Field emitter electrostatics: a review with special emphasis on modern high-precision finite-element modelling.

This review of the quantitative electrostatics of field emitters, covering analytical, numerical and 'fitted formula' approaches, is thought the first of its kind in the 100 years of the subject. The review relates chiefly to situations where emitters operate in an electronically ideal manner, and zero-current electrostatics is applicable. Terminology is carefully described and is 'polarity independent', so that the review applies to both field electron and field ion emitters. It also applies more generally to charged, pointed electron-conductors-which exhibit the 'electrostatic lightning-rod effect', but are poorly discussed in general electricity and magnetism literature. Modern electron-conductor electrostatics is an application of the chemical thermodynamics and statistical mechanics of electrons. In related theory, the primary role of classical electrostatic potentials (rather than fields) becomes apparent. Space and time limitations have meant that the review cannot be comprehensive in both detail and scope. Rather, it focuses chiefly on the electrostatics of two common basic emitter forms: the needle-shaped emitters used in traditional projection technologies; and the post-shaped emitters often used in modelling large-area multi-emitter electron sources. In the post-on-plane context, we consider in detail both the electrostatics of the single post and the interaction between two identical posts that occurs as a result of electrostatic depolarization (often called 'screening' or 'shielding'). Core to the review are discussions of the 'minimum domain dimensions' method for implementing effective finite-element-method electrostatic simulations, and of the variant of this that leads to very precise estimates of dimensionless field enhancement factors (error typically less than 0.001% in simple situations where analytical comparisons exist). Brief outline discussions, and some core references, are given for each of many 'related considerations' that are relevant to the electrostatic situations, methods and results described. Many areas of field emitter electrostatics are suggested where further research and/or separate mini-reviews would probably be useful.

Properties of blade-like field emitters.

Blade-Like Field Emitters (BFE), as defined here, are emitters expanded in one direction, forming a sharp emitting edge instead of a sharp tip. These structures have four main advantages compared to their needle counterparts, i.e., they are mechanically firmer, are better electrical and thermal conductors, and provide a larger emission area. We focus on the optimization of the last of these. We evaluate the emission properties of three types of BFEs, which we short-named hSoC-blade, HCP-blade and Elli-blade. Each is built from the expansion of a hemisphere-on-a-cone (hSoC), hemisphere-on-a-cylindrical-post (HCP) and an ellipsoidal (Elli) emitter, respectively. The characteristics of the field enhancement factor, the local electrostatic field distribution on each blades' edges and their notional area (An) of emission as a function of the expansion length are described. Finally, we point out how to improve the edge of the HCP-blade to obtain the optimal profile, which yield the largest An.

Physics-based derivation of a formula for the mutual depolarization of two post-like field emitters.

Recent analyses of the apex field enhancement factor (FEF) for many forms of field emitter have revealed that the depolarization effect is more persistent with respect to the separation between the emitters than originally assumed. It has been shown that, at sufficiently large separations, the fractional reduction of the FEF decays with the inverse cube power of separation, rather than exponentially. The behavior of the fractional reduction of the FEF encompassing both the range of technological interest [Formula: see text] (c being the separation and h is the height of the emitters) and large separations ([Formula: see text]) has not been predicted by the existing formulas in field emission literature, for post-like emitters of any shape. In this work, we use first principles to derive a simple two-parameter formula for fractional reduction that can be useful for experimentalists for modeling and interpreting the FEFs for small clusters of emitters or arrays at separations of interest. For the structures tested, the agreement between numerical and analytical data is ∼1%.

Numerical analysis of the notional area in cold field electron emission from arrays.

The notional area of field emission is an important parameter to correlate characteristic current density to the emission current, linking field emission theories to experimental observations. Recently, it has been reported that the notional area of emission contributes to the high brightness of large diameter emitters. Thus, it is necessary to understand how the notional area of emission depends on physical and geometrical parameters. In this work, we carried out numerical simulations to evaluate the notional area, A n, considering cold field electron emission from a hemisphere on a cylindrical post (HCP) emitter in an array. An HCP is suitable to model classically carbon nanotubes or carbon nanofibres-like emitters. We provide the dependence of A n on a wide range of physical and geometrical parameters, namely: the separation between the HCP emitters, the aspect ratio, radius, local work function and the macroscopic emission current. We explain the behavior of A n as a function of these parameters and show in which cases A n can be considered nearly constant. Our numerical results are within the framework of the standard Fowler-Nordheim (FN) theory and can simplify the modeling of the field emission phenomenon, because it directly relates simulation predictions to the currents observable experimentally. Also, this work provides information for experimentalists that can be useful to check the validity of the Schottky-Nordheim (SN) barrier upon the elementary FN theory.

Evidence of universal inverse-third power law for the shielding-induced fractional decrease in apex field enhancement factor at large spacings: a response via accurate Laplace-type calculations.

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.

Close proximity electrostatic effect from small clusters of emitters.

Using a numerical simulation based on the finite-element technique, this work investigates the field emission properties from clusters of a few emitters at close proximity, by analyzing the properties of the maximum local field enhancement factor ([Formula: see text]) and the corresponding emission current. At short distances between the emitters, we show the existence of a nonintuitive behavior, which consists of the increasing of [Formula: see text] as the distance c between the emitters decreases. Here we investigate this phenomenon for clusters with 2, 3, 4 and 7 identical emitters and study the influence of the proximity effect in the emission current, considering the role of the aspect ratio of the individual emitters. Importantly, our results show that peripheral emitters with high aspect-ratios in large clusters can, in principle, significantly increase the emitted current as a consequence only of the close proximity electrostatic effect (CPEE). This phenomenon can be seen as a physical mechanism to produce self-oscillations of individual emitters. We discuss new insights for understanding the nature of self-oscillations in emitters based on the CPEE, including applications to nanometric oscillators.

Mechanically stable nanostructures with desirable characteristic field enhancement factors: a response from scale invariance in electrostatics.

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 to describe the conditions to produce structures mechanically robust with desirable field enhancement factor (FEF). We modeled a substrate tip with a height h 1, radius r 1 and characteristic FEF [Formula: see text], and a top nanostructure with a height h 2, radius [Formula: see text] and FEF [Formula: see text], for both hemispheres on post-like structures. The nanostructure mounted on the support substrate tip then has a characteristic FEF, [Formula: see text]. Defining the relative difference [Formula: see text], where [Formula: see text] corresponds to the reference FEF for a hemisphere of the post structure with a radius [Formula: see text] and height [Formula: see text], our results show, from a numerical solution of Laplace's equation using a finite element scheme, a scaling [Formula: see text], where [Formula: see text] and [Formula: see text]. Given a characteristic variable u c, for [Formula: see text], we found a power law [Formula: see text], with [Formula: see text]. For [Formula: see text], [Formula: see text], which led to conditions where [Formula: see text]. As a consequence of scale invariance, it is possible to derive a simple expression for [Formula: see text] 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.

Field emission from non-uniform carbon nanotube arrays.

Regular arrays of carbon nanotubes (CNTs) are frequently used in studies on field emission. However, non-uniformities are always present like dispersions in height, radius, and position. In this report, we describe the effect of these non-uniformities in the overall emission current by simulation. We show that non-uniform arrays can be modeled as a perfect array multiplied by a factor that is a function of the CNTs spacing.