Editing Field electron emission (section)

=== 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.