Validation of the Parallel PIC-FEM Method

For validating the PIC-FEM code, two benchmark problems are presented, which are the quasi one-dimensional DC and RF gas discharge plasma. Gas discharge plasmas are benchmarked because they find well-established use in practical industrial applications, such as surface modification, lasers, lighting, etc. In addition, one-dimensional DC discharge is a prototype of all discharge simulation. Before simulating these plasma systems, the elementary gas discharge plasma physics is introduced. Then, the simulation conditions are described and the results are compared with experimental data and previous simulation wherever available.

4.4.1 Quasi One-dimensional DC Gas Discharge Plasma

A schematic picture of the elementary glow discharge processes is presented in Fig. 4.12. When a constant potential difference is applied between the cathode and anode, the ions are accelerated by the electric field in front of the cathode sheath and collide with the cathode electrode. Then the secondary electrons are emitted from the

cathode electrode, which are accelerated toward bulk region by electric field in front of the cathode sheath and collide with neutral species. This leads to many important collisions for sustaining plasma, such as ionization, excitation, elastic scattering, etc.

It is clear that the secondary electrons emission play an essential role for sustaining the DC gas discharge plasma. The main structure of the DC glow discharge plasma is shown in Fig. 4.13.It shows that there are many regions in DC glow discharge plasma, which are cathode dark space (CDS), negative glow (NG), Faraday dark space (FDS), the positive column (PC), and anode zone (AZ). However, when the distance between cathode and anode is short, there are only CDS, NG, and AZ formed in DC glow discharge plasma [Bogaerts et. al., 2002]. Here, the mechanism of each region is not described in detail for the brevity purposes.

Simulation Conditions

Consider the discharge sustained between two parallel electrodes by 40mm under an operation argon pressure of 42mtorr. The cathode and anode potentials are set to –1000Volts and 0Volts, respectively. The computational domain is divided with the cell size 0.2mm (≈λ_D) using an unstructured mesh. Initially, the spatial distributions of the ion and electron number densities are assumed uniform to each other. The electron timestep is 0.5×10⁻¹⁰s and the number of sub-cycling is 10. The particle velocities are sampled from the Maxwellian distribution at a corresponding

temperature, e.g., T_i =232Kfor ions and KT_e =0.5eV for electrons, where K is the Boltzmann constant. The secondary electron emission coefficient is 0.3. The initial velocity of the emitted electrons is assumed to be zero. The ions and electrons incident on the solid surfaces are always neutralized.

Simulation Results

The potential and electric field are shown in Fig. 4.14(a) and Fig. 4.14(b), respectively. The plasma potential is nearly constant and slightly positive (≈10V ) and hence, the electric field is very small in the bulk region. The ion and electron number densities are shown in Fig. 4.15. The net charge density is shown in Fig. 4.16.

One can easily recognize the cathode and anode sheaths. The ion and electron kinetic energies are shown in Fig. 4.17. Ions are rapidly accelerated in the sheath, reaching a velocity of about 10⁴m /s before impinging on the cathode. The ion energy distribution function (IEDF) onto the cathode surface has been sampled in the course of simulation as shown in Fig. 4.18. IEDF is falling off exponentially with energy, which demonstrates a good match with the theoretical predictions, e.g., [Serikov. and Nanbu, 1997], and [Abril et. al., 1983].

4.4.2 Quasi One-dimensional RF Gas Discharge Plasma

When one or both of the electrodes are non-conductive materials, the electrodes should be applied with an alternating voltage. The frequency of the alternating

voltages is typically in the radio-frequency (RF) range with a most common value of 13.56 MHz. With this applied alternating voltage, each electrode will act alternately as the cathode and anode in order to eliminate the charge accumulation on insulator electrodes. For RF gas discharge plasma, the electrons will follow the instantaneous electric fields, however, the ions can only follow time-averaged electric fields produced by the applied RF frequency. This totally different behavior can be easily explained by the different masses of ions and electrons.

Simulation Conditions

Consider the discharge sustained between two parallel electrodes by 20mm under an operation argon pressure of 50mtorr. The cathode potential is in the following,

ft V_rf π

φ =− cos2 (4.21)

Where f is the frequency, V_rf =500Volts is the amplitude. The conventional frequency is 13.56MHz. The computational domain is divided with the cell size 0.1mm (≈0.5λ_D) using an unstructured mesh. Initially, the spatial distributions of the ion and electron number densities are assumed uniform to each other. The electron

timestep is 3.695×10⁻¹¹s and the number of sub-cycling is 10. The particle velocities are sampled from the Maxwellian distribution at a corresponding temperature, e.g., T_i =232Kfor ions and KT_e =0.5eV for electrons, where K is the Boltzmann constant. The secondary electron emission coefficient is 0. The ions

and electrons incident on the solid surfaces are always neutralized.

Simulation Results

The potential are shown in Fig. 4.19. The plasma potential is nearly constant and positive (≈0.5V_rf ) and hence, the electric field is very small in the bulk region. The

ion and electron number densities are shown in Fig. 4.20. One can easily recognize the cathode and anode sheaths. The ion and electron kinetic energies are shown in Fig.

4.21. The electron energy probability function (EEPF) of two different pressures in

the bulk region has been sampled in the course of simulation as shown in Fig. 4.22. In Fig. 4.22(a) (50 mtorr), EEPF shows a weakly bi-Maxwellian distribution (TL=1.58 eV, TH=2.58 eV), while Fig. 4.22(b) (20 mtorr) shows strong bi-Maxwellian distribution (TL=0.833 eV, TH=3.264 eV), which is comparable with previous studies under similar simulation conditions, e.g., [Godyak et. al., 1992], [Mahony et. al., 1999], [Raizer et. al., 1995], [Turner et. al., 1993], and [Vahedi et. al., 1993]. At low

pressures the bi-Maxwellian EEPF revealing the stochastic electron heating mechanism, leading to the formation of cold bulk and oscillating hot tail electrons, which demonstrates a good match with the experimental data.