Sheath expansion of two-dimensional grid electrodes subjected to short pulses of negative high-voltage

Abstract

Sheath expansion was investigated for two-dimensional (2D) grid electrodes which consist of a periodic array of cylindrical electrodes when short pulses of negative high-voltage were applied to the electrodes immersed in plasmas. In the sheath expansion model, a geometric function which describes the electrode system is crucial to numerically calculate the temporal evolution of a sheath boundary. In this paper, the 2D geometric function of grid electrodes was obtained by using XOOPIC (particle-in-cell) simulation. When the ratio between the diameter of cylindrical electrodes and grid spacing is fixed, we found that the geometric functions and the temporal evolutions of the sheath boundary for grid electrodes are identical in normalized coordinates. The numerical calculation results of the temporal evolutions of the sheath boundary showed reasonable agreements with the experimental measurements carried out in argon plasmas produced by hot filament discharges with neutral gas pressure of ∼0.4 mTorr and plasma density in the order of 10 10 cm −3 in a multi-dipole device.

Normalized ion distribution function in expanding sheaths of 2D grid electrodes

“…When particles collided with the boundary electrodes, they were lost. These simulation conditions showed good agreement with the experimental results of transient sheath expansion of grid electrodes in our previous work [20]. Since transient sheath expansions are determined by ion distributions (Child-Langmuir currents) in expanding sheaths [1,2], the simulation conditions described above were a reasonable choice for properly simulating ion distributions in expanding collisionless sheaths.…”

“…In many cases, the ion distributions near the surface of electrodes were mainly studied for industrial applications such as the modification of the surface properties of materials using energetic ions. However, for grid electrodes, which are frequently used in basic plasma research [17] and industrial plasma applications [14,15,18,19], ion distributions in the entire region of an expanding sheath are important for investigating the characteristics of sheath expansion [20] and pseudo wave generation [17].…”

“…days), so generalization of simulation results is required to quickly obtain the ion distributions for various changes in inputs, such as the plasma parameters, voltage pulses applied to the electrodes, and the size of the grid electrodes. In our previous work [20], we generalized the sheath expansion of 2D grid electrodes by using normalized coordinates. When the opening ratio η is fixed, it was found that the temporal evolutions of the sheath boundary of 2D grid electrodes in normalized coordinates are independent of the plasma parameters, voltage pulses applied to the electrodes, and the size of the grid electrodes [20].…”

“…In our previous work [20], we generalized the sheath expansion of 2D grid electrodes by using normalized coordinates. When the opening ratio η is fixed, it was found that the temporal evolutions of the sheath boundary of 2D grid electrodes in normalized coordinates are independent of the plasma parameters, voltage pulses applied to the electrodes, and the size of the grid electrodes [20]. The normalized coordinates are given by [2,20]…”

“…A generalization similar to the case of sheath expansions can be obtained for ion distribution functions in the same normalized coordinates. For a given opening ratio of grid electrodes, the geometries of expanding sheaths in normalized coordinates are identical at the same normalized time regardless of the plasma parameters, voltage pulses applied to the electrodes, and the size of grid the electrodes [20]. By applying the single species steady-state Vlasov equation under these boundary conditions (see figure 1 for the schematics of the boundary conditions), it can be derived that normalized ion distribution functions in normalized coordinates are also identical at the same normalized time θ.…”