Development of Fowler-Nordheim theory for a spherical field emitter

Fowler-Nordheim (F-N) theory has been extended to use a potential distribution approximating that outside a hemisphere supported on a shank, instead of planar field as widely used hitherto. The extended theory includes an effective angle of emission and a supply factor (relative to free-electron supply) that can be calculated when other parameters are known. If single values of the F-N slope and intercept are available, compatible sets of parameters including emitter radius, surface field, solid angle of emission, and supply factor can be deduced. Specific values can be estimated for these parameters when the change of slope over the experimental range is known with sufficient accuracy.

Figure 1

Emitter outline and equipotentials (relative to Fermi level) in the barrier region including image potential, as given by (4) with $x=x_0\cos \left(\theta \right)$, $x_0=0.5$, $\kappa =0.288$, $\varphi =5\mathrm{eV}$ $({\mathrm{F}}_0=10\mathrm{V}{\mathrm{nm}}^{-1},\mathrm{a}=1\mathrm{nm})$.

Figure 2

Functions (a) $[f_1(x,\kappa )+\kappa ]$ and (b) $w_1(x,\kappa )$, both for five values of $\kappa$.

Figure 3

$S_2$ as a function of $V_a^{-1}$ and $\kappa$ for condition $S_2=-9000\mathrm{V}$ at $V_a^{-1}=0.0016$.