| Title | Determination of Optimal Polynomial Regression Function to Decompose On-Die Systematic and Random Variations |
| Author | *Takashi Sato, Hiroyuki Ueyama, Noriaki Nakayama, Kazuya Masu (Tokyo Institute of Technology, Japan) |
| Page | pp. 518 - 523 |
| Keyword | process variation , log-likelihood estimate, AIC, model selection |
| Abstract | A procedure that decomposes measured
parametric device variation into systematic and random components is
studied by considering the decomposition process as selecting the most
suitable model for describing on-die spatial variation trend. In order
to maximize model predictability, the log-likelihood estimate called
corrected Akaike information criterion is adopted. Depending on on-die
contours of underlying systematic variation, necessary and sufficient
complexity of the systematic regression model is objectively and
adaptively determined. The proposed procedure is applied to 90-nm
threshold voltage data and found the low order polynomials describe
systematic variation very well. Designing cost-effective variation
monitoring circuits as well as appropriate model determination of on-die
variation are hence facilitated.} |
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| Title | Within-Die Process Variations: How Accurately Can They Be Statistically Modeled? |
| Author | Brendan Hargreaves, Henrik Hult, *Sherief Reda (Brown Univ., United States) |
| Page | pp. 524 - 530 |
| Keyword | process variations, statistical modeling |
| Abstract | Within-die process variations arise during integrated circuit (IC) fabrication in the sub-100nm regime. These variations are of paramount concern as they deviate the performance of ICs from their designers’ original intent. These deviations reduce the parametric yield and revenues from integrated circuit fabrication. In this paper we provide a complete treatment to the subject of within-die variations. We propose a scan-chain based system, vMeter, to extract within-die variations in an automated fashion. We implement our system in a sample of 90nm chips, and collect the within-die variations data. Then we propose a number of novel statistical analysis techniques that accurately model the within-die variation trends and capture the spatial correlations. We propose the use of maximum-likelihood techniques to find the required parameters to fit the model to the data. The accuracy of our models is statistically verified through residual analysis and variograms. Using our successful modeling technique, we propose a procedure to generate synthetic within-die
variation patterns that mimic, or imitate, real silicon data. |
| PDF file |
| Title | Chebyshev Affine Arithmetic Based Parametric Yield Prediction Under Limited Descriptions of Uncertainty |
| Author | Jin Sun, Yue Huang (The University of Arizona, United States), Jun Li (Anova Solutions, United States), *Janet M. Wang (The University of Arizona, United States) |
| Page | pp. 531 - 536 |
| Keyword | Chebyshev Affine Arithmetic, Process Variations, Limited Description of Uncertainty, Dependency Bounds |
| Abstract | In modern circuit design, it is difficult to provide reliable parametric yield prediction since the real distribution of process data is hard to measure. Most existing approaches are not able to handle the uncertain distribution property coming from the process data. Other approaches are inadequate considering correlations among the parameters. This paper suggests a new approach that not only takes care of the correlations among distributions but also provides a low cost and efficient computation scheme. The proposed method approximates the parameter variations with Chebyshev Affine Arithmetics (CAA) to capture both the uncertainty and the nonlinearity in Cumulative Distribution Functions (CDF). The CAA based probabilistic presentation describes both fully and partially specified process and environmental parameters. Thus we are capable of predicting probability bounds for leakage consumption under unknown dependency assumption among variations. The end result is the chip level parametric yield estimation based on leakage prediction. The experimental results demonstrate that the new approach provides reliable bound estimation while leads to 20% yield improvement comparing with interval analysis. |
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