A reformulated general thermal-field emission equation

Abstract

Theoretical models of thermal, field, and thermal-field emission of electrons generally require the evaluation of the Gamow, or tunneling, factor. A rapid means to do so is developed using shape factor methods for general potentials, but, in particular, for the image charge (or Schottky-Nordheim) barrier from which the Fowler-Nordheim (field emission) and Richardson-Laue-Dushman (thermal emission) equations are derived. The shape factor method provides greater accuracy than the elliptical integral functions that the electron emission equations conventionally rely on and eliminates the need to use them. The mild overprediction of the field emission current by the Fowler-Nordheim equation at very high fields is corrected. Undesired behaviors associated with prior versions of the general Thermal-Field equation in the thermal-field regime are eliminated by properly identifying the maximum of the current density integrand and accounting for its shape. The methodology to rapidly identify the current density maximum in general is demonstrated for various combinations of field, temperature, and work function, and the improvements demonstrated. The utility of the reformulated methodology for the simulation of electron beams and the ability to characterize cathode configurations of technological interest are discussed.