Editing Field electron emission (section)

=== Comments ===
A historical note is necessary. The idea that the Schottky–Nordheim barrier needed a correction factor, as in eq. ({{EquationNote|9}}), was introduced by Nordheim in 1928,<ref name=n28b/> but his mathematical analysis of the factor was incorrect. A new (correct) function was introduced by Burgess, [[Herbert Kroemer|Kroemer]] and Houston<ref name=BKH53>{{cite journal|doi=10.1103/PhysRev.90.515|title=Corrected Values of Fowler–Nordheim Field Emission Functions v(y) and s(y)|year=1953|last1=Burgess|first1=R. E.|last2=Houston|first2=J. M.|journal=Physical Review|volume=90|page=515|last3=Houston|first3=J.|bibcode = 1953PhRv...90..515B|issue=4 }}</ref> in 1953, and its mathematics was developed further by Murphy and Good in 1956.<ref name=MG56>{{cite journal|doi=10.1103/PhysRev.102.1464|title=Thermionic Emission, Field Emission, and the Transition Region|year=1956|last1=Murphy|first1=E. L.|last2=Good|first2=R. H.|journal=Physical Review|volume=102|pages=1464–1473|bibcode = 1956PhRv..102.1464M|issue=6 }}</ref> This corrected function, sometimes known as a "special field emission elliptic function", was expressed as a function of a mathematical variable ''y'' known as the "Nordheim parameter". Only recently (2006 to 2008) has it been realized that, mathematically, it is much better to use the variable ''ℓ''{{prime}} {{nowrap|1=( = ''y''<sup>2</sup>)}}. And only recently has it been possible to complete the definition of ''ν''(''ℓ''{{prime}}) by developing and proving the validity of an exact series expansion for this function (by starting from known special-case solutions of the Gauss [[hypergeometric differential equation]]). Also, approximation ({{EquationNote|11}}) has been found only recently. Approximation ({{EquationNote|11}}) outperforms, and will presumably eventually displace, all older approximations of equivalent complexity. These recent developments, and their implications, will probably have a significant impact on field emission research in due course.

The following summary brings these results together. For tunneling well below the top of a well-behaved barrier of reasonable height, the escape probability {{nowrap|''D''(''h'', ''F'')}} is given formally by:

{{NumBlk|:|<math>D(h, F) \approx P\exp\left[-\frac{\nu(h, F) bh^{3/2}}{F}\right], </math>|{{EquationRef|15}}}}

where {{nowrap|''ν''(''h'', ''F'')}} is a correction factor that in general has to be found by numerical integration. For the special case of a Schottky–Nordheim barrier, an analytical result exists and {{nowrap|''ν''(''h'', ''F'')}} is given by ''ν''(''f<sub>h</sub>''), as discussed above; approximation (11) for ''ν''(''f<sub>h</sub>'') is more than sufficient for all technological purposes. The pre-factor ''P'' is also in principle a function of ''h'' and (maybe) ''F'', but for the simple physical models discussed here it is usually satisfactory to make the approximation ''P''&nbsp;=&nbsp;1. The exact triangular barrier is a special case where the Schrödinger equation can be solved exactly, as was done by Fowler and Nordheim;<ref name="Fowler1928"/> for this physically unrealistic case, ''ν''(''f<sub>h</sub>'')&nbsp;=&nbsp;1, and an analytical approximation for ''P'' exists.

The approach described here was originally developed to describe Fowler&ndash;Nordheim tunneling from smooth, classically flat, planar emitting surfaces. It is adequate for smooth, classical curved surfaces of radii down to about 10 to 20&nbsp;nm. It can be adapted to surfaces of sharper radius, but quantities such as ''ν'' and ''D'' then become significant functions of the parameter(s) used to describe the surface curvature. When the emitter is so sharp that atomic-level detail cannot be neglected, and/or the tunneling barrier is thicker than the emitter-apex dimensions, then a more sophisticated approach is desirable.

As noted at the beginning, the effects of the atomic structure of materials are disregarded in the relatively simple treatments of field electron emission discussed here. Taking atomic structure properly into account is a very difficult problem, and only limited progress has been made.<ref name=mo84/> However, it seems probable that the main influences on the theory of Fowler–Nordheim tunneling will (in effect) be to change the values of ''P'' and ''ν'' in eq. (15), by amounts that cannot easily be estimated at present.

All these remarks apply in principle to Fowler&nbsp;Nordheim tunneling from any conductor where (before tunneling) the electrons may be treated as in [[travelling wave|travelling-wave states]]. The approach may be adapted to apply (approximately) to situations where the electrons are initially in localized states at or very close inside the emitting surface, but this is beyond the scope of this article.