Editing Thermionic emission (section)

== Richardson's law ==

Following [[J. J. Thomson#Discovery of the electron|J. J. Thomson's identification of the electron]] in 1897, the British physicist [[Owen Willans Richardson]] began work on the topic that he later called "thermionic emission". He received a [[Nobel Prize in Physics]] in 1928 "for his work on the thermionic phenomenon and especially for the discovery of the law named after him".

From [[band theory]], there are one or two electrons per [[atom]] in a solid that are free to move from atom to atom. This is sometimes collectively referred to as a "sea of electrons". Their velocities follow a statistical distribution, rather than being uniform, and occasionally an electron will have enough velocity to exit the metal without being pulled back in. The minimum amount of energy needed for an electron to leave a surface is called the [[work function]]. The work function is characteristic of the material and for most metals is on the order of several [[electronvolt]]s (eV). Thermionic currents can be increased by decreasing the work function. This often-desired goal can be achieved by applying various oxide coatings to the wire.

In 1901 [[Owen Willans Richardson|Richardson]] published the results of his experiments: the current from a heated wire seemed to depend exponentially on the temperature of the wire with a mathematical form similar to the modified [[Arrhenius equation]], <math>T^{1/2} \mathrm{e}^{-b/T}</math>.<ref>{{cite journal | journal = Proceedings of the Cambridge Philosophical Society | author = O. W. Richardson | year = 1901 | url = https://www.biodiversitylibrary.org/item/108210#page/314/mode/1up | pages=286–295 | volume=11 | title = On the negative radiation from hot platinum}}</ref> Later, he proposed that the emission law should have the mathematical form<ref>While the empirical data favoured both the <math>T^{1/2} \mathrm{e}^{-b/T}</math> and <math>T^{2} \mathrm{e}^{-c/T}</math> forms, Richardson preferred the latter, stating that it was theoretically better founded.{{cite book | title = The emission of electricity from hot bodies, 2nd edition | author = Owen Willans Richardson | year = 1921 | pages=63–64 | url = https://archive.org/details/emissionelectricity00richrich/page/62/mode/2up}}</ref>
: <math>J = A_{\mathrm{G}} T^2 \mathrm{e}^{-W \over k T}</math>
where ''J'' is the emission [[current density]], ''T'' is the temperature of the metal, ''W'' is the [[work function]] of the metal, ''k'' is the [[Boltzmann constant]], and ''A''<sub>G</sub> is a parameter discussed next.

In the period 1911 to 1930, as physical understanding of the behaviour of electrons in metals increased, various theoretical expressions (based on different physical assumptions) were put forward for ''A''<sub>G</sub>, by Richardson, [[Saul Dushman]], [[Ralph H. Fowler]], [[Arnold Sommerfeld]] and [[Lothar Wolfgang Nordheim]]. Over 60 years later, there is still no consensus among interested theoreticians as to the exact expression of ''A''<sub>G</sub>, but there is agreement that ''A''<sub>G</sub> must be written in the form:
: <math> A_{\mathrm{G}} = \; \lambda_{\mathrm{R}} A_0 </math>
where ''λ''<sub>R</sub> is a material-specific correction factor that is typically of order 0.5, and ''A''<sub>0</sub> is a universal constant given by<ref name="Crowell">{{ cite journal | last1 = Crowell |first1 = C. R. | year = 1965 | title = The Richardson constant for thermionic emission in Schottky barrier diodes | journal = [[Solid-State Electronics]] | volume = 8 |issue = 4 |pages = 395–399 | bibcode = 1965SSEle...8..395C | doi = 10.1016/0038-1101(65)90116-4 }}</ref>
: <math>A_0 = {4 \pi m k^2 q_\text{e} \over h^3} = 1.20173 \times 10^6\,\mathrm{A{\cdot}m^{-2}{\cdot}K^{-2}}</math>
where <math>m</math> and <math>-q_\text{e}</math> are the mass and [[elementary charge|charge]] of an electron, respectively, and <math>h</math> is the [[Planck constant]].

In fact, by about 1930 there was agreement that, due to the wave-like nature of electrons, some proportion ''r''<sub>av</sub> of the outgoing electrons would be reflected as they reached the emitter surface, so the emission current density would be reduced, and ''λ''<sub>R</sub> would have the value {{nowrap|1 − ''r''<sub>av</sub>}}. Thus, one sometimes sees the thermionic emission equation written in the form:
: <math>J = (1-r_{\mathrm{av}})\lambda_\text{B} A_0 T^2 \mathrm{e}^{-W \over k T}</math>.

However, a modern theoretical treatment by Modinos assumes that the [[band theory|band-structure]] of the emitting material must also be taken into account. This would introduce a second correction factor ''λ''<sub>B</sub> into ''λ''<sub>R</sub>, giving <math> A_{\mathrm{G}} = \lambda_{\mathrm{B}} (1-r_{\mathrm{av}}) A_0 </math>. Experimental values for the "generalized" coefficient ''A''<sub>G</sub> are generally of the order of magnitude of ''A''<sub>0</sub>, but do differ significantly as between different emitting materials, and can differ as between different [[crystallographic face]]s of the same material. At least qualitatively, these experimental differences can be explained as due to differences in the value of ''λ''<sub>R</sub>.

Considerable confusion exists in the literature of this area because: (1) many sources do not distinguish between ''A''<sub>G</sub> and ''A''<sub>0</sub>, but just use the symbol ''A'' (and sometimes the name "Richardson constant") indiscriminately; (2) equations with and without the correction factor here denoted by ''λ''<sub>R</sub> are both given the same name; and (3) a variety of names exist for these equations, including "Richardson equation", "Dushman's equation", "Richardson–Dushman equation" and "Richardson–Laue–Dushman equation". In the literature, the elementary equation is sometimes given in circumstances where the generalized equation would be more appropriate, and this in itself can cause confusion. To avoid misunderstandings, the meaning of any "A-like" symbol should always be explicitly defined in terms of the more fundamental quantities involved.

Because of the exponential function, the current increases rapidly with temperature when ''kT'' is less than ''W''.{{explain|reason="This statement does not make sense because T is in the *denominator* of the exponent."|date=May 2023}} (For essentially every material, melting occurs well before {{nowrap|1=''kT'' = ''W''}}.)

The thermionic emission law has been recently revised for 2D materials in various models.<ref>{{ cite journal | author1 = S. J. Liang and L. K. Ang | journal = [[Physical Review Applied]] | title = Electron Thermionic Emission from Graphene and a Thermionic Energy Converter | date = January 2015 | volume = 3 | issue = 1 | pages = 014002 | doi = 10.1103/PhysRevApplied.3.014002 | arxiv = 1501.05056 | bibcode = 2015PhRvP...3a4002L | s2cid = 55920889}}</ref><ref>{{ cite journal | author1 = Y. S. Ang, H. Y. Yang and L. K. Ang | journal = [[Physical Review Letters]] | title = Universal scaling in nanoscale lateral Schottky heterostructures | date = August 2018 | volume = 121 | issue = 5 | pages = 056802 | doi = 10.1103/PhysRevLett.121.056802 | pmid = 30118283 | arxiv = 1803.01771 | s2cid = 206314695 }}</ref><ref>{{ cite journal | author1 = Y. S. Ang, Xueyi Chen, Chuan Tan and L. K. Ang | journal = [[Physical Review Applied]] | title = Generalized high-energy thermionic electron injection at graphene interface | date = July 2019 | volume = 12 | issue = 1 | pages = 014057 | doi = 10.1103/PhysRevApplied.12.014057|arxiv = 1907.07393 | bibcode = 2019PhRvP..12a4057A | s2cid = 197430947 }}</ref>