Failure Rate Calculation: Extending JESD74/JESD74A to Any Sample Size Siyuan Frank Yang, Wei-Ting Kary Chien Semiconductor Manufacturing International Corporation (SMIC) Corporate Quality-Reliability Center 18 Zhangjiang Road, Shanghai, China 201203 (
[email protected]) Abstract - The failure rate has been an important index in product reliability. Practitioners in microelectronics reliability have been using JEDEC standards to determine whether a product will pass the requirement of a prespecified failure rate. The limitation of the current
mention the possible large relative errors for small sample sizes. This constitutes the need to introduce a correct method for any sample size to practitioners. This paper introduces the exact method using F distribution, discusses the details of the sample size dependence, shows the magnitude of the errors when the sample size is not sufficiently large, and introduces a replacement factor universal for any sample size. The calculation of the replacement factor with the exact F method can be implemented using formula in Excel; which is practical and very easy for practitioners to use to accurately determine the UCL.
method used by JESD74 and its revision JESD74A in determining the upper confidence limit for failure rate is pointed out and discussed. Very large relative errors such as 40% have been shown for certain sample sizes which are not sufficiently large enough. This paper provides practitioners an exact method to calculate the confidence bounds of failure rates and therefore makes JESD74 and its revision JESD74A complete to any sample size Keywords - Failure rate, confidence limits, χ 2 method, exact F method, sample size, accelerated life test
II. STATISTICS FOR FAILURE RATE AND ITS CONFIDENCE LIMITS The hazard function, or sometimes called (instantaneous) failure rate at age t, h(t), is defined as the percentage of products (or components) that failed per unit time within a short time interval Δt from t to t + Δt. The following is the definition of failure rate [3], which is commonly seen in many reliability statistics textbooks.
I. INTRODUCTION Reliability implies time dependence of failure with time counting from the moment when the product is put into use (or operation). Failure rate is an important index of the reliability of a product. Measurements of early life reliability of microelectronic products are typically performed during product qualification. These measurements are carried out to assess the reliability performance over the most critical period in operation use. JESD74 published in April 2000 [1], and its revision JESD74A published in February 2007 [2] by JEDEC solid-state technology association, are the related JEDEC standards. JEDEC solid-state technology association (formerly known as Joint Electron Device Engineering Council) is the well-known standards resource for the world semiconductor industry (a semiconductor engineering standardization body of the Electronic Industries Alliance (EIA)). These two JEDEC documents have the same title of “Early Life Failure Rate Calculation Procedure for Electronic Components”, which are aimed to define a procedure for calculating the early life failure rate of electronic components using accelerated life testing. One of the important aspects is to determine the upper confidence limit (UCL) at certain conference level for the estimated failure rate. JESD74 and its revision JESD74A provided a formula to determine the UCL of the early life failure rate of a product, using accelerated testing, whose failure rate is constant over time. However, this formula is only applicable to very large sample sizes. Unfortunately both JEDEC standards did not mention the large sample size requirement, not to
978-1-4244-4870-8/09/$26.00 ©2009 IEEE
h(t ) =
f (t ) f (t ) = R(t ) 1 − F (t )
(1)
where F(t) is the cumulative distribution function (CDF), and
f (t ) =
dF (t ) is the probability density dt
function (pdf). R(t) is the reliability function and R(t)=1F(t). The above definition can be re-written as below with clearer physical meaning for failure rate.
− dR(t ) / dt − ΔR(t ) / Δt ≅ (2) R(t ) R(t ) − ΔR(t ) where is the percentage of the products that R(t ) h(t ) =
failed between t and t + Δt.
− ΔR(t ) is a binomial R(t ) ˆ parameter for failure proportion p. Its point estimator p The failure percentage
is.
pˆ =
204
f N
(3)
Proceedings of the 2009 IEEE IEEM
and the corresponding point estimator of instantaneous failure rate hˆ(t ) at t is
pˆ f hˆ(t ) = = Δt N * Δt
the same time
condensed tables of
(4)
And the lower confidence limit (two sided)
pLCL =
where
λˆ (t ) =
(5)
(6)
Fα [n1 , n2 ] is the upper α percentage point of
the F distribution with degrees of freedom of n1 and n2. For one-side upper 100(1-α)% confidence limit pucl ,
λ=
we replace α/2 with α, and we then have
pucl
( f + 1) Fα [ 2( f + 1),2( N − f )] = ( N − f ) + ( f + 1) Fα [ 2( f + 1),2( N − f )]
The percentages of the F distribution,
where (7)
Fα [n1 , n2 ] , can
χ12−α , d
χ2
χα2 ,d
(10)
pucl can be from either Eq. (7) or Eq. (8).
λ = χ c2, d /(2 * A * N * t A )
(11)
where tA is the stress time, same as Δt in Eq. (10) and
χ c2, d
is the probability point of the
χ2
distribution
with d degrees of freedom when the upper tail probability is (1-c). JESD74 and JESD74A provided a table of
χ c2, d
with various failure numbers from 0 to 10 and also seven confidence levels from 50% to 99%. Eq. (11) is the same as Eq. (10) when pucl is obtained
(8)
from Eq. (8). However, both JESD74 and JESD74A did not mention the assumption of the large sample size applied to Eq. (8) or (11). Therefore practitioners could misuse Eq. (11) for any sample size, which could result in large errors when the sample size is not sufficiently large.
distribution with d degrees of freedom
when the upper tail probability is α. The advantage of Eq. (8) is that the degree of freedom of
pucl / Δt A
Both JESD74 and its revision JESD74A used the following equation to determine the upper c%-confidence limit of the failure rate at the operation condition, λ, as follow.
2N 2 where d = 2 f + 2 and χ1−α , d is the probability point of the
(9)
III. THE EARLY LIFE FAILURE RATE CALCULATION PROCEDURE FROM JESD74/74A
be found from tables in many statistics textbooks or handbooks, such as CRC Handbook of Tables for Probability and Statistics [5]. However, most tables only listed the highest degrees of freedom up to 120. For realistic sample size N and possible failure number f, the degrees of freedom exceed 120 most of time. Therefore it is inconvenient to employ the exact method using the F distribution when the sample size N is large. Since in most cases the sample size N is very large and the failure number f is small, we can approximate the binomial distribution with Poisson distribution when f/N is near zero. Under this approximation, Grosh then derived the one-side upper 100(1-α)% confidence limit pucl for practical uses:
pucl =
hˆ(t ) pˆ / Δt = A A
where A = AT * AV is the acceleration factor for the ELF (Early-Life-Failure) test, which is a product of AT (High Temperature Acceleration Factor) and AV (High Voltage Acceleration Factor). If we use λ (same symbol as that JEDEC 74 and 74A used) to define the one-side upper 100(1-α)% confidence limit of λ(t), we then have
pLCL is:
f f + ( N − f + 1) Fα / 2 [2( N − f + 1),2 f ]
distribution from most basic
rate h(t) at a stressing level to the failure rate λˆ (t ) at the regular operation condition [6] since the number of failures (f) is obtained after stresses tests such as under a higher voltage or a higher temperature than regular operation conditions.
(5).
( f + 1) Fα / 2 [2( f + 1),2( N − f )] ( N − f ) + ( f + 1) Fα / 2 [2( f + 1),2( N − f )]
χ
is easily obtained from any 2
statistics textbooks. For the accelerated reliability tests that JESD74 and its revision JESD74A have dealt with, we use the conversion expressed in the following Eq. (9) to convert the failure
where N is the sample size of devices (or microelectronic components) which survived at moment t and f is the number of devices failed during the time period Δt, i.e. from t to t+Δt. The confidence limits for the binomial parameter are discussed with great statistical details in the book by Grosh [4]. Grosh derived the exact formula for upper 100(1-α)% confidence limit (two sided) pUCL as in Eq.
pUCL =
χα2 ,d
Therefore, we propose to replace the current
χ 2 method
in JESD74 and JESD74A with the exact F method in Eq. (7).
is a function of only the failure number f, which
is normally quite small comparing with sample size N. At
205
Proceedings of the 2009 IEEE IEEM
statistical tables in regular textbooks or handbooks. Here we use the formula in Excel. For the upper α percentage point of the F distribution with degrees of freedom n1 and n2, Fα [ n1 , n2 ] , we have
In next section, we use both Eq. (7) and Eq. (8) to calculate the one-side upper 100(1-α)% confidence limit pucl and show the relative errors resulted from the
χ 2 method for different sample sizes, failure numbers,
Fα [n1 , n2 ] = FINV (α , n1 , n2 ) , where FINV is the formula used in Excel to give the F values. For example, for f=2, c=60% (i.e. α = 1-c=0.4), and N=10000,
and confidence levels.
Fα [2( f + 1),2( N − f )] = F0.4 [ 2 * ( 2 + 1), 2 * (10000 − 2)]
IV. THE COMPARISON OF THE PROCEDURES FROM JESD74 AND JESD74A WITH THE EXACT METHOD
= F0.4 [6,19996 ] = FINV [0.4,6,19996 ] = 1.035 Therefore, R = F
For the exact method using F distribution, we obtain the one-side upper 100(1-α)% confidence limit of failure rate at operation condition using Eq. (12) derived from Eq. (10) and Eq. (7): λ=
For the probability point (
χ2
hˆ(t ) pˆ f = = A A * Δt A * N * t A
Therefore we have RCHI =
(14)
this case. In Table 1 below, we list the Replacement Factors for both the exact F method and the approximate χ 2 method, with different sample size (N = 10 to 10000), different failure number (f = 0 to 20) and 90% confidence level. We define the relative error as the difference between the exact F method and the χ 2 method relative to the exact F method. From Table 1, we find the relative error can range from zero to 43% within the given ranges of sample sizes and failure numbers in this table. Nowadays, the life tests of certain chips (e.g., high-end CPU’s) are expensive and this leads to the use of a small sample size. On the other hand, we may have quite a few rejects from the cycling tests for high memory-capacity (e.g., 4GB) NAND FLASH, for example. For these two cases, we will have a large relative error if we use the approximate χ 2 method. Hence, it is important for practitioners to
to binomial, we have the one-side upper 100(1-α)% confidence limit of failure rate at operation condition from Eq. (10) and Eq. (8): χ2 /2 p RCHI (15) = λ = ucl = c, d A * Δt A * N * t A A * N * t A
χ c2,d / 2
χ c2,d / 2 = 6.210/2 = 3.105,
which is exactly the same as the results with the exact method and is also the same as in Table 7.1-1 in JESD74. For a different sample size N = 20, but same f = 2 and same c = 60%, we have RF = 3.020 but RCHI is still 3.105 since RCHI does not depend on sample size N. The χ 2 method over estimates the confidence limit by 2.8% in
(13)
Comparing Eq. (12) and Eq. (14), we can name RF as the replacement factor for f to convert the point estimate in Eq. (14) to the one-side upper 100(1-α)% confidence limit with the exact method using F distribution in Eq. (12). For the one-side upper 100(1-α)% confidence limit using χ 2 distribution based on the Poisson approximation
where RCHI =
value. For example, for f=2, and c=60%, we have
= CHIINV [(1 − 60%), (2 * 2 + 2)] = CHIINV (0.4,6) = 6.210
The point estimate of the failure rate at operation condition in Eq. (9) can be further expressed as
λˆ (t ) =
χ2
χ c2,d = CHIINV [(1 − c), d ] = CHIINV [(1 − c), (2 f + 2)]
(12)
( f + 1) Fα [2( f + 1),2( N − f )] *N ( N − f ) + ( f + 1) Fα [2( f + 1),2( N − f )]
) of the
where CHIINV is the formula used in Excel to give the
( f + 1) Fα [2( f + 1),2( N − f )] 1 * N]* ( ) =[ ( N − f ) + ( f + 1) Fα [2( f + 1),2( N − f )] A * N * tA
F
χ c2, d
distribution with d degrees of freedom when the upper tail probability is (1-c), we have χ c2, d = CHIINV [(1 − c), d ] ,
( f + 1) Fα [2( f + 1),2( N − f )] 1 pucl * = A * Δt ( N − f ) + ( f + 1) Fα [2( f + 1),2( N − f )] A * t A
RF = A * N * tA where R =
(2 + 1) *1.035 *10000 = 3.105 (10000 − 2) + (2 + 1) *1.035
(16)
Comparing Eq. (15) with the point estimate of Eq. (14), we name RCHI as the replacement factor for f to convert the point estimate to the one-side upper 100(1α)% confidence limit using χ 2 distribution based on the
know the error and how to use exact F method since they might need to determine the confidence limits with the sample size and failure numbers that are not applicable for the χ 2 method. However, both JESD74 and its revision
Poisson approximation to binomial. The modern computing techniques in regular software, such as the very popular Excel, have made it very easy to determine the percentage point of F and χ 2 distributions
JESD74A did not mention the possible error and did not introduce the exact F method. Table 1: Replacement factors for both the exact F method and the χ 2 method with 90% confidence level.
for a very wide range of degrees of freedom and any percentages that could not be available by searching
206
Proceedings of the 2009 IEEE IEEM
f
N 10000 600 100 50 30 20 10
χ
2
F F F F F F F
Met hod Met hod Met hod Met hod Met hod Met hod Met hod Met hod
0 Replacement Fact or 2. 30 2. 30 2. 28 2. 28 2. 25 2. 22 2. 17 2. 06
f
N 10000 600 100 50 30 20 10
χ
2
F F F F F F F
Met hod Met hod Met hod Met hod Met hod Met hod Met hod Met hod
χ
2
F F F F F F F
Met hod Met hod Met hod Met hod Met hod Met hod Met hod Met hod
Replacement Fact or 3. 89 3. 89 3. 88 3. 83 3. 78 3. 71 3. 62 3. 37
Rel at i ve Er r or ( %) 0. 00 0. 02 0. 33 2. 03 4. 12 7. 01 10. 82 23. 78
Replacement Fact or 9. 27 9. 27 9. 24 9. 08 8. 88 8. 62 8. 30 7. 33
Rel at i ve Er r or ( %) 0. 00 0. 02 0. 42 2. 56 5. 27 9. 15 14. 53 37. 44
Replacement Fact or 14. 21 14. 20 14. 14 13. 84 13. 46 12. 96 12. 30 9. 90
4 Replacement Fact or 7. 99 7. 99 7. 97 7. 83 7. 68 7. 47 7. 21 6. 46
f
N 10000 600 100 50 30 20 10
1 Rel at i ve Er r or ( %) 0. 00 0. 11 0. 99 1. 16 2. 32 3. 89 5. 87 11. 99
Replacement Fact or 5. 32 5. 32 5. 31 5. 23 5. 15 5. 03 4. 90 4. 50
Rel at i ve Er r or ( %) 0. 00 0. 02 0. 36 2. 18 4. 43 7. 58 11. 77 26. 59
Rel at i ve Er r or ( %) 0. 00 0. 03 0. 44 2. 68 5. 53 9. 65 15. 45 43. 56
Replacement Fact or 6. 68 6. 68 6. 66 6. 56 6. 44 6. 28 6. 08 5. 52
Replacement Fact or 10. 53 10. 53 10. 49 10. 29 10. 06 9. 74 9. 35 8. 12
Rel at i ve Er r or ( %) 0. 00 0. 02 0. 38 2. 31 4. 73 8. 12 12. 70 29. 64
Replacement Fact or 11. 77 11. 77 11. 72 11. 49 11. 21 10. 83 10. 36 8. 84
10 Replacement Fact or 15. 41 15. 40 15. 34 14. 99 14. 56 13. 99 13. 24 N/ A
Rel at i ve Er r or ( %) 0. 00 0. 03 0. 45 2. 79 5. 78 10. 14 16. 39 N/ A
20 Replacement Rel at i ve Fact or Er r or ( %) 27. 05 0. 00 27. 04 0. 04 26. 89 0. 59 26. 07 3. 75 25. 03 8. 07 23. 45 15. 34 N/ A N/ A N/ A N/ A
6
9
V. CONCLUSION
Rel at i ve Er r or ( %) 0. 00 0. 02 0. 31 1. 86 3. 77 6. 40 9. 81 21. 09 7 Rel at i ve Er r or ( %) 0. 00 0. 02 0. 40 2. 44 5. 01 8. 65 13. 61 33. 13
REFERENCES
Failure rate is an important index in product reliability. The limitation of the current
χ
2
[1] JEDEC Solid State Technology Association, JEDEC STANDARD: JESD74 - Early Life Failure Rate Calculation Procedure for Electronic Components, 2500 Wilson Boulevard, Arlington, VA, 2000, page 7. [2] JEDEC Solid State Technology Association, JEDEC STANDARD: JESD74A - Early Life Failure Rate Calculation Procedure for Semiconductor Components, 2500 Wilson Boulevard, Arlington, VA, 2007, pp. 29. [3] Wayne B. Nelson, Applied Life Data Analysis, WileyInterscience, 1982, page 25. [4] Doris L. Grosh, A Primer of Reliability Theory, John Wiley & Sons, New York, 1989, pp. 239-244. [5] William H. Beyer, CRC Handbook of Tables for Probability and Statistics, The Chemical Rubber Co, Cleveland, Ohio, 1996, pp. 240-257. [6] Wayne B. Nelson, Accelerated Testing: Statistical Models, Test Plans, and Data Analyses, John Wiley & Sons, New York, 1990, pp. 154.
method in
determining upper confidence limit for early life failure rate for not sufficiently large sample size is pointed out and discussed. The exact method using F distribution is introduced to extend JESD74 and its revision JESD74A to any sample size. The relative errors up to ~40% have been shown in this paper with the failure numbers same as those used in the tables of both JESD74 and its revision JESD74A if small sample sizes are used. However both JESD74 and its revision JESD74A did not mention this limitation of the current
3 Rel at i ve Er r or ( %) 0. 00 0. 02 0. 28 1. 68 3. 39 5. 72 8. 72 18. 38
5
8 Replacement Fact or 12. 99 12. 99 12. 94 12. 67 12. 34 11. 91 11. 35 9. 45
2 Rel at i ve Er r or ( %) 0. 00 0. 01 0. 24 1. 45 2. 93 4. 93 7. 47 15. 47
χ 2 method
for small sample
sizes. It is necessary practitioners know the sample size dependence on failure rate calculation because, sometimes, the relative errors may be larger than what is acceptable at, e.g., a smaller N or a larger f in some reliability testing such as the life tests of certain chips and the cycling tests for high memory-capacity (e.g., 4GB) NAND FLASH and so on.
207
