Editing Field electron emission (section)

==== Zero-temperature form ====
Current density is best measured in A/m<sup>2</sup>. The total current density emitted from a small uniform region can be obtained by integrating the total energy distribution ''j''(''ε'') with respect to total electron energy ''ε''. At zero temperature, the [[Fermi–Dirac statistics|Fermi–Dirac distribution function]] {{nowrap|1=''f''<sub>FD</sub> = 1}} for {{nowrap|''ε'' < 0}}, and {{nowrap|1=''f''<sub>FD</sub> = 0}} for {{nowrap|1=''ε'' > 0}}. So the ECD at 0&nbsp;K, ''J''<sub>0</sub>, is given from eq. (18) by

{{NumBlk|:|<math> J_0 = z_{\mathrm{S}} d_{\mathrm{F}} D_{\mathrm{F}} \int_{-\infty}^{0} \exp(\epsilon /
d_{\mathrm{F}}) \; \mathrm{d} \epsilon \; = \; z_{\mathrm{S}} {d_{\mathrm{F}}}^2 D_{\mathrm{F}} \; = \; Z_{\mathrm{F}} D_{\mathrm{F}}, </math>|{{EquationRef|23}}}}

where <math> Z_{\mathrm{F}} \; [=z_{\mathrm{S}} {d_{\mathrm{F}}}^2] </math> is the ''effective supply for state F'', and is defined by this equation. Strictly, the lower limit of the integral should be −''K''<sub>F</sub>, where ''K''<sub>F</sub> is the [[Fermi energy]]; but if ''d''<sub>F</sub> is very much less than ''K''<sub>F</sub> (which is always the case for a metal) then no significant contribution to the integral comes from energies below ''K''<sub>F</sub>, and it can formally be extended to&nbsp;–∞.

Result (23) can be given a simple and useful physical interpretation by referring to Fig. 1. The electron state at point "F" on the diagram ("state F") is the "forwards moving state at the Fermi level" (i.e., it describes a Fermi-level electron moving normal to and towards the emitter surface). At 0&nbsp;K, an electron in this state sees a barrier of unreduced height ''φ'', and has an escape probability ''D''<sub>F</sub> that is higher than that for any other occupied electron state. So it is convenient to write ''J''<sub>0</sub> as ''Z''<sub>F</sub>''D''<sub>F</sub>, where the "effective supply" ''Z''<sub>F</sub> is the current density that would have to be carried by state F inside the metal if all of the emission came out of state F.

In practice, the current density mainly comes out of a group of states close in energy to state F, most of which lie within the heavily shaded area in the energy-space diagram. Since, for a [[free electron model|free-electron model]], the contribution to the current density is directly proportional to the area in energy space (with the Sommerfeld supply density ''z''<sub>S</sub> as the constant of proportionality), it is useful to think of the ECD as drawn from electron states in an area of size ''d''<sub>F</sub><sup>2</sup> (measured in eV<sup>2</sup>) in the energy-space diagram. That is, it is useful to think of the ECD as drawn from states in the heavily shaded area in Fig. 1. (This approximation gets slowly worse as temperature increases.)

''Z''<sub>F</sub> can also be written in the form:

{{NumBlk|:|<math> Z_{\mathrm{F}} =z_{\mathrm{S}} {d_{\mathrm{F}}}^2= {\lambda_d}^2 (z_{\mathrm{S}} e^2 g^{-2}) \phi^{-1} F^2 = {\lambda_d}^2 a \phi^{-1} F^2, </math>|{{EquationRef|24}}}}

where the universal constant ''a'', sometimes called the ''First Fowler–Nordheim Constant'', is given by

{{NumBlk|:|<math> a = z_{\mathrm{S}} e^2 g^{-2} = e^3 /8 \pi h_{\mathrm{P}} \approx \; 1.541434 \times 10^{-6} \; \mathrm{A \; eV} \; {\mathrm{V}}^{-2}. </math>|{{EquationRef|25}}}}

This shows clearly that the pre-exponential factor ''aφ''<sup>−1</sup>''F''<sup>2</sup>, that appears in Fowler–Nordheim-type equations, relates to the effective supply of electrons to the emitter surface, in a free-electron model.