Black's equation

Black's Equation is a mathematical model for the mean time to failure (MTTF) of a semiconductor circuit due to electromigration: a phenomenon of molecular rearrangement (movement) in the solid phase caused by an electromagnetic field.

The equation is:^[1]

$MTTF = Aj^{-n} e^\left(\frac{Q}{kT}\right)$

$A$ is a constant
$j$ is the current density
$n$ is a model parameter
$Q$ is the activation energy in eV (electron volts)
$k$ is Boltzmann constant
$T$ is the absolute temperature in K

The model is abstract, not based on a specific physical model, but flexibly describes the failure rate dependence on the temperature, the electrical stress, and the specific technology and materials. More adequately described as descriptive than prescriptive, the values for A, n, and Q are found by fitting the model to experimental data.

The model's value is that it maps experimental data taken at elevated temperature and stress levels in short periods of time to expected component failure rates under actual operating conditions. This data is obtained by running high temperature operating life (HTOL) testing.

Predicted semiconductor failure rates of millions or billions of hours should be used with caution because experimental correlation is all but impossible due to the predominance of other failure modes.

References

1. R. L. de Orio. Dissertation: "Electromigration Modeling and Simulation". 2010. 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.

• Black, J.R. (1969). "Electromigration - A Brief Survey and Some Recent Results". IEEE Transaction on Electron Devices (IEEE). ED-16 (4): 338.