Fundamental roles of extreme-value distributions in dielectric breakdown and memory applications (minimum-value versus maximum-value statistics)

In this paper, a thorough review of minimum- and maximum-value statistical distributions is provided. Using the Weibull model (statistics of minima) and the Gumbel model (statistics of maxima) along with the respective scaling properties of their scale-factor and distribution-percentile with device area (size), the application of these two types of extreme-value distributions to dielectric breakdown (BD) and memory operations is discussed. In the case of dielectric breakdown, the minimum-value distribution (the Weibull model) provides an indispensable tool to establish a valid voltage/field acceleration model from experimental perspectives. On the other hand, recent advances in the introduction of maximum-value distribution (the Gumbel model) overcomes the shortcomings of the conventional practice of adopting the normal distribution to characterize memory functional operations and provides much needed mathematical rigor and physical insight particularly for the rapid growing field of resistive random-access memory devices.