Approximate analytical modeling of silicon internal inductance of VLSI interconnects

In previous work, a closed form expressions for the series line impedance per unit length of a single and multiple coupled interconnects on lossy silicon substrate were presented. In this paper, the induced current density distribution inside silicon substrate and quasi-static Green's function approach were used to derive the frequency-dependent closed-form expression for the internal inductance per unit length accociated with the silicon substrate. With this expression, we can quantitatively determine what percentage of the total inductance per unit length is associated with the internal silicon substrate inductance for a given frequency. The effect of the substrate resistivity on the inductance of on-chip interconnects was examined. It was shown that depending on the frequency, large changes in the inductance can occur due to this effect.