Black's equation is an empirical model to estimate the MTTF (mean time to failure) of a wire, taking electromigration into consideration. Since then, the formula has been getting popularity in semiconductor industry.

\begin{eqnarray} MTTF = AJ^{-n}e^{\frac{E_{a}}{kT}} \end{eqnarray}

where $A$ is a constant, $J$ is the current density ($A/cm^2$), $n$ is a model parameter, $E_{a}$ is the activation energy ($eV$), $k$ is the Boltzmann constant, and $T$ is the temperature ($K$).

Since the current density $J$ is obtained as the ratio of current and cross-sectional area of the wire, and since $A$ is a constant which is obtained based on metal line properties and is cross section area dependent, are these two parameters ($A$ and $J$) correlated factors or not really? I am confused because, based on their definitions, they sound to be dependent, however, this dependence is never considered in relevant researches. Essentially, people try to widen the wires to decrease the current density but at the same time they use a same constant $A$ for different wire widths. Especially, when $n$ is considered as $2$, their correlation will be very significant.

P.S. I also did a quick research and noticed that the values for $n$ and essentially $A$ are found by fitting the model to experimental data.


I'm not in the field, but some comments/opinions:

The implicit dependence of $A(x,y)$ and $J(x,z)$ on $x$ isn't particularly relevant if the other value $y$ is used to match $J(x,z)$ to the experiment. They even point out the ad hoc'ness of the model in the article.

And I notice the similarity of the expression to the one used to justify chemical reaction rates via statistical physics and collision models of varying sophistication, namely the Arrhenius equation. The form of the expression comes about by guessing some cross section behaviour $S(E)$ (computing it being the job of microscopic scale physicists) and averaging this over a equilibrium distribution $$\int_{E_{\text{activation}}}^\infty S(E)\ \text{e}^{E/kT}\text d E=\dots$$ Depending on $S(E)$, this integral will be nasty, but the form $$\dots=S_{\text{new}}(E)\ \text{e}^{E_{\text{activation}}/kT}$$

is to be expected.

This is to motivate my pov that here you got some solid and for modeling purposes, you should take $A\cdot J^n$ to be the experience for $S_{\text{new}}$ to depend monomial on the current.


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