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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, , and corresponding standard deviation, , it generates a lognormal distribution of grain sizes according to

(5.12) 
The angles between the grain boundaries' planes and the line surface at the top follow a normal distribution,

(5.13) 
where
is the median value of the angles.
Taking a radom number,
, uniformly distributed, the grain sizes and grain boundary angles are determined by calculating , so that the inverse relation,

(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 bamboolike 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.

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R. L. de Orio: Electromigration Modeling and Simulation