A Novel Expression of Spatial Correlation with Random Curved Surface

Shin-ichi OHKAWA  Masakazu AOKI  Hiroo MASUDA  Yasuaki INOUE 

Publication
C - Abstracts of IEICE TRANSACTIONS on Electronics (Japanese Edition)  Vol.J92-C  No.3  pp.75-84
Publication Date: 2009/03/01
Online ISSN: 1881-0217
Print ISSN: 1345-2827
Type of Manuscript: PAPER
Category: 
Keyword: 
LSI,  design,  variation,  random curved surface,  Gaussian distribution, 

Full Text(in Japanese): PDF(1.6MB)

Summary: 
In sub-100 nm LSI, variability design is one of the indispensable issues for SoC products. We have developed a test structure for precise measurement of WID (Within Die) device variation. Also a new technique has been proposed to decompose the WID variation into systematic and random ones. It is clarified that the random part shows pure Gaussian distribution, however, the systematic part shows specific spatial distribution, which can be characterized with correlation length. However, it is impractical that whole correlation coefficient is considered among millions of devices or circuit. We newly propose a concept of "random curved surface" to describe the spatial characteristics. As mentioned above, variation of the random part shows Gaussian distribution. Similarly the variation of systematic part can be fully characterized with random curved surface. In this paper, we propose the idea and verify it by introducing sophisticate two-dimensional Legendre polynomial formula. This concept of random curved surface is clarified to be very accurate in modeling spatial character found in systematic WID variation.