Min-log approach to modeling dielectric breakdown data

The process of dielectric breakdown in thin films has been studied extensively in the literature. Typically, these processes are modeled in terms of stochastic quantities related to a Weibull distribution. In many cases the direct Weibull approach is not capable of explaining the observed times to dielectric breakdown (TDDB), leading to the necessity to introduce more complex models. This, in turn, leads to considerable complications in the process of modeling and analyzing this phenomenon. In this article we present an approach to analysis of TDDB data based on the assumption that a sample can be viewed as a collection of competing cells, where the same stochastic process of degradation is taking place in the individual cells. Every cell, therefore, has its individual time to failure, and the cell having the shortest time is the one that actually causes the failure. In many cases, this is the only lifetime that is actually observable, as the sample and the ongoing processes in the cells could be affected by the dielectric discharge in one of them. We consider the situation where times of breakdown of individual cells can be modeled by a lognormal distribution and develop an approach based on the finite-sample distribution of minimum. This model leads to a relatively simple explanation of the TDDB data, including both low and high percentiles. We develop procedures for inference based on complete or right-censored TDDB data and illustrate its application for data obtained in the course of stress-based experiments.