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

Abstract

This review of the quantitative electrostatics of field emitters, covering analytical, numerical and “fitted formula” approaches, is thought to be 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 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 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 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” 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.

Field emitter electrostatics: efficient improved simulation technique for highly precise calculation of field enhancement factors

“…The larger the lateral dimensions of the simulation box, then the smaller will be the depolarization effects and the smaller will be the inaccuracy. A similar argument (but not quite the same) applies to the simulation-box height (see [19]). Large simulation boxes require additional computing resource, so there is scope for a trade-off between accuracy and box dimensions.…”

“…Note that in some geometries, where no known analytical solution exists, it is necessary to replace the "analytical" value by a numerically calculated "highly precise" value. Also, note that the convention being used here differs in places from that used in our recent review [19], and in these cases the change results in a change in the sign of num . The revised definition here is more conventional, and thus may be clearer.…”

“…A macroscopic ES field is a field associated with the overall geometry of the system under discussion. There are several different kinds of macroscopic field (see [19]). In this article we are interested in the macroscopic ES field between the two parallel plates, in the absence of the post and in the absence of any effects that would be induced by its presence.…”

“…Note that, as compared with some of our previous papers, our terminology and notation here have become more precise. The terms "plate field" and "plate FEF" (and related notation) have replaced the terms "macroscopic field" and "macroscopic FEF" used in older papers, and the designation "macroscopic" has now been given a wider meaning (see [19]).…”

All field emission experiments are noisy, {\ldots} are any meaningful?

“…Just to have a rough idea of the size of the tip, we can use the Gomer formula 36 β = 1/5r (see also ref. 37 for an excellent review on the subject) where r is the radius of curvature of the emitter. This radius at the apex is then equal to 126.8 ±0.2nm.…”