Editing Field electron emission (section)

== Total-energy distribution ==
The energy distribution of the emitted electrons is important both for scientific experiments that use the emitted electron energy distribution to probe aspects of the emitter [[surface physics]]<ref name=GP73/> and for the field emission sources used in electron beam instruments such as [[electron microscopes]].<ref name=HK96/> In the latter case, the "width" (in energy) of the distribution influences how finely the beam can be focused.

The theoretical explanation here follows the approach of Forbes.<ref name=F04>{{cite journal|doi=10.1002/sia.1900|title=Use of energy-space diagrams in free-electron models of field electron emission|year=2004|last1=Forbes|first1=Richard G.|journal=Surface and Interface Analysis|volume=36|pages=395–401|issue=56|s2cid=121435016 }}</ref> If ''ε'' denotes the total electron energy relative to the emitter Fermi level, and ''K''<sub>p</sub> denotes the [[kinetic energy]] of the electron parallel to the emitter surface, then the electron's ''normal energy'' ''ε''<sub>n</sub> (sometimes called its "forwards energy") is defined by

{{NumBlk|:|<math> \; \epsilon_{\mathrm{n} } = \epsilon - K_{\mathrm{p} }. </math>|{{EquationRef|16}}}}

Two types of theoretical energy distribution are recognized: the ''normal-energy distribution'' (NED), which shows how the energy ''ε''<sub>n</sub> is distributed immediately after emission (i.e., immediately outside the tunneling barrier); and the ''total-energy distribution'', which shows how the total energy ''ε'' is distributed. When the emitter Fermi level is used as the reference zero level, both ''ε'' and ''ε''<sub>n</sub> can be either positive or negative.

Energy analysis experiments have been made on field emitters since the 1930s. However, only in the late 1950s was it realized (by Young and Mueller<ref name=Y59/> [,YM58]) that these experiments always measured the total energy distribution, which is now usually denoted by ''j''(''ε''). This is also true (or nearly true) when the emission comes from a small field enhancing protrusion on an otherwise flat surface.<ref name=GP73/>

To see how the total energy distribution can be calculated within the framework of a [[free electron model|Sommerfeld free-electron-type model]], look at the ''P-T energy-space diagram'' (P-T="parallel-total").
[[Image:P-T_Energy_Space (cropped).jpg|thumb|upright=1.5|none|P-T energy-space diagram, showing the region in P-T energy space where traveling-wave electron states exist.]]
This shows the "parallel kinetic energy" ''K''<sub>p</sub> on the horizontal axis and the total energy ''ε'' on the vertical axis. An electron inside the bulk metal usually has values of ''K''<sub>p</sub> and ''ε'' that lie within the lightly shaded area. It can be shown that each element d''ε''d''K''<sub>p</sub> of this energy space makes a contribution <math> z_{\mathrm{S}} f_{\mathrm{FD}} \mathrm{d}{\it{\epsilon}} \mathrm{d} K_{\mathrm{p}} </math> to the electron current density incident on the inside of the emitter boundary.<ref name=F04/> Here, ''z''<sub>S</sub> is the universal constant (called here the ''[[Arnold Sommerfeld|Sommerfeld]] supply density''):

{{NumBlk|:|<math>z_{\mathrm{S}}=4\mathrm{\pi}em / h_{\mathrm{P}}^3 = 1.618311 \times 10^{14} \, \rm{A} \, m^{-2} \, eV^{-2}, </math>|{{EquationRef|17}}}}

and <math> f_{\mathrm{FD}} </math> is the [[Fermi–Dirac statistics|Fermi–Dirac distribution function]]:

{{NumBlk|:|<math> \, f_{\mathrm{FD} } (\epsilon) = 1/[1 + \exp(\epsilon / k_{\mathrm{B} }T)], </math>|{{EquationRef|18}}}}

where ''T'' is [[thermodynamic temperature]] and ''k''<sub>B</sub> is the [[Boltzmann constant]].

This element of incident current density sees a barrier of height ''h'' given by:

{{NumBlk|:|<math> \, h=\phi - \epsilon + K_{\mathrm{p} } </math>|{{EquationRef|19a}}}}

The corresponding escape probability is {{nowrap|''D''(''h'', ''F'')}}: this may be expanded (approximately) in the form<ref name=F04/>

{{NumBlk|:|<math> D(h,F) \approx D_{\mathrm{F}} \; \exp(\epsilon / d_{\mathrm{F}}) \; \exp(-K_{\mathrm{p}} / d_{\mathrm{F}}) , </math>|{{EquationRef|19b}}}}

where ''D''<sub>F</sub> is the escape probability for a barrier of unreduced height equal to the [[work function|local work-function ''φ'']]. Hence, the element d''ε''d''K''<sub>p</sub> makes a contribution <math>z_{\mathrm{S}} f_{\mathrm{FD}} D \mathrm{d} {\it{\epsilon}} \mathrm{d} K_{\mathrm{p}} </math> to the emission current density, and the total contribution made by incident electrons with energies in the elementary range d''ε'' is thus

{{NumBlk|:|<math> j(\epsilon) \mathrm{d} \epsilon = z_{\mathrm{S}} f_{\mathrm{FD}} \left[ \int D \mathrm{d} K_{\mathrm{p}} \right] \mathrm{d} \epsilon =
z_{\mathrm{S}} f_{\mathrm{FD}} D_{\mathrm{F}} \exp(\epsilon / d_{\mathrm{F}}) \left[ \int_{0}^{\infty} \exp(-K_{\mathrm{p}} / d_{\mathrm{F}}) \; \mathrm{d} K_{\mathrm{p}} \right] \mathrm{d} \epsilon ,</math>|{{EquationRef|20}}}}

where the integral is in principle taken along the strip shown in the diagram, but can in practice be extended to ∞ when the decay-width ''d''<sub>F</sub> is very much less than the [[Fermi energy]] ''K''<sub>F</sub> (which is always the case for a metal). The outcome of the integration can be written:

{{NumBlk|:|<math> \, j(\epsilon) = z_{\mathrm{S}} d_{\mathrm{F}} D_{\mathrm{F}} f_{\mathrm{FD}}(\epsilon) \exp(\epsilon / d_{\mathrm{F}}) = j_{\mathrm{F}} f_{\mathrm{FD}}(\epsilon) \exp (\epsilon / d_{\mathrm{F}}), </math>|{{EquationRef|21}}}}

where <math> d_{\mathrm{F}} </math> and <math> D_{\mathrm{F}} </math> are values appropriate to a barrier of unreduced height ''h'' equal to the local work function ''φ'', and <math> j_{\mathrm{F}} [ \,= z_{\mathrm{S}} d_{\mathrm{F}} D_{\mathrm{F}} ] </math> is defined by this equation.

For a given emitter, with a given field applied to it, <math> j_{\mathrm{F}} </math> is independent of ''F'', so eq. (21) shows that the shape of the distribution (as ''ε'' increases from a negative value well below the Fermi level) is a rising exponential, multiplied by the FD distribution function. This generates the familiar distribution shape first predicted by Young.<ref name=Y59/> At low temperatures, <math> f_{\mathrm{FD}} (\epsilon) </math> goes sharply from 1 to 0 in the vicinity of the Fermi level, and the [[Full width at half maximum|FWHM]] of the distribution is given by:

{{NumBlk|:|<math> \mathrm{FWHM} \, = d_{\mathrm{F}} \ln (2) \approx 0.693 \, d_{\mathrm{F}}. </math>|{{EquationRef|22}}}}

The fact that experimental CFE total energy distributions have this basic shape is a good experimental confirmation that electrons in metals obey [[Fermi–Dirac statistics]].