Trade-off between the electrostatic efficiency and mechanical stability of two-stage field emitter structures

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 γ_C. We define the electrostatic efficiency as $ηR=(γC−γ1)/(γ3−γ1)$⁠, 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 $ηR=f(u≡λθ−n)$⁠, where $λ≡h2/h1, θ≡r1/r2$⁠, 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 γ_C 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.