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

We study the transmission coefficient of one-dimensional barriers that are relevant to field-emission problems. We compare, in particular, the results provided by the simple Jeffreys-Wentzel-Kramers-Brillouin(JWKB) approximation, the continued-fraction technique and the transfer-matrix methodology for the electronic transmission through square, triangular and Schottky-Nordheim barriers (the Schottky-Nordheim barrier is often used in models of field emission from flat metals). For conditions that are typical of field emission (Fermi energy of 10eV, work function of 4.5eV and field strength of 5Vnm ), it is shown that the simple JWKB approximation must be completed by an effective prefactor P in order to match the exact quantum-mechanical result. This prefactor takes typical values around 3.4 for square barriers, 1.8 for triangular barriers and 0.84 for the Schottky-Nordheim barrier. For fields F between 1 and 10Vnm and for work functions between 1 and 5eV, the prefactor P to consider in the case of the Schottky-Nordheim barrier actually ranges between 0.28 and 0.98. This study hence demonstrates that the Fowler-Nordheim equation (in its standard form that accounts for the image interaction and that actually relies on the simple JWKB approximation) overestimates the current emitted from a flat metal by a factor that may be of the order of 2-3 for the conditions considered in this work. The study thus confirms Forbes's opinion that this prefactor should be reintegrated in field-emission theories.