Kinetic simulation of the ideal multipole resonance probe

Abstract

Active plasma resonance spectroscopy (APRS) is a process-compatible plasma diagnostic method, which utilizes the natural ability of plasmas to resonate on or near the electron plasma frequency. The multipole resonance probe (MRP) is a particular design of APRS that has a high degree of geometric and electric symmetry. The principle of the MRP can be described on the basis of an idealized geometry that is particularly suited for theoretical investigations. In a pressure regime of a few Pa or lower, kinetic effects become important, which cannot be predicted by the Drude model. Therefore, in this paper, a dynamic model of the interaction of the idealized MRP with a plasma is established. The proposed scheme reveals the kinetic behavior of the plasma that is able to explain the influence of kinetic effects on the resonance structure. Similar to particle-in-cell, the spectral kinetic method iteratively determines the electric field at each particle position, however, without employing any numerical grids. The optimized analytical model ensures the high efficiency of the simulation. Eventually, the presented work is expected to cover the limitation of the Drude model, especially for the determination of the pure collisionless damping caused by kinetic effects. A formula to determine the electron temperature from the half-width [Formula: see text] is proposed.

The multipole resonance probe: simultaneous determination of electron density and electron temperature using spectral kinetic simulation

“…As described in our prior work [24], spectral kinetics is a particle based simulation scheme, similar to the PIC/MCC algorithm. The simulation input are the voltages u n (t) applied to the electrodes E n , while the recorded output is given by the resulting electrode charges Q n (t), with n ∈ {1, 2}.…”

“…The field solver calculates the electric potential Φ(r) in dipole projection. (See [24] for details and appendix A for a short review.) G(r, r ) is the Green function, Φ S (r) describes the influence of the immobile surface charges, and the Ψ n (r) represent the effect of the voltages applied to the electrodes,…”

“…The size of the simulation domain is set accordingly to represent infinity, which is in the range r ∈ (1,10). Depending on the investigated plasma (T e , n e ), by solving the Poisson-Boltzmann equation, the stationary state can be calculated as the initial condition of simulation [24]. Therefore, the static density profiles for ions and electrons are determined numerically, in addition to the static surface charge.…”

“…Unfortunately, explicit evaluation needs large numerical resources, especially at low pressure. Recently, we instead proposed a spectral kinetic method, i.e., a particlebased scheme similar to particle-in-cell/Monte Carlo collisions (PIC/MCC) but without a numerical grid [24]. In this study, we will expand our investigations.…”