Uniform macroscopic current in concentric field emitter architectures

Concentric-ring field emitter architectures are relevant to vacuum micro- and nanoelectronic applications. In view of field electron emission applications for these ring-emitters, a central question is to determine under what geometric conditions these structures, when subjected to an external electrostatic field Eappl, can sustain a uniform macroscopic current despite the electrostatic depolarization that couples neighboring rings. We address this question by computing stationary configurations for a cathode composed of concentric emitting rings of width ω and spacing c, with innermost height h1⋆ and each kth ring located at radius ρk. A robust finite-element-based numerical procedure is developed to obtain the optimized heights hk⋆ that sustain a prescribed uniform arc current. Using the dimensionless spacing s=c/h1⋆ and the inner-ring aspect ratio 2h1⋆/ω, we analyze the normalized profiles hk⋆/h1⋆ as functions of ρk/h1⋆. Our results reveal a systematic curvature transition governed by the interplay between the isolated-emitter characteristic field and the depolarization correction: for s≲1, depolarization decreases rapidly with the radial position ρk and all profiles remain concave down, whereas for s≳1, the slower radial decay of depolarization generates an inner convex region followed by an outer concave region. This geometric transition sets the nontrivial dependence of Eappl on spacing and aspect ratio, showing that uniform emission cannot be inferred from geometry alone without explicit optimization. For moderate inner-ring aspect ratios (2h1⋆/ω=15−20), Eappl(s) exhibits a maximum at sm≃0.35, increasing the required field by 4%–5% relative to the closely spaced regime and by about 35% relative to large spacings where depolarization becomes negligible. For taller inner rings (2h1⋆/ω≳40), curvature dominates and Eappl(s) is strictly monotonic over the explored spacing range. Therefore, we identify (2h1⋆/ω,s) as the key geometric parameter pair governing uniform-current operation and provide quantitative criteria for selecting the macroscopic field Eappl required to deliver specified macroscopic currents in concentric cold-cathode architectures.