6.3 Capacitance, Resistance and Thermal Simulators

In Section 6.1 we concluded that only numerical approaches can be used to fully and accurately characterize interconnect structures. There are several different methods reported in literature to perform these quite complex tasks. The most significant are:

• Finite differences methods [72],[73];
• Boundary elements methods [74];
• Finite-element methods [68,75,76,77];
• Multipole algorithms [78,79,80] and
• Stochastic methods [81].

All these methods can with a high accuracy solve the Laplace equation, which is a fundamental step to calculate the capacitances between arbitrary three-dimensional structures. The first three methods, when using optimized algorithms, achieve about the same precision for a given computational effort. Their drawback is that the required computer resources are very high (both in memory consumption and CPU time) which forces them to be used only in local problems and never at full integrated circuit level.

The last two methods try, by different mechanisms, to overcome the above mentioned drawback. Multipole acceleration techniques substantially speed up simulations based on the boundary element method. This is achieved by simplifying the system of equations that need to be solved in the boundary element method with an order of $(n \times n)$, where n is the number of panels used for discretization. The basic idea is either to ignore or to approximate small contributions to the potential coefficient matrix from pairs of panels that are far away from each other. Using this technique the system to be solved becomes of order $(n \times m)$ where m is the number of conductors (much smaller than n).

The last method uses non-deterministic techniques to statistically estimate the electrical field only in regions where it is really needed. They also take advantage of the fact that statistical errors in the electrical field tend to cancel during Gaussian-surface integration. For estimating one capacitance pair, a number of N random trajectories starting at one given node and terminating in the other one is checked to compute partial capacitances that are added together. The accuracy of the final value depends on the number of trajectories (or random walks) computed, but a relatively low number ($5 \cdot 10^3$ for typical conductors) can give an accuracy as low as 10%. However, for improving the accuracy of this method to less than 5-10% the number of trajectories start to explode [42] and no significant gain is obtained.

A severe problem of both techniques is, as the above referred papers reported, they assume a cube partitioning of the space. In fact, the algorithms were optimized for this kind of discretization, that is much less efficient than a tetrahedral description for arbitrary interconnect structures. The examples presented there always show parallelepiped structures, which we demonstrated to be a bad estimation of real ones. Thus, as we seek the maximum accuracy these two promising methods for medium precision capacitance extraction of large layout areas reveal not to be the best choice for maximum accuracy. As finite-element methods are the most general and easily extended to the resistance and thermal problems, it was decided to use them in the extracting tools.

Once the structures are gridded, we use SCAP and , finite element based simulators for calculating capacitances and resistances (or performing thermal analysis). Transient electrical simulation is also possible, which is of interest in cases where a very accurate estimate of delay times and crosstalk between arbitrary shape lines is necessary. In these cases lumped models based on the extracted resistances and capacitances can simply not achieve a sufficiently high precision.