Simulation of the Field Emission from a Hemisphere-on-Post Nanowire: Revision of the Fowler–Nordheim Formula in Terms of Nonuniformity and Resistance

Abstract

The electric field on a hemisphere-on-post nanowire is numerically calculated using the finite element method (FEM). The FEM calculation results show that the field is sufficiently strong for extracting a significant field emission current only in a small area at the top of the hemisphere, while the contribution to the field emission from the other part of the hemisphere and the flank side of the cylinder is negligible owing to the rapid drop of the electric field. Both the local current density at the top of the hemisphere ([Formula: see text]) and the average current density across the nanowire cross-section ([Formula: see text]) are calculated and the [Formula: see text]-to-[Formula: see text] ratio ([Formula: see text]) is introduced to reflect the nonuniformity of the field emission. An empirical formula with proper parameters that can best fit the simulation results is derived for describing the dependence of [Formula: see text] on the macroscopic electric field ([Formula: see text]). As a result, the [Formula: see text]–[Formula: see text] relationship is attained and the revision to the traditional Fowler–Nordheim (FN) formula caused by the nonuniformity of field distribution is found in both the pre-exponent part and the exponent part, so that the deviation of the FN plots from linearity often observed in experiments is partly accounted for. Moreover, the resistance at the emitter-substrate interface is shown to cause saturation in the field emission current and a downward bending of the FN plot in the high-field region.