5.3 Example

In the following, mesh adaptation techniques are demonstrated on a typical three-dimensional interconnect structure, shown in Figure 5.8 with trapezoidal tantalum (Ta) covered copper (Cu) lines, horizontal silicon carbide (SiC) etch stop layers embedded in some low- $\kappa$ material [102]. For the generation of the initial unstructured mesh, GMSH^5.1 was used, which is a finite element mesh generator (primarily Delaunay) with built-in pre- and post-processing facilities [103]. This mesh generator allows to control the initial spatial resolution of the simulation domain quite well, which is important for the coarsest mesh state.

The finest mesh of the initial structure was set at the narrowest region of the Cu line. This is in the domain of the interconnect via, because one can expect the highest current density in this part of the structure as depicted in Figure 5.8(b) and Figure 5.9(a). The mesh of the horizontal etch stop layers is kept mostly anisotropic in order to keep the total amount of mesh elements as small as possible. The coarsest mesh regions are set in the domain of silicon dioxide, since these parts are of lower importance for the numerical analysis.

The electrical field and temperature distribution for the test circuit are given in Figure 5.9(a) and Figure 5.9(b), respectively. The calculations have been carried out with the SMART ANALYSES PROGRAMS-Package [104] developed at the Institute for Microlectronics.

As mesh adaptation example a typical void movement is chosen, where the void was formed from a Cu-cap-layer interface near a Cu grain boundary, cf. the schematic overview given in Figure 5.1(a). Figure 5.10(a) shows an initial void formation with a fine mesh near the refined diffuse void-metal interface. During the transient simulation the void moves towards the interconnect via and the mesh is modified dynamically as depicted in Figure 5.10(b).

The refinement procedure generates a finer mesh on the moving metal-void interface to resolve the interface function properly. At least three mesh points are used, cf. Figure 5.5, according to the discussion given in Section 5.2.2. After movement of the void as depicted in Figure 5.10(b), the initial mesh density near the grain boundary is again recovered by the hierarchical mesh refinement-coarsement scheme (see Section 5.2.5), which keeps the number of overall mesh elements small.

In practice, the recursive refinement procedure shows a quite robust behavior being important for the number of overall mesh elements, which has a direct impact on computational costs, memory usage, and simulation time, for the actual void formation and movement simulation.

**Figure 5.8:** Three-dimensional domain for the interconnect electromigration simulation with trapezoid shaped tantalum (Ta) covered copper (Cu) lines, round conical via, and horizontal silicon carbide (SiC) etch stop layers embedded in some low- $\kappa$ material.
$\begin{figure} % latex2html id marker 5737 \setcounter{subfigure}{0} \center \s... ...eps,width=0.8\textwidth,height=0.59\textwidth}} \vspace{-0.6cm} \end{figure*}$

**Figure 5.9:** Three-dimensional interconnect electromigration simulation domain with trapezoidal shaped tantalum ( ) covered copper ( ) lines, round conical via, and horizontal silicon carbide ( ) etch stop layers embedded in some low- $\kappa$ material.
$\begin{figure}\setcounter{subfigure}{0} \center \subfigure[Electric field in th... ...re =pics/em_temp.eps2,width=0.8\textwidth,height=0.59\textwidth}}\end{figure}$

**Figure 5.10:** During void formation and movement a dynamic mesh adaptation scheme is used to guarantee a good spatial resolution of the void copper interface. Copper grain boundaries are taken into account and an appropriately fine mesh was computed.
$\begin{figure}\setcounter{subfigure}{0} \centering \subfigure[Void is formed ne... ... {\epsfig{figure =pics/em_void_pos2.eps2, width=0.93\textwidth}}\end{figure}$

5.3 Example

Footnotes