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

Abstract

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 10 eV, work function of 4.5 eV and field strength of 5 V nm(-1)), it is shown that the simple JWKB approximation must be completed by an effective prefactor P(eff) 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 10 V nm(-1) and for work functions φ between 1 and 5 eV, the prefactor P(eff) 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.

Numerical testing of the Fowler–Nordheim equation for the electronic field emission from a flat metal and proposition for an improved equation

“…As shown in previous work, 14 the simple JWKB approximation overestimates this transmission by a typical factor between 1.0 and 3.5 for the conditions considered in this work ͑the highest deviations correspond to small work functions and to high electric fields F͒. As shown in previous work, 14 the simple JWKB approximation overestimates this transmission by a typical factor between 1.0 and 3.5 for the conditions considered in this work ͑the highest deviations correspond to small work functions and to high electric fields F͒.…”

“…3,7,14,18 For typical metals, we have E F = 10 eV and = 4.5 eV. This is the general framework used in our previous work and the theory presented in this section is actually valid for a threedimensional barrier.…”

“…14 Since the standard FN theory is so widely used by the field-emission community, 15 it is important to deal with any issue that may question its validity. As argued by Forbes, 13 this approximation does not provide the exact result for this transmission so that the standard Fowler-Nordheim equation should actually be completed by a prefactor in order to account for the discrepancy between the transmission provided by the simple JWKB approximation and a more exact quantum-mechanical approach.…”

Extraction of emission parameters for large-area field emitters, using a technically complete Fowler–Nordheim-type equation

“…[28,[36][37][38][39][40][41] provides a modern reformulation of SN barrier theory that aims to be easier to understand, but contains some typographical errors. In addition to those already corrected [23], eq.…”

Exact solutions for the field electron emission achieved from a flat metal using the standard Fowler–Nordheim equation with a correction factor that accounts for the electric field, the work function, and the Fermi energy of the emitter

“…13,14 Since the standard FN theory relies on this JWKB approximation, it is also necessary to include a correction factor MG in the ͑modified͒ Murphy-Good expression, J MG = MG ϫ ͓k B T / d / sin͑k B T / d͔͒ ϫ at −2 −1 F 2 exp͓−bv 3/2 / F͔, in order to match the exact result. 13,14 Since the standard FN theory relies on this JWKB approximation, it is also necessary to include a correction factor MG in the ͑modified͒ Murphy-Good expression, J MG = MG ϫ ͓k B T / d / sin͑k B T / d͔͒ ϫ at −2 −1 F 2 exp͓−bv 3/2 / F͔, in order to match the exact result.…”