Accuracy of the Explicit Energy-Conserving Particle-in-Cell Method for Under-resolved Simulations of Capacitively Coupled Discharges

The trend in capacitively coupled plasma (CCP) discharges for semiconductor etching, is towards lower pressures, allowing for increased ion distribution function anisotropy and therefore improved etch performance. Modelling these discharges is challenging since the low pressure necessitates a kinetic treatment, and the commonly used particle-in-cell (PIC) technique suffers from the serious constraint of requiring resolution of the electron Debye length. It was recently shown that the explicit “energy-conserving” PIC scheme [1], could be applied accurately in plasmas dominated by ambipolar electric fields [2]. This method completely alleviates numerical restriction on cell size, allowing for cells much larger than the Debye length without triggering the finite-grid instability.

We perform one-dimensional simulations of a large (30cm) Helium CCP discharge using both the momentum and energy-conserving PIC methods and consider the fully resolved case and progressively underresolved cases. The energy-conserving method proves to be far more accurate than the traditional method in underresolved cases and only begins to reduce in accuracy when the cell sizes become comparable to the sheath size. When considering the gains available in higher dimensions we see this algorithm as a viable technique to significantly reduce the cost of plasma simulations relevant to industry.

[1] H. R. Lewis, J. of Comp. Phys., 6(1), 136-141 (1970)

[2] D. C. Barnes & L. Chacón, Comp. Phys. Comm., 258, 107560 (2021)