Discrepancies and universality in the fractional reduction of the apex-field enhancement factor considering small clusters of field emitters

Abstract

Numerical simulations are important when assessing the many characteristics of field emission related phenomena. In small simulation domains, the electrostatic effect from the boundaries is known to influence the calculated apex field enhancement factor (FEF) of the emitter, but no established dependence has been reported at present. In this work, we report the dependence of the lateral size, L, and the height, H, of the simulation domain on the apex-FEF of a single conducting ellipsoidal emitter. Firstly, we analyze the error, ε, in the calculation of the apex-FEF as a function of H and L. Importantly, our results show that the effects of H and L on ε are scale invariant, allowing one to predict ε for ratios L/h and H/h, where h is the height of the emitter. Next, we analyze the fractional change of the apex-FEF, δ, from a single emitter, γ1, and a pair, γ2. We show that small relative errors in γ1 (i.e., ε ≈ 0.5%), due to the finite domain size, are sufficient to alter the functional dependence δ(c), where c is the distance from the emitters in the pair. We show that δ(c) obeys a recently proposed power law decay [R. G. Forbes, J. of Appl. Phys. 120, 054302 (2016)], at sufficient large distances in the limit of infinite domain size (ε = 0, say), in contrast to a long time established exponential decay [J.-M. Bonard et al. Advanced Materials 13, 184 (2001)]. We also shown that this functional dependence is respected for various systems which includes infinity arrays and small clusters of emitters with different shapes. Thus, power law functional dependence, −δ ∼ c −m , with m = 3, is suggested to be a universal signature of the charge-blunting (CB) effect in small clusters or arrays, at sufficient large distances between emitters with any shape. These results explain the origin of the discrepancies in the literature and improves the scientific understanding of the field electron emission theory, for accurate characterization of emitters in small clusters or arrays. Finally, our results reinforce that the consequences of CB for a small cluster of emitters are also expected for infinity arrays.