Dependence of optimal spacing on applied field in ungated field emitter arrays

In ungated field emitter arrays, the field enhancement factor β of each emitter tip is reduced below the value it would have in isolation due to the presence of adjacent emitters, an effect known as shielding or screening. Reducing the distance b between emitters increases the density of emission sites, but also reduces the emission per site, leading to the existence of an optimal spacing that maximizes the array current. Most researchers have identified that this optimal spacing is comparable to the emitter height h, although there is disagreement about the exact optimization. Here, we develop a procedure to determine the dependence of this optimal spacing on the applied electric field. It is shown that the nature of this dependence is governed by the shape of the β(b) curve, and that for typical curves, the optimal value of the emitter spacing b decreases as the applied field increases.