Work function: Difference between revisions

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{{short description|Type of energy}}

In [[solid-state physics]], the '''work function''' (sometimes speltspelled '''workfunction''') is the minimum [[thermodynamic work]] (i.e., energy) needed to remove an [[electron]] from a solid to a point in the [[vacuum]] immediately outside the solid surface. Here "immediately" means that the final electron position is far from the surface on the atomic scale, but still too close to the solid to be influenced by ambient electric fields in the vacuum.

The work function is not a characteristic of a bulk material, but rather a property of the surface of the material (depending on crystal face and contamination).

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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''_F}} 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 discharge materials|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''_F / ''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).

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;[[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 [[Schottky–Mott rule]], are based on the thought experiment of two materials coming together in vacuum, such that the surfaces charge up and adjust their work functions to become equal just before contact. In reality these work function heuristics are inaccurate due to their neglect of numerous microscopic effects. However, they provide a convenient estimate until the true value can be determined by experiment.<ref>[[Herbert Kroemer]], "[https://www.nobelprize.org/nobel_prizes/physics/laureates/2000/kroemer-lecture.html Quasi-Electric Fields and Band Offsets: Teaching Electrons New Tricks]" Nobel lecture</ref><ref>{{cite web|url=http://academic.brooklyn.cuny.edu/physics/tung/Schottky/systematics.htm|title=Barrier Height Correlations and Systematics|website=academic.brooklyn.cuny.edu|access-date=11 April 2018}}</ref>

;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|access-date=11 April 2018|archive-date=29 December 2016|archive-url=https://web.archive.org/web/20161229160647/http://venables.asu.edu/qmms/PROJ/metal1a.html|url-status=dead}}</ref>

;[[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 ==

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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''_e. 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''_c is the collector's thermionic work function, Δ''V''_ce is the applied collector–emitter voltage, and Δ''V''_S is the [[Seebeck effect|Seebeck voltage]] in the hot emitter (the influence of Δ''V''_S is often omitted, as it is a small contribution of order 10 mV).

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| 4.36 – 4.95

| align="right" |[[Sodium|Na]]

| 2.3675

| align="right" |[[Niobium|Nb]]

| 3.95 – 4.87

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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. VariousHowever, various trends have, however, been identified. The work function tends to be smaller for metals with an open lattice,{{what|date=October 2020|reason=open not defined}} and larger for metals in which the atoms are closely packed. It is somewhat higher on dense crystal faces than open crystal faces, also depending on [[surface reconstruction]]s for the given crystal face.

=== Surface dipole ===

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=== Temperature dependence of the electron work function ===

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 the 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_\betatext{B}</math> is constant throughout the change.

== References ==

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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|url= https://semanticscholar.org/paper/e8c0a0669e82f16e291ffe0fe19b56d9af668688}}

== 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]