IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 54, NO. 9, SEPTEMBER 2007 2531 Compact Spreading Resistance Model for Rectangular Contacts on Uniform and Epitaxial Substrates Simon Kristiansson, Student Member, IEEE, Fredrik Ingvarson, Member,IEEE, and Kjell O. Jeppson, Senior Member, IEEE Abstract—We present a compact analytical spreading resistance model for substrate noise coupling analysis. The model can handle rectangular contacts on uniform substrates of finite thickness with a grounded backplane. In contrast to previously published compact models, the model does not require extraction of fitting parameters. The model is also scalable with the resistivity and thickness of the substrate, and with the contact size. The model is verified with extensive finite-element calculations, and the accu- racy is shown to be good. We also show that the model can predict the spreading resistance on epitaxial substrates. Index Terms—Elliptic integral, rectangular contact, spreading resistance modeling, substrate noise coupling. I. INTRODUCTION C ONCERNS of having substrate noise affecting sensitive circuitry in system-on-chips have increased in recent years (see, for example, [1]). Therefore, in contrast to tradi- tional postlayout noise coupling analysis, there is an increasing need for techniques to predict substrate noise coupling early in the design flow. For prelayout noise analysis, networks of lumped resistances are commonly used for estimating the coupling between different parts of the chip, and simple models of these substrate resistances are needed (see, for example, [2]). It has been shown that the resistance between two surface contacts saturates as the contact separation increases [3]. In a circuit model of the substrate, the two contacts can then be modeled as connected via a global substrate node. This common node can be a backplane on the chip or the heavily doped bulk layer in epitaxial substrates. In epitaxial substrates, the resistance between two coplanar contacts saturates more rapidly with the distance between them [4]. Thus, for substrate noise coupling analysis, a very important circuit element is the resistance between a contact on the silicon surface and the common node (see Fig. 1). Calculating this spreading resistance is a mixed boundary value problem that is difficult to solve exactly. For example, the exact solution for the spreading resistance of a circular contact is expressed in semi-infinite integral form and requires solving two dual integral equations (see the monograph of Sneddon [5]). The Manuscript received December 6, 2006; revised May 24, 2007. The review of this paper was arranged by Editor C. Jungemann. The authors are with the Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96 Göteborg, Sweden (e-mail: simon.kristiansson@chalmers.se; fredrik.ingvarson@chalmers.se; kjell.jeppson@chalmers.se). Digital Object Identifier 10.1109/TED.2007.902689 Fig. 1. Geometry for modeling the spreading resistance between a rectangular contact and an infinite backplane. The rectangular contact is approximated by an elliptic contact. Fig. 2. Cross-sectional views of the two modeled substrate types. (a) Uniform substrate. (b) Epitaxial substrate. common way of dealing with mixed boundary value problems in spreading resistance calculations is to assume a certain cur- rent distribution in the contact and thus reduce the problem to a more easily solvable homogeneous boundary value problem. Typically, these approximate solutions are expressed in semi- infinite integral form, as for the circular contacts in [6]. The spreading resistance of rectangular contacts was modeled in this way by Deferm et al. [7]. Unfortunately, there are convergence problems in the integrals used, which makes this solution less useful. Instead of models expressed as infinite series or integrals, compact models are preferable for circuit analysis. One of the earliest (for substrate noise coupling analysis) compact models for the spreading resistance of rectangular contacts on epitaxial substrates, see Fig. 2(b), was presented by Su et al. [4]. Their model can be reformulated as R epi = ρ epi t epi f 1 WL + f 2 2(L + W )+ f 3 (1) 0018-9383/$25.00 © 2007 IEEE