Characterization of sharp nanometric field-emitting electron sources down to the 4 nm apex radius limit

The analysis and characterization of field-emission (FE) devices have been performed during the last 50 years by a variety of methods like the more accurate analytical methods using curvilinear coordinates, the particle-in-cell method, and the shape factor method. The oldest and most simple to implement, which is still used today, is the tangent method based on the Murphy–Good equation, an equation applicable in principle to planar emitting surfaces and in practice to low curvature surfaces. However, many modern emitters have apex radii of curvature R of 10–20 nm, and yet, their analysis and characterization are still performed, in many circumstances, the same way. In this paper, we analyze such nanometric FE devices by extending the tangent method using as an emission law the generalized Fowler–Nordheim equation of Kyritsakis and Xanthakis, Proc. R. Soc. A 471, 20140811 (2014), which is valid down to R = 4 nm. Then, we obtain the new slope and intercept correction factors appropriate for nanometric R. Upon examination of six FE experiments with nanometric R, we show that, if the traditional Murphy–Good equation is still employed instead of Kyritsakis and Xanthakis, Proc. R. Soc. A 471, 20140811 (2014), the errors in the enhancement factor β are only 10%–20%, but the errors in the effective emission area may be up to 400% and even up to an order of magnitude. Furthermore, we estimate the limiting curvature of emitting surfaces at which the Murphy–Good law ceases to be valid. This limit is obtained by looking not only at the emitted current density but also at the effective area as deduced by the extended tangent method for nanometric emitters and is estimated to be around R = 30 nm.