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
$\gamma s$
.