Tradeoff between proximity and aspect-ratio in optimizing the field enhancement factor of large area field emitters

The apex-field enhancement factor (aFEF) is regarded as a good parameter to characterize field electron emission (FE) devices. It quantifies how much the emitter sharp tip locally magnifies the applied external electrostatic field and enhances its FE properties. In a single tip form, the aFEF increases with the emitter's aspect ratio. However, when in a cluster or an array, the aFEF decreases due to depolarization fields from neighboring emitters. Hence, in large arrays, there is a tradeoff between the height and density to maximize its aFEF. There are experimental and theoretical works that analyze FE from carbon nanotube (CNT) arrays with fixed spacing $(c)$, fixed radii $(r)$, and varying height $(h)$. Those works discuss the existence of an aspect-ratio $(h/r)$ for which a maximum FEF is achieved. Here, we found for an infinite square array of identical hemisphere-on-cylindrical-post emitters a universal behavior, in which $\gamma$ scales as $\gamma\sim\gamma s(h/c)^{0.84}$, valid for aspect-ratios $h/r\gtrsim 50$ and $h\lesssim 0.7c$. For $h \gtrsim 0.7c$, the aFEF simply saturate at $\gamma s$.