  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.
