1.4.3 Black's Equation

The dependence of the MTTF of the lognormal lifetime distribution on the acceleration parameters of temperature and current density is normally described by an Arrhenius-like semi-empirical equation originally formulated by Black [12,13,14] and then modified by Blair [15] in the form

\[MTTF =Aj^{-n} \ \text{exp}\left(\cfrac{E_\text{a}}{k_\text{B}T}\right)\] , (1.1)

where A is a constant which contains several physical and geometrical properties of the materials, n is the current density exponent, E_a is the activation energy for electromigration-induced failure, and k_B is the Boltzmann constant. Black's original empirical work predicted a failure time following a j^-2 dependence [12], while later experimental studies estimated values of n generally lying between 1 and 3 [138].

The estimation of the current density exponent can be determined from the results of the electromigration accelerated tests by taking the natural logarithm of equation (1.1) as follows

\[\begin{equation} \ln MTTF =\ln A - n \ln j + \cfrac{E_\text{a}}{k_\text{B}T}. \end{equation}\] , (1.2)

The sample estimate for the current density exponent n can be obtained from a linear regression analysis for one independent variable [52]. The analysis requires lifetime data from several electromigration tests conducted at different stress conditions of current density, while the stress temperature of the test structures is kept constant. A plot of the MTTF as a function of current density represents data points aligned along a straight line, which is the best fit to the data, as depicted in 1.11. The angular coefficient of the line determines the current density exponent n.

Figure 1.11: Time to failure dependence on current density illustrates the behavior of equation (1.2) for a constant test temperature. The line indicates the fitting to the data from which the determination of the current density exponent is obtained from its angular coefficient.

It has been shown that a current density exponent n value close to 2 indicates a lognormal lifetime distribution, which corresponds to a nucleation time dominated failure [138,36,63], while a value close to 1 indicates that the dominant phase of the electromigration failure time is the void evolution [51,104]. At a constant given temperature, Black's equation generalizes to

\[\begin{equation} MTTF =t_\text{N}+t_\text{E}=Bj^{-2}+Cj^{-1}, \end{equation}\] , (1.3)

where B and C are materials constants, t_N is the void nucleation time, and t_E is the void evolution time. The model in equation (1.3) assumes that the phases which dominate the electromigration-induced failure under accelerated test conditions are the same of those under device operating conditions [79].