Editing Field electron emission (section)

==== 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.