The dimensional effect of the space charge on the self-consistent electric field at the cathode surface

This work is devoted to a theoretical study of the effect of the space charge of emitted electrons on the electric field strength near the surface of field emitter and a point microprotrusion on a field emission cathode by using a two-dimensional axisymmetric problem statement. Based on the particle-in-cell method, a model has been developed and self-consistent calculations of the electric field strength and field emission characteristics has been performed for the range of emitter tip radius from ~10^-4 to ~10^-7 cm. For the geometry under investigation it has been shown that the space-charge screening of the external field is substantially less pronounced for emitters whose tip radii are comparable to the size of the region where the space charge is mostly localized. As for emitters having nanometer tip radii, their current-voltage characteristics remain linear in the Fowler-Nordheim coordinates up to field emission current densities of ~10⁹ A/cm².