Editing Electromigration (section)

=== Electromigration reliability of a wire (Black's equation) ===
{{main | Black's equation }}

At the end of the 1960s J. R. Black developed an empirical model to estimate the [[MTTF]] (mean time to failure) of a wire, taking electromigration into consideration. Since then, the formula has gained popularity in the semiconductor industry:<ref name="Black" /><ref>{{cite book |title=Handbook of multilevel metallization for integrated circuits: materials, technology, and applications |first1=Syd R. |last1=Wilson |first2=Clarence J. |last2=Tracy |first3=John L. |last3=Freeman |publisher=William Andrew |year=1993 |isbn=978-0-8155-1340-7 |page=607 |url=https://books.google.com/books?id=jHeN7KYkj28C}}, [https://books.google.com/books?id=jHeN7KYkj28C&pg=PA607 Page 607, equation 24]</ref>

:<math>\text{MTTF} = \frac{A}{J^n} \exp{\left(\frac{E_\text{a}}{k T}\right)}</math>.

Here <math>A</math> is a constant based on the cross-sectional area of the interconnect, <math>J</math> is the current density, <math>E_\text{a}</math> is the [[activation energy]] (e.g. 0.7 eV for grain boundary diffusion in aluminum), <math>k</math> is the [[Boltzmann constant]], <math>T</math> is the temperature in [[kelvin]]s, and <math>n</math> a scaling factor (usually set to 2 according to Black).<ref name="Black" /> The temperature of the conductor appears in the exponent, i.e. it strongly affects the MTTF of the interconnect. For an interconnect of a given construction to remain reliable as the temperature rises, the current density within the conductor must be reduced. However, as interconnect technology advances at the nanometer scale, the validity of Black's equation becomes increasingly questionable.