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

TL;DR

This study examines the validity of the cosine law for electric field variation on curved field-emitter tips in the presence of space charge, finding it remains accurate within 3% deviation in weak space charge regimes.

Contribution

It demonstrates that the cosine law can be reliably used in PIC simulations for curved emitters under weak space charge conditions, extending its applicability beyond flat surfaces.

Findings

Cosine law deviation is less than 3% for E_P/E_L ≥ 0.9.

Linear relation between field ratio and normalized current observed.

Field enhancement factor may need inclusion in curved emitter models.

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 is employed for this study. It is shown that if $E_P$ and $E_L$ be the field at the apex of tip with and without space charge respectively, then for $\vartheta=E_P/E_L \geq 0.9$, 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 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 $\vartheta$ and normalized current $\zeta$ for curved emitters in this regime is also investigated. A linear relation, $\vartheta=1 - \delta \zeta$ (where $\delta$ is a constant), similar to that obtained theoretically for flat emitting surfaces is observed but the value of $\delta$ indicates that the extension of the theory for curved emitters may require incorporation of the field enhancement factor.