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→Work functions of elements: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.26.380 |
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== Definition ==
The work function {{math|''W''}} for a given surface is defined by the difference<ref name="Kittel">{{cite book |title=[[Introduction to Solid State Physics]] |edition=7th |last1=Kittel |first1=Charles |author-link1=Charles Kittel |date=
:<math>W = -e\phi - E_{\rm F}, </math>
where {{math|−''e''}} is the charge of an [[electron]], {{math|''ϕ''}} is the [[electrostatic potential]] in the vacuum nearby the surface, and {{math|''E''<sub>F</sub>}} is the [[Fermi level]] ([[electrochemical potential]] of electrons) inside the material. The term {{math|−''eϕ''}} is the energy of an electron at rest in the vacuum nearby the surface.
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[[File:Work function mismatch gold aluminum.svg|thumb|300 px|Plot of electron energy levels against position, in a gold-vacuum-aluminium system. The two metals depicted here are in complete thermodynamic equilibrium. However, the vacuum [[electrostatic potential]] {{math|''ϕ''}} is not flat due to a difference in work function.]]
In practice, one directly controls {{math|''E''<sub>F</sub>}} by the voltage applied to the material through electrodes, and the work function is generally a fixed characteristic of the surface material. Consequently, this means that when a [[voltage]] is applied to a material, the electrostatic potential {{math|''ϕ''}} produced in the vacuum will be somewhat lower than the applied voltage, the difference depending on the work function of the material surface. Rearranging the above equation, one has
:<math>\phi = V - \frac{W}{e}</math>
where {{math|''V'' {{=}} −''E''<sub>F</sub> / ''e''}} is the voltage of the material (as measured by a [[voltmeter]], through an attached electrode), relative to an [[electrical ground]] that is defined as having zero Fermi level. The fact that {{math|''ϕ''}} depends on the material surface means that the space between two dissimilar conductors will have a built-in [[electric field]], when those conductors are in total equilibrium with each other (electrically shorted to each other, and with equal temperatures)
The work function refers to removal of an electron to a position that is far enough from the surface (many nm) that the force between the electron and its [[Method of image charges|image charge]] in the surface can be neglected.<ref name="Kittel" /> The electron must also be close to the surface compared to the nearest edge of a crystal facet, or to any other change in the surface structure, such as a change in the material composition, surface coating or reconstruction. The built-in electric field that results from these structures, and any other ambient electric field present in the vacuum are excluded in defining the work function.<ref name="Gersten 2001">{{cite book | last=Gersten | first=Joel | title=The physics and chemistry of materials | publisher=Wiley | publication-place=New York | year=2001 | isbn=978-0-471-05794-9 | oclc=46538642}}</ref>
== Applications ==
;[[Thermionic emission]]: In thermionic [[electron gun]]s, the work function and temperature of the [[hot cathode]] are critical parameters in determining the amount of current that can be emitted. [[Tungsten]], the common choice for vacuum tube filaments, can survive to high temperatures but its emission is somewhat limited due to its relatively high work function (approximately 4.5 eV). By coating the tungsten with a substance of lower work function (e.g., [[thorium]] or [[barium oxide]]), the emission can be greatly increased. This prolongs the lifetime of the filament by allowing operation at lower temperatures (for more information, see [[hot cathode]]).
;[[Band bending]] models in solid-state electronics: The behavior of a solid-state device is strongly dependent on the size of various [[Schottky barrier]]s and [[heterojunction|band offset]]s in the junctions of differing materials, such as metals, semiconductors, and insulators. Some commonly used heuristic approaches to predict the band alignment between materials, such as [[Anderson's rule]] and the [[
;Equilibrium electric fields in vacuum chambers: Variation in work function between different surfaces causes a non-uniform electrostatic potential in the vacuum. Even on an ostensibly uniform surface, variations in {{math|''W''}} known as patch potentials are always present due to microscopic inhomogeneities. Patch potentials have disrupted sensitive apparatus that rely on a perfectly uniform vacuum, such as [[Casimir force]] experiments<ref>{{Cite journal | doi = 10.1103/PhysRevA.85.012504| title = Modeling electrostatic patch effects in Casimir force measurements| journal = Physical Review A| volume = 85| issue = 1| pages = 012504| year = 2012| last1 = Behunin | first1 = R. O.| last2 = Intravaia | first2 = F.| last3 = Dalvit | first3 = D. A. R.| last4 = Neto | first4 = P. A. M. | last5 = Reynaud | first5 = S.|arxiv = 1108.1761 |bibcode = 2012PhRvA..85a2504B | s2cid = 119248753}}</ref> and the [[Gravity Probe B]] experiment.<ref>{{Cite journal | doi = 10.1103/Physics.4.43| title = Finally, results from Gravity Probe B| journal = Physics| volume = 4| issue = 43| pages = 43| year = 2011| last1 = Will | first1 = C. M. |arxiv = 1106.1198 |bibcode = 2011PhyOJ...4...43W | s2cid = 119237335}}</ref> Critical apparatus may have surfaces covered with molybdenum, which shows low variations in work function between different crystal faces.<ref name="venables">{{cite web|url=http://venables.asu.edu/qmms/PROJ/metal1a.html|title=Metal surfaces 1a|website=venables.asu.edu|
;[[Contact electrification]]: If two conducting surfaces are moved relative to each other, and there is potential difference in the space between them, then an electric current will be driven. This is because the [[surface charge]] on a conductor depends on the magnitude of the electric field, which in turn depends on the distance between the surfaces. The externally observed electrical effects are largest when the conductors are separated by the smallest distance without touching (once brought into contact, the charge will instead flow internally through the junction between the conductors). Since two conductors in equilibrium can have a built-in potential difference due to work function differences, this means that bringing dissimilar conductors into contact, or pulling them apart, will drive electric currents. These contact currents can damage sensitive microelectronic circuitry and occur even when the conductors would be grounded in the absence of motion.<ref>{{Cite journal | last1 = Thomas Iii | first1 = S. W. | last2 = Vella | first2 = S. J. | last3 = Dickey | first3 = M. D. | last4 = Kaufman | first4 = G. K. | last5 = Whitesides | first5 = G. M. | title = Controlling the Kinetics of Contact Electrification with Patterned Surfaces | doi = 10.1021/ja902862b | journal = Journal of the American Chemical Society | volume = 131 | issue = 25 | pages = 8746–8747 | year = 2009 | pmid = 19499916| bibcode = 2009JAChS.131.8746T | citeseerx = 10.1.1.670.4392 }}</ref>
== 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>▼
▲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 ===
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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''
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''
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.
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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|
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.
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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''
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''
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''
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"/>
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== Work functions of elements ==
The work function depends on the configurations of atoms at the surface of the material. For example, on polycrystalline silver the work function is 4.26 eV, but on silver crystals it varies for different crystal faces as [[Miller index|(100) face]]: 4.64 eV, [[Miller index|(110) face]]: 4.52 eV, [[Miller index|(111) face]]: 4.74 eV.<ref>{{Cite journal | last1 = Dweydari | first1 = A. W. | last2 = Mee | first2 = C. H. B. | doi = 10.1002/pssa.2210270126 | title = Work function measurements on (100) and (110) surfaces of silver | journal = Physica Status Solidi A | volume = 27 | issue = 1 | pages = 223 | year = 1975 |bibcode = 1975PSSAR..27..223D }}</ref> Ranges for typical surfaces are shown in the table below.<ref>CRC Handbook of Chemistry and Physics version 2008, p.
{| class="wikitable" reference
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| 4.36 – 4.95
| align="right" |[[Sodium|Na]]
| 2.
| align="right" |[[Niobium|Nb]]
| 3.95 – 4.87
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| 4.3
| align="right" |[[Tungsten|W]]
| 4.32 –
| align="right" |[[Yttrium|Y]]
| 3.1
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== Physical factors that determine the work function ==
Due to the complications described in the modelling section below, it is difficult to theoretically predict the work function with accuracy.
=== Surface dipole ===
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In a [[semiconductor]], the work function is sensitive to the [[doping (semiconductor)|doping level]] at the surface of the semiconductor. Since the doping near the surface can also be [[field effect (semiconductor)|controlled by electric fields]], the work function of a semiconductor is also sensitive to the electric field in the vacuum.
The reason for the dependence is that, typically, the vacuum level and the conduction band edge retain a fixed spacing independent of doping. This spacing is called the [[electron affinity]] (note that this has a different meaning than the electron affinity of chemistry); in silicon for example the electron affinity is 4.05 eV.<ref>{{cite web|url=http://www.virginiasemi.com/pdf/generalpropertiessi62002.pdf|title=The General Properties of Si, Ge, SiGe, SiO2 and Si3N4 |author=Virginia Semiconductor|date=June 2002|
:<math> W = E_{\rm EA} + E_{\rm C} - E_{\rm F}</math>
where ''E''<sub>C</sub> is taken at the surface.
From this one might expect that by doping the bulk of the semiconductor, the work function can be tuned. In reality, however, the energies of the bands near the surface are often pinned to the Fermi level, due to the influence of [[surface state]]s.<ref>{{cite web|url=http://academic.brooklyn.cuny.edu/physics/tung/Schottky/surface.htm|title=Semiconductor Free Surfaces|website=academic.brooklyn.cuny.edu|
=== Theoretical models of metal work functions ===
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The jellium model is only a partial explanation, as its predictions still show significant deviation from real work functions. More recent models have focused on including more accurate forms of [[electron exchange]] and correlation effects, as well as including the crystal face dependence (this requires the inclusion of the actual atomic lattice, something that is neglected in the jellium model).<ref name="venables"/><ref>{{cite book | isbn = 9780080536347 | title = Metal Surface Electron Physics | last1 = Kiejna | first1 = A. | last2 = Wojciechowski | first2 = K.F. | date = 1996 | publisher = [[Elsevier]] }}</ref>
=== Temperature dependence of the electron work function ===
==References==▼
The electron behavior in metals varies with temperature and is largely reflected by the electron work function. A theoretical model for predicting the temperature dependence of the electron work function, developed by Rahemi et al. <ref>{{cite journal|last1=Rahemi|first1=Reza|last2=Li|first2=Dongyang | title=Variation in electron work function with temperature and its effect on Young's modulus of metals| journal=Scripta Materialia| date=April 2015|volume=99|issue=2015|pages=41–44 | doi=10.1016/j.scriptamat.2014.11.022 | arxiv=1503.08250|s2cid=118420968 }}</ref> explains the underlying mechanism and predicts this temperature dependence for various crystal structures via calculable and measurable parameters. In general, as the temperature increases, the EWF decreases via <math display="inline">\varphi(T)=\varphi_0-\gamma\frac{(k_\text{B}T)^2}{\varphi_0}</math> and <math>\gamma</math> is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). <math>\varphi_0</math> is the electron work function at T=0 and <math>k_\text{B}</math> is constant throughout the change.
▲== References ==
{{Reflist}}
== Further reading ==
* {{Cite book|title=Solid State Physics|last1= Ashcroft |last2= Mermin|publisher= Thomson Learning, Inc.|date= 1976}}
* {{Cite book|last1=Goldstein|first1= Newbury|display-authors=etal |date=2003|title= Scanning Electron Microscopy and X-Ray Microanalysis|publisher= Springer|___location=New York}}
For a quick reference to values of work function of the elements:
* {{cite journal|first=Herbert B.|last= Michaelson|title=The work function of the elements and its periodicity|journal=J. Appl. Phys. |volume=48|issue= 11|page=4729 |date=1977|bibcode = 1977JAP....48.4729M |doi = 10.1063/1.323539 |s2cid= 122357835
== External links ==
* [http://repositories.tdl.org/ttu-ir/bitstream/handle/2346/21434/Vela_Russell_Thesis.pdf?sequence=1 Work function of polymeric insulators (Table 2.1)]
* [http://www3.ntu.edu.sg/home/ecqsun/rtf/SSC-WF.pdf Work function of diamond and doped carbon] {{Webarchive|url=https://web.archive.org/web/20120629102537/http://www3.ntu.edu.sg/home/ecqsun/rtf/SSC-WF.pdf |date=2012-06-29 }}
* [http://www.pulsedpower.net/Info/WorkFunctions.htm Work functions of common metals]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/tables/photoelec.html Work functions of various metals for the photoelectric effect]
* [http://academic.brooklyn.cuny.edu/physics/tung/Schottky/surface.htm Physics of free surfaces of semiconductors]
{{Thermionic_valves}}
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