Analyzing the distribution of threshold voltage degradation in nanoscale transistors by using reaction-diffusion and percolation theory

Abstract

Continued scaling of transistors into the nanoscale regime has led to large device-to-device variation in transistor characteristics. These variations reflect differences in substrate doping, channel length, interface and/or oxide defects, etc. among various transistors. In this paper, we develop a theory for the statistical distribution of threshold voltage degradation ( V T ) due to the Negative Bias Temperature Instability (NBTI). First, we model the time dynamics of interface defects within the Reaction-Diffusion (R-D) framework and calculate the statistics of interface defect using Markov Chain Monte-Carlo method. We show that the generation and annealing of interface defects are strongly correlated and that the statistics of interface defect at a given stress time (N IT @t STS ) follows a skew-normal distribution. Second, we explore the differential effect of the spatial distribution of interface defects in nanoscale transistors pre-populated with a discrete number of randomly placed substrate dopants. We model the effect of spatial distribution of defects using a percolative network and demonstrate that the distribution of threshold voltage degradation for a single additional interface defect, i.e., V T @ N IT = 1, is exponential, with a fraction of transistors having V T ∼ 0. Finally, we obtain the statis-Electronic supplementary material The online version of this article (tics of V T @t STS by convolving the statistics of N IT @t STS with that of V T @ N IT = 1. The resultant statistics of V T @t STS compares favorably with a broad range of experiments reported in the NBTI literature.

Investigation of stochastic implementation of reaction diffusion (RD) models for NBTI related interface trap generation

“…Some effort has already been made to study the stochastic N HT process in small area devices [11]- [13]. However, since NBTI also results in N IT generation, its dynamics in small area devices needs to be explored [10]. For large area devices, ΔN IT is modeled using the popular Reaction-Diffusion (RD) model, originally proposed in [14] and later modified in [6], [15]- [18], and extensively used in [1], [3]- [5] to successfully predict measured data from wide variety of devices and experimental conditions.…”

The relevance of deeply-scaled FET threshold voltage shifts for operation lifetimes

“…The resulting significant increase in both time-zero and time-dependent variability of many degradation mechanisms is thus best understood in terms of the impact of individual (charged) defects. This "bottom-up" approach to device reliability is already being advocated by several groups [1][2][3][4][5][6].…”

Asymptotic expansions for moments of skew-normal extremes

“…Its applications are too many to list. Some applications of the skew normal distribution that have appeared in the past year alone include: the distribution of threshold voltage degradation in nanoscale transistors by using reaction-diffusion and percolation theory (Islam and Alam, 2011); population structure of Schima superba in Qingliangfeng National Nature Reserve (Liu et al, 2011); rain height models to predict fading due to wet snow on terrestrial links (Paulson and Al-Mreri, 2011); modeling of seasonal rainfall in Africa (Siebert and Ward, 2011); modeling of HIV viral loads (Bandyopadhyay et al, 2012); multisite flooding hazard assessment in the Upper Mississippi River (Ghizzoni et al, 2012); modeling of diabetic macular Edema data (Mansourian et al, 2012); risks of macroeconomic forecasts (Pinheiro and Esteves, 2012); modeling of current account balance data (Saez et al, 2012); automated neonatal EEG classification (Temko et al, 2012).…”