5.6.1 Microstructure Generation

In order to include the grain distribution into the numerical simulations, a microstructure generation tool has been developed. Given a specific interconnect structure and providing the tool with a median grain size, $x_0$, and corresponding standard deviation, $\sigma$, it generates a lognormal distribution of grain sizes according to

$$pdf(x) = \frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{(\ln x-\ln x_0)^2}{2\sigma^2}\right].$$ (5.12)

The angles between the grain boundaries' planes and the line surface at the top follow a normal distribution,

$$pdf(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{(x-x_0)^2}{2\sigma^2}\right],$$ (5.13)

where $x_0 = 90^\circ$ is the median value of the angles. Taking a radom number, $y \in [0,1]$, uniformly distributed, the grain sizes and grain boundary angles are determined by calculating $z$, so that the inverse relation,

$$y = \int_{-\infty}^{z} pdf(x)\ dx,$$ (5.14)

holds. Once the grain sizes and angles are determined, the interconnect line is cut along its length by the planes which form the grain boundaries. A typical microstructure generated by such a procedure is shown in Figure 5.35. In this way, the microstructure generation tool yields a simple bamboo-like line.

Figure 5.35: Typical microstructure generated from the procedure described above. The grain sizes follow a lognormal distribution, and the angles of the grain boundaries in relation to the top line surface follow a normal distribution.