Abstract—This paper proposes a fitting method to approximate the mixtures of various sloped-tail Gamma distribution characterizing the random telegraph noises (RTN) by an adaptive segmentation Gaussian mixtures model (GMM). The concepts central to the proposed method are 1) adaptive segmentation of the long-heavy tailed distributions such that the log-likelihood of GMM in each partition is maximized and 2) copy and paste with an adequate weight into each partition. This allows the fitting model to apply various bounded tail distribution even with multiple convex and concave folding curves. It is verified that the proposed method can reduce the error of the fail-bit predictions by 2-orders of magnitude while reducing the iterations for EM step convergence to 1/16 at the interest point of the fail probability of 10 -12 which corresponds to the design point to realize a 99.9% yield of 1Gbit chips. Index Terms—Mixtures of Gaussian, random telegraph noise, em algorithm, heavy-tail distribution, long-tail distribution, fail-bit analysis, static random access memory, guard band design. I. INTRODUCTION The approximation-error of the tails of random telegraph noise (RTN) distribution will become an unprecedentedly crucial challenge resulting from the fact that: (1) its error directly leads to the error of the guard band (GB) design required to avoid the out of spec after shipped to the market, and (2) tails of RTN distribution will become much longer than that of random-dopant-fluctuation (RDF) which is the conventional dominant factor of the whole margin-variations and the convolution results of the two will be more affected by the RTN than the RDF, as can be seen in Fig. 1. Since the increasing paces of variation-amplitude Vth are differently dependent on the MOSFET channel-size (LW) like the below expressions of (1) and (2), the Vth increasing paces of RTN is a 1.4x faster than that of RDF if assuming the LW is scaled down to 0.5 every process generation, as shown in Fig. 1(a). (1) (2) where AVt (RDF) and AVt (RTN) are Pelgrom coefficients for RDF and RTN, respectively. Manuscript received December 15, 2012; revised February 23, 2013. This work was supported in part by MEXT/JSPS KAKENHI Grant Number of 23560424 and grant from Information Sceience Laboratory of Fukuoka Institute of Technology. The authors are with the Information Intelligent System Fukuoka Institute of Technology, 3-30-1, Wajiro-Higashi, Higashi-ku, Fukuoka, Japan (e-mail: bd12002@ bene.fit.ac.jp, yamauchi@fit.ac.jp). This means that RTN will soon exceed RDF and becomes a dominant factor of whole margin variations, as shown in Fig. 1(a). According to the references [1]-[5], there will come the time soon around 15nm scaled CMOS era. (a) (b) Fig. 1. (a) Trend of variation amplitude of RTN and RDF (b) Comparisons of distributions of convolution results between 3 cases of assuming RTN for 40nm, 16nm, and 7nm class device scaling. (1) RTN<RDF, (2) RTN=RDF, (3) RTN>RDF. RTN will dominate the whole variations. To make clear the issues we will discuss in this paper, the concepts of what will happen at that time are shown in Figs. 1-2. Fig. 1(b) illustrates the probability density functions for RDF, RTN1(40nm), RTN2(<16nm) and RTN3 (<7nm), and its convolution results, respectively. It is worth mentioning that the distribution-shape of the convolution results obey the Gaussian when RTN<RDF and changes to follow the combinations of Gamma and Gaussian distributions when RTN=RDF, and finally becomes dominated by Gamma distribution of RTN when RTN>RDF, respectively [4]-[5]. The tails on the both sides of the distribution are asymmetrical and are differently influenced by longer-tail Gamma-RTN for right side and shorter tail Gaussian-RDF for left side and, respectively, as shown in Fig. 1(b). Since the interest area for the GB design is on the right side, i.e., in less margin zone, it can be seen that the approximation-error of the RTN distribution directly leads to estimation-error of fail-bit counts (FBC). The conventional 0 -4 -8 -12 +2 0 -6 -10 -16 0 -4 -8 -12 0 -4 -8 -12 RTN1 RTN2 RTN3 RDF RDF RDF (1) RTN << RDF (2) RTN = RDF (3) RTN > RDF Margin scale x -2 -4 -8 -12 -14 +4 +6 pdf pdf pdf (less mar (more margin) Still Keeps Gaussian! It is just shifted. Tail distribution is dominated by RTN Tail is just obeyed to RTN distribution Convolution of RTN1 with RDF Convolution of RTN2 with RDF Convolution of RTN3 with RDF Adaptive Segmentation Gaussian Mixtures Models for Approximating to Drastically Scaled-Various Sloped Long-Tail RTN Distributions Worawit Somha and Hiroyuki Yamauchi LW (RTN) AVt (RTN) ΔVth LW (RDF) AVt (RDF) ΔVth ∝ ∝ International Journal of Future Computer and Communication, Vol. 2, No. 5, October 2013 407 DOI: 10.7763/IJFCC.2013.V2.195