View source for Thermionic emission

{{short description|Thermally induced flow of charge carriers from a surface}}
{{redirect-distinguish|Thermion|fermion}}
[[File:Thermionic filament.jpg|thumb|right|250px|Closeup of the [[electrical filament|filament]] in a [[mercury-vapor lamp|low pressure mercury]] [[gas-discharge lamp]] showing white thermionic emission mix coating on the central portion of the coil. Typically made of a mixture of [[barium]], [[strontium]] and [[calcium]] [[oxide]]s, the coating is [[sputtering|sputtered]] away through normal use, eventually resulting in lamp failure.]]
[[File:Edison light bulb with plate.jpg|thumb|One of the bulbs with which Edison discovered thermionic emission. It consists of an evacuated glass light bulb containing a [[carbon]] [[electrical filament|filament]] ''(hairpin shape)'', with an additional metal plate attached to wires emerging from the base. [[Electron]]s released by the filament were attracted to the plate when it had a positive voltage.]]

'''Thermionic emission''' (also known as '''thermal electron emission''' or the '''Edison effect''') is the liberation of [[electron]]s from an [[electrode]] by virtue of its [[temperature]] (releasing of energy supplied by [[heat]]). This occurs because the [[thermal energy]] given to the [[charge carrier]] overcomes the [[work function]] of the material. The charge carriers can be electrons or [[ion source|ions]], and in older literature are sometimes referred to as ''thermions''. After emission, a charge that is equal in magnitude and opposite in sign to the total charge emitted is initially left behind in the emitting region. But if the emitter is connected to a battery, the charge left behind is neutralized by charge supplied by the battery as the emitted charge carriers move away from the emitter, and finally the emitter will be in the same state as it was before emission.

The classical example of thermionic emission is that of electrons from a [[hot cathode]] into a [[vacuum]] in a [[vacuum tube]]. The hot cathode can be a metal filament, a coated metal filament, or a separate structure of metal or carbides or borides of transition metals. Vacuum emission from metals tends to become significant only for temperatures over {{convert|1000|K|C F}}.

This process is crucially important in the operation of a variety of electronic devices and can be used for [[electricity generation]] (such as [[thermionic converter]]s and [[electrodynamic tether]]s) or cooling. The magnitude of the charge flow increases dramatically with increasing temperature.

The term 'thermionic emission' is now also used to refer to any thermally-excited charge emission process, even when the charge is emitted from one [[solid-state physics|solid-state]] region into another.

== History ==
[[File:EdisonEffect.svg|frame|The Edison effect in a diode tube. A diode tube is connected in two configurations; one has a flow of electrons and the other does not. (The arrows represent electron current, not [[conventional current]].)]]

Because the [[electron]] was not identified as a separate physical particle until the work of [[J. J. Thomson]] in 1897, the word "electron" was not used when discussing experiments that took place before this date.

The phenomenon was initially reported in 1853 by [[Edmond Becquerel]].<ref>{{cite journal |last1=Becquerel |first1=Edmond |title=Reserches sur la conductibilité électrique des gaz à des températures élevées |journal=Comptes Rendus |date=1853 |volume=37 |pages=20–24 |url=https://www.biodiversitylibrary.org/item/16564#page/30/mode/1up |trans-title=Researches on the electrical conductivity of gases at high temperatures |language=French}}
* Extract translated into English: {{cite journal |last1=Becquerel |first1=E. |title=Researches on the electrical conductivity of gases at high temperatures |journal=Philosophical Magazine |date=1853 |volume=6 |pages=456–457 |url=https://www.biodiversitylibrary.org/item/20076#page/470/mode/1up |series=4th series}}</ref><ref>{{ cite thesis |last=Paxton |first=William Francis |title=Thermionic Electron Emission Properties of Nitrogen-Incorporated Polycrystalline Diamond Films |type=PhD dissertation |publisher=Vanderbilt University |date=18 April 2013 |url=https://etd.library.vanderbilt.edu/etd-03262013-131559 |access-date=2022-12-16 |archive-url=https://web.archive.org/web/20161123053917/http://etd.library.vanderbilt.edu/available/etd-03262013-131559/unrestricted/Paxton_Thesis.pdf |archive-date=2016-11-23 |hdl=1803/11438}}</ref><ref>{{ cite web |url=https://www.britannica.com/technology/thermionic-power-converter |title=Thermionic power converter |website=Encyclopedia Britannica |access-date=2016-11-22 |url-status=live |archive-url=https://web.archive.org/web/20161123055013/https://www.britannica.com/technology/thermionic-power-converter |archive-date=2016-11-23}}</ref> It was rediscovered in 1873 by [[Frederick Guthrie]] in Britain.<ref>{{ cite journal |first=Frederick |last=Guthrie |author-link=Frederick Guthrie |date=October 1873 |url=https://books.google.com/books?id=U08wAAAAIAAJ&pg=PA257 |title=On a relation between heat and static electricity |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |series=4th |volume=46 |issue= 306|pages=257–266 |url-status=live |archive-url=https://web.archive.org/web/20180113210540/https://books.google.com/books?id=U08wAAAAIAAJ&pg=PA257 |archive-date=2018-01-13 |doi=10.1080/14786447308640935}}</ref><ref>{{ cite journal |first=Frederick |last=Guthrie |author-link=Frederick Guthrie |date=February 13, 1873 |url=https://books.google.com/books?id=GLk9AQAAMAAJ&pg=PA168 |title=On a new relation between heat and electricity |journal=Proceedings of the Royal Society of London |volume=21 |issue= 139–147|pages=168–169 |doi=10.1098/rspl.1872.0037 |url-status=live |archive-url=https://web.archive.org/web/20180113210540/https://books.google.com/books?id=GLk9AQAAMAAJ&pg=PA168 |archive-date=January 13, 2018 |doi-access=free}}</ref> While doing work on charged objects, Guthrie discovered that a red-hot iron sphere with a negative charge would lose its charge (by somehow discharging it into air). He also found that this did not happen if the sphere had a positive charge.<ref>{{ cite book | last1 = Richardson | first1 = O. W. | year = 2003 | title = Thermionic Emission from Hot Bodies | url = https://books.google.com/books?id=PrbOIoMnxnwC&pg=PA196 | pages = 196 | publisher = [[Wexford College Press]] | isbn = 978-1-929148-10-3 | url-status = live | archive-url = https://web.archive.org/web/20131231221553/http://books.google.com/books?id=PrbOIoMnxnwC&pg=PA196 | archive-date = 2013-12-31 }}</ref> Other early contributors included [[Johann Wilhelm Hittorf]] (1869–1883),<ref>{{ cite journal |first=W. |last=Hittorf |author-link=Johann Wilhelm Hittorf |year=1869 |url=http://babel.hathitrust.org/cgi/pt?id=uc1.b4433674;view=1up;seq=25 |title=Ueber die Electricitätsleitung der Gase |language=de |trans-title=On electrical conduction of gases |journal=Annalen der Physik und Chemie |series=2nd series |volume=136 |issue=1 |pages=1–31 |doi=10.1002/andp.18692120102 |bibcode=1869AnP...212....1H}}</ref><ref>{{ cite journal |first=W. |last=Hittorf |author-link=Johann Wilhelm Hittorf |year=1869 |url=http://babel.hathitrust.org/cgi/pt?id=uc1.b4433674;view=1up;seq=221 |title=Ueber die Electricitätsleitung der Gase |language=de |trans-title=On electrical conduction of gases |journal=Annalen der Physik und Chemie |series=2nd series |volume=136 |issue= 2|pages=197–234 |doi=10.1002/andp.18692120203 |bibcode=1869AnP...212..197H}}</ref><ref>{{ cite journal |first=W. |last=Hittorf |author-link=Johann Wilhelm Hittorf |year=1874 |url=http://gallica.bnf.fr/ark:/12148/bpt6k149954/f452.image.langEN |title=Ueber die Electricitätsleitung der Gase |language=de |trans-title=On electrical conduction of gases |journal=Annalen der Physik und Chemie |volume=Jubalband (anniversary volume) |pages=430–445 |url-status=live |archive-url=https://web.archive.org/web/20180113210541/http://gallica.bnf.fr/ark:/12148/bpt6k149954/f452.image.langEN |archive-date=2018-01-13}}</ref><ref>{{ cite journal |first=W. |last=Hittorf |author-link=Johann Wilhelm Hittorf |year=1879 |url=http://babel.hathitrust.org/cgi/pt?id=wu.89048352421;view=1up;seq=575 |title=Ueber die Electricitätsleitung der Gase |language=de |trans-title=On electrical conduction of gases |journal=Annalen der Physik und Chemie |series=3rd series |volume=7 |issue= 8|pages=553–631 |doi=10.1002/andp.18792430804 |bibcode=1879AnP...243..553H}}</ref><ref>{{ cite journal |first=W. |last=Hittorf |author-link=Johann Wilhelm Hittorf |year=1883 |url=http://babel.hathitrust.org/cgi/pt?id=wu.89048352561;view=1up;seq=727 |title=Ueber die Electricitätsleitung der Gase |language=de |trans-title=On electrical conduction of gases |journal=Annalen der Physik und Chemie |series=3rd series |volume=20 |issue= 12|pages=705–755 |doi=10.1002/andp.18832561214}}</ref><ref>{{ cite journal |first=W. |last=Hittorf |author-link=Johann Wilhelm Hittorf |year=1884 |url=http://babel.hathitrust.org/cgi/pt?id=wu.89048352587;view=1up;seq=106 |title=Ueber die Electricitätsleitung der Gase |language=de |trans-title=On electrical conduction of gases |journal=Annalen der Physik und Chemie |series=3rd series |volume=21 |issue= 1|pages=90–139 |doi=10.1002/andp.18842570105 |bibcode=1884AnP...257...90H}}</ref> [[Eugen Goldstein]] (1885),<ref>E. Goldstein (1885) [https://books.google.com/books?id=yhVbAAAAYAAJ&pg=PA79 "Ueber electrische Leitung in Vacuum"] {{webarchive|url=https://web.archive.org/web/20180113210540/https://books.google.com/books?id=yhVbAAAAYAAJ&pg=PA79 |date=2018-01-13}} (On electric conduction in vacuum) ''Annalen der Physik und Chemie'', 3rd series, ''24'': 79–92.</ref> and [[Julius Elster]] and [[Hans Friedrich Geitel]] (1882–1889).<ref>Elster and Geitel (1882) [http://babel.hathitrust.org/cgi/pt?id=wu.89048352488;view=1up;seq=211 "Ueber die Electricität der Flamme"] (On the electricity of flames), ''Annalen der Physik und Chemie'', 3rd series, ''16'': 193–222.</ref><ref>Elster and Geitel (1883) [http://babel.hathitrust.org/cgi/pt?id=wu.89048352546;view=1up;seq=612 "Ueber Electricitätserregung beim Contact von Gasen und glühenden Körpern"] (On the generation of electricity by the contact of gases and incandescent bodies), ''Annalen der Physik und Chemie'', 3rd series, ''19'': 588–624.</ref><ref>Elster and Geitel (1885) [http://babel.hathitrust.org/cgi/pt?id=uc1.b4433723;view=1up;seq=15 "Ueber die unipolare Leitung erhitzter Gase"] (On the unipolar conductivity of heated gases") ''Annalen der Physik und Chemie'', 3rd series, ''26'': 1–9.</ref><ref>Elster and Geitel (1887) [http://babel.hathitrust.org/cgi/pt?id=uc1.b4433728;view=1up;seq=123 "Ueber die Electrisirung der Gase durch glühende Körper"] (On the electrification of gases by incandescent bodies") ''Annalen der Physik und Chemie'', 3rd series, ''31'': 109–127.</ref><ref>Elster and Geitel (1889) [http://babel.hathitrust.org/cgi/pt?id=uc1.b4433733;view=1up;seq=333 "Ueber die Electricitätserregung beim Contact verdünnter Gase mit galvanisch glühenden Drähten"] (On the generation of electricity by contact of rarefied gas with electrically heated wires) ''Annalen der Physik und Chemie'', 3rd series, ''37'': 315–329.</ref>

The effect was rediscovered again by [[Thomas Edison]] on February 13, 1880, while he was trying to discover the reason for breakage of lamp filaments and uneven blackening (darkest near the positive terminal of the filament) of the bulbs in his [[incandescent lamp]]s.

Edison built several experimental lamp bulbs with an extra wire, metal plate, or foil inside the bulb that was separate from the filament and thus could serve as an electrode. He connected a [[galvanometer]], a device used to measure [[electric current]] (the flow of charge), to the output of the extra metal electrode. If the foil was put at a negative [[electric potential]] relative to the filament, there was no measurable current between the filament and the foil. When the foil was raised to a positive potential relative to the filament, there could be a significant current between the filament through the vacuum to the foil if the filament was heated sufficiently (by its own external power source).

We now know that the filament was emitting electrons, which were attracted to a positively charged foil, but not to a negatively charged one. This one-way current was called the ''Edison effect'' (although the term is occasionally used to refer to thermionic emission itself). He found that the current emitted by the hot filament increased rapidly with increasing voltage, and filed a patent application for a voltage-regulating device using the effect on November 15, 1883 (U.S. patent 307,031,<ref>{{ cite patent|inventor1-first=Thomas A. |inventor1-last=Edison |inventorlink1=Thomas Edison |title=Electrical indicator |country=US |number=307031 |fdate=1883-11-15 |pubdate= 1884-10-21}}</ref> the first US patent for an electronic device). He found that sufficient current would pass through the device to operate a telegraph sounder. This was exhibited at the [[International Electrical Exposition]] in Philadelphia in September 1884. [[William Preece]], a British scientist, took back with him several of the Edison effect bulbs. He presented a paper on them in 1885, where he referred to thermionic emission as the "Edison effect."<ref>{{ cite journal |first=William Henry |last=Preece |author-link=William Preece |year=1885 |url=https://books.google.com/books?id=xmdDAAAAYAAJ&pg=PA219 |title=On a peculiar behaviour of glow lamps when raised to high incandescence |journal=Proceedings of the Royal Society of London |volume=38 |issue= 235–238|pages=219–230 |doi=10.1098/rspl.1884.0093 |url-status=live |archive-url=https://web.archive.org/web/20140626213555/http://books.google.com/books?id=xmdDAAAAYAAJ&pg=PA219 |archive-date=2014-06-26 |doi-access=free}} Preece coins the term the "Edison effect" on page 229.</ref><ref name=Josephson>{{ cite book | last1 = Josephson | first1 = M. | year = 1959 | title = Edison | url = https://archive.org/details/edisonbiography00jose | url-access = registration | publisher = [[McGraw-Hill]] | isbn = 978-0-07-033046-7 }}</ref> The British physicist [[John Ambrose Fleming]], working for the British "Wireless Telegraphy" Company, discovered that the Edison effect could be used to detect radio waves. Fleming went on to develop the two-element [[vacuum tube]] known as the [[diode#Vacuum tube diodes|diode]], which he patented on November 16, 1904.<ref>Provisional specification for a thermionic valve was lodged on November 16, 1904. In this document, Fleming coined the British term "valve" for what in North America is called a "vacuum tube": "The means I employ for this purpose consists in the insertion in the circuit of the alternating current of an appliance which permits only the passage of electric current in one direction and constitutes therefore an electrical valve."</ref><ref>{{ cite patent|inventor1-first=John Ambrose |inventor1-last=Fleming |inventor1-link=John Ambrose Fleming |title=Improvements in instruments for detecting and measuring alternating electric currents |country=GB |number=190424850 |fdate=1905-08-15 |pubdate=1905-09-21}}</ref><ref>{{ cite patent|inventor1-first=John Ambrose |inventor1-last=Fleming |inventor1-link=John Ambrose Fleming |title=Instrument for converting alternating electric currents into continuous currents |country=US |number=803684 |fdate=1905-04-19 |pubdate=1905-11-07}}</ref>

The thermionic diode can also be configured as a device that converts a heat difference to electric power directly without moving parts (a [[thermionic converter]], a type of [[heat engine]]).

== Richardson's law ==

Following 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. 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 [[Arrhenius equation]].<ref>O. W. Richardson (1901) [https://books.google.com/books?id=QAUPAAAAIAAJ&pg=PA286 "On the negative radiation from hot platinum,"] ''Philosophical of the Cambridge Philosophical Society'', ''11'': 286-295.</ref> Later, he proposed that the emission law should have the mathematical form<ref name="Crowell"/>{{failed verification|date=May 2022}}
:<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_e \over h^3} = 1.20173 \times 10^6\,\mathrm{A\,m^{-2}\,K^{-2}}</math>

where ''m'' and <math>-q_e</math> are the mass and [[elementary charge|charge]] of an electron, respectively, and ''h'' is [[Planck's 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 (1-''r''<sub>av</sub>). Thus, one sometimes sees the thermionic emission equation written in the form:

:<math>J = (1-r_{\mathrm{av}})\lambda_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 ''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>

== Schottky emission ==
{{Main|Schottky effect}}
[[File:Schottky-Emitter 01.jpg|thumb|Schottky-emitter electron source of an [[electron microscope]]]]

In electron emission devices, especially [[electron gun]]s, the thermionic electron emitter will be biased negative relative to its surroundings. This creates an electric field of magnitude ''E'' at the emitter surface. Without the field, the surface barrier seen by an escaping Fermi-level electron has height ''W'' equal to the local work-function. The electric field lowers the surface barrier by an amount Δ''W'', and increases the emission current. This is known as the ''Schottky effect'' (named for [[Walter H. Schottky]]) or field enhanced thermionic emission. It can be modeled by a simple modification of the Richardson equation, by replacing ''W'' by (''W''&nbsp;−&nbsp;Δ''W''). This gives the equation<ref>{{ cite journal | last1 = Kiziroglou |first1 = M. E. | last2 = Li |first2 = X. | last3 = Zhukov |first3 = A. A. | last4 = De Groot |first4 = P. A. J. | last5 = De Groot |first5 = C. H. | year = 2008 | title = Thermionic field emission at electrodeposited Ni-Si Schottky barriers | journal = [[Solid-State Electronics]] | volume = 52 |issue = 7 |pages = 1032–1038 | bibcode = 2008SSEle..52.1032K | doi = 10.1016/j.sse.2008.03.002
|url = https://eprints.soton.ac.uk/143941/1/Thermionic_field_emission_at_electrodeposited_Ni%25E2%2580%2593Si_Schottky_barriers.pdf }}</ref><ref name=Orloff>{{ cite book | last1 = Orloff | first1 = J. | year = 2008 | chapter = Schottky emission | chapter-url = https://books.google.com/books?id=y0FF19lud5YC&pg=PA6 | title = Handbook of Charged Particle Optics | pages = 5–6 | edition = 2nd | publisher = [[CRC Press]] | isbn = 978-1-4200-4554-3 | url-status = live | archive-url = https://web.archive.org/web/20170117111100/https://books.google.it/books?id=y0FF19lud5YC&pg=PA6&lpg=PA6&dq= | archive-date = 2017-01-17 }}</ref>

:<math>J (E,T,W) = A_{\mathrm{G}} T^2 e^{ - (W - \Delta W) \over k T}</math>
:<math>\Delta W = \sqrt{{q_e}^3 E \over 4\pi \epsilon_0},</math>

where ''ε''<sub>0</sub> is the electric constant (also, formerly, called the [[vacuum permittivity]]).

Electron emission that takes place in the field-and-temperature-regime where this modified equation applies is often called [[Schottky emission]]. This equation is relatively accurate for electric field strengths lower than about 10<sup>8</sup> V&nbsp; m<sup>−1</sup>. For electric field strengths higher than 10<sup>8</sup> V&nbsp;m<sup>−1</sup>, so-called [[field electron emission|Fowler-Nordheim (FN) tunneling]] begins to contribute significant emission current. In this regime, the combined effects of field-enhanced thermionic and field emission can be modeled by the Murphy-Good equation for thermo-field (T-F) emission.<ref>{{ cite journal | last1 = Murphy |first1 = E. L. | last2 = Good |first2 = G. H. | year = 1956 | title = Thermionic Emission, Field Emission, and the Transition Region | journal = [[Physical Review]] | volume = 102 |issue = 6 |pages = 1464–1473 | bibcode = 1956PhRv..102.1464M | doi = 10.1103/PhysRev.102.1464 }}</ref> At even higher fields, FN tunneling becomes the dominant electron emission mechanism, and the emitter operates in the so-called [[field electron emission|"cold field electron emission (CFE)"]] regime.

Thermionic emission can also be enhanced by interaction with other forms of excitation such as light.<ref>{{ cite journal | last1 = Mal'Shukov |first1 = A. G. | last2 = Chao |first2 = K. A. | year = 2001 | title = Opto-Thermionic Refrigeration in Semiconductor Heterostructures | journal = [[Physical Review Letters]] | volume = 86 |issue = 24|pages = 5570–5573 | bibcode = 2001PhRvL..86.5570M | doi = 10.1103/PhysRevLett.86.5570 |pmid = 11415303 }}</ref> For example, excited Cs-vapours in thermionic converters form clusters of Cs-[[Rydberg matter]] which yield a decrease of collector emitting work function from 1.5 eV to 1.0–0.7 eV. Due to long-lived nature of [[Rydberg matter]] this low work function remains low which essentially increases the low-temperature converter's efficiency.<ref>{{ cite journal | last1 = Svensson |first1 = R. | last2 = Holmlid |first2 = L. | year = 1992 | title = Very low work function surfaces from condensed excited states: Rydber matter of cesium | journal = [[Surface Science (journal)|Surface Science]] | volume = 269/270 |pages = 695–699 | bibcode = 1992SurSc.269..695S | doi = 10.1016/0039-6028(92)91335-9 }}</ref>

== Photon-enhanced thermionic emission ==

Photon-enhanced thermionic emission (PETE) is a process developed by scientists at [[Stanford University]] that harnesses both the light and heat of the sun to generate electricity and increases the efficiency of solar power production by more than twice the current levels. The device developed for the process reaches peak efficiency above 200&nbsp;°C, while most silicon [[solar cell]]s become inert after reaching 100&nbsp;°C. Such devices work best in [[parabolic dish]] collectors, which reach temperatures up to 800&nbsp;°C. Although the team used a [[gallium nitride]] semiconductor in its proof-of-concept device, it claims that the use of [[gallium arsenide]] can increase the device's efficiency to 55–60 percent, nearly triple that of existing systems,<ref>{{ cite news | last = Bergeron | first = L. | title = New solar energy conversion process discovered by Stanford engineers could revamp solar power production | url = http://news.stanford.edu/news/2010/august/new-solar-method-080210.html | work = Stanford Report | date = 2 August 2010 | access-date = 2010-08-04 | url-status = live | archive-url = https://web.archive.org/web/20110411040334/http://news.stanford.edu/news/2010/august/new-solar-method-080210.html | archive-date = 11 April 2011 }}</ref><ref>{{ cite journal | last1 = Schwede | first1 = J. W. | year = 2010 | title = Photon-enhanced thermionic emission for solar concentrator systems | journal = [[Nature Materials]] | volume = 9 | issue = 9 | pages = 762–767 | bibcode = 2010NatMa...9..762S | doi = 10.1038/nmat2814 | pmid = 20676086 | display-authors = etal }}</ref> and 12–17 percent more than existing 43 percent multi-junction solar cells.<ref>{{ cite journal | last1 = Green | first1 = M. A. | last2 = Emery | first2 = K. | last3 = Hishikawa | first3 = Y. | last4 = Warta | first4 = W. | year = 2011 | title = Solar cell efficiency tables (version 37) | journal = [[Progress in Photovoltaics: Research and Applications]] | volume = 19 | issue = 1 | pages = 84 | doi = 10.1002/pip.1088 | s2cid = 97915368 | doi-access = free }}</ref><ref>{{ cite journal | last1 = Ang | first1 = Yee Sin | last2 = Ang | first2 = L. K. | year = 2016 | title = Current-Temperature Scaling for a Schottky Interface with Nonparabolic Energy Dispersion | journal = [[Physical Review Applied]] | volume = 6 |issue = 3 | pages = 034013 | doi = 10.1103/PhysRevApplied.6.034013 | arxiv = 1609.00460 | bibcode = 2016PhRvP...6c4013A | s2cid = 119221695 }}</ref>

== See also ==

* [[Space charge]]

== References ==

{{refs}}

== External links ==

* [http://www.john-a-harper.com/tubes201/ How vacuum tubes really work with a section on thermionic emission, with equations], john-a-harper.com.
* [https://www.nobelprize.org/uploads/2018/06/richardson-lecture.pdf Thermionic Phenomena and the Laws which Govern Them], Owen Richardson's Nobel lecture on thermionics. nobelprize.org. December 12, 1929. (PDF)
* [http://www.physics.csbsju.edu/lab/thermionic.pdf Derivations of thermionic emission equations from an undergraduate lab] {{Webarchive|url=https://web.archive.org/web/20120205211720/http://www.physics.csbsju.edu/lab/thermionic.pdf |date=2012-02-05 }}, csbsju.edu.

{{Thermionic valves}}
{{Thomas Edison}}

[[Category:Atomic physics]]
[[Category:Electricity]]
[[Category:Energy conversion]]
[[Category:Vacuum tubes]]
[[Category:Thomas Edison]]