Magnetostatic Solvers for Conformal Finite Difference Time Domian Particle-In-Cell Simulations: Permanent Magnets

Magnetic circuit designs are crucial to operations of many vacuum electronics devices and plasma devices, such as magnetrons and magnetron sputtering, respectively. For industrial plasma processing of materials, the magnetron sputtering process is most widely used and the corresponding deposition or coating quality is enhanced and controlled by a well-tailored magnetic field distribution. Although the conformal finite-difference time-domain (CFDTD) particle-in-cell (PIC) method as implemented in VSim is suitable to model magnetron sputtering and study its behavior, the magnetic circuit design relies on a third part tool such as Maxwell or COMSOL based on a finite element method (FEM). For a full simulation of the magnetron sputtering, one has to export the magnetic circuit design from the FEM program and import it into the CFDTD-PIC simulation for studying particle-wave interaction as different meshes in the codes are used. In addition to tedious file format conversion and importation, a translation of representations between different numerical gridding systems has to be done. In this work, we propose the magnetostatic (MS) solvers newly developed in VSim for conducting the magnetic circuit designs as the same finite difference grids are used, avoiding the troubles caused by an export/import process or additional errors due to a translation/interpolation. For modeling a permanent magnet, we solve the scalar potential in a Poisson equation with well-defined spatially distributed magnetization and permeability. We have carefully benchmarked the 2-D cylindrical and 3-D Cartesian MS solvers against the FEM simulations using the magnetic circuit designs available in the literature of magnetron sputtering, giving good agreement. The detailed formulation and demonstrations will be presented and discussed.