Editing Field electron emission (section)

== Cold field electron emission ==

=== Fowler–Nordheim-type equations ===

==== Introduction ====
Fowler–Nordheim-type equations, in the ''J''–''F'' form, are (approximate) theoretical equations derived to describe the local current density ''J'' emitted from the internal electron states in the conduction band of a bulk metal. The ''emission current density'' (ECD) ''J'' for some small uniform region of an emitting surface is usually expressed as a function {{nowrap|''J''(''φ'', ''F'')}} of the local work-function ''φ'' and the local barrier field ''F'' that characterize the small region. For sharply curved surfaces, ''J'' may also depend on the parameter(s) used to describe the surface curvature.

Owing to the physical assumptions made in the original derivation,<ref name="Fowler1928"/> the term ''Fowler–Nordheim-type equation'' has long been used only for equations that describe the ECD at zero temperature. However, it is better to allow this name to include the slightly modified equations (discussed below) that are valid for finite temperatures within the CFE emission regime.

==== Zero-temperature form ====
Current density is best measured in A/m<sup>2</sup>. The total current density emitted from a small uniform region can be obtained by integrating the total energy distribution ''j''(''ε'') with respect to total electron energy ''ε''. At zero temperature, the [[Fermi–Dirac statistics|Fermi–Dirac distribution function]] {{nowrap|1=''f''<sub>FD</sub> = 1}} for {{nowrap|''ε'' < 0}}, and {{nowrap|1=''f''<sub>FD</sub> = 0}} for {{nowrap|1=''ε'' > 0}}. So the ECD at 0&nbsp;K, ''J''<sub>0</sub>, is given from eq. (18) by

{{NumBlk|:|<math> J_0 = z_{\mathrm{S}} d_{\mathrm{F}} D_{\mathrm{F}} \int_{-\infty}^{0} \exp(\epsilon /
d_{\mathrm{F}}) \; \mathrm{d} \epsilon \; = \; z_{\mathrm{S}} {d_{\mathrm{F}}}^2 D_{\mathrm{F}} \; = \; Z_{\mathrm{F}} D_{\mathrm{F}}, </math>|{{EquationRef|23}}}}

where <math> Z_{\mathrm{F}} \; [=z_{\mathrm{S}} {d_{\mathrm{F}}}^2] </math> is the ''effective supply for state F'', and is defined by this equation. Strictly, the lower limit of the integral should be −''K''<sub>F</sub>, where ''K''<sub>F</sub> is the [[Fermi energy]]; but if ''d''<sub>F</sub> is very much less than ''K''<sub>F</sub> (which is always the case for a metal) then no significant contribution to the integral comes from energies below ''K''<sub>F</sub>, and it can formally be extended to&nbsp;–∞.

Result (23) can be given a simple and useful physical interpretation by referring to Fig. 1. The electron state at point "F" on the diagram ("state F") is the "forwards moving state at the Fermi level" (i.e., it describes a Fermi-level electron moving normal to and towards the emitter surface). At 0&nbsp;K, an electron in this state sees a barrier of unreduced height ''φ'', and has an escape probability ''D''<sub>F</sub> that is higher than that for any other occupied electron state. So it is convenient to write ''J''<sub>0</sub> as ''Z''<sub>F</sub>''D''<sub>F</sub>, where the "effective supply" ''Z''<sub>F</sub> is the current density that would have to be carried by state F inside the metal if all of the emission came out of state F.

In practice, the current density mainly comes out of a group of states close in energy to state F, most of which lie within the heavily shaded area in the energy-space diagram. Since, for a [[free electron model|free-electron model]], the contribution to the current density is directly proportional to the area in energy space (with the Sommerfeld supply density ''z''<sub>S</sub> as the constant of proportionality), it is useful to think of the ECD as drawn from electron states in an area of size ''d''<sub>F</sub><sup>2</sup> (measured in eV<sup>2</sup>) in the energy-space diagram. That is, it is useful to think of the ECD as drawn from states in the heavily shaded area in Fig. 1. (This approximation gets slowly worse as temperature increases.)

''Z''<sub>F</sub> can also be written in the form:

{{NumBlk|:|<math> Z_{\mathrm{F}} =z_{\mathrm{S}} {d_{\mathrm{F}}}^2= {\lambda_d}^2 (z_{\mathrm{S}} e^2 g^{-2}) \phi^{-1} F^2 = {\lambda_d}^2 a \phi^{-1} F^2, </math>|{{EquationRef|24}}}}

where the universal constant ''a'', sometimes called the ''First Fowler–Nordheim Constant'', is given by

{{NumBlk|:|<math> a = z_{\mathrm{S}} e^2 g^{-2} = e^3 /8 \pi h_{\mathrm{P}} \approx \; 1.541434 \times 10^{-6} \; \mathrm{A \; eV} \; {\mathrm{V}}^{-2}. </math>|{{EquationRef|25}}}}

This shows clearly that the pre-exponential factor ''aφ''<sup>−1</sup>''F''<sup>2</sup>, that appears in Fowler–Nordheim-type equations, relates to the effective supply of electrons to the emitter surface, in a free-electron model.

==== Non-zero temperatures ====
To obtain a result valid for non-zero temperature, we note from eq. (23) that ''z''<sub>S</sub>''d''<sub>F</sub>''D''<sub>F</sub> = ''J''<sub>0</sub>/''d''<sub>F</sub>. So when eq. (21) is integrated at non-zero temperature, then – on making this substitution, and inserting the explicit form of the Fermi–Dirac distribution function – the ECD ''J'' can be written in the form:

{{NumBlk|:|<math> J=J_0 \int_{-\infty}^{\infty} \frac{\exp(\epsilon / d_{\mathrm{F}})}{1 + \exp [(\epsilon/d_{\mathrm{F}})(d_{\mathrm{F}}/k_{\mathrm{B}} T)]} \mathrm{d}(\epsilon/ d_{\mathrm{F}}) = \lambda_T J_0 ,</math>|{{EquationRef|26}}}}

where ''λ''<sub>''T''</sub> is a temperature correction factor given by the integral. The integral can be transformed, by writing <math> w = d_{\mathrm{F}}/k_{\mathrm{B}}T </math> and <math> x=\epsilon/d_{\mathrm{F}} </math>, and then <math> u = \exp(x) </math>, into the standard result:<ref>{{cite book|author=Gradshteyn and Rhyzhik|title=Tables of Integrals, Series and Products|year=1980|publisher=Academic, New York|bibcode=1980tisp.book.....G }} see formula 3.241 (2), with ''μ''=1</ref>

{{NumBlk|:|<math> \int_{-\infty}^{\infty} \frac{ {\mathrm{e}}^x } { 1+ {\mathrm{e}}^{wx} } \mathrm{d}x = \int_{0}^{\infty} \frac{\mathrm{d}u}{1+u^w} = \frac{\pi} {w\sin(\pi/w)}. </math>|{{EquationRef|27}}}}

This is valid for {{nowrap|''w'' > 1}} (i.e., {{nowrap|''d''<sub>F</sub>/''k''<sub>B</sub>''T'' > 1}}). Hence – for temperatures such that {{nowrap|''k''<sub>B</sub>''T'' < ''d''<sub>F</sub>}}:

{{NumBlk|:|<math> \lambda_T = \frac{\pi k_{\mathrm{B}} T/d_{\mathrm{F}} }{ \sin(\pi k_{\mathrm{B}} T / d_{\mathrm{F}})} \approx 1 + \frac{1}{6} \left( {\frac{\pi k_{\mathrm{B}} T}{ d_{\mathrm{F}}}} \right) ^2, </math>|{{EquationRef|28}}}}

where the expansion is valid only if ({{nowrap|π''k''<sub>B</sub>''T'' / ''d''<sub>F</sub>) ≪ 1}}. An example value (for {{nowrap|1=''φ'' = 4.5 eV}}, {{nowrap|1=''F'' = 5 V/nm}}, {{nowrap|1=''T'' = 300 K}}) is {{nowrap|1=''λ''<sub>''T''</sub> = 1.024}}. Normal thinking has been that, in the CFE regime, ''λ''<sub>''T''</sub> is always small in comparison with other uncertainties, and that it is usually unnecessary to explicitly include it in formulae for the current density at room temperature.

The emission regimes for metals are, in practice, defined, by the ranges of barrier field ''F'' and temperature ''T'' for which a given family of emission equations is mathematically adequate. When the barrier field ''F'' is high enough for the CFE regime to be operating for metal emission at 0&nbsp;K, then the condition {{nowrap|''k''<sub>B</sub>''T'' < ''d''<sub>F</sub>}} provides a formal upper bound (in temperature) to the CFE emission regime. However, it has been argued that (due to approximations made elsewhere in the derivation) the condition {{nowrap|''k''<sub>B</sub>''T'' < 0.7''d''<sub>F</sub>}} is a better working limit: this corresponds to a ''λ''<sub>''T''</sub>-value of around 1.09, and (for the example case) an upper temperature limit on the CFE regime of around 1770&nbsp;K. This limit is a function of barrier field.<ref name=mo84/><ref name=MG56/>

Note that result (28) here applies for a barrier of any shape (though ''d''<sub>F</sub> will be different for different barriers).

==== Physically complete Fowler–Nordheim-type equation ====
Result (23) also leads to some understanding of what happens when atomic-level effects are taken into account, and the [[band structure|band-structure]] is no longer free-electron like. Due to the presence of the atomic ion-cores, the surface barrier, and also the electron [[wave function|wave-functions]] at the surface, will be different. This will affect the values of the correction factor <math>\nu</math>, the prefactor ''P'', and (to a limited extent) the correction factor ''λ''<sub>''d''</sub>. These changes will, in turn, affect the values of the parameter ''D''<sub>F</sub> and (to a limited extent) the parameter ''d''<sub>F</sub>. For a real metal, the supply density will vary with position in energy space, and the value at point "F" may be different from the Sommerfeld supply density. We can take account of this effect by introducing an electronic-band-structure correction factor ''λ''<sub>B</sub> into eq. (23). Modinos has discussed how this factor might be calculated: he estimates that it is most likely to be between 0.1 and 1; it might lie outside these limits but is most unlikely to lie outside the range {{nowrap|0.01 < ''λ''<sub>B</sub> < 10}}.<ref name=Mo01>{{cite journal|doi=10.1016/S0038-1101(00)00218-5|title=Theoretical analysis of field emission data|year=2001|last1=Modinos|first1=A|journal=Solid-State Electronics|volume=45|pages=809–816|bibcode=2001SSEle..45..809M|issue=6}}</ref>

By defining an overall supply correction factor ''λ''<sub>''Z''</sub> equal to {{nowrap|''λ''<sub>''T''</sub> ''λ''<sub>B</sub> ''λ''<sub>''d''</sub><sup>2</sup>}}, and combining equations above, we reach the so-called physically complete Fowler–Nordheim-type equation:<ref name=F08b>{{cite journal|doi=10.1116/1.2827505|title=Physics of generalized Fowler–Nordheim-type equations|year=2008|last1=Forbes|first1=Richard G.|journal=Journal of Vacuum Science and Technology B|volume=26|page=788|bibcode = 2008JVSTB..26..788F|issue=2 |s2cid=20219379}}</ref>

{{NumBlk|:|<math> J \;= \lambda_Z a \phi^{-1} F^2 P_{\mathrm{F}} \exp[- \nu_{\mathrm{F}} b \phi^{3/2} / F ], </math>|{{EquationRef|29}}}}

where <math>{\nu}_{\mathrm{F}}</math> [= <math>{\nu}_{\mathrm{F}}</math>(''φ'', ''F'')] is the exponent correction factor for a barrier of unreduced height ''φ''. This is the most general equation of the Fowler–Nordheim type. Other equations in the family are obtained by substituting specific expressions for the three correction factors <math>{\nu}_{\mathrm{F}}</math>, ''P''<sub>F</sub> and ''λ''<sub>''Z''</sub> it contains. The so-called elementary Fowler–Nordheim-type equation, that appears in undergraduate textbook discussions of field emission, is obtained by putting {{nowrap|''λ''<sub>''Z''</sub> → 1}}, {{nowrap|''P''<sub>F</sub> → 1}}, {{nowrap|<math>{\nu}_{\mathrm{F}}</math> → 1}}; this does not yield good quantitative predictions because it makes the barrier stronger than it is in physical reality. The so-called standard Fowler–Nordheim-type equation, originally developed by Murphy and Good,<ref name=MG56/> and much used in past literature, is obtained by putting {{nowrap|''λ''<sub>''Z''</sub> → ''t''<sub>F</sub><sup>−2</sup>}}, {{nowrap|''P''<sub>F</sub> → 1}}, {{nowrap|<math>{\nu}_{\mathrm{F}}</math> → ''v''<sub>F</sub>}}, where ''v''<sub>F</sub> is ''v''(''f''), where ''f'' is the value of ''f''<sub>''h''</sub> obtained by putting {{nowrap|1=''h'' = ''φ''}}, and ''t''<sub>F</sub> is a related parameter (of value close to unity).<ref name=fd07/>

Within the more complete theory described here, the factor ''t''<sub>F</sub><sup>−2</sup> is a component part of the correction factor ''λ''<sub>''d''</sub><sup>2</sup> [see,<ref name=F08c/> and note that ''λ''<sub>''d''</sub><sup>2</sup> is denoted by ''λ''<sub>''D''</sub> there]. There is no significant value in continuing the separate identification of ''t''<sub>F</sub><sup>−2</sup>. Probably, in the present state of knowledge, the best approximation for simple Fowler–Nordheim-type equation based modeling of CFE from metals is obtained by putting {{nowrap|''λ''<sub>''Z''</sub> → 1}}, {{nowrap|''P''<sub>F</sub> → 1}}, {{nowrap|<math>{\nu}_{\mathrm{F}}</math> → ''v''(''f'')}}. This re-generates the Fowler–Nordheim-type equation used by Dyke and Dolan in 1956, and can be called the "simplified standard Fowler–Nordheim-type equation".

==== Recommended form for simple Fowler–Nordheim-type calculations ====
Explicitly, this recommended ''simplified standard Fowler–Nordheim-type equation'', and associated formulae, are:

{{NumBlk|:|<math> J = \; a {\phi^{-1}} F^2 \exp[- v(f) \;b \phi^{3/2} / F ], </math>|{{EquationRef|30a}}}}

{{NumBlk|:|<math> a \approx \; 1.541434 \times 10^{-6} \; \mathrm{A \; eV} \; {\mathrm{V}}^{-2};\;\;\;\;\; b \approx 6.830890 \; {\mathrm{eV}}^{-3/2} \; \mathrm{V} \; {\mathrm{nm}}^{-1}, </math>|{{EquationRef|30b}}}}

{{NumBlk|:|<math> v(f) \approx 1 - f + (1/6) f \ln f </math>|{{EquationRef|30c}}}}

{{NumBlk|:|<math> f = \; F/F_{\phi} = (e^3 / 4 \pi \epsilon_0) (F/ {\phi}^2) = (1.439964 \; {\mathrm{eV}}^2 \; {\mathrm{V}}^{-1} \; \mathrm{nm}) (F/ {\phi}^2). </math>|{{EquationRef|30d}}}}

where ''F''<sub>''φ''</sub> here is the field needed to reduce to zero a Schottky–Nordheim barrier of unreduced height equal to the local work-function ''φ'', and ''f'' is the scaled barrier field for a Schottky–Nordheim barrier of unreduced height ''φ''. [This quantity ''f'' could have been written more exactly as ''f''<sub>''φ''</sub><sup>SN</sup>, but it makes this Fowler–Nordheim-type equation look less cluttered if the convention is adopted that simple ''f'' means the quantity denoted by ''f''<sub>''φ''</sub><sup>SN</sup> in,<ref name=fd07/> eq. (2.16).] For the example case ({{nowrap|1=''φ'' = 4.5 eV}}, {{nowrap|1=''F'' = 5 V/nm}}), {{nowrap|1=''f'' ≈ 0.36}} and {{nowrap|''v''(''f'') ≈ 0.58}}; practical ranges for these parameters are discussed further in.<ref name=F08a>{{cite journal|doi=10.1116/1.2834563|title=Description of field emission current/voltage characteristics in terms of scaled barrier field values (f-values)|year=2008|last1=Forbes|first1=Richard G.|journal=Journal of Vacuum Science and Technology B|volume=26|issue=1|page=209|bibcode = 2008JVSTB..26..209F }}</ref>

Note that the variable ''f'' (the scaled barrier field) is not the same as the variable ''y'' (the Nordheim parameter) extensively used in past field emission literature, and that "''v''(''f'')" does NOT have the same mathematical meaning and values as the quantity "''v''(''y'')" that appears in field emission literature. In the context of the revised theory described here, formulae for ''v''(''y''), and tables of values for ''v''(''y'') should be disregarded, or treated as values of ''v''(''f''<sup>1/2</sup>). If more exact values for ''v''(''f'') are required, then<ref name=fd07/> provides formulae that give values for ''v''(''f'') to an absolute mathematical accuracy of better than 8×10<sup>−10</sup>. However, approximation formula (30c) above, which yields values correct to within an absolute mathematical accuracy of better 0.0025, should gives values sufficiently accurate for all technological purposes.<ref name=fd07/>

==== Comments ====
A historical note on methods of deriving Fowler–Nordheim-type equations is necessary. There are several possible approaches to deriving these equations, using [[free electron model|free-electron theory]]. The approach used here was introduced by Forbes in 2004 and may be described as "integrating via the total energy distribution, using the parallel kinetic energy ''K''<sub>p</sub> as the first variable of integration".<ref name=F04/> Basically, it is a free-electron equivalent of the Modinos procedure<ref name=mo84/><ref name=Mo01/> (in a more advanced quantum-mechanical treatment) of "integrating over the surface Brillouin zone". By contrast, the free-electron treatments of CFE by Young in 1959,<ref name=Y59/> Gadzuk and Plummer in 1973<ref name=GP73/> and Modinos in 1984,<ref name=mo84/> also integrate via the total energy distribution, but use the normal energy ''ε''<sub>n</sub> (or a related quantity) as the first variable of integration.

There is also an older approach, based on a seminal paper by Nordheim in 1928,<ref>{{cite journal|author=L.W. Nordheim|journal=Z. Phys.|volume=46|year=1928|pages=833–855|bibcode = 1928ZPhy...46..833N |doi = 10.1007/BF01391020|title=Zur Theorie der thermischen Emission und der Reflexion von Elektronen an Metallen|issue=11–12 |s2cid=119880921}}</ref> that formulates the problem differently and then uses first ''K''<sub>p</sub> and then ''ε''<sub>n</sub> (or a related quantity) as the variables of integration: this is known as "integrating via the normal-energy distribution". This approach continues to be used by some authors. Although it has some advantages, particularly when discussing resonance phenomena, it requires integration of the Fermi–Dirac distribution function in the first stage of integration: for non-free-electron-like electronic band-structures this can lead to very complex and error-prone mathematics (as in the work of Stratton on [[semiconductors]]).<ref name=St62>{{cite journal|doi=10.1103/PhysRev.125.67|title=Theory of Field Emission from Semiconductors|year=1962|last1=Stratton|first1=Robert|journal=Physical Review|volume=125|issue=1|pages=67–82|bibcode = 1962PhRv..125...67S }}</ref> Further, integrating via the normal-energy distribution does not generate experimentally measured electron energy distributions.

In general, the approach used here seems easier to understand, and leads to simpler mathematics.

It is also closer in principle to the more sophisticated approaches used when dealing with real bulk crystalline solids, where the first step is either to integrate contributions to the ECD over [[constant energy surface]]s in a [[wave vector|wave-vector]] space ('''''k'''''-space),<ref name=GP73/> or to integrate contributions over the relevant surface Brillouin zone.<ref name=mo84/> The Forbes approach is equivalent either to integrating over a spherical surface in '''''k'''''-space, using the variable ''K''<sub>p</sub> to define a ring-like integration element that has cylindrical symmetry about an axis in a direction normal to the emitting surface, or to integrating over an (extended) surface Brillouin zone using circular-ring elements.

=== 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>.

=== 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.

=== Modified equations for nanometrically sharp emitters ===
Most of the theoretical derivations in the field emission theory are done under the assumption that the barrier takes the Schottky–Nordheim form eq. (3). However, this barrier form is not valid for emitters with radii of curvature ''R'' comparable to the length of the tunnelling barrier. The latter depends on the work function and the field, but in cases of practical interest, the SN barrier approximation can be considered valid for emitters with radii {{nowrap|''R'' > 20 nm}}, as explained in the next paragraph.

The main assumption of the SN barrier approximation is that the electrostatic potential term takes the linear form <math>\Phi = Fx</math> in the tunnelling region. The latter has been proved to hold only if <math>x \ll R</math>.<ref name=KX>{{cite journal|doi=10.1098/rspa.2014.0811|title=Derivation of a generalized Fowler–Nordheim equation for nanoscopic field-emitters|year=2015|last1=Kyritsakis|first1=A.|last2=Xanthakis|first2=J. P. |journal=Proceedings of the Royal Society A|volume=471|issue=2174|page=20140811|bibcode=2015RSPSA.47140811K|doi-access=free}}</ref> Therefore, if the tunnelling region has a length <math>L</math>, <math>x<L</math> for all <math>x</math> that determines the tunnelling process; thus if <math>L \ll R</math> eq. (1) holds and the SN barrier approximation is valid. If the tunnelling probability is high enough to produce measurable field emission, L does not exceed 1–2&nbsp;nm. Hence, the SN barrier is valid for emitters with radii of the order of some tens of nm.

However, modern emitters are much sharper than this, with radii that of the order of a few nm. Therefore, the standard FN equation, or any version of it that assumes the SN barrier, leads to significant errors for such sharp emitters. This has been both shown theoretically<ref>{{cite journal|doi=10.1063/1.106257|title=Derivation of a generalized Fowler–Nordheim equation for nanoscopic field-emitters|year=1991|last1=He|first1=J.|last2=Cutler|first2=P. H.|journal=Applied Physics Letters|volume=59|issue=13|page=1644|bibcode=1991ApPhL..59.1644H}}</ref><ref>{{cite journal|doi=10.1116/1.589929|title=Deviations from the Fowler–Nordheim theory and peculiarities of field electron emission from small-scale objects|year=1998|last1=Fursey|first1=G. N.|last2=Glazanov|first2=D. V.|journal=Journal of Vacuum Science and Technology B|volume=16|issue=2|page=910|bibcode=1998JVSTB..16..910F}}</ref> and confirmed experimentally.<ref>{{cite journal|doi=10.1103/PhysRevB.87.115436|title=Scale invariance of a diodelike tunnel junction|year=2013|last1=Cabrera|first1=H.|journal=Physical Review B|volume=87|issue=11|page=115436|display-authors=etal|arxiv=1303.4985|bibcode=2013PhRvB..87k5436C|s2cid=118361236}}</ref>

The above problem was tackled by Kyritsakis and Xanthakis,<ref name=KX/> who generalized the SN barrier by including the electrostatic effects of the emitter curvature. The general barrier form for an emitter with radius of average curvature <math>R</math> (inverse of the average of the two principal curvatures) can be [[asymptotic expansion|asymptotically expanded]] as<ref>{{Cite journal |last=Kyritsakis |first=Andreas |date=2023-03-21 |title=General form of the tunneling barrier for nanometrically sharp electron emitters |url=https://doi.org/10.1063/5.0144608 |journal=Journal of Applied Physics |volume=133 |issue=11 |pages=113302 |doi=10.1063/5.0144608 |arxiv=2207.06263 |bibcode=2023JAP...133k3302K |s2cid=256390628 |issn=0021-8979}}</ref> {{NumBlk|:|<math> M^{KX} (x) = h - eFx \left[ 1-\frac{x}{R} + O \left(\frac{x}{R} \right)^2 \right] - \frac{e^2}{16\pi \epsilon_0 x} \left[ 1-\frac{x}{2R} + O \left(\frac{x}{R} \right)^2 \right] </math>.|{{EquationRef|43}}}}

After neglecting all <math>O(x/R)^2</math> terms, and employing the [[WKB approximation|JWKB approximation]] (4) for this barrier, the Gamow exponent takes a form that generalizes eq. (5) 
{{NumBlk|:|<math> G(h,F,R) = \frac{b h^{3/2}}{F} \left(v(f) + \omega(f)\frac{h}{eFR} \right) </math>|{{EquationRef|44}}}}
where <math>f</math> is defined by (30d), <math>v(f)</math> is given by (30c) and <math>\omega(f)</math> is a new function that can be approximated in a similar manner as (30c) (there are typographical mistakes in ref.,<ref name="KX" /> corrected here): 
{{NumBlk|:|<math>\omega(f) \approx \frac{4}{5} - \frac{7}{40} f  + \frac{1}{200} \log(f) . </math>|{{EquationRef|45}}}}

Given the expression for the Gamow exponent as a function of the field-free barrier height <math>h</math>, the emitted current density for cold field emission can be obtained from eq. (23). It yields 
{{NumBlk|:|<math> J = a \frac{F^2}{\phi} \left(\frac{1}{\lambda_d(f)} + \frac{\phi}{eFR} \psi(f) \right)^{-2} \exp\left[-\frac{b \phi^{3/2}}{F} \left(v(f) + \omega(f)\frac{\phi}{eFR} \right) \right] </math> |{{EquationRef|46}}}}
where the functions <math>\lambda_d(f)</math> and <math>\psi(f)</math> are defined as
{{NumBlk|:|<math>\frac{1}{\lambda_d(f)} \equiv v(f) - \frac{4}{3}f \frac{\partial v}{\partial f} \approx 1 + \frac{f}{9} - \frac{f \log(f)}{22} </math>|{{EquationRef|47a}}}}
and 
{{NumBlk|:|<math>\psi(f) \equiv \frac{5}{3} \omega(f) - \frac{4}{3}f\frac{\partial\omega}{\partial f} \approx \frac{4}{3} - \frac{f}{15} - \frac{f \log(f)}{1200} </math>|{{EquationRef|47b}}}}
In equation (46), for completeness purposes, ''λ''<sub>''d''</sub> is not approximated by unity as in (29) and (30a), although for most practical cases it is a very good approximation. Apart from this, equations (43), (44) and (46) coincide with the corresponding ones of the standard Fowler–Nordheim theory (3), (9), and (30a), in the limit {{nowrap|''R'' → ∞}}; this is expected since the former equations generalise the latter.

Finally, note that the above analysis is asymptotic in the limit {{nowrap|''L'' ≪ ''R''}}, similarly to the standard Fowler–Nordheim theory using the SN barrier. However, the addition of the quadratic terms renders it significantly more accurate for emitters with radii of curvature in the range ~&nbsp;5–20&nbsp;nm. For sharper emitters there is no general approximation for the current density. In order to obtain the current density, one has to calculate the electrostatic potential and evaluate the [[WKB approximation|JWKB integral]] numerically. For this purpose, scientific computing software has been developed (see e.g. [https://getelec.org GETELEC]<ref>{{cite journal|doi=10.1016/j.commatsci.2016.11.010|title=A general computational method for electron emission and thermal effects in field emitting nanotips|year=2017|last1=Kyritsakis|first1=A.|last2=Djurabekova|first2=F. |journal=Computational Materials Science|volume=128|page=15|arxiv=1609.02364|s2cid=11369516}}</ref>).

=== Empirical CFE ''i''–''V'' equation ===
At the present stage of CFE theory development, it is important to make a distinction between theoretical CFE equations and an empirical CFE equation. The former are derived from condensed matter physics (albeit in contexts where their detailed development is difficult). An empirical CFE equation, on the other hand, simply attempts to represent the actual experimental form of the dependence of current ''i'' on voltage&nbsp;''V''.

In the 1920s, empirical equations were used to find the power of ''V'' that appeared in the exponent of a semi-logarithmic equation assumed to describe experimental CFE results. In 1928, theory and experiment were brought together to show that (except, possibly, for very sharp emitters) this power is ''V''<sup>−1</sup>. It has recently been suggested that CFE experiments should now be carried out to try to find the power (''κ'') of ''V'' in the pre-exponential of the following empirical CFE equation:<ref name=F08e>{{cite journal|doi=10.1063/1.2918446|title=Call for experimental test of a revised mathematical form for empirical field emission current-voltage characteristics|year=2008|last1=Forbes|first1=Richard G.|journal=Applied Physics Letters|volume=92|page=193105|bibcode = 2008ApPhL..92s3105F|issue=19 |url=http://epubs.surrey.ac.uk/391/1/fulltext.pdf}}</ref>

{{NumBlk|:|<math> i = \; C V^{\kappa} \exp[-B/V],  </math>|{{EquationRef|48}}}}

where ''B'', ''C'' and ''κ'' are treated as constants.

From eq. (42) it is readily shown that

{{NumBlk|:|<math> - \mathrm{d}\ln i / \mathrm{d} (1/V) = \; \kappa V + B, </math>|{{EquationRef|49}}}}

In the 1920s, experimental techniques could not distinguish between the results {{nowrap|1=''κ'' = 0}} (assumed by Millikan and Laurtisen)<ref name=Millikan/> and {{nowrap|1=''κ'' = 2}} (predicted by the original Fowler–Nordheim-type equation).<ref name="Fowler1928"/> However, it should now be possible to make reasonably accurate measurements of dlni/d(1/V) (if necessary by using [[lock-in amplifier]]/phase-sensitive detection techniques and computer-controlled equipment), and to derive ''κ'' from the slope of an appropriate data plot.<ref name=SBHW76/>

Following the discovery of approximation (30b), it is now very clear that – even for CFE from bulk metals – the value {{nowrap|1=''κ'' = 2}} is not expected. This can be shown as follows. Using eq. (30c) above, a dimensionless parameter ''η'' may be defined by

{{NumBlk|:|<math> \eta = b \phi^{3/2} / F_{\phi} = \; (b e^3 / 4 \pi \epsilon_0) {\phi}^{-1/2} \approx 9.836239 \;\; (\mathrm{eV} / \phi)^{1/2}. </math>|{{EquationRef|50}}}}

For {{nowrap|1=''φ'' = 4.50 eV}}, this parameter has the value {{nowrap|1=''η'' = 4.64}}. Since {{nowrap|1=''f'' = ''F''/''F''<sub>''φ''</sub>}} and ''v''(''f'') is given by eq (30b), the exponent in the simplified standard Fowler–Nordheim-type equation (30) can be written in an alternative form and then expanded as follows:<ref name=fd07/>

{{NumBlk|:|<math> \exp [-v(f) \; b {\phi}^{3/2} / F] \; = \;\exp[-v(f) \; \eta /f] \; \approx \; {\mathrm{e}}^{\eta} f^{-\eta/6} \exp[- \eta /f] \; = \; {\mathrm{e}}^{\eta} f^{-\eta/6} \exp[-b {\phi}^{3/2} /F ]. </math>|{{EquationRef|51}}}}

Provided that the conversion factor ''β'' is independent of voltage, the parameter ''f'' has the alternative definition {{nowrap|1=''f'' = ''V''/''V''<sub>''φ''</sub>}}, where ''V''<sub>''φ''</sub> is the voltage needed, in a particular experimental system, to reduce the height of a Schottky–Nordheim barrier from ''φ'' to zero. Thus, it is clear that the factor ''v''(''f'') in the ''exponent'' of the theoretical equation (30) gives rise to additional ''V''-dependence in the ''pre-exponential'' of the empirical equation. Thus, (for effects due to the Schottky–Nordheim barrier, and for an emitter with {{nowrap|1=''φ'' = 4.5 eV}}) we obtain the prediction:

{{NumBlk|:|<math> \kappa \approx 2 - \eta / 6 = 2 - 0.77 = 1.23. </math>|{{EquationRef|52}}}}

Since there may also be voltage dependence in other factors in a Fowler–Nordheim-type equation, in particular in the notional emission area<ref name=AH39/> ''A''<sub>r</sub> and in the local work-function, it is not necessarily expected that ''κ'' for CFE from a metal of local work-function 4.5&nbsp;eV should have the value ''κ'' = 1.23, but there is certainly no reason to expect that it will have the original Fowler–Nordheim value {{nowrap|1=''κ'' = 2}}.<ref name=Je99>{{cite journal|doi=10.1063/1.369584|title=Exchange-correlation, dipole, and image charge potentials for electron sources: Temperature and field variation of the barrier height|year=1999|last1=Jensen|first1=K. L.|journal=Journal of Applied Physics|volume=85|page=2667|bibcode = 1999JAP....85.2667J|issue=5 |doi-access=free}}</ref>

A first experimental test of this proposal has been carried out by Kirk, who used a slightly more complex form of data analysis to find a value 1.36 for his parameter&nbsp;''κ''. His parameter ''κ'' is very similar to, but not quite the same as, the parameter ''κ'' used here, but nevertheless his results do appear to confirm the potential usefulness of this form of analysis.<ref>T. Kirk, 21st Intern. Vacuum Nanoelectronics Conf., Wrocław, July 2008.</ref>

Use of the empirical CFE equation (42), and the measurement of ''κ'', may be of particular use for non-metals. Strictly, Fowler–Nordheim-type equations apply only to emission from the conduction band of bulk [[crystalline]] solids. However, empirical equations of form (42) should apply to all materials (though, conceivably, modification might be needed for very sharp emitters). It seems very likely that one way in which CFE equations for newer materials may differ from Fowler–Nordheim-type equations is that these CFE equations may have a different power of ''F'' (or ''V'') in their pre-exponentials. Measurements of ''κ'' might provide some experimental indication of this.