3.7.2 Mean Time To Failure

The failure distribution of microelectronic devices follows a WEIBULL^3.5 distribution. At the beginning, early failure may occur due to fabrication faults. At the end of the device's life time, failure is mostly due to material fatigue of one or more parts of the device. To determine the reliability especially for metals, BLACK [215,216] has proposed a modified ARRHENIUS law, which can be arbitrarily adapted and calibrated to special failure mechanisms. BLACK's equation reads

$$\mathrm{\textsf{\sc mttf}} {=}A \vert{\mathbf{{J}}}\vert^{-n} \exp{\left(-\frac{{\mathcal{E}_{\mathrm{A}}}}{{k_{\mathrm{B}}}T}\right)},$$ (3.32)

where the local mean time to failure (MTTF) is inversely proportional to a certain power $n$ of the local current density and directly proportional to $\exp(-{\mathcal{E}_{\mathrm{A}}}/{{k_{\mathrm{B}}}T})$ , with ${\mathcal{E}_{\mathrm{A}}}$ as the activation energy. The proportionallity constant $A$ is geometry-dependent, and must be determined by measurements for different interconnect geometries.

However, BLACK's equation describes an empirical observed material behavior and is thus not valid for arbitrary use. It requires a separate calibration for each different failure mechanism. Nevertheless, BLACK's equation is still commonly used to estimate the reliability with respect to the mean time to failure [5,33,217].