Approximate universality in the electric field variation on a field-emitter tip in the presence of space charge

Abstract

The electric field at the surface of a curved emitter is necessary to calculate the field emission current. For smooth parabolic emitting tips where space charge is negligible, variation of the electric field at the surface is known to follow the generalized cosine law. Here, we investigate the validity of the cosine law in the regime, where space charge due to emitted electrons is important. Particle-in-cell (PIC) simulations with an emission algorithm based on the cosine law are employed for this study. It is shown that if EP and EL be the field at the apex of tip with and without space charge, respectively, then for ϑ=EP/EL≥0.9, the average relative deviation of the electric field from the cosine law is less than 3% over the endcap. Thus, an emission scheme based on the cosine law may be used in PIC simulations of field emission of electrons from curved emitter tips in the weak space charge regime. The relation between ϑ and normalized current ζ for curved emitters in this regime is also investigated. A linear relation, ϑ=1−δζ (where δ is a constant), similar to that obtained theoretically for flat emitting surfaces is observed, but the value of δ indicates that the extension of the theory for curved emitters may require incorporation of the field enhancement factor.