Moving Adaptive Unstructured 3-D Meshes in Semiconductor Process Modeling Applications

The next generation of semiconductor process and device modeling codes will require 3-Dmesh capabilities including moving volume and surface grids, adaptive mesh refinement andadaptive mesh smoothing. To illustrate the value of these techniques, a time dependent processsimulation model was constructed using analytic functions to return time dependentdopant concentration and time dependent SiO2 volume and surface velocities. Adaptive meshrefinement and adaptive mesh smoothing techniques were used to resolve the moving borondopant diffusion front in the Si substrate. The adaptive mesh smoothing technique involvesminimizing the L2 norm of the gradient of the error between the true dopant concentrationand the piecewise linear approximation over the tetrahedral mesh thus assuring that the meshis optimal for representing evolving solution gradients. Also implemented is constrainedboundary smoothing, wherein the moving SiO2/Si interface is represented by moving nodesthat correctly track the interface motion, and which use their remaining degrees of freedom tominimize the aforementioned error norm. Thus, optimal tetrahedral shape and alignment isobtained even in the neighborhood of a moving boundary. If desired, a topological “reconnection”step maintains a Delaunay mesh at all times. The combination of adaptive refinement,adaptive smoothing, and mesh reconnection gives excellent front tracking, feature resolution,and grid quality for finite volume/finite element computation