6.3.2 Simulation Results

The simulation results are evaluated by visualization of the electric field using VTK. In the presented example the simulation area consists of a SiO2 rectangular parallelepiped with the conductors inside as shown in Fig. <6.23>. The faces parallel to the $yz$ plane are defined as periodic boundary. At the remaining outer faces homogeneous Neumann boundary conditions for the electrostatic potential are applied. The simulated potential and the corresponding iso faces with periodic boundary conditions are shown in Fig. <6.25> and Fig. <6.26>. The electric field is equivalent to the electric field in an inner single cell of the original interconnect bus structure. That is as if the simulation domain would be copied repeatedly in $x$ and $-x$ direction by the length of its $x$ dimension. The electric field looks like as if one boundary parallel to $yz$ plane would be directly connected to the opposite one. The stamp of the electrodes which are lying on the one of the periodic faces can be seen on the other periodic face. In order to visualize the effect of the periodic boundary conditions, the same structure has been simulated with homogeneous Neumann boundary conditions (natural boundary conditions) for comparison. The obtained electric potential distribution is displayed in Fig. <6.27> and Fig. <6.28>. In this case the effect of the opposite electrodes is no longer seen on the side walls and the iso surfaces are now perpendicular to the boundary. The field in the simulation area is as if the simulation area would be mirrored with respect to these boundaries.

Originally the $z$ dimension of the simulation region is longer than this one shown in the figures. For visualization the simulation area is cut off perpendicularly to the $z$ axis and closely to the electrodes to minimize the mirronig effect of the homogeneous Neumann boundary conditions at the cutting planes, similarly to Subsection 4.1.5.

As expected the calculated capacitance between the electrodes in the small area from Fig. <6.23> is different for the different boundary conditions applied. The capacitance with the periodic boundary is 1.33 times the capacitance with homogeneous Neumann boundary. The capacitance of the whole area from Fig. <6.22> is the capacitance of the small simulation domain multiplied by the number of all small simulation domains needed to construct the complete structure.

Figure 6.22:
An interconnect bus.

Figure 6.23:
The simulation area.

Figure 6.24: The electrodes in the simulation area.

Figure 6.25: The electric potential distribution with x periodicity.

Figure 6.26: The iso faces of the electric potential distribution with x periodicity.

Figure 6.27: The electric potential distribution without periodicity.

Figure 6.28: The iso faces of the electric potential distribution without periodicity.