Variational capacitance modeling using orthogonal polynomial method

Abstract

In this paper, we propose a novel statistical capacitance extraction method for interconnects considering process variations. The new method, called statCap, is based on the spectral stochastic method where orthogonal polynomials are used to represent the statistical processes in a deterministic way. We first show how the variational potential coefficient matrix is represented in a first-order form using Taylor expansion and orthogonal decomposition. Then an augmented potential coefficient matrix, which consists of the coefficients of the polynomials, is derived. After that, corresponding augmented system is solved to obtain the variational capacitance values in the orthogonal polynomial form. Experimental results show that our method is two orders of magnitude faster than the recently proposed statistical capacitance extraction method based on the spectral stochastic collocation approach [18] and many orders of magnitude faster than the Monte Carlo method for several practical interconnect structures.

Variational capacitance extraction of on-chip interconnects based on continuous surface model

“…Another important character of on-chip variation is the spatial correlation. Thus, a spatially correlated multivariate Gaussian distribution is often assumed for the geometry parameter variations [3][4][5][6][7][8][9].…”

“…Based on a 3-D boundary element method (BEM) capacitance solver, a perturbation method was proposed in [5] to generate a quadratic model for capacitance variation. Two efficient methods were then proposed to obtain the quadratic variational capacitance with the Hermite polynomial chaos [6,7]. The spectral stochastic collocation method (SSCM) in [6] employs the techniques of homogenous chaos expansion and sparse grid quadrature to obtain the coefficients in the quadratic expression.…”

“…The spectral stochastic collocation method (SSCM) in [6] employs the techniques of homogenous chaos expansion and sparse grid quadrature to obtain the coefficients in the quadratic expression. The method in [7] employs the spectral stochastic method, and then solves augmented potential coefficient matrices. Considering only the first-order variation of potential coefficients, this method is very efficient, but not promising on the accuracy.…”

“…Although the methods in [5][6][7] generate quadratic variation model of capacitance, which has merit for subsequent statistical circuit analysis, they all consider a simplified model of geometric variations. The variational geometry is constructed by disturbing the panels on nominal conductor surface, which produces an incomplete surface.…”

“…This surface discontinuity obviously deviates from the actual situation, and is likely to induce large error for the interconnect capacitance under nanometer technology process. Another shortage of [5][6][7] is that they do not consider the actual chip-level capacitance extraction. Recently, techniques considering the window-based chip-level capacitance extraction were proposed in [9].…”