Local Electric Field Variation in the Field Emission Process

Tunnelling probabilities in field emission processes are usually calculated assuming flat emitting surfaces. This corresponds to consider the electric field as a constant along the potential barrier. Previous work have shown the importance of taking into account image effects but neglecting local field variations. Even when the local field has been taken into account, a proper three dimensional calculation of the transmission coefficient was not performed. We discuss previous calculation models, their limitations and conclusions, comparing them with our model results. We have determined the transmission coefficient using a 3-D numerical integration process. Considering a completely planar surface and another one with an hemispherical protrusion superimposed on it, we could compare the relative influence of either the local field variation along the potential barrier and image effects. These results lead to the conclusion that corrections due to the local field variation are of a magnitude that deserves to be considered, together as those resulting from the image potential.