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

Field emission from metallic emitters is generally described by the Fowler-Nordheim [F-N] theory, which is based on a planar model of the tip with a classical image correction. Within the free electron model and the WKB approximation, the planar tip model leads to the well-known Fowler-Nordheim equation, which predicts that a plot of log J/F² versus 1/F, where J is the current density and F, the field, should be a straight line within the narrow range of field strengths of typical field emission experiments, 3 - 5V/nm. This has been experimentally confirmed for conventional emitters, (i.e., electrolytically etched tips with radii {greater-than or approximate}50 nm). Field emitters fabricated with today's new techniques are much sharper with radii of curvature of the order of nm's 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 vs. 1/V, do not exhibit the straight line behavior predicted by the Fowler-Nordheim theory. In addition, there is a dramatic increase in the tunneling current for a fixed external bias, V, relative to the Fowler-Nordheim 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 field electron energy distributions and on electron emission in high fields and temperatures. These results continue to differ with the predictions of the Fowler-Nordheim equation for the same emitter models. Therefore, the adequacy of a β-factor in the conventional planar model Fowler-Nordheim equation to account for emitter curvature is examined. It is demonstrated that even a β-modified Fowler-Nordheim equation is not valid when applied to sharp emitters (r_t{less-than or approximate} 10nm) and will lead to spurious results when extracting information such as work function, field values or emitting area from experimental F-N curves. The explanation for this is discussed, and an approximate analytic expression for the J(V) characteristics of a prototype sharp emitter is derived which exhibits explicitly the dependence of the current density on field, tip geometry andmaterial parameters.