The consequences of dependence between the formal area efficiency and the macroscopic electric field on linearity behavior in Fowler-Nordheim plots

TL;DR

This paper provides a theoretical explanation for the crossover in linearity observed in Fowler-Nordheim plots of cold field electron emission, linking it to the dependence of emitter area efficiency on the macroscopic electric field and emitter non-uniformity.

Contribution

It introduces a model connecting the field-dependent emitter area efficiency with local emitter properties to explain FN plot slope changes, supported by numerical simulations.

Findings

Reproduces two slope regimes in FN plots for high electric fields.

Shows the impact of emitter non-uniformity on FN plot behavior.

Supports theoretical model with numerical Laplace equation solutions.

Abstract

This work presents a theoretical explanation for a crossover in the linear behavior in Fowler-Nordheim (FN) plots based on cold field electron emission (CFE) experimental data. It is characterized by a clear change in the decay rate of usually single-slope FN plots, and has been reported when non-uniform nano-emitters are subject to high macroscopic electric field $F_M$. We assume that the number of emitting spots, which defines an apparent formal area efficiency of CFE surfaces, depends on the macroscopic electric field. Non-uniformity is described by local enhancement factors $\left\{\gamma_j\right\}$, which are randomly assigned to each distinct emitter of a conducting CFE surface, from a discrete probability distribution $\rho{(\gamma_{j})}$, with $j=1,2$. It is assumed that $\rho{(\gamma_{1})} < \rho{(\gamma_{2})}$, and that $\gamma_{1} > \gamma_{2}$. The local current density is evaluated by considering a usual Schottky-Nordheim barrier. The results reproduce the two distinct slope regimes in FN plots when $F_M \in $ $[2,20]$ V/$\mu$m and are analyzed by taking into account the apparent formal area efficiency, the distribution $\rho$, and the slopes in the corresponding FN plot. Finally, we remark that our results from numerical solution of Laplace's equation, for an array of conducting nano-emitters with uniform apex radii $50$ nm but different local height, supports our theoretical assumptions and could used in orthodox CFE experiments to test our predictions.