Black's Equation for MTTF Due to Electromigration
Key Takeaways

Electromigration is a primary cause of failure in integrated circuits and is described by Black’s equation.

This empirical equation is based on an Arrhenius model and only accounts for a single diffusion mechanism.

There are some drawbacks to Black’s equation that should be considered when taking failure measurements during accelerated testing.
These integrated circuits will have some mean time to failure, which can be estimated using Black’s equation
When an IC fails, it doesn’t usually do so by catching on fire or exploding. The chip could simply stop working because of an internal failure. One common source of failure in an interconnect is electromigration, a phenomenon referring to the literal movement of atoms in an interconnect as current flows. This eventually leads to an open circuit along an interconnect, leading to partial or total failure of the system.
So what are the physical processes involved in electromigration leading to failure, and how can they be conveniently summarized to predict failure? Failure along a given interconnect can be nicely summarized using Black’s equation, which provides the mean time to failure (MTTF) of an interconnect in an IC. With ICs constantly getting smaller and packed with more functions, designers should understand the different processes involved in electromigration to better understand the meaning of Black’s equation and predict MTTF.
Black’s Equation for MTTF
Black’s equation is quite simple in that it roughly relates the geometry of an interconnect and the current density flowing on the wire to the MTTF value. This equation is empirical; it is not derived from first principles, although it does describe MTTF in an interconnect in terms of an Ahrrenius process for diffusion. Black’s equation for MTTF in an interconnect is shown below:
Black’s equation relating mean time to failure (MTTF) to current density and an Ahrrenius process for electromigration
The above equation is defined as a proportion where the proportionality constant is taken as a function of the area. In general, an interconnect with a larger area will have a longer MTTF value. The other parameters are defined as follows:

J: current density in the interconnect. For pulsed currents or AC currents, use the average current density.

N: a scaling factor, which usually ranges between 1 and 2 depending on the diffusion process (see below).

k: Boltzmann’s constant (1.38·1023 J/K).

T: temperature of the interconnect (in K).

Ea: activation energy (in J).
Note that Ea can also be stated in units of eV, thus k also needs to be converted to eV using the electron charge. There is good reason for this, as it allows the activation energy to be compared to the voltage across the interconnect.
The Ea value depends on the material being used in the interconnect and the dominant diffusion process that governs electromigration. There are three primary diffusion processes that can occur during electromigration:

Surface diffusion. In this process, diffusion occurs along the outer surface of the conductor.

Grain boundary diffusion. Here, diffusion occurs along grain boundaries between crystallites in the conductor.

Bulk diffusion. In bulk diffusion, atoms migrate within grains and can travel across grain boundaries.
These different mechanisms are dominant in different metals with lower activation energies. By “dominates,” we mean that a particular diffusion process has lower activation energy than the other two processes. For example, grain boundary diffusion dominates in aluminum with Ea = 0.4 to 0.5 eV [source], whereas surface diffusion dominates in copper with Ea = 0.7 to 0.9 eV [source]. In general, bulk diffusion will have the highest activation energy compared to diffusion along grain boundaries and along the surface of the conductor.
There are some important takeaways from Black’s equation. First, a larger current density leads to a shorter MTTF. Second, the proportionality constant is generally proportional to some power of the interconnect crosssectional area, so using a larger crosssectional area will provide larger MTTF. Finally, when the interconnect reaches a higher temperature, the MTTF is lower. This last point is quite important in two aspects: accelerated testing and thermal runaway in an interconnect.
Measurements of MTTF and Accelerated Testing
MTTF is easily determined statistically by measuring MTTF from multiple interconnects. The parameters in Black’s equation can then be determined by plotting the statistically determined MTTF value against the RHS of Black’s equation on a loglog scale. A linear fit to the data provides the proportionality constant and activation energy for a given N value.
For an integrated circuit with an MTTF of around 10 years, how can you determine the true MTTF value in a much shorter amount of time? This is done by measuring MTTF at a higher temperature and current density. Basically, following the same ideas used in accelerated testing. Once the MTTF from the accelerated test is calculated, the following equation can be used to calculate the MTTF at lower temperature and current density:
Accelerated testing with Black’s equation compares the ratio of the accelerated MTTF to the true MTTF
Because different diffusion processes can occur simultaneously, care must be taken that the conditions in the accelerated test match those in the lower temperature/current density test. If done correctly, these accelerated tests provide a useful measure of MTTF for a single diffusion process. This requirement to maintain a single dominant diffusion process in accelerated tests reveals some fundamental drawbacks to Black’s equation.
Where Does Black’s Equation Fall Short?
Black’s equation is a very useful empirical equation for describing the relationship between MTTF and current density. However, Black’s equation falls short in some key areas:

Assumes constant temperature. Black’s equation assumes that the interconnect is always at the same temperature. This is often not a realistic assumption. As electromigration begins, the crosssectional area decreases. Thus, the current density increases, and the temperature increases due to IR drop. This thermal runaway process reduces the MTTF value.

Only assumes a single diffusion process. This assumption is fine for lowlevel interconnects made of copper or aluminum (copper is primarily used in modern ICs). Black’s equation only describes the lowest energy electromigration process (bulk, grain boundary, or surface diffusion) and cannot describe multiple processes simultaneously. For other conductive materials etched in semiconductors, surface and grain boundary diffusion may occur simultaneously.
Working through the electromigration process can be tedious without proper tools and team understandings for the necessity of MTTF. Ensure that your designs are well taken care of — whether thermally, electromagnetically, or in any other reliable capacity — by having the analysis tools you can trust.
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