In this work many new numerical techniques and algorithms for topography simulation, especially for large three-dimensional geometries, have been presented. The combination of modern LS techniques, such as the sparse field method and the H-RLE data structure, lead to a fast optimal scaling surface evolution algorithm. Novel iterators for accessing the H-RLE data structure have been implemented. They enable fast serial processing of the H-RLE data structure with proper incorporation of boundary conditions. The developed iterators are used for the implementation of the sparse field method and for the realization of Boolean operations. Furthermore, efficient algorithms for testing unidirectional visibility and connectivity, which are useful for certain process models, have been described. In addition, a new way to handle multiple material regions, including thin layers, using a multi-LS description has been presented. The LS framework was further enhanced by adapting the H-RLE data structure for parallelization on shared-memory machines.

General surface kinetics models require the calculation of the particle transport in order to obtain surface velocities. Since ballistic particle transport can be assumed for many processes, ray tracing techniques, such as spatial subdivision, can be applied. A new data structure using neighbor links arrays was suggested, which appeared to be very suitable and efficient for topography simulation. Furthermore, ray tracing was directly applied to the implicit LS representation of the surface. Rates are directly calculated for surface points using tangential disks. Hence, no explicit surface representation is required, which saves memory and computation time.

Contrary to the conventional direct integration approach used for surface rate calculation, ray tracing enables the incorporation of higher order reemissions as well as effects which depend on the direction and energy of incident particles. At the same time, a reduced computational complexity is obtained and parallelization is straightforward. Furthermore, ray tracing allows the definition of a simple interface, which enables an easy implementation of new process models. Together with the LS framework, a powerful and efficient topography simulator was created, which can be used for a large variety of processes. As a demonstration, a selection of process models reported in the literature has been implemented and used for various three-dimensional applications.

For future work, it might be interesting to develop methods and algorithms, which are able to convert back the LS representation into a volume mesh of high quality, which can then be used for subsequent simulation of process steps requiring a volume mesh, e.g. diffusion. A promising approach for meshing an implicit surface is described in [18,82,129]. The idea is to start from a regular tetrahedral mesh which is refined near the surface. Tetrahedra outside of the surface are removed, and the remaining tetrahedral mesh is adapted to the surface by a relaxation procedure.

As already mentioned in Section 2.2.2, some processes might require the incorporation of electrostatic interactions. Ion incidence may charge the surface, which leads to an electrostatic field that deflects ion trajectories. The effect of surface charging has already been fully incorporated in two-dimensional MC simulations [104]. However, the accuracy of three-dimensional simulations is still limited by the high computational costs [94,97]. To speed up the calculation, it might be possible to extend the presented ray tracing techniques as follows: First, the electric field is calculated at all corners of each subbox of the spatial subdivision. This enables a multi-linear interpolation of the field within each subbox. Then, the ion trajectory through a subbox can be calculated using a standard leapfrog method [92]. If necessary, the spatial subdivision needs to be refined at regions with large field gradients.

Otmar Ertl: Numerical Methods for Topography Simulation