Analytical proof of Schottky Conjecture for multi-stage field emitters

TL;DR

This paper provides an analytical proof of the Schottky Conjecture for multi-stage field emitters with various protrusion shapes, using conformal mapping techniques, and clarifies conditions under which the conjecture holds.

Contribution

It offers the first analytical proof of the Schottky Conjecture for multi-stage emitters with arbitrary shapes and no self-similarity requirement, extending previous numerical validations.

Findings

Schottky Conjecture holds when each protrusion is much larger than the one above.

The proof applies to rectangular, trapezoidal, and triangular protrusions.

Self-similarity between stages is not necessary for the conjecture's validity.

Abstract

Schottky Conjecture is analytically proved for multi-stage field emitters consisting on the superposition of rectangular or trapezoidal protrusions on a line under some specific limit. The case in which a triangular protrusion is present on the top of each emitter is also considered as an extension of the model. The results presented here are obtained via Schwarz-Christoffel conformal mapping and reinforce the validity of Schottky Conjecture when each protrusion is much larger than the ones above it, even when an arbitrary number of stages is considered. Moreover, it is showed that it is not necessary to require self-similarity between each of the stages in order to ensure the validity of the conjecture under the appropriate limits.