Modeling and Prediction of Coupling Effects in Nanoscale VLSI Circuits Yiorgos I. Bontzios 1 , Michael G. Dimopoulos 2 , Alkis A. Hatzopoulos 1 1 Dept. of Electrical & Computer Eng., Aristotle Univ. of Thessaloniki, Thessaloniki 54124, Greece 2 TIMA Laboratory (CNRS-INP-UJF), 46 avenue Félix Viallet, 38031 GRENOBLE Cedex France E-mail: gmpontzi@auth.gr, Michael.Dimopoulos@imag.fr, aldimitr@auth.gr, alkis@eng.auth.gr Abstract—In this work, three approaches are presented for computing the resistive and capacitive coupling in VLSI circuits. The proposed methods are expressed in closed-form and are fast, accurate, scalable, and technology independent. They are validated against simulation and measurement data obtained by a fabricated chip. In all cases, the agreement is close. Keywords-coupling; noise; VLSI; Closed-form, CMOS I. INTRODUCTION In the deep submicrometer era, the feature size scales down, while at the same time, the chip’s complexity, speed, and clock frequencies increase. Serious impediment to this trend is the noise coupling among chip components, the correct prediction of which is imperative; otherwise designers are obliged to exaggerate any spacing. Thus, the demand for techniques accurately predicting the behavior of the chip, considering the coupling effects, is constantly rising. The accurate prediction of coupling is only possible by solving the electromagnetic (EM) problem [1]. Dedicated tools have been emerged [2], [3] utilizing the most common methods (FEM/BEM) for numerically solving the EM problem. However, they are not widely accepted among designers. Trying to overcome this drawback, several approaches have been reported which propose closed-form equations formulae [4], [5]. However, since most of them rely on fitting techniques their range of validity is limited. This work proposes alternative approaches for predicting the coupling effects among chip components. The resulted equations are directly obtained from the physical understanding of the problem. No fitting techniques are imposed in any stage of their derivation. Therefore, no measurement or simulation data are prerequisite. Hence, all methods are fast, accurate, scalable, and technology independent. This work is organized as follows. First, the proposed methods are described in detail, in Section II. Next, in Section III, these methods are applied on typical test cases encountered in practice and validated against simulation and measurement data. Finally, this paper is concluded in Section IV. II. PROPOSED METHODS Three different methods are proposed in this work for the prediction of the coupling mechanism. The first (RCCG) is based on the geometry of the coupling mechanism, the second on a memetic algorithm and the third (ESR) on the analytical solution of the Laplace equation that describes the problem. A. RCCG method The RCCG (Resistive and Capacitive Coupling exploiting Geometry constraints) method [6] is based on the geometry of the electric field lines produced by the coupled structures and on a primary principle of physics—the law of least energy. The super-position of three simple and symmetric structures is used to model any structure. Since RCCG is based on geometric calculations, it is generally scalable and technology independent. The shape of the current flow lines is approximated in each case by a combination of simple geometric shapes. Any 3D structure in general is treated as the superposition of simpler cases and the minimum resistance (or maximum capacitance) of each is calculated. Three geometric shapes are used for approximating any structure, which are the simplest possible, resulting thus to simpler equations and faster simulation times. The first is the circular disk above infinite (ground) plane, the second is the square above infinite (ground) plane and the third is the rectangle above infinite (ground) plane [6]. The case of the circular disk is illustrated below. (a) (b) Figure 1. Circular disk over an infinite plane (a) Electric field lines. (b) Assumed shape The assumed shape is constituted by a truncated cone of height p h and radius of the upper and lower bases c r and p r , respectively, followed by a cylinder of radius p r , as shown in Fig. 1(b). Consequently, the set of parameters required to be determined for this case consists of p r and p h . The analysis followed for the capacitive coupling calculation is the same as for the resistive one. The difference lies in the fact that the shape of the current flow lines which