Editing Field electron emission (section)

=== Modified equations for large-area emitters ===
The equations in the preceding section apply to all field emitters operating in the CFE regime. However, further developments are useful for large-area emitters that contain many individual emission sites.

For such emitters, the notional emission area will nearly always be very very{{Clarify|date=June 2009|reason=was repeated 'very' intended?}} much less than the apparent "macroscopic" geometrical area (''A''<sub>M</sub>) of the physical emitter as observed visually. A dimensionless parameter ''α''<sub>r</sub>, ''the area efficiency of emission'', can be defined by

{{NumBlk|:|<math> A_{\mathrm{r}} = \; \alpha_{\mathrm{r}} A_{\mathrm{M}}. </math>|{{EquationRef|34}}}}

Also, a "macroscopic" (or "mean") emission current density ''J''<sub>M</sub> (averaged over the geometrical area ''A''<sub>M</sub> of the emitter) can be defined, and related to the reference current density ''J''<sub>r</sub> used above, by

{{NumBlk|:|<math> J_{\mathrm{M}} = \; i/A_{\mathrm{M}} = \alpha_{\mathrm{r}} (i /A_{\mathrm{r}}) = \alpha_{\mathrm{r}} J_{\mathrm{r}}. </math>|{{EquationRef|35}}}}

This leads to the following "large-area versions" of the simplified standard Fowler–Nordheim-type equation:

{{NumBlk|:|<math> J_{\mathrm{M}} = \alpha_{\mathrm{r}} a {\phi^{-1}} F^2 \exp[- v(f) \;b \phi^{3/2} / F ], </math>|{{EquationRef|36}}}}

{{NumBlk|:|<math> i = \; \alpha_{\mathrm{r}} A_{\mathrm{M}} a {\phi^{-1}} {\beta}^2 V^2 \exp[- v(f) \;b \phi^{3/2} / \beta V ], </math>|{{EquationRef|37}}}}

Both these equations contain the area efficiency of emission ''α''<sub>r</sub>. For any given emitter this parameter has a value that is usually not well known. In general, ''α''<sub>r</sub> varies greatly as between different emitter materials, and as between different specimens of the same material prepared and processed in different ways. Values in the range 10<sup>−10</sup> to 10<sup>−6</sup> appear to be likely, and values outside this range may be possible.

The presence of ''α''<sub>r</sub> in eq. (36) accounts for the difference between the macroscopic current densities often cited in the literature (typically 10 A/m<sup>2</sup> for many forms of large-area emitter other than [[Spindt tip|Spindt arrays]]<ref name=SBHW76/>) and the local current densities at the actual emission sites, which can vary widely but which are thought to be generally of the order of 10<sup>9</sup>&nbsp;A/m<sup>2</sup>, or possibly slightly less.

A significant part of the technological literature on large-area emitters fails to make clear distinctions between local and macroscopic current densities, or between notional emission area ''A''<sub>r</sub> and macroscopic area ''A''<sub>M</sub>, and/or omits the parameter ''α''<sub>r</sub> from cited equations. Care is necessary in order to avoid errors of interpretation.

It is also sometimes convenient to split the conversion factor ''β''<sub>r</sub> into a "macroscopic part" that relates to the overall geometry of the emitter and its surroundings, and a "local part" that relates to the ability of the very-local structure of the emitter surface to enhance the electric field. This is usually done by defining a "macroscopic field" ''F''<sub>M</sub> that is the field that would be present at the emitting site in the absence of the local structure that causes enhancement. This field ''F''<sub>M</sub> is related to the applied voltage by a "voltage-to-macroscopic-field conversion factor" ''β''<sub>M</sub> defined by:

{{NumBlk|:|<math> F_{\mathrm{M}} = \; \beta_{\mathrm{M}} V. </math>|{{EquationRef|38}}}}

In the common case of a system comprising two parallel plates, separated by a distance ''W'', with emitting nanostructures created on one of them, {{nowrap|1=''β''<sub>M</sub> = 1/''W''}}.

A "field enhancement factor" ''γ'' is then defined and related to the values of ''β''<sub>r</sub> and ''β''<sub>M</sub> by

{{NumBlk|:|<math> \gamma = \; F_{\mathrm{r}} / F_{\mathrm{M}} = \beta_{\mathrm{r}} / \beta_{\mathrm{M}}. </math>|{{EquationRef|39}}}}

With eq. (31), this generates the following formulae:

{{NumBlk|:|<math> F = \; \gamma F_{\mathrm{M}} = \beta V ;</math>|{{EquationRef|40}}}}
{{NumBlk|:|<math> \beta = \; \beta_{\mathrm{M}} \gamma ;</math>|{{EquationRef|41}}}}

where, in accordance with the usual convention, the suffix "r" has now been dropped from parameters relating to the reference point. Formulae exist for the estimation of ''γ'', using [[electrostatics|classical electrostatics]], for a variety of emitter shapes, in particular the "hemisphere on a post".<ref name=FEV01>{{cite journal|doi=10.1016/S0304-3991(02)00297-8|title=Some comments on models for field enhancement|year=2003|last1=Forbes|first1=R|journal=Ultramicroscopy|volume=95|pages=57–65|pmid=12535545|last2=Edgcombe|first2=CJ|last3=Valdrè|first3=U|issue=1–4}}</ref>

Equation (40) implies that versions of Fowler–Nordheim-type equations can be written where either ''F'' or ''βV'' is everywhere replaced by <math>\gamma F_{\mathrm{M}}</math>. This is often done in technological applications where the primary interest is in the field enhancing properties of the local emitter nanostructure. However, in some past work, failure to make a clear distinction between barrier field ''F'' and macroscopic field ''F''<sub>M</sub> has caused confusion or error.

More generally, the aims in technological development of large-area field emitters are to enhance the uniformity of emission by increasing the value of the area efficiency of emission ''α''<sub>r</sub>, and to reduce the "onset" voltage at which significant emission occurs, by increasing the value of ''β''. Eq. (41) shows that this can be done in two ways: either by trying to develop "high-''γ''" nanostructures, or by changing the overall geometry of the system so that ''β''<sub>M</sub> is increased. Various trade-offs and constraints exist.

In practice, although the definition of macroscopic field used above is the commonest one, other (differently defined) types of macroscopic field and field enhancement factor are used in the literature, particularly in connection with the use of probes to investigate the ''i''–''V'' characteristics of individual emitters.<ref>{{cite journal|doi=10.1063/1.1989443|title=Interpretation of enhancement factor in nonplanar field emitters|year=2005|last1=Smith|first1=R. C.|last2=Forrest|first2=R. D.|last3=Carey|first3=J. D.|last4=Hsu|first4=W. K.|last5=Silva|first5=S. R. P.|journal=Applied Physics Letters|volume=87|issue=1|page=013111|bibcode = 2005ApPhL..87a3111S |url=http://epubs.surrey.ac.uk/153/1/fulltext.pdf}}</ref>

In technological contexts field-emission data are often plotted using (a particular definition of) ''F''<sub>M</sub> or 1/''F''<sub>M</sub> as the ''x''-coordinate. However, for scientific analysis it usually better not to pre-manipulate the experimental data, but to plot the raw measured ''i''–''V'' data directly. Values of technological parameters such as (the various forms of) ''γ'' can then be obtained from the fitted parameters of the ''i''–''V'' data plot (see below), using the relevant definitions.