Nanostructures mechanically stable with desirable characteristic field enhancement factor: a response from scale invariance in electrostatics

This work presents an accurate numerical study of the electrostatics of systems formed by individual nanostructure mounted on support substrate tip, a theoretical prototype for applications in field electron emission or for construction of tips in probe microscopy requiring high resolution. We modeled substrate tip with height $h_1$, radius $r_{1}$ and characteristic field enhancement factor (FEF) $\gamma_1$, and the top nanostructure with height $h_2$, radius $r_{2}<r_{1}$ and FEF $\gamma_2$, both hemisphere on post-like structures. Then, nanostructure mounted on support substrate tip has characteristic FEF, $\gamma_{C}$. Defining the relative difference $\eta_R \equiv (\gamma_{C} - \gamma_{1})/ (\gamma_{3} - \gamma_{1})$, where $\gamma_{3}$ corresponds to reference FEF for a hemisphere on post structure with radius $r_3=r_2$ and height $h_3=h_1 + h_2$, our results suggest, from numerical solution of Laplace's equation using a finite element scheme, a scaling $\eta_R = f(u\equiv\lambda\theta^{-1})$, where $\lambda\equiv h_{2}/h_1$ and $\theta \equiv r_1/r_2$. Given a characteristic variable $u_{c}$, we found, for $u \ll u_{c}$, a power law $\eta_{R} \sim u^{\kappa}$ with $\kappa \approx 0.55$. For $u \gg u_{c}$, $\eta_{R} \approx 1$ providing conditions where $\gamma_C \rightarrow \gamma_3$. As a consequence of scaling invariance, it's possible to derive a simple expression for $\gamma_C$, being possible to predict conditions to produce related systems with a desirable FEF and that, at the same time, are mechanically stable by presence of substrate tip. Finally, we also discuss the validity of Schottky's conjecture (SC) for these systems showing that, while to obey SC is a indicative of scale invariance, the opposite is not necessarily satisfied. This suggest that a careful analysis must be done before attribute the SC as a origin of giant FEF in experiments.