Does the Black's equation work for immortal wires / interconnections (or only for mortal ones)? I'm trying to understand whether the Black's equation is only true for mortal wire / interconnections or it is also applicable to immortal wires (essentially when its $jL$ product is less than $jL_{blech}$ where $j$ and $L$ are the current density and the length of the wire respectively).
Thanks,
 A: While I'm not an expert on electromigration, it seems obvious to me that the answer is "Black's equation does not apply to very short wires".


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*Black's equation is an empirical model to estimate the MTTF (mean time to failure) of a wire due to electromigration. The equation does not mention length, and so it predicts the same MTTF no matter what the length of the wire.

*Wikipedia says of the Blech length "Any wire that has a length below this limit will not fail by electromigration."

*Black's equation predicts that even very short wires will fail due to electromigration, which contradicts the discovery of Mr. Bleck that such wires actually don't fail due to electromigration.
Therefore


*

*Black's equation doesn't apply to wires shorter than the Blech length.


My understanding is that Black's equation is a very useful and practical rule of thumb that was used long before anyone understood the physical mechanisms of eletromigration, and is still used today.
Much like Newton's second law ("F=ma") is still used today, even though "F=ma" is known to be incorrect over very small distances or at very high velocities.
R.L. de Orio et. al. have have developed physics-based models to estimate MTTF of a wire.
As you might have expected, those models give approximately the same MTTF as Black's equation for long wires in the kind of environment Mr. Black worked in, and a much longer MTTF ("immortal") for very short wires.
R.L. de Orio, H. Ceric, S. Selberherr.
"Physically based models of electromigration: From Black’s equation to modern TCAD models"
Microelectronics Reliability journal.
2010.
