3.5 Re-Meshing and Boundary Processing

The requirements on grids of process- and device simulators are very different. On one hand the process simulation grid has to follow steep gradients in the doping profile and must resolve internal interfaces like $SiO_2$/$Si$ with an accuracy sufficient to model segregation and dopant transport across interfaces [124],[125]. Furthermore, it must be able to adapt to changes in the boundary occurring during process steps like oxidation or etching. These changes can lead even to topology changes (like etching a hole through one layer converting one segment into two separated segments) [87],[88],[126]. On the other hand the device simulation grid must resolve mainly the fields of physical quantities occurring during a device simulation, like carrier concentrations or electrostatic potential. Thus the grid obtained by the process simulation is normally not suitable to get sufficiently accurate results for device simulations.
Therefore, re-meshing of the resulting structure after process simulation is mandatory. Re-meshing is a very sensitive process, since it includes interpolation of the dopant distributions on a new grid [127],[128]. Care has to be taken, not to increase the errors in the mesh representation of the dopant field [129]. Especially the boundaries where currents occur during the device operation have to represented appropriately by the mesh. A proper grid for carrier transport simulation should follow the current flows during device operation (like a refined channel in a MOS structure for simulation of the transfer characteristics of a MOS device).
There are three strategies to generate a suitable grid. A straight forward and very stable method is the generation of the grid based on dopant gradient criteria. For certain applications these methods have big disadvantages, because they do not generate a high resolution in areas where necessary. For instance the channel, of a MOS transistor shows no steep dopant gradient and is thus not well resolved by a standard gradient criteria based grid. One workaround is to define a dedicated refinement region with a finer resolution in these areas. However, this approach is not suitable for automatic re-meshing of different device types.
A newer approach is the generation of boundary conforming meshes [130]. In this approach the boundaries of a certain material segment are the starting conditions for a mesh generation by offsetting mesh lines from the boundary by a certain distance, which increases as the mesh lines propagate into the material segment. This method yields excellent results for resolving critical channel areas as discussed above. However, it generates unwanted mesh points in areas which are not related to the active region of devices. Therefore to suppress this generation, a lot of user interaction has to be performed, do define the interesting segments in the structure.
A more general and automatic approach is a two step strategy to generate a suitable device simulation grid. First a coarse grid is used with the device simulator to obtain a coarse representation of the physical fields in the device. Based on this solution a physical field is chosen as the refinement criteria for a second iteration of the re-meshing process. For a CMOS device an appropriate field would give the carrier density in the device. One Drawback of this approach is the possibility of big errors in the initial solution of the physical fields, which could lead to convergence problems in the second iteration of the re-meshing. Generally the two step strategy demands more calculation time. For big two-dimensional grids or three-dimensional grids this method can, however, lead to much faster device simulations because of improved convergence. Some problems cannot even be solved with a simple one-step approach. Figure 3.8 shows the algorithm of both methods including the possible re-meshing criterion.

Figure 3.8: Strategies for re-meshing structures generated by process simulation. (a) Straight forward single step method (b) Two-step method

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(b)

A simple example of the refinement of a process simulation mesh of a CMOS transistor which has to be simulated for drain to substrate breakdown is shown in Figure 3.9.

Figure 3.9: Comparison of automatically and manually refined meshes (a) Initial coarse grid (b) 1st iteration (c) 2nd iteration (d) manually refined grid

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In Figure 3.9(a) a very coarse initial grid interpolated from the process simulation grid is shown. This grid is used for a first analysis of the junction breakdown between the drain and the substrate of the NMOS transistor. The resulting electric field inside the device extracted when the reverse current between drain an substrate reaches a certain level is used with gradient refinement criteria for the first iteration shown in Figure 3.9(b). The results of a device simulation with this grid are used for a 2nd iteration with the same criteria. The resulting mesh is shown in Figure 3.9(c). It can be seen clearly from this figure, that the fine mesh follows the field distribution in the breakdown situation very smoothly. As a comparison a mesh which is created by manually placed refinement boxes is shown in Figure 3.9(d). The electrical characteristics of the reverse biased drain/substrate diode for these different grids can be seen in Figure 3.10.

Figure 3.10: Comparison of the different meshes in terms of device simulation results

The initial grid gives a moderately inaccurate result (the simulation was stopped when the ionization integral inside the structure reached unity to speed up the simulation). The iterative and the manually optimized meshes give nearly identical results. However, to set up such a mesh manually human interaction is necessary, which is not suitable for automatic simulation flows. Therefore the approach to generate the mesh refinement based on initial device simulations is the method of choice for a stable automated mesh generation.
Drawbacks of this method can be an increased mesh node count, because the refinement is not restricted to certain parts of the device.