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 (ΔV 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., ΔV T @ΔN IT =1, is exponential, with a fraction of transistors having ΔV T ~0. Finally, we obtain the statistics of ΔV T @t STS by convolving the statistics of N IT @t STS with that of ΔV T @ΔN IT =1. The resultant statistics of ΔV T @t STS compares favorably with a broad range of experiments reported in the NBTI literature.