Quantum limits to the electron field emission from tapered conductive sheets

Fueled by a strong demand in various practical areas, the development of electron sources has reached new heights in recent years, with respect to both intensity and quality of the produced electron beams. In this context, field-emission (FE) cathodes play a crucial role for many special applications. To obtain strong and stable FE currents with relatively small anode voltages, sharper conductive structures have been introduced that may focus the electric field on small areas of their tips up to huge values of the order of 10 V/nm. Carbon nanotubes (CNTs) are well known examples of such structures [1]. Tapered CNTs and even conical graphene nanoneedles have been considered and studied as potential FE sources [2,3]. However, ultra-sharp features present in a quantum structure may determine specific configurations for the allowed electronic energy levels in the whole structure, mostly in devices with coherent transport of electrons. For example, ultra-sharp terminations of tapered CNTs may actually restrict the allowed states for the conduction electrons. As a consequence, the electron population on such sharp features may be lower than in other, more bulky, parts of the structure. Thus, the field enhancement that normally occurs on high-curvature parts of conductive surfaces may be hindered or even compensated by the lack of conduction electrons due to local tight spatial confinement [2].