1 Black's Law

To extrapolate electromigration failure times measured under accelerated test conditions to operating conditions, a semi-empirical model proposed by Black [75] has been widely used in which the time to failure $t_f$ is related to the temperature $T$ and the current density $J$ as

$$t_f = \frac{A}{J^n} (E_a/kT)$$ (215)

The current density exponent $n$ and activation energy $E_a$ are determined by experiment. The time to failure distribution according to Black's equation is displayed in Figure 4.1. In addition to this model, Black also observed that using an interconnect metal with a larger grain size leads to increased lifetimes [75].

Figure 4.1: Time to failure as a function of current density and temperature, as predicted by Black's equation.