Methodology for creating “rejuvenating” devices

PII:

Microelectron. Reliab., Vol. 38, No. 4, pp. 531±537, 1998 # 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0026-2714/98 $19.00 + 0.00 S0026-2714(97)00229-1

METHODOLOGY FOR CREATING ``REJUVENATING'' DEVICES E. M. BASKIN Institute of Electromechanics, 101000 Moscow, Glavpochta, P.O. Box 496, Russia (Received 20 December 1996; in revised form 12 November 1997) AbstractÐIn this paper we discuss the properties and characteristics of rejuvenating elements. By applying a developed method for expanding a distribution density function into generalized orthogonal Laguerre polynomes, the failure rate function of semiconductor devices has been investigated. It is shown that with this methodology one can create rejuvenating devices. # 1998 Elsevier Science Ltd. All rights reserved.

The second type is training of devices by means of hardening. The given devices may be made up from devices of an identical quality. The hardening process creates a development wear obstacle. Moreover, the hardening is followed by a decrease in velocity of wear. It is easy to show that standard laws of reliability describe distribution failures for the types of element mentioned above. RTÐa gamma and Weibull distribution form parameters of less than 1.0. Therefore, a variation coecient (v) will be more than 1.0. ATÐa gamma and Weibull distribution form parameters of more than 1.0. Therefore, a variation coecient will be less than 1.0. QATÐa mixture of exponential distribution with Weibull distribution to form parameters of more than 1.0 etc. QRTÐa lognormal distribution, a logarithmic distribution of Weibull, a logarithmic gamma distribution and Bernstein distribution. Unlike rejuvenating elements, the ageing ones appear to have an upper limit to their reliability or lifetime. Because rejuvenating elements seem to have no such limit, there naturally appears a question as to whether such elements do exist really, and if so, how to detect them. As the reliability of devices is now very high, the main method used to work out and to research a new device is an accelerated test. The foundation of the mathematical theory of rejuvenating elements was proposed in Ref. [2]. Ways of achieving long lifetimes (or mean residual lives) were discussed in Ref. [1]. However, to prove the feasibility of rejuvenating elements, one needs a special apparatus to ascertain that

INTRODUCTION

Mechanical devices have been intensively developed and used for several millennia in the history of human civilization. The ®rst to appear were wheels, chariots, carts, lifting machines etc. Until the invention of semiconductors there was a period of ageing (AT) [or quasiageing (QAT)] elements and systems, where failures were due to wear and fatigue. The appearance and mass application of semiconductor devices in the second half of the twentieth century marked a changeover from a period of ageing elements to a period of mixed technology using both ageing (rolling and sliding friction bearings, steam engines, internal combustion engines, electrical motors, reductors, relays, capacitors, electrical magnets, semiconductor devices etc.) as well as rejuvenating (RT) (or quasirejuvenating (QRT)) elements (magnetic suspension bearings, electronic relays and switches, semiconductor devices, and so on). The criterion to distinguish a given element is the behaviour of the failure rate function, l(t), at 0 < t < 1: RTÐl(t) should be monotonically decreasing up to zero at a large enough t; ATÐl(t) should be monotonically increasing at a large enough t; QRTÐl(t) is an arbitrary behaviour at t>0 and monotonically decreasing up to zero at a large enough t; QATÐl(t) is an arbitrary behaviour at t>0 and monotonically increasing at a large enough t. According to Ref. [1], the processes of hardening and running-in due to training occur in the rejuvenating elements. Therefore, it is necessary to distinguish the two types of training: The ®rst type is burning, in which it is used as a test of a device lot, made up from devices of a di€erent quality.

lim l…t† ˆ 0

t41

531

532

E. M. Baskin

for a given element. In addition, a theory of existence for such elements is also required.

distribution Pa,l(x) = 1 ÿ Fa,l(x) one may write fa,l(x) in the following form: fa;l …x† ˆ l
…l
x†a
eÿlx


ESTIMATION OF FAILURE RATE FUNCTION

ÿlx

f…x†  fa;l …x† ˆ l
…l
x†
e




2 X

ck


kˆ0

Lak …l


x† …1†

where a> ÿ 1. Lak …l


x† ˆ

k X iˆ0

Ckÿi k‡a …ÿl

Z

i


x† =i!

ck ˆ 1=G…1 ‡ a†=


Cak‡a




mi ˆ k X iˆ0

0

1

i

x f…x†
dx

i Ckÿi k‡a …ÿl†

bk
…l
x†k :

…2†

kˆ0

In the course of statistical research the following conditions are very important. The class of distribution used should be rather wide. It is desirable to include in it the majority of distributions and their combinations known in a reliability theory. The suggested method of an approximation of the laws of a reliability should be consistent, i.e. with an increase in the sample volume, the given approximation should aspire to a general population. The velocity of convergence of the given approximation should be rather high. Mathematical statistics use the following methods to construct a failure rate function: distribution histograms; standard probability papers for distributions; polynomial approximations; Pearson's curve approximations; Johnson's curve approximations: Histograms and standard probability papers are essentially manual ways of representing l(t), and as such, are rather inaccurate and unreliable. Pearson and Johnson approximations are based on lower moments; and are consequently inaccurate as well. One of the most accurate approaches to construct l(t) and to carry out the foregoing conditions is to use a computer program to expand distribution density functions into generalized orthogonal Laguerre polynomes. This method was developed in the early 1970 s in Russia [3±7] and later used in the West [8, 9]. It is known that most of the reliability laws are determined at the interval of time (0, 1). It is marked by means of s continuously di€erentiated probability density function (PDF) class determined at (0, 1). It is convenient to seek an approximation for an arbitrary f(x)e S by means of orthogonal polynomes determined at (0, 1) also, through generalized orthogonal Laguerre polynomes: a

2 X


mi =i!

In order to obtain an equation for time-to-failure

Whence it follows Pa;l …x† ˆ 1 ÿ Fa;l …x† ˆ 1 ÿ

2 X

Z bk


kˆ0

lx 0

xk‡a
eÿx
dx:

The equations for bk will have the following forms: b0 ˆ c0 ‡

2 X

cs


sˆ1

"

s Y …1 ‡a =i† iˆ1

bk ˆ …ÿ1†k =k!
ck ‡

2ÿk X

ck‡s


sˆ1

s Y …1 ‡ k=i† iˆ1

#


…1 ‡a =…k ‡ i†† k > 1 c0 ˆ 1=G…1 ‡ a†: One may get bk, substituting ck from Equation (1): b0 ˆ1=G…1 ‡ a†  ‰3 ‡ 5=2  a ‡ a2 =2 ÿ l  m1  …3 ‡ a† ‡ l2
m2 =2Š b1 ˆ ÿ 1G…1 ‡ a†  ‰3 ‡ a ÿ l  m1  …5 ‡ 2  a†=…1 ‡ a† ‡ l2
m2 =…1 ‡ a†Š b2 ˆ1=2=G…1 ‡ a†  ‰1 ÿ 2  lm1  =…1 ‡ a† ‡ l2
m2 =…1 ‡ a†=…2 ‡ a†Š: Then it is easy to get the estimation of failure rate function la;l …x† ˆ fa;l …x†=Pa;l …x†: In order to ascertain that fa,l(x) is PDF, it is necessary to execute the following two conditions: Z 1: 2:

0

1

fa;l …x†
dx ˆ 1; fa;l …x†  0;

at a > ÿ1; l  0 at x  0:

It is easy to demonstrate that a ®rst condition takes place for any a> ÿ 1 and le 0. From Equation (1) it follows that fa,l(x)e 0 when bk(l
x) will be positive for all x e0. A last condition will take place in the following cases: Dis > 0 and Dis < 0 and

bk > 0; b0 > 0

for k ˆ 0; 1; 2

where Dis = b1ÿ4
b0
b2. The distribution parameters are as follows: aÐ form parameter; lÐscale parameter, m1 and m2 are the ®rst and the second distribution moments, respectively.

Creating ``rejuvenating'' devices

The research has shown that the smaller a variation coecient (v) the more a form parameter and vice versa. Hence it follows that a physical meaning for a form parameter is the characteristics of timeto-failure scatter. It is possible to get the following necessary conditions for that element belonging to the classes AT or RT: An element belonging to class AT at a e 0 and to class RT at ÿ1 < a E 0. It is clear that the given conditions are not sucient. To assess parameters, one uses the least squares method by minimizing the following di€erences in smoothing statistics represented for various testing schemes: s n X Za;l ˆ min 1=n  ‰Fa;l …ti † ÿ Fe …ti †Š2 : …3† a;l m1 ;m2

iˆ1

where Fe(t) is the empirical DF. A computer program, Ravik3, has been developed which allows one to process the results of various testing schemes including repeatedly censured data as well as to treat the statistics represented by quantiles. A program to smooth test results, but only for testing scheme NUN, has been proposed [8]. Unfortunately, this lacks appraisal of the accuracy of approximation distribution functions. The accuracy of approximation is of paramount importance because the possibility of approximating the density distribution or the conditional density distribution are being investigated. Approximation accuracy depends on the convergence rate of an incomplete gamma-function.

533

To illustrate the possible inaccuracies due to manual methods, three examples have been considered. Also as a criterion of choice to seek distribution functions (DF), criterion type 3 is considered: s n X ZF ˆ min 1=n …4† ‰F…t† ÿ Fe …t†Š2 F…t†

iˆ1

where F(t)Ða arbitrary DF, and besides Fa,l(t), F(t)E S. In Ref. [10] the data relating to distribution function (DF) of 20 laying systems tested for 1089 h has been treated using the physical model. In accordance with it the lognormal distribution (LN) has been shown to apply and its parameters determined. It was shown that (LN) satis®ed the criterion W [22]. Determines in accordance with Equation (4) the value of criterion for (LN). It is 0.0363. An LN distribution is known to be a distribution whose failure rate approaches zero with a light. Processing these statistics with the Ravik3 program indicates the failure rate was increasing for a large enough t (Fig. 1). The distribution parameters [see formula (1)] were as follows: a = ÿ 0.062501, lH=l  T = 4.826251, m1H=m1/Tm=0.2777, variation coecient v = 1.511, Tm=1089 h. In accordance with (3) the value of criterion for Fa,l(t) is 0.0208, that is smaller, than the value of criterion for (LN), ZlLN=0.0363. The lifetime data of Boeing-720 type aircraft air conditioners has been treated in Ref. [21]. The number of units were N = 213. Using the histogram

Fig. 1. Failure rate function of the laying systems.

534

E. M. Baskin

Fig. 2. Failure rate function of the Boeing-720 type aircraft airconditioners.

approach, and in addition the theorem, it has been demonstrated that a DF with a decreasing failure rate function occurred. Let us apply once more the Ravik3 program. The result was a DF with the failure rate function for a bathtub with the following parameters: m1H=0.1527, v = 1.158, a = 0, lH=8.06, Tm=603 hrs (Fig. 2).

The lifetime data of Plastic Microcircuit MC 1741P in an epoxy package has been treated in Ref. [11]. Using standard probability papers, it has been demonstrated that a lognormal distribution occurs. Once again applying the Ravik3 program (Fig. 3), the distribution parameters were as follows: a = 18, lH=30.05, v = 0.301, m1H=0.7218, Tm=2648 h.

Fig. 3. Failure rate function of the plastic microcircuit MC 1741 P in epoxy package.

Creating ``rejuvenating'' devices

After a DF has been determined by some method, its ability to describe properly the existing statistical data must be tested using statistical criteria (Kholmogorov, o2, etc.). Note that the reliability of the existing criteria is rather poor as the above cited examples indicate. ANALYSIS OF ACCELERATED TESTING SEMICONDUCTOR DEVICES

Before describing how to create a rejuvenating element, we treat available testing data for semiconductor devices (SD) (Table 1) using the smoothing method (1). Data relating to 21 devices (10 di€erent types) have been used for various values of accelerating factor (temperature) and for di€ering levels of censoring procedure. It has been shown that 14

535

devices belong to quasirejuvenating or rejuvenating types of reliability law (QRT or RT) while seven belong to quasiageing or ageing types (QAT or AT). The criterion to distinguish QRT, RT, QAT, AT is the behaviour of l(t) at (0 < t < 1). As far as a ®tting procedure occurring at (0 E tE tp) then tp and p, together with a sample size, characterize an assurance to distinguish laws of reliability. The smaller value of p, the greater assurance. So, SD may display both types of reliability behaviour. It is only reasonable to ask for the cause of this di€erence. Inspection of Table 1 demonstrates that even devices of the same trade mark exhibit di€erent laws of reliability. For example, IMPATT diodes tested at 256 and 2808C exhibit quite di€erent behaviour: RT at lower temperature

Table 1. Distribution functions of SD failures Type of element CMOS-4007

Temperature (8C) 200

HF transist. NPN

85

HF transist. PNP

85

CMOS

100

CMOS

110

CMOS

125

CMOS

150

IMPATT

256

IMPATT Transmis. optoelec.

280

module

20 50 70

Power thyristor

Ð

ECL-100 k

125 175 225

IC

140 180

Component board 2N559

250 70 100±200

tp

Type of reliability law

t0.3 t0.67 t0.54 t0.8 t0.55 t0.4 t0.5 t0.35 t0.1 t0.4 t0.15 t0.12 t0.77 t0.65 t0.47 t0.43 t0.5 t0.46 t0.44 t0.86 t0.17

*tp=Tm where tp is the censored quantile of level p. Tm is the maximum duration of the SD test.

QRT QRT QRT AT QAT AT AT RT QRT RT RT RT QRT RT AT AT QRT QAT RT QRT QRT

Distribution parameters (1) a = ÿ 0.45, lH=6.07, m1H=0.397, m2H=0.26, Tm=450 h, a = ÿ 0.6, lH=3.085, m1H=0.723, m2H=0.7814, Tm=10,000 h, a = ÿ 0.35, lH=2.09, m1H=1.152, m2H=2.106, Tm=5000 h, a = 2.5, lH=4.08, m1H=1.504, m2H=2.641, Tm=10,000 h, a = 1.0, lH=4.08, m1H=1.122, m2H=1.52, Tm=5000 h, a = 2.0, lH=6.07, m1H=0.944, m2H=1.042, Tm=1600 h, a = 13, lH=18.01, m1H=0.994, m2H=1.057, Tm=200 h, a = ÿ 0.15, lH=2.09, m1H=0.8127, m2H=1.24, Tm=1800 h, a = 2.0, lH=8.06, m1H=0.5098, m2H=0.395, Tm=600 h, a = ÿ 0.05, lH=2.09, m1H=0.9984, m2H=2.055, Tm=5623 h, a = ÿ 0.5, lH=3.085, m1H=0.5794, m2H=0.75, Tm=5012 h, a = ÿ 0.35, lH=6.07, m1H=0.2101, m2H=0.102, Tm=1000 h, a = ÿ 0.4, lH=1.095, m1H=2.091, m2H=6.8, Tm=500 h, a = ÿ 0.725, lH=2.51, m1H=0.907, m2H=1.499, Tm=2000 h, a = 2.5, lH=8.76, m1H=0.5734, m2H=0.437, Tm=1000 h, a = 20, lH=61.25, m1H=0.375, m2H=0.1406, Tm=500 h, a = ÿ 0.125, lH=25.01, m1H=0.1289, m2H=0.0218, Tm=1140 h, a = 2.5, lH=10.01, m1H=0.3926, m2H=0.178, Tm=380 h, a = 0, lH=2.51, m1H=0.3517, m2H=0.139, Tm=58 h, a = ÿ 0.9, lH=1.095, m1H=1.589, m2H=4.367, Tm=1000 h, a = ÿ 0.25, lH=8.06, m1H=0.3017, v = 0.71, Tm=28 weeks.

Reference [13] [14] [14] [15] [15] [15] [15] [16] [16] [17] [17] [17] [18] [19] [19] [19] [19] [19] [19] [13] [12]

536

E. M. Baskin

and AT at higher one. Similarly, ECL-100 k logics with emitter connections belong to RT at 1258C, while at 175 and 2258C, they belong to AT. It may be concluded that raising the temperature may transform RT to AT. Evidently, this phenomenon is a direct result of changing failure physics or the appearance of new types of failures in the device under consideration. But it appears that temperature and possibly other load factors are not the only reasons to change the failure law of SD. Let us consider the test results of very-large-scaleintegration (VLSI) reported in Ref. [20]. This work studied electromigration e€ects in VLSI failure statistics. The ever increasing tendency for a more compact VLSI is accompanied by more stringent reliability requirements to interconnections. To meet these demands, the Al metallic interconnections are provided with a sublayer of refractory metals. A high reliability of these interconnections is closely related to minimizing electromigration at operating conditions. Meanwhile, investigations indicate that multilayer metallization feature processes which if neglected may not only reduce reliability but also cause crossover so that RT of VLSI could give way to AT one. Indeed, let us consider the accelerated testing of multilayer metallizations using method (1) to process the results (see Table 2). For a double layer metallization the criterion of failure due to electromigration was once its resistance rose by 1±10%; for a three-layer metallization the criterion of failure is a rise of 1.5± 10%, respectively. The following conclusions have been drawn: 1. the upper protective layer of TiN increases the reliability of a three-layer metallization compared with a two-layer one (t0.9=199 h up to t0.9=8.2 h, respectively); 2. an additional TiN layer results in a QRT of failure for a three-layer metallization as opposed to a QAT for a double layer case. Table 2. Distribution functions of multilayer metallization failures Metallization type at temperature 2008C and circuit density 3.0  106 cm

Type of reliability law

Double-layerÐTin/Al

QAT

Three-layerÐTin/Al/TIB

QRT

Distribution parameters [Equation (1)] a = 1.0 lH=8.06 m1=618 h v = 0.647 t0.9=82 h a = ÿ 0.05 lH=7.065 m1=510 h v = 0.557 t0.9=199 h

So a VLSI with a three-layer metallization falls into a category of rejuvenating elements. In a recent paper [21], semiconductor failures due to electromigration are shown to be of a lognormal type, thus a QRT takes place. But, as indicated above, the situation seems to be much more complicated. Accelerated testing of semiconductor devices suggests that: 1. SD features both ageing as well as rejuvenating failure statistics; 2. ageing is caused by a number of factors such as (a) high loads (temperature, current density, humidity etc.) (b) SD structure imperfections (double layer rather than a three-layer metallization etc.); 3. rejuvenating statistics appears once optimal loads or a properly developed SD are used. Further, according to Ref. [1] these kinds of statistics occur if processes of consolidation prevail. METHODOLOGY FOR CREATING REJUVENATING ELEMENTS

Now it becomes clear that the key element in creating rejuvenating elements is a smoothing procedure (1) to treat accelerated testing results. It allows the failure rate of an element to be determined with acceptable accuracy as well as the type of distribution. The following procedures are to be used when developing or testing SD: 1. To specify the load factors for accelerated testing. The number of samples and time-transforming functions (model) may be applied as recommended in Refs [7, 8]. 2. To carry out the accelerated testing at the speci®ed values of load factors. 3. To process the test results by means of the method (1). If accelerating factors were chosen properly, while structure and technology of SD were correct, we are bound to deal with RT (or QRT) statistics. Once AT (or QAT) is encountered, the following causes should be considered: . too high a value for the load factor may have been used; . insucient development; . improper technology. To progress further, item 3 should be dealt with conclusively which means that a sort RT (or QRT) is obtained. (4) After satisfactory results under speci®ed acceleration factors, structure and technology have been achieved, the distribution failure function needs to be calculated for normal values of accelerated factors based on the technique presented in Ref. [5].

Creating ``rejuvenating'' devices CONCLUSIONS

The above analysis of SD failure rates suggests that: 1. all SD belong to one of the two groups: ageing or rejuvenating elements; 2. the exponential failure rate accepted now as the mainstream reliability law seems to be only a rough model used to assess reliability of complex systems because of its inherent simplicity; 3. a new SD widely developed now forms a basis for rejuvenating elements; 4. the main approach to measure the ability of elements to rejuvenate is the smoothing technique given in Equation (1); 5. the proposed methodology for creating and exploring the category of rejuvenating elements serves as a foundation for a new generation of absolutely reliable devices with an unlimited operational life.

AcknowledgementsÐThe author is grateful to Dr Andrey Tutnev for his assistance in translating. REFERENCES 1. Gertsbah, I. B. and Kordonsky, X. B., Physics of failures, Sovetskoe Radio, Moscow, 1966, (in Russian). 2. Barlow, R. E. and Proscan, F., Mathematical Theory of Reliability, John Wiley, New York, London, Sydney, 1964. 3. Baskin, E. M., Izvestija Academy of Science USSR, Technical Cybernetic, Moscow, 1971, 5, 79±82 (in Russian). 4. Baskin, E. M., Izvestija Academy of Science USSR, Technical Cybernetic, Moscow, 1973, 5, 90±93 (in Russian).

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5. Baskin, E. M., Izvestija Academy of Science USSR, Technical Cybernetic, Moscow, 1988, 3, 102±108 (in Russian). 6. Baskin, E. M., VNIIEM Proceedings, Moscow, 1979, 43, 104±112 (in Russian). 7. Baskin, E. M. and Givartovskaja, N. A., Izvestija VUZs, Electromechanics, 1982, 6, 700±709 (in Russian). 8. Biernat, J., Jarnicky, J., Kaplon, K. and Kuras, A., Transactions on Power Systems, 1992, 7(2), 656± 664. 9. Maciejewski, H., Microelectronics Reliability, 1995, 35, 1047±1051. 10. Hahn, G. J. and Shapiro, S. S., Statistical Models in Engineering, Wiley, New York, 1967. 11. Lucoudes, N., in Reliability & Quality Handbook, Edition Motorola Semiconductor Products Sector, 1993. 12. Peck, D. S., Semiconductor Reliability Predictions from Life Distribution Data, in Proceedings of the Conference on Reliability of Semiconductor Devices, ed. J. E. Shwop. Elizabeth, New Jersey, 1961, pp. 78±100. 13. Moltoft, J., Microelectronics Reliability, 1983, 23, 489± 500. 14. Brambilla, E. and Brambilla, P., Microelectronics Reliability, 1983, 23, 577±585. 15. Stojadinovic, N. D., Microelectronics Reliability, 1983, 23, 609±707. 16. Sinnadurai, F. N., Microelectronics Reliability, 1981, 21, 209±219. 17. Shimotolscis, R. A. and Janchenco, A. D., Reliability and Control of Quality, 1987, 11, 53±57 (in Russian). 18. Shpez, V. L., Reliability and Control of Quality, 1984, 8, 11±16 (in Russian). 19. Baskin, E. M., Reliability and Control of Quality, 1987, 6, 14±18 (in Russian). 20. Onoda, H., IEEE Transactions Electronic Devices, 1993, 40, 1614±1620. 21. Smy, T. and Winterton, S. S. et al., Microelectronics Reliability, 1994, 34, 1047±1056. 22. Shapiro, S. S. and Wilk, M. B., Biometrika, 1965, 52, 591.