Editing Field electron emission (section)

=== 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.