5.2 Adapted Grid Generation

A high grid density in directions where simulation values change rapidly must be guaranteed to achieve an accurate numerical solution. As in the above example, the resolution of the $1/(y+d)^{2}$ dependency of the current density must be guaranteed and therefore the grid density under the gate oxide must be a fraction of the thickness of the oxide, whereas along the channel the grid density can be much cruder. An aspect ratio of about $1:100$ is not rare.

In general, increasing the number of points increases the accuracy of the solution (neglecting numerical errors). However, especially within three-dimensional simulations an isotropic increase of grid points will cause intolerable memory consumption and calculation times. Therefore, a compromise of these effects must be a main goal.

In areas where simulated values are strongly affected (also affected with high nonlinearities), the point density must be high and so the tetrahedrons have to be split into smaller ones. A two-dimensional grid example, which shows the refinement procedure, can bee seen in Figure 5.5. The right grid is refined with a global grid density criterion turned on. An example for such a global criterion is a maximal area constrain for all grid elements. As seen in Figure 5.5(b), the resulting triangles become nearly equilateral. Applying the same grid refinement method for three dimensions, the expected tetrahedrons will become nearly equilateral, too.

Figure 5.5: Refinement of a two-dimensional triangular grid.

Unrefined grid

        

Refined grid with drawn grid points of the unrefined version.

As the required grid density is directionally dependent (anisotropic) -- like in the transistor example -- the density in one direction is sufficient, but in another, it can be too high. In this direction the material parameters and unknowns do not change so rapidly and the density of discretization along this direction does not need to be so fine. With global refinement we have an undesirable high amount of grid points where the simulation must be performed and the memory and time consumption will be unnecessarily high. Therefore, investigations of producing grids of desired structure are necessary.

Due to the discretization of the differential equations and especially of the boundary conditions, orthogonal and boundary conform meshes are desired. Matrix entries and the discretization errors caused by Neumann conditions are decimated, which are achieved for instance between the gate oxide and the semiconductor segment. In general, orthogonal grid components in and across main directions of the current densities also decimate the errors caused by the Box Integration method inside the structures.

Therefore, a new method for three-dimensional grid generation was developed in the scope of this thesis. The methodologies of this method are described in the following sections.