The role of Hurst exponent on cold field electron emission from conducting materials: from electric field distribution to Fowler-Nordheim plots

Abstract: This work considers the effects of the Hurst exponent ($H$) on the local electric field distribution and the slope of the Fowler-Nordheim (FN) plot when considering the cold field electron emission properties of rough Large-Area Conducting Field Emitter Surfaces (LACFESs). A LACFES is represented by a self-affine Weierstrass-Mandelbrot function in a given spatial direction. For $0.1 \leqslant H < 0.5$, the local electric field distribution exhibits two clear exponential regimes. Moreover, a scaling between the macroscopic current density ($J_M$) and the characteristic kernel current density ($J_{kC}$), $J_{M} \sim [J_{kC}]^{\beta_{H}}$, with an H-dependent exponent $\beta_{H} > 1$, has been found. This feature, which is less pronounced (but not absent) in the range where more smooth surfaces have been found ($0.5 \leqslant H \leqslant 0.9$), is a consequence of the dependency between the area efficiency of emission of a LACFES and the macroscopic electric field, which is often neglected in the interpretation of cold field electron emission experiments. Considering the recent developments in orthodox field emission theory, we show that the exponent $\beta_{H}$ must be considered when calculating the slope characterization parameter (SCP) and thus provides a relevant method of more precisely extracting the characteristic field enhancement factor from the slope of the FN plot.