4 An Explanation of Black's Law

Today a semi-empirical model equation (4.5) is widely used to extrapolate interconnects' time to failure under accelerated test conditions compared to operating conditions. Based on experimental observations [3], the value of the current density exponent $n$ in (4.5) is found to be $n=1$ for the nucleation failure mechanism in which failure is dominated by the time required to build-up a critical stress or a critical vacancy concentration. For the void-growth mechanism, in which failure is determined by the growth of the void to the critical size, the current density exponent is $n=2$.

Considering the nucleation failure mechanism for arbitrary given critical vacancy concentration $C_V^f$, Shatzkes and Lloyd [80] derived following the equation for the time to failure,

$$t_f=2\frac{C_V^f}{D_V^0}\Bigl(\frac{kT}{Z^*e\gamma_T}\Bigr)^2 \frac{1}{J^2} (E_a/kT),$$ (241)

with $D_V^0$ the pre-exponential factor for grain-boundary self-diffusivity. The derivation of (4.31) is based on a discussion similar to that given in Section 4.4.1.