We measure the current vs voltage (I-V) characteristics of a diodelike tunnel junction consisting of a sharp metallic tip placed at a variable distance d from a planar collector and emitting electrons via electric-field assisted emission. All curves collapse onto one single graph when I is plotted as a function of the single scaling variable Vd^{-\lambda}, d being varied from a few mm to a few nm, i.e., by about six orders of magnitude. We provide an argument that finds the exponent {\lambda} within the singular behavior inherent to the electrostatics of a sharp tip. A simulation of the tunneling barrier for a realistic tip reproduces both the scaling behavior and the small but significant deviations from scaling observed experimentally.Comment: 6 pages, 6 figures. Accepted for publication in Physical Review

In this paper, we derive analytically from first principles a generalized Fowler-Nordheim (FN) type equation that takes into account the curvature of a nanoscopic emitter and is generally applicable to any emitter shape provided that the emitter is a good conductor and no field-dependent changes in emitter geometry occur. The traditional FN equation is shown to be a limiting case of our equation in the limit of emitters of large radii of curvature R. Experimental confirmation of the validity of our equation is given by the data of three different groups. Upon applying our equation to experimental FN plots complying with the above limitations, one may deduce (i) R and (ii) standard field emission parameters-e.g. enhancement factor-with better accuracy than by using the FN equation.

We present a method for the calculation of the local electric field at the surface of a nanoscopic emitting structure. The method is here applied to carbon nanotubes (NT) where symmetry makes the application of the method easier. The NT is simulated as a cylindrical array of touching spheres, each sphere representing an atom of the tube. The electrostatic potential is written as a linear combination of the potentials produced by each of the spheres. We calculate the local electric field and the corresponding enhancement factor γ for both open and closed nanotubes. For a closed NT we find for γ a simple polynomial expression in terms of the ratio of the height h of the tube to its radius R, which for h/R<40 reduces to a frequently quoted formula of γ. For an open single-wall NT we find that γ is three times greater than that of a single-wall NT of the same h/R. As the thickness of the wall increases this difference diminishes. From these results one may deduce a possible explanation as to why in some experiments a closed NT emits more current than a corresponding open one while in other experiments the opposite holds true.

The phenomenon of low-macroscopic-field (LMF) electron emission is sometimes exhibited by thin films of non-metallic materials. Both wide-band-gap semiconductors and hopping conductors such as amorphous carbon exhibit this phenomenon, and its origin has been under discussion for some years. This paper first uses a simple theory to discuss field penetration into hopping conductors with a high density of localised states near the charge-neutrality level. This theory is physically similar to widely used existing theories, in that it assumes continuous charge distributions and uses the one-dimensional Poisson's equation. The theory suggests that little field penetration occurs, and that models of LMF emission based on uniform barrier lowering are not viable. Unexpected self-consistency difficulties are also created for some types of local-field-enhancement model. A re-assessment of the validity of this simple field-penetration theory suggests the following. Contrary to a basic assumption of this type of theory, the granularity of the screening charge distribution is a dominant feature of the physics associated with LMF emission from nanoscale-sized emission sites on thin films of hopping conductors. The authors think that the screening is carried out by discrete charges in localised states, although details are not clear. This conclusion is potentially very far-reaching. In particular, new forms of field-penetration theory may be required before definitive decisions can be made about the self-consistency of some models of local field enhancement.

We have calculated the electric field around and on the surface of an open thick-wall carbon nanotube (CNT) of height h, external radius R, and wall thickness w. To accomplish that we simulate the CNT as a vertical array of touching toroids, each of external radius R and cross section radius w/2, and then we express the problem in toroidal coordinates. From our calculations we obtain the enhancement factor γ as a function of h, R, and w. By fitting to our numerical results we obtain an empirical but simple formula for γ, which extrapolates to that of a closed CNT in the limiting case of w=R.