2 Estimating the Void Growth Time and Resistance Change

An initial void with some small radius $r_0$ is placed at some characteristic position inside the three-dimensional interconnect structure (Figure 4.15). Since most of the fatal voids are nucleated in the vicinity or in the area of interconnect vias we consider in particular these cases. The configurable initial void volume is $V_0$ which is smaller than $4 \pi r_0^3/3$ because the void area is confined by sphere and boundary of the interconnect (Figure 4.15). Starting from the initial void radius $r_0$, the void radius is gradually incremented $r_0, r_1=r_0+\Delta r_1, r_2=r_1+\Delta r_2,...$, with $\Delta r_1\geq \Delta r_2\geq ...\geq\Delta r_n$. For each void radius the electrical field in the interconnect structure is calculated by means of the finite element method using a diffuse interface approach. To obtain the distribution of the electrical potential inside the interconnects the Poisson equation (4.52) has to be solved.

In order to obtain sufficient accuracy the scalar field $\phi(x,y,z,t)$ must be resolved on a locally refined mesh (Figure 4.16). For an electrical field calculated in such a way, the resistance of the interconnect via is also calculated [76,91]. With growing void size the resistance increases. The whole process is stopped when a void radius is reached for which $100\times (R_{actual}/R_{initial}-1)> 20 \%$.

Figure 4.15: Position of the growing void. Initial position and volume are chosen on the basis of experimental results.

The primary driving force of material transport at the void surface is electromigration proportional to the tangent component of the vector current density. Since the diffuse interface approach for the calculation of the current density ensures physical behavior of the electrical field in the vicinity of the isolating void, the normal component of the current density on the void surface is always zero and we can apply the formula

$$J_{m,i}=2\frac{\int_{V}\Vert\mathbf{J}\Vert[1-\phi^2_i(x,y,z)] dV}{\int_{V}[1-\phi^2_i(x,y,z)] dV},$$ (273)

for the average current density over a void with radius $r_i$. (4.63) expresses the averaging of the current density weighted with finite element volume inside the interconnect. Since $\phi_i(x,y,z)=1$ in metal and $\phi_i(x,y,z)=-1$ in the void area, the term $1-\phi^2_i(x,y,z)$ is non-zero only in void-metal interface area.

Figure 4.16: Typical current density distribution picture in the vicinity of the spherical void. The red area marks peak values of the current density. The mesh is locally refined in the void-metal interface.

The evolution of the void is caused by material transport on the void surface and in the vicinity of the void surface. The mass conservation law gives the mean propagation velocity $v_i$ of the evolving void-metal interface

$$v_i=\frac{D_V}{kT\rho(\phi)}eZ^{*} J_{m,i},$$ (274)

here $D_v$ is the vacancy diffusivity and $Z^{*}$ effective charge number of vacancies. (4.64) is valid for all void shapes.

Figure 4.17: Change of the average current density and via resistance depending on the void radius.

As we can see from Figure 4.17, the average current density on the void surface increases with the void size. Both, current density and resistance, exhibit a very similar dynamic behavior. The dynamical resistance increase is in accordance with the measurement results presented in [52]. Compared with the earlier result [92], which assumes cubical void shapes, our approach enables more realistic simulations. An open question is how to use the obtained average current density (Figure 4.17) for the estimation of the void growing time ($t_E$) up to the critical void size. In [92] a simple formula is applied

$$t_{E}=\frac{V_c-V_0}{v_m A_s}.$$ (275)

In this equation $V_c$ is the critical void size, $V_0$ is the initial void size, $A_s$ is the cross section of the interconnect in the vicinity of the growing void, and $v_m$ is the mean velocity of the evolving void-metal interface. However, this formula is only valid in the case of cubical void which is a very rough approximation of the real situation. According to newer experimental results [8] the real void shape is significantly better approximated by a spherical sector. In this case $t_E$ can be estimated as

$$t_{E}=\sum_{i}\frac{\Delta r_{i+1}}{v_{i}},$$ (276)

assuming that for sufficiently small $\Delta r_{i+1}$, the void radius grows from $r_i$ to $r_{i+1}$ with a constant velocity $v_i$.

As we can see from ( $\ref{est0}$), the velocity $v_i$ depends on the vacancy diffusivity $D_v$ which itself has significantly varying values depending on the diffusion path. The electromigration assisted self-diffusion of copper is a complex process which includes simultaneous diffusion through the crystal bulk, along grain boundaries, along the copper/barrier interfaces, and along the copper/cap-layer interface. Therefore, the diffusivity applied in ( $\ref{est0}$) must be a cumulative value (see equation 4.12). For a feasible estimation of $t_E$, reliable, experimentally determined values for all relevant diffusivities are needed and these are until now unfortunately not available [55,8].