Plasma Particle Simulation Weighting Factor
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Solution Overview
Problem
Plasma particle simulations face significant computational challenges due to large statistical errors near solid walls in plasma processing apparatuses, leading to solution divergence and inaccurate plasma distribution predictions, especially in regions with varying plasma particle densities and electric fields.
Innovation Solution
A method of plasma particle simulation that adjusts the weighting factor for superparticles based on their proximity to solid walls, using trigonometric functions to maintain an appropriate number of superparticles and prevent statistical errors, while also allowing for probabilistic adjustments during cell-to-cell movement, ensuring accurate behavior calculations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If the space within the housing chamber is divided into a plurality of cells with constant plasma particle density assumption, then the calculation is simplified, but the number of plasma particles in cells near the central axis and solid wall surfaces decreases, leading to larger statistical errors
Solution Approach 1:
The patent applies local quality by setting different weighting factors for superparticles in different spatial regions. Specifically, cells are divided into three regions (first region near central axis, second region in bulk, third region near solid wall surfaces) with progressively increasing weighting factors. This allows the simulation to maintain appropriate superparticle numbers in regions where plasma particle density varies significantly, thereby reducing statistical errors locally while keeping the overall calculation manageable.
2Measurement precision
If the weighting factor is increased to maintain the number of superparticles in cells near the central axis, then the statistical error decreases, but the computational load increases
Solution Approach 1:
Instead of uniformly increasing the weighting factor throughout the chamber, the patent applies local quality by assigning different weighting factors to different regions. The weighting factor is increased only in the first region (near central axis) and third region (near solid wall surfaces) where plasma particle density is low, while maintaining a lower weighting factor in the second region (bulk plasma) where particle density is high. This localized adjustment reduces statistical errors where needed without unnecessarily increasing computational load elsewhere.
3Device complexity
If a uniform weighting factor is applied to all cells, then the calculation is simple, but the number of superparticles becomes insufficient in regions with low plasma particle density, causing solution divergence
Solution Approach 1:
The patent resolves the contradiction between calculation simplicity and solution convergence by implementing local quality through region-specific weighting factors. By dividing the housing chamber into three distinct regions and assigning appropriate weighting factors to each, the simulation maintains sufficient superparticle numbers in low-density regions (ensuring convergence) while keeping the calculation structure organized and manageable through clear regional definitions.
4Manufacturing precision
If the number of cells near solid wall surfaces is increased to resolve sheath structure, then the spatial resolution improves, but the number of plasma particles in each cell decreases, leading to larger statistical errors
Solution Approach 1:
The patent addresses this contradiction by applying local quality through region-specific weighting factors. In the third region (near solid wall surfaces) where cells are divided finely to resolve sheath structure, the weighting factor is set to be larger than in the bulk region. This compensates for the reduced number of plasma particles in each small cell, maintaining sufficient superparticle numbers and reducing statistical errors while preserving the high spatial resolution needed for sheath analysis.
Data Source
AI summary
A method of plasma particle simulation capable of preventing solution divergence. A space within a housing chamber of a plasma processing apparatus is divided into a plurality of cells. A weighting factor corresponding to the number of plasma particles represented by a superparticle is set in each of the divided cells. Superparticles are set in each of the divided cells using plasma particles contained in the divided cell and the set weighting factor. The behavior of the superparticles in each of the divided cells is calculated. The weighting factor becomes smaller as the divided cell is located closer to a solid wall surface of the housing chamber.