General scaling laws of space charge effects in field emission

The suppression of field electron emission by space charge (SC) is described for planar diodes by the one dimensional (1D) theory. Here we generalize in 3D by deriving the scaling behavior of field suppression in the weak SC regime. We propose a corrected equivalent planar diode model, which describes the SC effects for any geometry in terms of the 1D theory, utilizing a correction factor that adjusts the scaling characteristics of the field suppression. We validate our theory by comparing it to both numerical calculations and existing experimental data, either of which can be used to obtain the correction factor.