Statistical study of on-chip spiral inductors using integral equation based full- wave electromagnetic analysis

This paper presents a statistical study of the various parameters of interest of an on- chip spiral inductor in the 10 GHz range using the PMCHWT formulation [1]. Variability in the conductivity of the CMOS grade Silicon substrate and trace metal widths are assumed. We show that EM analysis of these inductors can accurately help in understanding the impact of process variations especially at high frequencies. The use of the PMCHWT formulation also permits analysis of the electrical properties of the substrate that are particularly important at higher clock and carrier frequencies. To enable fast extraction of statistics, a simple 2D Lagrange interpolation technique is used to approximate the surfaces of the quantities of interest and then a Montecarlo operation is carried out on the interpolated 2D polynomial. We extract the probability density functions (PDFs) of the effective resistance (R), the inductance (L) and the quality factor (Q) of a spiral on a lossy Silicon substrate.