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# 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.

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

$A$ 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
$w$ is the width of the metal wire

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. Such 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

* cite web
last = Sjøthun
first = Sverre
title = Semiconductor Electromigration In-Depth
work =
date = 2005-12-15
format =
doi =
accessdate = 2006-11-14

* cite journal
last = Black
first = J.R.
title = Electromigration - A Brief Survey and Some Recent Results
journal = IEEE Transaction on Electron Devices
volume = ED-16
issue = 4
pages = 338
date = 1969
publisher = IEEE

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