Minimal domain size necessary to simulate the field enhancement factor numerically with specified precision

In the literature about field emission, finite elements and finite differences techniques are being increasingly employed to understand the local field enhancement factor (FEF) via numerical simulations. In theoretical analyses, it is usual to consider the emitter as isolated, i.e., a single tip field emitter infinitely far from any physical boundary, except the substrate. However, simulation domains must be finite and the simulation boundaries influence the electrostatic potential distribution. In either finite elements or finite differences techniques, there is a systematic error ( ϵ) in the FEF caused by the finite size of the simulation domain. It is attempting to oversize the domain to avoid any influence from the boundaries; however, the computation might become memory and time consuming, especially in full three dimensional analyses. In this work, we provide the minimum width and height of the simulation domain necessary to evaluate the FEF with ϵ being the desired tolerance. The minimum width ( A) and the height ( B) are given relative to the height of the emitter ( h), that is, ( A / h ) min × ( B / h ) min necessary to simulate isolated emitters on a substrate. The authors also provide the ( B / h ) min to simulate arrays and the ( A / h ) min to simulate an emitter between an anode–cathode planar capacitor. At last, they present the formulae to obtain the minimal domain size to simulate clusters of emitters with precision ϵ tol. The formulae account for ellipsoidal emitters and hemisphere on cylindrical posts. In the latter case, where an analytical solution is not known at present, the results are expected to produce an unprecedented numerical accuracy in the corresponding local FEF.