Deviation Of The Spatial Correlation. Chip Non-parametric Estimation Methods

Along with the rapid development of the modern semiconductor technologies, the scaling down of transistors according to moore' law has resulted in great achievements in chip performance, semiconductor technology enters the nanometer regime. Restricted by technology, The complex nano-technology has introduced the problem of unavoidable process variations. These process variations lead to geometric and electrical variations in the characteristics of the devices and interconnects. As a result, the performances of the fabricated circuits are degraded from the design specifications, the manufacturing yield is lost. Yield loss problem has become a bottleneck in nanometer chip design. Therefore, it's important to build process variation models and analyze it when designs.In the nanometer regime, the intra-die variations, which characterize the process variations within a single die, become predominant in total process variations. Intra-die variations often show spatially correlated patterns, which means devices that are closely placed tend to possess similar characteristics than those further apart. Spatial correlation is defined to describe the degree to which the values of device characteristics are related to each other as a function of spatial distances of the devices. Traditional methods employ parametric functions according to silicon measurement data, but these functions are too smooth to describe actual spatial correlations; also a study found that the spatial correlation of intra-die variation in different directions has different correlation which is called anisotropy, while traditional methods assume isotropy condition. Nonparametric method is researched, the common method-polynomial fitting method is not perfect because it can't avoid high-degree-item fluctuating and can't ensure monotonicity.In this paper we propose a nonparametric estimation method, using B-spine as the basis function of the correlation function, considering anisotropy situation. We describe how to accomplish B-spine curve fitting with least square method and maximum likelihood estimation method, and how to add constraints to ensure positive definiteness and monotonicity of the functions. Experiments is also made to compare with polynomial fitting method with data from computer simulation.