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

Abstract

The apex-field enhancement factor (aFEF) is regarded as a meaningful parameter to characterize field electron emission (FE) devices. If experimentally extracted from orthodox current–voltage characteristics, this parameter reliably quantifies how much the emitter's sharp tip locally magnifies the applied external electrostatic field. Many experimental works have reported FE from arrays of carbon nanotubes with fixed spacing (c) between nearest-neighbor emitters, fixed apex-radii (r), and various height (h) and claimed the existence of an aspect ratio (h/r) at which a local maximum effective aFEF is achieved. Hereafter, it is shown that those results are not consistent with simulations using basic electrostatics for both finite or infinite regular square arrays of emitters. Quite interestingly, our results show that the aFEF (γa), for an emitter in an infinity regular array, simply saturates at γs for h≳0.7c. Additionally, we found a universal behavior in which γa scales as γa∼γs(h/c)0.84, when h≲0.7c, for h/r≳50. These results provide a practical rule for the design of large arrays of field emitters, which can be used to build FE nanoelectromechanical resonators with both mechanical strength and reduced Joule losses.