Efficient Direct Boundary Element Method for Resistance Extraction of Substrate With Arbitrary Doping Profile

It is important to model the substrate coupling for mixed-signal or RF circuit designs. In this paper, a direct boundary element method (DBEM) and related efficient techniques are presented to calculate the coupling resistances for three-dimensional substrate structure. First, a nonuniform meshing scheme is presented to reduce boundary elements while preserving accuracy. Then, the unknowns on top medium surface are removed from the discretized linear system of DBEM with a matrix reduction technique. The third technique is applying the quasi-multiple medium idea (W. Yu, Z. Wang, and J. Gu, "Fast capacitance extraction of actual 3-D VLSI interconnects using quasi-multiple medium accelerated BEM," IEEE Trans. Microwave Theory Tech., vol. 51, no. 1, pp. 109-199, Jan. 2003), which greatly reduces the expense of matrix reduction and makes the final coefficient matrix much sparser. With these proposed techniques, the linear equation system is largely condensed and sparsified and then solved with a preconditioned generalized minimum residual solver for multiple right-hand sides to get the whole resistance matrix. Numerical experiments on typical substrates with various doping profiles show the high accuracy of the DBEM-based method. The authors also compared the DBEM method with the Green\´s function methods accelerated by discrete cosine transform or eigendecomposition techniques. The results show that the DBEM-based method is several times or tens of times faster than the other two. At the same time, the DBEM method has no difficulty in handling substrates with more complex than stratified doping profiles, which is a large advantage over the existing methods