A universal formula for the field enhancement factor

The field enhancement factor (FEF) is an important quantity in field emission calculations since the tunneling electron current depends very sensitively on its magnitude. The exact dependence of FEF on the emitter height $h$, the radius of curvature at the apex $R_a$, as well as the shape of the emitter base is still largely unknown. In this work, a universal formula for the field enhancement factor is derived. It depends on the ratio $h/R_a$ and has the form $\gamma_a = (2h/R_a)/[\alpha_1 \log(4h/R_a) - \alpha_2 ]$ where $\alpha_1$, $\alpha_2$ depend on the charge distribution on the emitter. Numerical results show that a simpler form $\gamma_a = (2h/R_a)/[\log(4h/R_a) - \alpha]$ is equally valid with $\alpha$ depending on the class of emitter and indicative of the shielding by the emitter-base. For the hyperboloid, conical and ellipsoid emitters, the value of $\alpha$ is $0, 0.88$ and $2$ while for the cylindrical base where shielding is minimum, $\alpha \simeq 2.6$.