Editing Field electron emission (section)

==== 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".