We have developed an algorithm for numerical solution of the Schroedinger equation in the case of complex surface potential barriers characteristic for field-emission experiments and calculated the transparency of the barriers for the case of metal tip-cathodes covered by superthin diamond coatings. A comparison has been made between the energy dependence of the transmission coefficients calculated within the classical local electrostatics approach and that accounting for non-local effects. The transparency dependence on the external electric field, film thickness, electron effective masses in the substrate and the film as well as barrier shape influence has been analyzed. The formation of a triangle-like quantum gap for electrons in the film region leads to the appearance of additional resonances in the transparency spectrum at large external fields. It has been shown, that energy of these resonances corresponds to quasistationary states of electrons in such gaps, i.e. resonant tunneling at specific energies and electric fields in the system.
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