A modified reliability expression for the electromigration time-to-failure

Microelectronics and Reliability, Vol. 14, pp. 43l to 433. Pergamon Press, 1975. Printed in Great Britain A MODIFIED RELIABILITY EXPRESSION FOR THE ELECTROMIGRATION TIME-TO-FAILURE A. BOBBIO and O. SARACCO Istituto Elettrotecnico Nazionale "Galileo Ferraris" Torino, Italy AImtraet--Electromigration life data reported by different authors show that a single exponent for the dependence of the mean time-to-failure (MTF) on the current density cannot be defined. A modified expression is proposed in which, through the self heating, the current density is directly inserted in the exponential term. Furthermore a simple relationship between the mean rate of the resistance change and the time-tofailure of each specimen is presented. Electromigration accelerated life tests have been widely carried out in recent years in order to provide a simple relationship between the mean time-to-failure (MTF) and the applied stress. However a wide spread has been noticed, as reviewed recently in [1], among the experimental results of different authors. An Arrhenius like dependence of the MTF on the temperature has been always observed with activation energies varying from 0.5 to 0"8 eV, while for the current density dependence a single power law (of the type MTF oc j-~) has been already proposed. Experimental values of the exponent n have been found generally in the range of 1 to 3, but larger values, up to 10, have been also reported. In any case the assumption of a constant exponent n implies that the weight of the current density on the degradation rate should be unaffected by the current density level itself. Recently detailed physical models, which consider electromigration damage as due to the growth of localized voids acting as sinks for the diffusing vacancies, have been presented. These models, although differing in the shape of the growing cavity, spherical [2], cylindrical [3] and crack shaped cavities [4] have been considered, provide a very similar behaviour for the dependence of MTF on the stress conditions, particularly showing the role of the Joule heating on the degradation rate. In agreement with the kinetics models a modified reliability formula is suggested in the following. In this formula the linearity between MTF and current density, as provided by the diffusion equation of the Huntington and Grone's theory [5] is maintained but a current density dependence is directly inserted in the temperature term through the self heating. Let us consider the simple equations that relate the conductor resistance, the temperature and the dissipated power: R = RA(1 + c t ( T - Ta)), (1) T = TA + bW. (2) Equations (1) and (2) can be combined to yield an expression of the temperature increase above the ambient, due to the self heating, as a function of j2: AT(j2) M&R 14/5--42 bRaj2 1/$2 _ 6Rao:j 2 , (3) 431 where b is the thermal resistance, R a the electrical resistance at the ambient temperature TA, ~ the temperature coefficient of resistance and S the film cross sectional area. All the parameters appearing in (3) are easily measurable by preliminary testing. Thus the proposed reliability formula may be written as: MTF - AJ exp K(T.4 TU2) , (4) where A is a constant characteristic of the specimen, ~bthe activation energy and K the Boltzmann constant. The suggested relationship takes into account, through the function AT(j2), tile properties of heat dissipation of the device, and its geometrical dimensions; moreover it puts explicitly into evidence that the deviations from linearity can be ascribed to the overheating of the film with respect to the environment in which the device operates and are mainly related to the thermal resistance b of the film which can be evaluated directly. In Fig. 1 the conductor life is plotted on a log-scale vs. the current density with the environmental temperature as parameter. Calculations have been performed assuming for the various parameters the following values derived from previous experiments [2]. ~b = 0"6 eV Ra = 3"25 f~ K = 1'380 x 10-16 erg/°C A = 0'05 b = 197-6°C/W S =(15 x 1-6) = 24#m 2 = 0.00449oc- 1 The constant A, which has been obtained fitting experimental results, lies in the range of values reported by Black [6]. The discrepancies already observed in the literature about the dependence of MTF on the current density may be justified examining (4) and Fig. 1; in fact it appears that a single exponent n cannot be univocally determined since the degradation rate is a function both of the applied stress, and of the thermal properties of the conductor. It is also evident that extrapolation of accelerated test data using a typical value n = 2 yields too 432 A. BOBBIOand O. SARA('(:() 108 I(? • I0 10 3 o~ LD ~6 * • E×perlmental points "m"a'~l~, C y l i n d r i c a l 2cu io d Crackand~~ c 2 .c l0 IO T c o ~: Jc id v 10 I0" IO; Life, F- IOt> ,'/* hr Fig, 2. Mean rate of change in resistance at 5 % vs. specimen life-time. Experimental points are obtained from three different series of measurements: • TA = 70°C; j = 1"2 x 106 A/cm 2 • Z A - 4 0 ° C ; j = 0"9 x 106 A/cm 2 • TA - 60°C; j - 0"8 x 106 A/cm 2 IO 103 104 Current 105 density, 406 A/cm FOr 2 Fig. 1. Mean time-to-failure vs. current density with the environmental temperature TA as parameter. optimistic values of the conductor life at normal stress levels. It should still be noted that the suggested formula neglects the formation of an hot spot in the weakest section during the last stages of the conductor life; this effect could be included adding in the temperature expression terms of higher order in j ; these terms, however, are difficult to evaluate experimentally. For reliability purposes it is of interest to define a useful life, that is a time during which the device can correctly operate. In thin film conductors the useful life may be conveniently referred to the conductor resistance change, which is an easily measurable parameter related to the device wear-out. For this reason we have applied the electromigration damage models to find the relation between the rate of the change in resistance in the first stages of the film degradation and the time-to-failure. The following expression has been derived: TF = C (5) where T F is the time-to-failure of each specimen, [ ( 1 / R o X d R / d t ) ] y % the mean rate of the resistance change calculated at a typical level y%, and C is a constant. The general character of (5) lies in the fact that the constant C is quite independent on the applied stress and on specimen characteristics, for all the considered theoretical models. The results are plotted in Fig. 2 for a mean rate of the resistance change calculated for instance at 5 °' q). The straight lines obtained from the cylindrical, spherical and crack models correspond to a value of the constant C equal to 0.1 [3]. Available experimental points in the range of accelerated tests lie very closely on the theoretical curve. This formula can provide in any case a rough estimate of the time-to-failure of a thin film conductor through resistance change measurements, that is by non destructive experiments. But vice versa if a useful life level, related to the change in resistance is established, it can be evaluated from accelerated life measurements. CONCLUSIONS A modified reliability formula is presented, in which the role of the self-heating is emphasized, inserting directly the current density dependence in the temperature term. Extrapolation of accelerated life data to normal stress levels through this formula leads to less optimistic values of the conductor life. The introduction of the thermal function AT(j 2) may be very useful in the statistical analysis of the life test results. In fact, as it has been experimentally proved [7], the spread in the time-to-failure distributions are in a large extent to be ascribed to the spread in the distribution of the specimen thermal resistances, i.e. in the distribution of the true operating temperatures. Finally a simple theoretical relationship which connects the time-to-failure with the mean rate of resistance change, and which agrees with experimental results, is reported. By means of this relationship a useful life level may be determined from accelerated experiments. A Modified Reliability Expression for the Electromigration Time-to-Failure REFERENCES 1. L. Braun, Microelectron. & Reliab. 13, 215 (1974). 2. A. Bobbio, A. Ferro and O. Saracco, IEEE Trans. Reliab. R-23, 194 (1974). 3. A. Bobbio, A. Ferro and O. Saracco, Proc. "II Congr~s National de Fiabilit6", Perros-Guirec, France (1974). 433 4. R. A. Sigsbee, J. appl. Phys. 44, 2533 (1973). 5. H. B. Huntingtone and A. R. Grone, J. Phys. Chem. Solids 20, 76 (1961). 6. J. R. Black, IEEE Trans. Electron Devices ED 16, 3381 (1969). 7, A. Bobbio and O. Saracco, Thin Solid Films 17, S-13 (1973).