The maximum principle is the most important property of solutions to diffusion equations. Violation of the maximum principle by the applied discretization scheme is the cause for severe numerical instabilities: the emergence of negative concentrations and, in the nonlinear case, the deterioration of the convergence of the Newton iteration. We compare finite volumes (FV) and finite elements (FE) in three dimensions with respect to the constraints they impose on the mesh to achieve a discrete maximum principle. Distinctive mesh examples and simulations are presented to clarify the mutual relationship of the resulting constraints: Delaunay meshes guarantee a maximum principle for FV, while the recently introduced dihedral angle criterion is the natural constraint for FE. By constructing a mesh which fulfills the dihedral angle criterion but is not Delaunay we illustrate the different scope of both criteria. Due to the lack of meshing strategies tuned for the dihedral angle criterion we argue for the use of FV schemes in three-dimensional diffusion modeling.

For wafer sizes in state of-the-art semiconductor manufacturing ranging up to 300 mm, the uniformity of processes across the wafer becomes a very important issue. We present a fully three-dimensional model for the feature scale simulation of continuum transport and reaction determined high-pressure chemical vapor deposition processes suitable for the investigation of such nonuniformities. The newly developed three-dimensional approach combines topography simulation, meshing, and finite element method tools, and allows simulations over arbitrary geometries such as structures resulting from nonuniform underlying physical vapor deposition films. This enables the examination of film profile variations across the wafer for multistep processes consisting of low- and high-pressure parts such as Ti/TiN/W plug-fills, Additionally, the model allows a very flexible formulation of the involved gas chemistry and surface reactions and can easily be extended to process chemistries including gas phase reactions of precursors as observed in deposition of silicon dioxide from tetraethylorthosilicate (TEOS). We show simulation examples for a tungsten deposition process, which is applied as last step in a Ti/TiN/W plug-fill. For filling of an L-shaped trench, we show the transition from transport to reaction limited process conditions.