Ring-shaped images as a result of non-uniform field emission from capped carbon nanotubes

Ring-shaped images may appear under high field emission conditions for very thin carbon nanotubes (CNTs). Such image patterns cannot be explained by the corresponding field enhancement only. A model for electron field emission from the CNT is developed. The model refers to a capped nanotube (with cylindrical body and hemispherical cap). It is assumed that for high emission currents/high local temperatures, part of the electrons behave as quasi-free. As a result, the spatial confinement quantization of their states appears. The Schrodinger equation for the single electron can be solved separately on the cylindrical and spherical parts of the structure and the corresponding solutions can be connected smoothly at the circular intersection of the two regions. Many electronic states that are possible on the two regions separately turn out to be forbidden for the capped nanotube. The selection of the possible electronic states under the aforementioned complex conditions is determined by the geometric parameters of the tube, namely the ratio between its length and diameter. The occupation of the allowed one- electron states is considered as governed by the usual Fermi statistics. Together with the quantum probability of finding an electron in some specified area of the surface, this gives the electron distribution on the tube, which is one of the key factors determining the electron field emission from the CNT. Another key factor is the applied extraction field. The extraction field has been numerically computed using Simion nanotube-on-post diode configuration.