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 influences the electrostatic potential distribution. In either finite elements or finite differences techniques, there is a systematic error ( ) 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 at the desired tolerance. The minimum width (A) and 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. We 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, we present the formulae to obtain the minimal domain size to simulate clusters of emitters with precision tol . Our formulae account for ellipsoidal emitters and hemisphere on cylindrical posts. In the latter case, where an analytical solution is not known at present, our results are expected to produce an unprecedented numerical accuracy in the corresponding local FEF.

This work presents an accurate numerical study of the electrostatics of a system formed by individual nanostructures mounted on support substrate tips, which provides a theoretical prototype for applications in field electron emission or for the construction of tips in probe microscopy that requires high resolution. The aim is describe the conditions to produce structures mechanically robust with desirable field enhancement factor (FEF). We modeled a substrate tip with a height h1, radius r1 and characteristic FEF γ1, and a top nanostructure with a height h2, radius r2 < r1 and FEF γ2, for both hemispheres on post-like structures. The nanostructure mounted on the support substrate tip then has a characteristic FEF, γC . Defining the relative difference ηR = (γC − γ1)/(γ3 − γ1), where γ3 corresponds to the reference FEF for a hemisphere of the post structure with a radius r3 = r2 and height h3 = h1 + h2, our results show, from a numerical solution of Laplace's equation using a finite element scheme, a scaling ηR = f (u ≡ λθ −1 ), where λ ≡ h2/h1 and θ = r1/r2. Given a characteristic variable uc, for u uc, we found a power law ηR ∼ u κ , with κ ≈ 0.55. For u uc, ηR → 1, which led to conditions where γC → γ3. As a consequence of scale invariance, it is possible to derive a simple expression for γC and to predict the conditions needed to produce related systems with a desirable FEF that are robust owing to the presence of the substrate tip. Finally, we discuss the validity of Schottky's conjecture (SC) for these systems, showing that, while to obey SC is indicative of scale invariance, the opposite is not necessarily true. This result suggests that a careful analysis must be performed before attributing SC as an origin of giant FEF in experiments.Producing nanostructures that allow one to amplify the applied electric field in their vicinity and which are mechanically stable remains an engineering challenge. This can be observed already a long time ago in the pioneer work by Gomer who discuss a method for growing metal whiskers in a modified field emission tube [1]. In fact, the issue of mechanical stability requires a solution for the degradation and failure of nanostructures that occurs during field electron emission at or near the substrate emitter contact [2] and for the self-mechanical oscillations that occur during field electron emission measurements [3,4] or from electrostatic interactions [5]. In particular, a method to study the self-oscillations of a nanostructure mounted on a macroscopic frame requires using a laser beam to excite the sample; subsequently, a second laser beam is then used to register the amplitude of vibrations at a certain point from the object [6].Applications of these nanostructures mounted on tip devices include carbon nanotubes (CNTs) mounted on a support tip, which can be used as an electron source in a high-resolution electron beam. The latter acquires properties such as a stable emitted current and high brightness [7]. Moreover, due to screening effects [8], there is a tendency to...

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, [Formula: see text], and a pair, [Formula: see text]. We show that small relative errors in [Formula: see text] (i.e. [Formula: see text]), due to the finite domain size, are sufficient to alter the functional dependence [Formula: see text], where c is the distance from the emitters in the pair. We show that [Formula: see text] obeys a recently proposed power law decay (Forbes 2016 J. Appl. Phys. 120 054302), at sufficiently large distances in the limit of infinite domain size ([Formula: see text], say), which is not observed when using a long time established exponential decay (Bonard et al 2001 Adv. Mater. 13 184) or a more sophisticated fitting formula proposed recently by Harris et al (2015 AIP Adv. 5 087182). We show that the inverse-third power law functional dependence is respected for various systems like infinity arrays and small clusters of emitters with different shapes. Thus, [Formula: see text], with m  =  3, is suggested to be a universal signature of the charge-blunting effect in small clusters or arrays, at sufficient large distances between emitters with any shape. These results improve the physical understanding of the field electron emission theory to accurately characterize emitters in small clusters or arrays.