Computation of capacitance matrix for integrated circuit interconnects using semi-analytic Green's function method

Abstract

In this paper a method for computing the capacitance matrix of a multiconductor interconnect line in a multilayered dielectric region is presented. The number of conductors and dielectric layers are arbitrary. The conductors are infinitesimally thin, and can be placed anywhere in the structure. The formulation is obtained by using a semi-analytic Green's function for multilayer structures, which is integrated to a series expansion, valid for uniform charge distribution on the conductors. In addition, the quasi-analytical evaluation of the entries of the Galerkin matrix leads to a very efficient and accurate computer code. Computed results are given for some cases of the integrated circuit interconnects to show the advantages and simplicity of our procedure as compared to the methods available in the literature.

Introduction

As the density, complexity, and speed of VLSI circuits are continuing to increase, the management of the on-chip interconnects becomes of paramount concern to the IC designer, especially with respect to internal parasitics parameters immunity. To optimize electrical properties of the integrated circuits, the estimation of the capacitance matrix of multilayer and multiconductor interconnects in very high-speed IC must be investigated. Coplanar and non-coplanar parallel multiconductor lines are the basic interconnect units in practice. The knowledge of the self and coupling capacitances can also help the designer to optimize the layout of the circuit. In some cases, knowledge of the coupling parameters is absolutely critical. For example, coupling coefficients are essential in predicting the amount of cross-talk noise in high-density coupled interconnect lines. Knowledge of capacitive and inductive coupling between neighboring lines may also be used in the design of many high-frequency circuits that rely positively on coupling.

In this paper, we will extend the technique presented in [1], [2], [3] for the analysis of multiconductor multilayer interconnect structures. In the previous work, the inductance and capacitance matrices were found by using the boundary integral equation method. This involves solving the appropriate quasi-static integral equation using the Green's function approach. The evaluation of the Green's function was limited to the top layer, and this limited the conductor locations to the top layer. We will extend this work by deriving a more general quasi-static Green's function of the structure. By deriving the Green's function for multilayer dielectric structures and by allowing evaluation in any layer, we can place conductors anywhere in the layered structure, and therefore solve for the capacitance matrix for an arbitrary arrangement of interconnect conductors.

A number of authors have analyzed the transmission line characteristics of multiconductor multilayer dielectric planar or non-planar structures, by using a full-wave or quasi-TEM analysis. Full-wave methods, except when applied to zero-thickness printed lines, usually demand a lot of computer time. Our approach is based on a quasi-TEM solution because the computational accuracy and efficiency are major concerns from the application point of view, which makes it useful to develop fast interconnect solvers. Recently, the measured equation of invariance (MEI) and on-surface MEI methods have been used in the capacitance matrix evaluation of high-speed IC interconnects [4], [5]. The computed results compared with the moment method [6], [7] are within 2–4% for the investigated examples. For interconnection lines with zero- or non-zero thickness, as is the case in VLSI and MCMs, another typical method is based on the total charge Green's function approach presented in [7]. This method is very flexible but consumes a large amount of computing time and requires a large memory. In [10], the concept of equivalent transmission lines is introduced into the method of lines to extract the capacitance matrix of a multiconductor multilayered interconnect structure. In principle, one might use the finite-difference [8] or finite-element [9] techniques but they are both memory- and time-consuming. Solutions based on the integral equation formulations seem to be well suited for many practical cases – specifically for application in microelectronic interconnect structures – if the goal is to get high accuracy with low computational cost.

In this paper, we propose a new method that combines the advantages of different formulations, using a dielectric Green's function for multilayered interconnect structures, in order to build up a simple and accurate computer solver for infinitesimally thin conductors embedded in a multilayered planar dielectric medium. The method is based on solving the space-domain integral equation for the charge distribution on the interconnect lines. Since the dimensions of the interconnect lines are very small (μm) we assume that charge is distributed uniformly on the conductors. In addition, the computation of the elements of the Galerkin matrix is performed in a very efficient way by using closed-form integration. It should be pointed out that our procedure depends on the dielectric Green's function, whereas the total charge and volumetric methods do not. As the final result, we have build a simple computer code which is accurate and efficient for fast computation even on a PC unit.