# 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,

• I am sorry but I do not understand what a mortal/immortal wire is. Can you give a link? Commented Jul 15, 2013 at 6:45
• @annav Refer to the section "wire length" synopsys.com/Tools/Verification/CapsuleModule/… Commented Jul 15, 2013 at 7:33
• thanks, but completely out of my data base. I think this should be asked in the electronics.stackexchange if it gets no response here. Commented Jul 15, 2013 at 10:57
• "immortal" is not a technical term. I think you are asking "why does electromigration not effect wires shorter than the Blech length?" (To which I do not, unfortunately, know the answer.) Commented Jul 15, 2013 at 11:54
• @WanderingLogic Thanks for contribution, but what I am asking is, whether the Black's equation gives a correct Mean Time To Failure (MTTF) estimation for the wires which do not suffer from electromigration (e.i. the wires with the $jL$ product smaller than $jL_{blech}$). Commented Jul 15, 2013 at 23:02

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".

• 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.