Sheath thickness evaluation for collisionless or weakly collisional bounded plasmas

The widely used Child-Langmuir law for sheath thickness evaluation in semi-infinite collisionless plasmas makes the assumptions of quasi-neutrality (n_e=n_i) and zero electric field intensity E=0 at the sheath edge, as well as applying the Bohm criterion for ions entering the sheath. However, through a whole region fluid model, Poisson´s equation has been solved numerically for the steady-state solution through the sheath and presheath without these assumptions. With the sheath edge defined, as in the Child-Langmuir law, at the place where the ion velocity is equal to the Bohm velocity, the sheath thickness of a bounded collisionless or weakly collisional plasma has been found with this model in some cases to be much larger than that obtained with the Child-Langmuir Law. The sheath thickness discrepancy is significant under conditions found in low pressure high density plasma (HDP) tools for plasma processing. Results presented indicate that the sheath thickness is very sensitive to the electric field and space charge density at the sheath edge. The electric field and space charge density can be successfully estimated by an intermediate scale matching method, and are used to derive a modified expression for the potential in the sheath that can be solved numerically for sheath thickness. With these results, the matching problem, arising when sheath and plasma are modeled separately, can be overcome