Contents 1. Introduction
The digital signal is a kind of pulse with a finite rise time and a finite fall time. The frequency components of the pulse can be decomposed of spectrums have three different propagation modes dependent of a silicon substrate resistivity and an operation frequency. The silicon substrate resistivity oftoday's IC circuits is not so low as to be the skin effect mode. Further, since it is not so high as to be the dielectric quasi-TEM mode, digital signal propagates with the slow wave mode (which dominates on the most substrates for contemporary used signal frequencies) rather than with the dielectric quasi-TEM mode [1], [3]. Therefore, as modeled by Hasegawa [4], in the slow wave mode, the transmission line effects of the signal line can also be modeled with an RLC circuit. In the slow wave mode, while the electric field can penetrate the oxide layer, it does not penetrate the silicon substrate. In contrast, the magnetic field can penetrate both the oxide layer and the silicon substrate. Therefore, the signal line capacitance is nearly equal to the oxide capacitance. In contrast, the inductance cannot be simply modeled, as is capacitance. Since the magnetic fields penetrate the silicon substrate, the return currents come back through the silicon substrate. And the silicon substrate effect must be taken into account to calculate the self- and mutual inductances (or impedances) [2], [3], [23], [24]. In [5], [24] have been reported that the most of the magnetic energy is confined within a vertical distance depending on the geometrical structure of the system ground path. For the loop composed of the signal conductor and the silicon substrate, an effective ground plane for the inter-linked magnetic flux determination can be attributed. Since the substrate effect as an effective ground plane is a function of a substrate doping concentration and an operating frequency, it is inherently very difficult to analytically figure it out [6]. Nevertheless, if the partial inductance concept is introduced, it can be approximately calculated. The frequency-variant transmission line parameters were extracted [9], [10] and some special structures were numerically analyzed [11]–[13]. In [6] a new analytic model of MIS coplanar line where two groundlines are connected to the silicon substrate have been suggested. In [20], [21] CAD-oriented equivalent-circuit models for single and coupled interconnects on lossy silicon substrate, based on an efficient quasi-static. spectral domain approach, is presented. The propagation mode of the frequency components of a pulsed signal on an IC interconnect may changes with the operation frequency and substrate resistivity, thereby making the system behavior sophisticated [14]–[18], [22]. As another attribute of the system, the parametric variations due to the silicon substrate and current return path impedance of the IC interconnect structure make the system behavior much more difficult. However, in spite of such difficulties, highly accurate closed-form expressions for frequency-dependent impedance of IC interconnects considering lossy silicon substrate effects [23] becomes essential for the high-performance VLSI circuit design.
