Analytic Assessment of the Significant Emission Area and Integrated Enhancement Factor for Ellipsoidal Electron Field Emitters

A lower bound to the field emitter surface area that significantly contributes to the overall field emitter current is found for ellipsoidal and spheroidal emitters. Equations are also given for the upper bound on the spheroidal emitter´s integrated enhancement factor β_I, which is defined as the enhancement factor obtained by curve fitting field emission data to the Fowler-Nordheim equation. Both bounds are highly dependent on the applied field over common experimental field ranges. For a large range of emitter geometric aspect ratios and applied fields, a succinct empirically determined equation set is provided to determine a lower bound on the spheroidal emitter significant emission area that provides for all but 0.0001% of the total emission current. At extremely high applied fields, β_I saturates to a constant value. The saturation value is suggested as an alternative nonfield-dependent “terminal” field enhancement factor to be used in evaluating emitter performance. An alternate expression for the integrated enhancement factor obtained using the Fowler-Nordheim slope method (SM) β_I,_SM is given and is shown to differ from β_I. An equation is presented to determine under what conditions is the SM´s usage valid for spheroid emitters to obtain the “terminal” field enhancement factor.