What Is Electromigration MTTF and How Do You Calculate It with Black's Equation?

Electromigration is one of the most common failure mechanisms in modern integrated circuits and interconnects. As current densities increase with shrinking device geometries, metal atoms migrate along the direction of electron flow, eventually causing voids or hillocks that lead to open or short circuits. To predict how long a device will survive under these conditions, reliability engineers use the electromigration Mean Time To Failure (MTTF), typically modeled by Black’s equation.

What It Is

Electromigration MTTF is a statistical estimate of the median time to failure for a population of conductors subjected to a given current density and temperature. It is not a fixed lifetime but a predictive figure derived from accelerated testing and physical models. The most widely accepted model is Black’s equation, which relates MTTF to current density, temperature, and material properties. This model is referenced in JEDEC JEP122, the standard for failure mechanisms and models in semiconductor devices.

How It Works: Black’s Equation

Black’s equation expresses MTTF as:

\[
MTTF = \frac{A}{j^n} \exp\left(\frac{E_a}{k T}\right)
\]

Where:

• MTTF = median time to failure (hours)
• A = constant that depends on the geometry, material, and microstructure of the conductor (empirical)
• j = current density (A/cm²)
• n = current density exponent (typically 2 for void-nucleation-dominated failure, but can range from 1 to 3 depending on the failure mode)
• E_a = activation energy for electromigration (eV), typically 0.6–1.0 eV for aluminum and copper interconnects
• k = Boltzmann’s constant (8.617 × 10⁻⁵ eV/K)
• T = absolute temperature (Kelvin)

The equation shows that MTTF decreases sharply with higher current density (inverse power law) and exponentially with higher temperature (Arrhenius behavior). To use it, you need empirically determined values for A, n, and E_a, which come from accelerated life tests on representative samples.

A Worked Illustrative Example

Example data (illustrative only):

Suppose a copper interconnect has the following empirically determined parameters:

• A = 1.0 × 10⁵ h·(A/cm²)²
• n = 2
• E_a = 0.8 eV
• Operating current density, j = 1.0 × 10⁶ A/cm²
• Operating temperature, T = 105 °C = 378 K

First, calculate the exponential term:

\[
\frac{E_a}{k T} = \frac{0.8}{8.617 \times 10^{-5} \times 378} = \frac{0.8}{0.03257} \approx 24.56
\]

\[
\exp(24.56) \approx 4.6 \times 10^{10}
\]

Now compute MTTF:

\[
MTTF = \frac{1.0 \times 10^5}{(1.0 \times 10^6)^2} \times \frac{1}{4.6 \times 10^{10}}
\]

\[
MTTF = \frac{1.0 \times 10^5}{1.0 \times 10^{12}} \times \frac{1}{4.6 \times 10^{10}} = 1.0 \times 10^{-7} \times 2.17 \times 10^{-11}
\]

\[
MTTF \approx 2.2 \times 10^{-18} \text{ hours}
\]

This result is unrealistically low because the chosen A value is not representative of real copper interconnects. In practice, A is calibrated from test data and is typically orders of magnitude larger. This example only demonstrates the calculation procedure — always use empirically fitted parameters from your own accelerated tests or from published process-specific data.

Common Pitfalls

• Using incorrect n value: Assuming n = 2 universally can misestimate lifetime. For bamboo-like or single-crystal structures, n may be closer to 1; for void-growth-dominated failures, n can be higher. Verify against your failure analysis.
• Ignoring temperature variations: Black’s equation requires absolute temperature. A small error in T significantly affects the exponential term — a 10 °C error can change MTTF by several times.
• Treating A as a universal constant: A is process- and geometry-specific. Do not reuse values from different metallization systems or line widths without validation.
• Confusing median with minimum life: MTTF is a median. For design, you should also consider the statistical distribution (typically lognormal) and calculate lower percentiles for safety margins.

Closing

Electromigration MTTF is a powerful predictive tool when applied correctly. To avoid manual calculation errors and quickly explore how changes in current density or temperature affect lifetime, use the free interactive calculator at https://www.6sq.com/tools/em_mttf/. It implements Black’s equation and lets you input your own empirical parameters for immediate results.