Plasma Fluid Modeling Transient Stochastic Transformation
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Solution Overview
Problem
Current plasma modeling and simulation methods for complex plasma systems, such as dual-frequency CCP etching systems, require extensive computational time, often taking several hours to days, due to the need to solve multiple frequency scales and complex chemistry, limiting real-time process optimization and hardware development.
Innovation Solution
A computer method that approximates transient sheath behavior using a probability function-based time-domain simulation, allowing for the simulation of plasma density distribution in response to variables with different time bases, reducing computational time by employing stochastic transformation and random sampling to achieve a steady-state plasma density distribution.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If full transient numerical simulation is performed to model plasma density distribution in dual-frequency CCP systems, then accuracy and reliability of plasma behavior prediction is improved, but computational time increases to several hours or days
Solution Approach 1:
The patent segments the plasma simulation problem into two distinct parts: (1) a simplified steady-state model that provides accurate plasma density distribution, and (2) a separate sheath model that accounts for transient effects. This segmentation allows each model to be optimized independently, achieving both accuracy and computational efficiency without requiring full transient simulation of the entire plasma system.
Solution Approach 2:
The patent changes the temporal parameter of the simulation by solving the plasma bulk equations under steady-state conditions rather than transient conditions. The time-dependent terms are retained only in the sheath region equations, effectively separating the time scales of the plasma bulk and sheath regions. This parameter change reduces computational time while maintaining accuracy for the relevant plasma properties.
2Reliability
If multiple frequency scales (e.g., 100 MHz and 1-2 MHz) are solved simultaneously in DF-CCP etching system, then completeness of plasma process modeling is improved, but computational complexity and time increase drastically
Solution Approach 1:
The patent segments the multi-frequency plasma system into a plasma bulk region and a sheath region, where each region is modeled with appropriate physics. The plasma bulk is modeled with time-averaged equations that capture the effect of multiple frequencies without resolving each frequency explicitly. The sheath region is modeled separately with transient equations that capture the high-frequency dynamics. This segmentation reduces computational complexity while maintaining modeling completeness.
Solution Approach 2:
The patent transitions from a time-domain multi-frequency solution to a spatial-domain solution by solving the plasma bulk equations in a time-averaged steady state. The temporal dimension is effectively removed from the bulk plasma equations, and the effects of multiple frequencies are captured through spatial distributions of plasma parameters. This dimensional change significantly reduces computational complexity.
3Manufacturing precision
If high resolution reactor geometry and complex chemistry are modeled, then manufacturing precision of plasma process control is improved, but computational time increases to several days
Solution Approach 1:
The patent segments the complex plasma chemistry and transport equations into a steady-state plasma bulk model and a transient sheath model. The complex chemistry reactions are solved in the steady-state bulk plasma where time derivatives are neglected, reducing the computational burden. High-resolution geometry is maintained in the spatial discretization, but the removal of time-dependent terms from the bulk equations dramatically reduces computational time while preserving manufacturing precision for plasma process control.
Data Source
AI summary
The invention relates to the simulation method and apparatus used in plasma modeling. It includes a method to transform transient formulations of the phenomenological plasma model into a quasi-stochastic spatial formulation. Specifically, the invention aids in decreasing computational time for the modeling of plasma in a plasma processing system, particularly those involving two different time-based parameters. The invention is particularly described in connection with plasma simulations used for the optimization dual-frequency capacitively-coupled plasma etching systems.