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

For field emission (FE), it is widely expected that its emitting current density $J$ will become space-charge-limited current (SCLC) due the built-up of charge in-transit 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 In this paper, we reveal a peculiar finding in which this expected two-stage transition (from FE to SCLC) is no longer valid for FE not obeying the traditional Fowler-Nordheim (FN) law. %Such effect arises when the non-FN based emitters fails to inject sufficient %charge current at high voltage to sustain the SCLC. By employing a generalized FN scaling of $\ln\left(J/V^k\right) \propto - 1/V$, we show the existence of a \emph{critical exponent} $k_c \equiv 3/2$ where unusual behaviours occur for $k < k_c$: (a) Only FE at small $D$ (no transition to SCLC even at infinitely large $V$), and (b) Three-stage transition from FE first to SCLC then back to FE at large $D$. For any $k > k_c$, the conventional two-stage transition from FE to SCLC will always occur for all $D$, which also includes the conventional FN law at $k$ = 2. Using various unconventional FE models with $k \neq 2$, 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.