具有理想特征场增强因子的机械稳定的纳米结构：静电学中尺度不变性的响应
Nanostructures mechanically stable with desirable characteristic field enhancement factor: a response from scale invariance in electrostatics
摘要：
这项工作提出了一个精确的数值研究的静电系统由安装在支持衬底尖端，理论原型的应用程序在场电子发射或建设的尖端在探针显微镜需要高分辨率的单个纳米结构。我们用高度$h_1$、半径$r_{1}$和特征场增强因子（FEF）$γ_1$对衬底尖端进行建模，并且用高度$h_2$、半径$r_{2}<r_{1}$和FEF $γ_2$对顶部纳米结构进行建模，两个半球都在柱状结构上。然后，安装在支撑衬底尖端上的纳米结构具有特征FEF、$γ_{C}$。定义相对差$η_R \equiv (γ_{C} - γ_{1})/ (γ_{3} - γ_{1})$，其中$γ_{3}$对应于具有半径$r_3=r_2$和高度$h_3=h_1 + h_2$的柱结构上的半球的参考FEF，我们的结果表明，从使用有限元方案的拉普拉斯方程的数值解，缩放$η_R = f(u\equivλθ^{-1})$，其中$λ\equiv h_{2}/h_1$和$θ\equiv r_1/r_2$。给定一个特征变量$u_{c}$，我们发现，对于$u \ll u_{c}$，一个幂律$η_{R} \sim u^κ$与$κ\approx 0.55$。对于$u \gg u_{c}$，$η_{R} \approx 1$提供了$γ_C \rightarrow γ_3$ .作为缩放不变性的结果，可以导出$γ_C$的简单表达式，可以预测产生具有期望的FEF的相关系统的条件，并且同时，通过存在衬底尖端而机械稳定。最后，我们还讨论了这些系统的肖特基猜想（SC）的有效性表明，虽然遵守SC是一个指标的规模不变性，相反的是不一定满意。这表明，在实验中将SC作为巨型FEF的起源之前，必须进行仔细的分析。
英文摘要：
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) $γ_1$, and the top nanostructure with height $h_2$, radius $r_{2}<r_{1}$ and FEF $γ_2$, both hemisphere on post-like structures. Then, nanostructure mounted on support substrate tip has characteristic FEF, $γ_{C}$. Defining the relative difference $η_R \equiv (γ_{C} - γ_{1})/ (γ_{3} - γ_{1})$, where $γ_{3}$ 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 $η_R = f(u\equivλθ^{-1})$, where $λ\equiv h_{2}/h_1$ and $θ\equiv r_1/r_2$. Given a characteristic variable $u_{c}$, we found, for $u \ll u_{c}$, a power law $η_{R} \sim u^κ$ with $κ\approx 0.55$. For $u \gg u_{c}$, $η_{R} \approx 1$ providing conditions where $γ_C \rightarrow γ_3$. As a consequence of scaling invariance, it's possible to derive a simple expression for $γ_C$, 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.
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