Work function: Difference between revisions

Revision as of 08:26, 2 December 2020

Revision as of 21:48, 6 December 2020

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== Definition ==

The work function {{math|''W''}} for a given surface is defined by the difference<ref>{{cite book |title=[[Introduction to Solid State Physics]] |edition=7th |last1=Kittel |first1=Charles |authorlink1author-link1=Charles Kittel |date= <!--replace this comment with the publication year--> |publisher=Wiley }}</ref>

:<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''_F}} 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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;[[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|author=|date=|website=academic.brooklyn.cuny.edu|accessdate=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|author=|date=|website=venables.asu.edu|accessdate=11 April 2018}}</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| pmc = | citeseerx = 10.1.1.670.4392 }}</ref>

== Measurement ==

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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 | pmid = | pmc = |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| pmc = |bibcode = 2018RScI...89d3702F}}</ref>

=== Methods based on thermionic emission ===

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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|author=|date=|website=www.virginia.edu|accessdate=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.

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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 | pmid = | pmc = |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.&nbsp;12–114.</ref>

{| class="wikitable" reference

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where ''E''_C 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|author=|date=|website=academic.brooklyn.cuny.edu|accessdate=11 April 2018}}</ref> If there is a large density of surface states, then the work function of the semiconductor will show a very weak dependence on doping or electric field.<ref>{{Cite journal | last1 = Bardeen | first1 = J. | title = Surface States and Rectification at a Metal Semi-Conductor Contact | doi = 10.1103/PhysRev.71.717 | journal = Physical Review | volume = 71 | issue = 10 | pages = 717–727 | year = 1947 | pmid = | pmc = |bibcode = 1947PhRv...71..717B }}</ref>

=== Theoretical models of metal work functions ===

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Theoretical modeling of the work function is difficult, as an accurate model requires a careful treatment of both electronic [[many-body problem|many body effects]] and [[surface chemistry]]; both of these topics are already complex in their own right.

One of the earliest successful models for metal work function trends was the [[jellium]] model,<ref>{{Cite journal | last1 = Lang | first1 = N. | last2 = Kohn | first2 = W. | doi = 10.1103/PhysRevB.3.1215 | title = Theory of Metal Surfaces: Work Function | journal = Physical Review B | volume = 3 | issue = 4 | pages = 1215 | year = 1971 | pmid = | pmc = |bibcode = 1971PhRvB...3.1215L }}</ref> which allowed for oscillations in electronic density nearby the abrupt surface (these are similar to [[Friedel oscillation]]s) as well as the tail of electron density extending outside the surface. This model showed why the density of conduction electrons (as represented by the [[Wigner–Seitz radius]] ''r_s'') is an important parameter in determining work function.

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>