In modern microelectronics the interconnects represent a complicated electromagnetic system which is essentially characterized by parameters like capacitance, resistance, and inductance at different frequencies. These parameters are extracted from the electric and magnetic fields after solving the Maxwell equations. This is usually based on numerical methods like finite element analysis which is probably one of the most superior simulation techniques in electromagnetics. Inductive effects have been neglected in integrated circuit design for a long time. That is on one side, because they become noticeable only with very high frequencies, and on the other side probably, because its computation is much more complex than those of the capacitances or direct current (DC) resistances. Usually capacitance and DC resistance are obtained by finite element analysis with scalar interpolation functions. For three-dimensional inductance and high frequency resistance calculation another kind of interpolation functions, the so called edge or vector interpolation functions have to be introduced.
The procedures presented in this work bear mostly on the computation of inductance and resistance of arbitrarily shaped three-dimensional interconnect structures in microelectronics at frequencies at which skin effect can be observed. The extreme small dimensions of modern microelectronics topology give the opportunity to use the quasi-magnetostatic case, assuming the so-called dominant magnetic field model. The resulting partial differential equation system arising from the Maxwell equations is solved in the frequency domain and provides the time-harmonic magnetic field distribution and the current density distribution in the domain of interest at the given frequency. Thereby induction, skin effect, and proximity effect are taken into account. As a consequence the inductance and the resistance of the interconnect structure in the simulation domain at a specific frequency is extracted from the numerically calculated fields' distributions. Working in the frequency domain gives no disadvantage as it appears. Each time dependent function can be expressed in terms of time-harmonic parts using Fourier analysis.
In Chapter 3 a brief introduction to the finite element method is given. The basic principle is illustrated by the weighted residual and by Galerkin's method. Since the mathematical models in this work contain scalar and vector fields, both, the scalar and the vector finite element interpolation is addressed. A detailed explanation of the finite element assembling is discussed in Chapter 4, where the concept for scalar finite elements is presented. The governing mathematical expressions are derived and given for the two- and three-dimensional case. Chapter 5 is devoted to the formulation of vector finite element analysis. Some basic applications of electromagnetics for the vector interpolation are presented. Again the corresponding general algorithms and formulas for two- and three-dimensional assembling are explained. The simulation domain discretization is discussed to interfere the importance of the properties of the generated mesh for further analysis steps.
Typical three-dimensional applications occurring in microelectronics for vector and scalar finite element techniques are presented in Chapter 6. The theory from the previous chapters is used for magnetic field and current density or electrostatic potential distribution calculation in different regions for given electromagnetic problems. Consequently the parameters of interest are extracted. The fields are visualized to demonstrate the properness of the numerical analysis. This chapter starts with an inductance and resistance extraction example for a specific structure, which also provides analytical formulas for the parameters calculation. Thus the method is evaluated (at least for this structure) by comparison between the numerically simulated and analytically calculated results. This example addresses also the parameter extraction from the numerically calculated fields by the power in the simulation domain. After that an on-chip spiral inductor as typically used in microelectronics is analyzed. Another interesting application is the simulation of periodic structures. In such cases by applying so-called periodic boundary conditions it is possible to simulate only small substructures which represent the geometrical period of the entire structure. Thus the simulation resources and the computation time can be significantly optimized.
Related topics are covered in the appendices. In Appendix A the integration domain transformation is explained, which is commonly used for the derivation of the finite element assembling expressions. Appendix B and Appendix C describe a possible discretization of the Neumann boundary term naturally arising after the so called weak formulation in the finite element analysis. It corresponds with the Neumann boundary condition given on the Neumann boundary. Normally the Neumann boundary is a part of or the entire outer closure of the simulation domain and the corresponding Neumann condition is assumed to be zero -- homogeneous Neumann boundary condition. This is almost legitimate also for open or infinite regions considering the fact that the field quantities decrease to zero with increasing distance to the source. However as discussed in Subsection 4.1.5 applying the homogeneous Neumann boundary to open regions distorts the results. In such cases hybrid techniques like a combination of finite elements and boundary integral methods can be addressed, where the discretization of the Neumann boundary term is required.