Stress testing and reliability

Reliability Engineering 14 (1986) 247 255

Stress Testing and Reliability

P. W. Hale IBM (UK) Ltd, PO Box 30, Spango Valley, Greenock, Scotland, Great Britain (Received: 8 October 1985)

ABSTRACT Two empirical equations giving the test strength of constant temperature and thermal cycling environments are compared .[or effectiveness in precipitating defects in electronic components. Analysis of the equations shows that they can be put into a Jorm which has a theoretical basis although the structure still remains empirical.

1

INTRODUCTION

The two principal methods of thermal stressing applied to electronic components are high temperature soak and rapid thermal cycling. These two methods have been used extensively over the past few years but little has been done to give some theoretical comparison between the two methods. Recently, work done by the Hughes' Aircraft Corporation x has been published in which the test strength of each of these environments has been related to the parameters employed. While these equations are empirical they can, in fact, be changed into a form which indicates the thermal reaction effect given by the Arrhenius equation. Following this line of analysis the equations can be compared for their effectiveness and relative acceleration of electronic defects. A version of this paper was presented at the 5th National Reliability Engineering Conference--Reliability '85, 10-12 July 1985, Birmingham, UK, and is reproduced by kind permission of the organisers. 247 Reliability Engineering 0143-8174/86/$03.50 © ElsevierApplied Science Publishers Ltd, England, 1986. Printed in Great Britain

248

P. W. Hale

2

THEORETICAL BACKGROUND

The most commonly used equation for predicting acceleration of defects is the one derived by Arrhenius: )~ = A e x p ( - E / K T ) where 2 is the failure rate, A is a constant, E is the activation energy of the failure mechanism in question (eV), K is Boltzmann's constant and T is the absolute temperature (K). It is important to understand that the failure rate given by this equation applies only to failure mechanisms having the specified activation energy. It is also important to understand that a specific failure type will have a distribution of activation energy centred about some mean value. In practical terms this means that a given failure type could have a range of relative accelerations in the stress environments. If an integrated circuit having inherent defects is exposed to a high temperature environment for a time t, then the total number of defects precipitated will be F~ = A

fo

exp ( - E/KT) dt

It is assumed that the defects have the same or greater activation energy than E and the temperature remains constant: then

F~ = A t e x p ( - E / K T )

(1)

Suppose now the case of thermal cycling is considered where the temperature excursion is duplicated for each cycle and the upper and lower limits remain the same as shown in Fig. 1. In this case the total number of defects in time t will be

Fc =A

exp(-E/KT(t))dt

where the temperature term is now a function of time. No matter which form the cycling takes this integral becomes impossible to solve analytically. However, progress can be made by making the following assumptions. The value of the integral over one cycle will be finite and could be represented by the effect associated with exposure at some constant effective temperature Tp For one cycle we could therefore write F,.(1 ) = A exp( - E/KTE)

Stress testing and reliability

249

TMIN

Fig. 1.

If each cycle is identical, then for N cycles the total number of failures will be F c -- A Nexp ( - E / K T E ) If the cycles are all identical then the number of cycles will be directly proportional to the total exposure time and Fc = Bt exp ( - E / K T E )

(2)

where B is a constant. It should be noted that T E is a function of the two temperature extremes which is different from the average. The form of eqns (1) and (2) is very similar and the relevance of this will be developed during the analysis of the Hughes' equations. 3

TEST S T R E N G T H

The notion of test strength is an important one in reliability testing since it represents the probability that a given stress environment will precipitate a defect if it is present. The effectiveness of the stress environment will clearly depend on the failure mechanisms extant in the circuitry and consequently the associated activation energies. It is not in general possible to predict in advance the nature or quantity of the failure mechanisms that will be present in a new piece of circuitry and an approach to this problem will be discussed later. Before looking at the Hughes' equations themselves it is worthwhile considering what one would expect an equation for test strength to look like. In particular, because the test strength represents a probability that a given stress environment will precipitate defects in a given time, it must be some function of the total number of defects found after that time. In mathematical terms r =f(F) where r is the test strength, F is the total number of failures in time t and f is a function having a maximum value of unity.

250

P. W. Hah'

In addition to this it is clear that a single stress environment will not, in general, lead to the precipitation of all defects. The reason for this is that the physical chemistry of the defect mechanisms is diverse and some of these mechanisms will not be activated in a single stress environment. Accordingly the test strength equation would be modified to read

r =aJ(F) where a < 1. Taking note ofeqns (1) and (2) in the previous section the test strength equations for constant temperature (r~) and thermal cycling (zc) would be expected to take the following forms

rs = a f (At exp ( - E / K T )

(3)

~ = b f ( Bt exp ( - E/ KTE)

(4)

where a, A, b and B are constants. Further reference to the equations will be made during the analysis of the Hughes' formulae.

4

THE HUGHES' EQUATIONS

Work published by R A D C 1 gives two equations developed by the Hughes' Aircraft Corporation reflecting the test strengths of constant high temperature and thermal cycling environments. These two equations are

T s = 0"611 - exp { - Rtexp (0.0122T) }]

E' exp{

where: R =2'63 x 10 -~ T = absolute temperature (K) t = exposure time (h) N = number of thermal cycles dO/dt = cycling rate (°C/min) TA = (10H -- 251 + l0 L -- 251 + 50)/2 + 273 0 n = high temperature of cycle (°C) 0L = lOW temperature of cycle (°C) OH > 0L, 0 k < 25 °C

Stress testingand reliability

251

At first glance these equations appear to have little in c o m m o n with the preceding theory, or one another. However, some analysis, taking into account the experimental conditions under which the equations were derived, can help to clarify their relation to the theory. In particular, the embedded exponential term can be treated as follows: exp

(aT) = exp (a(O + 273))

0 in °C

=

+ 2 x 273a

= exp(546a)exp ( - 2 7 3 a ( 1 - ~ 3 ) ) i.e.

ex.,a,

O))

(5)

Now 1 T

D

1 0+273 1

(o) 1(1_0)

273 1 +

applying the binomial approximation

It is quite legitimate to apply this approximation since the m a x i m u m temperature used to derive the equations was 60 °C. The resulting error of the approximation is less than 5 ~o. This leads to 0 273 273 T Substituting in eqn (5) gives /-2732a\

exp(aT)= Rexp~-

= R exp ( where E is a constant.

~

.)

E/KT)

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P. W. Hah'

Applying this approximation to the two equations leads to T~ = 0.611 - exp { - R't exp ( - E/KT)}]

Comparing the constant temperature equation with eqn (3) in the previous section it can be seen that a =0.6 and the functional form of the equation is 1-exp(-()) It may be deduced that F~ = R't exp ( - E / K T )

(6)

represents the total number of failures occurring in time t and this is completely consistent with the ideas formulated in eqn (3). The thermal cycling equation is similar in form but further analysis will reveal a more appropriate and surprising aspect. The definition of average temperature TA given earlier T A =(10 n - 25[ + [0L - 251 + 50)/2 + 2 7 3 presents problems in manipulation. A more comprehensive definition would be T,~ --

O H - 0L

2

+ 273

If it is assumed that the lower temperature 0 e is always less than zero T A may be rewritten as follows T A = ( 0 H - 25 - 0 e + 2 5 + 50)/2 + 2 7 3 _ OH - OL

2 = T.+25 whence T~ = TA--25

+ 25 + 273

Stress testing and reliability

253

Using this definition of average temperature the number of cycles N can be eliminated as follows. The cycle time tc may be defined as follows tC --

2(0 H - 0L) dO/dt

tc -

2 (OH - 0 e) 60 dO/dt

minutes

i.e. hours

Assuming N identical cycles, the total exposure time t is given by t = Ntc 2N~O n - 0L) 60 dO/dt

But 0a __ On - - 0 L

2 so that t-

4NOa 60(dO/dt)

whence N-

15t(dO/dt) Oa

Substituting for N in the thermal cycling equation gives [

[" 67.5

, [d0\2

_ E/KTA)}]

Comparing this with eqn (4) in the previous section reveals b =0.8 and 67"5 R,t(dO~2 Fc = 0, \~j exp ( - E/KTA)

(7)

The functional form is also identical with the constant temperature

254

P. W. Hah'

equation as was expected. The more interesting aspect of this equation is the presence of the squared cycling term which demonstrates how powerful thermal cycling can be when high cycling rates are used. A comparison between total number of failures may now be made directly by division as follows:

2 = (67"5/Oa)R't(dO/dt)2 exp ( - E/kTA) R't exp ( - E/KT) 67.5(dO/dt) 2 exp ( - E/KTA) o,, exp ( - E/ K T ) In the particular case where T a and T are chosen to be the same 67.5

;

(d0"] 2

\dr/

Since 0 a will, in general, be less than 67.5 the overall effectiveness of cycling will be more than the square of the cycling rate. In particular if OH = 60 °C 0 L = - 20 °C 0~ = 40 °C dO .... dt

5 °C/rain

then 2 = 42

5

FURTHER CONSIDERATIONS

The activation energy E in the Hughes' equation is around 0.1 eV and this is extremely low by normal standards. This could be the result of the electronics used as the basis for the experiments. Certainly the work was performed on circuitry having very high reliability components. In addition to this a certain amount of curve fitting was involved and this could also have led to error in the calculation of the constants. In any event the target failure mechanisms addressed by the equations will be those in the lower activation energy category and the use of them will therefore present something of a worst case.

Stress testing and reliability

255

Another point that is worth making is that certain failure mechanisms will be addressed only by the continuous application of heat and will not be affected by thermal cycling. The converse is also true. Accordingly, it is necessary to use both forms of stress environment in order to be sure that the failure mechanisms that are addressable thermally are all made to occur. Under these circumstances the overall test strength is given by 1 -

(1

-

~)(1

-

~)

where it is assumed that the mechanisms addressed by both stress environments are mututally exclusive.

6

CONCLUSIONS

The analysis indicates that the two equations have the appropriate parameters embedded in them and the expected general form. More surprising perhaps is the presence of the squared cycling rate term which emphasises the power of this kind of testing. General experience with electronic products shows that the failure mechanisms precipitated by these two environments are almost identical and there is therefore a strong case for cycling over constant temperature in view of the greater level of acceleration.

REFERENCES 1. RADC TR-81-87, Burn in: Which environmental stress screens should be used, Rome Air Development Center, NY. 2. RADC TR-78-55, Electronic screening and debugging techniques, Rome Air Development Center, NY.