A comparative study of the electron transmission through one-dimensional barriers relevantto field-emission problems

Abstract

We study the transmission coefficient of one-dimensional barriers that are relevant tofield-emission problems. We compare, in particular, the results provided by the simpleJeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation, the continued-fractiontechnique and the transfer-matrix methodology for the electronic transmission throughsquare, triangular and Schottky–Nordheim barriers (the Schottky–Nordheim barrier isoften used in models of field emission from flat metals). For conditions that are typical offield emission (Fermi energy of 10 eV, work function of 4.5 eV and field strength of5 V nm − 1), it is shown that the simple JWKB approximation must be completed by an effective prefactorPeff inorder to match the exact quantum-mechanical result. This prefactor takes typical values around 3.4for square barriers, 1.8 for triangular barriers and 0.84 for the Schottky–Nordheim barrier. For fieldsF between 1 and10 V nm − 1 and for work functionsϕ between 1 and5 eV, the prefactor Peff to consider in the case of the Schottky–Nordheim barrier actually ranges between 0.28 and0.98. This study hence demonstrates that the Fowler–Nordheim equation (in its standardform that accounts for the image interaction and that actually relies on the simple JWKBapproximation) overestimates the current emitted from a flat metal by a factor that may beof the order of 2–3 for the conditions considered in this work. The study thus confirmsForbes’s opinion that this prefactor should be reintegrated in field-emission theories.