Abstract

Inductive effects in microelectronics' interconnect structures are of raising importance because of the growing length, lowered resistances due to utilization of copper, and increasing clock frequencies. Especially pronounced are these effects in global busses and power distribution lines. Inductances influence delay, signal form due to switching noise, the tendency for ringing, and increase crosstalk. One main problem for the extraction of inductances is the fact that inductances are a function of closed loops, but a priori the current path is not completely known. Thus the concept of partial inductances is applied.

This thesis deals with different methods for the numerical calculation of inductances in interconnect structures. Two of these methods are based on the numerical integration of the Neumann formula, the third on an approach via the magnetic vector potential. All of them use the finite element method for the precalculation of the current density in the interconnects. The geometrical modeling is done with an unstructured tetrahedral mesh to gain high flexibility and to ensure a latter integration of the process flow. Hence, some simplifications compared to the real geometry, as other published approaches do implicitly, are obsolete. For example, other approaches are not able to handle more general structures, only geometries can be treated, which are build by prisms elements.

The first implemented method utilizes a scheme of different integration formulae based on triangles and tetrahedrons. The second evaluates the Neumann formula with the Monte Carlo method. This is performed by efficient localization of the elements for the random point coordinates to compute this integral. Classical implementation of the Monte Carlo method, where the whole geometry has to be hunted for the associated element loses efficiency. The third method is based on a rigorous approach using the magnetic vector potential to take a steady state electromagnetic analysis that can be extended for time dependent problems.


C. Harlander: Numerische Berechnung von Induktivitšten in dreidimensionalen Verdrahtungsstrukturen