Microphysical boundary condition for the electron kinetics of a plasma

TL;DR

This work derives and implements a microphysical, energy- and angle-dependent boundary condition for electron kinetics at plasma walls by embedding the Surface Scattering Kernel into a Legendre-expanded Boltzmann solver. It provides Marshak-like boundary relations for backscattering at the anode and a prescribed influx at the cathode, implemented with a Crank–Nicolson–type scheme and modest computational overhead. Numerical tests with a Si wall facing Ar, He, and O2 plasmas show meaningful changes in bulk and sheath distributions compared to elastic or absorber boundaries, especially when secondary-electron processes are significant. An effective, energy- and angle-dependent reflection coefficient is introduced to bridge to simpler models, emphasizing the importance of microphysical wall interactions for accurate discharge simulations across pressures and gas types.

Abstract

We derive and implement a suitable boundary condition for the kinetic description of the electrons inside a plasma, which takes into account microphysical processes inside the wall. It is based on the surface scattering kernel, which describes the scattering cascade of the electron in the solid and the excitation of secondary electrons. The resulting boundary condition is inelastic, angle- and energy-dependent. The implementation for a Boltzmann equation solved by a Legendre polynomial expansion method is presented, elucidating the modest additional computational cost of the new boundary condition. Results, indicating the influence of the inelasticity, are shown for the example of a silicon wall facing argon, helium and oxygen plasmas, but the described construction is also valid for other materials. An effective reflection coefficient is defined to compare the results with previously used boundary conditions.

Figures (12)

• Figure 1: Sketch of the geometry of the modeled discharge. The discharge has plane electrodes, a rotational symmetry around the $z$-axis and is axially inhomogeneous. At the cathode the influx of electrons with a given energy profile is used to construct a boundary condition, whereas at the anode the influx into the system is related to the outflux via the SSK.
• Figure 2: The derivate $R_n^{2\rho+1}$ of the SSK is shown for the calculated energy range and one index combination. For other indices, the same energy-dependencies occur, but the sign and the magnitude may change severely.
• Figure 3: The total electron density is plotted over the position $z$ for parameter set 1 (Ar).
• Figure 4: The total electron flux in the z-direction is plotted over the position $z$ for parameter set 1 (Ar).
• Figure 5: The electron density at the anode ($z=d=10$ cm) is shown on a logarithmic scale. The middle and the right plot are zoomed in parts of the first one.