Theory of electron emission in high fields from atomically sharp emitters: Validity of the Fowler–Nordheim equation

Abstract

Field emission from metallic emitters is generally described by the Fowler–Nordheim (FN) theory, which is based on a planar model of the tip with a classical image correction. Within the free-electron model and the Wentzel–Kramers–Brillouin approximation, the planar tip model leads to the well-known FN equation. The form of this equation predicts that a plot of log J/F2 versus 1/F, where J is the current density and F is the field, should be a straight line within the rather narrow region of field strengths of typical field emission experiments, 3–5 V/nm. This has been experimentally confirmed for conventional emitters, (i.e., electrolytically etched tips with radii ≳50 nm). However, field emitters fabricated with today’s new techniques are much sharper with radii of curvature of the order of nanometers or even the size of a single atom. Hence, the local geometry of the tip may become an important factor in the electron emission process. To investigate the effects of the shape and/or size on emission, the authors, in a recent series of papers, studied the dependence of the current–voltage characteristics on the local geometry of pointed emitters. It was found that the calculated results, plotted as log J/V2 versus 1/V, do not exhibit the straight line behavior predicted by the FN theory. In addition, there is a dramatic increase in the tunneling current for a fixed external bias, V, relative to the FN result for a planar model of the tip with the same bias voltage. Using the exact current integral additional results have been obtained exhibiting the effects of emitter curvature on electron emission in high fields and temperatures. These results continue to differ with the predictions of the β-modified FN equation. Therefore, the adequacy of a β factor in the conventional planar model FN equation to account for emitter curvature is examined. It is demonstrated that even the use of β-modified FN equation is not valid when applied to sharp emitters (rt≤10 nm) and will lead to spurious results when extracting information such as field values or emitting area from experimental FN curves. The explanation for this is discussed.