Editing Field electron emission (section)

=== Escape probability ===
For an electron approaching a given barrier from the inside, the ''probability of escape'' (or "[[transmission coefficient]]" or "penetration coefficient") is a function of ''h'' and ''F'', and is denoted by {{nowrap|''D''(''h'', ''F'')}}. The primary aim of tunneling theory is to calculate {{nowrap|''D''(''h'', ''F'')}}. For physically realistic barrier models, such as the Schottky–Nordheim barrier, the Schrödinger equation cannot be solved exactly in any simple way. The following so-called "semi-classical" approach can be used. A parameter {{nowrap|''G''(''h'', ''F'')}} can be defined by the [[WKB|JWKB (Jeffreys-Wentzel-Kramers-Brillouin)]] integral:<ref>{{cite journal|author=H. Jeffreys|journal=Proceedings of the London Mathematical Society |volume=23|year=1924|pages=428–436| doi = 10.1112/plms/s2-23.1.428|title=On Certain Approximate Solutions of Lineae Differential Equations of the Second Order}}</ref>

{{NumBlk|:|<math>G(h, F) = g\int M^{1/2}\mbox{d}x, </math>|{{EquationRef|4}}}}

where the integral is taken across the barrier (i.e., across the region where ''M''&nbsp;>&nbsp;0), and the parameter ''g'' is a universal constant given by

{{NumBlk|:|<math> g \,= 2\sqrt{2m}/\hbar \approx 10.24624 \; {\rm{eV}}^{-1/2}\; {\rm{nm}}^{-1}. </math>|{{EquationRef|5}}}}

Forbes has re-arranged a result proved by Fröman and Fröman, to show that, formally – in a one-dimensional treatment – the exact solution for ''D'' can be written<ref name=F08c>{{cite journal|doi=10.1063/1.2937077|title=On the need for a tunneling pre-factor in Fowler–Nordheim tunneling theory|year=2008|last1=Forbes|first1=Richard G.|journal=Journal of Applied Physics|volume=103|bibcode = 2008JAP...103k4911F|issue=11 |pages=114911–114911–8|url=http://epubs.surrey.ac.uk/22/1/fulltext.pdf}}</ref>

{{NumBlk|:|<math>\,D = \frac{P\mathrm{e}^{-G}}{1 + P\mathrm{e}^{-G}}, </math>|{{EquationRef|6}}}}

where the ''tunneling pre-factor'' ''P'' can in principle be evaluated by complicated iterative integrations along a path in [[Complex Hilbert space|complex space]], but is ≈&nbsp;1 for simple models.<ref name=F08c/><ref>H. Fröman and P.O. Fröman, "JWKB approximation: contributions to the theory" (North-Holland, Amsterdam, 1965).</ref> In the CFE regime we have (by definition) ''G''&nbsp;≫&nbsp;1. So eq. (6) reduces to the so-called simple [[WKB|JWKB]] formula:

{{NumBlk|:|<math>D\approx P \mathrm{e}^{-G} \approx \mathrm{e}^{-G}. </math>|{{EquationRef|7}}}}

For the exact triangular barrier, putting eq.&nbsp;({{EquationNote|2}}) into eq.&nbsp;({{EquationNote|4}}) yields {{nowrap|1=''G''<sup>ET</sup> = ''bh''<sup>3/2</sup>/''F''}}, where

{{NumBlk|:|<math> b = \frac{2g}{3e} = \frac{4\sqrt{2 m}}{3e\hbar} \approx 6.830890 \; {\mathrm{eV}}^{-3/2} \; \mathrm{V} \; {\mathrm{nm}}^{-1}. </math>|{{EquationRef|8}}}}

This parameter ''b'' is a universal constant sometimes called the ''second Fowler&ndash;Nordheim constant''. For barriers of other shapes, we write

{{NumBlk|:|<math>G(h, F) = \nu(h, F) G^{\mathrm{ET}} = \nu(h, F)b h^{3/2}/F, </math>|{{EquationRef|9}}}}

where {{nowrap|''ν''(''h'', ''F'')}} is a correction factor that determined by [[numerical integration]] of eq.&nbsp;({{EquationNote|4}}).