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

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 $\left\{γ_j\right\}$, which are randomly assigned to each distinct emitter of a conducting CFE surface, from a discrete probability distribution $ρ{(γ_{j})}$, with $j=1,2$. It is assumed that $ρ{(γ_{1})} < ρ{(γ_{2})}$, and that $γ_{1} > γ_{2}$. 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/$μ$m and are analyzed by taking into account the apparent formal area efficiency, the distribution $ρ$, 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.