn-in Time ( t b ) Time ( t b , ~ ~ ) (at 40°C) (at Tb = 125°C) 0.0 0.0 83.3 2,000.0 166.7 4,000.0 250.0 6,000.0 333.3 8,000.0 416.7 10,000.0 500.0 12,000 0 583 3 14,000.0

MRL

Total

TotalCost -

(P(tb))

cost

9 P(db)

(cb(t;t,,r,))

12,958.2 26,927.1 36,077.3 44,962.9 50,393.2 53,612.3 56,953.6 60.404.4

403.03 368.05 356.01 347.96 356.09 369.14 382.19 399.46

370.64 300.74 265.82 235.56 230.10* (min) 235.11 239.81 248.45

U=F(t), O h:; then construct hi = ht = 2 / ( & &);hi = hp,i # 2 , 3 . 2) If h:,i E [l, k], satisfy P1, then these become the h t . 3) If not, perform another PAV step; thus, say that h: = hi > hi; then construct hz = hg = hi = 3/(& RZ 6'3);h: = h:,a # 2,3,4. 4) If h:, i E [l,k ] , satisfy P1, then these become the h:. 5 ) Otherwise, go to 1. To start the algorithm, from (4) and P1, 0, is set to N t A t / f z . Algorithm 2 can be applied for the more general case.

+

+ +

A%= (1 + :)A2 A 2

hz

( 2 i - l){log F ( z ( , ) ;ij)

2=1

n

;=E?.

IV. APPLICATIONS

2=1

For the Weibull distribution, the statistic ( A i ) is

After observing a U-shaped system failure rate, we perform an exponentiality test on the failure times of each segment so that t L 1 and t L z can be found. The PAV algorithm is applied to have the failure rate unimodalized because the simulated failure rates usually are n o t x e l l U-shaped. The modified failure rate between t L 1 and t L , is hC.

A. The Weighted Factors where

A

and

fi

Uncertainties about the actual environmental stress to be encountered as well as the properties of a component can be modeled by the stress-strength model

satisfy

Reliability = P[Y > 21 = 3

1/P

A=(@) which can be solved by the Newton-Raphson method through iteration by using

for the mth iteration.

= /[1

-

s

F ( y ) dG(y)

G(z)] d F ( z )

where: 2 the random variable of stress; Y the random variable of strength; F ( z ) the CDF of stress 2; G(y) the CDF of strength Y . One way to incorporate the stress-strength model into generic failure rates of electronic components, which are available in many electronic handbooks and standards (like MIL-std-883C and Bellcore manual [2]), is the use of stress factors. Many stress factors are introduced in [2], [7]. After careful evaluation, four stress factors


466

are chosen by Chien and Kuo [4] to adjust the kth segment

the failure rate in

hk,

h f ,= h k T E 7 l T T S T Q

(9)

where T E (the environmental stress factor), T T (the temperature stress factor), T S (the electric stress factor) and T Q (the quality factor) are chosen to fit the situation where the product is used.

B. Total Cost Through field justifications, Kuo [9] formulated the s-expected cost at time t

cB(t;t h , T b ) = CS

$-

B

+

tb,TbCu

+ ehCs + ( 1 + 1)etCf

where B bum-in set-up cost; CS system developing cost; cs shop repair cost; cs field repair cost; cv bum-in variable cost, in $/hr; eh s-expected number of failures during bum-in; et s-expected number of failures at time t ; 1 ratio used to calculate the cost of the loss of credibility; t b , T b system bum-in time if burned-in at T t C ; and e b , et can be obtained from number of failures = (time) x ( h f , ) . The cost factors can be obtained from managerial records (for example, from sale, test, and customer service departments).

C. Warranty Plan The importance of warranty plan can be seen from [8] which reports that IBM paid $2.4 billion in warranty costs in 1989. The warranty duration is normally set to be one year by most electronic manufacturers under the assumption that the product will not deteriorate before it is shipped out to the customers. If the product wears-out after t h within a year, say, t~~ - t h < 8000 h, it is better to choose a shorter warranty duration. If there is any trade-in policy for the product, a used system will have less salvage value if the time that has been used since it is sold is greater than t L z - t b . The expected cost incurred in the warranty duration hence becomes

+B + + vewcw

C B ( t ; t h , T b )=CS

th,TbCu

+

ebCs

+ (1 + h c . f (10)

where cw repair cost if a failure is reported by a customer; e , s-expected number of failures occurred during warranty period; 7 percentage of a failure to be claimed during the warranty period.

D. Optimal Burn-In Time The MRL of a system surviving time tb, the corresponding bum-in time if burn-in occurs at system ambient temperature, is defined as

p ( t h ) E E [ X - thlx > t b ]

.c

exp ( - H ( z ) ) d z

-

exp ( - - H ( t b ) )

(11) ’

Larger MRL is preferred. However, more have to be invested to achieve a larger MRL. The optimal bum-in time can be determined by subtracting the MRL weighted by a time-proportional gain, g, from the total cost; that is, the optimal burn-in time can be found from mint, ( C B ( t~h ;, T b ) - g P ( t h ) ) .

(12)

V. EXAMPLE To illustrate the proposed method, consider a six-component series system. Suppose that the lives of six components ( r = 6) follow Weibull distribuQons. The parameters of each component are summarized in Table I (data from [9] and [2]). Note that component #5, which is used to take care of all non-IC components [9], is the one with CFR throughout its lifetime. The failure times are listed in Table 11. Take At = 2000 h and TSIM= 60000 h; that is, k = 30. The N k , f k , h,(t), and h’,(t) are summarized in Table 111. Fig. 2 depicts the system failure rate from simulation. From the exponentiality test, the results of which are shown in the 7th column of Table 111, we found that t~~ = 8000 h and t~~ = 50000 h. Sometimes it is hard to find t~~ and t L z if there is no significant trend indicating the start andlor the end of the CFR. One may then select a higher confidence level to find an observable trend. A second way is to increase the sample size, which will not be beneficial if the system being tested is expensive. The previous example is tested under the 95% confidence level and the decisions for tL1 and t L z are made only when three consecutive nonexponential segments are inferred. Table I shows that the CFR region should begin at 14000 h (max, T f ) . However, t~~ is estimated to be 8000 h. Simlar situation can be found for t ~The~ reason . for this phenomena is that the changing rate of a Weibull failure rate is very small when t is comparatively small for /3 > 1 and when t is large for p < 1; the corresponding X can be used to estimate the size of t to make the failure rate increase insignificantly. In this case, t~~and t~~ will be more accurate if the number of components increases or many different components are in the system, which means the parameters (both X and p) of the component differ greatly from one another. The Weibull testing outcomes are listed in the last column of Table 111. The Weibull testing is not significant in the example. One possible explanation is that the sample size in each segment is not large enough. Large numbers of failure times are expected to give a better estimation of the and fi in (8) because they are derived by the numerical method. After applying the PAV algorithm, the hb(t)’s are also drawn in Fig. 2. Attention should be paid on segment 21 which shows nd failure. If fk = 0 , 8 k is approximated by the largest 19 in the corresponding section. For instance, if there is no failure in the kth segment and it is found later that this segment belongs to the CFR section, then the largest 0 in CFR will be applied to the kth segment for a conservative estimate. One possible remedy for f k = 0 is increasing the sample size, which is especially recommended when f k = 0 occurs in the neighborhood of t L 1 and t L z . However, f k = 0 can be neglected if a trend (the failure rate is increasing or is decreasing) is detected. After considering the operational profile and the required quality of the product, engineers set TE = 1.0 (minor environmental stress), TT = 1.0 (operated under 40°C [2]), TS = 1.3 ( ~ 6 0 % electric stress), T Q = 1.5 (the defective proportion is 0.003 [12]). The failure rate in each segment is adjusted by (9). Suppose the system can be bumed-in at 125°C ( T b = 125°C) and it follows Curve #7 in [4]; that is, the accelerated factor is equal to 24. In other words, bum-in the system for 1 h is equivalent to use it for 24 h in the operational profile. The MRL’s under different system bum-in times are calculated from (1 1) and shown in Table IV. Let Cs = 200,B = 1,c, = 0 . 0 0 0 2 , ~=~ 0.1,cf = lO,c, = 15,g = 0.0025,Z = 3 (high penalty), t = 44000 h (-5 years), 7 = 0.75, and warranty duration is 10000 h ( ~ 1 . years). 1 The total costs under different burn-in times are derived by (10) and they are shown in the fourth column of Table IV. From (12), whose outcomes are in the last column of Table IV, the optimal bum-in time is set to be 333.3 h (at T b = 125”C), and the total cost is $356.09.

IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL.

9, NO.

VI. CONCLUSIONS AND DISCUSSIONS This paper presents a nonparametric approach for the optimal burnin strategy for the first time. This approach used at the system level further enhances the bum-in capability that considers the incompatibility factors existing at component, subsystem, and system levels. The constant failure rate after the PAV adjustment (0.00006581 11) is very close to the actual failure rate (0.0000737035) derived directly from the given parameters of each component by (2). The weighted factors can be used to compensate for this difference. The CDF of the series system can also be derived from the adjusted failure rate. For example, F(47000) = 0.990049 from (2) and F(47700) = 0.982716; the difference is less than 1%. To compensate for the limited field data, At may be different for each segment, that is, At can be replaced by At,; choose larger At, to contain more samples in a segment.

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[9] W. Kuo, “Reliability enhancement through optimal bum-in,’’ IEEE Trans. Rel., vol. R-33, pp. 145-156, June 1984. [lo] W. Kuo and Y. Kuo, “Facing the headaches of early failures: A stateof-the-art review of bum-in decisions,” Proc. IEEE, vol. 71, no. 11, pp. 1257-1266, 1983. [ 111 M. Lee, “Strong consistent modified maximum likelihood estimation of U-shaped hazard function,” unpublished Ph.D dissertation, Dept. Statistics, Iowa State Univ., Ames, 1987. [ 121 P. D. T. O’Connor, “Microelectronic system reliability prediction,” IEEE Trans. Rel., vol. R-32, no. 1, pp. 9-13, Apr. 1983. [13] C. W. Whitbeck and L. M. Leemis, “Component vs. system bum-in techniques for electronic equipment,” IEEE Trans. Rel., vol. 38, no. 2, pp. 206-209, June 1989.

A Model for Radial Yield Degradation as a Function of Chip Size

VII. APPENDIX

David Teets

Algorithm 1:

For (IC = 1; k 5 Testingperiod; IC++) FailFlag = 0; minFailTime = k a t 1; For ( i = 1;i 5 CurveNumber; i++) switch(WhichPart) DFR: FailTime = WeibullFail(Af , Pp); CFR: FailTime = ExponentialFail(hF) +kAt; IFR: FailTime = min[WeibullFail(A:, Pf )

+

+

+TD T C ,

+

ExponentialFail( h:) kAt]; If ( k a t 5 FailTime< ( k 1)A t ) minFailTime = FailTime; Failmag = 1; If (FailFlag = 1) FailNumber[k]++; Message “ S y s t e m f a i l s a t minFailTime.”

+

ACKNOWLEDGMENT The authors would like to thank the referees for their suggestions. REFERENCES R. E. Barlow, D. J. Bartholomew, J. M. Bremner, and H. D. Brunk, Statistical Inference Under Order Restrictions: The Theory and Application of Isotonic Regression. New York Wiley, 1972. Reliab. Predictions Procedure Electron. Equipment, Bellcore Tech. Ref. TR-NWT-000332, no. 3, 1990. M. Campbell, “Monitored bum-in improves VLSI IC reliability,” Comput. Des., pp. 143-146, Apr. 1985. W. T. K. Chien and W. Kuo, “Optimal bum-in simulation on highly integrated circuit systems,” IIE Trans., vol. 24, no. 5 , pp. 3343, Nov.

1992. M. A. Stephens, “Tests based on EDF statistics,” in Goodness-of-Fit Techniques, R. B. D’Agostino and M. A. Stephens, Eds. New York Marcel-Dekker, 1986. D. L. Denton and D. M. Blythe, “The impact of bum-in on IC reliability,” J. Environ. Sci., pp. 19-23, Jan./Feb. 1986. N. B. Fuqua, Reliability Engineering for Electronic Design. New York Marcel-Dekker, 1987. W. Klein, “On market-driven quality,” Think, vol. 4, pp. 26-29, 1990.

Abstract-Yield data was collected from a total of 928 200-mm silicon wafers which were processed using a 1 p m CMOS technology. Each wafer was patterned with one of four chips varying in area from 17 mm’ to 132 mm2. Wafers with like chips were binned together into a single grand composite wafer for each of the four chip sizes. The yield was subsequently plotted as a function of radius, and a mathematical expression was empirically fit to the radial yield plots. The coefficients from each of the fitted expressions were then used to form a generalized expression for radial yield degradation as a function of chip size. The method of normalization, and the algorithm used to generate the radial yield plots will be discussed. An explanation of the data is offered based on geometrical considerations. The radial yield dependence is then incorporated into more traditional yield models.

I. INTRODUCTION Edge clustering of defects, or radial yield degradation, has been a widely observed phenomenon in semiconductor manufacturing [ 11-[7]. Explanations of the radial yield degradation in the literature have ranged from epitaxial mound type defects [l], to slip lines caused by thermal gradients in furnaces [4], to physical damage caused by wafer handling [4], to diffusion pipes [5],to electrostatic attraction of particles to the wafer carriers [7]. There are also other processing factors that might play a role in radial yield degradation: many IC manufacturing tools have a center to edge uniformity dependence. Spin coaters, plasma etchers, and diffusion furnaces are all examples of equipment with just such a dependence. In addition, there are physical constraints that may play a role in radial yield degradation: defects caused by edge bead removal (a solvent applied to the outside edge of wafers to remove unwanted photoresist), particles from wafer clamping mechanisms, particles from wafer carriers, and microloading (a localized increase or decrease in plasma etch rate or develop rate due to localized geometries), are all certainly a function of radius. Various techniques have been used to model radial yield degradation. Yanagawa [2] fit an exponential curve to a plot of defect density versus radius. He subsequently analyzed yield numerically using computer simulated yield plots for different values of the Manuscript received October 15, 1995; revised March 17, 1990. The author is with Intel Corporation, Phoenix, AZ 85048 USA. Publisher Item Identifier S 0894-6507(96)05685-0.

0894-6507/96$05.00 0 1996 IEEE