1. Field of Invention
The present invention relates to integrated circuits and more particularly to modeling integrated circuits with multiple layers.
2. Description of Related Art
The full-wave electromagnetic solver has become an essential part in the design flow for radio frequency integrated circuits (RFICs) serving as both a design and a verification tool. A three-dimensional broadband full-wave solver based on the integral equation (IE) method with the pre-corrected Fast Fourier Transform (FFT) acceleration has been developed to fulfill these functions [1].
The multilayer Green's function evaluation is essential to the performance of an IE solver. Various approaches have been developed to accelerate the efficient computation of multilayer Green's functions, such as the discrete complex image method (DCIM) [2], the fast Hankel transform (FHT) [3], and the window function based method [4]. The DCIM has been widely employed and improved since its debut. Two-level DCIM [5] was developed to split the original path into two. By sampling the paths at different rates, the method improved the efficiency and accuracy comparing the original one-level algorithm. Generalized surface wave pole extraction was developed to extend the DCIM to general multilayer media [6].
For on-chip applications, the semiconductor process must first be examined. The vertical cross-section of the metallization layers of a typical four-metal-layer semiconductor process is shown in FIG. 2. In order to fill the IE matrix entries efficiently, a 3D Green's function database is deployed [1]. The database is populated with the value of the Green's function between several source points at z′ and observation points at z for various horizontal distances ρ. Even with the aid of such a database, the rapid evaluation of the Green's function for each matrix entry is critical. Note that the distance between the adjacent layers is only a few microns or less, which causes the spectral kernel of Green's functions to decay slowly. Consequently, number of sampling points has to be increased to accurately capture the whole spectrum, which will slow down the DCIM dramatically because the generalized pencil of function (GPOF) method [9] in the DCIM has computation time proportional to O(Ns3), Ns is the number of sampling points. For that reason, accelerating convergence of the spectral kernel is necessary before applying the GPOF.
Thus, there is a need for improved methods for evaluating Green's functions for modeling multilayer integrated circuits.