Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing

Abstract

We present a numerical algorithm for solving partial differential equations on irregular domains with moving interfaces. Instead of the typical approach of solving in a larger rectangular domain, our approach performs most calculations only in the desired domain. To do so efficiently, we have developed a one-sided multigrid method to solve the corresponding large sparse linear systems. Our focus is on the simulation of the electrodeposition process in semiconductor manufacturing in both two and three dimensions. Our goal is to track the position of the interface between the metal and the electrolyte as the features are filled and to determine which initial configurations and physical parameters lead to superfilling. We begin by motivating the set of equations which model the electrodeposition process. Building on existing models for superconformal electrodeposition, we develop a model which naturally arises from a conservation law form of surface additive evolution. We then introduce several numerical algorithms, including a conservative material transport level set method and our multigrid method for one-sided diffusion equations. We then analyze the accuracy of our numerical methods. Finally, we compare our result with experiment over a wide range of physical parameters.