Editing Work function (section)

=== Methods based on thermionic emission ===

The work function is important in the theory of [[thermionic emission]], where thermal fluctuations provide enough energy to "evaporate" electrons out of a hot material (called the 'emitter') into the vacuum. If these electrons are absorbed by another, cooler material (called the ''collector'') then a measurable [[electric current]] will be observed. Thermionic emission can be used to measure the work function of both the hot emitter and cold collector. Generally, these measurements involve fitting to [[Richardson's law]], and so they must be carried out in a low temperature and low current regime where [[space charge]] effects are absent.

[[File:Thermionic diode forward bias.svg|thumb|300px|Energy level diagrams for [[thermionic diode]] in ''forward bias'' configuration, used to extract all hot electrons coming out from the emitter's surface. The barrier is the vacuum near emitter surface.]]

In order to move from the hot emitter to the vacuum, an electron's energy must exceed the emitter Fermi level by an amount
:<math>E_{\rm barrier} = W_{\rm e}</math>
determined simply by the thermionic work function of the emitter.
If an electric field is applied towards the surface of the emitter, then all of the escaping electrons will be accelerated away from the emitter and absorbed into whichever material is applying the electric field.
According to [[Richardson's law]] the emitted [[current density]] (per unit area of emitter), ''J''<sub>e</sub> (A/m<sup>2</sup>), is related to the absolute [[temperature]] ''T''<sub>e</sub> of the emitter by the equation:
:<math>J_{\rm e} = -A_{\rm e} T_{\rm e}^2 e^{-E_{\rm barrier} / k T_{\rm e}}</math>
where ''k'' is the [[Boltzmann constant]] and the proportionality constant ''A''<sub>e</sub> is the [[Richardson's constant]] of the emitter.
In this case, the dependence of ''J''<sub>e</sub> on ''T''<sub>e</sub> can be fitted to yield ''W''<sub>e</sub>.

==== Work function of cold electron collector ====
[[File:Thermionic diode reverse bias.svg|thumb|300px|Energy level diagrams for [[thermionic diode]] in ''retarding potential'' configuration. The barrier is the vacuum near collector surface.]]

The same setup can be used to instead measure the work function in the collector, simply by adjusting the applied voltage.
If an electric field is applied ''away from'' the emitter instead, then most of the electrons coming from the emitter will simply be reflected back to the emitter. Only the highest energy electrons will have enough energy to reach the collector, and the height of the potential barrier in this case depends on the collector's work function, rather than the emitter's.

The current is still governed by Richardson's law. However, in this case the barrier height does not depend on ''W''<sub>e</sub>. The barrier height now depends on the work function of the collector, as well as any additional applied voltages:<ref>G.L. Kulcinski, "Thermionic Energy
Conversion" [http://fti.neep.wisc.edu/neep602/SPRING00/lecture9.pdf] {{Webarchive|url=https://web.archive.org/web/20171117230631/http://fti.neep.wisc.edu/neep602/SPRING00/lecture9.pdf|date=2017-11-17}}</ref>
:<math>E_{\rm barrier} = W_{\rm c} - e (\Delta V_{\rm ce} - \Delta V_{\rm S})</math>
where ''W''<sub>c</sub> is the collector's thermionic work function, Δ''V''<sub>ce</sub> is the applied collector–emitter voltage, and Δ''V''<sub>S</sub> is the [[Seebeck effect|Seebeck voltage]] in the hot emitter (the influence of Δ''V''<sub>S</sub> is often omitted, as it is a small contribution of order 10 mV).
The resulting current density ''J''<sub>c</sub> through the collector (per unit of collector area) is again given by [[Richardson's Law]], except now
:<math>J_{\rm c} = A T_{\rm e}^2 e^{-E_{\rm barrier}/kT_{\rm e}} </math>
where ''A'' is a Richardson-type constant that depends on the collector material but may also depend on the emitter material, and the diode geometry.
In this case, the dependence of ''J''<sub>c</sub> on ''T''<sub>e</sub>, or on Δ''V''<sub>ce</sub>, can be fitted to yield ''W''<sub>c</sub>.

This '''retarding potential method''' is one of the simplest and oldest methods of measuring work functions, and is advantageous since the measured material (collector) is not required to survive high temperatures.