Potential formation in front of an electron emitting electrode immersed in a plasma that contains a monoenergetic electron beam

Abstract

A one-dimensional fluid model of the sheath formation in front of a large, planar electron emitting electrode (collector) immersed in a plasma that contains a monoenergetic electron beam is presented. Expressions for the Bohm criterion, the total electric current to the collector, and for the zero electric field at the collector are derived. When there is no electron emission, the model predicts, in some cases, up to three different solutions. The low and the high solutions correspond to the fact that the presheath potential drop can be determined either by thermal or by the beam electrons. The middle solution between them has no physical meaning. When the electron emission is space charge limited, the model may have up to five solutions because the low solution can sometimes split into three parts. The current-voltage characteristic of the collector that exhibits triple floating potential is calculated and it shows qualitative similarity with experimental ones [C.-H. Nam et al., J. Appl. Phys. 63, 5674 (1988)]. When the emission is below the space charge limit, the model again exhibits up to five solutions, which are the low, middle, high, and two additional ”singularity solutions.” These two appear because of the singularity in the Bohm criterion. Regions of validity of the low and high solutions are determined from numerical solutions of the Poisson equation [T. Gyergyek et al., Plasma Sources Sci. Technol. 18, 035001 (2009)] and with the maximum positive ion flux test [J. I. Fernández Palop et al., J. Appl. Phys. 91, 2587 (2002)]. In the case of no or small emission, both methods give the same results. If the emission is space charge limited, the numerical solution method is much more reliable.