Editing Field electron emission (section)

== Fowler–Nordheim plots and Millikan–Lauritsen plots ==
The original theoretical equation derived by Fowler and Nordheim<ref name="Fowler1928"/> has, for the last 80 years, influenced the way that experimental CFE data has been plotted and analyzed. In the very widely used Fowler–Nordheim plot, as introduced by Stern ''et al.'' in 1929,<ref name=sgf29/> the quantity ln{''i''/''V''<sup>2</sup>} is plotted against 1/''V''. The original thinking was that (as predicted by the original or the elementary Fowler–Nordheim-type equation) this would generate an exact straight line of slope&nbsp;''S''<sub>FN</sub>. ''S''<sub>FN</sub> would be related to the parameters that appear in the exponent of a Fowler–Nordheim-type equation of ''i''–''V'' form by:

{{NumBlk|:|<math>S_{\mathrm{FN}} = \; - b {\phi}^{3/2} / \beta. </math>|{{EquationRef|47}}}}

Hence, knowledge of ''φ'' would allow ''β'' to be determined, or vice versa.

[In principle, in system geometries where there is local field-enhancing nanostructure present, and the macroscopic conversion factor ''β''<sub>M</sub> can be determined, knowledge of ''β'' then allows the value of the emitter's effective field enhancement factor'' γ'' to be determined from the formula {{nowrap|1=''γ'' = ''β''/''β''<sub>M</sub>}}. In the common case of a film emitter generated on one plate of a two-plate arrangement with plate-separation ''W'' (so {{nowrap|1=''β''<sub>M</sub> = 1/''W''}}) then

{{NumBlk|:|<math> \gamma = \; \beta W. </math>|{{EquationRef|48}}}}

Nowadays, this is one of the most likely applications of Fowler–Nordheim plots.]

It subsequently became clear that the original thinking above is strictly correct only for the physically unrealistic situation of a flat emitter and an exact triangular barrier. For real emitters and real barriers a "slope correction factor" ''σ''<sub>FN</sub> has to be introduced, yielding the revised formula

{{NumBlk|:|<math>S_{\mathrm{FN}} = \; - \sigma_{\mathrm{FN}} b {\phi}^{3/2} / \beta. </math>|{{EquationRef|49}}}}

The value of ''σ''<sub>FN</sub> will, in principle, be influenced by any parameter in the physically complete Fowler–Nordheim-type equation for ''i''(''V'') that has a voltage dependence.

At present, the only parameter that is considered important is the correction factor <math> \nu_{\mathrm{F}} </math> relating to the barrier shape, and the only barrier for which there is any well-established detailed theory is the Schottky–Nordheim barrier. In this case, ''σ''<sub>FN</sub> is given by a mathematical function called ''s''. This function ''s'' was first tabulated correctly (as a function of the Nordheim parameter ''y'') by Burgess, [[Herbert Kroemer|Kroemer]] and Houston in 1953;<ref name=BKH53/> and a modern treatment that gives ''s'' as function of the scaled barrier field ''f'' for a Schottky–Nordheim barrier is given in.<ref name=fd07/> However, it has long been clear that, for practical emitter operation, the value of ''s'' lies in the range 0.9 to 1.

In practice, due to the extra complexity involved in taking the slope correction factor into detailed account, many authors (in effect) put {{nowrap|1=''σ''<sub>FN</sub> = 1}} in eq. (49), thereby generating a systematic error in their estimated values of ''β'' and/or ''γ'', thought usually to be around&nbsp;5%.

However, empirical equation (42) – which in principle is more general than Fowler–Nordheim-type equations – brings with it possible new ways of analyzing field emission ''i''–''V'' data. In general, it may be assumed that the parameter ''B'' in the empirical equation is related to the unreduced height ''H'' of some characteristic barrier seen by tunneling electrons by

{{NumBlk|:|<math> B = \; b H^{3/2} / \beta. </math>|{{EquationRef|50}}}}

(In most cases, but not necessarily all, ''H'' would be equal to the local work-function; certainly this is true for metals.) The issue is how to determine the value of ''B'' by experiment. There are two obvious ways. (1) Suppose that eq. (43) can be used to determine a reasonably accurate experimental value of&nbsp;''κ'', from the slope of a plot of form [−dln{''i''}/d(1/''V'') vs. ''V'']. In this case, a second plot, of ln(''i'')/''V''<sup>''κ''</sup> vs. 1/''V'', should be an exact straight line of slope&nbsp;−''B''. This approach should be the most accurate way of determining&nbsp;''B''.

(2) Alternatively, if the value of ''κ'' is not exactly known, and cannot be accurately measured, but can be estimated or guessed, then a value for'' B'' can be derived from a plot of the form [ln{''i''} vs. 1/''V'']. This is the form of plot used by Millikan and Lauritsen in 1928. Rearranging eq. (43) gives

{{NumBlk|:|<math> B = \; - \mathrm{d}\ln (i) / \mathrm{d} (1/V) - \kappa (1/V). </math>|{{EquationRef|51}}}}

Thus, ''B'' can be determined, to a good degree of approximation, by determining the mean slope of a Millikan–Lauritsen plot over some range of values of 1/''V'', and by applying a correction, using the value of 1/''V'' at the midpoint of the range and an assumed value of&nbsp;''κ''.

The main advantages of using a Millikan–Lauritsen plot, and this form of correction procedure, rather than a Fowler–Nordheim plot and a slope correction factor, are seen to be the following. (1) The plotting procedure is marginally more straightforward. (2) The correction involves a physical parameter (''V'') that is a measured quantity, rather than a physical parameter (''f'') that has to be calculated [in order to then calculate a value of ''s''(''f'') or, more generally ''σ''<sub>FN</sub>(''f'')]. (3) Both the parameter ''κ'' itself, and the correction procedure, are more transparent (and more readily understood) than the Fowler–Nordheim-plot equivalents. (4) This procedure takes into account all physical effects that influence the value of ''κ'', whereas the Fowler–Nordheim-plot correction procedure (in the form in which it has been carried out for the last 50 years) takes into account only those effects associated with barrier shape – assuming, furthermore, that this shape is that of a Schottky–Nordheim barrier. (5) There is a cleaner separation of theoretical and technological concerns: theoreticians will be interested in establishing what information any measured values of ''κ'' provide about CFE theory; but experimentalists can simply use measured values of ''κ'' to make more accurate estimates (if needed) of field enhancement factors.{{Citation needed|reason=Reliable source needed for the whole paragraph|date=February 2016}}

This correction procedure for Millikan–Lauritsen plots will become easier to apply when a sufficient number of measurements of ''κ'' have been made, and a better idea is available of what typical values actually are. At present, it seems probable that for most materials ''κ'' will lie in the range {{nowrap|−1 < ''κ'' < 3}}.{{Citation needed|reason=Reliable source needed for the whole sentence|date=February 2016}}