Analytic model of the energy distribution function for highly energetic electrons in magnetron plasmas

TL;DR

This paper presents an analytical model for the energy distribution of highly energetic electrons in magnetron plasmas, validated by numerical simulations, aiding the understanding of electron kinetics in plasma discharges.

Contribution

It introduces a steady-state analytical solution for hot electron distribution considering Coulomb and inelastic collisions, applicable to practical time-dependent plasma discharges.

Findings

Analytical distribution matches numerical simulations for HiPIMS parameters.

Model applicable to various practical discharges despite steady-state assumption.

Provides a basis to incorporate kinetic effects into Maxwellian-based plasma models.

Abstract

This paper analyzes a situation which is common for magnetized technical plasmas such as dc magnetron discharges and HiPIMS systems, where secondary electrons enter the plasma after being accelerated in the cathode fall and encounter a nearly uniform bulk. An analytic calculation of the distribution function of hot electrons is presented; these are described as an initially monoenergetic beam that slows down by Coulomb collisions with a Maxwellian distribution of bulk (cold) electrons, and by inelastic collisions with neutrals. Although this analytical solution is based on a steady-state assumption, a comparison of the characteristic time-scales suggests that it may be applicable to a variety of practical time-dependent discharges, and it may be used to introduce kinetic effects into models based on the hypothesis of Maxwellian electrons. The results are verified for parameters appropriate to HiPIMS discharges, by means of time-dependent and fully-kinetic numerical calculations.