Self-consistent kinetic modeling of low-pressure inductively coupled radio-frequency discharges

Abstract

An efficient method for solving the spatially inhomogeneous Boltzmann equation in a two-term approximation for low-pressure inductively coupled plasmas has been developed. The electron distribution function ͑EDF͒, a function of total electron energy and two spatial coordinates, is found self-consistently with the static space-charge potential which is computed from a 2D fluid model, and the rf electric field profile which is calculated from the Maxwell equations. The EDF and the spatial distributions of the electron density, potential, temperature, ionization rate, and the inductive electric field are calculated and discussed.

Spatial profiles of a planar inductively coupled discharge in argon

“…Getting a self-consistent solution of the equations listed above is not a trivial task. The method used here is somewhat modified as compared with the one used in [19]. The procedure of how convergence is achieved in this work shall be sketched below.…”

“…Different approaches have been used to model the planar inductive discharge. These include fluid calculations [15], Monte Carlo simulations [16] and models with kinetic treatment of the electrons [8,9,17,18,19]. The kinetic models can be simplified by the non-local approximation [8,9], which is based on the experimentally well verified fact that at sufficiently low pressures the EDF of the variable ε (total energy, sum of kinetic and potential energy) is nearly constant in the discharge volume [20,21].…”

“…Here the electron-electron collision terms which are most important at low electron energies [22] are neglected in the kinetic equation for the high-energy part. Another group has solved the kinetic equation in the total energy range with spatial dependence [19]. Their calculations were performed in a parameter range where deviations from nonlocality in the high-energy part of the electron distribution are already significant.…”

Boltzmann equation and Monte Carlo analysis of the spatiotemporal electron relaxation in nonisothermal plasmas

“…One of the approaches consists of the solution of the electron Boltzmann equation. So far, the treatment of this kinetic equation is widely restricted to timedependent, but spatially homogeneous plasmas [1][2][3][4][5][6] or to space-dependent, but stationary plasmas [7][8][9][10][11][12][13][14][15][16][17]. In recent years the spatiotemporal description of the kinetics of electrons in spatially one-dimensional plasmas [18][19][20] became possible, too.…”

Progress of the Electron Kinetics in Spatial and Spatiotemporal Plasma Structures

“…Examples are the reduction of the dimensionality of the problem by applying the so-called non-local approach for the simplified description of the radial space charge confinement of the electrons in an ICP hybrid model [36] or the kinetic treatment only of the high energy part of the electrons by particle simulation and the use of rough assumptions on some transport properties of the electrons as involved in different hybrid models of ICP, CCP and cathode arrangements [37][38][39][40][41]. A certain progress with respect to the rigorous treatment of the two-dimensional kinetics could be recently reached by successfully developing techniques to analyse two-dimensional kinetic problems arising in low-pressure inductively coupled radio-frequency discharges [42] and in cylindrical glow discharge configurations [43,44]. Few basic aspects of the latter approach, its application to a two-dimensional relaxation study of the electron gas in the positive column of a dc glow discharge and some two-dimensional features of the electron gas properties obtained will be briefly considered in the succeeding presentation to illustrate the current possibilities in this field.…”