Statistics and physics of area-dependent dielectric breakdown: an overlooked yet inevitable natural phenomenon

The traditional inverse area-scaling law for dielectric breakdown (BD) has been recognized for more than a century since, but its physical basis is rarely discussed, aside from an earlier pioneering study. This may be due to the long-standing misconception that area-dependent BD is purely statistical, rather than acknowledging the transformation of a dielectric layer under electric stress from pristine to degraded and ultimately to fragmented because of BD. This unique area dependency distinguishes the BD mechanism from other failure mechanisms such as bias-temperature stability (BTI) and hot-carrier degradation (HCD). Recent experimental observations of reverse and diminished area-scaling phenomena in hydrogen-doped HfO 2 have prompted investigations into these effects presented in this article. Through probability density function (PDF) analysis, we examined the interplay between defect generation and annihilation from both statistical and physical viewpoints. Our findings show that time-dependent decay in annihilation centers is consistent with maximum-value statistics and reduces the likelihood of annihilation over time, thus supporting defect survival and altered BD area-scaling behaviors. Both defect creation and annihilation scale with area, making their combined dynamics compatible with a thermodynamic description of defect-generation–annihilation and leading to diverse area-scaling phenomena in conjunction with the joint minimum-value and maximum-value statistics. We show the traditional inverse area-scaling law is merely a special case assuming infinite annihilation times. Based on the agreement between our BD data using hydrogen-doped HfO 2 stacks and our defect-generation–annihilation model, we propose that intrinsic oxygen interstitials and extrinsic species such as hydrogen can act as annihilation centers. Notably, the estimated time span of the defect annihilation PDF aligns well with the estimated times from ∼1.5 ns to 1.5 μ s for hydrogen substitution into oxygen vacancies based on density functional theory (DFT) calculations. Overall, contrary to conventional wisdom, the root cause of BD area dependency is the most critical and inevitable for a comprehensive physical understanding of dielectric BD.