Quantum filtering of electron emission from ultrasharp tips

This paper derives an exact fully three-dimensional wave-function solution and presents representative calculations for the problem of an electron confined in an atomically sharp three-dimensional tip. The presented solution has important implications in technologies including self-collimated electron sources, ultrafine electron-beam lithography, electron holography, and cold cathode radio-frequency generation. The tip is modeled in the prolate-spheroidal coordinate system because the principal characteristics of an ultrasharp tip, including cone angle and tip radius, are naturally represented. This quantum-mechanical analysis corroborates several experimental observations on atomically sharp tips, including unexpectedly narrow energy bandwidth for the emitted electrons and pronounced collimation, or self-focusing, of the emitted electron beam. The three-dimensional analysis developed in this paper indicates that these experimental phenomena are attributable to a ``quantum filtering'' effect that arises in the geometric constriction characteristic of ultrasharp vacuum field emitters. The results of this paper suggest that the free-electron supply function, routinely invoked in classical field-emission analysis, may be an inaccurate approximation for the electronic states of ultrasharp tips having quantum-filtering characteristics. Surface plots of the electron wave-function density for various total electron energies and tip geometries demonstrate the tip selectivity with respect to electron states.