The electrostatic effects and mechanical stability of systems formed of nanostructures mounted on cylindrical/conical base structures were studied numerically using the finite element method. We modeled a base structure (lower-stage structure) with a height of h1, a base radius of r1, and a characteristic field enhancement factor (FEF) of γ1. The nanostructure on top (upper-stage structure) had a height of h2, a radius of r2 < r1, an FEF of γ2, and a hemisphere-on-post shape. The resulting two-stage system had a characteristic FEF of γC. We define the electrostatic efficiency as ηR=(γCγ1)/(γ3γ1), where γ3 is the reference FEF for a hemisphere-on-post structure of radius r3 = r2 and height h3 = h1 + h2. The results suggest a scaling of ηR=f(uλθn), where λh2/h1,θr1/r2, the exponent n depends on the geometry of the lower-stage structure, and u is a scale parameter of the two-stage system that arises from the scale-invariant nature of the electrostatic effects. Regarding the mechanical stability of the two-stage system, our results show that there are characteristic λ* and θ* values that result in the maximum mechanical stability. For a given relative difference δ between γC and γ3, our results suggest λ*θ*δα, where α ≈ 0.2 for both cylindrical and conical lower-stage structures. This result provides a relation between the electrostatic efficiency and the mechanical stability, allowing one to predict the necessary conditions for two-stage structures with the maximum sturdiness for a given FEF. This study, therefore, provides theoretical guidance for field electron emission applications, for the construction of needles for high-resolution probe microscopy, and for applications that require very high brightness but low emittance.

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