Transitional behaviors of fractional dimensional models of field-induced and space-charge limited emission from rough surfaces

Electron emission is an important physical phenomenon in both vacuum and solid-state devices. Previously, the effects of surface roughness or interface irregularity have been incorporated in the fractional Fowler-Nordheim (FNα) equation for field emission, the fractional Child-Langmuir (CLα) law for space-charge limited emission (SCLE) in vacuum, and the fractional Mott-Gurney (MGα) law for SCLE in the presence of carrier collisions using a fractional dimensional approach, where the fractional dimensional parameter ( 0 < α ≤ 1 ) defines the degree of roughness at electrode surface or irregularity of the gap medium. Here, the transitions between FNα, CLα, and MGα regimes have been investigated for electron emission from rough surfaces. It is shown that the fractional second-order or third-order nexuses, where at least two or three of the FNα, CLα, and MGα regimes converge, respectively, have a strong dependence on the surface conditions as captured by the parameter α. This work provides an important insight into the mechanism of electron emission in electron emitters where surface roughness is inevitable due to practical experimental fabrication and/or emitter surface degradation over time. The model shall serve as a versatile analytical tool to characterize the roughness by fitting the model to the experimental measurements.