Absence of Space-Charge-Limited Current in Unconventional Field Emission

Abstract

For field emission (FE), it is widely expected that its emitting current density $J$ will become space-charge-limited current (SCLC) within a gap spacing $D$ biased at sufficiently large voltage $V$. In this paper, we reveal a peculiar finding in which this expected two-stage transition (from FE to SCLC) is no longer valid for FE not obeying the traditional Fowler-Nordheim (FN) law. By employing a generalized FN scaling of $\mathrm{ln}(J/{E}_{s}^{k})\ensuremath{\propto}\ensuremath{-}1/{E}_{s}$, where ${E}_{s}$ is the surface electric field at the cathode, we show the existence of a critical exponent ${k}_{c}\ensuremath{\equiv}3/2$ where unusual behaviors occur for $k<{k}_{c}$: (a) only FE at small $D$ (no transition to SCLC possible) and (b) three-stage transition from FE first to SCLC then back to FE at large $D$. For $k>{k}_{c}$, the conventional two-stage transition from FE to SCLC will always occur for all $D$, including the classical case of $k$ = 2. Using various unconventional FE models with $k\ensuremath{\ne}\phantom{\rule{0.2em}{0ex}}2$, we specifically demonstrate these peculiar transitions. Under a normalized model, our findings uncover the rich interplay between the source-limited FE and bulk-limited SCLC over a wide range of operating conditions.