1 Two-Dimensional Void Evolution

In all following simulations a circle was chosen as initial void shape. The resolution of the parameter $\phi$ profile can be manipulated by setting parameter $n$ which is the mean number of triangles across the void-metal interface. On Fig. 4.8 initial grids for $n=1$ and $n=5$ are presented.

Figure 4.8: Initial grid refinements.

$n=1$

$n=5$

We consider a two-dimensional, stress free, electricaly conducting interconnect via.

Figure 4.9: Interconnect via with initial void.

$A$ constant voltage is applied between points $A$ and $B$ (Figure 4.9). At $B$ a refractory layer is assumed. Because of geometrical reasons there is current crowding in the adjacencies of the corners $C$ and $D$. The analytical solution of equation (4.52) has at these points actually a singularity [90,76].
High electrical field gradients in the area around the corner points increase overall error of the finite element scheme for the equation (4.52) which is overcome by applying an additional refinement of the basic mesh $T_h$ according to the local value of the electric field gradient (Figure 4.10).

Figure 4.10: Profile of the current density (in $A/m^2$) at the corners of the interconnect.

A fine triangulated belt area which is attached to the void-metal interface at the initial simulations step follows the interfacial area throughout the simulations whereby the interconnect area outside the interface is coarsened to the level of the basic grid $T_{h}$ (Figure 4.11).

Figure 4.11: Refined grid around the void in the proximity of the interconnect corner.

Figure 4.12: Void evolving through interconnect in the electric current direction

Figure 4.13: Time dependent resistance change during void evolution for the different initial void radius $r$.

In our simulations a void evolving through the straight part of the interconnect geometry exhibits similar shape changes as observed in earlier models [60,59]. There is also no significant fluctuation of the resistance during this period of interconnect evolution. The situation changes when the void evolves in the proximity of the interconnect corner (Figure 4.12). Due to current crowding in this area the influence of the electromigration force on the material transport on the void surface is more pronounced than the chemical potential gradient. This unbalance leads to higher asymmetry in the void shape then observed in the straight part of the interconnect (Figure 4.12). A void evolving in the proximity of the interconnect corner causes significant fluctuations in the interconnect resistance due to void asymmetry and position. The resistance change shows a charasteristic profile with two peaks and a valley (Figure 4.13). The extremes are more pronounced for the larger initial voids.
The capability of the applied adaptation scheme is also presented in the simulation of void collision with the interconnect refractory layer (Figure 4.14).
The time step $\Delta t$ for the numerical scheme (4.55)-(4.58) is fitted at the simulations begin taking into account inverse proportionality of the speed of the evolving void-metal interface to the initial void radius [60]:

$$\Delta t = \frac{\epsilon\pi r l}{2D_{s}\vert e\vert Z^{*}V_{0}}$$ (272)

Figure 4.14: Grid adaptation in the case of void collision with the refractory layer.

$l$ is the characteristical length of via geometry. An appropriate choice of the time step ensures that the evolving void-metal interface will stay inside the fine grid belt during the simulation. The dynamics of the evolving void-metal interface simulated with a the presented numerical scheme complies with the mass conservation law, the void area (where $\phi=-1$) remains approximately the same during the whole simulation. Notable area deviations during the simulation appear only, if a relatively large factor $\epsilon$ has been chosen. As scaling length we took $l=1 \mu m$ and for the initial void radius $r_0=0.035 l$, $r_1=0.045 l$, and $r_2=0.065 l$. Our simulations have shown that for all considered initial void radii, voids follow the electric current direction (Figure 4.13) and do not transform in slit or wedge like formations which have been found to be a main cause for a complete interconnect failure [54]. Already with $\epsilon = 0.003 l$ good approximations are achieved. The number of elements on the cross section of the void-metal interface was chosen between 6 and 10 with the interface width of $0.0015 l\pi$.