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.