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 Ra, as well as the shape of the emitter base are still largely unknown. In this work, a universal formula for the field enhancement factor is derived for a single emitter. It depends on the ratio h/Ra and has the form γa=(2h/Ra)/[α1ln(4h/Ra)−α2], where α1 and α2 depend on the charge distribution on the emitter. Numerical results show that a simpler form γa=(2h/Ra)/[ln(4h/Ra)−α] is equally valid with α depending on the emitter-base. Thus, for the hyperboloid, conical, and ellipsoid emitters, the value of α is 0, 0.88, and 2, while for the cylindrical base, α ≃ 2.6.