Field emission theory for an enhanced surface potential: a model for carbon field emitters

We propose a non-JWKB-based theory of electron field emission for carbon field emitters in which, for electrons with energy in the vicinity of the order of v to the Fermi level, the effective (1/x) surface potential is strongly enhanced. The model grossly violates the WKB validity criteria and necessitates an analytic treatment of the one-dimensional Schrodinger equation, which we first obtain. We determine v (which is field-dependent) from the wavefunction matching point close to the surface. For reasonable values of the surface parameters-work function phi approximate to 2-5 eV, electron affinity chi approximate to 2phi and an empirical electron loss factor sigma approximate to 10(-3) (and with no other adjustable parameters)-the theory provides an intriguing agreement with experimental data from carbon epoxy graphite composite (PFE) and certain graphitized carbon nanotube field emitters. We speculate on the surface potential enhancement, which can be interpreted as a massive (field-induced) dielectric effect of dynamic origin. This can be related via time-dependent perturbation theory to second-order non-linear polarizability enhancements at ultraviolet similar to3000 Angstrom wavelengths near the tunnelling region. Finally some exact mathematical results are included in the appendix for future reference.