Editing Field electron emission (section)

=== CFE theoretical equations ===
The preceding section explains how to derive Fowler–Nordheim-type equations. Strictly, these equations apply only to CFE from bulk metals. The ideas in the following sections apply to CFE more generally, but eq. (30) will be used to illustrate them.

For CFE, basic theoretical treatments provide a relationship between the local emission current density ''J'' and the local barrier field ''F'', at a local position on the emitting surface. Experiments measure the emission current ''i'' from some defined part of the emission surface, as a function of the voltage ''V'' applied to some counter-electrode. To relate these variables to'' J'' and ''F'', auxiliary equations are used.

The ''voltage-to-barrier-field conversion factor'' ''β'' is defined by:

{{NumBlk|:|<math> F = \; \beta V, </math>|{{EquationRef|31}}}}

The value of ''F'' varies from position to position on an emitter surface, and the value of ''β'' varies correspondingly.

For a metal emitter, the ''β''−value for a given position will be constant (independent of voltage) under the following conditions: (1) the apparatus is a "diode" arrangement, where the only electrodes present are the emitter and a set of "surroundings", all parts of which are at the same voltage; (2) no significant field-emitted vacuum [[space charge|space-charge]] (FEVSC) is present (this will be true except at very high emission current densities, around 10<sup>9</sup> A/m<sup>2</sup> or higher<ref name="Dyke1953"/><ref name=F08d>{{cite journal|doi=10.1063/1.2996005|title=Exact analysis of surface field reduction due to field-emitted vacuum space charge, in parallel-plane geometry, using simple dimensionless equations|year=2008|last1=Forbes|first1=Richard G.|journal=Journal of Applied Physics|volume=104|bibcode = 2008JAP...104h4303F|issue=8 |pages=084303–084303–10|url=http://epubs.surrey.ac.uk/307/1/fulltext.pdf}}</ref>); (3) no significant "patch fields" exist,<ref name=HN49/> as a result of non-uniformities in [[work function|local work-function]] (this is normally assumed to be true, but may not be in some circumstances). For non-metals, the physical effects called "field penetration" and "[[band bending]]" [M084] can make ''β'' a function of applied voltage, although – surprisingly – there are few studies of this effect.

The emission current density ''J'' varies from position to position across the emitter surface. The total emission current ''i'' from a defined part of the emitter is obtained by integrating ''J'' across this part. To obtain a simple equation for ''i''(''V''), the following procedure is used. A reference point "r" is selected within this part of the emitter surface (often the point at which the current density is highest), and the current density at this reference point is denoted by ''J''<sub>r</sub>. A parameter ''A''<sub>r</sub>, called the ''notional emission area'' (with respect to point "r"), is then defined by:

{{NumBlk|:|<math> i = A_{\mathrm{r}} J_{\mathrm{r}} = \int J \mathrm{d} A, </math>|{{EquationRef|32}}}}

where the integral is taken across the part of the emitter of interest.

This parameter ''A''<sub>r</sub> was introduced into CFE theory by Stern, Gossling and Fowler in 1929 (who called it a "weighted mean area").<ref name=sgf29/> For practical emitters, the emission current density used in Fowler–Nordheim-type equations is always the current density at some reference point (though this is usually not stated). Long-established convention denotes this reference current density by the simple symbol ''J'', and the corresponding local field and conversion factor by the simple symbols ''F'' and ''β'', without the subscript "r" used above; in what follows, this convention is used.

The notional emission area ''A''<sub>r</sub> will often be a function of the reference local field (and hence voltage),<ref name=AH39>{{cite journal|doi=10.1103/PhysRev.56.113|title=The Range and Validity of the Field Current Equation|year=1939|last1=Abbott|first1=F. R.|last2=Henderson|first2=Joseph E.|journal=Physical Review|volume=56|issue=1|pages=113–118|bibcode = 1939PhRv...56..113A }}</ref> and in some circumstances might be a significant function of temperature.

Because ''A''<sub>r</sub> has a mathematical definition, it does not necessarily correspond to the area from which emission is observed to occur from a single-point emitter in a [[field emission microscope|field electron (emission) microscope]]. With a large-area emitter, which contains many individual emission sites, ''A''<sub>r</sub> will nearly always be very very{{Clarify|date=June 2009|reason=was repeated 'very' intended?}} much less than the "macroscopic" geometrical area (''A''<sub>M</sub>) of the emitter as observed visually (see below).

Incorporating these auxiliary equations into eq. (30a) yields

{{NumBlk|:|<math> i = \; A_{\mathrm{r}} a {\phi^{-1}} {\beta}^2 V^2 \exp[- v(f) \;b \phi^{3/2} / \beta V ], </math>|{{EquationRef|33}}}}

This is the simplified standard Fowler–Nordheim-type equation, in ''i''–''V'' form. The corresponding "physically complete" equation is obtained by multiplying by ''λ''<sub>''Z''</sub>''P''<sub>F</sub>.