9 Summary and Outlook

The state of the art of numerical methods for topography simulation for Process TCAD was presented and discussed with a focus on computational aspects. In particular, different approaches for particle transport and surface advection were discussed. The level-set method was identified as predominant choice for surface advection for three-dimensional process simulation. Thus, from an implementation point of view, the starting position for any particle transport/flux calculation approach is a set of level-sets representing the material regions in the simulation domain. All common approaches for the particle transport, which all assume ballistic transport in the feature-scale region, rely on a vast number of ray-surface intersection tests to either perform visibility tests or to simulate the trajectories of particles in the simulation domain. This emphasizes that an indispensable requirement for a high performance particle transport/flux calculation approach is access to a highly efficient ray casting back end.

With these two requirements in mind, a simulation framework was developed to explore novel approaches for particle transport/flux calculations. The framework combines open-source third-party libraries for sparse volumetric data and ray tracing to store and advect the level-sets and to perform the ray-surface intersection tests. Using this framework, different approaches to reduce the computational workload of the particle transport/flux calculation were investigated.

The accuracy requirements for the ray-surface intersection tests was put in relation to the accuracy obtained with single-precision arithmetics for the ray casting: It is admissible to utilize single-precision arithmetics for the ray-surface intersection tests in practical process simulation scenarios. This is a fundamental finding as all particle transport methods benefit from the improved underlying performance.

As the surface advection is predominantly level-set-based, no explicit representation of the surfaces is available. Using two highly optimized open-source libraries (from the field of computer graphics) for ray casting on implicit (OpenVDB) and explicit (Embree) surfaces, it is shown that the overhead introduced by an extraction of a temporary polygonal mesh (in each time step of the simulation) is by far compensated by the performance gain obtained from the ray casting against the explicit surface.

To reduce the runtime of a bottom-up direct flux calculation two approaches were pursued: Firstly, the number of necessary visibility sampling directions (per integration point) is reduced by adaptively refining the sampling only around the boundary of the aperture regions using a hierarchical subdivision of the spherical directions. Secondly, the number of integration points on the surface is locally reduced (according to a freely definable application-specific condition) using an iterative partitioning scheme on the extracted polygonal surface mesh. Both approaches reduce the runtime of the direct flux calculation significantly. The accuracy of the result is not influenced by the first approach and the influence of the second approach is reasonably small.

All approaches are applicable for arbitrary three-dimensional geometries and were applied in conjunction to an etching simulation of a multi-material stack. The resulting overall simulation speedup is above 14 for a wide range of settings.

A further utilization of the application-specific refinement condition for the iterative partitioning has potential to increase the speedup – for example by introducing a material dependence, e.g., exploiting the fact that the demands for accuracy can differ greatly from material to material, or by introducing a dependence on the position in the domain, e.g., the accuracy demands can be tailored to simulations of high aspect ratio structures.

Introducing a dependence on the emission characteristics of the source for the adaptive visibility sampling and the subsequent integration potentially increases viability of this approach further. This is especially auspicious with regard to the integration accuracy for very directed sources.