Effect of the electron energy distribution function in plasma on the Bohm criterion and on the drop voltage through the sheath: Case of microwave expanding plasma

Abstract

This work is devoted to the study of the sheath expansion above a charged species collector immersed into the plasma in the general case of the electron energy distribution function ͑EEDF͒ ͑not necessarily a Boltzmann distributed function͒. In the first part, considering the general form of the electron kinetic energy distribution function ͑EEDF͒ in the plasma and assuming a one dimensional system, we propose an equation describing the limit value of the ion velocity at the sheath edge. Then, we extend this equation to the three dimensional problem. These results are discussed assuming Boltzmann distribution for the one dimensional system and Maxwell-Boltzmann distribution in the case of the three dimensional system. Both one and two temperature distributions are considered. Then, the method is applied to experimental results obtained in expanding microwave discharge sustained in argon. In the second part, we study the drop voltage through the sheath that is produced around a cylindrical collector biased at the floating potential. A theoretical model is given, and results are compared with experimental values measured by means of a cylindrical Langmuir probe immersed in microwave expanding plasma. The model agrees with experiments and points out the role of the EEDF profile on the ion velocity at the sheath edge and, consequently, on the drop voltage through the sheath.

Comments on the Bohm Criterion and on the Determination of the Ion Velocity at the Sheath Edge in Non‐Maxwellian Plasma

“…This condition can be used to determine the ion velocity at the sheath edge. This value is necessary to calculate the electron density at ion saturation of a negative biased Langmuir probe [3,4,5].…”

“…Generally, there are too large uncertainties on the distribution values, particularly at low electron energy. In a previous experimental work, it has been determined assuming that the condition n i (χ) ≥ n e (χ) is true anywhere in the sheath of the negative biased collector [4]. However, according to Equation 2, this condition is sufficient to have…”

“…We show that the conditions on the ions velocity defined at the sheath edge in the case of a Maxwellian distribution cannot be used in the case of other distributions like a Druyvesteyn distribution. According to the plasma parameters, we discuss the collisionless condition used in the model, then the condition n i (χ) ≥ n e (χ) in the sheath, used to determine the ion velocity at the sheath edge [3,4,5]. All these results are correlated to previous works [3,4,6,7].…”

“…Nowadays the criterion in its initial form defined for Maxwellian electrons, is largely used to describe the sheath produced in the various kinds of cold plasmas, including also non-Maxwellian plasmas. In recent publications [2,3,4,5], authors have pointed out problems due to this error. For example, wrong results have been obtained in the case of the measurements of the electron density by means of Langmuir probe at the ion saturation in non-Maxwellian plasmas.…”