Modeling and simulation are a crucial aid for reducing development cycle time and costs in modern semiconductor technology. New modeling concepts increasingly require long-term research performed in an interdisciplinary manner, and new numerical methods and algorithms are needed to implement these concepts.
The three-dimensional interconnect structure in the integrated circuits represents a difficult electromagnetic system. It includes many metalization layers with links for typically more than one million transistors, which are characterized by resistance, capacitance, and inductance parameters governing the electric signal behavior and supply. To extract these parameters an electromagnetic analysis is performed which is actually a problem of solving a set of Maxwell equations in the domain of interest with given boundary conditions. Unfortunately such boundary-value problems can be solved analytically only for a few special cases. In general, for arbitrary shaped domains numerical approximation methods like the finite element method are used.
The presented thesis treats the numerical calculation of three-dimensional electromagnetic fields using the finite element method based on vector and scalar shape functions followed by consequent parameter extraction. It begins with an explanation of the boundary value problems, weighted residual and Galerkin's method. The Galerkin's method belongs to the classical methods forming the basis of modern finite element analysis. Subsequently the vector finite elements are introduced, which are particular suitable for the description of the electric field and the magnetic field vector functions. The formulated edge based triangular elements for two-dimensional applications and the edge based tetrahedral elements for the three-dimensional ones are the foundation for all calculations in this thesis derived and explained in detail. A special focus is the careful definition and analysis of the numerical schemes describing the dominant magnetic field case. The following complex diffusion models are handled by a partial differential equation system which is numerically calculated using a combination of vector and scalar shape functions.
The developed models are implemented in the three-dimensional finite element simulation software SAP (Smart Analysis Programs). The simulation results demonstrate the physical plausibility of the applied models and numerical methods as well as the necessity of three-dimensional simulations.