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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

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

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

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

$\displaystyle y = \int_{-\infty}^{z} pdf(x)\ dx,$ (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.
\includegraphics[width=0.85\linewidth]{chapter_applications/Figures/gb_struct.eps}

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Next: 5.6.2 Simulation Approach Up: 5.6 Effect of Microstructure Previous: 5.6 Effect of Microstructure
R. L. de Orio: Electromigration Modeling and Simulation