Statistical extraction and modeling of 3-D inductance with spatial correlation

In this paper, we present a novel method for inductance extraction and modeling for interconnects considering process variations. The new method is based on the spectral stochastic method where orthogonal polynomials are used to represent the statistical processes in a deterministic way. Coefficients of the orthogonal polynomials are computed for the inductances. Statistical inductance values are then found using a fast multi-dimensional Gaussian quadrature method with sparse grid. To further improve the efficiency of the proposed method, a random variable reduction scheme is used. Given the interconnect wire variation parameters, the resulting method can derive the parameterized closed form of the inductance and its variation. We show that both partial and loop inductance variations can be significant given the width and height variations. This new approach can work with any existing inductance extraction tools to produce the variational inductance or impedance models. Experimental results show that our method is orders of magnitude faster than than the Monte Carlo method for several practical interconnect structures.