On the need for a tunneling pre-factor in Fowler–Nordheim tunneling theory

Abstract

This paper argues that a tunneling prefactor should appear in expressions for the tunneling probability D relevant to cold field electron emission ͑CFE͒ and in Fowler-Nordheim ͑FN͒ type equations. Except in the case of the "ideally smooth" parabolic barrier, a prefactor is always present for barriers where D can be found by exact solution of the Schrödinger equation. A review of the Jeffreys-Wentzel-Kramers-Brillouin ͑JWKB͒ approach to solving the Schrödinger equation shows that tunneling barriers should be classified according to whether they are weak or strong and ideally smooth or not: there are four different JWKB-type formulas, depending on the nature of the barrier. CFE tunneling barriers are not ideally smooth but since the 1950s have usually been analyzed using the JWKB formulas for ideally smooth barriers. These analyses, and the standard Murphy-Good FN-type equation, seem mathematically and physically incomplete. The FN-type equations currently used to describe CFE should be revised to explicitly include a tunneling prefactor. Some implications are explored.

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

“…[1], to derive a formula [eq. (16) below] that is used to estimate the characteristic field enhancement factor for a large-area field emitter (LAFE).…”

“…However, apart from the barrier-form difference, most of FN's assumptions were also used by MG. Further, MG used a JWKB-like approximation [26,27] that does not generate the tunnelling pre-factor that should physically be present [16,18,30], although a pre-exponential correction factor (t F -2 ) relating to analytical integration over emitter electron states does appear.…”

Exact analysis of surface field reduction due to field-emitted vacuum space charge, in parallel-plane geometry, using simple dimensionless equations

“…͑32͒, as F P approaches F b , are failure of the image potential energy to correctly represent correlation-and-exchange effects, 48,49,61,62 field-induced changes in surface wave functions that lead to a change in work function, 62 breakdown, when the barrier height is small, of the simple JefferiesWentzel-Kramers-Brillouin formula used in the derivation of Eq. ͑32͒, 59,63 and/or ͑close to F P = F b ͒ increasing incidence of thermally activated electron escape over the top of the barrier. As the references indicate, some of these effects have been investigated individually; but no integrated emission theory exists for this high-field intermediate regime.…”

“…Here is the local work function of the emitting surface, a͑Х1.541 434ϫ 10 −6 A eV V −2 ͒ and b͑Х6.830 890 eV −3/2 V nm −1 ͒ are the first and second FN constants as usually defined, 47 F is a physical correction factor associated with the barrier-shape model used, 58 P F is a tunnelling prefactor, 59 and Z is a physical correction factor associated with electron supply 58 to the barrier. To make calculations practicable, we used the so-called standard FN-type equation 46,47 in which the correction factors in Eq.…”

Analytical treatment of cold field electron emission from a nanowall emitter, including quantum confinement effects

“…where P is a 'transmission prefactor' discussed in Forbes (2008b). Except in a few special cases, calculating P requires specialist mathematical/numerical techniques.…”