Floating Sheath Potentials in Non-Maxwellian Plasmas

Abstract

Abstract-The floating sheath potential in a plasma having a Maxwellian electron distribution function is ec5 = -kTe In (a/b)/2 where Te is the electron temperature, a is the ratio of electron temperature to ion temperature, and b is the ratio of electron mass to ion mass. This expression is derived by equating the flux of electrons and ions to a surface in the plasma. Only electrons initially having an energy greater than -eo3 flow to the surface. These electrons are in the tail of the distribution, a region that differs significantly from a Maxwellian in many plasmas. An analysis is performed where the sheath potential is solved for using a two-temperature model for the electron distribution function. The two-temperature model accurately describes the distortion from a Maxweilian in the tail of the distribution function. The magnitude of the sheath potential calculated with the two-temperature distribution is significantly smaller than that obtained using a Maxwellian distribution, a result of the reduction in the relative abundance of energetic electrons in the tail of the distribution. greater than the sheath potential can be collected by the surface. These particles have energies many times the average electron energy and reside in the tail of the electron distribution. It is well known, though, that electron distribution functions differ from a Maxwellian most markedly at higher energies, especially for energies greater than the first inelastic threshold [2]. Typically, the electron distribution is depleted of electrons with energy greater than this value. Electrons energetic enough to excite the gas atoms or molecules, and do so, lose in energy a value at least equal to the threshold value for excitation. The electron typically rejoins the distribution at an energy below the threshold value. The electron distribution is therefore "cut off" at an energy given by the first excitation threshold. Since it is the more energetic electrons that are collected by a surface immersed in a plasma, assuming a Maxwellian electron distribution overestimates the electron flux to the surface, and hence overestimates the sheath potential.A complete description of the sheath region near a surface in a plasma requires one to simultaneously solve Poisson's and Boltzmann's equations for the electrons and ions in a region many Debye lengths thick adjacent to the surface. The effect of interest, that of the change in sheath properties resulting from a non-Maxwellian electron distribution, can be studied in some detail with a simpler analysis to be discussed here. In this analysis, a solution for the electron distribution function, called a two-temperature model, is used [3] - [5] . In this solution, the electron distribution function is assumed to be the continuation of two Maxwellian distributions with separate electron temperatures T1 and T2. For electron energies less than a cutoff energy Ec, typically equal to the first excitation threshold, the distribution is a Maxwellian with temperature T1. For electron energies greater t...

The use of Langmuir probes and optical emission spectroscopy to measure electron energy distribution functions in RF-generated argon plasmas

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“…significantly cooler [27,37,43]. Since only the tail electrons contribute to the electron flux leaving the plasma, and so the potential, it is the tail temperature that should be used in equation (16).…”