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The electric field in the vicinity of the top of an emitter with a profileconsisting of a triangular protrusion on an infinite line is analyticallyobtained when this system is under an external uniform electric field. The sameproblem is also studied when the profile features a two-stage system,consisting of a triangular protrusion centered on the top of a rectangular oneon a line. These problems are approached by using a Schwarz-Christoffelconformal mapping, and the validity of Schottky's conjecture (SC) is discussed.The authors provide an analytical proof of SC when the dimensions of theupper-stage structure are much smaller than those of the lower-stage structure,for large enough aspect ratios and considering that the field enhancementfactor (FEF) of the rectangular structure is evaluated on the center of the topof the structure, while the FEF of the triangular stage is evaluated near theupper corner of the protrusion. The numerical solution of our exact equationsshows that SC may remain valid even when both stages feature dimensions of thesame order of magnitude, reinforcing the validity of SC for multistage fieldemitters.

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