# Unexpected validity of Schottky's conjecture for two-stage field emitters: a response via Schwarz-Christoffel transformation

## TL;DR

This paper analytically investigates the electric field near two-stage field emitters using Schwarz-Christoffel mapping, demonstrating that Schottky's conjecture remains valid even when both stages are similarly sized, which was previously uncertain.

## Contribution

The authors provide an analytical proof of Schottky's conjecture for two-stage emitters with specific dimension conditions and show its validity through numerical solutions even when stages are comparable in size.

## Key findings

- Analytical proof of SC for small upper-stage dimensions
- SC remains valid for similar-sized stages based on numerical results
- Field enhancement factors are evaluated at specific emitter points

## Abstract

The electric field in the vicinity of the top of an emitter with a profile consisting of a triangular protrusion on an infinite line is analytically obtained when this system is under an external uniform electric field. The same problem is also studied when the profile features a two-stage system, consisting of a triangular protrusion centered on the top of a rectangular one on a line. These problems are approached by using a Schwarz-Christoffel conformal mapping, and the validity of Schottky's conjecture (SC) is discussed. The authors provide an analytical proof of SC when the dimensions of the upper-stage structure are much smaller than those of the lower-stage structure, for large enough aspect ratios and considering that the field enhancement factor (FEF) of the rectangular structure is evaluated on the center of the top of the structure, while the FEF of the triangular stage is evaluated near the upper corner of the protrusion. The numerical solution of our exact equations shows that SC may remain valid even when both stages feature dimensions of the same order of magnitude, reinforcing the validity of SC for multistage field emitters.