High frequencies in an integrated circuit (IC) affect both, the resistance and the inductance of the on-chip interconnects. These often as parasitics treated parameters cause longer signal rise, fall, and delay times and limit the maximum allowed frequency of modern ICs. However, as the operating frequencies increase, small inductors of high speed circuits can be also actively used. They can be even constructed on the chip. Thus the inductance of an on-chip interconnect line can be a disadvantage or very useful depending on the application. Of course the collateral resistance must also be considered. In each case it is necessary to investigate the structure of interest to obtain its inductance and resistance in order to estimate the impact on the entire electric circuit [104]. In the case of applications in radio frequency (RF) ICs such as voltage controlled oscillators or low noise amplifiers the inductance and the resistance of the on-chip inductors must be extensively investigated for the RF circuit design, performance optimization, and inductor quality factor. The frequency dependent inductance and resistance of wide on-chip interconnects must be captured to obtain the impact on power supply stability and signal delay.
Currently there are two major techniques for modeling of on-chip inductors: analytical compact modeling and numerical field calculation based modeling. In the case of a spiral inductor, where the models can be restricted to specific geometry classes, closed-form analytical models are very well suited for fast designs typical for the very early stage of the developing process [105,106]. However, analytical modeling of arbitrarily shaped three-dimensional structures is very complicated, if possible at all. Thus, analytical parameter extraction methods have only limited applicability. For final analysis prior to fabrication and for irregular inductor geometries numerical simulation methods normally based on solving the Maxwell equations provide the most accurate characterization. Moreover, the investigated interconnect structure can often be embedded in a small simulation region for which the optimized model of the dominant magnetic field can be used even at very high frequencies. The distributed vector and scalar fields must be extracted in structures which may consist of different inhomogeneous complex shaped three-dimensional regions, like splittings, widenings, and vertical connections. As a consequence, the vector and scalar finite element method for the qusi-magnetostatic case in the frequency domain on unstructured tetrahedral meshes [107,108] as described in Section 5.2 needs to be addressed.
In this application an optimized model for inductance and resistance analysis of an on-chip inductor at different frequencies is proposed. The model describes the proximity effect and the skin effect typically arising at higher frequencies as well. The three-dimensional finite element simulation software SAP (Smart Analysis Programs) [109] was extended to implement the developed model. Simulation results demonstrate the physical plausibility of the applied model and numerical methods, as well as the necessity of three-dimensional simulations.