Drift-Diffusion models for RF-CCPs at intermediate pressure: estimating transport coefficients*

RF-CCPs at gas pressures above 1 Torr are often simulated by fluid models using the drift-diffusion approximation, since Particle-in-Cell (PIC) models are too computationally intensive. Fluid models require the electron transport coefficients (mobility, μ, diffusion, D, energy mobility με and energy diffusion Dε), as well as rate constants (for ionisation, energy loss, etc.). These are calculated from the electron energy distribution function (EEDF), which is commonly estimated using local energy approximation or by simple Maxwellian distribution. The Local energy EEDF is calculated as a function of electron mean energy (<E>) using a Boltzmann solver (Loki). This assumption is well justified for homogeneous DC glow discharges, but its applicability to RF field is more questionable. For a Maxwellian EEDF, the coefficients and rates are directly calculated, with electron temperature determined by power balance equation during simulation. However, at intermediate pressures (1-10 Torr) the EEDF will deviate significantly from Maxwellian due to inelastic collisions. In this study, we compare these two approaches to a full PIC model in a 13.56 MHz discharge in 1 Torr Ar.

It's found that both fluid models underestimate the plasma density compared to PIC simulation, but Maxwellian one provides a better prediction. This is probably because the EEDF at sheath edge, where most electron heating and ionization happens, is more Maxwellian even at high pressure. Therefore, Maxwellian EEDF better estimates the electron heating and density.

For electron energy, the local energy model strongly overestimates, while the Maxwellian model underestimates it. Additionally, fluid model assumes a uniform EEDF throughout the plasma region, inducing errors on <E> profile. In PIC simulation, even though both electron heating and ionization localizes near sheath edge, <E> near boundary is not higher. This could be explained by the different EEDF near sheath and in bulk plasma: the EEDF near sheath is more Maxwellian, so there are more high energy electrons even with a lower <E>. However, in fluid model, the EEDF is fixed, so the electron heating and ionization peak could only be achieved by a higher <E>, resulting in a peak of it near the sheath edge.