Electrostatic effect on the characteristic field enhancement factors when two identical emitters are in close proximity

Abstract

Using numerical simulations based on finite-element techniques, this work investigates the electrostatic interaction (“mutual shielding”) between two emitters each modeled as a cylindrical post capped with a hemisphere (the “HCP” model). The interaction between two adjacent emitters has recently been investigated in [8, 11], using different models. Reference [8] used the (analytical) “floating spheres at emitter-plate potential (FSEEP)” model to study how the apex field enhancement factor (FEF) γa changed with the separation c of the sphere centers, for given emitter height l. With that model, the electron-thermodynamic effect called “mutual charge-blunting” is always the dominant influence and causes the fractional FEF change (−δ two ) (relative to large separation) to increase monotonically (γa to decrease monotonically) as separation decreases. In this work, we investigate these effects by considering the post's maximum FEF γ m , which occurs to one side of the post apex. For intermediate and large distances, our numerical results are similar to those reported in [8], namely (−δ two ) ∼ c-1 and (−δ two ) — c-3, respectively, corroborating the predictions of the FSEPP model. However, our numerical results show that, at very small emitter separations (c/l m goes through a minimum as c/l decreases further. This is qualitatively consistent with previous findings [11] made using a different (semi-analytical) approach and a differently shaped (“more conical”) emitter model. Our results suggest that this behavior is related to a change in shape of the equipotential surfaces above/near the post apexes, which is such that — as the two emitters get very close together — they tend towards behaving like a single emitter.