Editing Work function (section)

== Measurement ==

Certain physical phenomena are highly sensitive to the value of the work function. The observed data from these effects can be fitted to simplified theoretical models, allowing one to extract a value of the work function. These phenomenologically extracted work functions may be slightly different from the thermodynamic definition given above. For inhomogeneous surfaces, the work function varies from place to place, and different methods will yield different values of the typical "work function" as they average or select differently among the microscopic work functions.<ref name="pitfalls">{{Cite journal | last1 = Helander | first1 = M. G. | last2 = Greiner | first2 = M. T. | last3 = Wang | first3 = Z. B. | last4 = Lu | first4 = Z. H. | title = Pitfalls in measuring work function using photoelectron spectroscopy | doi = 10.1016/j.apsusc.2009.11.002 | journal = Applied Surface Science | volume = 256 | issue = 8 | pages = 2602 | year = 2010 |bibcode = 2010ApSS..256.2602H }}</ref>

Many techniques have been developed based on different physical effects to measure the electronic work function of a sample. One may distinguish between two groups of experimental methods for work function measurements: absolute and relative.

* Absolute methods employ electron emission from the sample induced by photon absorption (photoemission), by high temperature (thermionic emission), due to an electric field ([[field electron emission]]), or using [[quantum tunneling|electron tunnelling]].
* Relative methods make use of the [[contact potential difference]] between the sample and a reference electrode. Experimentally, either an anode current of a diode is used or the displacement current between the sample and reference, created by an artificial change in the capacitance between the two, is measured (the [[Kelvin probe force microscope|Kelvin Probe]] method, [[Kelvin probe force microscope]]). However, absolute work function values can be obtained if the tip is first calibrated against a reference sample.<ref name="calib">{{Cite journal | last1 = Fernández Garrillo | first1 = P. A. | last2 = Grévin | first2 = B. | last3 = Chevalier | first3 = N. | last4 = Borowik | first4 = Ł. | title = Calibrated work function mapping by Kelvin probe force microscopy | doi = 10.1063/1.5007619 | journal = Review of Scientific Instruments | volume = 89 | issue = 4 | pages = 043702 | year = 2018 | pmid =  29716375|bibcode = 2018RScI...89d3702F| url = https://hal.archives-ouvertes.fr/hal-02277068/file/Garrillo_2018_1.5007619.pdf }}</ref>

=== 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.

=== Methods based on photoemission ===

[[File:Photoelectric diode forward bias.svg|thumb|300 px|Photoelectric diode in ''forward bias'' configuration, used for measuring the work function ''W''<sub>e</sub> of the illuminated emitter.]]

The photoelectric work function is the minimum [[photon]] energy required to liberate an electron from a substance, in the [[photoelectric effect]].
If the photon's energy is greater than the substance's work function, [[photoelectric effect|photoelectric emission]] occurs and the electron is liberated from the surface.
Similar to the thermionic case described above, the liberated electrons can be extracted into a collector and produce a detectable current, if an electric field is applied into the surface of the emitter.
Excess photon energy results in a liberated electron with non-zero kinetic energy.
It is expected that the minimum [[photon energy]] <math> \hbar \omega </math> required to liberate an electron (and generate a current) is 
:<math>\hbar \omega = W_{\rm e} </math>
where ''W''<sub>e</sub> is the work function of the emitter.

Photoelectric measurements require a great deal of care, as an incorrectly designed experimental geometry can result in an erroneous measurement of work function.<ref name="pitfalls"/> This may be responsible for the large variation in work function values in scientific literature.
Moreover, the minimum energy can be misleading in materials where there are no actual electron states at the Fermi level that are available for excitation. For example, in a semiconductor the minimum photon energy would actually correspond to the [[valence band]] edge rather than work function.<ref>{{cite web|url=http://www.virginia.edu/ep/SurfaceScience/PEE.html|title=Photoelectron Emission|website=www.virginia.edu|access-date=11 April 2018}}</ref>

Of course, the photoelectric effect may be used in the retarding mode, as with the thermionic apparatus described above. In the retarding case, the dark collector's work function is measured instead.

=== Kelvin probe method ===
{{see also|Volta potential|Kelvin probe force microscope|Scanning Kelvin probe}}

[[File:Kelvin probe setup at flat vacuum.svg|thumb|300 px|Kelvin probe energy diagram at flat vacuum configuration, used for measuring work function difference between sample and probe.]]

The Kelvin probe technique relies on the detection of an electric field (gradient in ''ϕ'') between a sample material and probe material.
The electric field can be varied by the voltage Δ''V''<sub>sp</sub> that is applied to the probe relative to the sample.
If the voltage is chosen such that the electric field is eliminated (the flat vacuum condition), then
:<math>e\Delta V_{\rm sp} = W_{\rm s} - W_{\rm p}, \quad \text{when}~\phi~\text{is flat}.</math>
Since the experimenter controls and knows Δ''V''<sub>sp</sub>, then finding the flat vacuum condition gives directly the work function difference between the two materials.
The only question is, how to detect the flat vacuum condition?
Typically, the electric field is detected by varying the distance between the sample and probe. When the distance is changed but Δ''V''<sub>sp</sub> is held constant, a current will flow due to the change in [[capacitance]]. This current is proportional to the vacuum electric field, and so when the electric field is neutralized no current will flow.

Although the Kelvin probe technique only measures a work function difference, it is possible to obtain an absolute work function by first calibrating the probe against a reference material (with known work function) and then using the same probe to measure a desired sample.<ref name="calib"/>
The Kelvin probe technique can be used to obtain work function maps of a surface with extremely high spatial resolution, by using a sharp tip for the probe (see [[Kelvin probe force microscope]]).